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Separation of NP-Completeness Notions.

We use hypotheses of structural complexity theory to separate various NP-completeness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is $\mbox{${\leq}^{\rm P}_{\rm T}$}$-complete but not $\mbox{${\leq}^{\rm P}_{tt}$}$-complete. We provide fairly thorough analyses of the hypotheses that we introduce.

Introduction Ladner, Lynch, and Selman [LLS75] were the first to compare the strength of polynomial-time reducibilities. They showed, for the common polynomial-time reducibilities, Turing btt ), and many-one s means that # P r is properly stronger than # P s B, but the converse does not hold. In each case, the verifying sets belong to Ladner, Lynch, and Selman raised the obvious question of whether reducibilities differ on NP. If there exist sets A and B in NP (other than the empty set or S # ) such that A# P T B but A immediately. With this in mind, they conjectured that P #= NP implies that # P m differ on NP. In the intervening years, many results have explained the behavior of polynomial-time reducibilities within other complexity classes and have led to a complete understanding of the completeness notions that these reducibilities induce. For example, Ko and Moore [KM81] demonstrated the existence of # P T -complete sets for EXP that are not # P complete. Watanabe [Wat87] extended this result significantly, showing that # P btt -, tt -, and # P T -completeness for EXP are mutually different, while Homer, Kurtz, and Royer [KR93] proved that # P m - and # P 1-tt -completeness are identical. # Department of Computer Science and Engineering, University at Buffalo, Buffalo, NY 14260. Email: Department of Computer Science and Engineering, University at Buffalo, Buffalo, NY 14260. Email: selman@cse.buffalo.edu However, there have been few results comparing reducibilities within NP, and we have known very little concerning various notions of NP-completeness. It is surprising that no NP-complete problem has been discovered that requires anything other than many-one reducibility for proving its completeness. The first result to distinguish reducibilities within NP is an observation of Wilson in one of Selman's papers on p-selective sets [Sel82]. It is a corollary of results there that if NE# co-NE #= E, then there exist sets A and B belonging to NP such that A# P ptt B, B# P ptt denotes positive truth-table reducibility. Regarding completeness, Longpr- e and Young [LY90] proved that there are # P -complete sets for NP for which # P T -reductions to these sets are faster, but they did not prove that the completeness notions differ. The first to give technical evidence that # P -completeness for NP differ are Lutz and Mayordomo [LM96], who proved that if the p-measure of NP is not zero, then there exists a # P 3-tt -complete set that is not # P m - complete. Ambos-Spies and Bentzien [ASB00] extended this result significantly. They used an hypothesis of resource-bounded category theory that is weaker than that of Lutz and Mayordomo to separate nearly all NP-completeness notions for the bounded truth-table reducibilities. It has remained an open question as to whether we can separate NP-completeness notions without using hypotheses that involve essentially stochastic concepts. Furthermore, the only comparisons of reducibilities within NP known to date have been those just listed. Here we report some exciting new progress on these questions. Our main new result introduces a strong, but reasonable, hypothesis to prove existence of a # P T -complete set in NP that is not # P tt -complete. Our result is the first to provide evidence that # P tt -completeness is weaker than # P -completeness. Let Hypothesis H be the following assertion: There is a UP-machine M that accepts 0 # such that (i) no polynomial time-bounded Turing machine correctly computes infinitely many accepting computations of M, and (ii) for some e > 0, no 2 n e time-bounded Turing machine correctly computes all accepting computations of M. Hypothesis H is similar to, but seemingly stronger than, hypotheses considered by researchers previously, notably Fenner, Fortnow, Naik, and Rogers [FFNR96], Hemaspaan- dra, Rothe and Wechsung [HRW97], and Fortnow, Pavan, and Selman [FPS99]. This result is especially interesting because the measure theory and category theory techniques seem to be successful primarily for the nonadaptive reducibilities. Wewill prove an elegant characterization of the genericity hypothesis of Ambos-Spies and Bentzien and compare it with Hypothesis H. Here, somewhat informally, let us say this: The genericity hypothesis asserts existence of a set L in NP such that no 2 2n time-bounded Turing machine can correctly predict membership of infinitely many x in L from the initial characteristic sequence That is, L is almost-everywhere unpredictable within time 2 2n . Clearly such a set L is 2 2n -bi-immune. In contrast, we show that Hypothesis H holds if there is a set L in UP#co-UP such that L is P-bi-immune and L#0 # is not in DTIME(2 n e for some e > 0. Thus, we replace "almost-everywhere unpredictable" with P-bi-immunity and we lower the time bound from 2 2n to 2 n e , but we require L to belong to UP# co-UP rather than NP. We prove several other separations as well, and some with significantly weaker hy- potheses. For example, we prove that NP contains # P T -complete sets that are not # P m - complete, if NP# co-NP contains a set that is 2 n e -bi-immune, for some e > 0. Preliminaries We use standard notation for polynomial-time reductions [LLS75], and we assume that readers are familiar with Turing, # P T , and many-one, # P reducibilities. A set A is truth-table reducible to a set B (in symbols A # P tt B) if there exist polynomial-time computable functions g and h such that on input x, g(x) is a set of queries Q= {q 1 , q 2 , - , q k }, and x #A if and only if h(x,B(q 1 1. The function g is the truth-table generator and h is the truth-table evaluator. For a constant k > 0, A is k-truth-table reducible to B k-tt B) if for all x, and A is bounded-truth-table reducible to B (A# P there is a constant k > 0 such that A # P k-tt B. Given a polynomial-time reducibility # P r , recall that a set S is # P r -complete for NP if S # NP and every set in NP is # P r -reducible to S. Recall that a set L is p-selective if there exists a polynomial-time computable function such that for all x and y, f (x,y) # {x,y} and f (x,y) belongs to L, if either x # L or y # L [Sel79]. The function f is called a selector for L. Given a finite alphabet, let S w denote the set of all strings of infinite length of order type w. For r # S #S w , the standard left cut of r [Sel79, Sel82] is the set where < is the ordinary dictionary ordering of strings with 0 less than 1. It is obvious that every standard left cut is p-selective with selector f (x,y) =min(x,y). Given a p-selective set L such that the function f defined by f selector for L, we call f a min-selector for L. We will use the following simplified version of a lemma of Toda [Tod91]. be a p-selective set with a min-selector f . For any finite set Q there exists a string z # Q#} such that z}. The string z is called a "pivot" string. Now we review various notions related to almost-everywhere hardness. A language L is immune to a complexity class C , or C -immune, if L is infinite and no infinite subset of L belongs to C . A language L is bi-immune to a complexity class C , or C -bi-immune, if L is infinite, no infinite subset of L belongs to C , and no infinite subset of L belongs to C . A language is DTIME(T (n))-complex if L does not belong to DTIME(T (n)) almost everywhere; that is, every Turing machine M that accepts L runs in time greater than T (|x|), for all but finitely many words x. Balc- azar and Sch- oning [BS85] proved that for every time-constructible function T , L is DTIME(T (n))-complex if and only if L is bi-immune to DTIME(T (n)). Given a time bound T (n), a language L is T (n)-printable if there exists a T (n) time-bounded Turing machine that, on input 0 n , prints all elements of L#S =n [HY84]. A set S is T (n)-printable-immune if S is infinite and no infinite subset of S is T (n)-printable. In order to compare our hypotheses with the genericity hypothesis we describe time-bounded genericity [ASFH87]. For this purpose, we follow the exposition of Ambos-Spies, Neis, and Terwijn [ASNT96]. Given a set A and string x, is the n-th string in lexicographic order. We identify the initial segment A|z n with its characteristic sequence; i.e., A|z n =A(z condition is a set C # S # . A meets C if for some x, the characteristic sequence A|x #C. C is dense along A if for infinitely many strings x there exists i # {0,1} such that the concatenation (A|x)i #C. Then, the set A is DTIME(t(n))-generic if A meets every condition C#DTIME(t(n)) that is dense along A. To simplify the notation, we say that A is t(n)-generic if it is DTIME(t(n))- generic. Finally, we briefly describe the Kolmogorov complexity of a finite string. Later we will use this in an oracle construction. The interested reader should refer to Li and Vit- anyi [LV97] for an in-depth study. Fix a universal Turing machine U . Given a string x and a finite set the Kolmogorov complexity of x with respect to S is defined by 0, then K(x|S) is called the Kolmogorov complexity of x, denoted K(x). We will use time-bounded Kolmogorov complexity K t (x) also. For this definition, we require that U(p) runs in at most t(|x|) steps. 3 Separation Results Let Hypothesis H be the following assertion: Hypothesis H: There is a UP-machine M that accepts 0 # such that 1. no polynomial time-bounded Turing machine correctly computes infinitely many accepting computations of M, and 2. for some e > 0, no 2 n e time-bounded Turing machine correctly computes all accepting computations of M. Theorem 1 If Hypothesis H is true, then there exists a # P -complete language for NP that is not # P tt -complete for NP. Proof. Let M be a UP-machine that satisfies the conditions of Hypothesis H. For each a n be the unique accepting computation of M on 0 n , and let l |. Define the language Define the infinite string a = a 1 a 2 ., and define to be the standard left-cut of a. We define to be the disjoint union of L 1 and L 2 . We will prove that L is T -complete for NP but not # P T -complete for NP. Proof. It is clear that L belongs to NP. The following reduction witnesses that SAT# P Given an input string x, where use a binary search algorithm that queries L 2 to find a n . Then, note that x # SAT if and only if #x,a n # belongs to L 1 . Lemma 3 L is not # P tt -complete for NP. Proof. Assume that L is # P tt -complete for NP. Define the set | the i-th bit of a Clearly, S belongs to NP. Thus, by our assumption, there is a # P tt -reduction #g,h# from S to L. Given this reduction, we will derive a contradiction to Hypothesis H. Consider the following procedure A : 1. input 2. compute the sets Q 3. Let Q 1 be the set of all queries in Q to L 1 and let Q 2 be the set of all queries in Q to 4. If Q 1 contains a query #x,a t #, where t # n e , then output "Unsuccessful" and Print a t , else output "Successful". Observe that this procedure runs in polynomial time. We treat two cases, namely, either A (0 n ) is unsuccessful, for infinitely many n, or it is successful, for all but finitely many n. If the procedure A (0 n ) is unsuccessful for infinitely many n, then there is a polynomial time-bounded Turing machine that correctly computes infinitely many accepting computations of M, thereby contradicting Clause 1 of Hypothesis H. Proof. If A (0 n ) is unsuccessful, then it outputs a string a t such that t # n e . Hence, if A (0 n ) is unsuccessful for infinitely many n, then for infinitely many t there exists an n, outputs a t . The following procedure uses this observation to compute infinitely many accepting computations of M in polynomial time. do if A (0 j ) outputs a t then output a t and halt. The procedure runs in polynomial time because the procedure A (0 j ) runs in polynomial time. but finitely many n, then there is a 2 n e time-bounded Turing machine that correctly computes all accepting computations of M, thereby contradicting Clause 2 of Hypothesis H. Proof. We will demonstrate a procedure B such that for each n, if A (0 n ) is successful, then B on input 0 n outputs the accepting computation of M on 0 n in 2 n e time. If A (0 n ) is successful, then no member of the set Q 1 is of the form #x,a t # where t # n e . We begin our task with the following procedure C that for each query decides whether q # L 1 . 1. input 2. If z #= a t for some t, then #y, z# does not belong to L 1 ; (This can be determined in polynomial time.) 3. if z = a t , where t # n e , then #y, z# belongs to L 1 only if belongs to SAT. (Since t # n e this step can be done in time 2 n e Thus, C decides membership in L 1 for all queries q in Q 1 . Therefore, if for each query q in Q 2 , we can decide whether q belongs to L 2 , then the evaluator h can determine whether each input #0 n , belongs to S. That is, if for each query q in Q 2 , we can decide whether q belongs to L 2 , then we can compute a n . We can accomplish this using a standard proof technique for p-selective sets [HNOS96, Tod91]. Namely, since L 2 is a standard left- cut, by Lemma 1, there exists a pivot string z in Q 2 #} such that Q 2 #L 2 is the set of all strings in Q 2 that are less than or equal to z. We do not know which string is the pivot string, but there are only #Q 2 # choices, which is a polynomial number of choices. Thus, procedure B on input 0 n proceeds as follows to compute a n : For each possible choice of pivot and the output from procedure C , the evaluator h computes a possible value for each j-th bit of a n . There are only a polynomial number of possible choices of a n , because there are only a polynomial number of pivots. B verifies which choice is the correct accepting computation of M on 0 n , and outputs that value. Finally, we have only to note that the entire process can be carried out in 2 n e steps. This completes the proof of our claim, and of the theorem as well. Let Hypothesis H # be the following assertion: There is an NP-machine M that accepts 0 # such that for some 0 < e < 1, no time-bounded Turing machine correctly computes infinitely-many accepting computations of M. Theorem 2 If Hypothesis H # is true, then there exists a Turing complete language for NP that is not # P m -complete for NP. Proof. Let M be an NP-machine that satisfies the conditions of Hypothesis H # . For each a n be the lexicographically maximum accepting computation of M on 0 n , and let . Define the language an accepting computation of M on 0 m , Let a = a 1 a 2 a 3 -, and define It is easy to see, as in the previous argument, that L is # P T -complete for NP. In order to prove that L is not # P m -complete, we define the set | y is a prefix of an accepting computation of M on 0 n which belongs to NP, and assume there is a # P m -reduction f from S to L. Consider the procedure D in Figure 1: First we will analyze the running time and then we treat two cases, namely, either D (0 n ) is successful for infinitely many n, or it is unsuccessful for all but finitely many n. 3 The above procedure halts in O(l n 2 n e 2 /2 ) steps. Proof. Consider an iteration of the repeat loop. The most expensive step is the test of whether "z # SAT". This test occurs only when Hence we can decide whether z belongs to SAT in 2 n e 2 /2 steps. All other steps take polynomial time. Hence the time taken by the procedure is O(l the running time of procedure D is bounded by 2 n e for infinitely many n, then there is a 2 n e -time-bounded Turing machine that correctly computes infinitely many accepting computations of M. input Repeat l n times begin if both x 0 and x 1 are queries to L 2 then if x 0 # x 1 then y := y0 else y := y1 else {At least one of x 0 and x 1 is a query to L 1 {0,1} be the least index such that x b queries L 1 , and let x if u is not an accepting computation of M {thus, x b / then else {u is an accepting computation of M on 0 t then output "Unsuccessful," print u, and terminate else {t < n e then y := yb else {x b / output "Successful" and print y. Figure 1: Procedure D Proof. We demonstrate that if D is successful on an input 0 n , then the string that is printed is an accepting computation of M on 0 n . In order to accomplish this, we prove by induction that y is a prefix of an accepting computation of M on 0 n during every iteration of the repeat loop (i.e., a loop invariant). Initially when l this is true. Assume that y is a prefix of an accepting computation of M at the beginning of an iteration. Then, at least one of f (#0 n , must belong to L. If both x 0 and x 1 are queries to L 2 , then the smaller of x 0 and x 1 belongs to L 2 because L 2 is p-selective. Thus, in this case, the procedure extends y correctly. If at least one of x 0 and x 1 is a query to L 1 , then the procedure determines whether x b # L 1 , where x b is the query to L 1 with least index. If x b belongs to L, then #0 n , yb# S. Hence, yb is a prefix of an accepting computation. If # L, then x - b belongs to L, because at least one of x b or x - b belongs to L. Thus, in this case, y - b is a prefix of an accepting computation. This completes the induction argument. The loop repeats l n times. Therefore, the final value of y, which is the string that D prints, is an accepting computation. but finitely many n, then there is a 2 n e -time- bounded Turing machine that correctly computes infinitely many accepting computations of M. Proof. The proof is similar to the proof of Claim 1. The following procedure computes infinitely many accepting computations of M. input do if D (0 j ) outputs u and u is an accepting computation of M on 0 n then print u and terminate. The running time of this algorithm can be bounded as follows: The procedure D (0 j ) runs in time l steps. So the total running time is - n 1/e Since the cases treated both by Claims 4 and 5 demonstrate Turing machines that correctly compute infinitely many accepting computations of M in 2 n e time, we have a contradiction to Hypothesis H # . Thus L is not # P m -complete for NP. The following results give fine separations of polynomial time reducibilities in NP from significantly weaker hypotheses. Moreover, they follow readily from results in the literature Theorem 3 If there is a tally language in UP-P, then there exist two languages L 1 and in NP such that L 1 # P tt Proof. Let L be a tally language in UP-P. Let R be the polynomial-time computable relation associated with the language L. Define and i-th bit of w is one}. It is clear that L 1 is # P tt -reducible to L 2 . To see that L 2 is # P T -reducible to L 1 , implement a binary search algorithm that accesses L 1 to determine the unique witness w such that then find the i-th bit. Observe that L 2 is a sparse set. Ogihara and Watanabe [OW91] call L 1 the left set of L, and they and Homer and Longpr- e [HL94] proved for every L in NP that if the left set of L btt -reducible to a sparse set, then L is in P. Hence L 1 # btt L 2 . We now prove that Turing and truth-table reducibilities also differ in NP under the same hypothesis. Theorem 4 If there is a tally language in UP-P, then there exist two languages L 1 and in NP such that L 1 # P Proof. Hemaspaandra et al. [HNOS96] proved that the hypothesis implies existence of a tally language L in UP-P such that L is not # P tt -reducible to any p-selective set. In the same paper they also showed, given a tally language L in NP-P, how to obtain a p- selective set S such that L is # P T -reducible to S. Combing the two results we obtain the theorem. 4 Analysis of the Hypotheses This section contains a number of results that help us to understand the strength of Hypotheses H and H # . 1 The class of all languages that are # P T -equivalent to L 1 is a noncollapsing degree. 4.1 Comparisons With Other Complexity-Theoretic Assertions We begin with some equivalent formulations of these hypotheses, and then relate them to other complexity-theoretic assertions. The question of whether P contains a P-printable- immune set was studied by Allender and Rubinstein [AR88], and the equivalence of items 1 and 3 in the following theorem is similar to results of Hemaspaandra, Rothe, and Wechsung [HRW97] and Fortnow, Pavan, and Selman [FPS99]. The second item is similar to the the characterization of Grollmann and Selman [GS88] of one-one, one-way functions with the addition of the attribute almost-always one-way of Fortnow, Pavan, and Selman. Theorem 5 The following statements are equivalent: 1. There is a language L in P that contains exactly one string of every length such that L is P-printable-immune and, for some e > 0, L is not 2 n e -printable. 2. There exists a polynomial-bounded, one-one, function , such that f is almost-everywhere not computable in polynomial time, for some e > 0, f is not computable in time 2 n e , and the graph of f belongs to P. 3. Hypothesis H is true for some e > 0. Proof. Let L satisfy item one. Define the unique string of length n that belongs to L. Clearly, f us polynomial-bounded and one-one. The graph of f belongs to P, because L belongs to P. Suppose that M is a Turing machine that computes f and that runs in polynomial time on infinitely many inputs. Then, on these inputs, M prints L#S n . Similarly, f is not computable in time 2 n e Let f satisfy item two. Define a UP-machine M to accept 0 # as follows: On input 0 n , M guesses a string y of length within the polynomial-bound of f , and accepts if and only if The rest of the proof is clear. Let M be a UP-machine that satisfies item three, i.e., that satisfies the conditions of Hypothesis H. Let a n be the unique accepting computation of M on 0 n and let |a n r n be the rank of a n among all strings of length n l . Now, we define L as follows: Given a string x, if belongs to L if and only if x = a n . If (n-1) l < |x| < n l , then x belongs to L if and only if the rank of x (among all the string of length |x|) is r n-1 . It is clear that L # P and has exactly one string per each length. We claim that L is P-printable- immune and is not 2 n r -printable, where machine that prints infinitely many strings of L in polynomial time can be used to print infinitely many accepting computations of M in polynomial time. Thus L is P-printable-immune. Any machine that prints all the strings of L in 2 n r time can be used print all the accepting computations of M in 2 n e time. Thus L is not 2 n r -printable. We prove the following theorem similarly. Theorem 6 The following statements are equivalent 1. There is a language L in P that contains at least one string of every length such that, for some e > 0, L is 2 n e -printable-immune. 2. There is polynomial-bounded, multivalued function such that every refinement of f is almost-everywhere not computable in 2 n e -time, and the graph of f belongs to P. 3. Hypothesis H # holds for some e > 0. Next we compare our hypotheses with the following complexity-theoretic assertions: 1. For some e > 0, there is a P-bi-immune language L in UP#co-UP such that L#0 # is not in DTIME(2 n e 2. For some e > 0, there is language L in UP#co-UP such that L is not in DTIME(2 n e 3. For some e > 0, there is a 2 n e -bi-immune language in NP# co-NP. Theorem 7 Assertion 1 implies Hypothesis H and Hypothesis H implies Assertion 2. Proof. Let L be a language in UP# co-UP that satisfies Assertion 1. Define M to be the UP-machine that accepts 0 # as follows: On input 0 n , nondeterministically guess a string If w either witnesses that 0 n is in L or witnesses that 0 n is in L, then accept 0 n . It is immediate that M satisfies the conditions of Hypothesis H. To prove the second implication, let M a UP-machine that satisfies the conditions of Hypothesis H. Let a n denote the unique accepting computation of M on 0 n and define It is clear that L # UP#co-UP. If L # DTIME(2 n e then a binary search algorithm can correctly compute a n , for every n, in time 2 n e . This would contradict Hypothesis H. Hence, The discrete logarithm problem is an interesting possible witness for Assertion 2. The best known deterministic algorithm requires time greater than 2 3 [Gor93]. Thus, the discrete logarithm problem is a candidate witness for the noninclusion UP # co-UP # 3 . Corollary 1 If, for some e > 0, UP # co-UP has a 2 n e -bi-immune language, then # P completeness is different from # P tt -completeness for NP. Theorem 8 Assertion (3) implies Hypothesis H # . Corollary 2 If, for some e > 0, NP # co-NP has a 2 n e -bi-immune language, then # P completeness is different from # P m -completeness for NP. 4.2 Comparisons with Genericity The genericity hypothesis of Ambos-Spies and Bentzien [ASB00], which they used successfully to separate NP-completeness notions for the bounded-truth-table reducibilities, states that "NP contains an n 2 -generic language". Our next result enables us to compare this with our hypotheses. We say that a deterministic oracle Turing machine M is a predictor for a language L if for every input word x, M decides whether x # L with oracle L|x. L is predictable in time t(n) if there is a t(n) time-bounded predictor for L. We define a set L to be almost-everywhere unpredictable in time t(n) if every predictor for L requires more than t(n) time for all but finitely many x. This concept obviously implies DTIME(t(n))-complex almost everywhere, but the converse does not hold: Theorem 9 EXP contains languages that are DTIME(2 n )-complex but not almost-everywhere unpredictable in time 2 n . Now we state our characterization of t(n)-genericity. Theorem 10 Let t(n) be a polynomial. A decidable language L is t(n)-generic if and only if it is almost-everywhere unpredictable in time t(2 n -1). Proof. Assume that L is not almost-everywhere unpredictable in time t(2 n -1), and let M be a predictor for L that for infinitely many strings x runs in time t(2 n 1). Define a condition C so that the characteristic sequence (L|x)x #C #M with oracle L|x runs in time t(2 |x| -1) on input x. accepts x). Then, C is dense along L because M correctly predicts whether x # L for infinitely many x. It is easy to see that C # DTIME(t(n)). However, L is not t(n)-generic because we defined C so that L does not meet C. Assume that L is not t(n)-generic, and let C #DTIME(t(n)) be a condition that is dense along L such that L does not meet C. Let T be a deterministic Turing machine that halts on all inputs and accepts L. Define a predictor M for L to behave as follows on input x with oracle A|x: If (A|x)1 #C, then M rejects x, and if (A|x)0 #C, then M accepts x. If neither holds, then M determines membership in L by simulating T on x. Since L does not meet C, M is a predictor for L. Since C is dense along L and L does not meet C, for infinitely many x, either (A|x)1 #C or (A|x)0 #C, and in each of these cases, M runs for at most t(2 - 2 |x| ) steps. Since t(n) is polynomial function, by the linear speedup theorem [HS65], there is a Turing machine that is equivalent to M that runs in time t(2 |x| -1). Corollary 1 NP contains an n 2 -generic language if and only if NP contains a set that is almost-everywhere unpredictable in time 2 2n . By Theorem 8, Hypothesis H # holds if NP# co-NP contains a set that, for some e > 0, is -bi-immune. So, Hypothesis H # requires bi-immunity, which is weaker than almost-everywhere unpredictability, and the time-bound is reduced from 2 2n to 2 n e . On the other hand, we require the language to belong to NP# co-NP instead of NP. Similarly, when we consider Hypothesis H, we require the language to be P-bi-immune and not in DTIME(2 n e ), whereas now we require the language to be in UP# co-UP. Moreover, the conclusion of Theorem 1 is not known to follow from the genericity hypothesis. At the same time, we note that the genericity hypothesis separates several bounded-truth-table completeness notions in NP that do not seem obtainable from our hypotheses. 4.3 Relativization Theorem 11 There exists an oracle relative to which the polynomial hierarchy is infinite and Hypotheses H and H # both hold. Proof. Define Kolmogorov random strings r 0 , r 1 , . as follows: r n is the first string of length n such that Then, define the oracle Define M to be an oracle Turing machine that accept 0 # with oracle A as follows: On input guess a string y of length n. If y # A, then accept. M is a UP A -machine that accepts contains exactly one string of every length. Now we show that no 2 n e oracle Turing machine with oracle A, for any 0 < e < 1, correctly computes infinitely many accepting computations of M. Observe that relative to A, this implies both Hypotheses H and H # . Suppose otherwise, and let T be such an oracle Turing machine. The gist of the remainder of the proof is that we will show how to simulate T without using the oracle, and that will contradict the randomness or r n . Suppose that T A (0 n . Then we simulate this computation without using an oracle as follows: 1. Compute . Do this iteratively: Compute r i by running every program (with input strings r 0 , r 1 , . , r i-1 ) of length # i/2 for 2 i steps. Then r i is the first string of length i that is not output by any of these programs. Note that the total time for executing this step is 2. Simulate T on input 0 n , except replace all oracle queries q by the following rules: If |q| < l, answer using the previous computations. Otherwise, just answer "no." If the simulation is correct, then this procedure outputs r n without using the oracle. The running time of this procedure on input 0 n is 2 5n e +2 n e , which is less than 2 n . So, we can describe r n by a string of length O(logn), to wit, a description of T and 0 n . This contradicts the definition of r n . We need to show that the simulation is correct. The simulation can only be incorrect if |q| # l and be the first such query. This yields a short description of r m , given r 0 , r 1 , . , r l-1 . Namely, the description consists of the description of T (a constant), the description of 0 n (logn bits), and the description of the number j such that is the j-th query (at most n e ). Thus, the length of the description is O(n e ). Since that the length of the description of r m is less than m/2. The running time of T , given r 0 , r 1 , . , r l-1 , is 2 n e , which is less than 2 m . (The reason is that the first step in the simulation of T is not needed.) Therefore, the simulation is correct. Finally, because A is a sparse set, using results of Balc- azar et al. [BBS86], there is an oracle relative to which the hypotheses holds and the polynomial hierarchy is infinite. Hypothesis H fails relative to any oracle for which and Rogers [FR94] obtained an oracle relative to which NP #= co-NP and Hypothesis H # fails. We know of no oracle relative to which P #= NP and every # P T -complete set is # complete. 4.4 Extensions The extensions in this section are independently observed by Regan and Watanabe [RW01]. In Hypothesis H we can replace the UP-machine by an NP-machine under a stronger intractability assumption. Consider the following hypothesis: There is a NP-machine M that accepts 0 # such that 1. no probabilistic polynomial time-bounded Turing machine correctly outputs infinitely many accepting computations with non-trivial (inverse polynomial) probability, and 2. for some e > 0, no 2 n e time-bounded Turing machine correctly computes all accepting computations with non-trivial probability. We can prove that Turing completeness is different from truth-table completeness in NP under the above hypothesis. The proof uses the randomized reduction of Valiant and that isolates the accepting computations. We define L as in the proof of Theorem 2. Let i#v such that v is an accepting computation of M, and the ith bit of where v.r i denotes the inner product over GF[2]. Valiant and Vazirani showed that if we randomly pick r 1 , r 2 , - , r k , then with a non-trivial probability there exists exactly one accepting computation v of M whose inner product with each r i is 0. Thus, for a random choice of r 1 , - , r k , there is exactly one witness v for i#. The rest of the proof is similar to that of Theorem 1. We also note that we can replace the UP-machine in Hypothesis H with a FewP- machine. --R Separating NP-completeness under strong hypotheses Diagonalizations over polynomial time computable sets. Genericity and measure for exponential time. Resource bounded randomness and weakly complete problems. Relativizations of the P Completeness notions for nondeterministic complexity classes. On inverting onto functions. Distributionally hard languages. Discrete logarithms in GF(p) using the number field sieve. Complexity measures for public-key cryptosys- tems Easy sets and hard certificate schemes. On the computational complexity of algorithms. Computation times of NP sets of different densities. Completeness, approximation and density. A comparison of polynomial time re- ducibilities Cook versus karp-levin: Separating completeness notions if NP is not small On polynomial time bounded truth-table reducibility of NP sets to sparse sets Personal communication. Reductions on NP and P-selective sets On polynomial-time truth-table reducibilities of intractable sets to P-selective sets NP is as easy as detecting unique solutions. A comparison of polynomial time completeness notions. --TR --CTR A. Pavan , Alan L. Selman, Bi-immunity separates strong NP-completeness notions, Information and Computation, v.188 n.1, p.116-126, 10 January 2004 John M. Hitchcock , A. Pavan, Comparing reductions to NP-complete sets, Information and Computation, v.205 n.5, p.694-706, May, 2007 Christian Glaer , Alan L. Selman , Samik Sengupta, Reductions between disjoint NP-pairs, Information and Computation, v.200 n.2, p.247-267, 1 August 2005 Lane A. Hemaspaandra, SIGACT news complexity theory column 40, ACM SIGACT News, v.34 n.2, June 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

p-selectivity;truth-table completeness;turing completeness;p-genericity;many-one completeness

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A Constant-Factor Approximation Algorithm for Packet Routing and Balancing Local vs. Global Criteria.

We present the first constant-factor approximation algorithm for a fundamental problem: the store-and-forward packet routing problem on arbitrary networks. Furthermore, the queue sizes required at the edges are bounded by an absolute constant. Thus, this algorithm balances a global criterion (routing time) with a local criterion (maximum queue size) and shows how to get simultaneous good bounds for both. For this particular problem, approximating the routing time well, even without considering the queue sizes, was open. We then consider a class of such local vs. global problems in the context of covering integer programs and show how to improve the local criterion by a logarithmic factor by losing a constant factor in the global criterion.

Introduction . Recent research on approximation algorithms has focused a fair amount on bicriteria (or even multicriteria) minimization problems, attempting to simultaneously keep the values of two or more parameters "low" (see, e.g., [11, 21, 22, 29, 30, 32]). One motivation for this is that real-world problems often require such balancing. In this work, we consider a family of bicriteria problems that involve balancing a local capacity constraint (e.g., the maximum queue size at the links of a packet routing network, the maximum number of facilities per site in facility location) with a global criterion (resp., routing time, total cost of constructing the facilities). Since these global criteria are NP-hard to minimize even with no constraint on the local criterion, we shall seek good approximation algorithms. 1.1. Packet Routing. Our main result is a constant-factor approximation algorithm for store-and-forward packet routing, a fundamental routing problem in interconnection networks (see Leighton's book and survey [14, 15]); furthermore, the queue sizes will all be bounded by a constant. This packet routing problem has received considerable attention for more than 15 years, and is as follows: Definition 1.1 (Store-and-Forward Packet Routing). We are given an arbitrary N-node routing network (directed or undirected graph) G, and a set {1, 2, . , K} of packets which are initially resident (respectively) at the (multi-)set of nodes {s of G. Each packet k is a message that needs to be routed to some given destination node t k in G. We have to route each packet k from s k to t k , subject to: (i) each packet k must follow some path in G; (ii) each edge traversal takes one unit of time; (iii) no two packets can traverse the same edge at the same unit of time, and (iv) packets are only allowed to queue along the edges of # Bell Laboratories, Lucent Technologies, 600-700 Mountain Ave., Murray Hill, NJ 07974-0636, USA. Part of this work was done while at the School of Computing, National University of Singapore, Singapore 119260, and was supported in part by National University of Singapore Academic Research Fund Grants RP950662, RP960620, and RP970607. E-mail: srin@research.bell-labs.com. Dept. of Decision Sciences, National University of Singapore, Singapore 119260, Republic of Singapore. Supported in part by National University of Singapore Academic Research Fund Grant RP3970021, and a Fellowship from the Singapore-MIT Alliance Program in High-Performance Computation for Engineered Systems. E-mail: fbateocp@nus.edu.sg. G during the routing stage. There are no other constraints on the paths taken by the packets, i.e., they can be arbitrary paths in G. The NP-hard objective is to select a path for each packet and to coordinate the routing so that the elapsed time by which all packets have reached their destinations is minimized; i.e., we wish to keep this routing time as small as possible. Extensive research has been conducted on this problem: see [14, 15] and the references therein. The most desirable type of algorithm here would, in addition to keeping the routing time and queue sizes low, also be distributed: given a set of incoming packets and their (source, destination) values, any switch (node of G) decides what to do with them next, without any other knowledge of the (multi-)set This would be ideal for parallel computing. (Distributed algorithms in this context are also termed on-line algorithms in the literature.) Several such ingenious results are known for specific networks such as the mesh, butterfly, or hypercube. For instance, given any routing problem with N packets on an N-node butterfly, there is a randomized on-line routing algorithm that, with high probability, routes the packets in O(log N) time using O(1)-sized queues [28]. (We let e denote the base of the natural logarithm, and, for x > 0, lg x, ln x, and respectively denote log 2 x, log e x, and max{log e x, 1}. Also, Z+ will denote the set of non-negative Good on-line algorithms here, however, are not always feasible or required, for the following reasons: . A large body of research in routing is concerned with fault-tolerance: the possibility of G being a reasonable routing network when its nodes are subject to (e.g., random or worst-case) faults. See, e.g., Kaklamanis et al. [12], Leighton, Maggs & Sitaraman [18], and Cole, Maggs & Sitaraman [6]. In this case, we do not expect good on-line algorithms, since the fault-free subgraph G of G has an unpredictable structure. Indeed, a fair amount of research in this area, e.g., [6, 18], focuses on showing that - G is still a reasonably good routing network under certain fault models, assuming global information about {(s k , t k )} and the fault structure. . Ingenious on-line algorithms for specific networks such as the butterfly in the fault-free case [28] are only existentially (near-)optimal. For instance, the O(lg N) routing time of [28] is existentially optimal to within a constant factor, since there are families of routing instances that require #(lg N) time. However, the worst-case approximation ratio can be #(lg N ). It seems very hard (potentially impossible) to devise on-line algorithms that are near-optimal on each instance. . The routing problem can be considered as a variant of unit-demand multi-commodity flow where all arc capacities are the same, queuing is allowed, and where delivery time is also a crucial criterion. (Algorithms for this problem that require just O(1) queue sizes, such as ours, will also scale with network size.) For such flow problems, the routing problems often have to be run repeatedly. It is therefore reasonable to study o#-line approximation algo- rithms, i.e., e#cient algorithms that use the knowledge of the network and of and have a good approximation ratio. Furthermore, it seems like a di#cult problem to construct on-line routing algorithms for arbitrary networks, even with, say, a polylogarithmic approximation guar- antee. See Ostrovsky and Rabani [26] for good on-line packet scheduling algorithms, given the path to be traversed for each packet. By combining some new ideas with certain powerful results of Leighton, Maggs we present the first polynomial-time o#- line constant-factor approximation algorithm for the store-and-forward packet routing problem. Furthermore, the queue sizes of the edges are bounded by O(1). No approximation algorithms with a sub-logarithmic approximation guarantee were known for this problem, to the best of our knowledge. For instance, a result from the seminal work of Leighton & Rao [19] leads to routing algorithms that are existentially good. Their network embedding of G ensures that there is some routing instance on G for which their routing time is to within an O(lg N) factor of optimal, but no good worst-case performance guarantee is known. We may attempt randomized rounding on some suitable linear programming (LP) relaxation of the problem; however, apart from di#culties like controlling path lengths, it seems hard to get a constant-factor approximation using this approach, for families of instances where the LP optimal value grows as o(lg(N + K)). Our approach uses the rounding theorem of [13] to select the set of paths that will be used in the routing algorithm of [17]. The analysis involves an interesting trade-o# between the "dilation" criterion (maximum path length) and the "congestion" criterion (maximum number of paths using any edge). 1.2. Covering Integer Programs. Let v T denote the transpose of a (column) vector v. In the second part of the paper, we continue to address the problem of simultaneously obtaining good bounds on two criteria of a problem. We focus on the NP-hard family of covering integer programs (CIPs), which includes the well-known set cover problem. This class of problems exhibits features similar to our packet routing problem: the latter can be formulated as a covering problem with side packing constraints. In CIPs, the packing constraints are upper bound constraints on the variables. Definition 1.2 (Covering Integer Programs). Given A # [0, seeks to minimize c T subject to Ax # b, x # Z n for each j (the d j # Z+ are given integers). If A # {0, 1} m-n , then we assume w.l.o.g. that each b i is a positive integer. we may assume B # 1. A CIP is uncapacitated if It is well-known that the two assumptions above are without loss of generality. (i) If A # {0, 1} m-n , then we can clearly replace each b i by #b i #. (ii) Given a CIP with some A i,j > b i , we can normalize it by first setting A i,j := b i for each such (i, j), and then scaling A and b uniformly so that #k, (b k # 1 and max # A k,# 1). This is easily seen to result in an equivalent CIP. To motivate the model, we consider a concrete CIP example: a facility location problem that generalizes the set cover problem. Here, given a digraph G, we want to place facilities on the nodes suitably so that every node has at least B facilities in its out-neighborhood. Given a cost-per-facility c j of placing facilities at node j, we desire to place the facilities in a way that will minimize the total cost. It is easy to see that this NP-hard problem is a CIP, with the matrix A having only zeroes and ones. This problem illustrates one main reason for the constraints {x j # d j }: for reasons of capacity, security, or fault-tolerance (not many facilities will be damaged if, for instance, there is an accident/failure at a node), we may wish to upper bound the number of facilities that can be placed at individual sites. The more general problem of "file sharing" in a network has been studied by Naor & Roth [24], where again, the maximum load (number of facilities) per node is balanced with the global criterion of total construction cost. For similar reasons, CIPs typically include the constraints In fact, the case where d Dobson [7] and Fisher &Wolsey [8] study a natural greedy algorithm GA for CIPs. For a given CIP, let OPT denote the value of its optimal integral solution. We define shown in [8] that GA produces a solution of value at most OPT (1 each row of the linear system Ax # b is scaled so that the minimum nonzero entry in the row is at least 1, it is shown in [7] that GA's output is at most OPT (1 Another well-known approach to CIPs is to start with their LP relaxation, wherein each x j is allowed to be a real in the range [0, d j ]. Throughout, we shall let y # denote the LP optimum of a given CIP. Clearly, y # is a lower bound on OPT . Bertsimas & Vohra [5] conduct a detailed study of approximating CIPs and present an approximation algorithm which finds a feasible solution whose value is O(y # lg m) [5]. Previous work of this paper's first author [31] presents an algorithm that computes an x # Z n such that Ax # b and for some absolute constant a 0 > 0. 1 The bound "x j # d # j may not hold for all j, but we will have for all j that for a certain absolute constant a 1 > 0. A related result is presented in [24] for file-sharing If B is "large" (greater than a certain threshold), then these results significantly improve previous results in the "global" criterion of keeping c T compromising somewhat on the "local" capacity constraints {x j # d j }. This is a common approach in bicriteria approximation: losing a small amount in each criterion to keep the maximum such loss "low". In particular, if y # grows at least as fast as me -O(B) , then the output value here is O(y # ), while maintaining x the CIP is uncapacitated, then the above is a significant improvement if B is large.) We see from (1.2) that in the case where ln the maximum "violation" are bounded by constants, which is reasonable. Thus, we consider the case where ln however, the violation can be as high as 1 which is unsatisfactory. If it is not feasible (e.g., for capacity/fault-tolerance reasons) to deviate from the local constraints by this much, then even the gain in the global criterion (caused by the large value of B) will not help justify such a result. So, a natural question is: is it possible to lose a small amount in the global criterion, while losing much less in the local criterion (i.e., in in the case where ln this in the a#rmative. (a) For the important special case of unweighted CIPs (#j, c consider the case parameter #, 0 < # < 1, we present an algorithm that outputs an x with 1 Recall that To parse the term "ln note that it is me -B , and is #(1) otherwise. (ii) the objective function value is at most a 2 y # (1/(1-#)+(1/# 2 for an absolute constant a 2 > 0. Note the significant improvement over (1.1) and (1.2), particularly if # is a con- stant: by losing just a constant factor in the output value of the objective function, we have ensured that each x j /d j is bounded by a constant (at most 1/(1 - #) This is an improvement over the bound stated in (1.2). In our view, ensuring little loss in the local criterion here is quite important as it involves all the variables x j (e.g., all the nodes of a graph in facility location) and since may be required to be low due to physical and other constraints. (b) For the case where the coe#cient matrix A has only zeroes and ones and where a feasible solution (i.e., #j, x j # d j ) to a (possibly weighted) CIP is really required, we present an approximation algorithm with output value at most O(y # This works whether not. While incomparable with the results of [7, 8], this is better if y # is bigger than a certain threshold. This is also seen to be an improvement over the O(y # lg m) bound of [5] if y # m a , where a # (0, 1) is an absolute constant. Thus, this work presents improved local vs. global balancing for a family of prob- lems: the basic packet-routing problem (the first constant-factor approximation) and CIPs (gaining more than a constant factor in the local criterion while losing a constant factor in the global criterion). The structure of the rest of the paper is as follows. In -2, we discuss the algorithm for the packet routing problem, which consists mainly of three steps: (1) constructing and solving an LP relaxation (-2.1); (2) obtaining a set of routes via suitable rounding (-2.2); and (3) scheduling the packets (-2.3) using the algorithm of [17]. The nature of our LP relaxation also provides an interesting re-interpretation of our result, as shown by Theorem 2.4 in -2.3. We discuss in -2.4 an extension of our idea to a more general setting, where the routing problem is replaced by a canonical covering problem. In -3, we discuss our results for the general covering integer programs. We present our improved local vs. global balancing for unweighted CIPs in -3.1; the case where x j # d j is really required for all j is handled in -3.2, for the case where the coe#cient matrix has only zeroes and ones. (Note, for instance, that the coe#cient matrix has only zeroes and ones for the facility location problem discussed in -1.2.) 2. Approximating the Routing Time to within a Constant Factor. We refer the reader to the introduction for the definition and motivation for packet rout- ing. Leighton, Maggs & Rao, in a seminal paper, studied the issue of scheduling the movement of the packets given the path to be traversed by each packet [16]. They showed that the packets can be routed in time proportional to the sum of "conges- tion" and "dilation" of the paths selected for each packet. However, they did not address the issue of path selection; one motivation for their work is that paths can plausibly be selected using, e.g., the well-known "random intermediate destinations" idea [33, 34]. However, no general results on path selection, and hence on the time needed for packet routing, were known for arbitrary networks G. We address this issue here by studying the paths selection problem. Theorem 2.1. There are constants c # , c # > 0 such that the following holds. For any packet routing problem on any network, there is a set of paths and a corresponding schedule that can be constructed in polynomial time, such that the routing time is at most c # times the optimal. Furthermore, the maximum queue size at each edge is bounded by c # . We shall denote any path from s k to t k as an (s k , t k )-path. Given a (directed) path P , E(P ) will denote its set of (directed) edges. 2.1. A Linear Programming Relaxation. Consider any given packet routing problem. Let us consider any feasible solution for it, where packet k is routed on path denote the dilation of the paths selected, i.e., D is the length of a longest path among the P k . Clearly, the time to route all the packets is bounded below by D. Similarly, let C denote the congestion of the paths selected, i.e., the maximum number of packets that must traverse any single edge during the entire course of the routing. Clearly, C is also a lower bound on the time needed to route the packets. Let N denote the number of nodes in the network and K the number of packets in the problem. We now present a linear programming (LP) relaxation for the problem; some of the notation used in this relaxation is explained in the following paragraph. (ROUTING) min(C +D)/2 subject to: E(G). The vector x above is basically a "fractional flow" in G, where x k f denotes the amount of "flow" of packet k on edge f # E(G). The superscript k merely indexes a packet, and does not mean a kth power. The constraints "N k x model the requirement that for packet k, (i) a total of one unit of flow leaves s k and reaches and (ii) at all other nodes, the net inflow of the flow corresponding to packet k, equals the net outflow of the flow corresponding to packet k. For conciseness, we have avoided explicitly writing out this (obvious) set of constraints above. Constraints say that the "fractional congestion" on any edge f is at most C. Constraints (2.2) say that the "fractional dilation" f , is at most D. This is a somewhat novel way of relaxing path lengths to their fractional counterparts. It is easy to see that any path-selection scheme for the packets, i.e., any integral flow (where all the x k f are either 0 or 1) with congestion C and dilation D, satisfies the above system of inequalities. Thus, since C and D are both lower bounds on the length of the routing time for such a path-selection strategy, so is (C +D)/2. Hence, the optimum value of the LP is indeed a lower bound on the routing time for a given routing problem: it is indeed a relaxation. Note that the LP has polynomial size since it has "only" O(Km) variables and O(Km) constraints, where m denotes the number of edges in the network. Thus, it can be solved in polynomial time. Let {x, C , D} denote an optimal solution to the program. In -2.2, we will conduct a certain type of "filtering" on x. Section 2.3 will then construct a path for each packet, and then invoke the algorithm of [17] for packet scheduling. 2.2. Path Filtering. The main ideas now are to decompose x into a set of "flow paths" via the "flow decomposition" approach, and then to adapt the ideas in Lin- Vitter [20] to "filter" the flow paths by e#ectively eliminating all flow paths of length more than 2D. The reader is referred to Section 3.5 of [1] for the well-known flow decomposition approach. This approach e#ciently transforms x into a set of flow paths that satisfy the following conditions. For each packet k, we get a collection Q k of flows along each Q k has at most m paths. Let P k,i denote the ith path in Q k . P k,i has an associated flow value z k,i # 0, such that for each k, words, the unit flow from s k to t k has been decomposed into a convex combination of )-paths.) The total flow on any edge f will be at most C: z the inequality in (2.4) follows from (2.1). Also, let |P | denote the length of (i.e., the number of edges in) a path P . Importantly, the following bound will hold for each k: z k,i |P k,i with the inequality following from (2.2). The main idea now is to "filter" the flow paths so that only paths of length at most 2D remain. For each k, define z k,i . It is to easy to check via (2.5) that g k # 1/2 for each k. Thus, suppose we define new flow values {y k,i } as follows for each k: y if |P k,i | # 2D. We still have the property that we have a convex combination of flow values: # i y 1. Also, since g k # 1/2 for all k, we have y k,i # 2z k,i for all k, i. So, implies that the total flow on any edge f is at most 2C: Most importantly, by setting y all the "long" paths P k,i (those of length more than 2D), we have ensured that all the flow paths under consideration are of length at most O(D). We denote the collection of flow paths for packet k by P k . For ease of exposition, we will also let yP denote the flow value of any general flow path Remarks. We now point out two other LP relaxations which can be analyzed similarly, and which yield slightly better constants in the approximation guarantee. . It is possible to directly bound path-lengths in the LP relaxation so that filtering need not be applied; one can show that this improves the approximation guarantee somewhat. On the other hand, such an approach leads to a somewhat more complicated relaxation, and furthermore, binary search has to be applied to get the "optimal" path-length. This, in turn, entails potentially O(lg N) calls to an LP solver, which increases the running time. Thus, there is a trade-o# involved between the running time and the quality of approximation. . In our LP formulation, we could have used a variable W to stand for max{C, D} in place of C and D; the problem would have been to minimize W subject to the fractional congestion and dilation being at most W . Since W is a lower bound on the optimal routing time, this is indeed a relaxation; using our approach with this formulation leads to a slightly better constant in the quality of our approximation. Nevertheless, we have used our approach to make the relationship between C and D explicit. 2.3. Path Selection and Routing. Note that } is a fractional feasible solution to the following set of inequalities: To select one path from P k for each packet k, we need to modify the above fractional solution to an integral 0-1 solution. To ensure that the congestion does not increase by much, we shall use the following rounding algorithm of [13]: Theorem 2.2. ([13]) Let A be a real valued r - s matrix, and y be a real-valued s-vector. Let b be a real valued vector such that Ay = b and t be a positive real number such that, in every column of A, (i) the sum of all the positive entries is at most t and (ii) the sum of all the negative entries is at least -t. Then we can compute an integral vector y such that for every i, either y Furthermore, if y contains d non-zero components, the integral approximation can be obtained in time O(r 3 lg(1 To use Theorem 2.2, we first transform our linear system above to the equivalent system: The set of variables above is }. Note that yP # [0, 1] for all these variables. Furthermore, in this linear system, the positive column sum is bounded by the maximum length of the paths in P 1 #P K . Since each path in any P k is of length at most 2D due to our filtering, each positive column sum is at most 2D. Each negative column sum is clearly -2D. Thus, the parameter t for this linear system, in the notation of Theorem 2.2, can be taken to be 2D. Hence by Theorem 2.2, we can obtain in polynomial time an integral solution y satisfying For each packet k, by conditions (2.8) and (2.9), we have 1. (Note the crucial role of the strict inequality in (2.8).) Thus, for each packet k, we have selected at least one path from s k to t k , with length at most 2D; furthermore, the congestion is bounded by 2C+2D (from (2.7)). If there are two or more such )-paths, we can arbitrarily choose one among them, which of course cannot increase the congestion. The next step is to schedule the packets, given the set of paths selected for each packet. To this end, we use the following result of [17], which provides an algorithm for the existential result of [16]: Theorem 2.3. ([17]) For any set of packets with edge-simple paths having congestion c and dilation d, a routing schedule having length O(c d) and constant maximum queue size, can be found in random polynomial time. Applying this theorem to the paths selected from the previous stage, which have dilation d # 2D, we can route the packets in time D). Recall that (C + D)/2 is a lower bound on the length of the optimal schedule. Thus, we have presented a constant-factor approximation algorithm for the o#-line packet routing problem; furthermore, the queue-sizes are also bounded by an absolute constant, in the routing schedule produced. An interesting related point is that our LP relaxation is reasonable: its integrality gap (worst-case ratio between the optima of the integral and fractional versions) is bounded above by O(1). An Alternative View. There is an equivalent interesting interpretation of Theorem 2.1: Theorem 2.4. Suppose we have an arbitrary routing problem on an arbitrary graph let L be any non-negative parameter (e.g., O(1), O(lg n), O( # n)). K} be the set of source-destination pairs for the packets. Suppose we can construct a probability distribution D k on the (s k , t k )-paths for each k such that if we sample, for each packet k, an (s k , t k )-path from D k independently of the other packets, then we have: (a) for any edge e # E(G), the expected congestion on e is at most L, and (b) for each k, the expected length of the (s k , t k )-path chosen is at most L. Then, there is a choice of paths for each packet such that the congestion and dilation are O(L). Thus, the routing can be accomplished in O(L) time using constant-sized queues; such a routing can also be constructed o#-line in time polynomial in |V | and K. We remark that the converse of Theorem 2.4 is trivially true: if an O(L) time routing can be accomplished, we simply let D k place all the probability on the (s k , t k )- path used in such a routing. Proof of Theorem 2.4: Let # k P denote the probability measure of any (s k , t k )- path P under the distribution D k . Let supp(D k ) denote the support of D k , i.e., the set of (s k , t k )-paths on which D k places nonzero probability. The proof follows from the fact that for any (i, is a feasible solution to (ROUTING), with C, D replaced by L. Hence by our filter- round approach, we can construct one path for each packet k such that the congestion and dilation are O(L). As seen above, the path selection and routing strategies can be found in polynomial time. We consider the above interesting because many fault-tolerance algorithms use very involved ideas to construct a suitable (s k , t k )-path for (most) packets [6]. These paths will need to simultaneously have small lengths and lead to small edge congestion. Theorem 2.4 shows that much more relaxed approaches could work: a distribution that is "good" in expectation on individual elements (edges, paths) is su#cient. Recall that in many "discrete ham-sandwich theorems" (Beck & Spencer [4], Raghavan & Thompson [27]), it is easy to ensure good expectation on individual entities (e.g., the constraints of an integer program), but is much more di#cult to construct one solution that is simultaneously good on all these entities. Our result shows one natural situation where there is just a constant-factor loss in the process. 2.4. Extensions. The result above showing a constant integrality gap for packet routing, can be extended to a general family of combinatorial packing problems as follows. Let S k be the family of all the subsets of vertices S such that s k # S and S. Recall that the (s k , t k )-shortest path problem can be solved as an LP via the following covering formulation: c subject to: (i,j)#E: i#S,j / #S E(G). Constraint (2.10) expresses the idea that "flow" crossing each s-t cut is at least 1. The following is an alternative relaxation for the packet routing problem: (ROUTING-II) min(C +D)/2 subject to: (i,j)#E: i#S,j / #S We can use the method outlined in Section 2.1, 2,2 and 2.3 to show that the optimal solution of (ROUTING-II) is within a constant factor of the optimal routing time. A natural question that arises is whether the above conclusion holds for more general combinatorial packing problems. To address this question, we need to present an alternative (polyhedral) perspective of our (path) selection routine. First we recall some standard definitions from polyhedral combinatorics. Suppose we are given a finite set family F of subsets of N . For any S # N , let #S # {0, 1} n denote the incidence vector of S. We shall consider the problem is a weight function on the elements of N . Definition 2.5. ([25]) The blocking clutter of F is the family B(F), whose members are precisely those H # N that satisfy: P1. P2. Minimality: If H # is any proper subset of H, then H # violates property P1. A natural LP relaxation for (OPT Q is known as the blocking polyhedron of F . The following result is well-known and easy to check: F such that #i # F, x i # 1}. For several classes of clutters (set-systems), it is known that the extreme points of Q are the integral vectors that correspond to incidence vectors of elements in F . By Minkowski's Theorem [25], every element in Q can be expressed as a convex combination of the extreme points and extreme rays in Q. For blocking polyhedra, the set of rays is Suppose we have a generic integer programming problem that is similar to (ROUTING- II), except for the fact that for each k, (2.11) is replaced by the constraint F k can be any clutter that is well-characterized by its blocking polyhedron Q k (i.e., the extreme points of the blocking polyhedron Q k are incidence vectors of the elements in the clutter F k ). Thus, we have a generalization of (ROUTING-II): subject to: Note that the variables x are now indexed by elements of the set N . In the previously discussed special cases, the elements of N are edges, or pairs of nodes. The LP relaxation of (BLOCK) replaces the constraint (2.13) by Theorem 2.6. The optimum integral solution to (BLOCK) has a value that is at most a constant factor greater than the optimal value to its LP relaxation. Proof. Let denote an optimal solution to the LP relaxation. By Caratheodory's Theorem [25], for each fixed k, can be expressed as a convex combination of extreme points and extreme rays of the blocking polyhedron Q k . However, note that the objective function can only improve by decreasing the value of long as the solution remains feasible. Furthermore, the extreme rays of the blocking polyhedron correspond to vectors v with each v i non-negative. Thus, without loss of generality, we may assume that the LP optimum is lexicographically minimal. This ensures that the optimal solution can be expressed as a convex combination of the extreme points of the polyhedron alone. As seen above, the extreme points in this case are incidence vectors of elements of the k-th clutter (we use polyhedral language to let "k-th clutter" denote the set-system F k ). Let C and D denote the fractional "congestion" and fractional "dilation" of the optimal solution obtained by the LP relaxation of (BLOCK). Let A k 2 , . denote incidence vectors of the elements in the k-th clutter, and let A k (i) be the ith coordinate of A k . Then we have a convex combination, for each k: Thus, by constraints (2.12), # j:|A k By filtering out those A k j with size greater than 2D, we obtain a set of canonical objects for each k, whose sizes are at most 2D. By scaling the # k j by a suitable factor, we also obtain a new set of # k j such that . Using these canonical objects and {# k } as the input to Theorem 2.2, we obtain a set of objects (one from each clutter) such that the dilation is not more than 2D and the congestion not more than 2(C +D). Hence the solution obtained is at most O(1) times the LP optimum. Remark. As pointed out by one of the referees, it is not clear whether the lexicographically minimal optimal solution can be constructed in polynomial time. The above result is thus only about the quality of the LP relaxation. It would be nice to find the most general conditions under which the above can be turned into a polynomial-time approximation algorithm. 3. Improved Local vs. Global Balancing for Covering. Coupled with the results of [16, 17], our approximation algorithm for the routing time (a global crite- rion) also simultaneously kept the maximum queue size (a local capacity constraint) constant; our approach there implicitly uses the special structure of the cut covering formulation. We now continue the study of such balancing in the context of covering integer programs (CIPs). The reader is referred to -1.2 for the relevant definitions and history of CIPs. In -3.1, we will show how to approximate the global criterion well without losing much in the "local" constraints {x j # d j }. In -3.2, we present approximation algorithms for a subfamily of the CIPs where x j # d j is required for all j. One of the key tools used in -3.1 and -3.2 is Theorem 3.3, which builds on an earlier rounding approach (Theorem 3.2) of [31]. 3.1. Balancing Local with Global. The main result of -3.1 is Corollary 3.5. This result is concerned with unweighted CIPs, and the case where ln In this setting, Corollary 3.5 shows how the local capacity constraints can be violated much less in comparison with the results of [31], while keeping the objective function value within a constant factor of that of [31]. Let exp(x) denote e x ; given any non-negative integer k, let [k] denote the set {1, 2, . , k}. We start by reviewing the Cherno#-Hoe#ding bounds in Theorem 3.1. Let G(-, #) . Theorem 3.1. ([23]) Let X 1 , X 2 , . , X # be independent random variables, each taking values in [0, 1], We shall use the following easy fact: From now on, we will let {x [n]} be the set of values for the variables in an arbitrary feasible solution to the LP relaxation of the could be an optimal LP solution.) Let y Recall that the case handled well in [31]; thus we shall assume B. We now summarize the main result of [31] for CIPs as a theorem: Theorem 3.2. ([31]) For any given CIP, suppose we are given any 1 # < # such that holds. Then we can find in deterministic polynomial time, a vector of non-negative integers such that: (a) (Az) i # b i # for each i # [m], (b) y #, and (c) z j #x # j #d j # for each j # [n]. The next theorem presents a rounding algorithm by building on Theorem 3.2: Theorem 3.3. There are positive constants a 3 and a 4 such that the following holds. Given any parameter #, 0 < # < 1, let # be any value such that # (a 3 Then we can find in deterministic polynomial time, a vector non-negative integers such that: (a) (Az) i # b i #(1- #) for each i # [m], (b) c T z # a 4 y #, and (c) z j #x # j #d j # for each j # [n]. Remark. It will be shown in the proof of Theorem 3.3 that we can choose, for instance, a 2. Since there is a trade-o# between a 3 and a 4 that can be fine-tuned for particular applications, we have avoided using specific values for a 3 and a 4 in the statement of Theorem 3.3. The following simple proposition will also be useful: Proposition 3.4. If 0 < x < 1/e, then 1 - x > exp(-1.25x). Proof of Theorem 3.3: We choose a 2. In the notation of Theorem 3.2, we take #(1 - #) and Our goal is to validate (3.2); by (3.1), it su#ces to show that exp(-y Note that the left- and right-hand sides of (3.3) respectively decrease and increase with increasing #; thus, since # 0 it is enough to prove (3.3) for # 0 . We consider two cases. Case I. 1/e. So, Proposition 3.4 implies that in order to prove (3.3), it su#ces to show that i.e., that y # 2 # 1.25m exp(-1.5B). This is true from the facts that: (i) m/y # (which follows from the fact that ln(m/y # Case II. it su#ces to show that exp(-y Recall that 1. So, we have mB/y # > e, i.e., y # /(mB) < 1/e. Thus, The inequality follows from Proposition 3.4. So, to establish (3.4), we just need show that i.e., that e, which in turn follows from the facts that # 1. This completes the proof. Our required result is: Corollary 3.5. Given any unweighted CIP with any parameter #, 0 < # < 1, we can find in deterministic polynomial time, a vector non-negative integers such that: (a) Av # b, (b) a is an absolute constant, and Proof. Let #(a 3 z be as in the statement of Theorem 3.3. for each j. Conditions (a) and (c) are easy to check, given Theorem 3.3. Since the z j 's are all non-negative integers and since the CIP is unweighted condition (b) of Theorem 3.3 shows that at most a 4 y # of them can be nonzero. Thus, condition (b) follows since v j # z j /(#(1-#))+1 if z j > 0 and since v As mentioned in -1, this improves the value of [31] to O(1/(1-#)), while keeping (c T relatively small at O((1/# 2 )-ln (as long as # is a constant bounded away from 1). 3.2. Handling Stringent Constraints. We now handle the case where the constraints have to be satisfied and where the coe#cient matrix A has only zeroes and ones. Recall from -1 that there is a family of facility location problems where the coe#cient matrix has only zeroes and ones; this is an example of the CIPs to which the following results apply. We start with a technical lemma. Lemma 3.6. For any # > 0, the sum is at most u Proof. If be the highest index such that u r < #/e. Thus, s ur it follows that t r # u r ln(#/u r the last inequality follows from the fact that for any x # y such that x < #/e, The following simple proposition will also help. Proposition 3.7. For any # > 0 and # 1, Proof. The proposition is immediate if # e. Next note that for any a # e, the function g a decreases as x increases from 1 to infinity. So, if # e and e, then Finally, if # > e and # > e, then (ln Theorem 3.8. Suppose we are given a CIP with the matrix A having only zeroes and ones. In deterministic polynomial time, we can construct a feasible solution z to the CIP with z j # d j for each j, and such that the objective function value c T - z is O(y Proof. Let a 3 and a 4 be as in the proof of Theorem 3.3. Define a and, for any S # [n], y # Starting with S we construct a sequence of sets S 0 # S 1 # - as follows. Suppose we have constructed S . If S #, we stop; or else, if all j # S i satisfy a 5 stop. If not, define the proper subset S i+1 of S i to be {j # to be d j # x note that for all such j, t be the final set we construct. If S #, we do nothing more; since z j # x # j for all j, we will have Az # b as required. Also, it is easy to check that z j # d j for all j. So suppose S t #. Let we stopped at the non-empty set we see that #x # j # d j for all j # S t . Recall that for all j # S t , we have fixed the value of z j to be d j # x # j . Let w denote the vector of the remaining variables, i.e., the restriction of x # to S t . Let A # be the sub-matrix of A induced by the columns corresponding to S t . We will now focus on rounding each x # j (j # S t ) to a suitable non-negative integer z j # d j . for each i # [m], A since z j # x # j for all j # S t , we get A Since each b i and A i,j is an integer, so is each b # i . Suppose b # i # 0 for some i. Then, whatever non-negative integers z j we round the j # S t to, we will satisfy the constraint So, we can ignore such indices i and assume without loss of generality that B # . constraints corresponding to indices i with b # i # 0 can be retained as "dummy constraints".) Proposition 3.7 shows that i.e., that # (a 3 Thus, by Theorem 3.3, we can round each x # j (j # S t ) to some non-negative integer z j #x # j # d j in such a manner that the last inequality (i.e., that #(1 - # 1/2) follows from the fact that # a 5 # 2. So we can check that the final solution is indeed feasible. We only need to bound the objective function value, which we proceed to do now. We first bound Fix any i, 0 # i # t - 1. Recall that for each j # (S i - S i+1 ), we set z a Setting substituting (3.7) into (3.6), where gives the final objective function value. Else shows that This, in combination with (3.8) and Lemma 3.6, shows that This completes the proof. 4. Conclusion. In this paper, we analyze various classes of problems in the context of balancing global versus local criteria. Our main result is the first constant-factor approximation algorithm for the o#- line packet routing problem on arbitrary networks: for certain positive constants c # and c # , we show that given any packet routing problem, the routing time can e#ciently be approximated to within a factor of c # , while ensuring that all edge-queues are of size at most c # . Our result builds on the work of [16, 17], while exploiting an interesting trade-o# between a (hard) "congestion" criterion and an (easy) "dilation" criterion. Furthermore, we show that the result can be applied to a more general setting, by providing a polyhedral perspective of our technique. Our approach of appropriately using the rounding theorem of [13] has subsequently been applied by Bar-Noy, Guha, Naor & Schieber to develop approximation algorithms for a family of multi-casting problems [3]. It has also been applied for a family of routing problems by Andrews & Zhang [2]. The second major result in the paper improves upon a class of results in multi-criteria covering integer programs. We show that the local criterion of unweighted covering integer programs can be improved from an approximately logarithmic factor to a constant factor, with the global criterion not deteriorating by more than a constant factor (i.e., we maintain a logarithmic factor approximation). The third main result improves upon a well-known bound for covering integer pro- grams, in the case where the coe#cient matrix A has only zeroes and ones. We show that the approximation ratio can be improved from O(y # lg m) to O(y # Some open questions are as follows. It would be interesting to study our packet- routing algorithm empirically, and to fine-tune the algorithm based on experimental observation. It would also be useful to determine the best (constant) approximation possible in approximating the routing time. An intriguing open question is whether there is a distributed packet-routing algorithm with a constant-factor approximation guarantee. Finally, in the context of covering integer programs, can we approximate the objective function to within bounds such as ours, with (essentially) no violation of the local capacity constraints? Acknowledgements . We thank Bruce Maggs, the STOC 1997 program committee and referee(s), and the journal referees for their helpful comments. These have helped improve the quality of this paper a great deal. In particular, one of the journal referees simplified our original proof of Lemma 3.6. --R Network flows: theory Packet routing with arbitrary end-to-end delay requirements Integral approximation sequences. Rounding algorithms for covering problems. Routing on butterfly networks with random faults. On the greedy heuristic for continuous covering and packing problems. Correlational inequalities for partially ordered sets. Blocking Polyhedra. Scheduling to minimize average completion time: O Asymptotically tight bounds for computing with faulty arrays of processors. Global wire routing in two-dimensional arrays Introduction to Parallel Algorithms and Architectures: Arrays Methods for message routing in parallel machines. Packet routing and job-shop scheduling in O(congestion+dilation) steps Fast algorithms for finding O(congestion On the fault tolerance of some popular bounded-degree networks An approximate max-flow min-cut theorem for uniform multi-commodity flow problems with applications to approximation algorithms Scheduling n independent jobs on m uniform machines with both flow time and makespan objectives: a parametric approach. Randomized Algorithms. Optimal file sharing in distributed networks. Universal O(congestion Randomized rounding: a technique for provably good algorithms and algorithmic proofs. How to emulate shared memory. Improved approximation guarantees for packing and covering integer programs. On the existence of schedules that are near-optimal for both makespan and total weighted completion time A scheme for fast parallel communication. Universal schemes for parallel communication. --TR --CTR Stavros G. Kolliopoulos , Neal E. Young, Approximation algorithms for covering/packing integer programs, Journal of Computer and System Sciences, v.71 n.4, p.495-505, November 2005 Stavros G. Kolliopoulos, Approximating covering integer programs with multiplicity constraints, Discrete Applied Mathematics, v.129 n.2-3, p.461-473, 01 August Konstantin Andreev , Bruce M. Maggs , Adam Meyerson , Ramesh K. Sitaraman, Designing overlay multicast networks for streaming, Proceedings of the fifteenth annual ACM symposium on Parallel algorithms and architectures, June 07-09, 2003, San Diego, California, USA

linear programming;approximation algorithms;multicommodity flow;packet routing;rounding theorems;randomized algorithms;covering integer programs;discrete ham-sandwich theorems;randomized rounding

586898

On Bipartite Drawings and the Linear Arrangement Problem.

The bipartite crossing number problem is studied and a connection between this problem and the linear arrangement problem is established. A lower bound and an upper bound for the optimal number of crossings are derived, where the main terms are the optimal arrangement values. Two polynomial time approximation algorithms for the bipartite crossing number are obtained. The performance guarantees are O(log n) and O(log2 n) times the optimal, respectively, for a large class of bipartite graphs on n vertices. No polynomial time approximation algorithm which could generate a provably good solution had been known. For a tree, a formula is derived that expresses the optimal number of crossings in terms of the optimal value of the linear arrangement and the degrees, resulting in an O(n1.6) time algorithm for computing the bipartite crossing number.The problem of computing a maximum weight biplanar subgraph of an acyclic graph is also studied and a linear time algorithm for solving it is derived. No polynomial time algorithm for this problem was known, and the unweighted version of the problem had been known to be NP-hard, even for planar bipartite graphs of degree at most 3.

Introduction The planar crossing number problem calls for placing the vertices of a graph in the plane and drawing the edges with Jordan curves, so that the number of edge crossings is minimized. This problem has been extensively studied in graph theory [32], combinatorial geometry [22], and theory of VLSI [16]. In this paper we study the bipartite crossing number problem which is an important variation of the planar crossing number. Throughout this paper E) denotes a connected bipartite graph, are the two classes of independent vertices, and E is the edge set. We will assume that # The research of the first author was supported by NSF grant CCR-9528228. The research of the second and fourth authors was supported in part by the Alexander von Humboldt Foundation and by the Slovak Scientific Grant Agency grant No. 95/5305/277. Research of the third author was supported in part by the Hungarian NSF contracts T 016 358 and T 019 367, and by the NSF contract DMS 970 1211. A preliminary version of this paper was published at WADS'97. m. A bipartite drawing [13], or 2-layer drawing of G consists of placing the vertices of V 0 and V 1 into distinct points on two parallel lines and then drawing each edge using a straight line segment connecting the points representing the endvertices of the edge. Let bcr(G) denote the bipartite crossing number of G, that is, bcr(G) is the minimum number of edge crossings over all bipartite drawings of G. Computing bcr(G) is NP-hard [11] 1 but can be solved in polynomial time for bipartite permutation graphs [29]. The problem of obtaining nice multiple layer drawings of graphs (i.e. drawings with small number of crossings), has been extensively studied by the graph drawing, VLSI, and CAD communities [6, 7, 19, 30, 31]. In particular one of the most important aesthetic objectives in graph drawing is reducing the number of crossings [23]. Very recently J-unger and Mutzel, [14] and Mutzel [20] succeeded to employ integer programming methods in order to compute bcr(G) exactly, or to estimate it, nevertheless, these methods do not guarantee polynomial time convergence. In fact, although a O(log 4 n) times optimal polynomial time algorithm for approximating the planar crossing number of degree bounded graphs has been known [17], no polynomial time approximation algorithm whose performance is guaranteed has been previously known for approximating bcr(G). A nice result in this area is a fast polynomial time algorithm of Eades and Wormald [7] which approximates the bipartite crossing number by a factor of 3, when the positions of vertices in V 0 are fixed. In this paper we explore an important relationship between the bipartite drawings and the linear arrangement problem, which is another well-known problem in the theory of VLSI [4, 5, 15, 18, 28]. In particular, it is shown that for many graphs the order of magnitude for the optimal number of crossings is bounded from below, and above, respectively, by minimum degree times the optimal arrangement value, and by arboricity times the optimal arrangement value, where the arboricity of G is the minimum number of acyclic graphs that G can be decomposed to. Hence for a large class of graphs, it is possible to estimate bcr(G) in terms of the optimal arrangement value. Our general method for constructing the upper bound is shown to provide for an optimal solution and an exact formula, resulting to an O(n 1.6 computing bcr(G) when G is a tree. The presence of arboricity in our upper bound allows us to relate some important topological properties such as genus and page number, to bcr(G). In particular, our results easily imply that when G is "nearly planar", i.e. it either has bounded genus, or bounded page number, then, the asymptotic values of bcr(G), and the optimal arrangement are the same, provided that G is not too sparse. A direct consequence of our results is that for many graphs, the bipratite drawings with small sum of edge lenghts also have small bipartite crossings, and vis versa, and therefore, a suboptimal solution to the bipartite crossing number problem can be extracted from a suboptimal solution to the linear arrangement problem. Hence, we have derived here, the first polynomial time approximation algorithms for bcr(G), which perform within a multiplicative factor of O(log n log log n) from the optimal, for a large class of graphs. Moreover, we show here that the traditional divide and conquer paradigm in which the divide phase approximately bisects the graph, also obtains a provably good approximation, in polynomial time, for bcr(G) within a multiplicative factor of O(log 2 n) from the optimal, for a variety of graphs. Crucial to verifying the performance guarantee of the divide and conquer algorithm, is a lower bound of # G nb # (G)), derived here, for bcr(G), where b # (G), # < 1/2, and # G are the size of the #-bisection and minimum degree of G, respectively. This significantly improves Leighton's well-known lower bound of # b 23 (G)) [16] which was derived for the planar crossing number of degree bounded graphs. The class of graphs for which the performance of our approximation algorithms is guaranteed is very large, and in particular contains those regular graphs, degree bounded graphs, and genus bounded graphs, which are not too sparse. Another notable aspect of relating bcr(G) to the linear arrangement problem is that, both algorithms produce drawings with near optimal number of crossings in which the coordinates of all vertices are integers, so that the total edge length is also Technically speaking, the NP-hardness of the problem was proved for multigraphs, but it is widely assumed that it is also NP-hard for simple graphs. near optimal, with the same performance guarantee as for the number of crossings. We also study biplanar graphs. A bipartite graph E) is called a biplanar, if it has a bipartite drawing in which no two edges cross each other. Eades and Whitesides [8] have shown that the problem of determining largest biplanar subgraph is NP-hard even when G is planar, and the vertices in V 0 and V 1 have degrees at most 3 and 2, respectively. This raised the question of whether or not computing a largest biplanar subgraph can be done in polynomial time when G is acyclic [20]. In this paper we present a linear time dynamic programming algorithm for the weighted version of this problem in an acyclic graph. (The weighted version was first introduced by Mutzel [20].) Section 2 contains our general results regarding the relation between bcr(G) and the linear arrangement problem. Section 3 contains the applications, and includes several important observations, the bisection based lower bound for bcr(G), and the approximation algorithms. Finally, Section 4 contains our linear time algorithm for computing a largest biplanar subgraph of a tree. Linear arrangement and bipartite crossings We denote by d v the degree of v, and by d # v denote the number vertices adjacent to v of degree 1. We denote by # G the minimum degree of G. A bipartite drawing of G is obtained by: (i) placing the vertices of V 0 and V 1 into distinct points on two horizontal lines y 0 , y 1 , respectively, (ii) drawing each edge with one straight line segment which connects the points of y 0 and y 1 where the endvertices of the edge were placed. Hence, the order in which the vertices are placed on y 0 and y 1 will determine the drawing. Let DG be a bipartite drawing of G; when the context is clear, we omit the subscript G and write D. For any e # E, let bcr D (e) denote the number of crossings of the edge e with other edges. Edges sharing an endvertex do not count as crossing edges. Let bcr(D) denote the total number of crossings in D, i.e. The bipartite crossing number of G, denoted by bcr(G) is the minimum number of crossings of edges over all bipartite drawings of G. Clearly, We assume throughout this paper that the vertices of V 0 are placed on the line y 0 which is taken to be the x-axis, and vertices of V 1 are placed on the line y 1 which is taken to be the line For a vertex x-coordinate in the drawing D. We call the function the coordinate function of D. Throughout this paper, we often omit the y coordinates. Note that xD is not necessarily an injection, since for a # V 0 , and b # V 1 , we may have xD (a) = xD (b). Given an arbitrary graph E), and a real function define the length of f , as The linear arrangement problem is to find a bijection of minimum length. This minimum value is denoted by - L(G). E) and D be a bipartite drawing of G. Define the length of D to be In this section we derive a relation between the bipartite crossing number and the linear arrangement problem. Let D be a bipartite drawing of E) such that the vertices of V 0 are placed into the points (1, 0), (2, 0), ., (|V 0 |, 0). For its neighbors satisfying xD the median vertex of v, We say that D has the median property if the vertices of G have distinct x-coordinates and the x-coordinate of any vertex v in V 1 is larger than, but arbitrarily close to, xD (med(v)), with the restriction that if a vertex of odd degree and a vertex of even degree have the same median vertex, then the odd degree vertex has a smaller x-coordinate. Note that if D has the median property, then xD is an injection. When the bipartite drawing D does not have the median property, one can always convert it to a drawing which has the property, by first placing the vertices of V 0 in the same order in which they appear in D into the locations (1, 0), (2, 0), ., (|V 0 |, 0), and then placing each v # V 1 on a proper position so that the median property holds. Such a construction is called the median construction and was utilized by Eades and Wormald [7] to obtain the following remarkable result. Theorem 2.1 [7] Let E), and D be a bipartite drawing of G. If D # is obtained using the median construction from D, then E) and D be a bipartite drawing of G. Consider an edge let u be a vertex in V 0 # V 1 so that u / # {a, b}. We say e covers u in D, if the line parallel to the y axis passing through u has a point in common with the edge e. Thus for neither a nor b are covered by e. However, a vertex c # V 1 with xD (c) = xD (a) is covered by e. Let ND (e) denote the number of those vertices in V 1 which are covered by e in D. We will use the following two lemmas later. Lemma 2.1 For let D be a bipartite drawing of G. Recall that xD is the coordinate function of D. Then, the following hold. (i) Assume that xD (v) is an integer for all x # V 0 . Then, there is a bijection f so that for any e = ab # E, it holds (ii) Assume that D has the median property. Then for the bijection f # in (i), it holds d a d # a m. Proof. To prove (i), we construct f # by moving all vertices in V to integer locations. Formally, let be the order of vertices of V that we may have xD (w i since xD may not be an injection.) the proof of (i) easily follows. (In particular note that the factor +1 appears in the upper bound, since the end point of e which belongs to V 1 may not have an integer coordinate.) For (ii), let Assume x(a) > x(b), and let v be any vertex in V 1 covered by e in D. Since D has the median property, at least #d v /2# of vertices adjacent to v are separated from v in D by the straight line segment e. This means, in this case, that vertex v generates at least # G /2# G - 1)/2 crossings on e. Moreover, vertex v, even if it has degree 1, generates one crossing on e, since v and med(v) are separated by the line segment e in D. Thus G+1. Now assume xD (a) < xD (b), and let v be a vertex covered by e. Then, v generates at least d v - # dv crossings on e provided that v is not a vertex of degree 1 which is adjacent only to a. Consequently, in this case, bcr D (e) # (ND (e) - d # a We conclude that in either case, bcr D a , and consequently, using (i), To finish the proof of (ii) take the sum over all Lemma 2.2 Let E), and let D be a bipartite drawing of G which has the median property, then dv #2 with an arbitrary small # > 0. Proof. To prove the claim, let uv # E with has the median property, thus v is placed arbitrary close to u. So we may assume that |x D (v) - xD (u)| # This way the total sum of the contributions of all edges which are incident to a vertex of degree one in V 1 to L xD is at most |V 1 | # # and the claim follows. 2 We now prove the main result of this section. Theorem 2.2 Let L(G). Proof. Let D be a bipartite drawing of G. We will construct an appropriate bijection f {1, 2, ., n}. Let D # be a drawing which is obtained by applying the median construction to D. Let its neighbors with xD # i be an integer, 1 # i #d v /2#, and let u be a vertex in V 0 so that xD # Observe that u generates d u crossings on the edges u i v and u dv -i+1 v, if it is not adjacent to v. Similarly, u generates d u - 1 crossings on the edges u i v and u dv-i+1 v, if it is adjacent to v. Thus Note that D # has the median property, thus for and hence (1) implies Using (2) observe that, for (bcr Thus, using (3), when d v # 2 is even, we have dv (bcr dv Moreover, when d v # 2 is odd, we have, dv where the upper bound is obvious, and the lower bound holds since no vertex adjacent to v is between and u . Consequently, when d v # 2 is odd, we have, dv where the last line is obtained by observing that xD # (u Combining this with (3), for odd d v , we obtaindv dv We note that since (5) is weaker than (4), it must also hold when d v is even, and conclude by summing dv #2 dv #2 v . Using Lemma 2.2, we get v . (6) Consider the bijection f # in Part (ii) of Lemma 2.1. Then Observe that # G # 2 implies P v#V 0 Hence (6) implies v . (7) Observing that L f # - , and v , we obtain which finishes the proof. 2 Next, we investigate the cases for which the error term P v#V d 2 v can be eliminated from Theorem 2.2. Corollary 2.1 Let E) so that m # (1 # and # are positive constants. Then Proof. To prove the result we will first show that for any bipartite drawing D of G it holds, For now assume that (8) holds. It is easy to see that bcr(G) # m- 1+# m, we conclude that 1)bcr(G). Combining this inequality with (8), we obtain v , and thus and the claim follows from Theorem 2.2. To prove (8), let D be any bipartite drawing of G, and let v # V 0 so that d v - d # v # 2. Let be the set of vertices of degree at least 2 which are adjacent to v, and assume with no loss of generality that xD be an integer, 1 # i # dv -d # v and note that any vertex u generates at least one crossing on the edges and u dv-i+1 v. Thus bcr(vu 2 #, and therefore We conclude that by summing Similarly we can show that 2bcr(D) # ( P v#V 0 hence the claim follows. 2 Remarks. The conditions of Corollary 2.1, involving # and # are not restrictive at all. For instance, any bipartite graph of minimum degree at least 3, satisfies the conditions. We identify more additional graphs which satisfy these conditions in Section 3. 2.2 An upper bound We now derive an upper bound on bcr(G). We need the following obvious lemma. Lemma 2.3 Let D be a bipartite drawing of 1 be two edges which cross in D. Assume that |x D (v) - xD (u)| # |x D (a) - xD (b)|, then either a or b is covered by e in D. Moreover, if a is covered by e, then if b is covered by e, then |x D (a) - xD (v)| # |x D (v) - xD (u)|.Let VH and EH , denote the vertex set and the edge set of a subgraph H, of G. The arboricity of G, denoted by aG , is maxH # #, where the maximum is taken over all subgraphs H, with 2. Note that # G /2 # aG #G , where #G denotes the maximum degree of G. A well-known theorem of Nash-Williams [21] asserts that aG is the minimum number of edge disjoint acyclic subgraphs that edges of G can be decomposed to. Theorem 2.3 Let L(G). Proof. Consider a solution (not necessarily optimal) of the linear arrangement of G, realized by a bijection n}. The mapping f # induces an ordering of vertices of V y 0 . Lift up the vertices of V 1 into y 1 and draw the edges with respect to the new locations of these vertices to obtain a bipartite drawing D. Note that for this drawing D. Let I e to be the set all edges crossing e in D so that for any ab # I e , Observe that if any edge e # / # I e crosses e, then e # I e # . Hence, in this case the crossing of e and e # contributes one to |I e # |. We conclude that |I e |, and will show that |I e | # aG (4|x D (u) - xD (v)| 1). For ebe the set of all those vertices y of V 0 so that |x D (y) - xD (v)| # |x D (u) - xD (v)|. Similarly, let ebe the set of all those vertices y of V 1 so that |x D (y) - xD (u)| # |x D (u) - xD (v)|. Note that, since the coordinates of all vertices are integers. Therefore, we have 2. Let - observe that by Lemma 2.3, a # V e 1 . Consequently, |I e | # is the edge set of the induced subgraph of G on the vertex set V e by the definition of aG , and thus I e # aG (4L xD +m). To complete the proof we take f # to be the optimal solution to the linear arrangement problem, that is, 2.3 Bipartite crossings in trees We note that if aG is small, then, the gap between the upper bound and the lower bound in Theorems 2.2 and 2.3 is small, and hence, it is natural to investigate the case In fact, in this case the method in the proof of Theorem 2.3 provides for an optimal bipartite drawing. Theorem 2.4 Let T be a tree on the vertex set are the partite sets, and be a bijection utilizing the optimal solution to the linear arrangement problem. Let D # be a bipartite drawing constructed by the method of Theorem 2.3, that is, by lifting the vertices in V 1 into the line Proof. We prove the Theorem by induction on n. The result is true for 2. Let n # 3. Assume that the Theorem is true for all l-vertex trees, l < n, and let T be a tree on n vertices. We first show that the RHS of (11) is a lower bound on bcr(T ). We then show that bcr(D # ) equals to RHS of (11). Consider an optimal bipartite drawing D of T . It is not di#cult to see that one of the leftmost (rightmost) vertices is a leaf. Denote the left leaf by v 0 , the right leaf by v k , and let be the path between v 0 and v k . Note that P will cross any edge in T which is not incident to v i , path P will generate at least crossings, where c P counts exactly the number of edges in T (in D) which are not incident to any vertex on P . Deleting the edges of P we get trees T i , on the vertex set V rooted in 1. Consider the optimal bipartite drawings of T i , place them consecutively such that T i does not cross T j , for i #= j. Then draw the path P without self crossings such that v 0 (v k ) is placed to the left (right) of the drawing of T 1 (T k-1 ). Then clearly the number of crossings in this new drawings is P k-1 so we conclude that for otherwise D is not an optimal drawing. For any v # V , let d i denote the degree of v in T i ; applying the inductive hypothesis to T i , Now observe that for Consequently, where the last line is obtained by observing that j dv i -2 follows using (13) that Now consider the optimal linear arrangements of the trees T i , for place them consecutively in that order on a line, and the path P . Let g denote the bijection associated with this arrangement, then L 1. Using this fact (15) implies since L g # - To finish the proof we will show that bcr(D # ) equals to the RHS of (11). Consider an optimal linear arrangement f # of the tree T . It is not di#cult to see that, f #-1 (1) and f #-1 (n) are leaves, [25, 4]. Let be the path between v trees defined in the first part of the proof. Note that for the bijection g, described earlier, it holds thus we conclude that, and note that the above equation implies that P does not cross itself, in the arrangement associated with f # . It follows that P does not cross itself in the bipartite drawing D # . Let f # be the restriction of f # to V i , and D # i be the subdrawing in D # which is associated with 1. Note that However, it is easy to see that D # is obtained from f # by lifting the vertex set V i 1 to the line hence we can apply the induction hypothesis to D # i , to obtain Substituting c P its value from (12), and repeating the same steps used in deriving (15), we obtain To complete the proof use (16) in (18) and obtain, .Since the optimal linear arrangement of an n-vertex tree can be found in O(n 1.6 computing D # can also be done in O(n 1.6 ) time. Applications It is instructive to provide examples of graphs G for which L(G)). Consider any bipartite G with # G # 3 and # regular bipartite graph with # G # 3. Then, conditions of Corollary 2.1 are met, and thus by Theorem 2.3, L(G)). Moreover, consider any connected bipartite G of degree at most a constant k, with is fixed. Note that, d v - d # v # 1 for any v # V , since G is connected and is not a star, and thus, n. (Note that the star is excluded by the density condition , to obtain n # 1 . Hence this graph satisfies the conditions of Corollary 2.1, moreover, it is easy to see that aG # O(1), and we conclude using Theorem 2.3 that L(G)). 3.1 Bipartite crossings, bisection, genus, and page number The appearance of aG in the upper bound of Theorem 2.3 relates bcr(G) to other important topological properties of G such as genus of G, denoted by g G [32], and page number of G [1], denoted by p G . Observation 3.1 E), and assume that # G # 2 and m # (1 + #)n, for a fixed # > 0. L(G)), provided that Consequently, under the given conditions for G, if either L(G)). Proof. Assume that using Corollary 2.1 and Theorem 2.3, and observing that, O(1), we conclude that L(G)). (Note that, # G # 2, gives d # v for all v # V . ) To finish the proof, observe that implies that Next, we provide another application of our results, by deriving nontrivial upper bounds on the bipartite crossing number. Observation 3.2 Let E), with page number p G and genus g G . Then L(G). Proof. Since cr(G) # bcr(G) # 5a G L(G), by Theorem 2.3, we need to bound aG in terms of g G and G . Let H be a subgraph of G with the vertex set VH , |V H | # 2, and the edge set EH . Note that which verifies the upper bound involving p G . To finish the proof observe that is a lower bound on the genus of H, or g H [32]. Thus, H is at most (|V H | - 1) 2 /12 [32], it follows that for any subgraph H, p g G /12 # p g H /12 # , and consequently aG # 2 # Let 0 < # 1be a constant and denote by b # (G) size of the minimal #-bisection of G. That is, denotes a cut which partitions V into A and - A. Leighton [16] proved for any degree bounded graph G, the inequality (G)), where cr(G) is the planar crossing number of G. Another very interesting consequence of Theorem 2.2 is providing a stronger version of Leighton's result, for bcr(G). Theorem 3.1 in particular when G is regular, it holds Proof. The claim follows from the lower bound in Theorem 2.2 and the well-known observation that (G). (See for instance [12].) 2 Remarks. After proving Theorem 3.1, we discovered that a weaker version of this Theorem for degree bounded graphs can be obtained by a shorter proof which uses Menger's Theorem [27]. 3.2 Approximation algorithms Given a bipartite graph G, the bipartite arrangement problem is to find a bipartite drawing D of G with smallest L xD , or smallest length, so that the x coordinate of any vertex is an integer. We denote this minimum value by - L(G). Note that coordinate function xD , for a bipartite drawing need not to be an injection, since we may have xD (a) = xD (b), for a # V 0 , and b # V 1 . Thus, in general L(G). Our approximation algorithms in this section provide a bipartite drawing in which all vertices have integer coordinates, so that the number of crossings and at the same time the length of the drawing is small. We need the following Lemma giving a relation between - L(G). Lemma 3.1 For any connected bipartite graph E) it holds 4 . Proof. Let D be a bipartite drawing of G in which all x coordinates are integers. Let and note that ND (e) # |x D (a) - xD (b)|, since any vertex in V 0 # V 1 has an integer x coordinate. Let f # be the bijection in Part (i) in Lemma 2.1, then |f # (a) -f # (b)| # 2|x D (a) -xD (b)| + 1, and hence by taking the sum over all edges, we obtain L f # 2L xD +m. To prove the lemma, we claim that there are at least m-1edges so that xD (a) #= xD (b), and consequently L xD # m-1, which implies the result. To prove our claim, note that there are at most nedges ab, so that xD (a) = xD (b), and hence at least m- n m-1edges ab, with xD (a) #= xD (b), since G is connected and therefore has at least Even et al. [9] in a breakthrough result came up with polynomial time O(log n log log n) times optimal approximation algorithms for several NP-hard problems, including the linear arrangement problem. Combining their result with ours, we obtain the following. Theorem 3.2 Let E), and consider the drawing D (with integer coordinates) in Theorem 2.3 obtained form an approximate solution to the linear arrangement problem provided in [9]. Then L(G)). Moreover, if G meets the conditions in Corollary 2.1, then O(log n log log nbcr(G)), provided that # Proof. Note that L log n log log n) and thus the claim regarding L xD follows from Lemma 3.1. To finish the proof note that, Theorem 2.3 gives L(G)), and the claim regarding bcr(D) is verified by the application of Corollary 2.1, since # divide and conquer paradigm has been very popular in solving VLSI layout problems both in theory and also in practice. Indeed, the only known approximation algorithm for the planar crossing number is a simple divide and conquer algorithm in which the divide phase consists of approximately bisecting the graph [2]. This algorithm approximates cr(G)+n to within a factor of O(log 4 n) from the optimal, when G is degree bounded [17]. A similar algorithm approximates - L(G) to within a factor of O(log 2 n) from the optimal. To verify the quality of the approximate solutions, in general, one needs to show that the error term arising in the recurrence relations associated with the performance of algorithms are small compared to the value of the optimal solution. A nice algorithmic consequence of Theorem 3.1 is that the standard divide and conquer algorithm in which the divide phase consists of approximately bisecting the graph gives a good approximation for bcr(G) in polynomial time. The divide stage of our algorithm uses an approximation algorithm for bisecting a graph such as those in [10, 17]. These algorithms have a performance guarantee of O(log n) from the optimal [10, 17]. It should be noted that the lower bound of # b 23 (G)), although is su#cient to verify the the performance of the divide and conquer approximation algorithm for the planar crossing number, can not be used to show the quality of the approximation algorithm for bcr(G), since (as we will see) it does not bound from above the error term in our recurrence relation. Thus our lower bound of # n#G b 1(G)) is crucial to show the suboptimality of the solution. Theorem 3.3 Let A be a polynomial time 1/3-2/3 bisecting algorithm to approximate the bisection of a graph with a performance guarantee O(log n). Consider a divide and conquer algorithm which (a) recursively bisects the graph G, using A, (b) obtains the two bipartite drawings, and then (c) inserts the edges of the bisection between these two drawings. This divide and conquer algorithm generates, in polynomial time, a bipartite drawing D with integer coordinates, so that L L(G)). Moreover, if G meets the conditions in Corollary 2.1, then Proof. Assume that using A, we partition the graph G to 2 vertex disjoint subgraphs G 1 and G 2 recursively. Let - b(G) denote the number of those edges having one endpoint in the vertex set of G 1 , and the other in the vertex set of G 2 . Let DG 1 , and DG 2 be the bipartite drawings already obtained by the algorithm for G 1 and G 2 , respectively. Let D denote the drawing obtained for G. To show the claim regarding L xD , note that Since, we use the approximation algorithm A for bisecting we have - nb 1(G)), hence the error term in the recurrence relation is O(n log nb 1(G)). Moreover, 3 - consequently using Lemma 3.1, we obtain, 12 - 1(G)n. Thus the error term is O(log n - and consequently, which implies L L(G)). To verify the claim regarding bcr(D), note that Now observing that m # aGn, - nb 1(G)), and nb 1(G) # 3 - L(G), we obtain, log n) which implies Note that by Corollary 2.1, L(G)), and the claim follows. 2 Remarks. The method of Even et al. that we suggested to use in Theorem 3.2, although a theoretical breakthrough, requires the usage of specific interior point linear programming methods which may be computationally expensive or hard to code. Hence, the the divide and conquer approximation algorithm, although in theory, weaker than the method of Theorem 3.2, it may be easier to implement. Moreover, one may use very fast and simple heuristics developed by the VLSI and CAD communities [24] for graph bisection in the divide stage. Although, these heuristics do not produce provably sub-optimal solutions for bisecting a graph, they work well in practice, and are extremely fast. Therefore, one may anticipate that certain implementations of the divide and conquer algorithm are very fast and e#ective in practice. Note that since aG can be computed in polynomial time, the class of graphs with aG # c# G is recognizable in polynomial time, when c is a given constant. Hence, those graphs which meet the required conditions in Theorems 3.2, and 3.3 can be recognized in polynomial time. Also, note that many important graphs such those introduced in Section 3 meet the conditions, and hence for these graphs the performance of both approximation algorithms is guaranteed. Largest biplanar subgraphs in acyclic graphs be a tree and w ij be a weight assigned to each edge ij . For any B # E T , define the weight of B, denoted by w(B), to be the sum of weights for all edges in B. In this section we present a linear time algorithm to compute a biplanar subgraph of T of largest weight. A tree on at least 2 vertices is called a caterpillar if it consists of a path to which some vertices of degree 1 (leaves) are attached. We distinguish four categories of vertices in a caterpillar. First consider caterpillars which are not stars. They have a unique path connecting two internal vertices to which all leaves are attached to. We call this path the backbone of the caterpillar. The two endvertices of the backbone are called endbone vertices, internal vertices of the backbone are called midbone vertices. Leaves attached to endbones are called endleaves. Leaves attached to midbones are called midleaves. For a star with at least 3 vertices, the middle vertex is considered as endbone, the backbone path consists of this single endbone, and the leaves in the star are considered endleaves. If a star has two vertices, then we treat these vertices as endbones. be an unrooted tree and r # V T . Then, we view r as the root of T . Then any vertex will have a unique parent which is the first vertex on the path towards the root. For , the set of children of x, denoted by N x , are those vertices of T whose parent is x. For any we denote by T x the component of T , containing x, which is obtained after removing the parent of x from T . We define T r to be T . We use the notation B x for a biplanar subgraph of T x , x # V T , and treat B x as an edge set. We say that B x spans a vertex a, if there is an edge ab # B x . For x # V T , we define Our goal is to determine W (T r ). To achieve this goal, we define 5 additional related optimization problems as follows: x is not spanned by B x } . It is obvious that and therefore solving all 5 problems for T x determines W (T x ). For any leaf v set w 1 Finally, for u # N x , x # V T define, It is well-known and easy to show that a graph is biplanar i# it is a collection of vertex disjoint caterpillars. This is equivalent to saying that a graph is biplanar i# it does not contain a double claw which is a star on 3 vertices with all three edges subdivided. Therefore our problem is to find a maximum weight forest of caterpillars in an edge-weighted acyclic graph. We will use these facts in the next lemma, sometimes without explicitly referring to them. Lemma 4.1 y #Nx\{y} y #Nx\{y} y #Nx\{y} Proof Sketch. The basic idea for the recurrence relations is to describe how an optimal solution for in the trees rooted in N x . Indeed, (21), (22), and (25) are obvious. For (23), note that if x is an endbone in a maximum weight biplanar B x , then x is an endbone in a caterpillar C # B x . Consider the case that C is not a star. Since, x is an endbone of C, it has at least two neighbors in C, and all but one of its neighbors are leaves in C. Then exactly one neighbor y of x is an endbone or an endleaf in C \ {x}. This justifies the presence of the first two terms in the inner curly bracket. To justify the presence of the sum on y # , note that, in order to maximize the total weight of B x , we must attach y # N x \ {y} to C as a leaf, only if f(y # must include in B x , the maximum biplanar subgraph of T y # which has the total weight f(y # To justify the term P y#Nx f(y), consider the case that C is a star. Then we must attach any y # N x to C as a leaf only if we include in B x the maximum biplanar subgraph of T y . For (24), note that, if x is a midbone in a maximum weight B x , then x is a midbone of C # B x , and has 2 neighbors y 1 and y 2 in C. By deleting x from C, we obtain exactly two caterpillars C 1 and C 2 so that y i is either an endbone or an endleaf for C i , 2. Now follow an argument similar to (23) to finish the proof of (24) 2 Theorem 4.1 For an edge-weighted acyclic graph largest weight biplanar subgraph can be computed in O(|V T |) time. Proof Sketch. With no loss of generality assume that T is connected, otherwise we apply our arguments to the components of T . We select a root r for T , and then perform a post order traversal and show that we can compute w i (T x quantities are already known for the children of x. This is obvious for (20) and (25). For (21) and (22) the expressions in curly braces are easy to evaluate in linear time, if a maximizing y is known. So the issue is to find a maximizing y in linear time. It is easy to see that for (21) we look for y # N x which maximizes w xy we look for y # N x which maximizes all these can be computed in O(|N x |) time. For (23), it su#ces to show that a y # N x can be found in O(|N x |) time which maximizes To do so find note that Thus, to maximize w 4 (T x ), we should find y 1 , y 2 # N x , y 1 #= y 2 which give the largest two values for It is easy to maintain for every x not just the values w i (T x also the edge-set of B x which realizes this value, therefore, we can store the edge set of a largest biplanar subgraph as well. 2 Acknowledgment . The research of the second and fourth author was done while they were visiting Department of Mathematics and Informatics of University in Passau. They thank Prof. F.-J. Brandenburg for perfect work conditions and hospitality. A preliminary version of this paper was published at WADS'97 [26]. That version contained slight inaccuracies like missing error terms which are fixed in the current version. --R The book thickness of a graph A framework for solving VLSI layout problems The assignment heuristics for crossing reduction On optimal linear arrangements of trees Graph layout problems Algorithms for drawing graphs: an annotated bibliography Edge crossings in drawings of bipartite graphs Drawing graphs in 2 layers Fast Approximate Graph Partition Algorithms Crossing number is NP-complete Approximate algorithms for geometric embeddings in the plane with applications to parallel processing problems A new crossing number for bipartite graphs Exact and heuristic algorithm for 2-layer straight line crossing number Optimal linear labelings and eigenvalues of graphs Complexity issues in VLSI Combinatorial algorithms for integrated circuit layouts On the bipartite crossing number An alternative method to crossing minimization on hierarchical graphs Edge disjoint spanning trees of finite graphs Combinatorial Geometry Which aesthetic has the greatest e An introduction to VLSI physical design The optimal numbering of the vertices of a tree A minimum linear arrangement algorithm for undirected trees Discrete Applied Mathematics 19 Methods for visual understanding of hierarchical systems structures Crossing theory and hierarchy mapping Topological graph theory --TR --CTR Robert A. Hochberg , Matthias F. Stallmann, Optimal one-page tree embeddings in linear time, Information Processing Letters, v.87 n.2, p.59-66, 31 July Journal of Discrete Mathematics Staff, Research problems, Discrete Mathematics, v.257 n.2-3, p.599-624, 28 November Hillclimbing Algorithm for the Optimal Linear Arrangement Problem, Fundamenta Informaticae, v.68 n.4, p.333-356, December 2005 Matthias Stallmann , Franc Brglez , Debabrata Ghosh, Heuristics, Experimental Subjects, and Treatment Evaluation in Bigraph Crossing Minimization, Journal of Experimental Algorithmics (JEA), 6, p.8-es, 2001 Dimitrios M. Thilikos , Maria Serna , Hans L. Bodlaender, Cutwidth II: algorithms for partial w-trees of bounded degree, Journal of Algorithms, v.56 n.1, p.25-49, July 2005 Josep Daz , Jordi Petit , Maria Serna, A survey of graph layout problems, ACM Computing Surveys (CSUR), v.34 n.3, p.313-356, September 2002

approximation algorithms;biplanar graph;bipartite drawing;linear arrangement;bipartite crossing number

586901

Regular Languages are Testable with a Constant Number of Queries.

We continue the study of combinatorial property testing, initiated by Goldreich, Goldwasser, and Ron in [J. ACM, 45 (1998), pp. 653--750]. The subject of this paper is testing regular languages. Our main result is as follows. For a regular language $L\in \{0,1\}^*$ and an integer n there exists a randomized algorithm which always accepts a word w of length n if $w\in L$ and rejects it with high probability if $w$ has to be modified in at least $\epsilon n$ positions to create a word in L. The algorithm queries $\tilde{O}(1/\epsilon)$ bits of w. This query complexity is shown to be optimal up to a factor polylogarithmic in $1/\epsilon$. We also discuss the testability of more complex languages and show, in particular, that the query complexity required for testing context-free languages cannot be bounded by any function of $\epsilon$. The problem of testing regular languages can be viewed as a part of a very general approach, seeking to probe testability of properties defined by logical means.

Introduction Property testing deals with the question of deciding whether a given input x satises a prescribed property P or is \far" from any input satisfying it. Let P be a property, i.e. a non-empty family of binary words. A word w of length n is called -far from satisfying P , if no word w 0 of the same length, which diers from w in no more than n places, satises P . An -test for P is a randomized algorithm, which given the quantity n and the ability to make queries about the value of any desired bit of an input word w of length n, distinguishes with probability at least 2=3 between the case of w 2 P and A preliminary version of this paper appeared in the Proceedings of the 40 th Symposium on Foundation of Computer Science y Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel, and AT&T Labs{Research, Florham Park, NJ 07932, USA. Email: noga@math.tau.ac.il. Research supported by a USA Israeli BSF grant, by a grant from the Israel Science Foundation and by the Hermann Minkowski Minerva Center for Geometry at Tel Aviv University. z Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: krivelev@math.tau.ac.il Part of this research was performed when this author was with DIMACS Center, Rutgers University, Piscataway NJ, 08854, USA and AT&T Labs{Research, Florham Park, NJ 07932, USA. Research supported in part by a DIMACS Postdoctoral Fellowship. x Department of Computer Science, University of Haifa, Haifa, Israel. E-mail: ilan@cs.haifa.ac.il. Part of this research was performed when this author was visiting AT& T Labs { Research, Florham Park, NJ 07932, USA. { School of Mathematics, Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA. E-mail: szegedy@math.ias.edu. Part of this research was performed when this author was with AT&T Labs{Research, Florham Park, NJ 07932, USA. the case of w being -far from satisfying P . Finally, we say that property P is (c; )-testable if for every > 0 there exists an -test for P whose total number of queries is bounded by c. Property testing was dened by Goldreich et. al [7] (inspired by [13]). It emerges naturally in the context of PAC learning, program checking [6, 3, 10, 13], probabilistically checkable proofs [2] and approximation algorithms [7]. In [7], the authors mainly consider graph properties, such as bipartiteness and show (among other things) the quite surprising fact that testing bipartiteness can be done by randomly testing a polynomial in 1= number of edges of the graph, answering the above question with constant probability of failure. They also raise the question of obtaining general results as to when there is, for every > 0, an -test for a property using queries (i.e c is a function of but independent of n) with constant probability of failure. We call properties of this type -testable. So far, such answers are quite sparse; some interesting examples are given in [7], several additional ones can be obtained by applying the Regularity Lemma as we show in a subsequent paper [1]. In this paper we address testability of formal languages (see [8] as a general reference). A language is a property which is usually viewed as a sequence of Boolean functions f Our main result states that all regular languages are -testable with query complexity only ~ O(1=). We also show that this complexity is optimal up to a factor poly-logarithmic in 1=. This positive result cannot be extended to context-free languages, for there is an example of a very simple context-free language which is not testable. Since regular languages can be characterized using second order monadic logic, we thus obtain a large set of logically dened objects which are testable. In [1] we provide testable graph properties described by logical means as well. These results indicate a strong interrelation between testability and logic. Although our result on regular languages can be viewed as a separate result having no logical bearing at all, our opinion is that logic does provide the right context for testability problems, which may lead to the discovery of further classes of testable properties. The rest of this paper is organized as follows. In Section 2 we present the proof of the main result showing that every regular language is testable. In Section 3 we show that the upper bound of ~ O(1=) for the query complexity of testing regular languages, obtained in Theorem 1, is tight up to a poly-logarithmic factor. Section 4 is devoted to the discussion of testability of context-free languages. There we show in particular that there exist non-testable context-free languages. We also discuss testability of the Dyck languages. The nal Section 5 contains some concluding remarks and outlines new research directions. Testing Regular Languages In this section we prove the main result of the paper, namely that regular languages are ( ~ O( 1 testable. As this result is asymptotic, we assume that n is big enough with respect to 1 (and with respect to any other constant that depends only on the xed language we are working with). All logarithms are binary unless stated explicitly otherwise. We start by recalling the standard denition of a regular language, based on nite automata. This denition is convenient for algorithmic purposes. Denition 2.1 A deterministic nite automaton (DFA) M over f0; 1g with states is given by a function with a set F Q. One of the states, q 1 is called the initial state. The states belonging to the set F are called accepting states, - is called the transition function. We can extend the transition function - to f0; 1g recursively as follows. Let denote the empty word. Then Thus, if M starts in a state q and processes string u, then it ends up in a state -(q; u). We then say that M accepts a word u if -(q rejects u means that -(q 1 Finally, the language accepted by M , denoted by LM , is the set of all u 2 f0; 1g accepted by M . We use the following denition of regular languages: Denition 2.2 A language is regular i there exists a nite automaton that accepts it. Therefore, we assume in this section that a regular language L is given by its automaton M so that A word w of length n denes a sequence of states (q ) in the following natural way: q and for 1 j n, This sequence describes how the automaton M moves while reading w. Later in the paper we will occasionally refer to this sequence as the traversal path of w. A nite automaton M denes a directed graph G(M) by V g. The period g(G) of a directed graph G is the greatest common divisor of cycle lengths in G. If G is acyclic, we set We will use the following lemma about directed graphs. Lemma 2.3 Let E) be a nonempty, strongly connected directed graph with a nite period g(G). Then there exist a partition V which does not exceed 3jV j 2 such 1. For every 0 1 and for every the length of every directed path from u to v in G is (j i) mod 2. For every 0 1 and for every and for every integer r m, if (mod g), then there exists a directed path from u to v in G of length r. Proof. To prove part 1, x an arbitrary vertex z 2 V and for each 0 i g 1, let V i be the set of all those vertices which are reachable from v by a directed, (not necessarily simple), path of length g. Note that since any closed (directed) walk in G is a disjoint union of cycles, the length of each such walk is divisible by g. This implies that the sets V i are pairwise disjoint. Indeed, assume this is false and suppose w lies in V i \ V j with i 6= j. As G is strongly connected there is a path p 1 from w to z, and by denition there is a path p 2 of length i mod g from z to w as well as a path p 3 of length mod g from z to w. Now the number of edges of either 3 is not divisible by g, which is impossible. Therefore the sets V i form, indeed, a partition of V . For the union of any (directed) path from z to u with a (directed) path from u to v forms a path from z to v, and as any such path must have length j mod g the assertion of part 1 follows. We next prove part 2. Consider any set of positive integers fa i g whose greatest common divisor is g. It is well known that there is a smallest number t such that every integer s t which is divisible by g is a linear combination with non-negative integer coe-cients of the numbers a i . Moreover, it is known (see [9], [5]), that t is smaller than the square of the maximal number a i . Fix a closed (directed) walk in G, that visits all vertices and whose length is at most jV j 2 . (This is easily obtained by numbering the vertices of G arbitrarily as by concatenating directed paths from v i to v i+1 for each 0 i k 1, where the indices are taken modulo k). Associate now the set of cycle lengths in this walk with the set of positive integers fa i g as above. Then, following this closed walk and traversing each directed cycle as many times as desired, we conclude that every integer which is divisible by g and exceeds 2jV j 2 is a length of a closed walk passing through all vertices of the graph. Given, now, a vertex and an integer r > 3jV (j i) mod g, x a shortest path p from u to v, and note that its length l satises l = (j i) mod g and l < jV j( jV j 2 ). Adding to p a closed walk of length r l from v to itself we obtain the required path, completing the proof. 2 We call the constant m from the above lemma the reachability constant of G and denote it by m(G). In the sequel we assume that m is divisible by g. If LM \ f0; 1g testing algorithm can reject any input without reading it at all. Therefore, we can assume that we are in the non-trivial case LM \ f0; 1g n 6= ;. We now introduce a key denition for the sequel: Denition 2.4 Given a word w 2 f0; 1g n , a sub-word (run) w 0 of w starting at position i is called feasible for language LM , if there exists a state q 2 Q such that q is reachable from q 1 in G in exactly steps and there is a path of length n (jw in G from the state -(q; w 0 ) to at least one of the accepting states. Otherwise, w 0 is called Of course, nding an infeasible run in w proves that w 62 L. Our aim is to show that if a given word w of length n is far from any word of length n in L, then many short runs of w are infeasible. Thus a choice of a small number of random runs of w almost surely contains an infeasible run. First we treat the following basic case: Denition 2.5 We call an automaton M 'essentially strongly connected' if 1. M has a unique accepting state q acc ; 2. The set of states of the automaton, Q, can be partitioned into two parts, C and D so that the subgraph of G(M) induced on C is strongly connected; no edges in G(M) go from D to C (but edges can go from C to D). (Note that D may be empty.) Lemma 2.6 Assume that the language contains some words of length n, and that M is essentially strongly connected with C and D being the partition of the states of M as in Denition 2.5. Let m be the reachability constant of G[C]. Assume also that n 64m log(4m=). Then if for a word w of length exists an integer 1 i log(4m=) such that the number of infeasible runs of w of length 2 i+1 is at least 2 i 4 n Proof. Our intention is to construct a sequence (R j ) j=1;::: of disjoint infeasible runs, each being minimal in the sense that each of its prexes is feasible, and so that each is a subword of the given word w. We then show that we can concatenate these subwords to form a word in the language that is not too far from w ('not too far' will essentially depend on the number of runs that we have constructed). This in turn will show that if dist(w; L) n then there is a lower bound on the number of these infeasible runs. For reasons to become obvious later we also want these runs to be in the interval [m A natural way to construct such a sequence is to repeat the following procedure starting from 1 be the shortest infeasible run starting from w[m + 1] and ending before there is no such run we stop. Assume that we have constructed so ending at w[c j 1 ], next we construct R j by taking the minimal infeasible run starting at w[c and ending before w[n m+ 1]. Again if there is no such run we stop. Assume we have constructed in this way runs R 1 ; :::; R h . Note that each run is a subword of w, the runs are pairwise disjoint and their concatenation in order forms a (continuous) subword of w. Also, note that by the denition of each run R j being minimal infeasible, its prex R ( obtained by discarding the last bit of R j is feasible. This, in turn, implies that R 0 j which is obtained from R j by ipping its last bit is feasible. In addition, by Denition 2.4, this means that for each R 0 there is a state and such that q i j is reachable from q 1 in c Next we inductively construct a word w 2 L such that dist(w; w ) hm+ 2m+ 2. Assuming that dist(w; L) n this will imply a lower bound on h. The general idea is to 'glue' together the R 0 h, each being feasible and yet very close to a subword of w (except for the last bit in each). The only concern is to glue the pieces together so that as a whole word it will be feasible. This will require an extra change of m bits per run, plus some additional 2m bits at the end of the word. We maintain during the induction that for we construct is feasible starting from position 1, and it ends in position c j . For the base case, let c to be any word of length m which is feasible starting from position 1. Assume we have already dened a word w from position 1 and ending in position c j 1 . Let -(q As both p j and q i j are reachable from q 1 by a path of length c j 1 , according to Lemma 2.3 we can change the last m bits in w j 1 so that we get a word u j for which -(q 1 ; . We now dene w j as a concatenation of u j and R 0 . Let w h be the nal word that is dened in this way, ending at place c h . Now the reason we have stopped with R h is either that there is no infeasible run starting at c h + 1, in which case, changing the last m bits of w h and concatenating to it the remaining su-x of w (that starts at position c h exactly as in the case of adding R 0 yields the required w . The other possible reason for stopping growing R h is when there is a minimal infeasible run that start at c h ends after position n m+ 1. Let R be that run, and let R 0 be the run obtained by ipping the last bit of R. As was the case with any R 0 is feasible from position c h + 1. Hence there is a feasible word u of which R 0 is a prex, u is of length and so that -(q i h . We can construct w from w h and u exactly as we have constructed w form w h and the su-x of w in the previous case. By the denition of w , w 2 L. Following the inductive construction of w it follows that for 1. Then to get from w h to w we concatenate R 0 which is either a subword of w (in the rst case previously discussed) or it is a subword of w where one bit was changed (in the second case), following by changing m bits at the end of w h and possibly additional m bits at the end of u. Therefore dist(w; w ) hm 2, as we claimed. Recalling that dist(w; L) n, we conclude that h n 2 last inequality is by our assumptions that n 64m log(4m=)). This already shows that if dist(w; L) n then there are n) many disjoint infeasible runs in w. However, we need a stronger dependence as stated in the lemma. We achieve this in the following way. Let log(4m=). For 1 i a, denote by s i the number of runs in fR j g h whose length falls in the interval [2 P a h n=(4m) n=(4m). Therefore there exists an index i for which s i n=(4am). Consider all infeasible runs R j with jR that if a run contains an infeasible sub-run then it is infeasible by itself. Now, each infeasible run of length between 2 contained in at least runs of length 2 i+1 , except maybe, for the rst two and the last two runs (these with the two smallest j's and these with the two largest j's). As R j are disjoint, each infeasible run of length contains at most three of the R j s of length at least 2 1. Thus, we a get a total of at least runs of length at most 2 i+1 . By our assumption on the parameters this number is: am log(4m=) , as claimed. 2 Now our aim is to reduce the general case to the above described case. For a given DFA M with a graph by C(G) the graph of components of G, whose vertices correspond to maximalby inclusion strongly connected components of G and whose directed edges connect components of G, which are connected by some edge in G. Note that some of the vertices of C(G) may represent single vertices of G with no self loops, that do not belong to any strongly connected subgraph of G with at least two vertices. All other components have non empty paths inside them and will be called truly connected. From now on we reserve k for the number of vertices of C(G) and set may assume that all vertices of G are reachable from the initial state q 1 . Then C(G) is an acyclic graph in which there exists a directed path from a component C 1 , containing q 1 , to every other component. runs over all truly connected components of G, corresponding to vertices of C(G). We will assume in the sequel that the following relation are satised between the parameters: Condition (*) 2k 64m log 8mk . log(1=) < 1 clearly, for any xed k; m; l for small enough and n large enough condition (*) holds. Our next step is to describe how a word w 2 LM of length n can move along the automaton. If a word w belongs to L, it traverses G starting from q 1 and ending in one of the accepting states. Accordingly, w traverses C(G) starting from C 1 and ending in a component containing an accepting state. For this reason, we call a path A in C(G) admissible, if it starts at C 1 and ends at a component with an accepting state. Given an admissible path in C(G), a sequence of pairs of vertices of G (states of M) is called an admissible sequence of portals if it satises the following restrictions: 1. for every 1 j t; 2. 3. t is an accepting state of M ); 4. For every 2 j t one has (p 2 The idea behind the above denition of admissible portals is simple: Given an admissible path A, an admissible sequence P of portals denes how a word w 2 L moves from one strongly connected component of A to the next one, starting from the initial state q 1 and ending in an accepting state. The are the rst and last states that are traversed in C i j Now, given an admissible path A and a corresponding admissible sequence P of portals, we say that an increasing sequence of integers forms an admissible partition with respect to (A; P ) if the following holds: 1. 2. for every 1 j t, there exists a path from p 1 j to p 2 of length n j+1 3. The meaning of the partition j=1 is as follows. If w 2 L and w traverses M in accordance with t, the value of n j indicates that w arrives to component C for the rst time after n j bits. For convenience we also set n 1. Thus, for each 1 j t, the word w stays in C i j in the interval [n that it is possible in principle that for a given admissible path A and a corresponding admissible sequence of portals P there is no corresponding admissible partition (this could happen if the path A and the set of portals P correspond to no word of length n). A triplet (A; is an admissible path, P is a corresponding admissible sequence of portals and is a corresponding admissible partition, will be called an admissible triplet. It is clear from the denition of an admissible triplet that a word w 2 L traverses G in accordance with a scenario suggested by one of the admissible triplets. Therefore, in order to get convinced that w 62 L, it is enough to check that w does not t any admissible triplet. Fix an admissible triplet (A; . For t, we dene a language L j that contains all words that traverse in M from p 1 j to p 2 . This is done formally by dening an automaton M j as follows: The set of states of M j is obtained by adding to a new state f j . The initial state of M j and its unique accepting state are p 1 respectively. For each and 2 f0; 1g, if - M (q; , we set - M j We Namely, in M j all transitions within C remain the same. All transitions going to other components now go to f j which has a loop to itself. Thus, M j is essentially strongly connected as in Denition 2.5 with g. Then L j is the language accepted by M j . Given the xed admissible triplet (A; word w of length sub-words of setting t. Note that jw Namely, if w were to path through M according to the partition then the substring w j corresponds to the portion of the traversal path of w that lies within the component C Lemma 2.7 Let (A; be an admissible triplet , where . Let w be a word of length n satisfying dist(w; L) n. Dene languages (L and words (w as described above. Then there exists an index j, 1 j t, for which dist(w k . Proof. Assume this is not the case. Let j=1 be the partition and recall that t k. For every be a word of length n j+1 n j 1 for which (the empty word). Also, for 1 j t 1 choose j 2 f0; 1g so that - M (p 2 j+1 . Then by construction the word w belongs to L and dist(w; w { a contradiction.Now we present a key idea of the proof. Ideally, we would like to test whether an input word w of length n ts any admissible triplet. In the positive case, i.e. when w 2 LM , the traversal path of w in M denes naturally an admissible triplet which w will obviously t. In the negative case, i.e. when dist(w; L) n, Lemma 2.7 implies that for every admissible triplet (A; P; ), at least one of the sub-words w j is very far from the corresponding language L j . Then by Lemma 2.6 w j contains many short infeasible runs, and thus sampling a small number of random runs will catch one of them with high probability. However, the problem is that the total number of admissible triplets clearly depends on n, which makes the task of applying directly the union bound on the probability of not catching an infeasible run impossible. We circumvent this di-culty in the following way. We place evenly in a bounded number (depending only on and the parameters of M) of transition intervals T s of a bounded length and postulate that a transition between components of C(G) should happen inside these transition intervals. Then we show that if w 2 L, it can be modied slightly to meet this restriction, whereas if dist(w; L) n, for any choice of such an admissible triplet, w is far from tting it. As the number of admissible triplets under consideration is bounded by a function of only, we can apply the union bound to estimate the probability of failure. Recall that runs over all truly connected components of G, corresponding to vertices of C(G). Let log(1=)=. We place S transition intervals s=1 evenly in [n], where the length of each transition interval T s is jT s m). For . ALGORITHM Input: a word w of length 1. For each 1 i log(8km=) choose r i random runs in w of length 2 i+1 2. For each admissible triplet (A; j=1 such that for all 2 j t one has do the following: Form the automata M j , 1 j t, as described above. Discard those chosen runs which end or begin at place p for which jp n j j n=(128km log(1=)). Namely, those runs which have one of their ends closer than n=(128km log(1=)) from some For each remaining run R, if R falls between n j and n j+1 , check whether it is feasible for the automaton M j starting at b n is the rst coordinate of R in w. Namely, is the place where R starts relative to n j , which is the the place w \enters" M j . 3. If for some admissible triplet all checked runs turned out to be feasible, output "YES". Otherwise (i.e, in the case where for all admissible triplets at least one infeasible run has been found) output "NO". Lemma 2.8 If dist(w; L) n, then the above algorithm outputs "NO" with probability at least 3=4. If w 2 L, then the algorithm always outputs "YES". Proof. The proof contains two independent parts, in the rst we consider the case of an input w with dist(w; L) n, on which the algorithm should answer 'NO' (with high probability). The other part treats the case where w 2 L, for which the algorithm should answer 'YES'. Let us rst assume that dist(w; L) n. The number of admissible triplets (A; partition points fall into the union of transition intervals can be estimated from above by (rst choose an admissible path in C(G), the number of admissible paths is at most 2 k as any subset of vertices of C(G) denes at most one path spanning it; then choose portals, the total number of chosen portals is at most 2k, therefore there are at most jV j 2k possible choices for portals; then for a xed there are at most SjT s j choices for each n j , where 2 j t and t k). For satisfying condition (*) and S as above, this expression is at most (1=) 2k . Thus we need to check at most (1=) 2k admissible triplets. Let be an admissible triplet satisfying the restriction formulated in Step 2 of the above algorithm. Write . Then the triplet denes automata and languages (L as described before. By Lemma 2.7 for some 1 j t one has n=(2k). Then by Lemma 2.6 there exists an i, 1 i log(8km=) so that contains at least (2 i 4 n=(2km log(8km=)) runs of length 2 i+1 . At most of them may touch the last bits of the interval [n 1], and at most of them may touch the rst bits of this interval. Hence there are at least 2 i 6 n=(km log(1=)) 2 of them that touch neither the rst nor the last n=(128km log(1=)) bits of the interval Obviously, if a random sample contains one of these infeasible runs, then it provides a certicate for the fact that w does not t this admissible triplet. A random sample of r i runs of length 2 i+1 misses all of these infeasible runs with probability at most 2k Thus by the union bound we conclude that in this case a random sample does not contain a "witness" for each feasible triplet with probability at most 1=4. This completes the proof for the case of dist(w; L) n. We now address the case for which w 2 L. We need to show that in this case the algorithm answers 'YES'. For this is is enough to show that if w 2 L, then there exists an admissible triplet which passes successfully the test of the above algorithm. A traversal of w in M naturally denes a triplet (A; as follows: are components from C(G), ordered according to the order of their traversal by w; is the rst (resp. the last) state of C visited by w; set to be the rst time w enters while traversing M . However, this partition does not necessarily meet the requirement stated in Step 2 of the algorithm: In the true traversal of w in M the transitions from C i j to C i j+1 might occur outside the transition intervals T s . We show that the desired triplet can be obtained from the actual triplet, modifying only the third component of it. This modied triplet would then correspond to a dierent word w (which is quite close to w) that makes all the transitions inside the postulated transition intervals. In addition, we will take care that no query is made to bits in which w 0 diers from w. Hence, the algorithm will actually be consistent with both. This is in fact the reason for discarding the runs that are too close to some n j in Step 2 of the algorithm. Intuitively, this is done as follows: Assume n j is not in a transition interval, then we either make the traversal in longer so to end in p 2 in a transition interval, or we shorten the traversal in C so to enter a transition interval, depending on where the closest transition interval is. Formally this is done as follows. Dene a new partition choose a transition interval T s closest to n j . If C is a truly connected component, we choose n 0 j as the leftmost coordinate in T s satisfying the following restrictions: (a) n 0 is a singleton without loops we set n 0 such an n 0 exists. Finally, we set Note that the obtained triplet (A; is truly connected. As there exists a path from p 1 j to p 2 of length n j+1 n j 1, there also exists a path of length n 0 j 1. This implies the admissibility of 0 and hence the admissibility of (A; Let now R be a run of w inside [n 0 j+1 n=(128km log(1=))] and let b be its rst coordinate. Since we placed S transition intervals fT s g evenly in [n], we have jn 0 +m). Therefore, R falls also completely inside [n remark at this point that the purpose of discarding marginal runs at Step 2 of the algorithm is to achieve that each one of the remaining runs will fall completely not only within [n 0 j+1 ], but also within As we will see immediately this guarantees that R will be feasible for the corresponding automaton M j . Without this deletion, with positive probability one of the sampled runs R may start in a place where w is in C and end in a place where w is in C i j , thus making it impossible to attribute R to one particular automaton M j . Therefore, with positive probability the algorithm would fail in the positive case. Discarding marginal runs allows us to get a one-sided error algorithm). As w 2 L, there exists a state q 2 C so that -(q; R) 2 C . Also, q is reachable from p 1 (the initial state of C steps (b is the rst coordinate of R). According to the choice of n 0 j we have is the period of C . But then by Lemma 2.3 q is reachable from p 1 in m) steps. This shows that R is feasible for M j , starting at b n 0 1. Thus, if w 2 L, the above algorithm always outputs "YES". 2 Finally, the number of bits of w queried by our algorithm is at most log(8km=) X log(8km=) X We have thus proven the following theorem. Theorem 1 For every regular language L, every integer n and every small enough > 0, there exists a one-sided error -testing algorithm for L\ f0; 1g n , whose query complexity is c log 3 (1=)=, where the constant c > 0 depends only on L. A nal note about the dependence of the complexity on the parameters is in place here. In the proof M is considered xed, as the algorithm is tailored for a xed given language. However, in the calculation above we have kept the dependence of the query complexity on the parameters of M explicit. One has to take in mind though that the estimates hold only when condition (*) holds. In particular we require (third item in (*)), that 1=( Another note is about the running time of the algorithm (rather then just its query complexity). The dominating term in Step 1 and the rst two subsets of Step 2 of the algorithm is the query complexity. In the last substeps, each run has to be checked against M j . Each such check involves checking whether there is a word u and a word v (of suitable lengths) so that uRv 2 L. Checking whether there are such u; v is done directly by Lemma 2.3 in case the length of u and v are longer than m, or by checking all words if one of them is shorter than m. 3 Lower bound for regular languages In many testability questions, it is quite natural to expect a lower bound of order 1= for the query complexity of testing. This is usually proven by taking a positive example of size n and perturbing it in randomly chosen n places to create a negative instance which is hard to distinguish from the positive one. Regular languages are not an exception in this respect, as shown by the next proposition and its fairly simple proof. Proposition 1 Let L be the regular language over the alphabet f0; 1g dened by 1g. For any n an -test for L \ f0; 1g n has query complexity at least 1 3 . Proof. Our proof is based on the following reformulation of the renowned principle of Yao [14], saying that if there exists a probability distribution on the union of positive and negative examples such that any deterministic testing algorithm of query complexity d is correct with probability less than 2/3 for an input randomly chosen from according to this distribution, then d is a lower bound on the query complexity of any randomized testing algorithm. Dene a distribution on the set of positive and negative instances of length n as follows. The word gets probability 1=2. Next we partition the index set [1; n] into , each of size n, and for each 1 i t give probability 1=(2t) to the vector y i created from 1 n by ipping all bits in I i from 1 to 0. Note that dist(y are negative instances. Now we apply the above mentioned principle of Yao. Let A be a deterministic -testing algorithm with query complexity d. If A is incorrect on the word 1 n , then it is already incorrect with probability at least 1=2. Otherwise, it should accept the input if all d tested bits equal to 1. Therefore it accepts as well at least t d of the inputs y i . This shows that A gives an incorrect answer with probability at least (t d)=(2t) < 1=3, implying d > t=3. 2. The main idea of the proof of the above proposition can be used to get an =) lower bound on the query complexity of testing any non-trivial regular language, with a natural denition of non-trivial. This is proven in the next proposition. A somewhat paradoxical feature of its proof is that our main positive result (Theorem 1) and its proof are used here to get a negative result. For a language L let L Denition 3.1 A language L is non-trivial if there exists a constant 0 < 0 < 1, so that for innitely many values of n the set L n is non-empty, and there exists a word w 2 f0; 1g n so that dist(w; L n ) 0 n. Proposition 2 Let L be a non-trivial regular language. Then for all su-ciently small > 0, any -testing algorithm for L requires queries. Proof. The proof here is essentially a generalization of the proof of Proposition 1. We thus present it in a somewhat abridged form. Let n be large enough. Assume L n 6= ;, and w 2 f0; 1g n is such that dist(w; L n ) 0 n. We may clearly assume that the constant 0 is as small as needed for our purposes. Our main result, Theorem 1, and its proof imply that with probability at least 2=3, a random choice of a set of runs, built as described at Step 1 of the testing algorithm of Theorem 1, and having total length ~ the algorithm to reject w. As we have noticed, the testing algorithm has one sided error, i.e., it always accepts a word from L. Thus, if we choose a random set of runs as above, it will cause to reject w with probability 2/3 and it will not coincide with any word u 2 L n (for otherwise, it would reject u too). Each such random set of runs is just a random set of intervals in ng (of length as dened in Step 1 of the testing algorithm) of total length bounded by ~ that two such random sets intersect with probability ~ n)). Therefore if we choose ~ n) such subsets at random, then we expect that ~ O( 2 n) pairs of them will intersect, and that 2/3 of the members will reject w. This implies that there exists a family S of ~ disjoint sets of runs so that for each member of S, no word of L n coincides with w on this set. Fix now 0 and let > 0 be small enough compared to 0 . We partition the family S into , each of cardinality n, where the constant c depends on 0 only and is thus independent of . Let u be a word in L n . For each 1 i t, the word w i is obtained from u by changing the bits of u, corresponding to S i , to those from w. It follows then that Indeed, to transform w i into a word in L n , at least one bit has to be changed in every member of S i . Now, as in the proof of Proposition 1, we dene a probability distribution on the union of positive and negative examples. The word u gets probability 1=2, and each one of the t words w probability 1=(2t). A simple argument, essentially identical to that in the proof of Proposition 1, shows that any deterministic algorithm needs to query at least 3 =) bits of the input word to be successful with probability at least 2=3 on the dened probability distribution. Applying Yao's principle, we get the desired result. 2 4 Testability of context-free languages Having essentially completed the analysis of testability of regular languages, it is quite natural to try to make one step further and to address testability of the much more complex class of context-free languages (see, e.g., [8] for a background information). It turns out that the general situation changes drastically here as compared to the case of regular languages. We show that there exist quite simple context-free languages which are not -testable. Then we turn our attention to one particular family of context-free languages { the so-called Dyck languages. We prove that the rst language in this family, testable in time polynomial in 1=, while all other languages in the family are already non-testable. All relevant denitions and proofs follow. 4.1 Some context-free languages are non-testable As we have already mentioned, not all context-free languages are testable. This is proven in the following proposition. Theorem 2 Any -testing algorithm for the context-free language the reversal of a word w, requires n) queries in order to have error of at most 1=3. Proof. Let n be divisible by 6. We again dene a distribution D on the union of positive and negative inputs in the following way. A negative instance is chosen uniformly at random from among all negative instances (i.e. those words w 2 f0; 1g n which are at distance at least n from L). We refer to this distribution as N . Positive instances are generated according to a distribution P dened as follows: we pick uniformly at random an integer k in the interval [n=6 and then select a positive example uniformly among words vv R uu R with k. Finally the distribution D on all inputs is dened as follows: with probability 1/2 we choose a positive input according to P and with probability 1=2 we choose a negative input according to N . We note that a positive instance is actually a pair (k; w) (the same word w may be generated using dierent k's). We use the above mentioned Yao's principle again. Let A be a deterministic -testing algorithm for L. We show that for any such A, if its maximum number of queries is n), then its expected error with respect to D is at least 1 A be such an algorithm. We can view A as a binary decision tree, where each node represents a query to a certain place, and the two outgoing edges, labeled with 0 or 1, represent possible answers. Each leaf of A represents the end of a possible computation, and is labeled 'positive' or `negative' according to the decision of the algorithm. Tracing the path from the root to a node of A, we can associate with each node t of A a pair (Q t ng is a set of queries to the input word, and f is a vector of answers received by the algorithm. We may obviously assume that A is a full binary tree of height d and has thus 2 d leaves. Then jQ for each leaf t of A. We will use the following notation. For a subset Q ng and a function f with f on Qg ; with f on Qg ; is the set of all negative (resp. positive) instances of length n consistent with the pair (Q; f ). Also, if D is a probability distribution on the set of binary strings of length n and is a subset, we dene Pr D w2E Pr D [w]. be the set of all leaves of A labeled 'positive', let T 0 be the set of all leaves of T labeled 'negative'. Then the total error of the algorithm A on the distribution D is Pr The theorem follows from the following two claims. 4.1 For every subset Q ng of cardinality Pr D [E (Q; f )] 4.2 For every subset Q ng of cardinality n) and for every function f Pr Based on Claims 4.1, 4.2, we can estimate the error of the algorithm A by Pr The theorem follows. 2 We now present the proofs of Claims 4.1 and 4.2. Proof of Claim 4.1: Notice rst that L has at most 2 n=2 n=2 words of length n (rst choose a word of length n=2 and then cut it into two parts v and u, thus getting a word the number of words of length n at distance less than n from L is at most jL \ f0; 1g n j log(1=)n . We get It follows then from the denition of D that Pr D [E (Q; f Proof of Claim 4.2: It follows from the denition of the distribution D that for a word w 2 L\f0; 1g n , Pr D Recall that f) is the set of words in L for which are consistent with f on the set of queries Q, Hence, Pr Now observe that for each of the d pairs of places in Q there are at most two choices of k, for which the pair is symmetric with respect to k or to n=2 + k. This implies that for n=6 2 choices of k, the set Q does not contain a pair symmetric with respect to k or n=2+k. For each such k, Therefore, Pr As a concluding remark to this subsection we would like to note that in the next subsection (Theorem we will give another proof to the fact that not all context-free languages are testable by showing the non-testability of the Dyck language D 2 . However, we preferred to give Theorem 2 as well due to the following reasons. First, the language discussed in Theorem 2 is simpler and more natural than the Dyck language D 2 . Secondly, the lower bound of Theorem 2 is better than that of Theorem 4. The proofs of these two theorems have many common points, so the reader may view Theorem 2 as a "warm-up" for Theorem 4. 4.2 Testability of the Dyck languages It would be extremely nice to determine exactly which context-free languages are testable. At present we seem to be very far from fullling this task. However, we are able to solve this question completely for one family of context-free languages { the so called Dyck languages. For an integer n 1, the Dyck language of order n, denoted by D n , is the language over the alphabet of 2n symbols grouped into n ordered pairs (a The language D n is dened by the following productions: 2. 3. where denotes the empty word. Though the words of D n are not binary according to the above denition, we can easily encode them and the grammar describing them using only 0's and 1's. Thus we may still assume that we are in the framework of languages over the binary alphabet. We can interpret D n as the language with n distinct pairs of brackets, where a word w belongs to D n i it forms a balanced bracket expression. The most basic and well known language in this family is D 1 , where we have only one pair of brackets. Dyck languages play an important role in the theory of context-free languages (see, e.g., [4] for a relevant discussion) and therefore the task of exploring their testability is interesting. Our rst goal in this subsection is to show that the language D 1 is testable. Let us introduce a suitable notation. First, for the sake of simplicity we denote the brackets a Assume that n is a large enough even number (obviously, for odd n we have D 1 \ f0; 1g there is nothing to test in this case). Let w be a binary word of length n. For 1 i n, we denote by x(w; i) the number of 0's in the rst i positions of w. Also, y(w; i) stands for the number of 1 0 s in the rst i positions of w. We have the following claims. 4.3 The word w belongs to D 1 if and only if the following two conditions hold: (a) x(w; i) Proof. Follows easily from the denition of D 1 , for example, by induction on the length of w. We omit a detailed proof. 2 Proof. Observe rst that by Claim 4.3 a word w is in D 1 if and only if we can partition its letters into pairwise disjoint pairs, so that the left letter in each pair is a zero, and the right letter is a one. Consider the bipartite graph, whose two classes of vertices are the set of indices i for which and the set of indices i for which respectively, where each i with connected to all assumption (a) and the defect form of Hall's theorem, this graph contains a matching of size at least y(w; n) s 1 . By assumption (b), y(w; n) n=2 s 2 =2. Therefore, there are at least n=2 s 2 =2 s 1 disjoint pairs of letters in w, where in each pair there is a zero on the left and a one on the right. Let us pair the remaining elements of w arbitrarily, where all pairs but at most one consist of either two 0's or two 1's. By changing, now, when needed, the left entry of each such pair to 0 and its right entry to 1 we obtain a word in D 1 , and the total number of changes performed is at most (s 2 completing the proof. 2 a) If for some 1 i n one has y(w; i) x(w; i) s, then dist(w; D 1 ) s=2; b) If Proof. Follows immediately from Claim 4.3. 2 We conclude from the above three claims that a word w is far from D 1 if and only if for some coordinate i it deviates signicantly from the necessary and su-cient conditions provided by Claim 4.4. This observation is used in the analysis of an algorithm for testing D 1 , proposed below. where C > 0 is a su-ciently large constant, whose value will be chosen later, and assume d is an even integer. In what follows we omit all oor and ceiling signs, to simplify the presentation. ALGORITHM Input: a word w of length 1. Choose a sample S of bits in the following way: For each bit of w, independently and with probability choose it to be in S. Then, if S contains more then d 'YES' without querying any bit. Else, 2. If dist(S; D 1 \ f0; 1g d 0 Lemma 4.6 The above algorithm outputs a correct answer with probability at least 2=3. Proof. As we have already mentioned, we set The proof contains two independent parts, in the rst we prove that the algorithm is correct (with probability and in the second part we prove that the algorithm has a bounded error for words w for which dist(w; D 1 ) n. Consider rst the positive case w 2 D 1 . Set assume for simplicity that t as well as n=t are integers. For 1 j t, let X j be the number of 0's in S, sampled from the interval [1; nj=t]. Let also Y j denote the number of 1's in S, sampled from the same interval. Both X j and Y j are binomial random variables with parameters x(w; nj=t) and p, and y(w; nj=t) and p, respectively. As w 2 D 1 , we get by Claim 4.3 that x(w; nj=t) y(w; nj=t), implying EX j EY j . Applying standard bounds on the tails of binomial distribution, we obtain: For . Note that EZ j np=t. Using similar argumentation as above, we get As w 2 D 1 , we have by Claim 4.3 x(w; Hence Finally, we have the following estimate on the distribution of the sample size jSj: Choosing C large enough and recalling the denition of t, we derive from (1){(4) that with probability at least 2=3 the following events hold simultaneously: 1. 2. 3. X t np 4. jSj np Assume that the above four conditions are satised. Then we claim that dist(S; D 1 ) < . Indeed, the rst two conditions guarantee that for all 1 i jSj we have y(S; i) x(S; i) =2+2np=t 2=3. The last two conditions provide x(S; jSj) y(S; Therefore, by Claim 4.4 our algorithm will accept w with probability at least 2=3, as required. This ends the rst part of the proof. Let us now consider the negative case. Assume that dist(w; D 1 \ f0; 1g n ) n. By Claim 4.4 we have then that at least one of the following two conditions holds: a) there exists an index 1 i n, for which y(w; i) x(w; i) n=2; b) x(w; n) y(w; n) n=2. In the former case, let X , Y be the number of 0's, 1's, respectively, of S, sampled from the interval [1; i]. Let also k be the number of elements from [1; i] chosen to S. Then are binomially distributed with parameters x(w; i) and p, and y(w; i) and p, respectively. It follows from the denition of i that EY EX np=2. But then we have Choosing the constant C to be su-ciently large and recalling the denitions of p and , we see that the above probability is at most 1=6. But if y(S; it follows from Claim 4.5 that If x(w; n) y(w; n) n=2, we obtain, using similar arguments: The above probability can be made at most 1=6 by the choice of C. But if x(S; jSj) y(S; jSj) 2, it follows from Claim 4.5 that dist(S; D 1 ) . Thus in both cases we obtain that our algorithm accepts w with probability at most 1=6. In addition, the algorithm may accept w (in each of the cases), when (rst item in the algorithm). However, by equation (4) this may be bounded by 1/6 (choosing C as in the rst part). Hence the algorithm rejects w with probability at least 2=3. This completes the proof of Lemma 4.6. 2. By Lemma 4.6 we have the following result about the testability of the Dyck language D 1 . Theorem 3 For every integer n and every small enough > 0, there exists an -testing algorithm for query complexity is C log(1=)= 2 for some absolute constant C > 0. The reader has possibly noticed one signicant dierence between the algorithm of Section 2 for testing regular languages and our algorithm for testing D 1 . While the algorithm for testing regular languages has a one-sided error, the algorithm of this section has a two-sided error. This is not a coincidence. We can show that there is no one-sided error algorithm for testing membership in D 1 , whose number of queries is bounded by a function of only. Indeed, assume that A is a one-sided error algorithm for testing D 1 . Consider its execution on the input word . It is easy to see that dist(u; D 1 ) n. Therefore, A must reject u with probability at least 2=3. Fix any sequence of coin tosses which makes A reject u and denote by Q the corresponding set of queried bits of u. We claim that if jQ\[1; n=2+n]j n=2 n, then there exists a word w of length n from D 1 , for which for all i 2 Q. To prove this claim, we may clearly assume that jQ \ [1; n=2 as follows. For we take the rst n indices i in [1; n=2 and set For the last n indices i in [1; n=2 the su-cient condition for the membership in D 1 , given by Claim 4.3. Indeed, at any point j in [1; n=2+ n] the number of 0's in the rst j bits of w is at least as large as the number of 1's. Also, for j n=2 Therefore w 2 D 1 . As A is assumed to be a one-sided error algorithm, it should always accept every But then we must have jQ \ [1; n=2 queries a linear in n number of bits. We have proven the following statement. Proposition 3 Any one-sided error -test for membership in D 1 queries n) bits on words of length n. Our next goal is to prove that all other Dyck languages, namely D k for all k 2 are non-testable. We will present a detailed proof of this statement only for 2, but this clearly implies the result for all k 3. For the sake of clarity of exposition we replace the symbols a in the denition of D 2 by respectively. Then D 2 is dened by the following context-free where is the empty word. Having in mind the above mentioned bracket interpretation of the Dyck languages, we will sometimes refer to 0; 2 as left brackets and to 1; 3 as right brackets. Note that we do not use an encoding of D 2 as a language over f0; 1g, but rather over an alphabet of size 4. Clearly, non-testability of D 2 as dened above will imply non-testability of any binary encoding of D 2 that is obtained by a xed binary encoding of f0; 1; 2; 3g. Theorem 4 The language D 2 is not -testable. Proof. Let n be a large enough integer, divisible by 8. We denote L Using Yao's principle, we assign a probability distribution on inputs of length n and show that any deterministic algorithm probing bits outputs an incorrect answer with probability 0:5 o(1). Both positive and negative words will be composed of three parts: The rst which is a sequence of matching 0=1 (brackets of the rst kind) followed by a sequence of 0=2 (left brackets) and a sequence of 1=3 (right brackets). Positive instances are generated according to the distribution P as follows: choose k uniformly at random in the range Given k, the word of length n is is of length n 2k generated by: for choose v[i] at random from 0; 2 and then set v[n 2k+1 Negative instances are chosen as follows: the process is very similar to the positive case except that we do not have the restriction on v[n 2k 1. Namely, we choose k at random in the range Given k, a word of length n is is of length n 2k generated by: choose v[i] at random from 0; 2 and for choose v[n 2k +1 i] at random from 1; 3. Let us denote by N the distribution at this stage. Note that the words that are generated may be of distance less than n from L n (in fact some words in L n are generated too). Hence we further condition N on the event that the word is of distance at least n from L n . The probability distribution over all inputs of length n is is now dened by choosing with probability 1/2 a positive instance, generated as above, and with probability 1/2 a negative instance, chosen according to the above described process. 4.7 The probability that an instance generated according to N is n-close to some word in L n is exponentially small in n. Proof. Fix k and let be a word of length n generated by N . For such xed k the three parts of w are the rst part of matching 0=1 of length 2k, the second part which is a random sequence of 0=2 of length n 2kand the third part which is a random sequence of 1=3 of length n 2k. Let us denote by these three disjoint sets of indices of w. We will bound from above the number of words w of length n of the form 2kwhich are at distance at most n from L n . First we choose the value of w on N 2 , which gives 2 n 2kpossibilities. Then we choose (at most) n bits of w to be changed to get a word from L n ( n choices) and set those bits (4 n possibilities). At this point, the only part of w still to be set is its value of N 3 , where we are allowed to use only right brackets 1; 3. The word to be obtained should belong to L n . It is easy to see that there is at most one way to complete the current word to a word in L n using right brackets only. Hence the number of such words altogether is at most 2 n 2k . The total number of words w of the form 0 and each such word gets the same probability in the distribution N . Therefore the probability that a word chosen according to N is n-close to L n can be estimated from above by n=4 ))n+2n n for small enough > 0 as promised. 2 d, be a xed set of places and let k be chosen uniformly at random in the range n=8; :::; n=4. Then S contains a pair i < j symmetric with respect to (n 2k)=2 with probability at most d 8 n . Proof. For each distinct pair there is a unique k for which are symmetric with respect to the above point. Hence the above probability is bounded by d 8 We now return to the proof of Theorem 4. Let A be an algorithm for testing L n that queries at most queries. As may assume that A is non-adaptive, namely, it queries some xed set of places S of size d (as every adaptive A can be made non adaptive by querying ahead at most 2 d possible queries dened by two possible branchings after each adaptive query. We then look at these queries as our S). For any possible set of answers f and an input the event that w is consistent with f on S. Let NoSym be the event that S contains no symmetric pair with respect to (n 2k)=2. Also, let F 0 denote all these f 's on which the algorithm answers 'NO' and let F 1 be all these f 's on which it answers 'YES'. Finally denote by (w positive) and (w negative) the events that a random w is a positive instance and a negative instance, respectively. The total error of the algorithm is However, given that S contains no symmetric pairs, for a xed f , Prob[f w ^ (w is negative)] is essentially equal to Prob[f w ^ (w is positive)] (these probabilities would be exactly equal if negative w would be generated according to N . Claim 4.7 asserts that N is exponentially close to the real distribution on negative instances). Hence each is of these probabilities is 0:5Prob[f w jNoSym] o(1). Plugging this into the sum above, and using Claim 4.8 we get that the error probability is bounded from below by Prob(NoSym) f (0:5 o(1))Prob[f w jNoSym] (1 d 8 Concluding remarks The main technical achievement of this paper is a proof of testability of regular languages. A possible continuation of the research is to describe other classes of testable languages and to formulate su-cient conditions for a context-free language to be testable (recall that in Theorem 2 we have shown that not all context-free languages are testable). One of the most natural ways to describe large classes of testable combinatorial properties is by putting some restrictions on the logical formulas that dene them. In particular we can restrict the arity of the participating relations, the number of quantier alternations, the order of the logical expression (rst order, second order), etc. The result of the present paper is an example to this approach, since regular languages are exactly those that can be expressed in second order monadic logic with a unary predicate and an embedded linear order. Another example can be found in a sequel of this paper [1], which addresses testability of graph properties dened by sentences in rst order logic with binary predicates, and which complements the class of graph properties shown to be testable by Goldreich et al [7]. Analogous results for predicates of higher arities would be desirable to obtain, but technical di-culties arise when the arity is greater than two. As a long term goal we propose a systematic study of the testability of logically dened classes. Since many dierent types of logical frameworks are known, to nd out which one is suited for this study is a challenge. Virtually all single problems that have been looked at so far have the perspective of being captured by a more general logically dened class with members that have the same testability properties. A very dierent avenue is to try to develop general combinatorial techniques for proving lower bounds for the query complexity of testing arbitrary properties, possibly by nding analogs to the block sensitivity [12] and the Fourier analysis [11] approaches for decision tree complexity. At present we have no candidates for combinatorial conditions that would be both necessary and su-cient for -testability. Acknowledgment . We would like to thank Oded Goldreich for helpful comments. We are also grateful to the anonymous referees for their careful reading. --R Proof veri Proof of a conjecture by Erd Property testing and its connections to learning and approximation. Introduction to Automata Theory A bound for a solution of a linear Diophantine problem New directions in testing On the degree of Boolean functions as real polynomials Robust characterization of polynomials with applications to program testing. Probabilistic computation --TR --CTR Michal Parnas , Dana Ron , Ronitt Rubinfeld, Testing membership in parenthesis languages, Random Structures & Algorithms, v.22 n.1, p.98-138, January Beate Bollig, A large lower bound on the query complexity of a simple boolean function, Information Processing Letters, v.95 n.4, p.423-428, 31 August 2005 Beate Bollig , Ingo Wegener, Functions that have read-once branching programs of quadratic size are not necessarily testable, Information Processing Letters, v.87 n.1, p.25-29, July Eldar Fischer, On the strength of comparisons in property testing, Information and Computation, v.189 n.1, p.107-116, 25 February 2004 Eldar Fischer , Eric Lehman , Ilan Newman , Sofya Raskhodnikova , Ronitt Rubinfeld , Alex Samorodnitsky, Monotonicity testing over general poset domains, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada Eli Ben-Sasson , Prahladh Harsha , Sofya Raskhodnikova, Some 3CNF properties are hard to test, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA Alon, Testing subgraphs in large graphs, Random Structures & Algorithms, v.21 n.3-4, p.359-370, October 2002 Eldar Fischer , Ilan Newman , Ji Sgall, Functions that have read-twice constant width branching programs are not necessarily testable, Random Structures & Algorithms, v.24 n.2, p.175-193, March 2004 Alon , Asaf Shapira, Every monotone graph property is testable, Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, May 22-24, 2005, Baltimore, MD, USA Asaf Shapira, A combinatorial characterization of the testable graph properties: it's all about regularity, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA Alon , Asaf Shapira, Testing subgraphs in directed graphs, Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, June 09-11, 2003, San Diego, CA, USA Alon , Asaf Shapira, A characterization of easily testable induced subgraphs, Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, January 11-14, 2004, New Orleans, Louisiana

regular languages;context-free languages;property testing

586907

The Combinatorial Structure of Wait-Free Solvable Tasks.

This paper presents a self-contained study of wait-free solvable tasks. A new necessary condition for wait-free solvability, based on a restricted set of executions, is proved. This set of executions induces a very simple-to-understand structure, which is used to prove tight bounds for k-set consensus and renaming. The framework is based on topology, but uses only elementary combinatorics, and, in contrast to previous works, does not rely on algebraic or geometric arguments.

Introduction This paper studies the tasks that can be solved by a wait-free protocol in shared-memory asynchronous systems. A shared-memory system consists of n that communicate by reading and writing shared variables; here we assume only atomic read/write registers. We also assume that processes are completely asynchronous, i.e., each process runs at a completely arbitrary speed. Processes start with inputs and, after performing some protocol, have to halt with some outputs. A task specifies the sets of outputs that are allowable for each assignment of inputs to processes. A protocol is wait-free if each process halts with an output within a finite number of its own steps, regardless of the behavior of other processes. A task is wait-free solvable if there exists a wait-free protocol that solves it. The study of wait-free solvable tasks has been central to the theory of distributed computing. Early research studied specific tasks and showed them to be solvable (e.g., approximate agreement [9], 2n-renaming [2], k-set consensus with at most k \Gamma 1 failures [7]) or unsolvable (e.g., consensus [10], n+1-renaming [2]). A necessary and sufficient condition for the solvability of a task in the presence of one process failure was presented in [3]. In 1993, a significant advancement ??? Supported by grant No. 92-0233 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and the fund for the promotion of research in the Technion. Email: hagit@cs.technion.ac.il. y Part of this work was done while visiting the MIT Laboratory for Computer Science, and the Cambridge Research Laboratory of DEC. Supported by CONACyT and DGAPA Projects, UNAM. Email: rajsbaum@servidor.unam.mx. was made in the understanding of this problem with [4, 17, 20]. This advancement yielded new impossibility results for k-set consensus ([4, 17, 20], and later [6, 14, 15]) and renaming ([17, 15]), as well as a necessary and sufficient condition for wait-free solvability ([17, 18]). Of particular interest was the use of topological notions to investigate the problem, suggested in [17, 20]. Yet, much of this development remained inaccessible to many researchers, since it relied on algebraic and geometric tools of topology. In this paper, we present a self-contained study of wait-free solvable tasks starting from first principles. We introduce a new necessary and sufficient condition for wait-free solvability. This condition is used to prove tight bounds on renaming and k-set consensus. It is also used to derive an extension of the necessary condition of [3]. Our approach borrows critical ideas from previous works in this area (especially, [4, 5, 17, 18, 20]), and integrates them into a unified framework. Below we discuss the relationships between our work and previous work. To provide a feeling for our results, we present the following rough description of key notions from combinatorial topology. A colored simplex is a set, in which each of the elements, called vertices, is colored with a process id. A colored complex is a collection of colored simplexes which is closed under containment. A mapping from the vertices of one colored complex to the vertices of another is simplicial if it maps a simplex to a simplex; it is color preserving if a vertex with id p i is mapped to vertex with id p i . Finally, a complex whose largest simplex contains m vertices is a pseudomanifold if every simplex with is contained in either one or two simplexes with m vertices. Precise definitions appear in Section 3; they do not rely on algebraic or geometric interpretations. The novel combinatorial concept we use is of a pseudomanifold being a divided image of a simplex. Very roughly, a pseudomanifold is a divided image of a simplex if it has the same boundary as the simplex. The divided image preserves some of (but not all) the topological structure of the simplex. We prove a new necessary condition for wait-free solvability (Corollary 13): if a task is wait-free solvable, then there exists a divided image of the complex of possible inputs; it is straightforward to see that the decisions made by the protocol must induce a simplicial map from this divided image to the complex of possible outputs which must agree with the task specification. We present a necessary and sufficient condition for wait-free solvability, i.e., a characterization of the wait-free solvable tasks. Consider a task, and a wait-free protocol that solves it. We explicitly show that a subset of the protocol's executions, called immediate snapshot executions [4, 20], induce a divided image of the complex of possible inputs. We use a solution for the participating set problem ([5]) to show that the above property is also sufficient. Namely, if there exists a simplicial map from a divided image induced by immediate snapshots executions to the output complex which agrees with the task, then the problem is wait-free solvable. We prove that the divided image induced by immediate snapshot executions is orientable. We then prove a combinatorial theorem which extends Sperner's Lemma (for orientable divided images). This theorem is the key to a completely combinatorial proof that M-renaming is wait-free solvable only if M - 2n. Using the basic Sperner's Lemma, we also show that k-set consensus is wait-free solvable only if k ? n. (These bounds are known to be tight, see [2] and [7], respectively.) Divided images play a role similar to spans (both the geometric version used in [17, 18, 14], and the algebraic version introduced in [15]). As discussed below (after Definition 1) divided images have weaker mathematical properties than geometric spans, in particular, they may have "holes". We show (in the full version of the paper) that an orientable divided image corresponds in a natural manner to an algebraic span. It was shown that such spans exist (in [17]), but this proof requires a combination of algebraic (homology theory) and geometric (subdivided simplexes) arguments. The existence of algebraic spans with certain properties imply impossibilities of set consensus and renaming [15], without relying on the more involved arguments of [17]. The necessary and sufficient condition we derive is not exactly the same as the one proved by Herlihy and Shavit in [18]. We explicitly construct a specific well-structured divided image (induced by immediate snapshot executions), while Herlihy and Shavit show that an arbitrary span exists ([17]). The notion of immediate snapshot executions was introduced in [4, 20]. The basic ideas needed to show that immediate snapshot executions induce a divided image already appeared in Borowsky and Gafni's paper [4]. However, they were interested in properties of immediate snapshot executions to prove the impossibility result for set consensus. It was not shown that they are orientable (a property used for the renaming impossibility) or that they induce an algebraic span (or our simpler combinatorial notion of a divided image), and no general conditions for wait-free solvability were derived from them. In the full version of this paper, we derive another necessary condition for wait-free solvability from Corollary 13, of a different nature. This condition is based on connectivity, and is therefore computable. This condition extends the condition for solvability in the presence of one failure [3]. It follows from [11, 16] that there is no computable necessary and sufficient condition for wait-free solvability. 2 Model of Computation Our model is standard and was used in many papers; we follow [1]. A system consists of n Each process is a deterministic state machine, with a possibly infinite number of states. We associate with each process a set of local states. Among the states of each process is a subset called the initial states and another subset called the output states. Processes communicate by means of a finite number of single-writer multi-reader atomic registers (also called shared variables). No assumption is made regarding the size of the registers, and therefore we may assume that each process p i has only one register R i . Each process p i has two atomic operations available to it: its entire state to R i . reads the shared variable R and returns its value v. A system configuration consists of the states of the processes and registers. Formally, a configuration C is a vector hs is the local state of process p i and v j is the value of the shared variable R j . Denote state Each shared variable may attain values from some domain which includes a special "undefined" value, ?. An initial configuration is a configuration in which every local state is an initial state and all shared variables are set to ?. We consider an interleaving model of concurrency, where executions are modeled as sequences of steps. Each step is performed by a single process. In each step, a process p i performs either a write i operation or a read i (R) operation, but not both, performs some local computation, and changes to its next local state. The next configuration is the result of these modifications. We assume that each process p i follows a local protocol P i that deterministically determines p i 's next step: P i determines whether p i is to write or read, and (in case of a read) which variable R to read, as a function of p i 's local state. If next state as a function of p i 's current state and the value v read from R. If p i writes R, then P i determines p i 's next state and as a function of p i 's current state. We assume that all local protocols are identical, i.e., depend only on the state, but not on the process id. A protocol is a collection P of local protocols P An event of p i is simply p i 's index i. A schedule is a finite or infinite sequence of events. An execution is a finite or infinite alternating sequence of configurations and events C is the initial configuration and C k is the result of applying the event j k to C k\Gamma1 , for all k - 1. The schedule of this execution is Given an execution and a process p i , the view of p i in ff, denoted ffji is the sequence state i (C 0 ); state i (C Intuitively, for example, if decides in ff without taking any steps, then the only information contained in ffji is p i 's initial state. A process p i is faulty in an infinite schedule oe if it takes a finite number of steps (i.e., has a finite number of events) in oe, and nonfaulty otherwise. These definitions also apply to executions by means of their schedules. We assume that each process has two special parts of its state, an input value and an output value. Initial configurations differ only in the input values of the processes. If we want to have a local protocol which depends on the process id, then the id has to be provided explicitly as part of the input. We assume that the output value is irrevocable, i.e., the protocol cannot over-write the output value. Note that in our definition processes do not halt; they decide by writing the output value, but continue to take steps (which are irrelevant). task \Delta has some domain I of input values and domain O of output values; specifies for each assignment of input values to the processes which output values can be written by the processes. A protocol solves \Delta if for any finite execution, the output values already written by the processes can be completed (in any infinite extension of the execution where all processes are nonfaulty) to output values for all processes that are allowable for the input values in the execution. The protocol is wait-free if every nonfaulty process eventually writes an output value. 3 Combinatorial Topology Concepts In this section, we introduce the basic topological concepts we use in this pa- per. Previous papers in this area, e.g., [8, 14, 17, 18, 20], used geometric or algebraic interpretations of topological structures; in contrast, our approach is purely combinatorial, abstracting ideas from [12, 19, 21]. Basic Notions: The basis of our definitions is the notion of a complex. A complex K is a collection of finite nonempty sets closed under containment; that is, if oe is an element of K, then every nonempty subset of oe is an element of K. A nonempty subset of oe is a face of oe. A face of oe is proper if it is not equal to oe. Each element of a complex is called a simplex. A complex K 0 is a subcomplex of a complex K if K 0 ' K. The dimension of a simplex oe, dim(oe), is the number of its elements minus one. A simplex of dimension m (with m+1 elements) is called an m-simplex. The dimension of a complex K is the maximum dimension of its simplexes; we only consider complexes of finite dimension. A complex of dimension m is called an m-complex. We sometimes use a superscript notation to denote the dimension of simplexes and complexes, e.g., oe m is an m-simplex and K m is an m-complex. The vertex set of K is the union of the 0-simplexes of K. We identify the vertex v and the 0-simplex fvg. Consider two complexes K and L. Let f be a function from the vertices of K to the vertices of L. f is simplicial if for every simplex fv of K, is a simplex of L. (Note that ff(v 0 treated as a set, since f need not be one-to-one and there may be repetitions.) This implies that a simplicial map f can be extended to all simplexes of K. Intuitively, a simplicial map f maps every simplex oe of K to a simplex f(oe) (perhaps of smaller dimension) of L. We extend f to a set of simplexes of K, S, by defining f(S) to be the set of simplexes f(oe) in L, where oe ranges over all simplexes of S. Clearly, if S is a subcomplex of K then f(S) is a subcomplex of L. Divided Images: An m-complex K m is full to dimension m if every simplex of K m is contained in some m-simplex of K m . Let K m be a complex full to dimension m. An (m \Gamma 1)-simplex of K m is external if it is contained in exactly one m-simplex; otherwise, it is internal. The boundary complex of K m , denoted bound(K m ), is the subcomplex containing all the faces of external simplexes of K m . Clearly, bound(K m ) is full to dimension Abusing notation, let bound(oe m ) be the set of (m \Gamma 1)-faces of simplex oe m . A complex K m is an m-pseudomanifold, if it is full to dimension m and every contained in either one or two m-simplexes. 5 An m-manifold is an m-pseudomanifold in which every (m \Gamma 1)-simplex is contained in two m-simplexes, i.e., it has no external simplexes. The following combinatorial definition will play a key role later when we cast the structure of a protocol in the topological framework. be a complex. A complex K m is a weak divided image of there exists a function / that assigns to each simplex of L m a subcomplex of K m , such that: 1. for every - 2 K m there exists a simplex oe 2 L m such that - 2 /(oe), 2. for every oe single vertex, and 3. for every oe; oe assume that K m is a divided image of L m if it also satisfies the following condition: 4. for every oe 2 L m , /(oe) is a dim(oe)-pseudomanifold with We say that K m is a divided image of L m under /. Intuitively, a divided image is obtained from L m by replacing each simplex of with a pseudomanifold, making sure that they "fit together" (in the sense of Condition 3). In addition, Condition 1 guarantees that / maps Condition 2 guarantees that / maps vertices of L m to vertices of K finally, Condition 4 guarantees that / preserves the dimension and the boundary of simplexes in L m . Fig. 1 shows an example of the divided image of a complex containing two simplexes. In the figure, solid lines show the boundary of L 2 and their image under /, in K 2 . Consider a set oe m and let M (oe m ) be the complex consisting of oe m and all its proper subsets; M (oe m ) is an m-pseudomanifold consisting of a single m-simplex and all its faces. Of particular importance for us is the case where K m is a divided image of M (oe m ). In this case, /(oe m Remark. The concept of a divided image is reminiscent of the notion of acyclic carrier 7 of [19], in that it associates subcomplexes of one complex to simplexes of another. Munkres uses acyclic carriers to study subdivisions, a fundamental concept of algebraic topology (cf. [19, 21]). However, divided images differ from subdivisions, even if the requirement of connectivity is added. For example, a 2-dimensional torus with a triangle removed from its surface is a divided image of a 2-simplex, since its boundary is a 1-dimensional triangle. However it is 5 In algebraic topology, pseudomanifolds are assumed to have additional properties, which we do not require for our applications. 6 Notice that bound(oe) is a set of simplexes, and /(bound(oe)) is the complex which is the union over these simplexes - of /(- ). 7 Not to be confused with the notion of carrier defined later. Fig. 1. K 2 is a divided image of L 2 under /. neither an acyclic carrier nor a subdivided simplex since it has "holes" (non- trivial homology groups). The next proposition states some simple properties of divided images; its proof is left to the full paper. Proposition2. Let K m be a divided image of L m under /. (i) For every oe, oe (ii) For every pair of j-simplexes oe j; oe j2 L m , if oe j6= oe j 2 , and oe j" oe j6= ;, then is a pseudomanifold of dimension strictly smaller than j. (iii) For every i-simplex oe divided image of M (oe i ) under (iv) A simplex - external if and only if for some external simplex oe The carrier of a simplex - 2 K m , denoted carr(- ), is the simplex oe 2 L m of smallest dimension such that - 2 /(oe). Intuitively, the carrier of a simplex - is the "smallest" simplex in L m which is mapped to - . By Definition 1(1), every simplex - 2 K m is in /(oe), for some oe 2 L m . By Proposition 2(ii), the carrier is unique. Therefore, the carrier is well-defined. Connectivity: For any j, 0 m, the j-graph of K m consists of one vertex for every j-simplex of K m , and there is an edge between two vertices if and only if their intersection is a (j \Gamma 1)-simplex of K m . K m is j-connected if its j-graph is connected; K m is 0-connected if it consists of a single vertex. Lemma 3. Let K m be a divided image of oe m under /. There exists a complex ~ /, a restriction of / to ~ K m is a divided image of oe m under ~ for every i ? 1, and every oe i 2 oe m . Colorings: A complex K is colored by associating a value from some set of colors with each of its vertices. A coloring is proper if different vertices in the same simplex have different colors. A simplicial preserving if for every vertex v of K, that if a coloring is proper and a simplicial map is color preserving, then for any simplex fv the vertices are different, i.e., f(oe) is of the same dimension as oe. Let K m be a divided image of L m . A simplicial map Sperner coloring if for every v 2 K m , -(v) 2 carr(v). Intuitively, - "folds" K m into L m with the requirement that each vertex of K m goes to a vertex of its carrier. The main combinatorial definition we use is: A complex K m is a (weak) chromatic divided image of L m , if it is a (weak) divided image of L m with a proper Sperner coloring -. Let K m be a divided image of M (oe m ). The next well-known lemma says that an odd number of m-simplexes of K m must go to oe m (and in particular, at least one simplex). This lemma is used in Section 7; it follows from the Index Lemma (Lemma 17), presented later. Lemma 5 (Sperner Lemma). Consider a divided image K m of M (oe m ) under /, and a Sperner coloring - : K There exists an odd number of Modeling Tasks and Protocols In this section we model distributed tasks using combinatorial topology; this is an adaptation of [17, 18] to our framework. Tasks: Denote ids ng. For some domain of values V , let P (V ) be the set of all pairs consisting of an id from ids and a value from V . For a domain of inputs I, an input complex, I n , is a complex that includes n-simplexes (i.e., subsets of n+1 elements) of P (I) and all their faces, such that the vertices in an n-simplex have different id fields. For a domain of outputs O, an output complex, O n , is defined similarly over O. That is, if (i; val) is a vertex of I n then val denotes an input value for process p i , while if (i; val) is a vertex of O n then val is an output value for process p i . Note that I n and O n are properly colored by the id fields, and are full to dimension n. In addition, each complex is colored (not necessarily properly) by the corresponding domain of values. Using the combinatorial topology notions, a task is identified with a triple I n is an input complex, O n is an output complex, and \Delta maps each n-simplex of I n to a non-empty set of n-simplexes in O n . We sometimes mention only \Delta when I n and O n are clear from the context. The simplexes in \Delta(oe n ) are the admissible output simplexes for oe n . Intuitively, if oe n is an input simplex and - n 2 \Delta(oe n ) is an admissible simplex, then - n is an admissible output configuration when the system starts with input oe n . We extend \Delta to simplexes of dimension smaller than n, i.e., for executions in which n processes or less take steps, as follows. Recall that it must be possible to complete the outputs of some processes in an execution to outputs for all processes that are allowed for the inputs of the execution. Therefore, \Delta maps an input simplex oe of dimension smaller than n to the faces of n-simplexes in \Delta(oe n ) with the same dimension and ids, for all input simplexes oe n that contain oe. Extended in this manner, \Delta(M (oe n )) is a subcomplex of O n . There is another variant of wait-free solvability, which allows to explicitly define \Delta for simplexes of dimension smaller than n. This can be captured in our model by adding as part of the input a bit that tells the process whether to participate or not. Non-participating processes are required to output some default value. Protocol Complexes: We say that a view of a process is final if the process has written an output. For an execution ff, the set f(0; views(ff). Given a protocol P, the protocol complex, P n , is defined over the final views reachable in executions of P, as follows. An n-simplex of final views is in P n if and only if it is views(ff) for some execution ff of P. In addition, P n contains all the faces of the n-simplexes. The protocol complex for an input n-simplex oe, P n (oe), is the subcomplex of P n containing all n-simplexes corresponding to executions of P where processes start with inputs oe, and all their faces. Intuitively, only if there exists an execution ff with initial values oe, such that the views of processes in - are the same as in ff. Note however that ff is not necessarily unique. The protocol complex, P n , is the union of the complexes P n (oe), over all input n-simplexes oe. If a protocol is wait-free then P n (oe) is finite, since a process terminates after a finite number of steps. Observe that the protocol complex depends not only on the possible interleavings of steps (schedules), but also on the transitions of processes and their local states. One can regard P n as colored with four colors-an id, an input value, a view, and an output value. Note that the ids coloring is proper. The protocol implies a decision map which specifies the output value for each final view of a process. When P solves \Delta it holds that if - 2 corresponds to an output simplex. Therefore, ffi P is simplicial and preserves the ids coloring. Furthermore, for any input n-simplex oe, ffi P (P n (oe)) is a complex. Since the protocol depends only on the input values, if two input n-simplexes oe, oe 0 , have the same input values, i.e., differ only by a permutation of the ids, then P n (oe) can be obtained from P n (oe 0 ) by applying the same permutation to the ids. Therefore, the decision map must be anonymous; i.e., ffi P (P n (oe)) determines ffi P (P n (oe 0 )). If the protocol has to depend on the ids, then they have to be given as part of the inputs. The above definitions imply: solves hI n ; O n ; \Deltai if and only if ffi P (P n (oe)) ' \Delta(M (oe)), for every n-simplex oe 2 I n . We say that agrees with \Delta. Round (a) Execution ff 1 . Round (b) Execution ff 2 . Fig. 2. Executions ff 1 and ff 2 are indistinguishable to p1 and p2 . This is the topological interpretation of the operational definition of a protocol solving a task (presented at the end of Section 2). 5 A Condition for Wait-Free Solvability In this section we define immediate snapshot executions and prove that the subcomplex they induce is a chromatic divided image of the input complex. This implies a necessary condition for tasks which are solvable by a wait-free protocol. This condition is also sufficient since immediate snapshot executions can be emulated in any execution. An immediate snapshot execution (in short, ISE) of a protocol is a sequence of rounds, defined as follows. Round k is specified by a concurrency class (called a block in [20]) of process ids, s k . The processes in s k are called active in round k. In round k, first each active process performs a write operation (in increasing order of ids), and then each active process reads all the registers, i.e., performs operations (in increasing order of ids). We assume that the concurrency class is always non-empty. It can be seen that, for a given protocol, an immediate snapshot execution, ff, is completely characterized by the sequence of concurrency classes. Therefore, we can write Immediate snapshot executions are of interest because they capture the computational power of the model. That is, a task \Delta is wait-free solvable if and only if there exists a wait-free protocol which solves \Delta in immediate snapshot executions (this is shown as part of the proof of Theorem 14 below). Although they are very well-structured, immediate snapshot executions still contain some degree of uncertainty, since a process does not know exactly which processes are active in the last round round. That is, if p i is active in round k and observes some other process p j to be active (i.e., perform a write), p i does not know whether p j is active in round k \Gamma 1 or in round k. Consider for example, Fig. 2. Only p 0 distinguishes between executions ff 1 and have the same views in both executions and cannot distinguish between them. However, as we prove below (in Proposition 8) this is the only uncertainty processes have in immediate snapshot executions. Denote the subcomplex of the protocol complex which contains all immediate snapshot executions by E n . For an input simplex oe n 2 I n , E n (oe n ) is the subcomplex of all immediate snapshot executions starting with oe n . Fig. 3. The ISE complex, when each process takes at most one step. We now show that if the protocol is wait-free and uses only read/write op- erations, then the ISE complex is a divided image of the input complex. This is done by defining a function / that assigns a subcomplex of E n to each simplex of I n . Fig. 3 contains an example of an immediate snapshot executions complex for a single input simplex. This is the complex where each process takes at most one step. Note that there are simplexes that correspond to the executions ff 1 and ff 2 from Fig. 2. Indeed, the vertices that correspond to p 1 and to p 2 are the same in these simplexes, i.e., p 1 and p 2 have the same views. First, we need a few simple definitions. For a simplex oe of I n , O n , or E n , let ids(oe) be the set of ids appearing in vertices of oe. For a simplex oe of I n or inputs(oe) be the set of pairs of inputs with corresponding ids appearing in vertices of oe. Finally, for a simplex oe of E n , let views(oe) be the set of views appearing in vertices of oe and let observed(oe) be the set of ids of processes whose operations appear in views(oe). 8 Intuitively, if p i is not in observed(oe), then the views in oe are the same as in an execution in which p i does not take a step. Notice that ids(- ) ' observed(- ), since a process always "observes itself." We can now define /. For oe 2 I n , /(oe) is the complex containing all simplexes faces. Notice that /(oe) is full to dimension dim(oe). A fact we use later is: Proposition6. For any - 2 E n and oe 2 I n , - 2 /(oe) if and only if ids(- Proof. If - 0 is a face of - , then ids(- 0 Thus, the definition of / implies that if - 2 /(oe) then ids(- 8 Recall that these ids are not known by the processes', unless explicitly given in the inputs. To make this definition concrete, a special part of the process' state captures its identity. We defer the exact details to the full version. ids(oe). Since the protocol is wait-free, there exists an execution in which all processes in only processes in ids(oe), and processes in ids(- ) have the same views as in - . Let - the simplex in E n that corresponds to this execution. Note that and is a face of -, the claim follows. ut We first show that the ISE complex is a weak divided image of the input complex. In fact, this property does not depend on the protocol being wait-free or on the type of memory operations used, i.e., that the protocol uses only atomic read/write operations. Lemma 7. E n is a weak chromatic divided image of I n under /. Proof. Clearly, the process ids are a proper Sperner coloring of E n . We proceed to prove that the three conditions of weak divided images (Definition 1) hold. Condition (1): Consider a simplex be such that - n . Then there is a simplex oe n 2 I n with ids(- n is a face of - n , - 2 /(oe n ). Condition (2) follows since the protocol is deterministic. Condition (3) follows from Proposition happens if and only if - We say that process p j is silent in an execution ff, if there exists a round for every round r - k. Intuitively, this means that no other process ever sees a step by p j . If p j is not silent in ff, then it is seen in some round. Formally, a process p j is seen in round k, and there exists some process p 0 . The last seen round of p j is the largest round k in which p j is seen. These definitions imply: Proposition8. Consider a finite immediate snapshot execution ff. If p j is not silent in ff, then k is the last seen round of p j in ff if and only if (a) s for every round r ? k, (b) s k 6= fp j g, and (c) either (i) g. As a consequence, we have the next lemma. Lemma 9. Consider an immediate snapshot execution complex E n . Let - i 1 be an i-simplex of E n corresponding to an execution ff, and p i 2 ids(- i). (i) If p i is not silent in ff, then there exists - i another i-simplex of E n , that differs only in p i 's view, corresponding to ff 0 . (ii) If there exists - i another i-simplex of E n , that differs only in p i 's view, corresponding to ff 0 , then p j is not silent in ff; ff 0 . If k is the last seen round of p j in ff, then, without loss of generality, p j is in the kth concurrency class of - i 1 and the kth concurrency class of - i 2 is fp j g. Lemma 10. For every simplex oe Proof. Let It follows from the definition of / that - To show that - notice that observed(- be a process id in ids(oe i sees a step by p j , and in - i , does not see a step by any process not in ids(oe i ), it follows that p j 's view is determined (because the protocol is deterministic). Namely, - i\Gamma1 is contained in a single i-simplex - i , and hence - is a face of - by the definition of /, - 2 bound(/(oe i )). The other direction of the proof is similar. Since - 2 bound(/(oe i that - is a face of some - This implies that - i\Gamma1 is a face of a single is not in observed(- (by Lemma 9(i)). Hence, observed(- It follows that - This implies that - 2 /(bound(oe i )). ut Intuitively, the next lemma implies that once we fix the views of all processes but one, the remaining process may have only one of two views, which correspond to the two options in Proposition 8(c). This shows that the uncertainty about another process is restricted to its last seen round. Lemma 11. For every simplex oe i 2 I n , /(oe i ) is an i-pseudomanifold. Proof. As noted before, /(oe i ) is full to dimension i. We show that any simplex contained in at most two i-simplexes. Let - i 2 /(oe i ) be such that - i\Gamma1 is a face of - i . Since - i\Gamma1 and - i are properly colored by the ids, there exists some id p j , such that p j appears in - i but not in - i\Gamma1 . In fact, any i- simplex of /(oe i includes p j . Let ff be the prefix of an execution with steps by processes in ids(oe i ), corresponding to - i . We can take such a prefix because There are two cases: Case 1: p j is silent in ff. Then observed(- does not see an id not in ids(oe i ), its view is determined. Hence, - i is unique. Case 2: p j is not silent in ff. Let k be the last seen round of p j in ff. Lemma 9(ii) implies that that there are only two possible views for compatible with the views in - g. ut By Lemma 7, E n is a weak chromatic divided image of I n . Lemma 11 and imply Condition (4) of Definition 1. Therefore, we have: Theorem12. E n is a chromatic divided image of I n under /. This implies the following necessary condition for wait-free solvability: \Deltai be a task. If there exists a wait-free protocol which solves this task then there exists a chromatic divided image E n of I n and a color-preserving (on ids), anonymous simplicial map ffi from E n to O n that agrees with \Delta. We now restrict our attention to full-information protocols, in which a process writes its whole state in every write to its shared register. The complex induced by immediate snapshot executions of the full-information protocol for some input complex I n is called the full divided image of I n . We have the following necessary and sufficient condition for wait-free solvability. Theorem 14. Let hI n ; O n ; \Deltai be a task. There exists a wait-free protocol which solves this task if and only if there exists an full divided image E n of I n and a color-preserving (on ids), anonymous simplicial map ffi from E n to O n that agrees with \Delta. Sketch of proof. Assume there exists a protocol P which solves \Delta. Without loss of generality, we may assume that in P each process operates by writing and then reading the registers R solves \Delta, it must solve \Delta in immediate snapshot executions. By Theorem 12, the ISE complex, E n , is a chromatic divided image of I n . Since the protocol can be simulated by a full-information protocol, the corresponding full divided image is also a chromatic divided image of I n . Clearly, ffi P is a color-preserving (on ids), anonymous simplicial map from to O n that agrees with \Delta. Assume there exists an full divided image E n of I n and a color-preserving (on ids), anonymous simplicial map ffi from E n to O n that agrees with \Delta. By using a protocol for the participating set problem ([5]), the immediate snapshot executions can be simulated in a full-information manner. Using ffi as the output rule of the protocol, we get the "only if" direction of the theorem. ut Remark. The above theorem ignores the issue of computability. Clearly, the sufficient condition requires that ffi is computable; furthermore, if a task is solvable then it implies a way to compute ffi . Therefore, we can add the requirement that is computable to the necessary and sufficient condition for wait-free solvability. The previous theorem provides a characterization of wait-free solvable tasks which depends only on the topological properties of hI To see if a task is solvable, when the input complex is finite, we produce all E-divided images of I n and check if a simplicial map ffi as required exists. Note that if we are interested only in protocols that are bounded wait-free, i.e., where the protocol has to hold within a predetermined number of steps N , then producing all E- divided images of the input complex (which is finite) is recursive. Orientability: We now show that the ISE complex, E n , is an orientable chromatic divided image. This is used to prove that it induces an algebraic span [15]. We leave the proof that an orientable chromatic divided image induces an algebraic span to the full paper, since obviously, it requires the definition of algebraic span, an algebraic concept of a different flavor from the rest of this paper. be an m-pseudomanifold. An orientation of a simplex is an equivalence class of orderings of its vertices, consisting of one particular ordering and all even permutations of it. If the vertices are colored with ids, we could consider the positive orientation to be the one in which the vertices are ordered Fig. 4. An oriented 2-pseudomanifold, with a coloring (in brackets). with the ids from small to large, and the negative to be the one where the two vertices with smallest ids are exchanged (each orientation together with all its even permutations). Denote by oe (i) the face of oe m in which the vertex with id i is removed; e.g., oe (1) is the face with ids f0; mg. An orientation of an m-simplex induces an orientation on each of its faces, oe (i) , according to the sign of (\Gamma1) i . For example, if oe 2 is oriented hv then the induced orientations are K m is orientable if there exists an orientation for each of its m-simplexes such that an m \Gamma 1-simplex contained in two m-simplexes gets opposite induced orientations. K m together with such an orientation is an oriented pseudoman- ifold. (See an example in Fig. 4 of a simple oriented 2-pseudomanifold and the induced orientations.) In the sequel, we sometimes use a combinatorial notion of orientability. In the full paper, we prove that the previous (usual) definition of orientability is equivalent to the combinatorial definition, for chromatic pseudomanifolds. Lemma 15. A chromatic pseudomanifold K m is orientable if and only if its m-simplexes can be partitioned into two disjoint classes, such that if two m- simplexes share an (m \Gamma 1)-face then they belong to different classes. We say that a chromatic divided image of M (oe m ) under /, K m , is orientable if, for every oe 2 M (oe m ), /(oe) is orientable. Theorem16. Let E n be a chromatic divided image of M (oe n ) under /, that corresponds to the ISE complex starting with input oe n , in which any processor takes the same number of steps in every execution. Then E n is orientable. Proof. Let oe i be a face of oe m . We explicitly partition the i-simplexes of /(oe i ) into two disjoint classes, positive and negative. Let the length of an immediate snapshot execution be the number of concurrency classes in it. An i-simplex - 2 /(oe i ) is in positive if the length of the immediate snapshot execution corresponding to - is even; otherwise, it is in neg- ative. Consider two i-simplexes, - iand - i, that share an (i \Gamma 1)-face, and let p j be the processor whose view is different. By Lemma 9, without loss of generality, is in the kth concurrency class of - iand the kth concurrency class of - iis is the last seen round of p j in - i. Furthermore, since the views of all other processors are exactly the same, it follows that the lengths of the corresponding executions differ exactly by one. Hence, the corresponding simplexes are in different classes, i.e., have different orientations. ut 6 The Number of Monochromatic Simplexes In this section we prove a combinatorial lemma about the number of monochromatic simplexes in any binary coloring of an orientable divided image; this lemma is used in the next section to show a lower bound on renaming. be an orientable, chromatic divided image of oe m under /. Fix an orientation of K m , and an induced orientation on its boundary. K m is symmetric if, for any two i-faces of oe, oe iand oe i, /(oe i) and /(oe i) are isomorphic, under a one-to-one simplicial map i that is order preserving on the ids: if v and w belong to the same simplex, and id(v) ! id(w) then binary coloring, b, of K m is symmetric, if This definition is motivated by the notion of comparison-based protocols for renaming, presented in the next section. be the number of monochromatic m-simplexes of K m , counted by orientation, i.e., an m-simplex is counted as +1 if it is positively oriented, otherwise, it is counted as \Gamma1. For example, if K m consists of just two m-simplexes, both monochromatic, then the count would be 0, since they would have opposite orientations, and hence one would count +1 and the other \Gamma1. The main theorem of this section states that, if K m is a symmetric, oriented chromatic divided image of oe m under /, with a symmetric binary coloring b, The proof of this theorem relies on the Index Lemma-a classical result of combinatorial topology, generalizing Sperner's Lemma (cf. [12, p. 201]). To state and prove the Index Lemma, we need the following definitions. Fix a coloring c of K m with mg. A k-simplex of K m is complete under c, if it is colored with k. The content, C, of c is the number of complete m-simplexes, counted by orientation. That is, a complete simplex - m is counted +1, if the order of the vertices given by the colors agrees with the orientation of counts +1 if the order given by the colors belongs to the equivalence class of orderings of the orientation, and else it counts \Gamma1. For example, the 2-simplex - 1 in Fig. 4 is ordered hv and the colors are under this order are h0; 1; 2i, hence, it would count +1. On the other hand, the 2-simplex - 2 in Fig. 4 is ordered and the colors are under this order are h1; 0; 2i, hence, it would count \Gamma1. The index, I, of c is the number of 1)-simplexes on the boundary of K m , also counted by orientation (the orientation induced by the unique m-simplex that contains it). where the (m \Gamma 1)-simplexes in each m-simplex are considered separately, and counted as +1 or \Gamma1, by their induced orientations. We argue that I and To prove that consider the following cases. If an (m \Gamma 1)-face is inter- nal, then it contributes 0 to S, since the contributions of the two m-simplexes containing it cancel each other. Obviously, an internal (m \Gamma 1)-face contributes 0 to I. An external (m \Gamma 1)-face in the boundary of K m is counted the same, or \Gamma1 by orientation, in both S and I. Therefore, To prove that consider an m-simplex - m , and look at the following cases. If - m contains two 1)-faces which are completely colored, then - m is not completely colored and contributes 0 to C. Note that - m contributes 0 also to S, since the contributions of the two faces cancel each other. If - m contains exactly one (m \Gamma 1)-face which is completely colored (with must be completely colored and contributes +1 or \Gamma1, by orientation, to C as well as to S. If - m does not contain any (m \Gamma 1)-face which is completely colored, then - m is not completely colored and therefore, it contributes 0 to C as well as to S. Finally, note that - m cannot contain more than two which are completely colored. ut Theorem18 (Binary Coloring Theorem). Let K m be a symmetric, oriented chromatic divided image of oe m under /, with a symmetric binary coloring b. Proof. Let ae be the simplicial map from oe m to itself that maps the vertex v whose id is i to the vertex whose id is (i (that is, the mapping the rotates the id's). In the rest of the proof, we assume that sub-indices are taken Define a coloring of K m , 1), for every v. Notice that an m-simplex, - m , is completely colored by c if and only if - m is monochromatic under b. Moreover, for every v, Let C and I be the content and index of K m under c. Clearly, I 6= 0. The proof is by induction on m. I 6= 0. For the induction step, we consider bound(K m ), and "squeeze" it, by using contractions. A contraction of bound(K m ) is obtained by identifying one of its vertices, v 0 , with another vertex, v, with the same color, and deleting any simplex containing both v and v 0 . Consider an internal (m \Gamma 2)-simplex, - m\Gamma2 2 bound(K m ), which is contained in two Its link vertices are v 1 , which is the vertex of - 1 not in - m\Gamma2 , and v 2 , which is the vertex of - 2 not in - m\Gamma2 . A binary coloring is irreducible if the link vertices of any internal (m \Gamma 2)-simplex simplex of /(oe (i) ) have different binary colors. The first stage of the proof applies a sequence of specific contractions to its coloring is irreducible, while preserving all other properties. The contractions we apply are symmetric contractions, in which we choose an internal (m \Gamma 2)-simplex, - m\Gamma2 2 /(oe (m) ), to which a contraction can be applied; that is, such that its two link vertices have the same binary coloring. We contract - m\Gamma2 and simplexes symmetric to it in /(oe (i) ), for all i. (This is a sequence of m+ 1 contractions.) Notice that the simplexes which are symmetric to - m\Gamma2 are also internal and their link vertices have the same binary coloring. A boundary is proper symmetric if it is the boundary of a symmetric, oriented chromatic divided image of oe m under /, with a symmetric binary coloring b. In the next claim we show that a symmetric contraction preserves all properties of a proper symmetric boundary. Assume we apply a symmetric contraction to bound(K m ), and get a complex bound 0 . Then bound 0 is a non-empty, proper symmetric boundary under Furthermore, I(bound 0 Proof. Given note that we have that Therefore, bound 0 is chromatic. Also, it is easy to see that the orientation on bound 0 is still well defined: two (m \Gamma 1)-simplexes that did not have an (m \Gamma 2)-face in common before the contraction will have it after the contraction, only if they differ in exactly one vertex, in addition to v 1 and Thus, two such simplexes have opposite orientations. By the definition of symmetric contraction, bound 0 remains symmetric. By induction hypothesis of the theorem, #mono(/(oe (i) )) 6= 0, for every i. Since a contraction removes simplexes with opposite orientations and the same binary colorings, #mono(/(oe (i) for every i, and This implies that bound 0 is non-empty. ut By Claim 19, for the rest of the proof, we can assume that /(bound(oe m non-empty, proper symmetric boundary with an irreducible binary coloring. Recall that 1)-simplexes on the boundary of K m are counted with the same sign by I. Proof. We first argue that every complete (m \Gamma 1)-simplex in /(oe (i) ) is counted with the same sign by I, for any i. To see this, assume, without loss of generality, that consider an colored with (the first component of a vertex is the id and the second is its binary color). Consider a path to any other (m \Gamma 1)-simplex - 2 colored with the same ids and colors; such a path must exist since, by Lemma 3, we can assume that /(oe (i) ) is 1)-connected. Notice that the colors assigned by c are the same in - 1 and will be counted by I. It remains to show that - 1 and have the same orientation and hence are counted with the same sign by I. Note that this path consists of a sequence of (m \Gamma 1)-simplexes, each sharing an (m\Gamma2)-face with the previous simplex, and differing from the previous simplex in exactly one binary color. Thus the path corresponds to a sequence of binary vectors, starting with the all 0's vector and ending with the all 0's vector, and each vector differing from the previous vector in exactly one binary color. That is, the path corresponds to a cycle in a hypercube graph. Since the hypercube graph is bipartite, the length of any cycle in it is even; therefore, the length of the path is even. Clearly, since the complex is oriented, consecutive simplexes on the path have different orientations. Since the length of the path is even, - 1 have the same orientation. Hence, - 1 and - 2 are counted with the same sign by I. Next, we show that complete (m \Gamma 1)-simplexes in different /(oe (i) )'s are also counted with the same sign by I. Again, without loss of generality, assume that counted by I. Note that the c coloring of - 1 is f(0; 0); (1; 1)g. We now show that any complete (m \Gamma 1)-simplex - 3 2 /(oe (i) ) will be counted with the same sign by I. Without loss of generality, assume - 3 is complete, with id's mg. Thus, the binary color of the vertex with process id m must be 1, in order to get the color 0 under c. This implies that the c coloring of - 3 is f(1; 1); (2; its binary coloring is 1)g. Consider the simplex - 2 2 /(oe (0) ), which is the image of - 1 under the symmetry map, ae. That is, 0)g. Consider a path in /(oe (0) ) and - 3 . Since the binary coloring vector of - 3 differs from the binary coloring vector of - 2 in exactly one position, the length of this path must be odd. Therefore, and - 3 must have different orientations. The c coloring of - 3 , f(1; 1); (2; rotated w.r.t. its ids, and hence the orderings of - 2 and - 3 agree (on the sign of the permutation) if and only if m is odd. E.g., if 0)g. Finally, the orientation of - 1 is (\Gamma1) m times the orientation of - 2 , since they are symmetric simplexes in /(oe (m) ) and /(oe (0) ). That is, the orientations of - 1 and - 2 agree when m is even, and disagree otherwise. Therefore, the orientations of - 1 and - 3 agree, and they are counted with the same sign by I. ut Since bound is non-empty and contains at least one simplex, Claim 20 implies I 6= 0, which proves the theorem. ut Applications In this section, we apply the condition for wait-free solvability presented earlier (Corollary 13) to derive two lower bounds, for renaming and for k-set consensus. The first lower bound also relies on Theorem 18, and therefore, on the fact that the chromatic divided image induced by immediate snapshot executions is orientable. In the full version of the paper we also derive another necessary condition, based on connectivity. 7.1 Renaming In the renaming task ([2]), processes start with an input value (original name) from a large domain and are required to decide on distinct output values (new names) from a domain which should be as small as possible. Clearly, the task is trivial if processes can access their id; in this case, process p i decides on i, which yields the smallest possible domain. To avoid trivial solutions, it is required that the processes and the protocol are anonymous [2]. That is, process p i with original name x executes the same protocol as process p j with original name x. Captured in our combinatorial topology language, the M-renaming task is the triple hD contains all subsets of some domain D (of original names) with different values, M n contains all subsets of [0::M ] (of new names) with different values, and \Delta maps each oe n 2 D n to all n-simplexes of M n . We use Theorem 12 and Theorem to prove that there is no wait-free anonymous protocol for the M-renaming task, if M - 2n \Gamma 1. The bound is tight, since there exists an anonymous wait-free protocol ([2]) for the 2n-renaming problem. Theorem 21. If M ! 2n, then there is no anonymous wait-free protocol that solves the M -renaming task. Proof. Assume, by way of contradiction, that P is a wait-free protocol for the M-renaming task, M - 2n \Gamma 1. Without loss of generality, we assume that every process executes the same number of steps. Also, P is comparison-based, i.e., the protocol produces the same outputs on inputs which are order-equivalent. (See Herlihy [13], who attributes this observation to Eli Gafni). Assume that assume that the original names are only between 0 and 2n. By Corollary 13, there exists a a chromatic full divided image of the input complex D n , be the decision map implied by P. By Theorem 16, S is orientable. Since the protocol is comparison-based and anonymous, it follows that for any two i-simplexes, oe iand oe iof D n , ffi P maps /(oe i) and /(oe i) to simplexes that have the same output values (perhaps with different process ids). be the binary coloring which is the parity of the new names assigned by Therefore, the assumption of Theorem is satisfied for S(oe n ), and therefore, at least one simplex of S(oe n ) is monochromatic under ffi 0 . On the other hand, note that the domain [0; 2n \Gamma 1] does not include n different odd names; similarly, the domain [0; 2n \Gamma 1] does not include n different even names. This implies that ffi 0 cannot color any simplex of S with all zeroes or with all ones; i.e., no simplex of S is monochromatic. A contradiction. ut 7.2 k-Set Consensus Intuitively, in the k-set consensus task ([7]), processes start with input values from some domain and are required to produce at most k different output values. To assure non-triviality, we require all output values to be input values of some processes. Captured in our combinatorial topology language, the k-set consensus task is the triple hD n ; D n ; \Deltai. D n is P (D), for some domain D, and \Delta maps each to the subset of n-simplexes in D n that contain at most k different values from the values in oe n . In the full version of the paper, we use Theorem 12 and Sperner's Lemma to prove that any wait-free protocol for this problem must have at least one execution in which k +1 different values are output. This implies: Theorem22. If k - n then there does not exists a wait-free protocol that solves the k-set consensus task. This bound is tight, by the protocol of [7]. This paper presents a study of wait-free solvability based on combinatorial topol- ogy. Informally, we have defined the notion of a chromatic divided image, and proved that a necessary condition for wait-free solvability is the existence of a simplicial chromatic mapping from a divided image of the inputs to the outputs that agrees with the problem specification. We were able to use theorems about combinatorial properties of divided images to derive tight lower bounds for renaming and k-set consensus. Our results do not use homology groups, whose computation may be complicated. We also derive a new necessary and sufficient condition, based on a specific, well structured chromatic divided image. Many questions remain open. First, it is of interest to find other applications to the necessary and sufficient condition presented here; in particular, can we derive interesting protocols from the sufficient condition? Second, there are several directions to extend our framework, e.g., to allow fewer than n failures (as was done for one failure in [3]), to handle other primitive objects besides read/write registers (cf. [14, 6]), and to incorporate on-going tasks. Acknowledgments We would like to thank Javier Bracho, Eli Gafni, Maurice Herlihy, Nir Shavit and Mark Tuttle for comments on the paper and very useful discussions. --R "Are Wait-Free Algorithms Fast?" "Renaming in an asynchronous environment," "A combinatorial characterization of the distributed 1.solvable tasks," "Generalized FLP impossibility result for t-resilient asynchronous computations," "Immediate atomic snapshots and fast renaming," "The implication of the Borowsky-Gafni simulation on the set consensus hierarchy," "More Choices Allow More Faults: Set Consensus Problems in Totally Asynchronous Systems," "A tight lower bound for k-set agreement," "Reaching Approximate Agreement in the Presence of Faults," "Impossibility of distributed commit with one faulty process," "3-processor tasks are undecidable," A Combinatorial Introduction to Topology A. Tutorial on "Set Consensus Using Arbitrary Objects," "Algebraic Spans," "On the Decidability of Distributed Decision Tasks," "The asynchronous computability theorem for t- resilient tasks," "A simple constructive computability theorem for wait-free computation," Elements of "Wait-free k-set agreement is impossible: The topology of public knowledge," --TR --CTR Faith Fich , Eric Ruppert, Hundreds of impossibility results for distributed computing, Distributed Computing, v.16 n.2-3, p.121-163, September

atomic read/write registers;set consensus;consensus;combinatorial topology;renaming;distributed systems;shared memory systems;wait-free solvable tasks

586915

Randomness, Computability, and Density.

We study effectively given positive reals (more specifically, computably enumerable reals) under a measure of relative randomness introduced by Solovay [manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1975] and studied by Calude, Hertling, Khoussainov, and Wang [Theoret. Comput. Sci., 255 (2001), pp. 125--149], Calude [Theoret. Comput. Sci., 271 (2002), pp. 3--14], Kucera and Slaman [SIAM J. Comput., 31 (2002), pp. 199--211], and Downey, Hirschfeldt, and LaForte [Mathematical Foundations of Computer Science 2001, Springer-Verlag, Berlin, 2001, pp. 316--327], among others. This measure is called domination or Solovay reducibility and is defined by saying that $\alpha$ dominates $\beta$ if there are a constant c and a partial computable function $\varphi$ such that for all positive rationals $q<\alpha$ we have $\varphi(q)\!\downarrow<\beta$ and $\beta- \varphi(q) \leqslant c(\alpha- q)$. The intuition is that an approximating sequence for $\alpha$ generates one for $\beta$ whose rate of convergence is not much slower than that of the original sequence. It is not hard to show that if $\alpha$ dominates $\beta$, then the initial segment complexity of $\alpha$ is at least that of $\beta$.In this paper we are concerned with structural properties of the degree structure generated by Solovay reducibility. We answer a natural question in this area of investigation by proving the density of the Solovay degrees. We also provide a new characterization of the random computably enumerable reals in terms of splittings in the Solovay degrees. Specifically, we show that the Solovay degrees of computably enumerable reals are dense, that any incomplete Solovay degree splits over any lesser degree, and that the join of any two incomplete Solovay degrees is incomplete, so that the complete Solovay degree does not split at all. The methodology is of some technical interest, since it includes a priority argument in which the injuries are themselves controlled by randomness considerations.

Introduction In this paper we are concerned with eectively generated reals in the interval (0; 1] and their relative randomness. In what follows, real and rational will mean positive real and positive rational, respectively. It will be convenient to work modulo 1, that is, identifying and we do this below without further comment. Our basic objects are reals that are limits of computable increasing sequences of ratio- nals. We call such reals computably enumerable (c.e.), though they have also been called recursively enumerable, left computable (by Ambos-Spies, Weihrauch, and Zheng [1]), and, together with the limits of computable decreasing sequences of rationals, semicom- putable. If, in addition to the existence of a computable increasing sequence q rationals with limit , there is a total computable function f such that q f(n) < 2 n for all n 2 !, then is called computable. These and related concepts have been widely studied. In addition to the papers and books mentioned elsewhere in this intro- duction, we may cite, among others, early work of Rice [24], Lachlan [19], Soare [27], and Cetin [8], and more recent papers by Ko [16, 17], Calude, Coles, Hertling, and Khoussainov [5], Ho [15], and Downey and LaForte [14]. A real is random if its dyadic expansion forms a random innite binary sequence (in the sense of, for instance, Martin-Lof [23]). Chaitin's number the halting probability of a universal self-delimiting computer, is a standard random c.e. real. (We will dene these concepts more formally below.) Many authors have studied and its properties, notably Chaitin [9, 10, 11] and Martin-Lof [23]. In the very long and widely circulated manuscript [30] (a fragment of which appeared in [31]), Solovay carefully investigated relationships between Martin-Lof- Chaitin prex-free complexity, Kolmogorov complexity, and properties of random languages and reals. See Chaitin [9] for an account of some of the results in this manuscript. Solovay discovered that several important properties of (whose denition is model- dependent) are shared by another class of reals he called lle e, whose denition is model-independent. To dene this class, he introduced the following reducibility relation among c.e. reals, called domination or Solovay reducibility. 1.1. Denition. Let and be c.e. reals. We say that dominates , and write 6 S , if there are a constant c and a partial computable function that for each rational q < we have '(q)#< and We write S if 6 S and 6 S . The notation 6 dom has sometimes been used instead of 6 S . Solovay reducibility is naturally associated with randomness because of the following fact, whose proof we sketch for completeness. Let H() denote the prex-free complexity of and K() its standard Kolmogorov complexity. (Most of the statements below also hold with K() in place of H( ). For the denitions and basic properties of H() and K( ), see Calude [3] and Li and Vitanyi [22]. Among the many works dealing with these and related topics, and in addition to those mentioned elsewhere in this paper, we may cite Solomono [28, 29], Kolmogorov [18], Levin [20, 21], Schnorr [25], and the expository article Calude and Chaitin [4].) We identify a real 2 (0; 1] with the innite binary string such that = 0:. (The fact that certain reals have two dierent dyadic expansions need not concern us here, since all such reals are rational.) 1.2. Theorem (Solovay [30]). Let 6 S be c.e. reals. There is a constant O(1) such that H( n) 6 H( n) +O(1) for all n 2 !. Proof sketch. We rst sketch the proof of the following lemma, implicit in [30] and noted by Calude, Hertling, Khoussainov, and Wang [6]. 1.3. Lemma. Let c 2 !. There is a constant O(1) such that, for all n > 1 and all binary strings ; of length n with j0: 0: j < c2 n , we have jH() H()j 6 O(1). The proof of the lemma is relatively simple. We can easily write a program P that, for each su-ciently long , generates the 2c binary strings 0 of length n with . For any binary strings ; of length n with j0: 0: j < c2 n , in order to compute it su-ces to know a program for and the position of on the list generated by P on input . Turning to the proof of the theorem, let ' and c be as in Denition 1.1. Let 0:( n). Since n is rational and n < 2 (n+1) , we have '( n ) < c2 (n+1) . Thus, by the lemma, H( n) = H('( n ))+O(1), and hence H( n) 6 H( n)+O(1). Solovay observed that dominates all c.e. reals, and Theorem 1.2 implies that if a c.e. real dominates all c.e. reals then it must be random. This led Solovay to dene a c.e. real to be -like if it dominates all c.e. reals. The point is that the denition of seems quite model-independent, as opposed to the model-dependent denition of This circle of ideas was completed recently by Slaman [26], who proved the converse to the fact that at e reals are random. 1.4. Theorem (Slaman). A c.e. real is random if and only if it is -like. It is natural to seek to understand the c.e. reals under Solovay reducibility. A useful characterization of this reducibility is given by the following lemma, which we prove in the next section. 1.5. Lemma. Let and be c.e. reals. Then 6 S if and only if for every computable sequence of rationals a 0 ; a a n there are a constant c and a computable sequence of rationals " c such that Phrased another way, Lemma 1.5 says that the c.e. reals dominated by a given c.e. real essentially correspond to splittings of under arithmetic addition. 1.6. Corollary. Let 6 S be c.e. reals. There is a c.e. real and a rational c such that Proof. Let a 0 ; a be a computable sequence of rationals such that a n . Let be as in Lemma 1.5. Dene each " n is less than c, the real is c.e., and of course . Solovay reducibility has a number of other beautiful interactions with arithmetic, as we now discuss. The relation 6 S is symmetric and transitive, and hence S is an equivalence relation on the c.e. reals. Thus we can dene the Solovay degree [] of a c.e. real as its S equivalence class. (When we mention Solovay degrees below, we always mean Solovay degrees of c.e. reals.) The Solovay degrees form an upper semilattice, with the join of [] and [] being [+]=[], a fact observed by Solovay and others ( is denitely not a join operation here). We note the following slight improvement of this result. Recall that an uppersemilattice U is distributive if for all a exist 1.7. Lemma. The Solovay degrees of c.e. reals form a distributive uppersemilattice with Proof. Suppose that 6 S of rationals such that 1. By Lemma 1.5, there are a constant c and a computable sequence of rationals " c such that n . Then This establishes distributivity. To see that the join in the Solovay degrees is given by addition, we again apply Lemma 1.5. Certainly, for any c.e. reals 0 and 1 we have and hence Conversely, suppose that 0 ; 1 6 . Let a 0 ; a be a computable sequence of rationals such that a n . For each there is a constant c i and a computable sequence of rationals " i n a n . Thus n is less than c 0 +c 1 , a nal application of Lemma 1.5 show that 0 The proof that the join in the Solovay degrees is also given by multiplication is a similar application of Lemma 1.5. There is a least Solovay degree, the degree of the computable reals, as well as a greatest one, the degree of For proofs of these facts and more on c.e. reals and Solovay reducibility, see for instance Chaitin [9, 10, 11], Calude, Hertling, Khoussainov, and Wang [6], Calude and Nies [7], Calude [2], Slaman [26], and Coles, Downey, and LaForte [12]. Despite the many attractive features of the Solovay degrees, their structure is largely unknown. Coles, Downey, and LaForte [12] have shown that this structure is very complicated by proving that it has an undecidable rst order theory. One question addressed in the present paper, open since Solovay's original 1974 notes, is whether the structure of the Solovay degrees is dense. Indeed, up to now, it was not known even whether there is a minimal Solovay degree. That is, intuitively, if a c.e. real is not computable, must there be a c.e. real that is also not computable, yet is strictly less random that ? In this paper, we show that the Solovay degrees of c.e. reals are dense. To do this we divide the proof into two parts. We prove that if < then there is a c.e. real with < S and we also prove that every incomplete Solovay degree splits over each lesser degree. The nonuniform nature of the argument is essential given the techniques we use, since, in the splitting case, we have a priority construction in which the control of the injuries is directly tied to the enumeration of The fact that if a c.e. real is Solovay- incomplete then must grow more slowly than is what allows us to succeed. (We will discuss this more fully in Section 3.) This unusual technique is of some technical interest, and clearly cannot be applied to proving upwards density, since in that case the top degree is itself. To prove upwards density, we use a dierent technique, taking advantage of the fact that, however we construct a c.e. real, it is automatically dominated by In light of these results, and further motivated by the general question of how randomness can be produced, it is natural to ask whether the complete Solovay degree can be split, or in other words, whether there exist nonrandom c.e. reals and such that We give a negative answer to this question, thus characterizing the random c.e. reals as those c.e. reals that cannot be written as the sum of two c.e. reals of lesser Solovay degrees. We remark that there are (non-c.e.) nonrandom reals whose sum is random; the following is an example of this phenomenon. Dene the real by letting is even and n) otherwise. (Here we identify a real with its dyadic expansion as above.) Dene the real by letting n) otherwise. Now and are clearly nonrandom, but is random. Before turning to the details of the paper, we point out that there are other reducibilities one can study in this context. Coles, Downey, and LaForte [12, 13] introduced one such reducibility, called sw-reducibility ; it is dened as follows. For sets of natural numbers A and B, we say that A 6 sw B if there are a computable procedure and a constant c such that A and the use of on argument x is bounded by x reals we say that 6 sw if there are sets A and B such that is the characteristic function of the set S. As in the case of Solovay reducibility, it is not di-cult to argue that if 6 sw then that is sw-complete. Furthermore, Coles, Downey, and LaForte [12] proved the analog of Slaman's theorem above in the case of sw-reducibility, namely that if a c.e. real is random then it is sw-complete. They also showed that Solovay reducibility and sw-reducibility are dierent, since there are c.e. reals , , , and - such that 6 S but sw and S -, and that there are no minimal sw-degrees of c.e. reals. 1.8. Question. Are the sw-degrees of c.e. reals dense? Ultimately, the basic reducibility we seek to understand is H-reducibility, where 6H if there is a constant O(1) such that H( n) 6 H( n) +O(1) for all n 2 !. Little is known about this directly. Preliminaries Fix a self-delimiting universal computer M . (That is, for all binary strings , if M()# then M( 0 )" for all 0 properly extending .) Then one can dene via M()# (The properties of relevant to this paper are independent of the choice of M .) The c.e. real is random in the canonical Martin-Lof sense. Recall that a Martin- Lof test is a uniformly c.e. sequence fV e : e > 0g of subsets of f0; 1g such that for all e > 0, where denotes the usual product measure on f0; 1g ! . The string 2 f0; 1g ! and the real 0: are random, or more precisely, 1-random, if =T for every Martin-Lof test fV e : e > 0g. An alternate characterization of the random reals can be given via the notion of a Solovay test. We give a somewhat nonstandard denition of this notion, which will be useful below. A Solovay test is a c.e. sequence fI of intervals with rational end-points such that is the length of the interval I. As Solovay [30] showed, a real is random if and only if is nite for every Solovay test The following lemma, implicit in [30] and proved in [12], provides an alternate characterization of Solovay reducibility, which is the one that we will use below. 2.1. Lemma. Let and be c.e. reals, and let increasing sequences of rationals converging to and , respectively. Then 6 S if and only if there are a constant d and a total computable function f such that for all Whenever we mention a c.e. real , we assume that we have chosen a computable increasing sequence converging to . The previous lemma guarantees that, in determining whether one c.e. real dominates another, the particular choice of such sequences is irrelevant. For convenience of notation, we adopt the convention that, for any c.e. real mentioned below, the expression s s 1 is equal to 0 when We will also make use of two more lemmas, the rst of which has Lemma 1.5 as a corollary. 2.2. Lemma. Let 6 S be c.e. reals and let be a computable increasing sequence of rationals converging to . There is a computable increasing sequence ^ of rationals converging to such that for some constant c and all s 2 !, Proof. Fix a computable increasing sequence rationals converging to , let d and f be as in Lemma 2.1, and let c > d be such that f(0) < c 0 . We may assume without loss of generality that f is increasing. There must be an s 0 > 0 for which f(s 0 ) f(0) < would have contradicting our choice of d and f . It is now easy to s0 so that ^ We can repeat the procedure in the previous paragraph with s 0 in place of 0 to obtain an s1 such that ^ Proceeding by recursion in this way, we dene a computable increasing sequence of rationals with the desired properties. We are now in a position to prove Lemma 1.5. 1.5. Lemma. Let and be c.e. reals. Then 6 S if and only if for every computable sequence of rationals a 0 ; a a n there are a constant c and a computable sequence of rationals " c such that Proof. The if direction is easy; we prove the only if direction. Suppose that 6 S . Given a computable sequence of rationals a 0 ; a i6n a i and apply Lemma 2.2 to obtain c and ^ that lemma. Dene " n . Now for all n 2 !, We nish this section with a simple lemma which will be quite useful below. 2.3. Lemma. Let be c.e. reals. The following hold for all total computable functions f and all k 2 !. 1. For each there is an s 2 ! such that either (a) t f(n) < k( t n ) for all t > s or 2. There are innitely many for which there is an s 2 ! such that t f(n) > Proof. If there are innitely many t 2 ! such that t f(n) 6 k( t n ) and innitely many which implies that S . If there are innitely many t 2 ! such that t f(n) 6 k( t n ) then So if this happens for all but nitely many n then 6 S . (The nitely many n for which f(n) > k( n ) can be brought into line by increasing the constant k.) 3 Main Results We now proceed with the proofs of our main results. We begin by showing that every incomplete Solovay degree can be split over any lesser Solovay degree. 3.1. Theorem. Let be c.e. reals. There are c.e. reals 0 and 1 such that and 0 Proof. We want to build 0 and 1 so that and the following requirement is satised for each 2: By Lemma 2.2 and the fact that =c S for any rational c, we may assume without loss of generality that 2( s (Recall our convention that 0 c.e. real .) In the absence of requirements of the form R 1 i;e;k , it is easy to satisfy simultaneously all requirements of the form R i;e;k : for each s 2 !, simply let i s and 1 i s . In the presence of requirements of the form R 1 i;e;k , however, we cannot aord to be quite so cavalier in our treatment of enough of has to be kept out of 1 i to guarantee that 1 i does not dominate . Most of the essential features of our construction are already present in the case of two requirements R i;e;k and R 1 i;e 0 ;k 0 , which we now discuss. We assume that R i;e;k has priority over R 1 i;e 0 ;k 0 and that both e and e 0 are total. We will think of the j as being built by adding amounts to them in stages. Thus j s will be the total amount added to j by the end of stage s. At each stage s we begin by adding s s 1 to the current value of each j ; in the limit, this ensures that j > S We will say that R i;e;k is satised through n at stage s if e (n)[s]# and s e (n) > ). The strategy for R i;e;k is to act whenever either it is not currently satised or the least number through which it is satised changes. Whenever this happens, R i;e;k initializes R 1 i;e 0 ;k 0 , which means that the amount of 2 that R 1 i;e 0 ;k 0 is allowed to funnel into i is reduced. More specically, once R 1 i;e 0 ;k 0 has been initialized for the mth time, the total amount that it is thenceforth allowed to put into i is reduced to The above strategy guarantees that if R 1 i;e 0 ;k 0 is initialized innitely often then the amount put into i by R 1 i;e 0 ;k 0 (which in this case is all that is put into i except for the coding of adds up to a computable real. In other words, i S < S . But it is not hard to argue, with the help of Lemma 2.3, that this means that there is a stage s after which R i;e;k is always satised and the least number through which it is satised does not change. So we conclude that R 1 i;e 0 ;k 0 is initialized only nitely often, and that R i;e;k is eventually permanently satised. This leaves us with the problem of designing a strategy for R 1 i;e 0 ;k 0 that respects the strategy for R i;e;k . The problem is one of timing. To simplify notation, let ^ s . Since R 1 i;e 0 ;k 0 is initialized only nitely often, there is a certain that it is allowed to put into i after the last time it is initialized. Thus waits until a stage s such that ^ adding nothing to i until such a stage is reached, then from that point on it can put all of ^ s into i , which of course guarantees its success. The problem is that, in the general construction, a strategy working with a quota 2 m cannot eectively nd an s such that ^ If it uses up its quota too soon, it may nd itself unsatised and unable to do anything about it. The key to solving this problem (and the reason for the hypothesis that < S is the observation that, since the much more slowly than the sequence ^ can be used to modulate the amount that R 1 i;e 0 ;k 0 puts into More specically, at a stage s, if R 1 i;e 0 ;k 0 's current quota is 2 m then it puts into i as much of ^ possible, subject to the constraint that the total amount put into i by R 1 i;e 0 ;k 0 since the last stage before stage s at which R 1 i;e 0 ;k 0 was initialized must not exceed 2 s . As we will see below, the fact that implies that there is a stage v after which R 1 i;e 0 ;k 0 is allowed to put in all of ^ In general, at a given stage s there will be several requirements, each with a certain amount that it wants (and is allowed) to direct into one of the j . We will work back- wards, starting with the weakest priority requirement that we are currently considering. This requirement will be allowed to direct as much of ^ wants (subject to its current quota, of course). If any of ^ then the next weakest priority strategy will be allowed to act, and so on up the line. We now proceed with the full construction. We say that R i;e;k has stronger priority We say that R i;e;k is satised through n at stage s if s be the least n through which R i;e;k is satised at stage s, if such an n exists, and let n i;e;k A stage s is e-expansionary if Let q be the last e-expansionary stage before stage s (or let there have been none). We say that R i;e;k requires attention at stage s if s is an e-expansionary stage and there is an r 2 [q; s) such that either n i;e;k r r 1 . If R i;e;k requires attention at stage s then we say that each requirement of weaker priority than R i;e;k is initialized at stage s. Each requirement R i;e;k has associated with it a c.e. real i;e;k , which records the amount put into 1 i for the sake of R i;e;k . We decide how to distribute at stage s as follows. 1. Let s s s 1 to the current value of each i . 2. Let i < 2 and be such that 2he; ki be the number of times R i;e;k has been initialized and let t be the last stage at which R i;e;k was initialized. Let (j+m) (It is not hard to check that is non-negative.) Add to " and to the current values of i;e;k and 1 i . 3. " to the current value of 0 and end the stage. Otherwise, decrease j by one and go to step 2. This completes the construction. Clearly, We now show by induction that each requirement initializes requirements of weaker priority only nitely often and is eventually satised. Assume by induction that R i;e;k is initialized only nitely often. Let be the number of times R i;e;k is initialized, and let t be the last stage at which R i;e;k is initialized. If e is not total then R i;e;k is vacuously satised and eventually stops initializing requirements of weaker priority, so we may assume that e is total. Now the following are clearly equivalent: 1. R i;e;k is satised, 2. lim s n i;e;k s exists and is nite, and 3. R i;e;k eventually stops requiring attention. Assume for a contradiction that R i;e;k requires attention innitely often. Since , part 2 of Lemma 2.3 implies that there are v > u > t such that for all w > v we Furthermore, by the way the amount added to i;e;k at a given stage is dened in step 2 of the construction, i;e;k (j+m) u and i;e;k Thus for all w > v, (j+m) (j+m) (j+m) From this we conclude that, after stage v, the reverse recursion performed at each stage never gets past j, and hence everything put into i after stage v is put in either to code or for the sake of requirements of weaker priority than R i;e;k . Let be the sum of all 1 i;e 0 ;k 0 such that R 1 i;e 0 ;k 0 has weaker priority than R i;e;k . Let s l > t be the lth stage at which R i;e;k requires attention. If R 1 i;e 0 ;k 0 is the pth requirement on the priority list and p > j then s l l and hence is computable. Putting together the results of the previous two paragraphs, we see that i 6 S Since , this means that It now follows from Lemma 2.3 that there is an such that R i;e;k is eventually permanently satised through n, and such that R i;e;k is eventually never satised through any n 0 < n. Thus lim s n i;e;k s exists and is nite, and hence R i;e;k is satised and eventually stops requiring attention. We now show that the Solovay degrees are upwards dense, which together with the previous result implies that they are dense. 3.2. Theorem. Let be a c.e. real. There is a c.e. real such that < . Proof. We want to build > S to satisfy the following requirements for each and As in the previous proof, the analysis of an appropriate two-strategy case will be enough to outline the essentials of the full construction. Let us consider the strategies S e;k and R e 0 ;k 0 , the former having priority over the latter. We assume that both e and are total. The strategy for S e;k is basically to make look like . At each point of the con- struction, R e 0 ;k 0 has a certain fraction of that it is allowed to put into . (This is in addition to the coding of into , of course.) We will say that S e;k is satised through n at stage s if e (n)# and s e (n) > k( s n ). Whenever either it is not currently satised or the least number through which it is satised changes, S e;k initializes R e 0 ;k 0 , which means that the fraction of that R e 0 ;k 0 is allowed to put into is reduced. As in the previous proof, if S e;k is not eventually permanently satised through some n then the amount put into by R e 0 ;k 0 is computable, and hence S . But, as before, this implies that there is a stage after which S e;k is permanently satised through some n and never again satised through any n 0 < n. Once this stage has been reached, R e 0 ;k 0 is free to code a xed fraction of into , and hence it too succeeds. We now proceed with the full construction. We say that a requirement X e;k has stronger priority than a requirement Y e 0 ;k 0 if either he; ki < he We say that R e;k is satised through n at stage s if e (n)# and s We say that S e;k is satised through n at stage s if e (n)# and s For a requirement X e;k , let n X e;k s be the least n through which X e;k is satised at stage s, if such an n exists, and let n X e;k As before, a stage s is e-expansionary if Let X e;k be a requirement and let q be the last e-expansionary stage before stage s (or there have been none). We say that requires attention at stage s if s is an e-expansionary stage and there is an r 2 [q; s) such that either n X e;k r r 1 . At stage s, proceed as follows. First add s s 1 to the current value of . If no requirement requires attention at stage s then end the stage. Otherwise, let X e;k be the strongest priority requirement requiring attention at stage s. We say that X e;k acts at stage s. If then initialize all weaker priority requirements and end the stage. be the number of times that R e;k has been initialized. If s is the rst stage at which R e;k acts after the last time it was initialized then let t be the last stage at which R e;k was initialized, and otherwise let t be the last stage at which R e;k acted. Add 2 (j+m)( s t ) to the current value of and end the stage. This completes the construction. Since is bounded by it is a well-dened c.e. real. Furthermore, We now show by induction that each requirement initializes requirements of weaker priority only nitely often and is eventually satised. Assume by induction that there is a stage u such that no requirement of stronger priority than X e;k requires attention after stage u. If e is not total then X e;k is vacuously satised and eventually stops requiring attention, so we may assume that e is total. Now the following are clearly equivalent: 1. X e;k is satised, 2. lim s n X e;k s exists and is nite, 3. X e;k eventually stops requiring attention, and 4. X e;k acts only nitely often. First suppose that be the number of times that R e;k is initialized. (Since R e;k is not initialized at any stage after stage u, this number is nite.) Suppose that R e;k acts innitely often. Then the total amount added to for the sake of R e;k is 2 (j+m) and hence S 2 (j+m) . It now follows from Lemma 2.3 that there is an such that R e;k is eventually permanently satised through n, and such that R e;k is eventually never satised through n 0 < n. Thus lim s n R e;k s exists and is nite, and hence R e;k is satised and eventually stops requiring attention. Now suppose that acts innitely often. If v > u is the mth stage at which S e;k acts then the total amount added to after stage v for purposes other than coding is bounded by . This means that S It now follows from Lemma 2.3 that there is an such that S e;k is eventually permanently satised through n, and such that S e;k is eventually never satised through n 0 < n. Thus s exists and is nite, and hence S e;k is satised and eventually stops requiring attention. Combining Theorems 3.1 and 3.2, we have the following result. 3.3. Theorem. The Solovay degrees of c.e. reals are dense. We nish by showing that the hypothesis that < in the statement of Theorem 3.1 is necessary. This fact will follow easily from a stronger result which shows that, despite the upwards density of the Solovay degrees, there is a sense in which the complete Solovay degree is very much above all other Solovay degrees. We begin with a lemma giving a su-cient condition for domination. 3.4. Lemma. Let f be an increasing total computable function and let k > 0 be a natural number. Let and be c.e. reals for which there are innitely many s 2 ! such that k( s ) > f(s) , but only nitely many s 2 ! such that k( t s ) > f(t) f(s) for all t > s. Then 6 S . Proof. By taking as an approximating sequence for , we may assume that f is the identity. By hypothesis, there is an r 2 ! such that for all s > r there is a t > s with . Furthermore, there is an s 0 > r such that k( s Given s i , let s i+1 be the least number greater than s i such that k( s i+1 Assuming by induction that k( s i , we have Thus s 0 < s 1 < is a computable sequence such that k( s i for all Now dene the computable function g by letting g(n) be the least s i that is greater than or equal to n. Then g(n) < k( g(n) ) 6 k( n ) for all n 2 !, and hence 6 S . 3.5. Theorem. Let and be c.e. reals, let f be an increasing total computable func- tion, and let k > 0 be a natural number. If is random and there are innitely many such that k( s ) > f(s) then is random. Proof. As in Lemma 3.4, we may assume that f is the identity. If is rational then we can replace it with a nonrational computable real 0 such that 0 0 s > s for all so we may assume that is not rational. We assume that is nonrandom and there are innitely many s 2 ! such that show that is nonrandom. The idea is to take a Solovay test !g such that 2 I i for innitely many use it to build a Solovay test !g such that 2 J i for innitely many i 2 !. Let Except in the trivial case in which S , Lemma 2.3 guarantees that U is 0 2 . Thus a rst attempt at building B could be to run the following procedure for all parallel. Look for the least t such that there is an s < t with s 2 U [t] and s 2 I i . If there is more than one number s with this property then choose the least among such numbers. Begin to add the intervals to B, continuing to do so as long as s remains in U and the approximation of remains in I i . If the approximation of leaves I i then end the procedure. If s leaves U , say at stage u, then repeat the procedure (only considering t > u, of course). If 2 I i then the variable s in the above procedure eventually assumes a value in U . For this value, k( s ) > s , from which it follows that k( u s ) > s for some u > s, and hence that must be in one of the intervals () added to B by the above procedure. Since is in innitely many of the I i , running the above procedure for all guarantees that is in innitely many of the intervals in B. The problem is that we also need the sum of the lengths of the intervals in B to be nite, and the above procedure gives no control over this sum, since it could easily be the case that we start working with some s, see it leave U at some stage t (at which point we have already added to B intervals whose lengths add up to t 1 s ), and then nd that the next s with which we have to work is much smaller than t. Since this could happen many times for each i 2 !, we would have no bound on the sum of the lengths of the intervals in B. This problem would be solved if we had an innite computable subset T of U . For each I i , we could look for an s 2 T such that s 2 I i , and then begin to add the intervals () to B, continuing to do so as long as the approximation of remained in I i . (Of course, in this easy setting, we could also simply add the single interval [ to B.) It is not hard to check that this would guarantee that if 2 I i then is in one of the intervals added to B, while also ensuring that the sum of the lengths of these intervals is less than or equal to k jI i j. Following this procedure for all would give us the desired Solovay test B. Unless 6 S , however, there is no innite computable so we use Lemma 3.4 to obtain the next best thing. Let If 6 S then is nonrandom, so, by Lemma 3.4, we may assume that S is innite. Note that k( s ) > s for all s 2 S. In fact, we may assume that k( s ) > s for all s 2 S, since if k( s dier by a rational amount, and hence is nonrandom. The set S is co-c.e. by denition, but it has an additional useful property. Let If s 2 S[t 1] S[t] then no u 2 (s; t) is in S, since for any such u we have In other words, if s leaves S at stage t then so do all numbers in (s; t). To construct B, we run the following procedure P i for all in parallel. Note that B is a sequence rather than a set, so we are allowed to add more than one copy of a given interval to B. 1. Look for an s 2 ! such that s 2 I i . 2. Let I i then terminate the procedure. 3. If and go to step 2. Otherwise, add the interval to B, increase t by one, and repeat step 3. This concludes the construction of B. We now show that the sum of the lengths of the intervals in B is nite and that is in innitely many of the intervals in B. For each i 2 !, let B i be the set of intervals added to B by P i and let l i be the sum of the lengths of the intervals in B i . If P i never leaves step 1 then eventually terminates then l i 6 k( t s ) for some s; t 2 ! such that s ; t 2 I i , and hence reaches step 3 and never terminates then 2 I i and l i 6 k( s ) for some s 2 ! such that s 2 I i , and hence again l i 6 k jI i j. Thus the sum of the lengths of the intervals in B is less than or equal to k To show that is in innitely many of the intervals in B, it is enough to show that, for each i 2 !, if 2 I i then is in one of the intervals in B i . Fix such that 2 I i . Since is not rational, u 2 I i for all su-ciently large must eventually reach step 3. By the properties of S discussed above, the variable s in the procedure P i eventually assumes a value in S. For this value, , from which it follows that k( u s ) > s for some u > s, and hence that must be in one of the intervals (), all of which are in B i . 3.6. Corollary. If 0 and 1 are c.e. reals such that 0 random then at least one of 0 and 1 is random. Proof. Let . For each s 2 !, either 3( 0 0 so for some i < 2 there are innitely many s 2 ! such that 3( i i Theorem 3.5, i is random. Combining Theorem 3.1 and Corollary 3.6, we have the following results, the second of which also depends on Theorem 1.4. 3.7. Theorem. A c.e. real is random if and only if it cannot be written as + for c.e. reals ; < S 3.8. Theorem. Let d be a Solovay degree. The following are equivalent: 1. d is incomplete. 2. d splits. 3. d splits over any lesser Solovay degree. --R Weakly computable real numbers A characterization of c. Information and Randomness Nature 400 Centre for Discrete Mathematics and Theoretical Computer Science Research Report Series 59 Algorithmic information theory Incompleteness theorems for random reals Randomness and reducibility I Presentations of computably enumerable reals On the de On the continued fraction representation of computable real numbers Three approaches to the quantitative de On the notion of a random sequence The various measures of the complexity of An Introduction to Kolmogorov Complexity and its Appli- cations Process complexity and e Randomness and recursive enumerability Cohesive sets and recursively enumerable Dedekind cuts Draft of a paper (or series of papers) on Chaitin's work --TR --CTR Rod Downey , Denis R. Hirschfeldt , Geoff LaForte, Undecidability of the structure of the Solovay degrees of c.e. reals, Journal of Computer and System Sciences, v.73 n.5, p.769-787, August, 2007

kolmogorov complexity;algorithmic information theory;randomness;solovay reducibility;computably enumerable reals

586923

A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs.

In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusion-free family of intervals. This problem has important applications in physical mapping of DNA.We give a near-optimal fully dynamic algorithm for this problem. It operates in O(log n) worst-case time per edge insertion or deletion. We prove a close lower bound of $\Omega(\log n/(\log\log n+\log b))$ amortized time per operation in the cell probe model with word-size b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time. As a byproduct of our algorithm, we solve in O(log n) worst-case time the problem of maintaining connectivity in a dynamically changing proper interval graph.

Introduction A graph G is called an interval graph if its vertices can be assigned to intervals on the real line so that two vertices are adjacent in G iff their intervals intersect. The set of intervals assigned to the vertices of G is called a realization of G. If the set of intervals can be chosen to be inclusion-free, then G is called a proper interval graph. Proper interval graphs have been studied extensively in the literature (cf. [7, 13]), and several linear time algorithms are known for their recognition and realization [2, 3]. This paper deals with the problem of recognizing and representing dynamically changing proper interval graphs. The input is a series of operations to be performed on a graph, where an operation is any of the following: Adding a vertex (along with the edges incident to it), deleting a vertex (and the edges incident to it), adding an edge and deleting an edge. The objective is to maintain a representation of the dynamic graph as long as it is a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph. In the incremental version of the problem, only addition operations are permitted, i.e., the operations include only the addition of a vertex and the addition of an edge. In the decremental version of the problem only deletion operations are allowed. The motivation for this problem comes from its application to physical mapping of DNA [1]. Physical mapping is the process of reconstructing the relative position of DNA fragments, called clones, along the target DNA molecule, prior to their sequencing, based on information about their pairwise overlaps. In some biological frameworks the set of clones is virtually inclusion-free - for example when all clones have a similar length (this is the case for instance for cosmid clones). In this case, the physical mapping problem can be modeled using proper interval graphs as follows. A graph G is built according to the biological data. Each clone is represented by a vertex and two vertices are adjacent iff their corresponding clones overlap. The physical mapping problem then translates to the problem of finding a realization of G, or determining that none exists. Had the overlap information been accurate, the two problems would have been equivalent. However, some biological techniques may occasionally lead to an incorrect conclusion about whether two clones intersect, and additional experiments may change the status of an intersection between two clones. The resulting changes to the corresponding graph are the deletion of an edge, or the addition of an edge. The set of clones is also subject to changes, such as adding new clones or deleting 'bad' clones (such as chimerics [14]). These translate into addition or deletion of vertices in the corresponding graph. Therefore, we would like to be able to dynamically change our graph, so as to reflect the changes in the biological data, as long as they allow us to construct a map, i.e., as long as the graph remains a proper interval graph. Several authors have studied the problem of dynamically recognizing and representing certain graph families. Hsu [10] has given an O(m+ n log n)-time incremental algorithm for recognizing interval graphs. (Throughout, we denote the number of vertices in the graph by n and the number of edges in it by m.) Deng, Hell and Huang [3] have given a linear-time incremental algorithm for recognizing and representing connected proper interval graphs This algorithm requires that the graph will remain connected throughout the modifications. In both algorithms [10, 3] only vertex increments are handled. Recently, Ibarra [11] found a fully dynamic algorithm for recognizing chordal graphs, which handles each edge operation in O(n) time, or alternatively, an edge deletion in O(n log n) time and an edge insertion in O(n= log n) time. Our results are as follows: For the general problem of recognizing and representing proper interval graphs we give a fully dynamic algorithm which handles each operation in time O(d log n), where d denotes the number of edges involved in the operation. Thus, in case a vertex is added or deleted, d equals its degree, and in case an edge is added or deleted, d = 1. Our algorithm builds on the representation of proper interval graphs given in [3]. We also prove a lower bound for this problem of \Omega (log n=(log log n+ log b)) amortized time per edge operation in the cell probe model of computation with word-size b [16]. It follows that our algorithm is nearly optimal (up to a factor of O(log log n)). For the incremental and the decremental versions of the problem we give optimal algorithms (up to a constant factor) which handle each operation in time O(d). For the incremental problem this generalizes the result of [3] to arbitrary instances. As a part of our general algorithm we give a fully dynamic procedure for maintaining connectivity in proper interval graphs. The procedure receives as input a sequence of operations each of which is a vertex addition or deletion, an edge addition or deletion, or a query whether two vertices are in the same connected component. It is assumed that the graph remains proper interval throughout the modifications, since otherwise our main algorithm detects that the graph is no longer a proper interval graph and halts. We show how to implement this procedure in O(log n) time per operation. In compar- ison, the best known algorithms for maintaining connectivity in general graphs require O(log 2 n) amortized time per operation [9], or O( n) worst-case (deterministic) time per operation [4]. We also show that the lower bound of Fredman and Henzinger [5] of \Omega (log n=(log log n+log b)) amortized time per operation (in the cell probe model with word-size b) for maintaining connectivity in general graphs, applies to the problem of maintaining connectivity in proper interval graphs. The paper is organized as follows: In section 2 we give the basic background and describe our representation of proper interval graphs and the realization it defines. In sections 3 and 4 we present the incremental algorithm. In section 5 we extend the incremental algorithm to a fully dynamic algorithm for proper interval graph recognition and representation. We also derive an optimal decremental algorithm. In section 6 we give a fully dynamic algorithm for maintaining connectivity in proper interval graphs. Finally, in section 7 we prove a lower bound on the amortized time per operation of a fully dynamic algorithm for recognizing proper interval graphs. For lack of space, some of the proofs and some of the algorithmic details are omitted. Preliminaries E) be a graph. We denote its set V of vertices also by V (G) and its set E of edges also by E(G). For a vertex and N [v] := N (v) [ fvg. Let R be an equivalence relation on V defined by uRv iff Each equivalence class of R is called a block of G. Note that every block of G is a complete subgraph of G. The size of a block is the number of vertices in it. Two blocks A and B are neighbors in G if some (and hence all) vertices a 2 A; b 2 B, are adjacent in G. A straight enumeration of G is a linear ordering \Phi of the blocks in G, such that for every block, the block and its neighboring blocks are consecutive in \Phi. l be an ordering of the blocks of G. For any 1 - we say that is ordered to the left of B j , and that B j is ordered to the right of B i . A chordless cycle is an induced cycle of length greater than 3. A claw is an induced K 1;3 . A graph is claw-free if it does not contain an induced claw. For basic definitions in graph theory see, e.g., [7]. The following are some useful facts about interval and proper interval graphs. Theorem 1. ([12]) An interval graph contains no chordless cycle. Theorem 2. ([15]) A graph is a proper interval graph iff it is interval and claw-free. Theorem 3. ([3]) A graph is a proper interval graph iff it has a straight enumeration. Lemma 1 ("The umbrella property"). Let \Phi be a straight enumeration of a connected proper interval graph G. If A; B and C are blocks of G, such that A and A is adjacent to C, then B is adjacent to A and to C (see figure 1). Fig. 1. The umbrella property Let G be a connected proper interval graph and let \Phi be a straight enumeration of G. It is shown in [3] that a connected proper interval graph has a unique straight enumeration up to its full reversal. Define the out-degree of a block B w.r.t. \Phi, denoted by o(B), as the number of neighbors of B which are ordered to its right in \Phi. We shall use the following representation: For each connected component of the dynamic graph we maintain a straight enumeration (in fact, for technical reasons we shall maintain both the enumeration and its reversal). The details of the data structure containing this information will be described below. This information implicitly defines a realization of the dynamic graph (cf. [3]) as follows: Assign to each vertex in block B i the interval ]. The out-degrees and hence the realization of the graph can be computed from our data structure in time O(n). 3 An Incremental Algorithm for Vertex Addition In the following two sections we describe an optimal incremental algorithm for recognizing and representing proper interval graphs. The algorithm receives as input a series of addition operations to be performed on a graph. Upon each operation the algorithm updates its representation of the graph and halts if the current graph is no longer a proper interval graph. The algorithm handles each operation in time O(d), where d denotes the number of edges involved in the operation. It is assumed that initially the graph is empty, or alternatively, that the representation of the initial graph is known. A contig of a connected proper interval graph G is a straight enumeration of G. The first and the last blocks of a contig are called end-blocks. The rest of the blocks are called inner-blocks. As mentioned above, each component of the dynamic graph has exactly two contigs (which are full reversals of each other) and both are maintained by the algorithm. Each operation involves updating the representation. (In the sequel we concentrate on describing only one of the two contigs for each component. The second contig is updated in a similar way.) 3.1 The Data Structure The following data is kept and updated by the algorithm: 1. For each vertex we keep the name of the block to which it belongs. 2. For each block we keep the following: (a) An end pointer which is null if the block is not an end-block of its contig, and otherwise points to the other end-block of that contig. (b) The size of the block. (c) Left and right near pointers, pointing to nearest neighbor blocks on the left and on the right respectively. (d) Left and right far pointers, pointing to farthest neighbor blocks on the left and on the right respectively. (e) Left and right self pointers, pointing to the block. (f) A counter. In the following we shall omit details about the obvious updates to the name of the block of a vertex and to the size of a block. During the execution of the algorithm we may need to update many far pointers pointing to a certain block, so that they point to another block. In order to be able to do that in O(1) time we use the technique of nested pointers: We make the far pointers point to a location whose content is the address of the block to which the far pointers should point. The role of this special location will be served by our self-pointers. The value of the left and right self-pointers of B is always the address of B. When we say that a certain left (right) far pointer points to B, we mean that it points to a left (right) self-pointer of B. Let A and B be blocks. In order to change all left (right) far pointers pointing to A so that they point to B, we require that no left (right) far pointer points to B. If this is the case, we simply exchange the left (right) self-pointer of A with the left (right) self-pointer of B. This means that: (1) The previous left (right) self-pointer of A is made to point to B, and the algorithm records it as the new left (right) self- pointer of B; (2) The previous left (right) self-pointer of B is made to point to A, and the algorithm records it as the new left (right) self-pointer of A. We shall use the following notation: For a block B we denote its address in the memory by &B. When we set a far pointer to point to a left or to a right self-pointer of B we will abbreviate and set it to &B. We denote the left and right near pointers of B by l (B) and N r (B) respectively. We denote the left and right far pointers of B by F l (B) and F r (B) respectively. We denote its end pointer by E(B). In the sequel we often refer to blocks by their addresses. For example, if A and B are blocks, and N r sometimes refer to B by N r (A). When it is clear from the context, we also use a name of a block to denote any vertex in that block. Given a contig \Phi we denote its reversal by \Phi R . In general when performing an operation, we denote the graph before the operation is carried out by G, and the graph after the operation is carried out by G 0 . 3.2 The Impact of a New Vertex In the following we describe the changes made to the representation of the graph in case G 0 is formed from G by the addition of a new vertex v of degree d. We also give some necessary and some sufficient conditions for deciding whether G 0 is proper interval. Let B be a block of G. We say that v is adjacent to B if v is adjacent to some vertex in B. We say that v is fully adjacent to B if v is adjacent to every vertex in B. We say that v is partially adjacent to B if v is adjacent to B but not fully adjacent to B. The following lemmas characterize, assuming that G 0 is proper interval, the adjacencies of the new vertex. Lemma 2. If G 0 is a proper interval graph then v can have neighbors in at most two connected components of G. Lemma 3. [3] Let C be a connected component of G containing neighbors of v. Let be a contig of C . Assume that G 0 is proper interval and let 1 - a ! k. Then the following properties are satisfied: 1. If v is adjacent to B a and to B c , then v is fully adjacent to B b . 2. If v is adjacent to B b and not fully adjacent to B a and to B c , then B a is not adjacent to B c . 3. If is adjacent to B b , then v is fully adjacent to B a or to One can view a contig \Phi of a connected proper interval graph C as a weak linear on the vertices of C, where x ! \Phi y iff the block containing x is ordered in \Phi to the left of the block containing y. We say that \Phi 0 is a refinement of \Phi if for every a contig can be reversed, we also allow complete reversal of \Phi). Lemma 4. If G is a connected induced subgraph of a proper interval graph G 0 , \Phi is a contig of G and \Phi 0 is a straight enumeration of G 0 , then \Phi 0 is a refinement of \Phi. Note, that whenever v is partially adjacent to a block B in G, then the addition of v will cause B to split into two blocks of G 0 , namely B nN (v) and B " N (v). Otherwise, if B is a block of G to which v is either fully adjacent or not adjacent, then B is also a block of G 0 . Corollary 1. If B is a block of G to which v is partially adjacent, then B n N (v) and occur consecutively in a straight enumeration of G 0 . Lemma 5. Let C be a connected component of G containing neighbors of v. Let the set of blocks in C which are adjacent to v be fB g. Assume that in a contig of . If G 0 is proper interval then the following properties are satisfied: are consecutive in C. 2. If k - 3 then v is fully adjacent to B 3. If v is adjacent to a single block B 1 in C, then B 1 is an end-block. 4. If v is adjacent to more than one block in C and has neighbors in another compo- nent, then B 1 is adjacent to B k , and one of B 1 or B k is an end-block to which v is fully adjacent, while the other is an inner-block. Proof. Claims 1 and 2 follow directly from part 1 of Lemma 3. Claim 3 follows from part 3 of Lemma 3. To prove the last part of the lemma let us denote the other component containing neighbors of v by D. Examine the induced connected subgraph H of G whose set of vertices is V (D). H is proper interval as an induced subgraph of G. It is composed of three types of blocks: Blocks whose vertices are from V (C), which we will call henceforth C-blocks; blocks whose vertices are from V (D), which we will call henceforth D-blocks; and fvg which is a block of H since fvg is not connected. All blocks of C remain intact in H, except B 1 and B k which might split into B j n N (v) and Surely in a contig of H, C-blocks must be ordered completely before or completely after D-blocks. Let \Phi denote a contig of H , in which C-blocks are ordered before D- blocks. Let X denote the rightmost C-block in \Phi. By the umbrella property, and moreover, X is adjacent to v. By Lemma 4, \Phi is a refinement of a contig of C. Hence, precisely, Therefore, one of B 1 or B k is an end-block. W.l.o.g. . Suppose to the contrary that v is not fully adjacent to B k . Then by Lemma 4 we have contradicting the umbrella property. B 1 must be adjacent to B k , or else G 0 contains a claw consisting of and a vertex from V (D)"N (v). It remains to show that B 1 is an inner-block. Suppose it is an end block. Since B 1 and B k are adjacent, C contains a single block contradiction. Thus, claim 4 is proved. 3.3 The Algorithm In our algorithm we rely on the incremental algorithm of Deng, Hell and Huang [3], which we call henceforth the DHH algorithm. This algorithm handles the insertion of a new vertex into a graph in O(d) time, provided that all its neighbors are in the same connected component, changing the straight enumeration of this component appropriately. We refer the reader to [3] for more details. We perform the following upon a request for adding a new vertex v. For each neighbor u of v we add one to the count of the block containing u. We call a block full if its counter equals its size, empty if its counter equals zero, and partial otherwise. In order to find a set of consecutive blocks which contain neighbors of v, we pick arbitrarily a neighbor of v and march down the enumeration of blocks to the left using the left near neighbor pointers. We continue till we hit an empty block or till we reach the end of the contig. We do the same to the right and this way we discover a maximal sequence of nonempty blocks in that component which contain neighbors of v. We call this maximal sequence a segment. Only the two extreme blocks of the segment are allowed to be partial or else we fail (by Lemma 5(2)). If the segment we found contains all neighbors of v then we can use the DHH algorithm in order to insert v into G, updating our internal data structure accordingly. Otherwise, by Lemmas 2 and 5(1) there could be only one more segment which contains neighbors of v. In that case, exactly one extreme block in each segment is an end-block to which v is fully adjacent (if the segment contains more than one block), and the two extreme blocks in each segment are adjacent, or else we fail (by Lemma 5(3,4)). We proceed as above to find a second segment containing neighbors of v. We can make sure that the two segments are from two different contigs by checking that their end-blocks do not point to each other. We also check that conditions 3 and 4 in Lemma 5 are satisfied. If the two segments do not cover all neighbors of v, we fail. If v is adjacent to vertices in two distinct components C and D, then we should merge their contigs. Let R be the two contigs of C. Let l ; \Psi R be the two contigs of D. The way the merge is performed depends on the blocks to which v is adjacent. If v is adjacent to B k and to B 0 by the umbrella property the two new contigs (up to refinements described below) are . In the following we describe the necessary changes to our data structure in case these are the new contigs. The three other cases are handled similarly. - Block enumeration: We merge the two enumerations of blocks and put a new block fvg in-between the two contigs. Let the leftmost block adjacent to v in the new ordering and let the rightmost block adjacent to v be B 0 . If is partial we split it into two blocks - in this order. If B 0 j is partial we split it into two blocks (v) in this order. - End pointers: We set E(B 1 l We then nullify the end pointers of B k and B 0 1 . Near pointers: We update N l and N l (B 0 In case B i was split we update N r ( - are made in case B 0 was split to the near pointers of B 0 j+1 . - Far pointers: If B i was split we set F l ( - the left self-pointer of B i with the left self-pointer of - was split we set F r ( - 1 and exchange the right self-pointer of j with the right self-pointer of - j . In addition, we set all right far pointers of and all left far pointers of B 0 j to &fvg (in O(d) time). Finally, we set F l . 4 An Incremental Algorithm for Edge Addition In this section we show how to handle the addition of a new edge (u; v) in O(1) time. We characterize the cases for which G proper interval and show how to efficiently detect them, and how to update our representation of the graph. Lemma 6. If u and v are in distinct components in G, then G 0 is proper interval iff u and v were in end-blocks of their respective contigs. Proof. To prove the 'only if' part let us examine the graph fug. H is proper interval as an induced subgraph of G. If G 0 is proper interval, then by Lemma must be in an end-block of its contig, since u is not adjacent to any other vertex in the component containing v. The same argument applies to u. To prove the 'if' part we give a straight enumeration of the new connected component containing u and v in G 0 . Denote by C and D the components containing u and v respectively. Let be a contig of C , such that l be a contig of D, such that l is a straight enumeration of the new component. We can check in O(1) time if u and v are in end-blocks of distinct contigs. If this is the case, we update our data structure according to the straight enumeration given in the proof of Lemma 6 in O(1) time. It remains to handle the case where u and v were in the same connected component C in G. If N then by the umbrella property it follows that C contains only three blocks which are merged into a single block in G 0 . In this case G 0 is proper interval and updates to the internal data structure are trivial. The following lemma analyses the case where N (u) 6= N (v). Lemma 7. Let be a contig of C , such that k. Assume that N (u) 6= N (v). Then G 0 is proper interval iff G. Proof. To prove the 'only if' part assume that G 0 is proper interval. Since B i and B j are not adjacent, F r (B i . Suppose to the contrary that . If in addition F l (B is a strict containment). As v and z are in distinct blocks, there exists a vertex b 2 N [v]n N [z]. But then, v; b; z; u induce a claw in G 0 , a contradiction. Hence, F l (B and so F r (B are in distinct blocks, either (u; y) 62 E(G) or there is a vertex a 2 N [u] n N [x] (or both). In the first case, v; u; x; y and the vertices of the shortest path from y to v induce a chordless cycle in G 0 . In the second case u; a; x; v induce a claw in G 0 . Hence, in both cases we arrive at a contradiction. The proof that F l (B To prove the 'if' part we shall provide a straight enumeration of C [ fu; vg. If we move v from B j to contained only v, F l (B move u from B i to B i+1 . If u was not moved and B i oe fug, we split B i into B i n fug; fug in this order. If v was not moved and B j oe fvg, we split B j into fvg; B j n fvg in this order. It is easy to see that the result is a straight enumeration of C [ fu; vg. We can check in O(1) time if the condition in Lemma 7 holds. If this is the case, we change our data structure so as to reflect the new straight enumeration given in the proof of Lemma 7. This can be done in O(1) time, in a similar fashion to the update technique described in Section 3.3. The details are omitted here. The following theorem summarizes the results of Sections 3 and 4. Theorem 4. The incremental proper interval graph representation problem is solvable in O(1) time per added edge. 5 The Fully Dynamic Algorithm In this section we give a fully dynamic algorithm for recognizing and representing proper interval graphs. The algorithm performs each operation in O(d log n) time, where d denotes the number of edges involved in the operation. It supports four types of operations: Adding a vertex, adding an edge, deleting a vertex and deleting an edge. It is based on the same ideas used in the incremental algorithm. The main difficulty in extending the incremental algorithm to handle all types of operations, is updating the end pointers of blocks when deletions are allowed. To bypass this problem we do not keep end pointers at all. Instead, we maintain the connected components of G, and use this information in our algorithm. In the next section we show how to maintain the connected components of G in O(log n) time per operation. We describe below how each operation is handled by the algorithm. 5.1 The Addition of a Vertex or an Edge These operations are handled in essentially the same way as done by the incremental algorithm. However, in order to check if the end-blocks of two segments are in distinct components, we query our data structure of connected components (in O(log n) time). Similarly, in order to check if the endpoints of an added edge are in distinct components, we check if their corresponding blocks are in distinct components (in O(log n) time). 5.2 The Deletion of a Vertex We show next how to update the contigs of G after deleting a vertex v of degree d. Note that G 0 is proper interval as an induced subgraph of G. Denote by X the block containing v. If X oe fvg, then the only change needed is to delete v. We hence concentrate on the case that fvg. We can find in O(d) time the segment of blocks which includes X and all its neighbors. Let the contig containing X be and let the blocks of the segment be We make the following updates: l, we check whether B i can be merged with B i\Gamma1 . If F l (B them by moving all vertices from B i to B i\Gamma1 (in O(d) time) and deleting B i . If l we act similarly w.r.t. B j and B j+1 . Finally, we delete B l . If are non-adjacent, then by the umbrella property they are no longer in the same connected component, and the contig should be split into two contigs, one ending at B l\Gamma1 and one beginning at B l+1 . were merged, we update N r (B updates should be made w.r.t. in case were merged. If the contig is split, we nullify N r (B l\Gamma1 ) and l (B l+1 ). Otherwise, we update N r (B were merged, we exchange the right self-pointer of B i with the right self-pointer of B i\Gamma1 . Similar changes should be made w.r.t. B j and . We also set all right far pointers previously pointing to B l , to &B all left far pointers previously pointing to B l , to &B l+1 (in O(d) time). Note that these updates take O(d) time and require no knowledge about the connected components of G. 5.3 The Deletion of an Edge Let (u; v) be an edge of G to be deleted. Let C denote the connected component of G containing u and v, and let be a contig of C. If into resulting in a straight enumeration of G 0 . Updates are trivial in this case. If N then one can show that G is a proper interval graph iff C was a clique, so again k = 1. We assume henceforth that k ? 1 and W.l.o.g. were far neighbors of each other, then we should split the contig into two contigs, one ending at B i and the other beginning at B j . Otherwise, updates to the straight enumeration are derived from the following lemma. Lemma 8. Let be a contig of C, such that k. Assume that N (u) 6= N (v). Then G 0 is proper interval iff F r (B i and F l (B G. Proof. Assume that G 0 is proper interval. We will show that F r (B i . The proof that F l (B are adjacent in G, F r (B i Suppose to the contrary that F r (B are in distinct blocks, either there is a vertex a 2 N [v] n N [x] or there is a vertex b 2 N [x] n N [v] (or both). In the first case, by the umbrella property (a; u) 2 E(G) and therefore u; x; v; a induce a chordless cycle in G 0 . In the second case, x; b; u; v induce a claw in G 0 . Hence, in both cases we arrive at a contradiction. To prove the opposite direction we give a straight enumeration of C n f(u; v)g. If we move u into B i\Gamma1 . If B i contained only u, F r (B j+1 . If u was not moved and fug in this order. If v was not moved and fvg in this order. The result is a contig of v)g. If the conditions of Lemma 8 are fulfilled, one has to update the data structure according to its proof. These updates require no knowledge about the connected components of G, and it can be shown that they take O(1) time. Hence, from Sections 5.2 and 5.3 we obtain the following result: Theorem 5. The decremental proper interval graph representation problem is solvable in O(1) time per removed edge. 6 Maintaining the Connected Components In this section we describe a fully dynamic algorithm for maintaining connectivity in a proper interval graph G in O(logn) time per operation. The algorithm receives as input a series of operations to be performed on a graph, which can be any of the follow- ing: Adding a vertex, adding an edge, deleting a vertex, deleting an edge or querying if two blocks are in the same connected component. The algorithm depends on a data structure which includes the blocks and the contigs of the graph. It hence interacts with the proper interval graph representation algorithm. In response to an update request, changes are made to the representation of the graph based on the structure of its connected components prior to the update. Only then are the connected components of the graph updated. Let us denote by B(G) the block graph of G, that is, a graph in which each vertex corresponds to a block of G and two vertices are adjacent iff their corresponding blocks are adjacent in G. The algorithm maintains a spanning forest F of B(G). In order to decide if two blocks are in the same connected component, the algorithm checks if they belong to the same tree in F . The key idea is to design F so that it can be efficiently updated upon a modification in G. We define the edges of F as follows: For every two vertices u and v in B(G), corresponding blocks are consecutive in a contig of G. Conse- quently, each tree in F is a path representing a contig. The crucial observation about F is that an addition or a deletion of a vertex or an edge in G induces O(1) modifications to the vertices and edges of F . This can be seen by noting that each modification of G induces O(1) updates to near pointers in our representation of G. It remains to show how to implement a spanning forest in which trees may be cut when an edge is deleted from F , linked when an edge is inserted to F , and which allows to query for each vertex to which tree does it belong. All these operations are supported by the ET-tree data structure of [8] in O(log n) time per operation. We are now ready to state our main result: Theorem 6. The fully dynamic proper interval graph representation problem is solvable in O(d log n) time per modification involving d edges. 7 The Lower Bound In this section we prove a lower bound of\Omega (log n=(log log n log b)) amortized time per edge operation for fully dynamic proper interval graph recognition in the cell probe model of computation with word-size b [16]. Fredman and Saks [6] proved a lower bound amortized time per operation for the following parity prefix sum (PPS) problem: Given an array of integers execute an arbitrary sequence of Add(t) and Sum(t) operations, where an Add(t) increases A[t] by 1, and Sum(t) returns ( 2. Fredman and Henzinger [5] showed that the same lower bound applies to the problem of maintaining connectivity in general graphs, by showing a reduction from a modified PPS problem, called helpful parity prefix sum, for which they proved the same lower bound. A slight change to their reduction yields the same lower bound for the problem of maintaining connectivity in proper interval graphs, as the graph built in the reduction is a union of two paths and therefore proper interval. Using a similar construction we can prove the following result: Theorem 7. Fully dynamic proper interval recognition takes\Omega (log n=(log log n log b)) amortized time per edge operation in the cell probe model with word-size b. Acknowledgments The first author gratefully acknowledges support from NSERC. The second author was supported in part by a grant from the Ministry of Science, Israel. The third author was supported by Eshkol scholarship from the Ministry of Science, Israel. --R Establishing the order of human chromosome-specific DNA fragments Simple linear time recognition of unit interval graphs. Recognition and representation of proper circular arc graphs. SIAM Journal on Computing Lower bounds for fully dynamic connectivity problems in graphs. The cell probe complexity of dynamic data structures. Algorithmic Graph Theory and Perfect Graphs. Randomized dynamic graph algorithms with polylogarithmic time per operation. A simple test for interval graphs. Fully dynamic algorithms for chordal graphs. Representation of a finite graph by a set of intervals on the real line. Indifference graphs. Recombinant DNA. Eigenschaften der Nerven homologisch einfacher Familien in R n Should tables be sorted. --TR --CTR Ron Shamir , Roded Sharan, A fully dynamic algorithm for modular decomposition and recognition of cographs, Discrete Applied Mathematics, v.136 n.2-3, p.329-340, 15 February 2004 C. Crespelle , C. Paul, Fully dynamic recognition algorithm and certificate for directed cograph, Discrete Applied Mathematics, v.154 n.12, p.1722-1741, 15 July 2006 Derek G. Corneil, A simple 3-sweep LBFS algorithm for the recognition of unit interval graphs, Discrete Applied Mathematics, v.138 n.3, p.371-379, 15 April 2004 Jrgen Bang-Jensen , Jing Huang , Louis Ibarra, Recognizing and representing proper interval graphs in parallel using merging and sorting, Discrete Applied Mathematics, v.155 n.4, p.442-456, February, 2007

proper interval graphs;lower bounds;graph algorithms;fully dynamic algorithms

586924

A Polynomial Time Approximation Scheme for General Multiprocessor Job Scheduling.

Recently, there have been considerable interests in the multiprocessor job scheduling problem, in which a job can be processed in parallel on one of several alternative subsets of processors. In this paper, a polynomial time approximation scheme is presented for the problem in which the number of processors in the system is a fixed constant. This result is the best possible because of the strong NP-hardness of the problem and is a significant improvement over the past results: the best previous result was an approximation algorithm of ratio $7/6 + \epsilon$ for 3-processor systems based on Goemans's algorithm for a restricted version of the problem.

Introduction . One of the assumption made in classical scheduling theory is that a job is always executed by one processor at a time. With the advances in parallel algorithms, this assumption may no longer be valid for job systems. For example, in semiconductor circuit design workforce planning, a design project is to be processed by a group of people. The project contains n jobs, and each job can be worked on by one of a set of alternatives, where each alternative consists of one or more persons in the group working simultaneously on the particular job. The processing time of each job depends on the subgroup of people being assigned to handle the job. Note that the same person may belong to several different subgroups. Now the question is how we can schedule the jobs so that the project can be finished as early as possible. Other applications include (i) the berth allocation problem [21] where a large vessel may occupy several berths for loading and unloading, (ii) diagnosable microprocessor systems [20] where a job must be performed on parallel processors in order to detect faults, (iii) manufacturing, where a job may need machines, tools, and people simultaneously, and (iv) scheduling a sequence of meetings where each meeting requires a certain group of people [11]. In the scheduling literature [17], this kind of problems are called multiprocessor job scheduling problems. Among the others, two types of multiprocessor job scheduling problems have been extensively studied [7, 22]. The first type is the Pm jfixjC max problem, in which the subset of processors and the processing time for parallel processing each job are fixed. The second type is a more general version, the Pm jsetjC max problem, in which each job may have a number of alternative processing modes and each processing mode specifies a subset of processors and the job processing time on that particular processor subset. The objective for both problems is to construct a scheduling of minimum makespan on the m-processor system for a given list of jobs. The jobs are supposed to be non-preemptive. Approximability of the multiprocessor job scheduling problems has been studied. The problem is a generalized version of the classical job scheduling prob- Department of Computer Science, Texas A&M University, College Station, Email: chen@cs.tamu.edu. Supported in part by the National Science Foundation under Grant CCR-9613805. y Department of Computer Science, Bucknell University, Lewisburg, Pennsylvania 17837, Email: amiranda@eg.bucknell.edu. J. CHEN and A. MIRANDA lem on a 2-processor system [13], thus it is NP-hard. Hoogeveen et al. [18] showed that the P 3 jfixjC max problem (thus also the P 3 jsetjC max problem) is NP-hard in the strong sense thus it does not have a fully polynomial time approximation scheme 5]). Blazewicz et al. [4] developed a polynomial time approximation algorithm of ratio 4=3 for the problem P 3 jfixjC max , which was improved later by Dell'Olmo et al. [10], who gave a polynomial time approximation algorithm of ratio 5=4 for the same problem. Both algorithms are based on the study of a special type of schedulings called normal schedulings. Goemans [14] further improved the algorithms by giving a polynomial time approximation algorithm of ratio 7=6 for the recently, Amoura et al. [1] developed a polynomial time approximation scheme for the problem Pm jfixjC max for every fixed integer m. Approximation algorithms for the Pm jsetjC max problem were not as successful as that for the Pm jfixjC max problem. Bianco et al. [3] presented a polynomial time approximation algorithm for the Pm jsetjC max problem whose approximation ratio is bounded by m. Chen and Lee [8] improved their algorithm by giving a polynomial time approximation algorithm for the Pm jsetjC max problem with an approximation showed that the problem P 3 jsetjC max can be approximated in polynomial time with a ratio 7=6 ffl. Before the present paper, it was unknown whether there is a polynomial time approximation algorithm with ratio c for the problem Pm jsetjC max , where c is a constant independent of the number m of processors in the system. In this paper, we present a polynomial time approximation scheme for the problem Pm jsetjC max . Our algorithm combines the techniques developed by Amoura et al. [1], who split jobs into large jobs and small jobs, and the techniques developed by Bell'Olmo et al. [10] and Goemans [14] on normal schedulings, plus the standard dynamic programming and scaling techniques. More precisely, based on a classification of large jobs and small jobs, we introduce the concept of (m; ffl)-canonical schedulings, which can be regarded as a generalization of the normal schedulings. We show that for any job list, there is an (m; ffl)-canonical scheduling whose makespan is very close to the optimal makespan. Then we show how this (m; ffl)-canonical scheduling can be approximated. Combining these two steps gives us a polynomial time approximation scheme for the Pm jsetjC max problem. Our result is the best possible in the following sense: because the problem Pm jsetjC max is NP-hard in the strong sense, it is unlikely that our algorithm can be further improved to a fully polynomial time approximation scheme [13]. More- over, the polynomial time approximation scheme cannot be extended to the more general problem P jsetjC max , in which the number m of processors in the system is given as a parameter in the input: it can be shown that there is a constant such that the problem P jsetjC max has no polynomial time approximation algorithms whose approximation ratio is bounded by n ffi [23]. The paper is organized as follows. Section 2 gives necessary background and preliminaries for the problem. In section 3 we introduce (m; ffl)-canonical schedulings and study their properties. Section 4 presents the polynomial time approximation scheme for the problem Pm jsetjC max , and section 5 concludes with some remarks and further research directions. 2. Preliminaries. The Pm jsetjC max problem is a scheduling problem minimizing the makespan for a set of jobs, each of which may have several alternative processing modes. More formally, an instance J of the problem Pm jsetjC max is a list of jobs: each job J i is associated with a list of alternative processing modes: Each processing mode (or simply mode) M ij is specified by a a subset of processors in the m-processor system and ij is an integer indicating the parallel processing time of the job J i on the processor In case there is no ambiguity, we also say that the processor set Q ij is a mode for the job J i . For each job J min i be the . The value min i will be called the minimum parallel processing time for the job J i . Given a list of jobs, a scheduling \Gamma(J ) of J on the m-processor system consists of two parts: (1) determination of a processing mode for each job J i in J ; and (2) determination of the starting execution time for each job under the assigned mode so that at any moment, each processor in the system is used for (maybe parallel) processing at most one job (assuming that the system starts at time The makespan of the scheduling \Gamma(J ) is the latest finishing time of a job in J under the scheduling \Gamma(J ). Let Opt(J ) denote the minimum makespan over all schedulings for J . The Pm jsetjC max problem is for a given instance J to construct a scheduling of makespan Opt(J ) for J . Let Pm be the set of the m processors in the m-processor system. A collection k g of k nonempty subsets of Pm is a k-partition of Pm if collection of subsets of Pm is a partition of Pm if it is a k-partition for some integer k 1. The total number Bm of different partitions of the set Pm is called the mth Bell number [16]. Using the formula of Comtet [9], we have A looser but simpler upper bound for Bm , Bm m!, can be easily proved by induction. Another combinatorial fact we need for analysis of our scheduling algorithm is the "cut-index" in a nonincreasing sequence of integers. Lemma 2.1. Let be a nonincreasing sequence of integers, let be a fixed integer and ffl ? 0 be an arbitrary real number. Then there is an (with respect to m and ffl) such that is an integer; and (2) for any subset T 0 of at most 3j 0 mBm integers t q in T with q ? j 0 , we have Proof. To simplify expressions, let b 1. Decompose the sum 4 J. CHEN and A. MIRANDA Since there are at most bm=fflc subsums A j larger than be the first subsum such that A k+1 (ffl=m) . Since the sum of the first b k+1 integers t q in T with q ? m is bounded by (ffl=m) (ffl=m) and the sequence is nonincreasing, we conclude that for any subset T 0 of T of at most 3j 0 mBm integers t q with q ? j 0 , we must have This completes the proof. For the nonincreasing sequence T of integers, we will denote by j m;ffl the smallest index that satisfies conditions (1) and (2) in Lemma 2.1. The index j m;ffl will be called the cut-index for the sequence T . 3. On (m; ffl)-canonical schedulings. In this section, we first assume that the mode assignment for each job in the instance J is decided, and discuss how we schedule the jobs in J under the mode assignment to the processor set Pm . By this assumption, the job list J is actually an instance for the Pm jfixjC max problem (recall that the Pm jfixjC max problem is the problem Pm jsetjC max with the restriction that every job in an instance has only one processing mode). be an instance for the Pm jfixjC max problem, where each job J i requires a fixed set Q i of processors for parallel execution with processing time loss of generality, assume that the processing time sequence is nonincreasing. For the fixed number m of processors in the system, and for an arbitrarily given real number ffl ? 0, let j m;ffl be the cut-index for the sequence T , as defined in Lemma 2.1. That is, is an integer bounded by bm=fflc, and for any subset T 0 of at most 3j m;ffl mBm integers t q in T with q ? j m;ffl , we have We split the job set J into two subsets: (1) The jobs in JL will be called large jobs and the jobs in JS will be called small jobs. Let \Gamma(J ) be a scheduling for the job set J . Consider the nondecreasing sequence of integers, where h, are the starting or finishing times of the j m;ffl large jobs in \Gamma(J ). A small job block in \Gamma(J ) consists of a subset P 0 ' Pm of processors and a time interval [ such that the subset of processors are exactly those that are executing large jobs in the time interval [ will be called the height and the processor set P 0 will be called the type of the small job block . Therefore, the subset P 0 of processors associated with the small job block are those processors that are either idle or used for executing small jobs in the time interval Note that the small job block can be of height 0 when The small job block of time interval [ is the latest finish time of a large job, will be called the "last small job block". Note that the last small job block has type Pm . Let be a small job block associated with a processor set P 0 and a time interval The small job block at any time moment in the time interval [ can be characterized uniquely as a collection [Q of pairwise disjoint subsets of the processor set P 0 such that at the time , for processors in the subset Q i are used for parallel execution on the same small job (thus, the subset is the subset of idle processors at time ). The collection [Q will be called the type of the time moment . A layer in the small job block is a such that all time moments between i and j are of the same type. The type of the layer is equal to the type of any time moment in the layer and the height of the layer is Let L 1 and L 2 be two layers in the small job block of types [Q respectively. We say that layer L 1 covers layer L 2 if fR g. In particular, if L 1 and L 2 are two consecutive layers in the small job block such that layer L 2 starts right after layer L 1 finishes and L 1 covers L 2 , then layer L 2 is actually a continuation of the layer L 1 with some of the small jobs finished. Definition 3.1. A floor oe in the small job block is a sequence fL of consecutive layers such that (1) for h, layer L i starts right after layer and (2) all small jobs interlacing layer in layer L 1 and all small jobs interlacing layer L h finish in layer L h . An example of a floor is given in Figure 1(a). Note that a small job block may not have any nonempty floor at all, as shown in Figure 1(b). Remark 1. There are a few important properties of floors in a small job block. Suppose that the layer L 1 starts at time 1 while layer L h finishes at time 2 . Then by property (2) in the definition, no small jobs cross the floor boundaries 1 and 2 . Therefore, the floor oe can be regarded as a single job that uses the processor set P 0 , starts at time 1 and finishes at time 2 . The height of the floor oe is defined to be which is equal to the sum of the heights of the layers L 1 Secondly, since all floors in the small job block are for the same processor subset P 0 and there are no small jobs crossing the starting and finishing times of any floors, the floors in the same small job block can be rearranged in any order but can still fit into the small job block without exceeding the height of the small job block. Finally, property (1) in the definition ensures that no matter how the small jobs in a floor are rearranged, a simple greedy algorithm is sufficient to refit the small jobs into the floor without exceeding the floor height. The greedy algorithm is based on the idea of the well-known Graham's scheduling algorithm for the classical job scheduling problem [15]. Definition 3.2. Let J be an instance of the problem Pm jfixjC max and let be any permutation of the jobs in J . The list scheduling algorithm based on the ordering is to schedule each job J i of mode Q i in J , following the ordering of , at the earliest time when the processor subset Q i becomes available. Lemma 3.3. Let J oe be the set of small jobs in the floor oe. The list scheduling algorithm based on any ordering of the jobs in J oe will always reconstruct the floor oe. Proof. Suppose that the first layer L 1 in the floor oe is of type According to property (1) in the definition, every job in J oe must have a mode Q i for some i. By the definition, each layer covers the layer L j , therefore, in the floor oe, no processor subset Q i can become idle before its final completion time. Now 6 J. CHEN and A. MIRANDA (b) c small job block (a) small job block c floor s Fig. 1. (a) A floor fL1 ; L2 ; L3g; (b) a small job block with no floor. since the subsets Q are pairwise disjoint, the jobs of mode Q i in J oe can be executed by the processor subset Q i in any order without changing the completion time of Q i . Therefore, regardless of the ordering of the jobs in J oe , as long as the list scheduling algorithm starts each job at its earliest possible time (thus no subset can become idle before its final completion time), the completion time for each subset will not be changed. Therefore, the list scheduling algorithm will construct a floor with exactly the same layers L 1 Definition 3.4. Let [Q be a partition of the processor subset P 0 . We say that we can assign the type [Q to a floor if the type of the layer L 1 is a subcollection of fQ g. Note that it is possible that we can assign two different types to the same floor as long as the type of the floor is a subcollection of the assigned floor types. For example, let be a partition of the processor subset P 0 . If the first layer L 1 in a floor oe is of type [Q then we can assign either type to the floor oe. Definition 3.5. A small job block is a tower if it is constituted by a sequence of floors such that we can assign types to the floors so that no two floors in the tower are of the same type. Note that since each floor type is a partition of the processor subset P 0 , a tower contains at most Bm floors, where Bm , the mth Bell number, is the number of different partitions of a set of m elements. In our discussion, we will be concentrating on schedulings of a special form, in the following sense. Definition 3.6. Let J be an instance of the problem Pm jfixjC max , which is divided into large job set JL and small job set JS as given in Equation (1) for a fixed fixed constant ffl ? 0. A scheduling \Gamma(J ) of J is (m; ffl)-canonical if every small job block in \Gamma(J ) is a tower. Remark 2. Note that in an (m; ffl)-canonical scheduling, no small jobs cross the boundary of a tower. Therefore, a tower of height t and associated with a processor set Q can be simply regarded as a job of mode (Q; t). We first show that an (m; ffl)-canonical scheduling \Gamma(J ) of J can be constructed by the list scheduling algorithm when large jobs and towers in \Gamma(J ) are given in a proper order. Lemma 3.7. Let \Gamma(J ) be an (m; ffl)-canonical scheduling for the job set J . Let be the sequence of the large jobs and towers in \Gamma(J ), ordered in terms of their starting times in \Gamma(J ). Then the list scheduling algorithm based on the ordering , which regards each tower as a single job, constructs a scheduling of J with makespan not larger than that of \Gamma(J ). Proof. Let be any prefix of the ordered sequence , where each J j is either a large job or a tower. Let \Gamma(J i ) be the scheduling of J i obtained from \Gamma(J ) by removing all large jobs and towers that are not in J i , and let \Gamma 0 (J i ) be the scheduling by the list scheduling algorithm on the jobs in J i . It suffices to prove that for all i, the completion time of any processor in \Gamma 0 (J i ) is not larger than the completion time of the same processor in \Gamma(J i ). We prove this by induction on i. The case Now suppose that the mode for the job (or tower) requires the processor subset Q i+1 for parallel processing time t i+1 . Let be the earliest time in the scheduling \Gamma(J i ) at which the processor subset Q available and let 0 be the earliest time in the scheduling \Gamma 0 (J i ) at which the processor subset available. The list scheduling algorithm will start J i+1 at time thus in the scheduling \Gamma 0 (J i+1 ), the completion time of each processor in the subset . On the other hand, in the scheduling \Gamma(J i+1 ), the job J i+1 cannot start earlier than since according to the definition of the ordered sequence , J i+1 cannot start until all jobs in J i have started. Therefore, in the scheduling \Gamma(J i+1 ), the completion time of each processor in Q i+1 is at least which is not smaller than since the inductive hypothesis assumes that 0 . Finally, for each processor not in the subset Q i+1 , the completion time in \Gamma 0 (J i+1 ) is equal to that in \Gamma 0 (J i ), which by the induction is not larger than that in \Gamma(J i ), which is equal to the completion time of the same processor in \Gamma(J i+1 ). Thus, once the ordering of large jobs and towers is decided, it is easy to construct a scheduling that is not worse than the given (m; ffl)-canonical scheduling. In the following, we will prove that for any instance J for the problem Pm jfixjC max , there is an (m; ffl)-canonical scheduling whose makespan is very close to the optimal makespan. Theorem 3.8. Let J be an instance for the problem Pm jfixjC max . Then for any ffl ? 0, there is an (m; ffl)-canonical scheduling \Gamma(J ) of J such that the makespan of \Gamma(J ) is bounded by (1 Proof. be an optimal scheduling of makespan Opt(J ) for J . We construct an (m; ffl)-canonical scheduling for J based on the optimal scheduling 8 J. CHEN and A. MIRANDA Let JL and JS be the set of large jobs and the set of small jobs in J , respectively, according to the definition in Equation (1). Consider a small job block in the scheduling Assume that the small job block is associated with a processor set P 0 of r processors, r m, and a time interval [ be the list of all partitions of the processor set P 0 , where We divide the layers in the small job block into groups, each corresponding to a partition of P 0 , as follows. A layer of type T 0 is put in the group corresponding to a partition T j if T 0 is a subcollection of T j . Note that a layer type T 0 may be a subcollection of more than one partition of P 0 . In this case, we put the layer arbitrarily into one and only one of the groups to ensure that each layer belongs to only one group. For each partition T j of P 0 , we construct a floor frame oe j whose type is T j and height is equal to the sum of heights of all layers belonging to the group corresponding to the partition T j . Note that so far we have not actually assigned any small jobs to any floor frames oe 1 yet. Moreover, since each layer belongs to exactly one of the groups, it is easy to see that the sum of the heights of the floor frames oe 1 is equal to the sum of the heights of all layers in the small job block , which is equal to the height of the small job block . The construction for the floor frames for the last small job block in \Gamma 1 (J ) is slightly different: for which we only group layers in which not all processors are idle. Thus, the sum of the heights of all floor frames in the last small job block is equal to is the latest finish time for some large job in the scheduling After the construction of the floor frames for each small job block in the scheduling the small jobs in JS to the floor frames using the following greedy method. For each small job J that requires a parallel processing by a processor subset Q, we assign J to an arbitrary floor frame oe in a small job block as long as the floor frame oe satisfies the following conditions: (1) the type of the floor frame oe contains the subset Q; and (2) adding the job J to oe does not exceed the height of the floor frame oe (if there are more than one floor frames satisfying these conditions, arbitrarily pick one of them). Note that we assign a job to a floor frame only when the mode of the job is contained in the type of the floor frame. Therefore, this assignment will never leave a "gap" between two jobs in the same floor frame. The above assignment of small jobs in JS to floor frames stops when none of the small jobs left in JS can be assigned to any of the floor frames according to the above rules. Now each floor frame becomes a floor. For each small job block in \Gamma 1 (J ), let S be the set of floor frames in . Since the height of a resulting floor is not larger than the height of the corresponding floor frame, the sum of the heights of the floors resulting from the floor frames in S is not larger than the height of the small job block . Therefore, we can put all these floors into the small job block (in an arbitrary order) to make a tower. Doing this for all small job blocks in \Gamma 1 (J ) gives an (m; ffl)-canonical scheduling the job set JL [ J 0 S is the set of small jobs that have been assigned to the floor frames in the above procedure. The makespan of the scheduling is bounded by Opt(J ). Now the only thing left is that we still need to schedule the small jobs that have not been assigned to any floor frames. Let J 00 S be the set of small jobs that are not assigned to any floor frames by the above procedure. We want to demonstrate that there are not many jobs in the set J 00 S . By the definition, the number of small job blocks in the scheduling \Gamma 1 (J ) is 2j m;ffl +1 3j m;ffl . Since each small job block is associated with at most m processors, the number of floor frames constructed in each small job block is bounded by Bm . Therefore, the total number of floor frames we constructed from the scheduling is bounded by 3Bm j m;ffl . Moreover, each floor type is a collection of at most m processor subsets. If the set J 00 contains more than 3mBm j m;ffl small jobs, then there must be a subset Q of processors such that the number of small jobs of mode Q in J 00 S is larger than the number of the constructed floor frames whose type contains the subset Q. be the set of floor frames whose type contains the subset Q. By our assignment rules, assigning any job of mode Q in J 00 S to a floor frame in would exceed the height of the corresponding floor frame. Since there are more than d small jobs of mode Q in J 00 S , the sum of processing times of all small jobs of mode Q in JS is larger than On the other hand, by our construction of the floor frames in each small job block , the sum of the heights of the floor frames in whose type contains Q should not be smaller than the sum of the heights of the layers in whose type contains Q. Summarizing this over all small job blocks, we conclude that the sum smaller than the sum of processing times of all small jobs of mode Q in JS (since each small job of mode Q must be contained in consecutive layers whose type contains Q). This derives a contradiction. The contradiction shows that there are at most 3mBm j m;ffl small jobs in the set J 00 S . Now we assign the small jobs in J 00 S to the floor frames in the last small job block in the scheduling \Gamma 2 (JL [J 0 S ). For each small job J of mode Q in J 00 S , we arbitrarily assign J to a floor frame whose type contains Q in the last small job block, even this assignment exceeds the height of the floor frame. Note that the last small job block is associated with the whole processor set Pm , so for any mode Q, there must be a floor frame in the last small job block whose type contains the processor subset Q. This procedure stops with all small jobs in J 00 S assigned to floor frames in the last small job block. It is easy to see that the resulting scheduling is an (m; ffl)-canonical scheduling of the original job set J . Moreover, since the makespan of the scheduling S ) is bounded by Opt(J ), the makespan of the (m; ffl)-canonical scheduling \Gamma 3 (J ) is bounded by where t(J) is the parallel processing time of the small job J . Since there are at most small jobs in the set J 00 S , by Lemma 2.1, It is easy to see that Opt(J ) ( Therefore, the makespan of the (m; ffl)- canonical scheduling \Gamma 3 (J ) is bounded by (1 This completes the proof of the theorem. Before we close this section, we introduce one more definition. Definition 3.9. Let oe be a floor of type are pairwise disjoint subsets of processors in the processor set Pm . Then each subset plus the height l is called a room in the floor oe, whose type is Q i . J. CHEN and A. MIRANDA 4. The approximation scheme. Now we come back to the original problem Pm jsetjC max . Recall that an instance J of the problem Pm jsetjC max is a set of jobs each job J i is given by a list of alternative processing modes in which the pair (Q specifies the parallel processing time t i;j of the job J i on the subset Q i;j of processors in the m-processor system. In order to describe our polynomial time approximation scheme for the problem, let us first discuss why this problem is more difficult than the classical job scheduling problem. In the classical job scheduling problem, each job is executed by one processor in the system. Therefore, the order of executions of jobs in each processor is not crucial: the running time of the processor is simply equal to the sum of the processing times of the jobs assigned to the processor. Therefore, the decision of which job should be assigned to which processor, in any order, will uniquely determine the makespan of the resulting scheduling. This makes it possible to use a dynamic programming approach that extends a scheduling for a subset of jobs to that for a larger subset. The situation in the general multiprocessor job scheduling problem Pm jsetjC max , on the other hand, is more complicated. In particular, the makespan of a scheduling depends not only on the assignment of processing modes to jobs, but also on the order in which the jobs are executed. Therefore, the techniques of extending a scheduling for a subset of jobs in the classical job scheduling problem are not directly applicable here. Theorem 3.8 shows that there is an (m; ffl)-canonical scheduling whose makespan is very close to the optimal makespan. Therefore, constructing a scheduling whose makespan is not larger than the makespan of a good (m; ffl)-canonical scheduling will give a good approximation to the optimal schedulings. Nice properties of an (m; ffl)-canonical scheduling are that within the same tower, the order of the floors does not affect the height of the tower, and that within the same floor, the order of the small jobs does not affect the height of the floor (see Remark 1 and Remark 2 in the previous section). Therefore, the only factor that affects the heights of towers and floors are the assignments of jobs to towers and floors. This makes it become possible, at least for small jobs, to apply the techniques in classical job scheduling problem to our current problem. This is described as follows. First suppose that we can somehow divide the job set J into large job set JL and small job set JS . Let us start with an (m; ffl)-canonical scheduling \Gamma(J ) of the set J . The scheduling \Gamma(J ) gives a nondecreasing sequence f of integers, where are the starting or finishing times of the j m;ffl large jobs in JL . Let the corresponding towers be g, where the tower j consists of a subset P 0 j of processors and the We suppose that the subset P 0 j of processors associated with each tower j is known, and that the large jobs and towers of the scheduling \Gamma(J ) are ordered into a sequence in terms of their starting times. However, we assume that the assignment of small jobs to the rooms of the scheduling \Gamma(J ) is unknown. We show how this information can be recovered. For each tower j associated with the processor set P 0 , the number of floors in the tower j is q r is the number of processors in the set P 0 . Let oe j;1 be the floors of all possible different types in the tower j . For each floor oe j;q , let fl j;q;1 jq be the rooms in the floor oe j;q , where r jq m. Therefore, the configuration of the small jobs in the (m; ffl)-canonical scheduling \Gamma(J ) Algorithm. Schedule-Small Input: The set JS of small jobs and an order of the large jobs and towers in \Gamma(J ) Output: A scheduling for the job set J 1. 2. for to nS do for each mode Q ij of the small job J 0 for each True such that the job J 0 under mode Q ij is addable to the room fl j;q;r 3. for each call the list scheduling algorithm based on the order to construct a scheduling for J in which the room fl j;q;r has running time t j;q;r for all t j;q;r 0; 4. return the scheduling constructed in step 3 with the minimum makespan. Fig. 2. Scheduling small jobs in floors can be specified by a ((2j m;ffl where t j;q;r specifies the running time of the room fl j;q;r (for index fj; q; rg for which the corresponding room fl j;q;r does not exists, we can simply set t Suppose that an upper bound T 0 for the running time of rooms is derived, then we can use a Boolean array D of (2j m;ffl dimensions to describe the configuration of a subset of small jobs in a scheduling: -z is the number of small jobs in J , such that if and only if there is a scheduling on the first i small jobs to the floors in \Gamma(J ) such that the running time of the room fl j;q;r is t j;q;r (recall that the running time of a room is dependent only on the assignment of small jobs to the room and independent of the order in which the small jobs are executed in the room). Initially, all array elements in the array Suppose that a configuration of a scheduling for the first small jobs is given (2) We say that the ith small job J 0 under mode Q i is addable to a room fl j;q;r in the configuration in (2) if the room fl j;q;r is of type Q i and adding the job J 0 i to the room does not exceed the upper bound T 0 of the running time of the room fl j;q;r . Now we are ready to present our dynamic programming algorithm for scheduling small jobs into the rooms in the (m; ffl)-canonical scheduling \Gamma(J ). The algorithm is given in Figure 2. Note that the algorithm Schedule-Small may not return an (m; ffl)-canonical scheduling for the job set J . In fact, there is no guarantee that the height of the towers constructed in the algorithm does not exceed the height of the corresponding towers J. CHEN and A. MIRANDA in the original (m; ffl)-canonical scheduling \Gamma(J ). However, we can show that the scheduling constructed by the algorithm Schedule-Small has its makespan bounded by the makespan of the original (m; ffl)-canonical scheduling \Gamma(J ). Lemma 4.1. For all i, 0 i nS , the array element if and only if there is a way to assign modes to the first i small jobs and arrange them into the rooms such that the room fl j;q;r has running time t j;q;r for all fj; q; rg. Proof. We prove the lemma by induction on i. The case can be easily verified. Suppose that there is a way W to assign modes to the first i small jobs and arrange them into the rooms such that the room fl j;q;r has running time t j;q;r for all rg. Suppose that W assigns the ith small job J 0 i of processing time t ij to the room of Removing the job J 0 i from W , we obtain a way that assigns modes to the first arrange them into the rooms such that the room fl j;q;r has running time t j;q;r for all fj; q; rg 6= and the room fl j0 ;q 0 ;r 0 has running time t j0 ;q 0 ;r 0 . By the inductive hypothesis, we have Now in the ith execution of the for loop in step 2 in the algorithm Schedule-Small, when the mode of the small job J 0 i is chosen to be Q ij with processing time t ij , the algorithm will assign The other direction of the lemma can be proved similarly. We omit it here. The above lemma gives us directly the following corollary. Corollary 4.2. If the sequence of large jobs and towers is ordered in terms of their starting times in the (m; ffl)-canonical scheduling \Gamma(J ), then the algorithm Schedule-Small constructs a scheduling for job set J whose makespan is bounded by the makespan of the (m; ffl)-canonical scheduling \Gamma(J ). Proof. Note that the (m; ffl)-canonical scheduling \Gamma(J ) gives a way to assign and arrange all small jobs in JS into the rooms. According to Lemma 4.1, the corresponding array element in the array D must have value True: For this array element, step 3 of the algorithm will construct the towers that have exactly the same types and heights as their corresponding towers in the (m; ffl)-canonical scheduling \Gamma(J ) (this may not give exactly the same assignment of small jobs to rooms. However, the running times of the corresponding rooms must be exactly the same). Now since the sequence is given in the order sorted by the starting times of the large jobs and towers in the (m; ffl)-canonical scheduling \Gamma(J ), by Lemma 3.7, the call in step 3 to the list scheduling algorithm based on the order and this configuration will construct a scheduling whose makespan is not larger than the makespan of the (m; ffl)-canonical scheduling \Gamma(J ). Finally, since step 4 of the algorithm returns the scheduling of the minimum makespan constructed in step 3, we conclude that the algorithm returns a scheduling whose makespan is not larger than the makespan of \Gamma(J ). We analyze the algorithm Schedule-Small. Lemma 4.3. Let T 0 be the upper bound used by the algorithm Schedule-Small on the running time of the rooms. Then the running time of the algorithm Schedule- Small is bounded by O(n2 m m;ffl T m;ffl Proof. The number nS of small jobs in JS is bounded by the total number n of jobs in J , each small job may have at most 2 different modes. Also as we indicated before, the number of rooms is bounded by the running time for each room is bounded by T 0 , for each fixed i, there cannot be more than T m;ffl Finally, for each we can check each of the m;ffl component values t j;q;r to decide if the job J 0 under is addable to the room fl j;q;r . In conclusion, the running time of step 2 in the algorithm Schedule-Small is bounded by O(n We will also attach the mode assignment and room assignment of the job J 0 to each element True. With this information, from a given configuration True, a corresponding scheduling for the small jobs in the rooms can be constructed easily by backtracking the dynamic programming procedure and its makespan can be computed in time m;ffl . Therefore, step 3 of the algorithm takes time In conclusion, the running time of the algorithm Schedule-Small is bounded by O(n2 m m;ffl T m;ffl We now discuss how an upper bound T 0 for the running time of rooms can be derived. Given an instance of the problem Pm jsetjC max and a positive real number ffl ? 0, where each job J i is specified by a list of alternative processing modes J Recall that g. Then the sum T obviously an upper bound on the makespan of the (m; ffl)-canonical schedulings for J (T 0 is the makespan of a straightforward scheduling that assigns each job J i the mode corresponding to min i then starts each job J i when the previous job J finishes. There- fore, if no (m; ffl)-canonical scheduling has makespan better than T 0 , we simply return this straightforward scheduling). In particular, the value is an upper bound for the running time for all rooms. Moreover, since the job set J takes at least T 0 amount of "work" (the work taken by a job is equal to the parallel processing time multiplied by the number of processors involved in this processing) and the system has m processors, the value T 0 also provides a lower bound for the optimal makespan Opt(J In order to apply algorithm Schedule-Small, we need to first decide how the set J is split into large job set JL and small job set JS , what are the modes for the large jobs, what are the types for the towers, and what are the ordering for the large jobs and towers on which the list scheduling algorithm can be applied. According to Lemma 2.1, the number of large jobs is of form j k bm=fflc, and by the definition, the number of towers is 2j m;ffl + 1. When m and ffl are fixed, the number of large jobs and the number of towers are both bounded by a constant. Therefore, we can use any brute force method to exhaustively try all possible cases. To achieve a polynomial time approximation scheme for the problem Pm jsetjC max , we combine the standard scaling techniques [19] with the concept of (m; ffl)-canonical schedulings, as follows. 14 J. CHEN and A. MIRANDA Algorithm. Approx-Scheme Input: An instance J for the problem Pm jsetjCmax and Output: A scheduling of J 2. let J 0 be the job set obtained by scaling the job set J by K; 3. for to bm=fflc do 3.1. for each subset J 0 L of j 0 jobs in J 0 3.2. for each mode assignment A to the jobs in J 0 3.3. for each possible sequence of 2j0 3.4. for each ordering of the j 0 jobs in J 0 L and the 2j0 call Schedule-Small on small job set J 0 S and the ordering to construct a scheduling for the job set J 0 (use T 0 as the upper bound for the running time of rooms); 4. be the scheduling constructed in step 3 with the minimum makespan; 5. replace each job J 0 by the corresponding job J i to obtain a scheduling \Gamma 0 (J ) for the job set J ; 6. return the job scheduling \Gamma 0 (J ). Fig. 3. The approximation scheme be an instance of the Pm jsetjC max problem, where J We let construct another instance n g for the problem, where J 0 That is, the jobs in J 0 are identical to those in J except that all processing times t ij are replaced by bt ij =Kc. We say that the job set J 0 is obtained from the job set J by scaling the processing times by K. We apply the algorithm discussed above to the instance J 0 to construct a scheduling for J 0 from which a scheduling for J is induced. The formal algorithm is presented in Figure 3. We explain how step 5 converts the scheduling \Gamma 0 (J 0 ) for the job set J 0 into a scheduling for the job set J . We first multiply the processing time and the starting time of each job J 0 i in the scheduling \Gamma 0 (J 0 ) by K (but keeping the processing mode). That is, for the job J 0 i of mode Q ij and processing time bt ij =Kc that starts at time i in \Gamma 0 (J 0 ), we replace it by a job J 00 i of mode Q ij and processing time K and let it start at time K i . This is equivalent to proportionally "expanding" the scheduling K. Now on this expansion of the scheduling \Gamma 0 (J 0 ), following the order in terms of their finish times, we do "correction" on processing times by increasing the processing time of each job J 00 i from K that this increase in processing time may cause many jobs in the scheduling to delay their starting time by units. In particular, this increase may cause the makespan of the scheduling to increase by units.) After the corrections on the processing time for all jobs in J , we obtain a scheduling \Gamma 0 (J ) for the job set J . Lemma 4.4. For fixed m 2 and ffi ? 0, The running time of the algorithm Approx-Scheme for the problem Pm jsetjC max is bounded by O(n m;ffl +j m;ffl +1 ), where Proof. Since the integer k is bounded by bm=fflc, the number j 0 of large jobs in J 0 is bounded by j . Therefore, there are at most ways to choose the large job set J 0 L . Since each job may have up to 2 alternative mode assignments, the total number of mode assignments to each large set J 0 L is bounded by Each tower is associated with a subset of the processor set Pm of m processors. Thus, each tower may be associated with different subsets of Pm . Therefore, the number of different sequences of up to 2j m;ffl +1 towers is bounded by . Finally, the number of permutations of the j 0 large jobs and 2j 0 +1 towers is (3j 0 +1)!. Summarizing all these together, we conclude that the number of times that the algorithm Schedule-Small is called is bounded by: O(bm=fflc When the algorithm Schedule-Small is applied to the job set J 0 S , the upper bound on the running time of the rooms is According to Lemma 4.3, each call to the algorithm Schedule-Small takes time where Combining Equations (3) and (4), and noting that m and ffi thus ffl are fixed constants, we conclude that the running time of the algorithm Approx-Scheme is bounded by O(n m;ffl +j m;ffl +1 ). Now we are ready to present our main theorem. Theorem 4.5. The algorithm Approx-Scheme is a polynomial time approximation scheme for the problem Pm jsetjC max . Proof. As proved in Lemma 4.4, the algorithm Approx-Scheme runs in polynomial time when m and ffi are fixed constants. Therefore, we only need to show that the makespan of the scheduling \Gamma 0 (J ) constructed by the algorithm Approx- Scheme for an instance J of the problem Pm jsetjC max is at most (1 times the optimal makespan Opt(J ) for the instance J . Again let Let \Gamma(J ) be an optimal scheduling of makespan Opt(J ). Under the scheduling \Gamma(J ), the mode assignments of the jobs are fixed. Thus, this particular mode assignment makes us able to split the job set J into large job set JL and small job set JS according the job processing time. According to Theorem 3.8, there is an (m; ffl)- canonical scheduling for the instance J , under the same mode assignments, such that the makespan of \Gamma 1 (J ) is bounded by (1 Consider a room fl j;q;r in the (m; ffl)-canonical scheduling \Gamma 1 (J ). Suppose that are the small jobs assigned to the room fl j;q;r by the scheduling \Gamma 1 (J ). Then is the processing time for the job J p i under which is the same as under \Gamma(J ). Thus we must have 0Therefore, under the same mode assignments (with processing time replaced by and the same room assignments, the corresponding scheduling \Gamma 1 (J 0 ) for the set J 0 has no rooms with running time exceeding T 0 . Thus, by Lemma 4.1, when step 3 of the algorithm Approx-Scheme loops to the stage in which the large job set and their mode assignments, the tower types, and the ordering of the large jobs and the towers all match that in the scheduling \Gamma 1 (J 0 ), the array element J. CHEN and A. MIRANDA corresponding to the room configurations of the scheduling must have value True. Thus, a scheduling \Gamma 0 based on this configuration is constructed and its makespan is calculated. Note that the scheduling \Gamma 0 may not be exactly the scheduling must have exactly the same makespan. Since step 4 of the algorithm Approx-Scheme picks the scheduling \Gamma 0 (J 0 ) that has the smallest makespan over all schedulings for J 0 constructed in step 3, we conclude that the makespan of the scheduling \Gamma 0 (J 0 ) is not larger than the makespan of the scheduling \Gamma 0 larger than the makespan of the scheduling As we described in the paragraph before Lemma 4.4, to obtain the corresponding scheduling for the job set J , we first expand the scheduling \Gamma 0 (J 0 ) by K (i.e., multiplying the job processing times and starting times in \Gamma 0 (J 0 ) by K). Let the resulting scheduling be \Gamma 0 (J 00 ). Similarly we expand the scheduling \Gamma 1 (J 0 ) by K to obtain a scheduling \Gamma 1 (J 00 ). The makespan of the scheduling \Gamma 0 (J 00 ) is not larger than the makespan of the scheduling \Gamma 1 (J 00 ) since they are obtained by proportionally expanding the schedulings respectively, by the same factor K. Moreover, the makespan of \Gamma 1 (J 00 ) is not larger than the makespan of the (m; ffl)- canonical scheduling This is because these two schedulings use the same large job set under the same mode assignment, the same small job set under the same mode assignment and room assignment, and the same order of large jobs and towers. The only difference is that the processing time t ij of each job J i in \Gamma 1 (J ) is replaced by a possibly smaller processing time K of the corresponding job J 00 In consequence, we conclude that the makespan of the scheduling \Gamma 0 (J 00 ) is not larger than the makespan of the (m; ffl)-canonical scheduling \Gamma 1 (J ), which is bounded by Finally, to obtain the scheduling \Gamma 0 (J ) for the job set J , we make corrections on the processing times of the jobs in the scheduling \Gamma 0 (J 00 ). More precisely, we replace the processing time K which is the processing time of the job J i in the job set J . Correcting the processing time for each job J 00 may make the makespan of the scheduling increase by Therefore, after the corrections of processing time for all jobs in J 00 , the makespan of the finally resulting scheduling \Gamma 0 (J ) for the job set J , constructed by the algorithm Approx-Scheme, is bounded by the makespan of Here we have used the fact that Opt(J ) T 0 =m. This completes the proof of the theorem. 5. Conclusion and remarks. In this paper, we have developed a polynomial time approximation scheme for the Pm jsetjC max problem for any fixed constant m. The result is achieved by combinations of the recent techniques developed in the area of multiprocessor job schedulings plus the classical dynamic programming and scaling techniques. Note that this result is a significant improvement over the previous results on the problem: no previous approximation algorithms for the problem Pm jsetjC max have their approximation ratio bounded by a constant that is independent of the number m of processors in the system. Our result also confirms a conjecture made by Amoura et al. [1]. In the following we make a few remarks on further work on the problem. The multiprocessor job scheduling problem seems an intrinsically difficult prob- lem. For example, if the number m of processors in the system is given as a variable in the input, then the problem becomes highly nonapproximable: there is a constant ffi such as no polynomial time approximation algorithm for the problem can have an approximation ratio smaller than n [23]. Observing this plus the difficulties in developing good approximation algorithms for the problem, people had suspected whether the Pm jsetjC max problem for some fixed m should be MAX-NP hard [8]. The present paper completely eliminates this possibility [2]. The current form of our polynomial time approximation scheme may not be practically useful, yet. Even for a small integer m and a reasonably small constant ffl, the time complexity of our algorithm is bounded by a polynomial of very high degree. On the other hand, our algorithm shows that there are very "normalized" schedul- ings whose makespan is close to the optimal ones, and that these "good" normalized schedulings can be constructed systematically. We are interested in investigating the tradeoff between the degree of this kind of normalization and the time complexity of approximation algorithms. In particular, we are interested in developing more practical polynomial time algorithms for systems with small number of processors, such as P 4 jsetjC max . Note that currently there is no known practical approximation algorithm for the P 4 jsetjC max problem whose approximation ratio is smaller than 2 (a ratio 2 approximation algorithm for the problem follows from Chen and Lee's recent work on general Pm jsetjC max problem [8]). Moreover, so far all approximation algorithms for the Pm jsetjC max problem are involved in the technique of dynamic programming, which in general results in algorithms of high complexity. Are there any other techniques that may avoid dynamic programming? Acknowledgement . The authors would like to thank Don Friesen and Frank Ruskey for their helpful discussions. --R Scheduling independent multiprocessor tasks Proof verification and the hardness of approximation problems Scheduling multiprocessor tasks on a dynamic configuration of dedicated processors Scheduling multiprocessor tasks on the three dedicated processors "Scheduling multiprocessor tasks on the three dedicated processors, Information Processing Letters 41, (1992), pp. 275-280." Scheduling multiprocessor tasks to minimize scheduling length Scheduling multiprocessor tasks - a survey General multiprocessor tasks scheduling Efficiency and effectiveness of normal schedules on three dedicated processors Simultaneous resource scheduling to minimize weighted flow times Complexity of scheduling parallel task systems Computers and Intractability: A Guide to the Theory of NP-Completeness An approximation algorithm for scheduling on three dedicated machines Bounds for certain multiprocessing anomalies Concrete Mathematics Approximation algorithms for scheduling Complexity of scheduling multi-processor tasks with prespecified processor allocations Fast approximation algorithms for the Knapsack and sum of subset problems An approximation algorithm for diagnostic test scheduling in multicomputer systems Scheduling multiprocessor tasks without prespecified processor allo- cations Current trends in deterministic scheduling Approximation algorithms in multiprocessor task scheduling --TR --CTR C. W. Duin , E. Van Sluis, On the Complexity of Adjacent Resource Scheduling, Journal of Scheduling, v.9 n.1, p.49-62, February 2006 Klaus Jansen , Lorant Porkolab, Polynomial time approximation schemes for general multiprocessor job shop scheduling, Journal of Algorithms, v.45 n.2, p.167-191, November 2002 Jianer Chen , Xiuzhen Huang , Iyad A. Kanj , Ge Xia, Polynomial time approximation schemes and parameterized complexity, Discrete Applied Mathematics, v.155 n.2, p.180-193, January, 2007

multiprocessor processing;polynomial time approximation scheme;job scheduling;approximation algorithm

586942

Linear Time Algorithms for Hamiltonian Problems on (Claw,Net)-Free Graphs.

We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that 2-connected claw-free graphs, containing an induced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Hamiltonian cycle. As a consequence, we obtain linear time algorithms for both problems if the input is restricted to (claw,net)-free graphs. These graphs enjoy those interesting structural properties.

Introduction . Hamiltonian properties of claw-free graphs have been studied extensively in the last couple of years. Di#erent approaches have been made, and a couple of interesting properties of claw-free graphs have been established (see [1, 2, 3, 5, 6, 13, 14, 15, 16, 19, 22, 23, 25, 26]). The purpose of this work is to consider the algorithmic problem of finding a Hamiltonian path or a Hamiltonian cycle e#ciently. It is not hard to show that both the Hamiltonian path problem and the Hamiltonian cycle problem are NP-complete, even when restricted to line graphs [28]. Hence, it is quite reasonable to ask whether one can find interesting subclasses of claw-free graphs for which e#cient algorithms for the above problems exist. Already in the eighties, Du#us, Jacobson, and Gould [12] defined the class of (claw,net)-free (CN-free) graphs, i.e., graphs that contain neither an induced claw nor an induced net (see Figure 1.1). Although this definition seems to be rather restrictive, the family of CN-free graphs contains a couple of graph families that are of interest in their own right. Examples of those families are unit interval graphs, claw-free asteroidal triple-free (AT-free) graphs, and proper circular arc graphs. In their paper [12], Du#us, Jacobson, and Gould showed that this class of graphs has the nice property that every connected CN-free graph contains a Hamiltonian path and every 2-connected CN-free graph contains a Hamiltonian cycle. Later, Shepherd [27] proved that there is an O(n 6 ) algorithm for finding such a Hamiltonian path/cycle in CN-free graphs. Note also that CN-free graphs are exactly the Hamiltonian-hereditary # Received by the editors June 23, 1999; accepted for publication (in revised form) January 13, 2000; published electronically November 28, 2000. An extended abstract of these results has been presented at the 25th International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci. 1665, Springer-Verlag, New York, 1999, pp. 364-376. http://www.siam.org/journals/sicomp/30-5/35777.html Fachbereich Informatik, Universit?t Rostock, A. Einstein Str. 21, D-18051 Rostock, Germany (ab@informatik.uni-rostock.de). # Department of Mathematics and Computer Science, Kent State University, Kent, OH 44242 (dragan@mcs.kent.edu). The research of this author was supported by the German National Science Foundation (DFG). Fachbereich Mathematik, Technische Universit?t Berlin, Stra-e des 17. Juni 136, D-10623 Berlin, Germany (ekoehler@math.TU-Berlin.DE). The research of this author was supported by the graduate program "Algorithmic Discrete Mathematics," grant GRK 219/2-97 of the German National Science Foundation (DFG). LINEAR ALGORITHMS FOR HAMILTONIAN PROBLEMS ON. 1663 graphs [10], i.e., the graphs for which every connected induced subgraph contains a Hamiltonian path. In this paper we give a constructive existence proof and present linear time algorithms for the Hamiltonian path and Hamiltonian cycle problems on CN-free graphs. The important structural property that we exploit for this is the existence of an induced dominating path in every connected CN-free graph (Theorem 2.3). The concept of a dominating path was first used by Corneil, Olariu, and Stewart [8] in the context of AT-free graphs. They also developed a simple linear time algorithm for finding such a path in every AT-free graph [7]. As we show in Theorem 2.3, for the class of CN-free graphs, a linear time algorithm for finding an induced dominating path exists as well. This property is of interest for our considerations since we prove that all claw-free graphs that contain an induced dominating path have a Hamiltonian path (Theorem 3.1). The proof implies that, given a dominating path, one can construct a Hamiltonian path for a claw-free graph in linear time. For 2-connected claw-free graphs, we show that the existence of a dominating pair is su#cient for the existence of a Hamiltonian cycle. dominating pair is a pair of vertices such that every induced path connecting them is a dominating path.) Again, given a dominating pair, one can construct a Hamiltonian cycle in linear time (Theorem 5.6). This already implies, for example, a linear time algorithm for finding a Hamiltonian cycle in claw-free AT-free graphs, since every AT-free graph contains a dominating pair and it can be found in linear time [9]. Unfortunately, CN-free graphs do not always have a dominating pair. For example, an induced cycle with more than six vertices is CN-free but does not have such a pair of vertices. Nevertheless, 2-connected CN-free graphs have another nice property: they have a good pair or an induced doubly dominating cycle. An induced doubly dominating cycle is an induced cycle such that every vertex of the graph is adjacent to at least two vertices of the cycle. A good pair is a pair of vertices, such that there exist two internally disjoint induced dominating paths connecting these vertices. We prove that the existence of an induced doubly dominating cycle or a good pair in a claw-free graph is su#cient for the existence of a Hamiltonian cycle (Theorems 5.1 and 5.5). Moreover, given an induced doubly dominating cycle or a good pair of a claw-free graph, a Hamiltonian cycle can be constructed in linear time. In section 4 we present an O(m algorithm which, for a given 2-connected CN-free graph, finds either a good pair or an induced doubly dominating cycle. For terms not defined here, we refer to [11, 17]. In this paper we consider finite connected undirected graphs E) without loops and multiple edges. The cardinality of the vertex set is denoted by n, whereas the cardinality of the edge set is denoted by m. A path is a sequence of vertices (v 0 , . , v l ) such that all v i are distinct and its length is l. An induced path is a path where cycle (k-cycle) is a path (v 0 , . , its length is k. An induced cycle is a cycle is an induced cycle of length k # 5. The distance dist(v, u) between vertices v and u is the smallest number of edges in a path joining v and u. The eccentricity ecc(v) of a vertex v is the maximum distance from v to any vertex in G. The diameter diam(G) of G is the maximum eccentricity of a vertex in G. A pair v, u of vertices of G with dist(v, is called a diametral pair. A. BRANDST - ADT, F. F. DRAGAN, AND E. K - OHLER a c d a c x y z Fig. 1.1. The claw K(a; b, c, d) and the net N(a, b, c; x, y, z). For every vertex we denote by N(v) the set of all neighbors of v, 1}. The closed neighborhood of v is defined by {v}. For a vertex v and a set of vertices S # V , the minimum distance between v and vertices of S is denoted by dist(v, S). The closed neighborhood N[S] of a set S # V is defined by We say that a set S # V dominates G if doubly dominates G if every vertex of G has at least two neighbors in S. An induced path of G which dominates G is called an induced dominating path. A shortest path of G which dominates G is called a dominating shortest path. Analogously one can define an induced dominating cycle of G. A dominating pair of G is a pair of vertices v, such that every induced path between v and u dominates G. A good pair of G is a pair of vertices v, u # V , such that there exist two internally disjoint induced dominating paths connecting v and u. The claw is the induced complete bipartite graph K 1,3 , and for simplicity, we refer to it by K(a; b, c, d) (see Figure 1.1). The net is the induced six-vertex graph y, z) shown in Figure 1.1. A graph is called CN-free or, equivalently, (claw, net)-free if it contains neither an induced claw nor an induced net. An asteroidal triple of G is a triple of pairwise nonadjacent vertices, such that for each pair of them there exists a path in G that does not contain any vertex in the neighborhood of the third one. A graph is called AT-free if it does not contain an asteroidal triple. Finally, a Hamiltonian path or Hamiltonian cycle of G is a path or cycle, respectively, containing all vertices of G. 2. Induced dominating path. In this section we give a constructive proof for the property that every connected CN-free graph contains an induced dominating path. In fact, we show that there is an algorithm that finds such a path in linear time. To prove the main theorem of this section we will need the following two lemmas. Lemma 2.1 (see [12]). Let be an induced path of a CN-free graph G, and let v be a vertex of G such that dist(v, P 2. Then any neighbor y of v with dist(y, P adjacent to x 1 or to x k . Lemma 2.2. Let P be an induced path connecting vertices v and u of a CN-free graph G. Let also s be a vertex of G such that s / Then 1. for every shortest path P # connecting v and s, P # 2. if there is an edge xy of G such that x # P \ {v} and y # P # \ {v}, then both x and y are neighbors of v. Proof. Let y be the vertex of P # \ {v} which is closest to s and has a neighbor x on P \ {v}; clearly, y #= s. Let s # , v # be the neighbors of y on the subpaths of P # connecting y with s and y with v, respectively. Since s # / # N[P ], by Lemma 2.1, vertex LINEAR ALGORITHMS FOR HAMILTONIAN PROBLEMS ON. 1665 y must be adjacent to v or to u. If yu # E, then v # u # E, too (otherwise, we have a claw K(y; s # , v # , u)). But now dist(v, u) # dist(v, dist(v, u), and a contradiction arises. Therefore, y is adjacent to v, and since y / the paths P and P # have only the vertex v in common. Moreover, to avoid a claw vertex x has to be adjacent to v. Theorem 2.3. Every connected CN-free graph G has an induced dominating path, and such a path can be found in O(n +m) time. Proof. Let G be a connected CN-free graph. One can construct an induced dominating path in G as follows. Take an arbitrary vertex v of G. Using breadth first search (BFS), find a vertex u with the largest distance from v and a shortest path P connecting u with v. Check whether this path P dominates G. If so, we are done. Now, assume that the set is not empty. Again, using BFS, find a vertex s in S with largest distance from v and a shortest path P # connecting v with s. Create a new path P # by joining P and P # in the following way: there is a chord xy between the paths P and P # (see Lemma 2.2), and P otherwise. By Lemma 2.2, the path P # is induced. It remains to show that this path dominates G. Assume there exists a vertex t # V \ N[P # ]. First, we claim that t is dominated neither by P nor by P # . Indeed, if t # (N[P necessarily tv # E and v / neighbors x # P and y # P # of v are adjacent. Therefore, we get a net N(v, y, x; t, s # , u # ), where s # and u # are the vertices at distance two from v on paths P # and P , respectively. Note that vertices s # , u # exist because dist(v, s) # 2. Thus, t is dominated neither by P nor by P # . Moreover, from the choice of u and s we have 2 # dist(v, t) # dist(v, s) # dist(v, u). Now let P # be a shortest path, connecting t with v, and let z be a neighbor of v on this path. Applying Lemma 2.2 twice (to P, P # and to P # , P # ), we obtain a subgraph of G depicted in Figure 2.1. We have three shortest paths P, P # , P # , each of length at least 2 and with only one common vertex v. These paths can have only chords of type zx, zy, xy. Any combination of them leads to a forbidden claw or net. This contradiction completes the proof of the theorem. Evidently, the method described above can be implemented to run in linear time. t u x z y s Fig. 2.1. It is not clear whether CN-free graphs can be recognized e#ciently. But, to apply our method for finding an induced dominating path in these graphs, we do not need to know in advance that a given graph G is CN-free. Actually, our method can be A. BRANDST - ADT, F. F. DRAGAN, AND E. K - OHLER applied to any graph G. It either finds an induced dominating path or returns either a claw or a net of G, showing that G is not CN-free. Corollary 2.4. There is a linear time algorithm that for a given (arbitrary) connected graph G either finds an induced dominating path or outputs an induced claw or an induced net of G. Proof. Let G be a graph. For an arbitrary vertex v of G, we find a vertex u with the largest distance from v and a shortest path P connecting u with v. If P dominates G, then we are done. Else, we find a vertex s # V \ N[P ] with the largest distance from v and a shortest path P # connecting v with s. If there are vertices in P # \ {v} which have a neighbor on P \ {v}, we take the vertex y that is closest to s and check whether y is adjacent to v and u. If it is adjacent neither to u nor to v, then G has a net or a claw (see the proof of Lemma 2.1). If yu # E or yv # E and a neighbor x of y on P \ {v} is not adjacent to v, then G has a claw (see Lemma 2.2). Now, if we did not yet find a forbidden subgraph, then the only possible chord between the paths P and P # is xy with xv, yv # E, and we can create an induced path P # as described in the proof of Theorem 2.3. Hence, it remains to check whether P # dominates G. If there exists a vertex t # V \ N[P # ], then again we will find a net or a claw in G (see Theorem 2.3). It is easy to see that the total time bound of all these operations is linear. 3. Hamiltonian path. In what follows we show that for claw-free graphs the existence of an induced dominating path is a su#cient condition for the existence of a Hamiltonian path. The proof for this result is constructive, implying that, given an induced dominating path, one can find a Hamiltonian path e#ciently. Theorem 3.1. Every connected claw-free graph G containing an induced dominating path has a Hamiltonian path. Moreover, given an induced dominating path, a Hamiltonian path of G can be constructed in linear time. Proof. Let E) be a connected claw-free graph and let be an induced dominating path of G. If and, since G is claw-free, there are no three independent vertices in G - {x 1 }. (By we denote a subgraph of G induced by vertices V \ {x 1 }.) If G - {x 1 } is not connected, then, again because G is claw-free, it consists of two cliques C 0 , C 1 and a Hamiltonian path of G can easily be constructed. If G - {x 1 } is connected, we can construct a Hamiltonian path as follows. First, we construct a maximal path vertices that are not in P 1 are neither connected to y 1 nor to y l . Let R be the set of all remaining vertices. If #, we are done. If there is any vertex in R, it follows that y 1 y l # E since otherwise there are three independent vertices in G- {x 1 }. Furthermore, any two vertices of R are joined by an edge, since otherwise they would form an independent triple with y 1 (and with y l as well). Hence, R induces a clique. Since G-{x 1 } is connected, there has to be an edge from a vertex R # R to some vertex y Now we can construct a Hamiltonian path P of G: R), where - R stands for an arbitrary permutation of the vertices of R \ {v R }. For first construct a Hamiltonian path P 2 for G described above, using x 1 as the dominating vertex. At least one endpoint of P 2 is adjacent to x 2 since if G # -{x 1 } is not connected, x 2 has to be adjacent to all vertices of either C 0 or C 1 (otherwise, there is a claw in G), and if G # - {x 1 } is connected, the construction gives a path ending in x 1 which is, of course, adjacent to x 2 . To construct a Hamiltonian path for the rest of the graph we define for each vertex x i (i # 2) of P a set of vertices Each set C i forms a clique of G since if two LINEAR ALGORITHMS FOR HAMILTONIAN PROBLEMS ON. 1667 vertices u, v # C i are not adjacent, then the set u, v, x i , x i-1 induces a claw. Hence we can construct a path P stands for an arbitrary permutation of the vertices of C i \{x i+1 }. This path P # is a Hamiltonian path of G because it obviously is a path, and, since P is a dominating path, each vertex of G has to be either on P , P 2 , or in one of the sets C i . For the case finding the connected components of G - {x 1 } and constructing the path P 1 can easily be done in linear time. For k # 2 we just have to make sure that the construction of the sets C i can be done in O(n+m), and this can be realized easily within the required time bound. Theorem 3.2. Every connected CN-free graph G has a Hamiltonian path, and such a path can be found in O(n +m) time. Proof. By Theorem 2.3, every connected CN-free graph has an induced dominating path P , and it can be found in linear time. Using the path P , by Theorem 3.1, one can construct a Hamiltonian path of G in linear time. Analogously to Corollary 2.4, we can state the following. Corollary 3.3. There is a linear time algorithm that for a given (arbitrary) connected graph G either finds a Hamiltonian path or outputs an induced claw or an induced net of G. Proof. The proof follows from Corollary 2.4 and the proof of Theorem 3.1. 4. Induced dominating cycle, dominating shortest path, or good pair. In this section we show that every 2-connected CN-free graph G has an induced doubly dominating cycle or a good pair. Moreover, we present an e#cient algorithm that, for a given 2-connected CN-free graph G, finds either a good pair or an induced doubly dominating cycle. Lemma 4.1. Every hole of a connected CN-free graph G dominates G. Corollary 4.2. Let H be a hole of a connected CN-free graph G. Every vertex of V \ H is adjacent to at least two vertices of H. A subgraph G # of G (doubly) dominates G if the vertex set of G # (doubly) dominates G. Lemma 4.3. Every induced subgraph of a connected CN-free graph G which is isomorphic to S 3 or S - 3 (see Figure 4.1) dominates G. a d c f e a d c f e Fig. 4.1. Proof. Let G contain an induced subgraph isomorphic to S - 3 , and assume that it does not dominate G. Then, there must be a vertex s such that dist(s, S - 2. Let x be a neighbor of s from N[S - 3 ]. If x is adjacent neither to a, nor to b, nor to c (see Figure 4.1), then G contains a claw (e.g., if xf # E, then a claw K(f Thus, without loss of generality, x has to be adjacent to a or b. 1668 A. BRANDST - ADT, F. F. DRAGAN, AND E. K - OHLER If xa # E, then x is adjacent neither to b nor to c, since otherwise we will get a claw (K(x; a, b, s) or K(x; a, c, s)). To avoid a net N(a, e, d; x, b, c) vertex x must be adjacent to e or d. But, if ex # E, then xd # E too. (Otherwise, we will have a claw Analogously, if xd # E, then also xe # E. Hence, x is adjacent to both e and d, and a net N(x, e, d; s, b, c) arises. Now, we may assume that x is adjacent to b and not to a, c. To avoid a claw K(b; x, e, f ), x must be adjacent to e or f . But again, xe # E if and only if xf # E. (Otherwise, we get a net N(x, b, e; s, f, a) or N(x, b, f ; s, e, c).) Hence x is adjacent to both e and f and a claw K(x; s, e, f) arises. Consequently, S - 3 dominates G. Similarly, every induced S 3 (if it exists) dominates G. Lemma 4.4. Let P be an induced path connecting vertices v and u of a connected CN-free graph G. Let s be a vertex of G such that s / has an induced doubly dominating cycle, and such a cycle can be found in linear time. Proof. Let P v and P u be shortest paths connecting vertex s with v and u, re- spectively. Both these paths as well as the path P have lengths at least 2. Since and there is a chord between P and P v , then it is unique and both its endvertices are adjacent to v. The same holds for P and P endvertices of the chord (if it exists) are adjacent to u. Now, without loss of generality, we suppose that dist(s, u) # dist(s, v). Then, from u / 2.2 we deduce that P u # P and P u at most one chord is possible, namely, the one with both endvertices adjacent to s. Consequently, we have constructed an induced subgraph of G shown in Figure 4.2 (only chords s # s # , v # v # and u # u # are possible). s Fig. 4.2. If the lengths of all three paths P, P v , P u are at least 3, then it is easy to see that G has a hole H k (k # 6). Furthermore, if at least one of these paths has length greater than or equal to 4, or two of them have lengths 3, then G must contain a hole It remains to consider two cases: lengths of both P v and P u are 2 and LINEAR ALGORITHMS FOR HAMILTONIAN PROBLEMS ON. 1669 the length of P is 3 or 2. Clearly, in both of these cases the graph G contains either a hole or an induced subgraph isomorphic to S - 3 or S 3 . By Corollary 4.2, every hole of G doubly dominates G. Let G contain an S - 3 with vertex labeling shown in Figure 4.1. We claim that the induced cycle (e, b, f, d, e) dominates G or G contains a hole H 6 . Indeed, if a vertex s of G does not belong to S - 3 , then, by Lemma 4.3, it is adjacent to a vertex of S - 3 . Suppose that s is adjacent to none of e, b, f, d. Then, without loss of generality, sa # E, and we obtain an induced subgraph of G isomorphic either to a net N(e, a, d; b, s, c) or to H depending on whether vertices s and c are adjacent. Hence, we may assume that (e, b, f, d, e) dominates G, and since G is claw-free, this cycle is doubly dominating. Now let G contain an S 3 with vertex labeling shown in Figure 4.1. We will show that every vertex of G is adjacent to at least two vertices of the cycle (e, f, d, e) or G contains a hole H 5 . Suppose vertex s of G is adjacent to none of e, d. Then, by Lemma 4.3, s is adjacent to at least one of a, b, c, f . Let sf # E. To avoid a claw, vertex s is adjacent to both b and c. But then a hole H arises. Assume that loss of generality, sa # E. To avoid a net N(a, e, d; s, b, c), s must be adjacent to b or c. In both cases a hole H 5 occurs. Clearly, the construction of an induced doubly dominating cycle of G given above takes linear time. Theorem 4.5. There is a linear time algorithm that, for a given connected CN- finds an induced doubly dominating cycle or gives a dominating shortest path of G. Proof. Let G be a connected CN-free graph. One can construct an induced doubly dominating cycle or a dominating shortest path of G as follows (compare with the proof of Theorem 2.3). Take an arbitrary vertex v of G. Find a vertex u with the largest distance from v and a shortest path P connecting u with v. Check whether dominates G. If so, we are done; P is a dominating shortest path of G. Assume now that the set is not empty. Find a vertex s in S with the largest distance from v and a shortest path P v connecting v with s. Create again a new path shortest paths P and P v as in the proof of Theorem 2.3. We have proven there that P # dominates G. Now let P u be a shortest path between s and u. If dist(s, u) # dist(v, u) or both dist(s, u) > dist(v, u) and v / 4.4 can be applied to get an induced doubly dominating cycle of G in linear time. Therefore, we may assume that dist(s, u) > dist(v, u) # dist(v, s) and v # N[P u ]. Now we show that the shortest path P u dominates G. If v lies on the path P u , then and we are done. Otherwise, let x be a neighbor of v in P u . Note that dist(v, s) > 1 and so x #= s, u. Since G is claw-free, v is adjacent to a neighbor x. Assume, without loss of generality, that x is closer to s than y. If we show that dist(v, by the proof of Theorem 2.3, the path P u will dominate G (as a path obtained by "joining" two shortest paths that connect v with u and v with s, respectively). By the triangle condition, we have dist(u, s) < dist(v, u)+dist(v, s) (strict inequality because Consequently, Since all our proofs were constructive, we can conclude the following. A. BRANDST - ADT, F. F. DRAGAN, AND E. K - OHLER Corollary 4.6. There is a linear time algorithm that, for a given (arbitrary) connected graph G, either finds an induced doubly dominating cycle, or gives a dominating shortest path, or outputs an induced claw or an induced net of G. Lemma 4.7. Let be a dominating shortest path of a graph G. Then max{ecc(v), ecc(u)} # diam(G) - 1. Proof. Let x, y be a diametral pair of vertices of G; that is, If both x and y are on P , then necessarily {x, and holds or, without loss of generality, and Finally, if both x and y are in N[P dist(x, y), then we may assume that at least one of x, y belongs to N(v), say, x. Hence, dist(x, y) A pair of vertices u, v of G with dist(u, called a pair of mutually furthest vertices. Corollary 4.8. For a graph G with a given dominating shortest path, a pair of mutually furthest vertices can be found in linear time. Proof. Let be a dominating shortest path of G with holds. Denote by x a vertex of G such that dist(v, ecc(v). Note that both the eccentricity of v and a vertex furthest from v can be found in linear time by BFS. Now, if then v, x are mutually furthest vertices of G. Else, ecc(x) > ecc(v) # diam(G) - 1 must hold and vertices x and y, where y is a vertex with dist(x, diametral pair of G; dist(x, In what follows we will use the fact that in a 2-connected graph every pair of vertices is joined by two internally disjoint paths. In order to actually find such a pair of paths, one can use Tarjan's linear time depth first search- (DFS)-algorithm for finding the blocks of a given graph. For the proof of Lemma 4.9, we refer to [21]. Lemma 4.9. Let G be a 2-connected graph, and let x, y be two di#erent nonadjacent vertices of G. Then one can construct in linear time two induced, internally disjoint paths, both joining x and y. Theorem 4.10. There is a linear time algorithm that, for a given 2-connected CN-free graph G, either finds an induced doubly dominating cycle or gives a good pair of G. Proof. By Theorem 4.5, we get either an induced doubly dominating cycle or a dominating shortest path of G in linear time. We show that, having a dominating shortest path of a 2-connected graph G, one can find in linear time a good pair or an induced doubly dominating cycle. By Corollary 4.8, we may assume that a pair x, y of mutually furthest vertices of G is given. Let also be two induced internally disjoint paths connecting x and y in G. They exist and can be found in linear time by Lemma 4.9 (clearly, we may assume that xy / # E, because otherwise together with a vertex z # V \ {x, y} will form a doubly dominating triangle). If one of these paths, say, P 1 , is not dominating, then there must be a vertex s # V \ N[P 1 ] . Since x, y are mutually furthest vertices of G, we have dist(s, x) # dist(x, y), dist(s, y) # dist(x, y). Hence, we are in the conditions of Lemma 4.4 and can find an induced doubly dominating cycle of G in linear time. Corollary 4.11. There is a linear time algorithm that, for a given (arbitrary) 2-connected graph G, either finds an induced doubly dominating cycle, or gives a good or outputs an induced claw or an induced net of G. LINEAR ALGORITHMS FOR HAMILTONIAN PROBLEMS ON. 1671 5. Hamiltonian cycle. In this section we prove that, for claw-free graphs, the existence of an induced doubly dominating cycle or a good pair is su#cient for the existence of a Hamiltonian cycle. The proofs are also constructive and imply linear time algorithms for finding a Hamiltonian cycle. Theorem 5.1. Every claw-free graph G that contains an induced doubly dominating cycle has a Hamiltonian cycle. Moreover, given an induced doubly dominating cycle, a Hamiltonian cycle of G can be constructed in linear time. Proof. Let be an induced doubly dominating cycle of G. As before, we define C k). Each set C i forms a clique of G; otherwise, we would have a claw. Furthermore, C the sets N[x 1 ], C 2 , . , C k-1 form a partition of the vertex set of G. Note that any vertex adjacent to x k and not to x j (1 < j < belongs to N[x 1 ], since the cycle DC is doubly dominating. Let G }) be the subgraph of G induced by If we show that there is a Hamiltonian path P in G # starting at a neighbor of x k and ending at a neighbor of x 2 , then we are done; the cycle Hamiltonian cycle of G (recall that stands for an arbitrary permutation of the vertices of C i \ {x i+1 }). G # is a connected graph, by Theorem 3.1, there exists a Hamiltonian path . Assume that x k s, x k t / # E. Then, to avoid a claw vertices s and t have to be adjacent, giving a new Hamiltonian path P # of G # starting at x 1 and ending at a vertex y. If y is adjacent neither to x k nor to x 2 , then a claw K(x 1 occurs. (Note that in case is adjacent to at least one of x k , x 2 because the cycle doubly dominating.) Without loss of generality, yx 2 # E and the path P # is a desired path of G # . So, we may assume that x k is adjacent to t or s. Analogously, x 2 is adjacent to one of t, s. If x k , x 2 are adjacent to di#erent vertices, then we are done; the path P # starts at a neighbor of x k and ends at a neighbor of x 2 . Otherwise, let both x k and x 2 be adjacent to t and not to s. Then a claw K(x 1 or we get a contradiction with the property of to be a doubly dominating cycle. Corollary 5.2. Every claw-free graph, containing an induced dominating cycle of length at least 4, has a Hamiltonian cycle, and, given that induced dominating cycle, one can construct a Hamiltonian cycle in linear time. E) be a graph, and let be an induced dominating path of G. P is called an enlargeable path if there is some vertex v in V \P that is either adjacent to x 1 or to x k but not to both of them and, additionally, to no other vertex in P . Consequently, an induced dominating path P is called nonenlargeable if such a vertex does not exist. Obviously, every graph G that has an induced dominating path has a nonenlargeable induced dominating path as well. Furthermore, given an induced dominating path P , one can find in linear time a nonenlargeable induced dominating path P # by simply scanning the neighborhood of both x 1 and x k . For the next theorem we will need an auxiliary result. Lemma 5.3. Let G be a claw-free graph, and let an induced nonenlargeable dominating path of G such that there is no vertex y in G with there is a Hamiltonian path in G that starts in x 1 1672 A. BRANDST - ADT, F. F. DRAGAN, AND E. K - OHLER Proof. Let is empty. Using the method described in the proof of Theorem 3.1, we can easily construct a path, starting in x 1 and ending in x k , that contains all vertices of C 2 , . , C k-1 . This implies that we have to worry only about how to insert the vertices of the neighborhood of x 1 into this path. We have to consider two cases. Case 1. consists of two connected components C 0 , C 1 . Since G is claw-free, both C 0 and C 1 induce cliques in G. Furthermore, x 2 is adjacent to all vertices of at least one of C 0 and C 1 , say, C 1 , because otherwise we have a claw in G. Let y be an arbitrary vertex of C 0 . Since P is nonenlargeable, y has at least one neighbor on P \{x 1 }, and let x j be the one with smallest index. By the preconditions of our lemma, j #= k. If j > 2, then y has to be adjacent to x j+1 as well, since y, x j-1 , x j+1 ) is a claw. Furthermore, y is adjacent to all vertices y, x j-1 , c j ) is a claw. Hence, when constructing the Hamiltonian path, we can simply add y to C j . Now we consider the set Y of all vertices y of C 0 with yx 2 # E. Suppose there is a vertex c 2 in C 2 with c 2 #= x 3 . If there is a vertex c 1 # C 1 that is nonadjacent to vertex there is an edge from every vertex c 0 # Y to c 2 ; otherwise, K(x a claw of G. This implies that we can construct a Hamiltonian path with the required properties. If, on the other hand, all vertices of C 1 are adjacent to all vertices of C 2 , we can construct such a path by starting in x 1 , traversing through Y , x proceeding as before. Now suppose that there is no vertex In this case either all vertices c 0 # Y or all vertices c 1 # C 1 have to be adjacent to x 3 , because otherwise K(x claw. Suppose, without loss of generality, that all vertices of Y are adjacent to x 3 . Then we construct the path by starting in x 1 , traversing through C 1 , x 2 , Y , x 3 , and proceeding as before. Case 2. induces a connected graph. If x 2 is not adjacent to any of the vertices in H, then H has to be a clique and we can apply the method described in case 1. Suppose now that x 2 is adjacent to some vertex in H. First, we construct a Hamiltonian path which is done as in the proof of Theorem 3.1, since there is no independent triple in H. Now we claim that either x 2 is adjacent to one of y 1 or y l , or P # does in fact induce a Hamiltonian cycle of H implying again the existence of a path with an end-vertex adjacent to x 2 . Indeed, suppose x 2 is not adjacent to any of the endvertices of P # . Then, since G is claw-free, y 1 has to be adjacent to y l , because otherwise K(x would induce a claw in G. Hence induces a Hamiltonian cycle in H. Using P # , we can easily construct a Hamiltonian path in N[x 1 ] starting in x 1 and ending in x 2 . The rest of the Hamiltonian path of G can be constructed as before. In fact, we can prove a slightly stronger result. Let Each of these sets forms a clique of G. Lemma 5.4. Let G be a claw-free graph, and let an induced dominating path of G such that there is no vertex y in G with N(y) also P be enlargeable but only to one end, e.g., assume that there exists an edge zb with z # C k-1 \ {x k } and b # B. Then there is LINEAR ALGORITHMS FOR HAMILTONIAN PROBLEMS ON. 1673 Proof. First, we can easily construct a path, starting in x 1 and ending in x k-1 , that contains all vertices of C 2 , . , C k-2 . Then we attach to this path a path which starts at x k-1 , goes through C k-1 , B using all their vertices, and ends in x k . Finally, we insert the vertices of the neighborhood of x 1 into the obtained path as we have done in the proof of Lemma 5.3. Theorem 5.5. Let G be a 2-connected claw-free graph with a good pair u, v. Then G has a Hamiltonian cycle and, given the corresponding induced dominating paths, one can construct a Hamiltonian cycle in linear time. Proof. Let be the induced dominating paths, corresponding to the good pair u, v. By the definition of a good both k and l are greater than 2. We may also assume that, for any induced dominating path exists such that together with y would form an induced dominating cycle of length at least 4, and we can apply Corollary 5.2 to construct a Hamiltonian cycle of G in linear time. Let A 1 be the set of vertices a 1 that are adjacent to x 1 but to no other vertex of be the set of vertices b 1 that are adjacent to x k but to no other vertex of are defined accordingly for P 2 . Of course, each of the sets A 1 , A 2 , forms a clique of G. First we assume that one of these paths, say, P 1 , is nonenlargeable, i.e., A In this case we do the following. We remove the inner vertices of P 2 from G and get the graph G- (P 2 ), where denotes the inner vertices of P 2 . Then, using we create a Hamiltonian path in G- (P 2 ) that starts at u and ends at v (Lemma 5.3), and we add (P 2 ) to this path to create a Hamiltonian cycle of G. We can use this method for creating a Hamiltonian cycle of G whenever we have two internally disjoint paths P, P # of G both connecting u with v such that one of them is an induced dominating and nonenlargeable path of the graph obtained from G by removing the inner vertices of the other path. Now we suppose that both paths P 1 , P 2 are enlargeable. Because of symmetry we have to consider the following three cases. Case 1. There exist a vertex a 1 # A 1 \ A 2 and a vertex In this case there must be edges from a 1 , b 1 to inner vertices y i , y j of P 2 . Con- sequently, we can form a new path P # 2 by starting in u and traversing through A 1 , is the subpath of P 2 between y i and y j . Evi- dently, contains all vertices of B 1 , A 1 and is internally disjoint from P 1 , which is nonenlargeable in G- (P # Case 2. there exists a vertex a 1 # A 1 \ A 2 . In this case none of the vertices of B #) has a neighbor in v. As G is 2-connected, for some vertex b # B there has to be a vertex dominates G and z / # B, vertex z must be adjacent to a vertex y # P 2 \ {v}. If z is only adjacent to y but to no other vertex of P 2 , then z necessarily belongs to A 2 and we can form a new path P #by starting in u, using all vertices of A 2 , B and ending in v. Again, P # 1 is internally disjoint from P 2 and P 2 is nonenlargeable in G- (P # then we can apply Corollary 5.2. Therefore, we may assume that z is adjacent to an inner vertex y of P 2 . Now, if there exists a vertex a 1 # A 1 \ A 2 , then a 1 is adjacent to some vertex y # of we can construct a new path P # 2 by using u, A 1 , y # , . , y, z, B, v. (If B was empty, then 2 ends at . , y # , . , y l-1 , v.) This path is internally disjoint from P 1 , which A. BRANDST - ADT, F. F. DRAGAN, AND E. K - OHLER is nonenlargeable in G - (P # then from the discussion above we may assume that either A := A is empty or there is a vertex z # V \ which is adjacent to a vertex of A and has a neighbor y # in (P 2 ). Hence, we can construct a path P # 2 by using u, A, z # , y # , . , y, z, B, v, which is internally disjoint from P 1 . (If z 2 is constructed by using u, A, z, B, v.) Case 3. A 2 is strictly contained in A 1 , and B 1 is strictly contained in B 2 . Consider vertices cliques C then we can construct a new path P # 2 by using This path is internally disjoint from P 1 , which is nonenlargeable in G- there must be a neighbor y # (P 2 ) of z # . If vertex b is adjacent to some vertex in C k-1 \{v}, then we construct a new path P # 2 by using u, A 1 , y # , . , v. It will be internally disjoint from P 1 , which is enlargeable only to one end (at x G-(P # We are now in the conditions of Lemma 5.4 and can construct a Hamiltonian path of G- (P # starts in u and ends in v. Adding (P # 2 ) to this path, we obtain a Hamiltonian cycle of G. So, we may assume that zz # / vertex z # A 1 \ A 2 and that vertex b is not adjacent to any vertex of C k-1 \ {v}. From this we conclude also that z / But since z / there must be a neighbor x j # (P 1 ) of z. We choose vertex with the smallest j. Clearly, 1 < j < k - 1 and z # C j . First we define a new induced path P # cliques We have z # A # 1 , since otherwise from the construction of P # would be adjacent to z, and that is impossible Note that vertex x j+1 is dominated by the path P 2 . If it is adjacent to only vertex v from P 2 , then arises. Therefore, x j+1 must be adjacent to an inner vertex y of P 2 . Now we define a new path P #by using u, A # 1 , y # , . , y, x j+1 , C j+1 , x j+2 , . , C k-1 , v. It is internally disjoint from 1 and contains all vertices of A # 1 and C 1). It is clear from the construction that the path P # 1 dominates the graph G - (P # (Every vertex which was not dominated by the path P # 1 in G belongs to some set C i (j It remains to show that the path P # 1 is nonenlargeable in G - (P # Assume by way of contradiction that it is enlargeable. Since A # 1 # (P # 2 ), this is possible only if be a vertex of B # 1 . Then p does not belong to B 1 , since otherwise it should be adjacent to z, which is contained in (P # are cliques, .) Now, from we conclude that the neighbors of p in P 1 \ {v} are only vertices from {x j+1 , . , x k-1 }, i.e., p belongs to a set C s for some s 1. Consequently, a contradiction to C s # (P # arises. It is not hard to see that the above method can be implemented to run in linear time. Theorem 5.6. Every 2-connected claw-free graph G that contains a dominating pair has a Hamiltonian cycle, and, given a dominating pair, a Hamiltonian cycle can be constructed in linear time. Proof. Let v, u be a dominating pair of a 2-connected graph G. If vu / by Lemma 4.9, there exist two internally disjoint induced paths connecting v and u. Both these paths dominate G, and, therefore, u, v is a good pair of G. Thus, the statement holds by Theorem 5.5. Now let vu # E. Define sets A := N(u) \ N[v], B := N(v) \ N[u], and S := # N(u). Since G is claw-free, the sets A and B are cliques of G. Notice also that LINEAR ALGORITHMS FOR HAMILTONIAN PROBLEMS ON. 1675 sets A, B, S, and {v, u} form a partition of the vertex set of G. If there is an edge ab in G such that a # A and b # B, then vertices a, u, v, b induce a 4-cycle which dominates G. Hence, we can apply Corollary 5.2 to get a Hamiltonian cycle of G. Therefore, assume that no such edge exists. But since G is 2-connected, there must be edges ax, by with x, y # S, a # A, and b # B. We distinguish between two cases. Let G S denote the subgraph of G induced by S. Case 1. G S is disconnected. Then, it consists of two cliques S 1 and S 2 . Now, if vertices x, y are in di#er- ent components of G S , say, x y, P S2\{y} , u) is a Hamiltonian cycle of G. (P M stands for an arbitrary permutation of the vertices of a set M .) If x, y are in one component, say, S 1 , then is a Hamiltonian cycle of G. Case 2. G S is connected. Then, by Theorem 3.1, there exists a Hamiltonian path G S . Assume that as, at / # E. Then, to avoid a claw K(u; a, s, vertices s and t have to be adjacent, giving a Hamiltonian cycle HC := (s, y 1 , . , y l , t, s) of G S . Vertices x and y split this cycle into two paths Hence, a is a Hamiltonian cycle of G. Now, we may assume that a is adjacent to s or t. Analogously, b is adjacent to one of t, s. If a, b are adjacent to di#erent vertices, say, as, bt # E, then is a Hamiltonian cycle of G. Finally, if a, b are adjacent only to s (similarly, to t), then (u, P \ {s}, v, P B\{b} , b, s, a, P A\{a} , u) is a Hamiltonian cycle of G. Theorem 5.7. Every 2-connected CN-free graph G has a Hamiltonian cycle, and such a cycle can be found in O(n +m) time. Proof. The proof follows from Theorems 4.10, 5.1, and 5.5. Corollary 5.8. There is a linear time algorithm that for a given (arbitrary) 2-connected graph G either finds a Hamiltonian cycle or outputs an induced claw or an induced net of G. Corollary 5.9. A Hamiltonian cycle of a 2-connected (claw,AT)-free graph can be found in O(n +m) time. Remark. Corollary 5.8 implies that every 2-connected unit interval graph has a Hamiltonian cycle, which is, of course, well known (see [24, 20]). The interesting di#erence of the above algorithm compared to the existing algorithms for this problem on unit interval graphs is that it does not require the creation of an interval model. It also follows from Corollaries 3.3 and 5.8 that both the Hamiltonian path problem and the Hamiltonian cycle problem are linear time solvable on proper circular arc graphs. Note that previously known algorithms for these problems had time bounds Fig. 5.1. Claw-free graph, containing a dominating pair and a net. 1676 A. BRANDST - ADT, F. F. DRAGAN, AND E. K - OHLER It should also be mentioned that Theorems 3.1 and 5.5 do cover a class of graphs that is not contained in the class of CN-free graphs. Figure 5.1 shows a graph that is claw-free, does contain a dominating/good pair and, consequently, a dominating path, but, obviously, it is neither AT-free nor net-free. --R Every 3-connected Minimal 2-connected non-hamiltonian claw-free graphs A linear time algorithm to compute a dominating path in an AT-free graph Discrete Math. time algorithms for dominating pairs in asteroidal triple-free graphs Graph Theory Forbidden subgraphs and the Hamiltonian theme Characterizing forbidden pairs for hamiltonian properties On Hamiltonian claw-free graphs Algorithmic Graph Theory and Perfect Graphs Local tournaments and proper circular arc graphs Finding hamiltonian circuits in interval graphs Hamiltonian cycles in 3-connected claw-free graphs An optimum Hamiltonicity in claw-free graphs Introduction to Graph Theory --TR --CTR Jou-Ming Chang, Induced matchings in asteroidal triple-free graphs, Discrete Applied Mathematics, v.132 n.1-3, p.67-78, 15 October Andreas Brandstdt , Feodor F. Dragan, On linear and circular structure of (claw, net)-free graphs, Discrete Applied Mathematics, v.129 n.2-3, p.285-303, 01 August Sun-Yuan Hsieh, An efficient parallel strategy for the two-fixed-endpoint Hamiltonian path problem on distance-hereditary graphs, Journal of Parallel and Distributed Computing, v.64 n.5, p.662-685, May 2004

hamiltonian path;hamiltonian cycle;linear time algorithms;dominating pair;claw-free graphs;net-free graphs;dominating path

586944

On the Determinization of Weighted Finite Automata.

We study the problem of constructing the deterministic equivalent of a nondeterministic weighted finite-state automaton (WFA). Determinization of WFAs has important applications in automatic speech recognition (ASR). We provide the first polynomial-time algorithm to test for the twins property, which determines if a WFA admits a deterministic equivalent. We also give upper bounds on the size of the deterministic equivalent; the bound is tight in the case of acyclic WFAs. Previously, Mohri presented a superpolynomial-time algorithm to test for the twins property, and he also gave an algorithm to determinize WFAs. He showed that the latter runs in time linear in the size of the output when a deterministic equivalent exists; otherwise, it does not terminate. Our bounds imply an upper bound on the running time of this algorithm.Given that WFAs can expand exponentially in size when determinized, we explore why those that occur in ASR tend to shrink when determinized. According to ASR folklore, this phenomenon is attributable solely to the fact that ASR WFAs have simple topology, in particular, that they are acyclic and layered. We introduce a very simple class of WFAs with this structure, but we show that the expansion under determinization depends on the transition weights: some weightings cause them to shrink, while others, including random weightings, cause them to expand exponentially. We provide experimental evidence that ASR WFAs exhibit this weight dependence. That they shrink when determinized, therefore, is a result of favorable weightings in addition to special topology. These analyses and observations have been used to design a new, approximate WFA determinization algorithm, reported in a separate paper along with experimental results showing that it achieves significant WFA size reduction with negligible impact on ASR performance.

Introduction Finite-state machines and their relation to rational functions and power series have been extensively studied [2, 3, 12, 16] and widely applied in fields ranging from image compression [9-11, 14] to natural language processing [17, 18, 24, 26]. A subclass of finite-state machines, the weighted finite-state automata (WFAs), has recently assumed new importance, because WFAs provide a powerful method for manipulating models of human language in automatic speech recognition (ASR) systems [19, 20]. This new re-search direction also raises a number of challenging algorithmic questions [5]. A weighted finite-state automaton (WFA) is a nondeterministic finite automaton (NFA), A, that has both an alphabet symbol and a weight, from some set K, on each transition. be a semiring. Then A together with R generates a partial function from strings to K: the value of an accepted string is the semiring sum over accepting paths of the semiring product of the weights along each accepting path. Such a partial function is a rational power series [25]. An important example in ASR is the set of WFAs with the min-sum semiring, which compute for each accepted string the minimum cost accepting path. In this paper, we study problems related to the determinization of WFAs. A deter- ministic, or sequential, WFA has at most one transition with a given input symbol out of each state. Not all rational power series can be generated by deterministic WFAs. A determinization algorithm takes as input a WFA and produces a deterministic WFA that generates the same rational power series, if one exists. The importance of determinization to ASR is well established [17, 19, 20]. As far as we know, Mohri [17] presented the first determinization procedure for WFAs, extending the seminal ideas of Choffrut [7, 8] and Weber and Klemm [27] regarding string-to-string transducers. Mohri gives a determinization procedure with three phases. First, A is converted to an equivalent unambiguous, trim WFA A t , using an algorithm analogous to one for NFAs [12]. (Unambiguous and trim are defined below.) Mohri then gives an algorithm, TT, that determines if A t has the twins property (also defined below). If A t does not have the twins property, then there is no deterministic equivalent of A. If A t has the twins property, a second algorithm of Mohri's, DTA, can be applied to A t to yield A 0 , a deterministic equivalent of A. Algorithm TT runs in O(m 4n 2 is the number of transitions and n the number of states in A t . Algorithm runs in time linear in the size of A 0 . Mohri observes that A 0 can be exponentially larger than A, because WFAs include classical NFAs. He gives no upper bound on the worst-case state-space expansion, however, and due to weights, the classical NFA upper bound does not apply. Finally, Mohri gives an algorithm that takes a deterministic WFA and outputs the minimum-size equivalent, deterministic WFA. In this paper, we present several results related to the determinization of WFAs. In Section 3 we give the first polynomial-time algorithm to test whether an unambiguous, WFA satisfies the twins property. It runs in O(m 2 n 6 ) time. We then provide a worst-case time complexity analysis of DTA. The number of states in the output deterministic WFA is at most 2 n(2 lg n+n 2 lg j\Sigmaj+1) , where \Sigma is the input alphabet. If the weights are rational, this bound becomes 2 n(2 lg n+1+min(n 2 lg j\Sigmaj;ae)) , where ae is the maximum bit-size of a weight. When the input WFA is acyclic, the bound becomes which is tight (up to constant factors) for any alphabet size. In Sections 4-6 we study questions motivated by the use of WFA determinization in ASR [19, 20]. Although determinization causes exponential state-space expansion in the worst case, in ASR systems the determinized WFAs are often smaller than the input WFAs [17]. This is fortuitous, because the performance of ASR systems depends directly on WFA size [19, 20]. We study why such size reductions occur. The folklore explanation within the ASR community credits special topology-the underlying directed graph, ignoring weights-for this phenomenon. ASR WFAs tend to be multipartite and acyclic. Such a WFA always admits a deterministic equivalent. In Section 4 we exhibit multi-partite, acyclic WFAs whose minimum equivalent deterministic WFAs are exponentially larger. In Section 5 we study a class of WFAs, RG, with a simple multi-partite, acyclic topology, such that in the absence of weights the deterministic equivalent is smaller. We show that for any A 2 RG and any i - n, there exists an assignment of weights to A such that the minimal equivalent deterministic WFA has states. Using ideas from universal hashing, we show that similar results hold when the weights are random i-bit numbers. We call a WFA weight-dependent if its expansion under determinization is strongly determined by its weights. We examined experimentally the effect of varying weights on actual WFAs from ASR applications. In Section 6 we give results of these experiments. Most of the ASR examples were weight-dependent. These experimental results together with the theory we develop show that the folklore explanation is insufficient: ASR WFAs shrink under determinization because both the topology and weighting tend to be favorable. Some of our results help explain the nature of WFAs from the algorithmic point of view, i.e., how weights assigned to the transitions of a WFA can affect the performance of algorithms manipulating it. Others relate directly to the theory of weighted automata. Definitions and Terminology Given a semiring (K; weighted finite automaton (WFA) is a tuple is the set of states, - q 2 Q is the initial state, \Sigma is the set of symbols, ffi ' Q \Theta \Sigma \Theta K \Theta Q is the set of transitions, and Q f ' Q is the set of final states. We assume throughout that j\Sigma j ? 1. A deterministic, or sequential, WFA has at most one transition WFA can have multiple transitions on a pair (q 1 ; oe), differing in target state q 2 . The problems examined in this paper are motivated primarily by ASR applications, which work with the min-sum semiring, Furthermore, some of the algorithms considered use subtraction, which the min-sum semiring admits. We thus limit further discussion to the min-sum semiring. Consider a sequence of transitions t induces string String w is accepted by t if q q and q ' 2 accepted by G if some t accepts w. Let c(t i be the weight of t i . The weight of t is (w) be the set of all sequences of transitions that accept string w. The weight of w is The weighted language of G is the set of weighted strings accepted by G: accepted by Gg : Intuitively, the weight on a transition of G can be seen as the "confidence" one has in taking that transition. The weights need not, however, satisfy stochastic constraints, as do the probabilistic automata introduced by Rabin [22]. Fix two states q and q 0 and a string v 2 \Sigma . Then c(q; v; q 0 ) is the minimum of taken over all transition sequences from q to q 0 generating v. We refer to c(q; v; q 0 ) as the optimal cost of generating v from q to q 0 . We generally abuse notation so that ffi(q; w) can represent the set of states reachable from state q 2 Q on string w 2 \Sigma . We extend the function ffi to strings in the usual way: q means that there is a sequence of transitions from q to q 0 generating v. The topology of G, top(G), is the projection -Q\Theta\Sigma \ThetaQ (ffi): i.e., the transitions of G without respect to the weights. We also refer to top(G) as the graph underlying G. A WFA is trim if every state appears in an accepting path for some string and no transition is weighted - 0 (1 in the min-sum semiring). A WFA is unambiguous if there is exactly one accepting path for each accepted string. Determinization of G is the problem of computing a deterministic WFA G 0 such that if such a G 0 exists. We denote the output of algorithm DTA by dta(G). We denote the minimal deterministic WFA accepting L(G) by min(G), if one exists. We say that G expands if dta(G) has more states and/or transitions than G. let the size of G be n m. We assume that each transition is labeled with exactly one symbol, so j\Sigma j - m. Recall that the weights of G are non-negative real numbers. Let C be the maximum weight. In the general case, weights are incommensurable real numbers, requiring "infinite precision." In the integer case, weights can be represented with bits. We denote the integral range [a; b] by [a; b] Z . The integer case extends to the case in which the weights are rationals requiring ae bits. We assume that in the integer and rational cases, weights are normalized to remove excess least-significant zero bits. For our analyses, we use the RAM model of computation as follows. In the general case, we charge constant time for each arithmetic-logic operation involving weights (which are real numbers). We refer to this model as the !-RAM [21]. The relevant parameters for our analyses are n, m, and j\Sigma j. In the integer case, we also use a RAM, except that each arithmetic-logic operation now takes O(ae) time. We refer to this model as the CO-RAM [1]. The relevant parameters for the analyses are n, m, j\Sigma j, and ae. 3 Determinization of WFAs 3.1 An Algorithm for Testing the Twins Property Definition 1. Two states, q and q 0 , of a WFA G are twins if 8(u; v) 2 (\Sigma ) 2 such that q 2 ffi( - the following holds: c(q; v; has the twins property if all pairs q; q 0 2 Q are twins. That is, if states q and q 0 are reachable from - q by a common string, then q and q 0 are twins only if any string that induces a cycle at each induces cycles of equal optimal cost. Note that two states having no cycle on a common string are twins. G be a trim, unambiguous WFA. G has the twins property if and only if 8(u; v) 2 (\Sigma ) 2 such that juvj - 2n the following holds: when there exist two states q and q 0 such that (i) fq; q 0 g ' ffi( - q; u), and (ii) q 2 ffi(q; v) and must follow. are analogous to those stated by Choffrut [7, 8] and (in different terms) by Weber and Klemm [27] to identify necessary and sufficient conditions for a string-to-string transducer to admit a sequential transducer realizing the same rational transduction. The proof techniques used for WFAs differ from those used to obtain analogous results for string-to-string transducers, however. In particular, the efficient algorithm we derive here to test a WFA for twins is not related to that of Weber and Klemm [27] for testing twins in string-to-string transducers. We define T - q;-q , a multi-partite, acyclic, labeled, weighted graph having 2n 2 layers, as follows. The root vertex comprises layer zero and corresponds to (-q; - q). For i ? 0, given the vertices at layer i \Gamma 1, we obtain the vertices at layer i as follows. Let u be a vertex at layer corresponding to (q 1 connected to u 0 , corresponding to (q 0 2 ), at layer i if and only if there are two distinct transitions and G. The arc connecting u to u 0 is labeled with a 2 \Sigma and has cost q;-q has at most 2n Let (q; q 0 ) i be the vertex corresponding to (q; q 0 at layer i of T - q;-q , if any. Let be the set of pairs of distinct states of G that are reachable from (-q; - q;-q . For each (q; q 0 analogously to T - q;-q . Fix two distinct states q and q 0 of G. Let (q; q 0 , be all the occurrences of (q; q 0 ) in T q;q 0 , excluding (q; q 0 ) 0 . This sequence may be empty. A symmetric sequence can be extracted from T q 0 ;q . We refer to these sequences as the common cycles sequences of (q; q 0 ). We say that q and q 0 satisfy the local twins property if and only if (a) their common cycles sequences are empty or (b) zero is the cost of (any) shortest path from (q; q 0 ) 0 to (q; q 0 and from (q to Lemma 2. Let G be a trim, unambiguous WFA. G satisfies the twins property if and only if (i) RT is empty or (ii) all (q; q 0 the local twins property. Proof (Sketch). We outline the proof for the sufficient condition. The only nontrivial case is when some states in RT satisfy the local twins property and their common cycles sequences are not empty. Let RT 0 be such a set. Assume that G does not satisfy the twins property. We derive a contradiction. Since RT 0 is not empty, we have that the set of pairs of states for which (i) and (ii) are satisfied in Lemma 1 is not empty. But since G does not satisfy the twins property, there must exist two states q and q 0 and a string 1, such that (i) both q and q 0 can be reached from the initial state of G through string u; (ii) q 2 ffi(q; v) and q loss of generality, assume that Now, one can show that (q; q 0 using the fact that G is unambiguous, one can show that there is exactly one path in T q;q 0 from the root to (q; q 0 ) jvj with cost the local twins property. To test whether a trim, unambiguous WFA has the twins property, we first compute q;-q and the set RT . For each pair of states (q; q 0 that has not yet been processed, we need only compute T q;q 0 and T q 0 ;q and their respective shortest path trees. Theorem 1. Let G be a trim unambiguous WFA. In the general case, whether G satisfies the twins property can be checked in O(m using the !-RAM. In the integer case, the bound becomes O(aem using the CO-RAM. 3.2 The In this section we describe the algorithm. We then give an upper bound on the size of the deterministic machines produced by the algorithm. The results of Section 5 below show that our upper bound is tight to within polynomial factors. Given WFA generalizes the classic power-set construction to construct deterministic WFA G 0 as follows. The start state of G 0 is f(-q; 0)g, which forms an initial queue P . While P 6= ;, pop state from P , where q 1g. The r i values encode path-length infor- mation, as follows. For each oe 2 \Sigma , let fq 0 m g be the set of states reachable by oe-transitions out of all the q i . For be the minimum of the weights of oe-transitions into q 0 j from the q i plus the respective g. Let q m. We add transition (q; oe; ae; q 0 ) to G 0 and push q 0 onto P if q 0 is new. This is the only oe-transition out of state q, so G 0 is deterministic. Let TG (w) be the set of sequences of transitions in G that accept a string w 2 \Sigma ; let t G 0 (w) be the (one) sequence of transitions in G 0 that accepts the same string. Mohri [17] shows that c(t G 0 let be the set of sequences of transitions in G from state - q to state q that induce string w. Again, let t G 0 (w) be the (one) sequence of transitions in G 0 that induces the same string; t G 0 (w) ends at some state f(q 1 such that some shows that c(t G Thus, each r i is a remainder that encodes the difference between the weight of the shortest path to some state that induces w in G and the weight of the path inducing w in G 0 . Hence at least one remainder in each state must be zero. 3.3 Analyzing We first bound the number of states in dta(G), denoted #dta(G). Theorem 2. If WFA G has the twins property, then in the general case; in the integer (or rational) case; and #dta(G) ! 2 n lg j\Sigmaj if G is acyclic, independent of any assumptions on weights. The acyclic bound is tight (up to constant factors) for any alphabet. Proof (Sketch). Let ~ R be the set of remainders in dta(G). Let R be the set of remainders r for which the following holds: 9w 2 \Sigma , jwj - and two states q 1 and q 2 , such that )j. The twins property implies that ~ R ' R. In the worst case, each i-state tuple from G will appear in dta(G), and there are j ~ i-tuples of remainders it can assume. (This over counts by including tuples without any zero remainders.) Therefore, #dta(G) - General Case: Each string of length at most n reach a pair of (not necessarily distinct) states in G. Therefore, jRj . Integer Case: The remainders in R are in [0; (n . Acyclic Case: #dta(G) is bounded by the number of strings in the weighted language accepted by G, which is bounded by j\Sigma j n . We discuss tightness in Section 5. Processing each tuple of state-remainders generated by time, excluding the cost of arithmetic and min operations, yielding the following. Theorem 3. Let G be a WFA satisfying the twins property. In the general case, takes O(j\Sigmaj(n on the !-RAM. In the (rational or) integer case, on the CO- RAM. In the acyclic case, takes O(j\Sigmaj(n +m)2 n lg j\Sigmaj ) time on the !-RAM and O(aej\Sigmaj(n +m)2 n lg j\Sigmaj ) time on the CO-RAM. We can use the above results to generate hard instances for any determinization algorithm. A reweighting function (or simply reweighting) f is such that, when applied to a WFA G, it preserves the topology of G but possibly changes the weights. We want to determine a reweighting f such that min(f(G)) exists and jmin(f(G))j is maximized among reweightings for which min(f(G)) exists. We restrict attention to the integer case and, without loss of generality, we assume that G is trim and unambiguous. Theorem 2 shows that for weights to affect the growth of dta(G), it must be that ae - n 2 lg j\Sigma j. Set ae To find the required reweight- ing, we simply consider all possible reweightings of G satisfying the twins property and requiring at most ae max bits. There are (2 ae possible reweightings, and it takes 2 O(n(2 lg n+(n 2 lg j\Sigmaj))) time to compute the expansion or decide that the resulting machine cannot be determinized, bounding the total time by 4 Hot Automata This section provides a family of acyclic, multi-partite WFAs that are hot: when de- terminized, they expand independently of the weights on their transitions. Given some alphabet an g, consider the language i.e., the set of all n-length strings that do not include all symbols from \Sigma . It is simple to obtain an acyclic, multi-partite NFA H of poly(n) size that accepts L. It is not hard to show that the minimal DFA accepting L has \Theta(2 n+lg n ) states. Furthermore, we can construct H so that these bounds hold for a binary alphabet. H corresponds to a WFA with all arcs weighted identically. Since acyclic WFAs satisfy the twins property, they can always be determinized. Altering the weights can only increase the expansion. Kintala and Wotschke [15] provide a set of NFAs that produces a hierarchy of expansion factors when determinized, providing additional examples of hot WFAs. 5 Weight-Dependent Automata In this section we study a simple family of WFAs with multi-partite, acyclic topology. We examine how various reweightings affect the size of the determinized equivalent. This family shrinks without weights, so any expansion is due to weighting. This study is related in spirit to previous works on measuring nondeterminism in finite automata [13,15]. Here, however, nondeterminism is encoded only in the weights. We first discuss the case of a binary alphabet and then generalize to arbitrary alphabets. 5.1 The Rail Graph We denote by RG(k) the k-layer rail graph. RG(k) has 2k g. There are arcs (0; See Fig. 1. RG(k) is 1)-partite and also has fixed in- and out-degrees. If we consider the strings induced by paths from 0 to either T k or B k , then the language of RG(k) is the set of strings LRG . The only nondeterministic choice is at the state 0, where either the top or bottom rail may be selected. Hence a string w can be accepted by one of two paths, one following the top rail and the other the bottom rail. a a a a a a a a a a Fig. 1. Topology of the k-layer rail graph. Technically, RG(k) is ambiguous. We can disambiguate RG(k) by adding transitions from T k and B k , each on a distinct symbol, to a new final state. Our results extend to this case. For clarity of presentation, we discuss the ambiguous rail graph. The rail graph is weight-dependent. In Section 5.2 we provide weightings such that produces the (k+1)-vertex trivial series-parallel graph: a graph on k+1 vertices, with transitions, on all symbols, only between vertices i and i+1, for 1 - i - k. On the other hand, in Section 5.3 we exhibit weightings for the rail graph that cause DTA to produce exponential state-space expansions. We also explore the relationship between the magnitude of the weights and the amount of expansion that is possible. In Section 5.4, we show that random weightings induce the behavior of worst-case weightings. Finally, in Section 5.5 we generalize the rail graph to arbitrary alphabets. 5.2 Weighting RG(k) Consider determinizing RG(k) with DTA. The set of states reachable on any string g. For a given weighting function c, let c T (w) denote the cost of accepting string w if the top path is taken; i.e., c T Analogously define c B (w) to be the corresponding cost along the bottom path. Let R(w) be the remainder vector for w, which is a pair of the form (0; c B 0). A state at layer 0 in the determinized WFA is labeled (fT string w leading to that state. Thus, two strings w 1 and w 2 of identical length lead to distinct states in the determinized version of the rail graph if and only if R(w 1 It is convenient simply to write (w). The sign of R(w) then determines which of the two forms (0; x) or (x; 0) of the remainder vector occurs. denote the weight on the top (rsp., bottom) arc labeled oe into vertex T Theorem 4. There is a reweighting f such that which consists of the series-parallel graph Proof. Any f for which suffices, since in this case R(w 1 g. In particular, giving zero weights suffices. 5.3 Worst-Case Weightings of RG(k) Theorem 5. For any j 2 [0; k] Z there is a reweighting f such that layers 0 through j of dta(f(RG(k))) form the complete binary tree on 2 vertices. Proof (Sketch). Choose any weighting such that Consider a pair of strings w identical length such that w 1 6= w 2 . The weighting ensures that R(w 1 Theorem 6. For any j 2 [0; k] Z there is a reweighting f such that layers 0 through the complete binary tree on vertices. Theorem 6, generalized by Theorem 10, shows that weight-dependence is not an artifact of DTA and that the acyclic bound of Theorem 2 is tight for binary alphabets. We now address the sensitivity of the size expansion to the magnitude of the weights, arguing that exponential state-space expansion requires exponentially big weights for the rail graph. (This means that the size expansion, while exponential in the number of states, is only super-polynomial in the number of bits.) Theorem 7. Let f be a reweighting. If jdta(f(RG(k)))j are required to encode f(RG(k)). Proof (Sketch). There must be\Omega remainders among the states at depth k in the determinized WFA, necessitating\Omega distinct permutations of the d k bits among them. Thus\Omega (k) weights must have similarly high-order bits set. Corollary 1. Let f be a reweighting. If jmin(f(RG(k)))j are required to encode f(RG(k)). 5.4 Random Weightings of RG(k) Theorem 8. Let G be RG(k) weighted with numbers chosen independently and uniformly at random from [1; denotes the expected value of the random variable X . Theorem 9. Let G be RG(k) weighted with logarithms of numbers chosen independently and uniformly at random from [1; The proofs of Theorems 8 and 9 use the observation that the random functions defined by RG are essentially universal hash functions [6] to bound sufficiently low the probability that the remainders of two distinct strings are equal. Theorem 9 is motivated by the fact that the weights of ASR WFAs are negated log probabilities. Extending RG(k) to Arbitrary Alphabets We can extend the rail graph to arbitrary alphabets, defining RG(r; k), the k-layer r- rail graph, as follows. RG(r; . Assume the alphabet is for all 1 - The subgraph induced by vertex 0 and vertices v i comprises rail i of RG(r; k). The subgraph induced by vertices and some j comprises layer j of RG(r; k). Vertex 0 comprises layer 0 of RG(r; k). Thus, RG(2; k) is the k-layer rail graph, RG(k), defined in Section 5.1. Let c(i; j; s) be the weight of the arc labeled s into vertex v i Theorems 4 and 5 generalize easily to the k-layer r-rail graphs. Theorem 6 generalizes to RG(r; follows, showing that the acyclic bound of Theorem 2 is tight for arbitrary alphabets. Theorem 10. For any j 2 [0; k] Z there is a reweighting f such that layers 0 through the complete r-ary tree on r j \Gamma1 vertices. Proof (Sketch). Choose the following weighting. Set c(i; '; for all 1 - Given two strings, w 1 6= w 2 , such that jw 1 j, we can show that w 1 and w 2 must lead to different vertices in any deterministic realization, D, of RG(r; k). Assume that w 1 and w 2 lead to the same vertex in D. Let c d (w) be the cost of string w in D. Given any suffix s of length k \Gamma ', we can show that c(w 1 c d (w 1 The right hand side is a fixed value, \Delta. Consider any position i - ' in which w 1 and w 2 differ. Denote the ith symbol of string w by w(i). Consider two suffixes, s 1 and s 2 , of length (i). Observe that the given weighting on RG(r; forces the minimum cost path for any string with some symbol oe in position j to follow rail (r\Gammaoe). Thus, We can use this to show that c(w 1 6 Experimental Observations on ASR WFAs To determine whether ASR WFAs manifest weight dependence, we experimented on 100 WFAs generated by the AT&T speech recognizer [23], using a grammar for the Air Travel Information System (ATIS), a standard test bed [4]. Each transition was labeled with a word and weighted by the recognizer with the negated log probability of realizing that transition out of the source state; we refer to these weights as speech weights. We determinized each WFA with its speech weights, with zero weights, and with weights assigned independently and uniformly at random from [0; 2 i \Gamma1] Z (for each 0 - could not be determinized with speech weights due to computational limitations, and it is omitted from the data. Figure 2(a) shows how many WFAs expanded when determinized with different weightings. Figure 2(b) classifies the 63 WFAs that expanded with at least one weight- ing. For each WFA, we took the weighting that produced maximal expansion. This was usually the 8-bit random weighting, although due to computational limitations we were unable to determinize some WFAs with large random weightings. The x-axis indicates the open interval within which the value lg(jdta(G)j=jGj) falls. The utility of determinization in ASR includes the reduction in size achieved with actual speech weights. In our sample, 82 WFAs shrank when determinized. For each, we computed the value lg(jGj=jdta(G)j), and we plot the results in Fig. 2(c). In Fig. 2(d), we examine the relationship between the value lg(jdta(G)j=jGj) and the number of bits used in random weights. We chose the ten WFAs with highest final expansion value and plotted lg(jdta(G)j=jGj) against the number of bits used. For reference the functions i are plotted, where i is the number of bits. Most speech zerosbit rndbit rndbit rndbit rndbit rndbit rndbit rndbit rnd Type of weighting2060 Number of WFAs that expand (a) Log base 2 of expansion factor515 Number of WFAs (b) Log base 2 of shrinkage Number of WFAs (c) Number of random bits26Log of expansion factor q0v004 (d) Fig. 2. Observations on ASR WFAs. of the WFAs exhibit subexponential growth as the number of bits increases, although some, like q0t063 have increased by 128 times even with four random bits. The WFA that could not be determinized with speech weights was "slightly hot," in that the determinized zero-weighted variant had 2.7% more arcs than the original WFA. The remaining ninety-nine WFAs shrank with zero weights: none was hot. If one expanded, it did so due to weights rather than topology. Figure 2(a) indicates that many of the WFAs have some degree of weight dependence Figure 2(d) suggests that random weights are a good way to estimate the degree to which a WFA is weight dependent. Note that the expansion factor is some superlin- possibly exponential, function of the number of random bits, suggesting that large, e.g., 32-bit, random weights should cause expansion if anything will. Analogous experiments on the minimized determinized WFAs yield results that are qualitatively the same, although fewer WFAs still expand after minimization. Hence weight dependence seems to be a fundamental property of these WFAs rather than an artifact of DTA. Acknowledgements . We thank Mehryar Mohri, Fernando Pereira, and Antonio Restivo for fruitful discussions. --R Network Flows: Theory Rational Series and Their Languages. Algorithmic aspects in speech recognition: An intro- duction Universal classes of hash functions. Une caracterisation des fonctions sequentielles et des fonctions sous- sequentielles en tant que relations rationnelles Finite automata computing real functions. On computational power of weighted finite automata. On measuring nondeterminism in regular languages. Arithmetic coding of weighted finite automata. Amounts of nondeterminism in finite automata. On the use of sequential transducers in natural language processing. Speech recognition by composition of weighted finite automata. Weighted rational transductions and their application to human language processing. Computational Geometry: An Introduction. Probabilistic automata. The AT&T Analyse Syntaxique Transformationelle du Francais par Transducteurs et Lexique- Grammaire Economy of description for single-valued transducers --TR --CTR Mark G. Eramian, Efficient simulation of nondeterministic weighted finite automata, Journal of Automata, Languages and Combinatorics, v.9 n.2-3, p.257-267, September 2004 Bjrn Borchardt, A pumping lemma and decidability problems for recognizable tree series, Acta Cybernetica, v.16 n.4, p.509-544, 2004 Julien Quint, On the equivalence of weighted finite-state transducers, Proceedings of the ACL 2004 on Interactive poster and demonstration sessions, p.23-es, July 21-26, 2004, Barcelona, Spain Manfred Droste , Dietrich Kuske, Skew and infinitary formal power series, Theoretical Computer Science, v.366 n.3, p.199-227, 20 November 2006 Bjrn Borchardt , Heiko Vogler, Determinization of finite state weighted tree automata, Journal of Automata, Languages and Combinatorics, v.8 n.3, p.417-463, 06/01/2003 Manfred Droste , Paul Gastin, Weighted automata and weighted logics, Theoretical Computer Science, v.380 n.1-2, p.69-86, June, 2007 Cyril Allauzen , Mehryar Mohri, Efficient algorithms for testing the twins property, Journal of Automata, Languages and Combinatorics, v.8 n.2, p.117-144, April Ines Klimann , Sylvain Lombardy , Jean Mairesse , Christophe Prieur, Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton, Theoretical Computer Science, v.327 n.3, p.349-373, 2 November 2004 Sylvain Lombardy , Jacques Sakarovitch, Sequential?, Theoretical Computer Science, v.356 n.1, p.224-244, 5 May 2006 Manfred Droste , Heiko Vogler, Weighted tree automata and weighted logics, Theoretical Computer Science, v.366 n.3, p.228-247, 20 November 2006

rational functions and power series;algorithms;speech recognition;weighted automata

586945

On the Relative Complexity of Resolution Refinements and Cutting Planes Proof Systems.

An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refutations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and cutting planes. In both cases only superpolynomial separations were known [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467; J. Johannsen, Inform. Process. Lett., 67 (1998), pp. 37--41; P. Clote and A. Setzer, in Proof Complexity and Feasible Arithmetics, Amer. Math. Soc., Providence, RI, 1998, pp. 93--117]. In order to prove these separations, the lower bounds on the depth of monotone circuits of Raz and McKenzie in [ Combinatorica, 19 (1999), pp. 403--435] are extended to monotone real circuits.An exponential separation is also proved between tree-like resolution and several refinements of resolution: negative resolution and regular resolution. Actually, this last separation also provides a separation between tree-like resolution and ordered resolution, and thus the corresponding superpolynomial separation of [A. Urquhart, Bull. Symbolic Logic, 1 (1995), pp. 425--467] is extended. Finally, an exponential separation between ordered resolution and unrestricted resolution (also negative resolution) is proved. Only a superpolynomial separation between ordered and unrestricted resolution was previously known [A. Goerdt, Ann. Math. Artificial Intelligence, 6 (1992), pp. 169--184].

Introduction The motivation to work on the proof length of propositional proof systems is double. First, by the work of Cook and Reckhow [15] we know that the claim that for every propositional proof system there is a class of tautologies with no polynomial-size proofs is equivalent to NP 6= co-NP . This connection explains the interest in developing combinatorial techniques to prove lower bounds for dierent proof systems. The second motivation comes from the interest in studying eciency issues in Automated Theorem Proving. The question is which proof systems have ecient algorithms to nd proofs. Actually the proof system most widely used for implementations is resolution or restrictions of resolution. Our work is relevant to both motivations. On one hand, all the separation results of this paper improve to exponential the previously known superpolynomial ones. On the other hand these exponential separations harden the known results showing ineciency of several widely used strategies for nding proofs, especially for the resolution system. A preliminary version of this paper appeared as [8] and as ECCC TR98-035. y Departament de Llenguatges i Sistemes Informatics, Universitat Politecnica de Catalunya, fbonet,esteban,galesig@lsi.upc.es z Partially supported by projects SPRIT 20244 ALCOM-IT, TIC 98-0410-C02-01 and PB98-0937-C04-03. x Partially supported by project KOALA:DGICYT:PB95-0787. { Supported by an European Community grant under the TMR project. k Institut fur Informatik, Ludwig-Maximilians-Universitat Munchen, jjohanns@informatik.uni-muenchen.de. Research of this author done at Department of Mathematics, University of California, San Diego, supported by DFG grant No. Jo 291/1-1. Haken [21] was the rst in proving exponential lower bounds for unrestricted resolution. He showed that the Pigeonhole Principle requires exponential-size resolution refutations. Later Urquhart [35] found another class of tautologies with the same property. Chvatal and Szemeredi [11] showed that in some sense, almost all classes of tautologies require exponential size resolution proofs (see [4, 5] for simplied proofs of these results). These exponential lower bounds are bad news for Automated Theorem Proving, since they mean that often the time used in nding proofs will be exponentially long in the size of the tautology, just because the shortest proofs are also exponentially long in the size of the tautology. A natural question then is what happens with the classes of tautologies with polynomial-size proofs. Can we nd these proofs eciently? Several authors [4, 12, 5] have given weakly exponential (in the minimal proof size) time algorithms for nding resolution proofs. The question obviously is whether these results can be improved or not. Formally, we say that a propositional proof system S is automatizable if there is an algorithm that for every tautology F , nds a proof of F in S in time polynomial in the length of the shortest proof of F in S. The only propositional proof systems that are known to be automatizable are algebraic proof systems like Hilbert's Nullstellensatz [2] and Polynomial Calculus [12]. On the other hand bounded-depth Frege proof systems are not automatizable, assuming factoring is hard [29, 10, 7]. Since Frege systems and Extended Frege systems polynomially simulate bounded-depth Frege systems, they are also not automatizable under the same assumptions. Note that automatizability is equivalent to the approximability to within a polynomial factor of the following optimization problem: Given a proof of some tautology, nd its minimal size proof . Iwama [23] and Alekhnovich et al. [1] show that it is NP -hard to approximate this problem to within a linear factor, for most of the commonly studied proof systems. Many strategies for nding resolution proofs are described in the literature, see e.g. Schoning's textbook [34]. One commonly used type of strategy is to reduce the search space by dening restricted versions of resolution that are still complete. Such restricted forms are commonly referred to as resolution renements. One particularly important resolution renement is tree-like resolution. Its importance stems from the close relationship between the complexity of tree-like resolution proofs and the runtime of a certain class of satisability testing algorithms, the so-called Algorithms (cf. [31, 3]). We prove an exponential separation between tree-like resolution and unrestricted resolution (Corollary 20), thus showing that nding tree-like resolution proofs is not an ecient strategy for nding resolution proofs. Until now only superpolynomial separations were known [36, 13]. In this paper, we consider three more of the most commonly used resolution renements: negative resolution, regular resolution and ordered resolution. We show an exponential separation between tree-like resolution and each one of the above restrictions (Corollary 20 for negative resolution and Corollary 23 for both regular and ordered resolution). Goerdt [19, 18, 20] gave several superpolynomial separations between unrestricted resolution and some renements, in particular he gave a superpolynomial separation between ordered resolution and unrestricted resolution. We improve this result by giving an exponential separation between ordered and negative resolution (Corollary 28), thus showing that unrestricted resolution can have an exponential speed-up over ordered resolution. The Cutting Planes proof system, CP from now on, is a refutation system based on manipulating integer linear inequalities. Exponential lower bounds for the size of CP refutations are already proven. Impagliazzo et al. [22] proved exponential lower bounds for tree-like CP . Bonet et al. [9] proved a lower bound for the subsystem CP , where the coecients appearing in the inequalities are polynomially bounded in the size of the formula being refuted. This is a very important result because all known CP refutations fulll this property. Finally, Pudlak [30] and Cook and Haken [14] gave general circuit complexity results from which exponential lower bounds for CP follow. To this day it is still unknown whether CP is more powerful than CP , i.e., whether it produces shorter proofs or not. Nothing is known about automatizability of CP proofs. Since there is an exponential speed-up of CP over resolution, it would be nice to nd an ecient algorithm for nding CP proofs. A question to ask is whether trying to nd tree-like CP proofs would be an ecient strategy for nding Cutting Planes proofs. Johannsen [24] gave a superpolynomial separation, with a lower bound of the log n ), between tree-like CP and dag-like CP (this was previously known for CP from [9]). Here we improve that separation to exponential (Corollary 20). This means that trying to nd tree-like proofs is also not a good strategy for nding proofs in CP . 1.1 Interpolation and Lower Bounds Interpolation is a technique for proving lower bounds for resolution and Cutting Planes systems. The name comes from a classical theorem of Mathematical Logic, the Craig's interpolation theorem. Krajcek [27] reformulated this classical theorem in order to use it to prove lower bounds for proof systems. Closely related ideas appeared previously in the mentioned works that gave lower bounds for fragments of CP ([22, 9]). The interpolation method translates proofs of certain formulas to circuits, preserving sizes. So it is a way to reduce the problem of proving proof complexity lower bounds to circuit complexity lower bounds. This is very important because in some cases there are strong circuit complexity lower bounds. In particular for monotone circuits, there are both size and depth lower bounds. The interpolation method works as follows. We consider a hard boolean function, i.e., one that requires exponential size monotone circuits to be computed. We dene a contradiction ~r), such that A(~p; ~q) expresses that ~ p is a minterm 1 of the function (the variables ~q describe the minterm), and B(~p; ~r) says that ~ p is a maxterm of the function (~r describes the max- term). We suppose that A(~p; ~q) ^B(~p;~r) has a subexponential size refutation in a system with the interpolation theorem (monotone version). Then by the theorem we can extract a subexponential size monotone circuit that computes the hard function. Since this is impossible, requires exponential size proofs. With the above method we can prove lower bounds for resolution and CP (see [9, 30, 27]). But to get an exponential lower bound for full CP we actually need an interpolation theorem that translates proofs into monotone real circuits [30]. See Section 3 for the denition of monotone real circuits and Theorem 17 for the interpolation theorem for Cutting Planes. The main body of this paper consists on exponential separations between tree-like and dag-like (general) versions of two proof systems, resolution and Cutting Planes. So far we have outlined the ideas in order to prove complexity lower bounds for systems like resolution and CP . In order to separate the tree-like versions from the general versions of the proof systems, we need to dene a contradiction that has polynomial size proofs in resolution and CP but for which we can prove exponential size lower bounds for the corresponding tree-like versions. The interpolation theorem applied on tree-like proofs gives rise to tree-like circuits (i.e., formu- las). Therefore we need exponential size lower bounds for monotone formulas, or equivalently, we need linear depth lower bounds for monotone circuits. Karchmer and Wigderson [26] proved an O(log 2 n) lower bound on the depth of monotone 1 Recall that a minterm (respectively a maxterm) of a boolean function f : f0; 1g n ! f0; 1g is a set of inputs such that for each y 2 f0; 1g n obtained from x by changing a bit from 1 to 0 (respectively by changing a bit from 0 to 1) it holds that circuits computing the st-connectivity function. Johannsen [24] extended this lower bound to real circuits, and using the interpolation theorem he proved a superpolynomial separation between tree-like and dag-like CP . Lower bounds on the depth of monotone boolean circuits of the order were given by Raz and McKenzie [32]. Here we extend their results to the case of monotone real circuits. Namely, we prove an lower bound on the depth of monotone real circuits computing a certain monotone function Gen n which is computable in polynomial time. This implies an lower bound on the size of monotone real formulas computing Gen n . Hence, by the interpolation theorem, we get the exponential separation of tree-like from dag-like Cutting Planes. The same ideas also separate tree-like from dag-like resolution. 1.2 Section Description The paper is organized as follows. In Section 2 we give the basic denitions of the proof systems considered in the paper. In Section 3 we dene monotone real circuits, and prove the depth lower bound for them. This is applied in Section 4 to prove the lower bounds for tree-like CP , giving exponential separations of tree-like CP from CP , tree-like resolution from resolution as well as from regular resolution, ordered resolution and negative resolution. Finally in Section 5 we prove the exponential lower bound for ordered resolution, separating it from negative resolution. We conclude by stating some open problems. 2 The Proof Systems Resolution is a refutation proof system for CNF formulas, which are represented as sets of clauses, i.e., disjunctions of literals. We identify clauses in which the same literals occur, multiple occurrences and the order of appearance are disregarded. The only inference rule is the resolution rule: C _ D The clause C _ D is called the resolvent , and we say that the variable x is eliminated in this inference. A resolution refutation of some set of clauses is a derivation of the empty clause from using the above inference rule. Resolution is a sound and complete refutation system, i.e., a set of clauses has a resolution refutation if and only if it is unsatisable. Several renements of the resolution proof system, i.e., restricted forms that are still complete, have appeared in the literature. In this paper we consider the following three: 1. The regular resolution system: Viewing the refutations as graph, in any path from the empty clause to any initial clause, no variable is eliminated twice. 2. The ordered 2 resolution system: There exists an arbitrary ordering of the variables in the formula being refuted, such that if a variable x is eliminated before a variable y on any path from an initial clause to the empty clause, then x is before y in the ordering. As no variable is eliminated twice on any path, ordered resolution is a restriction of regular resolution. 3. The negative resolution system (N-resolution for short): To apply the resolution rule, one of the two clauses should consists only of negative literals. In Goerdt's paper [18] and in the preliminary version [8] of this paper, this renement is called Davis-Putnam resolution. In the meantime, we have learned that it is more commonly known as ordered resolution. In a tree-like proof any line in the proof can be used only once as a premise. Should the same line be used twice, it must be rederived. A proof system that only produces tree-like proofs is called tree-like. Otherwise we will call it dag-like, or just skip the adjective. When nothing is said it is understood that the system is dag-like. There is an algorithm (see e.g. Urquhart [36]) that transforms a tree-like resolution proof into a possibly smaller regular tree-like resolution proof. Therefore tree-like resolution proofs of minimal size are regular. That means that under the viewpoint of proof system complexity, tree-like resolution and tree-like regular resolution are polynomially equivalent. The Cutting Planes proof system, CP for short, is a refutation system for CNF formulas, as resolution. It works with linear inequalities. The initial clauses are transformed into linear inequalities in the following way: _ _ A CP refutation of a set E of inequalities is a derivation of 0 1 from the inequalities in E and the axioms x 0 and x 1 for every variable x, using the CP rules which are basic algebraic manipulations, additions of two inequalities, multiplication of an inequality by a positive integer and the following division rule: i2I a i x i k i2I a i where b is a positive integer that evenly divides all a i , It can be shown that a set of inequalities has a CP -refutation i it has no f0; 1g-solution. Any assignment satisfying the original clauses is actually a f0; 1g-solution of the corresponding inequalities, provided that we assign the numerical value 1 to True and the value 0 to False. It is also well-known that CP can polynomially simulate resolution [16], and this simulation preserves tree-like proofs. 3 Monotone Real Circuits A monotone real circuit is a circuit of fan-in 2 computing with real numbers where every gate computes a nondecreasing real function. This class of circuits was introduced by Pudlak [30]. We require that monotone real circuits output 0 or 1 on every input of zeroes and ones only, so that they are a generalization of monotone boolean circuits. The depth and size of a monotone real circuit are dened as usual, and we call it a formula if every gate has fan-out at most 1. Lower bounds on the size of monotone real circuits were given by Pudlak [30], Cook and Haken [14] and Fu [17]. Rosenbloom [33] shows that they are strictly more powerful than monotone boolean circuits, since every slice function can be computed by a linear size, logarithmic depth monotone real circuit, whereas most slice functions require exponential size general boolean circuits. On the other hand, Jukna [25] gives a general lower bound criterion for monotone real circuits, and uses it to show that certain functions in P=poly require exponential size monotone real circuits. Hence the computing power of monotone real circuits and general boolean circuits is incomparable. For a monotone boolean function f , we denote by dR (f) the minimal depth of a monotone real circuit computing f , and by s R (f) the minimal size of a monotone real formula computing f . The method of proving lower bounds on the depth of monotone boolean circuits using communication complexity was used by Karchmer and Wigderson [26] to give an n) lower bound on the monotone depth of st-connectivity. Using the notion of real communication complexity introduced by Krajcek [28], Johannsen [24] proved the same lower bound for monotone real circuits. In the case of boolean circuits the Karchmer-Wigderson result was generalized by Raz and McKenzie [32]. Consider the monotone function Gen n of n 3 inputs t a;b;c , 1 a; b; c n is dened as follows: for c n, we dene the relation ' c (c is generated) recursively by or there are a; b n with ' a Finally Gen n ( ~ From now on we will write a; b ' c for t Raz and McKenzie [32] proved a lower bound of the on the depth of monotone boolean circuits computing Gen n . By a modication of their method we show that this result also holds for monotone real circuits: Theorem 1. For some > 0 and suciently large n dR (Gen n ) This section is dedicated entirely to the proof of the above theorem. In the next section (Section we will see how to use the lower bounds provided by Theorem 1 to obtain lower bounds for the complexity of proofs in resolution and Cutting Planes proof systems. 3.1 Real Communication Complexity Let R X Y Z be a multifunction, i.e. for every pair (x; y) 2 X Y , there is a z 2 Z with We view such a multifunction as a search problem, i.e., given input (x; y) 2 X Y , the goal is to nd a z 2 Z such that (x; A deterministic communication protocol P over XY Z species the exchange of information bits between two players, I and II, that receive as inputs respectively x 2 X and y 2 Y and nally agree on a value P (x; y) 2 Z such that (x; R. The deterministic communication complexity of R, CC(R), is the number of bits communicated between players I and II in the optimal protocol for R. A real communication protocol over X Y Z is executed by two players I and II who exchange information by simultaneously playing real numbers and then comparing them according to the natural order of R. This generalizes ordinary deterministic communication protocols in the following way: in order to communicate a bit, the sender plays this bit, while the receiver plays a constant between 0 and 1, so that he can determine the value of the bit from the outcome of the comparison. Formally, such a protocol P is specied by a binary tree, where each internal node v is labeled by two functions f I giving player I's move, and f II and each leaf is labeled by an element z 2 Z. On input (x; y) 2 X Y , the players construct a path through the tree according to the following rule: At node v, if f I (y), then the next node is the left son of v, otherwise the right son of v. The value P (x; y) computed by P on input (x; y) is the label of the leaf reached by this path. A real communication protocol P solves a search problem R X Y Z if for every (x; y) 2 holds. The real communication complexity CCR (R) of a search problem R is the minimal depth of a real communication protocol that solves R. For a natural number n, let [n] denote the set be a monotone boolean function, let X := f 1 (1) and Y := f 1 (0), and let the multifunction R f X Y [n] be dened by The Karchmer-Wigderson game for f is dened as follows: Player I receives an input x 2 X and Player II an input y 2 Y . They have to agree on a position i 2 [n] such that (x; Sometimes we will say that R f is the Karchmer-Wigderson game for the function f . There is a relation between the real communication complexity of R f and the depth of a monotone real circuit or the size of a monotone real formula computing f , similar to the boolean case: f be a monotone boolean function. Then CCR (R f ) dR (f) and CCR (R f ) log 3=2 s R (f) : For a proof see [28] or [24]. We will apply Lemma 2 to the boolean function Gen to prove a linear lower bound for dR (Gen) and an exponential lower bound for s R (Gen), from a lower bound for CCR (R Gen ). It is immediate to see that to establish Theorem 1 from Lemma 2, it suces to prove the following result: Theorem 3. For some > 0 and suciently large n CCR (R Genn ) Analogously to the case of [32], to prove Theorem 3 we will prove a more general result about real communication complexity. As in [32] we will introduce a class of special games, the DART games, and the measure of structured communication complexity. In the next subsection we prove that lower bounds for the real communication complexity of a relation R associated to a DART game can can be obtained proving lower bounds for the structured communication complexity of R (Theorem 4). 3.2 DART games and structured protocols Raz and McKenzie [32] introduced a special kind of communication games, called DART games, and a special class of communication protocols, the structured protocols, for solving them. For is the set of communication games specied by a relation R . I.e., the inputs for Player I are k-tuples of elements . I.e., the inputs for Player II are k-tuples of binary colorings y i of [m]. For all The relation R X Y Z dening the game only depends on e i.e., we can describe R(x; can be expressed as a DNF-Search-Problem, i.e., there exists a DNF- tautology FR dened over the variables e k such that Z is the set of terms of FR , and holds if and only if the term z is satised by the assignment (e A structured protocol for a DART game is a communication protocol for solving the search problem R, where player I gets input x 2 X, player II gets input y 2 Y , and in each round, player I reveals the value x i for some i, and II replies with y i . The structured communication complexity of R 2 DART(m; k), denoted by SC(R), is the minimal number of rounds in a structured protocol solving R. The main theorem of [32] showed that for suitable m and k, the deterministic communication complexity of a DART game cannot be much smaller than that of a structured protocol. We shall show the same for its real communication complexity. Obviously, a structured protocol solving R in r rounds can be simulated by a real communication protocol solving R in r (dlog me+1) rounds. Conversely, we will prove that the following holds: Theorem 4. Let m; k 2 N. For every relation R 2 DART(m; k), where m k 14 , CCR (R) SC(R) The proof of this theorem is the main technical result of this section and we dedicate to it the entire Subsection 3.3. As a rst corollary to Theorem 4, we observe that for DART games, real communication protocols are no more powerful than deterministic communication protocols. Corollary 5. Let m; k 2 N. For R 2 DART(m; Proof. CC(R) CCR (R) SC(R) m) CC(R)). In the rest of this subsection we show how to obtain the proof of Theorem 3 using Theorem 4. For g. Consider the communication game PyrGen(m; d) dened as follows: We regard the indices as elements of Pyr d , so that the inputs for the two players I and II in the PyrGen(m; d) game are respectively sequences of elements and we picture these as laid out in a pyramidal form with (1; 1) at the top and (d; j), 1 j d and the bottom. The goal of the game is to nd either an element colored 0 at the top of the pyramid, or an element colored 1 at the bottom of the pyramid, or an element colored 1 with the two elements below it colored 0. That is we have to nd indices (i; such that one of the following holds: 1. 2. y i;j 3. Observe that, setting e search problem can be dened as a DNF search problem given by the following DNF tautology: _ _ 1jd e d;j 3 Observe that w.l.o.g. we can assume that both players know the structure of the protocol of the game. Hence we can assume that at each round they both know what is the coordinate i of the inputs they have to talk about. Therefore they have no need to transmit the index i of this coordinate. Therefore, PyrGen(m; d) is a game in DART(m; d+1 ). The following reduction shows that the real communication complexity of the game PyrGen(m; d) is bounded by the real communication complexity of the Karchmer-Wigderson game for Gen n for a suitable n. The proof is taken from [32], we include it because we will have to refer to some details of it below. 2. Then CCR (PyrGen(m; d)) CCR (R Genn Proof. We prove that any protocol P solving the Karchmer-Wigderson game for Gen n can be used to solve the PyrGen(m; d) game. From their respective inputs for the PyrGen(m; d) game, Player I and II compute respectively a minterm and a maxterm 4 for Gen n and then apply the protocol We interpret the elements between 2 and n 1 as triples (i; j; k), where (i; Now player I computes from his input x : Pyr d ! [m] an input ~ t x to Gen n with Gen n ( ~ t x by setting the following: a 1;1 ; a 1;1 ' n a i+1;j ; a i+1;j+1 ' a i;j for (i; where a i;j := (i; j; x i;j ). This completely determines ~ t x . Likewise Player II computes from his input y : Pyr d ! (2 [m] ) a coloring col of the elements from [n] by setting From this, he computes an input ~ t y by setting a; b ' c i it is not the case that Obviously Gen Playing the Karchmer-Wigderson game for Gen n now yields a triple (a; b; c) such that a; b ' c in ~ t x and a; b 6' c in ~ t y . By denition of ~ t y , this means that by denition of ~ t x one of the following cases must hold: a d;j for some j d. By denition of col, y d;j . In this case, y 1;1 a i;j . Then we have y i;j In either case, the players have solved PyrGen(m; d) without any additional communication. A lower bound on the structured communication complexity of PyrGen(m; d) was proved in [32]: Lemma 7 (Raz/McKenzie [32]). SC(PyrGen(m; d)) d. A proof of Theorem 3 therefore follows immediately from the above results: 4 Recall the denition of minterm and maxterm from footnote 1. Proof of Theorem 3. Fix 28 . By Theorem 4 and Lemma 7, we get CCR (PyrGen(m; d)) d log m) . Recall that 2. Therefore Lemma 6 immediately implies the Theorem, taking 1From our Theorem 3 we obtain consequences for monotone real circuits analogous to those obtained in [32] for monotone boolean circuits. Denition (Pyramidal Generation). Let ~ t be an input to Gen n . We say that n is generated in a depth-d pyramidal fashion by ~ t if there is a mapping m : Pyr d ! [n] such that the following hold (recall that a; b ' c means t Observe that the reduction in the proof of Lemma 6 produces only inputs from Gen 1 n (1) which have the additional property that n is generated in a depth-d pyramidal fashion. Hence we can state the following strengthening of Theorem 1: Corollary 8. Let n; d and be as above. Every monotone real formula that outputs 1 on every input to Gen n for which n is generated in a depth-d pyramidal fashion, and outputs 0 on all inputs where Gen n is 0, has to be of size z n The other consequences drawn from Theorem 4 and Lemma 7 in [32] apply to monotone real circuits as well, e.g. we just state without proof the following result: Theorem 9. There are constants 0 < ; < 1 such that for every function d(n) n , there is a family of monotone functions f that can be computed by monotone boolean circuits of size n O(1) and depth d(n), but cannot be computed by monotone real circuits of depth less than d(n). The method also gives a simpler proof of the lower bounds in [24], in the same way as [32] simplies the lower bound of [26]. 3.3 Proof of Theorem 4: To prove Thm. 4, we rst need some combinatorial notions from [32] and some lemmas. Let A [m] k and 1 j k. For x be the number of 2 [m] such that A. Then we dene The following lemmas about these notions were proved in [32]: ([32]). For every A 0 A and 1 j k, Lemma 11 ([32]). Let 0 < < 1 be given. If for every 1 j k, AV every > 0 there is A 0 A with jA 0 j (1 )jAj and In particular, setting 14 , we get Corollary 12. If m k 14 and for every 1 j k, AV 14 , then there is A 0 A with 14 . This corollary is almost identical to the corresponding statement in [32]. Only the values of and have been slightly modied to improve the nal bound. For a relation R 2 DART(m; k), A X and B Y , let CCR (R; A; B) be the real communication complexity of R restricted to AB. Denition called an (; ; ')-game if the following conditions hold: 1. R 2 DART(m; k), 2. SC(R) ', 3. jAj 2 jXj and jBj 2 jY j, 4. T 14 . The following lemma and its proof are slightly dierent from the corresponding lemma in [32], because we use the strong notion of real communication complexity where [32] use ordinary communication complexity. The modication we apply is analogous to that introduced by Johannsen [24] to improve the result of Karchmer and Wigderson [26] to the case of real communication complexity. This modication will aect the proof of the rst point of the next lemma. Lemma 13. For every 1. if for every 1 j k, AV 14 , then there is an (+2; +1; ')-game (R with (R 2. if ' 1 and for some 1 j k, AV 14 , then there is an (R Proof of Lemma 13. For part 1, we rst show that CCR (R; A; B) > 0. Assume otherwise, then there is a term z in the DNF tautology FR dening R that is satised for every (x; y) 2 A B. Therefore y j denote the number of possible values of x j in elements of A, then this implies that jBj 2 mk . On the other hand, jBj 2 mk , hence it follows that , which is a contradiction since m 1 14 implies 14 . Let an optimal real communication protocol solving R restricted to AB be given. For a 2 A and b 2 B, let a and b be the real numbers played by I and II in the rst round on input a and b, respectively. W.l.o.g. we can assume that these are jAj distinct real numbers. Now consider a f0; 1g-matrix of size jAj jBj with columns indexed by the a and rows indexed by the b , both in increasing order, and where the entry in position ( a ; b ) is 1 if a > b and 0 if a b . Thus this entry determines the outcome of the rst round, when these numbers are played. It is now obvious that either the upper right quadrant or the lower left quadrant must form a monochromatic rectangle. Hence there are A A and B 0 B with jA j 1jAj and jB 0 j 1jBj such that R restricted to can be solved by a protocol with one round fewer than the original protocol. By Lemma 10 (1), AV DEG j 14 for every 1 j k, hence by Corollary 12 there is A 0 A with 4 jAj and T 14 . Thus (R; A is an For part 2 we proceed like in the proof of the corresponding lemma of [32], with the numbers slightly adjusted. Assume without loss of generality that k is the coordinate for which 14 . Let R 0 and R 1 be the restrictions of R in which the k-th coordinate xed to 0 and 1, respectively. Obviously, R 0 and R 1 are DART (m; k 1) relations, and therefore at least one of SC(R 0 ) and SC(R 1 ) is at least k 1. Assume without loss of generality that SC(R 0 ) k 1. We will prove that there are two sets A 0 such that the following properties hold: 14 (5) (R This means that there is a (+3 log m; +1; k 1)-game (R (R CCR (R; A; B) and this proves part 2 of Lemma 13. Given any set U [m] consider the sets AU [m] k 1 and BU (f0; 1g m associated to the set U by the following denition of [32]: there is an u 2 U such that there is a w 2 f0; 1g m such that The following two claims can be proved exactly as the corresponding Claims of [32] and we omit their proof. 14. For a random set U of size m 5 14 , with m 1000 14 , we have that 15. For a random set U of size m 5 14 , with m 1000 14 we have that 3 Moreover it is immediate to see that the same reduction used in Claim 6.3 of [32] also works for the case of real communication complexity. Therefore we get: 16. For every set U [m] (R Take a random set U which with probability greater than 1, satises both the properties of 14 and Claim 15, and dene A 0 := AU and B 0 := BU . This means that with probability at least 1both A Recall that jAj and that, by hypothesis on Part 2 of the lemma 14 . Therefore we have that This proves (3). For (4) observe that by Claim 15 we have The property (5) follows directly from Lemma 10 (2), and nally (6) follows from Claim 16. We nally end with the proof of Theorem 4 from Lemma 13. Proof of Theorem 4. Let k 2 N, k 1000. We prove that for any ; is such that CCR (R; A; B) ' log m4 Observe that by Denition of (; ; ')-game, when we have that Therefore CCR (R; A; (R). Moreover the right side of Equation 7 reduces to ' 493 m). Since by the same Denition ' SC(R), for we get the claim of the theorem: CCR (R) SC(R) m) To prove Equation 7, we proceed by induction on ' 1 and m 1=7 . In the base case ' < 1 (that is 7 , the inequality (7) is trivial, since the right hand side gets negative for large m. In the inductive step consider (R; A; B) be an (; ; ')-game, and assume that (7) holds for all . For sake of contradiction, suppose that CCR (R; A; B) < ' log m4 +. Then either for every 1 j k, AV 14 , and Lemma 13 gives an ( (R (R or for some 1 j k, AV 14 , then Lemma 13 gives an ( game (R (R log m4 log both contradicting the assumption. 4 Separation between tree-like and dag-like versions of Resolution and Cutting Planes Cutting Planes refutations are linked to monotone real circuits by the following interpolation theorem due to Pudlak: Theorem 17 (Pudlak [30]). Let ~ p; ~q; ~r be disjoint vectors of variables, and let A(~p; ~q) and B(~p; ~r) be sets of inequalities in the indicated variables such that the variables ~ p either have only nonnegative coecients in A(~p; ~q) or have only non-positive coecients in B(~p; ~r). Suppose there is a CP refutation R of A(~p; ~q) [ B(~p; ~r). Then there is a monotone real circuit C(~p) of size O(jRj) such that for any vector ~a 2 f0; 1g j~pj Furthermore, if R is tree-like, then C(~p) is a monotone real formula. In [30], only the relationship between resolution proof size and monotone real circuit size was stated. The fact that C(~p) is a monotone real formula if R is tree-like is not part of the original theorem, but can be directly obtained from the proof of the theorem in [30]. The reason is that the underlying graphs of the refutation and the circuit are the same. We now dene an unsatisable set of clauses related to the boolean function Gen n . Let n and d be natural numbers whose values will be xed below. Recall that Pyr d := f (i; g. For a given mapping m dening a pyramidal generation in the sense of the denition above, our unsatisable set of clauses will be the conjunction of two CNF, Gen(~p; ~q) and Col(~p; ~r). The clauses in Gen(~p; ~q) will encode the property that the inputs ~q dene a pyramidal generation, and therefore Gen 1. The clauses in Col(~p; ~r) will say that that the inputs ~r dene a coloring, so that Gen More precisely: the variables p a;b;c for a; b; c 2 [n] represent the input to Gen n ; variables q i;j;a for (i; d and a 2 [n] encode a pyramid, where the element a is assigned to the position (i; by the mapping m : Pyr d ! [n]; the variables r a for a 2 [n] represent a coloring of the elements by 1 such that 1 is colored 0, n is colored 1 and the elements colored 0 are closed under generation. The set Gen(~p; ~q) is given by (8) - (11), and Col(~p; ~r) by (12) - (14). _ 1an q i;j;a for (i; r a for a 2 [n] (12) a for a 2 [n] (13) r a _ r b _ r c for a; b; c 2 [n] (14) If Gen( ~ t; ~q) is satisable for a xed vector ~ t 2 f0; 1g n 3 , then n is generated in a depth-d pyramidal fashion, and if Col( ~ t; ~r) is satisable, then Gen( ~ the variables ~ p occur only positively in Gen(~p; ~q) and only negatively in Col(~p; ~r), Theorem 17 is applicable, and the formula obtained from this application satises the conditions of Corollary 8. Hence we can conclude: Theorem 18. Every tree-like CP refutation of the clauses Gen(~p; ~q) [ Col(~p; ~r) has to be of sizen ) , for some > 0. On the other hand, there are polynomial size dag-like resolution refutations of these clauses. Theorem 19. There are (dag-like) resolution refutations of size n O(1) of the clauses Gen(~p; ~q) [ Proof. First we resolve clauses (9) and (12) to get q d;j;c _ r c (15) Now we want to derive r c for every (i; downward from d to 1. The induction base is just (15). Now by induction we have r a and we resolve them against (14) to get r c for 1 a; b; c n and then resolve them against (11) and get for every 1 a; b n. All of these are then resolved against two instances of (8), and we get the desired q i;j;c _ r c for every 1 c n. Finally, we have in particular q 1;1;a _ r a for every 1 c n. We resolve them with (13) and get a;a;n for every 1 a n. These are resolved with (10) to get q 1;1;a for every 1 a n. Finally, this clause is resolved with another instance of (10) (the one with to get the empty clause. It is easy to check that the above refutation is an N-resolution refutation. The following corollary is an easy consequence of the above theorems and known simulation results. Corollary 20. The clauses Gen(~p; ~q) [ Col(~p; ~r) exponentially separate tree-like resolution from dag-like resolution and (dag-like) N-resolution as well as tree-like Cutting Planes from dag-like Cutting Planes. The resolution refutation of Gen(~p; ~q) [ Col(~p; ~r) that appears in the proof of Theorem 19 is not regular. We do not know whether Gen(~p; ~q) [ Col(~p; ~r) has polynomial size regular resolution refutations. To obtain a separation between tree-like resolution and regular resolution we will modify the clauses Col(~p; ~r). 4.1 Separation of tree-like CP from regular resolution The clauses Col(~p; ~r) are modied (and the modication called RCol(~p;~r)), so that Gen(~p; ~q) [ allow small regular resolutions, but in such a way that the lower bound proof still applies. We replace the variables r a by r a;i;D for a 2 [n], 1 i d and D 2 fL; Rg, giving the coloring of element a, with auxiliary indices i being a row in the pyramid and D distinguishing whether an element is used as a left or right predecessor in the generation process. The set RCol(~p;~r) is dened as follows: r a;d;D for a 2 [n] and D 2 fL; Rg (16) r a;i+1;L _ r b;i+1;R _ r a;i;D _ r a;i; r a;i;D _ r a;j;D for 1 Due to the clauses (19) and (20), the variables r a;i;D are equivalent for all values of the auxiliary indices D. Hence a satisfying assignment for RCol(~p;~r) still codes a coloring of [n] such that elements a with are colored 0, the elements b with b; b ' n are colored 1, and the 0-colored elements are closed under generation. Hence if RCol( ~ t; ~r) is satisable, then Gen( ~ Hence any interpolant for the clauses Gen(~p; ~q) [RCol(~p;~r) satises the assumptions of Corollary 8, and we can conclude Theorem 21. Tree-like CP refutations of the clauses Gen(~p; ~q) [ RCol(~p;~r) have to be of sizen ) . On the other hand, we have the following upper bound on (dag-like) regular resolution refutations of these clauses: Theorem 22. There are (dag-like) regular resolution refutations of the clauses Gen(~p; ~q)[RCol(~p; ~r) of size n O(1) . Proof. First we resolve clauses (9) and (16) to get q d;j;a _ r a;d;D (21) Rg. Next we resolve (10) and (17) to get for 1 a n and D 2 fL; Rg. Finally, from (11) and (18) we obtain Rg. Now we want to derive r a;i;D for every (i; induction on i downward from d to 1. The induction base is just (21). For the inductive step, resolve (23) against the clauses q i+1;j;a _ r a;i+1;L and q i+1;j+1;b _ which we have by induction, to give q i+1;j;a _ r c;i;D for every 1 a; b n. All of these are then resolved against two instances of (8), and we get the desired q i;j;c _ r c;i;D . Finally, we have in particular r a;1;L , which we resolve against (22) to get q 1;1;a for every a n. From these and an instance of(8) we get the empty clause. Note that the refutation given in the proof of Theorem 22 is actually a ordered refutation: It respects the following elimination order r 1;d Corollary 23. The clauses Gen(~p; ~q) [ RCol(~p;~r) exponentially separate the following proof sys- tems: Tree-like resolution from regular and ordered resolution. 5 Lower bound for ordered resolutions Goerdt [18] showed that ordered resolution is strictly weaker than unrestricted resolution, by giving a superpolynomial lower bound (of the order log log n )) for ordered resolutions of a certain family of clauses, which on the other hand has polynomial size unrestricted resolution refutations. In this section we improve this separation to an exponential one, in fact, we give an exponential separation of ordered resolution from N-resolution. To simplify the exposition, we apply the method of [18] to a set of clauses SP n;m expressing a combinatorial principle that we call the String-of-Pearls principle: From a bag of m pearls, which are colored red and blue, n pearls are chosen and placed on a string. The string-of-pearls principle SP n;m says that, if the rst pearl is red and the last one is blue, then there must be a blue pearl next to a red pearl somewhere on the string. SP n;m is given by an unsatisable set of clauses in variables p i;j and q j for where p i;j is intended to say that pearl j is at position i on the string, and q j means that pearl j is colored blue. The clauses forming SP n;m are: _ These rst three sets of clauses express that there is a unique pearl at each position. These last three sets of clauses express that the rst pearl is red, the last one is blue, and that a pearl sitting next to a red pearl is also colored red. The clauses SP n;m are a modied and simplied version of the clauses related to the st-connectivity problem that were introduced by Clote and Setzer [13]. We shall modify the clauses SP n;m in such a way as to make small ordered resolution refutations impossible, while still allowing for small unrestricted resolutions. The lower bound is then proved by a bottleneck counting argument similar to that used in [18], which is based on the original argument of Haken [21]. Note that the clauses (24) - (26) are similar to the clauses expressing the Pigeonhole Principle, which makes the bottleneck counting technique applicable in our situation. The set SP 0 n;m is obtained from SP n;m by adding additional literals to some of the clauses. First, the clauses (27) and (29) for 1 i < nand j 0 nare replaced by for every ' 2 [m], where ^{ := n 1. Similarly, the clauses (28) and (29) for n replaced by for every ' 2 [m], where now ^{ := 2j. All other clauses remain unchanged. The modied clauses n;m do not have an intuitive combinatorial interpretation dierent from the meaning of the original clauses SP n;m . The added literals only serve to make the clauses hard for ordered refutations. The idea is that, for the clauses (30)-(33) to be used as one would use the original (27)-(29) in a natural short, inductive proof (like the one given below), the additional literals ^{;' have to be removed rst. The positions ^{ are chosen in such a way that this cannot be done in a manner consistent with a global ordering of the variables. Theorem 24. The clauses SP 0 n;m have negative resolution refutations of size O(nm 2 ). Proof. We rst give a negative refutation of the clauses SP n;m , and then show how to modify them for SP 0 n;m . For every i 2 [n], we will derive the clauses p i;j ! [m] from SP n;m by a negative resolution derivation. For these are the clauses (27) from SP n;m . Inductively, assume we have derived p i;j and we want to derive p (i+1);j ! q j from these. Consider the clauses (29) of the form p i;j Using the inductive assumption, we derive from these the clauses p i;j that these are negative clauses. By a derivation of length m, we obtain p (i+1);j ! q j from these and the clause SP n;m . The whole derivation is of length O(m), and we need m of them, giving a total length of for the induction step. We end up with a derivation of the clauses p n;j ! of length O(nm 2 ). In another m steps we resolve these with the initial clauses (28), obtaining the singleton clauses Finally we derive a contradiction from these and the clauses Now we modify this refutation for the modied clauses SP 0 n;m . First, note that the original clauses (27) can be obtained from (30) by a negative derivation of length m. Next, we modify those places in the inductive step where the clauses (29) are used that have been modied. First, we resolve the modied clauses (31) resp. (33) with the inductive assumption, yielding the negative clauses These are then resolved with the clause after which we can continue as in the original refutation. In the places where the clauses (28) are used in the original refutation, we rst resolve (32) with the clauses p n;j ! yielding n;j , which can be resolved with to get the singleton clauses n;j as in the original refutation. In particular, there are polynomial size unrestricted resolution refutations of these clauses. The next theorem gives a lower bound for ordered resolution refutations of these clauses. Theorem 25. For suciently large n and m 9n, every ordered resolution refutation of the clauses SP 0 n;m contains at least 2 n clauses. Proof. For sake of simplicity, let n be divisible by 8, say nm+m be the number of variables, and let an ordering x of the variables be given, i.e., each x is one of the variables p i;j or q j . Let R be a ordered resolution refutation of SP 0 n;m respecting this elimination ordering, i.e., on every path through R the variables are eliminated in the prescribed order. We shall show that R contains at least k! dierent clauses, which is at least 2 n 8 (log n 5) for large n. For a position i 2 [n] and N , let S(i; ) be the set of those pearls j 2k such that p i;j is among the rst eliminated variables, i.e., be the unique position such that there is an index 0 with In other words, i 0 is the rst position for which k of the variables p i 0 ;j with j 2k are eliminated. Let the elements of S(i enumerated in increasing order for definiteness only, the order is irrelevant for the argument. For each 1 k, dene a position i by Note that i is the position ^{ appearing in the added literals in the modied clauses (31) for or (27), where in the rst case, respectively in the clauses (33) for in the second case. Further dene R := [2k] n S(i ; 0 ), i.e., R is the set of those pearls j 2k for which the variable eliminated later than any of the variables p i 0 ;j for 1 k. Note that for all by denition of i 0 . Denition. A critical assignment is an assignment that satises all the clauses of SP 0 n;m except for exactly one of the clauses (24). From a critical assignment , we dene the following data: The unique position i 2 [n] such that (p i ;j the gap of . A 1-1 mapping is the unique j 2 [m] such that (p i;j For every j 2 [m], we refer to the value (q j ) as the color of j, where we identify the value 0 with red and 1 with blue. A critical assignment is called 0-critical, if the gap is i each are colored blue (i.e., (q are colored red (i.e., (q j 1 Note that the positions and the pearls thus the notion of 0-critical assignment, only depend on the elimination order and not on the refutation R. As in other bottleneck counting arguments, the lower bound will now be proved in two steps: First, we show that there are many 0-critical assignments. Second, we will map each 0-critical assignment to a certain clause C in R, and then show that not too many dierent assignments can be mapped to the same clause C , thus there must be many of the clauses C . The rst goal, showing there are many 0-critical assignments, is reached with the following claim: 26. For every choice of pairwise distinct pearls b is a 0-critical assignment with m (i Proof of Claim 26. For those positions i such that m (i) is not dened yet, i.e. arbitrarily but consistently, i.e. choose an arbitrary 1-1 mapping from [n] n fi to [m] n fb g. This is always possible, since by assumption m 9k. Finally, color those pearls that are assigned to positions to the left of the gap red, and those that are assigned to positions to the right of the gap blue, i.e., set (q m (i) . The pearls are colored according to the requirement in the denition of a 0-critical assignment. Note that this does not result in a con ict even if some of the are among the because the positions are always on the correct side of the gap: if i 0 n, then k. The remaining pearls can be colored arbitrarily. Now we map 0-critical assignments to certain clauses in R. For a 0-critical assignment , let C be the rst clause in R such that does not satisfy C , and occurs in C This clause exists because determines a path through R from the clause to the empty clause, such that does not satisfy any clause on this path. The variables p i 0 ;j with j 2k are eliminated along that path, and are the rst among them in the elimination order. 27. Let be a 0-critical assignment and ' := m (i ). Then for every 1 k, the literal occurs in C . Proof of Claim 27. Let 0 be the assignment dened by 0 (p other variables x. As p i 0 ;j does not occur in C , 0 does not satisfy C either. There is exactly one clause in SP 0 n;m that is not satised by 0 , depending on where the gap i 0 is, this clause is The requirement for the coloring of the j in the denition of a 0-critical assignment entails that these clauses are not satised by 0 , and that all other clauses are satised by 0 . In any case, the literal p i ;' occurs in this clause, and there is a path through R leading from the clause in question to C , such that 0 does not satisfy any clause on that path. The variable that is eliminated in the last inference on that path must be one of the p i 0 ;j for 1 k, by the denition of C . Since ' 2 R , the variable p i ;' appears after p i 0 ;j in the elimination order, by the denition of R . Therefore p i ;' cannot have been eliminated on that path, so occurs in C . Finally we are ready to nish the proof of the theorem. Let ; be two 0-critical assignments such that ' := m (i so that (p i ;' 27, the literal occurs in C , therefore satises C , and hence C 6= C . By Claim 26, there are at least k! 0-critical assignments that dier in at least one of the values m (i ). Thus R contains at least k! distinct clauses of the form C . The following corollary is a direct consequence of Theorems 25 and 24. Corollary 28. The clauses SP 0 exponentially separate ordered resolution from unrestricted resolution and N-resolution. A modication similar to the one that transforms SP n;m into SP 0 n;m can also be applied to the clauses Gen(~p; ~q), yielding a set DPGen(~p; ~q). Then for the clauses DPGen(~p; ~q) [ Col(~p; ~r), an exponential lower bound for ordered resolutions can be proved by the method of Theorem 25 (this was presented in the conference version [8] of this paper). Also the N-resolution proofs of Theorem 19 can be modied for these clauses. Thus the clauses exponentially separate ordered from negative resolution as well. 6 Open Problems We would like to conclude by stating some open problems related to the topics of this paper. 1. For boolean circuits (monotone as well as general), circuit depth and formula size are essentially the same complexity measure, as they are exponentially related by the well-known Brent-Spira theorem. Is there an analogous theorem for monotone real circuits, i.e., is every monotone function f? This would be implied by the converse to Lemma 2, i.e., dR (f) CCR (R f ). Does this hold for every monotone function 2. The separation between tree-like and dag-like resolution was recently improved to a strongly exponential one, with a lower bound of the form 2 n= log n ([5, 6, 31]). Can we prove the same strong separation between tree-like and dag-like CP ? 3. A solution for the previous problem would follow from a strongly exponential separation of monotone real formula size from monotone circuit size. Such a strong separation is not even known for monotone boolean circuits. 4. Can the superpolynomial separations of regular and negative resolution from unrestricted resolution [19, 20] be improved to exponential as well? And is there an exponential speed-up of regular over ordered resolution? Acknowledgments We would like to thank Ran Raz for reading a previous version of this work and discovering an error, Andreas Goerdt for sending us copies of his papers, Sam Buss for helpful discussions and nally Peter Clote for suggesting us to work on resolution separations. --R Minimum propositional proof length is NP-hard to linearly approximate Short proofs are narrow Exponential separations between restricted resolution and cutting planes proof systems. Lower bounds for cutting planes proofs with small coe Using the Groebner basis algorithm to An exponential lower bound for the size of monotone real circuits. The relative e Lower bounds on sizes of cutting planes proofs for modular coloring principles. Unrestricted resolution versus N-resolution Regular resolution versus unrestricted resolution. The intractability of resolution. Upper and lower bounds for tree-like cutting planes proofs Complexity of Lower bounds for monotone real circuit depth and formula size and tree-like cutting planes Combinatorics of monotone computations. Monotone circuits for connectivity require super-logarithmic depth Separation of the monotone NC hierarchy. Monotone real circuits are more powerful than monotone boolean circuits. Hard examples for resolution. The complexity of propositional proofs. --TR --CTR Juan Luis Esteban , Jacobo Torn, A combinatorial characterization of treelike resolution space, Information Processing Letters, v.87 n.6, p.295-300, September Michael Alekhnovich , Jan Johannsen , Toniann Pitassi , Alasdair Urquhart, An exponential separation between regular and general resolution, Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, May 19-21, 2002, Montreal, Quebec, Canada 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 Paolo Liberatore, Complexity results on DPLL and resolution, ACM Transactions on Computational Logic (TOCL), v.7 n.1, p.84-107, January 2006 Jakob Nordstrm, Narrow proofs may be spacious: separating space and width in resolution, Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, May 21-23, 2006, Seattle, WA, USA 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 Robert Nieuwenhuis , Albert Oliveras , Cesare Tinelli, Solving SAT and SAT Modulo Theories: From an abstract Davis--Putnam--Logemann--Loveland procedure to DPLL(T), Journal of the ACM (JACM), v.53 n.6, p.937-977, November 2006 Henry Kautz , Bart Selman, The state of SAT, Discrete Applied Mathematics, v.155 n.12, p.1514-1524, June, 2007

resolution;proof complexity;cutting planes proof system;computational complexity;circuit complexity

586946

Randomness is Hard.

We study the set of incompressible strings for various resource bounded versions of Kolmogorov complexity. The resource bounded versions of Kolmogorov complexity we study are polynomial time CD complexity defined by Sipser, the nondeterministic variant CND due to Buhrman and Fortnow, and the polynomial space bounded Kolmogorov complexity CS introduced by Hartmanis. For all of these measures we define the set of random strings $\mathrm{R}^{\mathit{CD}}_t$, $\mathrm{R}^{\mathit{CND}}_t$, and $\mathrm{R}^{\mathit{CS}}_t$ as the set of strings $x$ such that $\mathit{CD}^t(x)$, $\mathit{CND}^t(x)$, and $\mathit{CS}^s(x)$ is greater than or equal to the length of $x$ for $s$ and $t$ polynomials. We show the following: $\mathrm{MA} \subseteq \mathrm{NP}^{\mathrm{R}^{\mathit{CD}}_t}$, where $\mathrm{MA}$ is the class of Merlin--Arthur games defined by Babai. $\mathrm{AM} \subseteq \mathrm{NP}^{\mathrm{R}^{\mathit{CND}}_t}$, where $\mathrm{AM}$ is the class of Arthur--Merlin games. $\mathrm{PSPACE} \subseteq \mathrm{NP}^{\mathrm{cR}^{\mathit{CS}}_s}$. In the last item $\mathrm{cR}^{\mathit{CS}}_s$ is the set of pairs $\langle x,y \rangle$ so that x is random given y. These results show that the set of random strings for various resource bounds is hard for complexity classes under nondeterministic reductions.This paper contrasts the earlier work of Buhrman and Mayordomo where they show that for polynomial time deterministic reductions the set of exponential time Kolmogorov random strings is not complete for EXP.

Introduction The holy grail of complexity theory is the separation of complexity classes like P, NP and PSPACE. It is well known that all of these classes possess complete sets and that it is thus sucient for a separation to show that a complete set of one class is not contained in the other. Therefore lots of eort was put into the study of complete sets. (See [BT98].) Kolmogorov [Lev94] however suggested to focus attention on sets which are not complete. His intuition was that complete sets possess a lot of \structure" that hinders a possible lower bound proof. He suggested to look at the set of time bounded Kolmogorov random strings. In this paper we will continue this line of research and study variants of this set. Kolmogorov complexity measures the \amount" of regularity in a string. Informally the Kolmogorov complexity of a string x, denoted as C(x), is the size of the smallest program that prints x and then stops. For any string x, C(x) is less than or equal to the length of x (up to some additive constant). Those strings for which it holds that C(x) is greater than or equal to the length of x are called incompressible or random. A simple counting argument shows that random strings exist. In the sixties, when the theory of Kolmogorov complexity was developed, Martin [Mar66] showed that the coRE set of Kolmogorov random strings is complete with respect to (resource unbounded) Turing reductions. Kummer [Kum96] has shown that this can be strengthened to show that this set is also truth-table complete. The resource bounded version of the random strings was rst studied by Ko [Ko91]. The polynomial time bounded Kolmogorov complexity C p (x), for p a polynomial is the smallest program that prints x in p(jxj) steps ([Har83]). Ko showed that there exists an oracle such that the set of random strings with respect to this time bounded Kolmogorov complexity is complete for coNP under strong nondeterministic polynomial time reduc- tions. He also constructed an oracle where this set is not complete for coNP under deterministic polynomial time Turing reductions. Buhrman and Mayordomo [BM95] considered the exponential time Kolmogorov random strings. The exponential time Kolmogorov complexity is the smallest program that prints x in t(jxj) steps for functions . They showed that the set of t(n) random strings is not deterministic polynomial time Turing hard for EXP. They showed that the class of sets that reduce to this set has p measure 0 and hence that this set is not even weakly hard for EXP. The results in this paper contrast those from Buhrman and Mayordomo. We show that the set of random strings is hard for various complexity classes under nondeterministic polynomial time reductions. We consider three well studied measures of Kolmogorov complexity that lie in between C p (x) and C t (x) for p a polynomial and . We consider the distinguishing complexity as introduced by Sipser [Sip83]. The distinguishing complexity, CD t (x), is the size of the smallest program that runs in time t(n) and accepts x and nothing else. We show that the set of random strings R CD xed polynomial is hard for MA under nondeterministic reductions. MA is the class of Merlin- Arthur games introduced by Babai [Bab85]. As an immediate consequence we obtain that BPP and NP BPP are in NP R CD t . Next we shift our attention to the nondeterministic distinguishing complexity which is dened as the size of the smallest nondeterministic algorithm that runs in time t(n) and accepts only x. We dene R CND xed polynomial. We show that AM NP R CND t where AM is the class of Arthur-Merlin games [Bab85]. It follows that the complement of the graph isomorphism problem, GI, is in NP R CND t and that if for some polynomial t, R CND The s(n) space bounded Kolmogorov complexity, CS s (xjy) is dened as the size of the smallest program that prints x, given y and uses at most s(jxj+ tape cells [Har83]. Likewise we dene cR CS for s(n) a polynomial. We show that PSPACE NP cR CS s . For the rst two results we use the oblivious sampler construction of Zuckerman [Zuc96], a lemma [BF97] that measures the size of sets in terms of CD complexity, and we prove a Lemma that shows that the rst bits of a random string are in a sense more random than the whole string. For the last result we make use of the interactive protocol [LFKN92, Sha92] for QBF. To show optimality of our results for relativizing techniques, we construct an oracle world where our rst result can not be improved to deterministic reductions. We show that there is an oracle such that BPP 6 P R CD t for any polynomial t. The construction of the oracle is an extension of the techniques developed by Beigel, Buhrman and Fortnow [BBF98]. Denitions and Notations We assume the reader familiar with standard notions in complexity theory as can be found e.g., in [BDG88]. Strings are elements of , where For a string s and integers n; m jsj we use the notation s[n::m] for the string consisting of the nth through mth bit of s. We use for the empty string. We also need the notion of an oblivious sampler from [Zuc96]. Denition 2.1 A universal (r; d; m; ; )-oblivious sampler is a deterministic algorithm which on input a uniformly random r-bit string outputs a sequence of points z any collection of d functions it is the case that Pr d (Where In our application of this denition, we will always use a single function f . Fix a universal Turing machine U , and a nondeterministic universal machine U n . All our results are independent of the particular choice of universal machine. For the denition of Kolmogorov complexity we need the fact that the universal machine can, on input p; y halt and output a string x. For the denition of distinguishing complexity below we need the fact that the universal machine on input p; x; y can either accept or reject. We also need resource bounded versions of this property. We dene the Kolmogorov complexity function C(xjy) (see [LV97]) by xg. We dene unconditional Kolmogorov complexity by Hartmanis dened a time bounded version of Kolmogorov complexity in [Har83], but resource bounded versions of Kolmogorov complexity date back as far as [Bar68]. (See also [LV97].) Sipser [Sip83] dened the distinguishing complexity CD t . We will need the following versions of resource bounded Kolmogorov complexity and distinguishing complexity. uses at most . (See [Har83].) (1) U(p; x; y) accepts rejects runs in at most (See [Sip83].) (1) U n (p; x; y) accepts rejects runs in at most (See [BF97].) For 0 < 1 we dene the following sets of strings of \maximal" CD p and CND p complexity. Note that for these sets are the sets mentioned in the introduction. In this case we will omit the and use R CD t and R CND t . We also dene the set of strings of maximal space bounded complexity. cR CS The c in the notation is to emphasize that randomness is conditional. Also, cR CS s technically is a set of pairs rather than a set of strings. The unconditional space bounded random strings would be R CS s g: We have no theorems concerning this set. The C-complexity of a string is always upperbounded by its length plus some constant depending only on the choice of the universal machine. The CD- and CND-complexity of a string are always upperbounded by the C- complexity of that string plus some constant depending again only on the particular choice of universal machine. All quantiers used in this paper are polynomially bounded. Often the particular polynomial is not important for the sequel or it is clear from the context and is omitted. Sometimes we need explicit bounds. Then the particular bound is given as a superscript to the quantier. E.g., we use 9 m y to denote \There exists a y with jyj m," or 8 =n x to denote \For all x of length n." The classes MA and AM are dened as follows. Denition 2.2 L 2 MA i there exists a jxj c time bounded machine M such 1. x 2 L =) 9yPr[M(x; 2. where r is chosen uniformly at random in f0; 1g jxj c there exists a jxj c time bounded machine M such that 1. x 2 L =) Pr[9yM(x; 2. where r is chosen uniformly at random in f0; 1g jxj c It is known that NP [ BPP MA AM PSPACE [Bab85]. Let #M(x) represent the number of accepting computations of a non-deterministic Turing machine M on input x. A language L is in P if there exists a polynomial time bounded nondeterministic Turing machine M such that for all x: Let g be any function. We say that advice function f is g-bounded if for all n it holds that jf(n)j g(n). In this paper we will only be interested in functions g that are polynomial. The notation sn T is used for strong nondeterministic Turing reductions, which are dened by A sn Distinguishing Complexity for Derandomization In this section we prove hardness of R CD t and R CND t for Arthur-Merlin and Merlin-Arthur games respectively under NP-reductions. Theorem 3.1 For any t with t(n) 2 !(n log n), MA NP R CD t . and Theorem 3.2 For any t with t(n) 2 !(n), AM NP R CND t . The proof of both theorems is roughly as follows. First guess a string of high CD poly -complexity, respectively CND poly -complexity. Next, we use the nondeterministic reductions once more to play the role of Merlin, and use the random string to derandomize Arthur. Note that this is not as straightforward as it might look. The randomness used by Arthur in interactive protocols is used for hiding and can in general not be substituted by computational randomness. The idea of using strings of high CD-complexity and Zuckerman's sampler derandomization stems from [BF00] (Section 8), which is the full version of [BF97]. Though they do not explicitly dene the set R CD t , they use the same approach to derandomize BPP computations there. The proof needs a string of high CD p respectively CND p complexity for some polynomial. We rst show that we can nondeterministically extract such a string from a longer string with high CD t complexity (respectively CND t complexity) for any xed t with t(n) 2 !(n log n). Lemma 3.3 Let f be such that f(n) < n, and let t, t 0 and T be such that T all suciently large s with CD t holds that CD t 0 Proof . Suppose for a contradiction that for any constant d 0 and innitely many s with CD t holds that CD t 0 log jf(jsj)j d 0 . Then for any such s there exists a program p s that runs in recognizes only s[1::f(jsj)] where jp s j < (f(jsj) 2 log jf(jsj)j d 0 . The following program then recognizes s and no other string. Input y Check that the rst f(jsj) bits of y equal using p s . (Assume jf(jsj)j is stored in the program for a cost of log jf(jsj)j bits.) Check that the last jsj f(jsj) bits of y equal bits are also stored in the program.) This program runs in time T Therefore it takes at most t(jsj) steps on U for all suciently large s [HS66]. We lose the log n factor here because our algorithm must run on a xed machine and the simulation is deterministic. The program's length is jp log jf(jsj)j d which is less than jsj for almost all s. Hence CD t (s) < jsj, which contradicts the assumption. 2 Corollary 3.4 For every polynomial n c , t 2 !(n log n) and suciently large string s with CD t c and s Proof . Take t 0 c and apply Lemma 3.3. 2 Lemma 3.3 and Corollary 3.4 have the following nondeterministic analogon Lemma 3.5 For every polynomial n c , t 2 !(n) and suciently large string s with CND t c and s Proof . The same proof applies, with a lemma similar to Lemma 3.3. How- ever, in the nondeterministic case the simulation costs only linear time [BGW70].Before we can proceed with the proof of the theorems, we also need some earlier results. We rst need the following Theorem from Zuckerman: Theorem 3.6 ([Zuc96]) There is a constant c such that for (m) and there exists a universal (r; d; m; ; )-oblivious sampler which runs in polynomial time and uses only bits and outputs We also need the following lemma by Buhrman and Fortnow: Lemma 3.7 ([BF97]) Let A be a set in P. For each string x 2 A =n it holds that CD p (x) 2 log(jjA =n jj) +O(log n) for some polynomial p. As noted in [BF97], an analogous lemma holds for CND p and NP. Lemma 3.8 ([BF97]) Let A be a set in NP. For each string x 2 A =n it holds that CND p (x) 2 log(jjA =n jj) +O(log n) for some polynomial p. From these results we can prove the theorems. If we, for Theorem 3.1, want to prove that an NP machine oracle with oracle R CD t can recognize a set A in MA then the positive side of the proof is easy if x in A then there exists a machine M and a string y such that a 2=3 fraction of the strings r of length jxj c makes M(x; accept. So an NP machine can certainly guess one such pair x; y as a \proof" for x 2 A. The negative side is harder. We will show that if 2 A and we substitute for r a string of high enough CD complexity (CND complexity for Theorem 3.2) then no y can make M(x; accept. To grasp the intuition behind the proof let us look at the much simplied example of a BPP machine M having a 1=3 error probability on input x and a string r of maximal unbounded Kolmogorov complexity. There are 2 jxj k possible computations on input x, where jxj k is the runtime of M . Suppose that M must accept x then at most a 1=3 fraction, i.e. at most 2 jxj c of these computations reject x. Each rejecting computation consists of a deterministic part described by M and x and a set of jxj c coin ips. such a set of coin ips with a binary string and we have that each rejecting computation uniquely identies a string of length jxj c . Call this set B. We would like to show by contradiction that a random string cannot be a member of this set, and hence that any random string, used as a sequence of coin ips, leads to a correct result. Any string in B is described by M , x and an index in B, which has length log jjBjj jxj c log 3. So far there are no grounds for a contradiction since a description consisting of these elements can have length greater than jxj c . However we can amplify the computation of M on input x by repetition and taking majority. Then, repeating the computation x times, blows up the number of incorrect computations to using x c+1 random bits. However, for large enough x a description of jxj plus or minus some additive constant depending on the the denition of Kolmogorov complexity used is smaller then jxj c+1 and thus will lead to a contradiction. Unfortunately, in our case the situation is a bit more complicated. The factor 2 in Lemma 3.7 renders standard amplifaction of randomized computation useless. Fortunately, Theorem 3.6 allows for a dierent type of amplication using much less random bits, so that the same type of argument can be used. We will now proceed to show how to t the amplication given by Theorem 3.6 to our situation. Lemma 3.9 1. Let L be a language in MA. For any constant k and any constant 0 < 1there exists a deterministic polynomial time bounded machine M such that: and r is chosen uniformly at random from the strings in f0; 1g (1+)(1+k)m 2. Let L be a language in AM. For any constant k and any constant 0 < there exists a deterministic polynomial time bounded machine M such that: (a) x 2 L =) Pr[9yM(x; and r is chosen uniformly at random from the strings in f0; 1g (1+)(1+k)m Proof . 1. Furer et al. showed that the fraction 2=3 (see Denition 2.2) can be replaced by 1 in [FGM + 89]. Now let ML be the deterministic polynomial time machine corresponding to L in Denition 2.2, adapted so that it can accept with probability 1 if x 2 L. Assume ML runs in This means that for ML the 9y and 8y in the denition can be assumed to be 9 n c y and 8 n c y respectively. Also, the random string may be assumed to be drawn uniformly at random from f0; 1g n c . To obtain the value 2 km in the second item, we use Theorem 3.6 with 1=6. For given x and y let f xy be the FP function that on input z computes ML (x; We use the oblivious sampler to get a good estimate for Ef xy . That is, we feed a random string of length (1+)(1+k)m in the oblivious sampler and it returns sample points z d on which we compute 1 d is the machine that computes this sum on input x, y and r and accepts i its value is greater than 1=2. If x 2 L there is a y such that Pr[ML (x; no matter which sample points are returned by the oblivious sampler. If y. With probability 1 the sample points returned by the oblivious sampler are such that d , so 1 d probability 2 km . 2 2. The proof is analogous to the proof of Part 1. We just explain the dierences. For the 1 in the rst item of the claim we can again refer to [FGM + 89], but now to Theorem 2(ii) of that paper. In this part ML is the deterministic polynomial time machine corresponding to the AM-language L and we dene the function f x : f0; 1g m 7! [0; 1] as the function that on input z computes 9 n c is an FP NP computable function. The sample points z z d that are returned in this case have the following properties. If x 2 L then no matter which string is returned as z i . That is for every possible sample point z i there is a y i such that ML (x; y for any set of sample points z that the sampler may return, there exists a such that ML (x; y than half of the sample points with probability 1 . That is Pr "d d is less than 2 km . So if we let M(x; r) be the deterministic polynomial time machine that uses r to generate d sample points and then interprets y as <y and counts the number of accepts of ML (x; y accepts if this number is greater than 1d we get exactly the desired result. 2 In the next lemma we show that a string of high enough CD poly (CND poly ) can be used to derandomize an MA (AM) protocol. Lemma 3.10 1. Let L be a language in MA and 0 < 1. There exists a deterministic polynomial time bounded machine M , a polynomial q, > 0 and integers k and c such that for almost all n and every r with 2. Let L be a language in AM and 0 < 1. There exists a deterministic polynomial time bounded machine M a polynomial q, > 0 and integers k and c such that for almost all n and every r with Proof . 1. Choose < and k > 6 . Let M be the deterministic polynomial time bounded machine corresponding to L, k and of Lemma 3.9, item 1. The polynomial n c will be the time bound of the machine witnessing L 2 MA of that same lemma. We will determine q later, but assume for now that r is a string of length (1 that CD q (r) jrj, and for ease of notation set Suppose x 2 L. Then it follows that there exists a y such that for all s of length (1 1. So in particular it holds that M(x; Suppose x 62 L. We have to show that for all y it is the case that Suppose that this is not true and let y 0 be such that A x;y 0 It follows that A x;y0 2 P by essentially a program that simulates M and has x and y 0 hardwired. (Although A x;y0 is nite and therefore trivially in P it is crucial here that the size of the polynomial program is roughly Because of the amplication of the MA protocol we have that: Since r 2 A x;y0 it follows by Lemma 3.7 that there is a polynomial p such On the other hand we chose r such that: CD q (r) jrj which gives a contradiction for q p. 2. Choose < and k > 5 . Let M be the deterministic polynomial time bounded machine corresponding to L, and k of Lemma 3.9, item 2. Again, n c will be the time bound of the machine now witnessing will be determined later. Assume for now that r is a string of length (1 c such that CND q (r) jrj. Suppose x 2 L. Then it follows that for all s there exists a y such that 1. So in particular there is a y r such that M(x; y r Suppose L. We have to show that 8yM(x; that this is not true. Dene A 1g. Then A x 2 NP by a program that has x hardwired, guesses a y and simulates . Because of the amplication of the AM protocol we have that jjA x jj 2 (1+)(1+k)m km . Since r 2 A x it follows by Lemma 3.8 there exists a polynomial p such that: On the other hand, we chose r such that: CND q (r) jrj which gives a contradiction whenever q p.The following corollary shows that a string of high enough CD poly complexity can be used to derandomize a BPP machine (See also Theorem 8.2 in [BF00]). Corollary 3.11 Let A be a set in BPP. For any > 0 there exists a polynomial time Turing machine M a polynomial q such that if CD q (r) jrj with then for all x of length n it holds that x 2 A () Proof of Theorem 3.1. Let A be a language in MA. Let q, M , and be as in Lemma 3.10, item 1. The nondeterministic reduction behaves as follows on input x of length n. First guess an s of size check that s 2 R CD t . Set accept if and only if there exists a y such that M(x; 1. By Corollary 3.4 it follows that CD q (r) jrj=2 and the correctness of the reductions follows directly from Lemma 3.10, item 1 with Proof of Theorem 3.2. This follows directly from Lemma 3.10, item 2. The NP-algorithm is analogous to the one above. Corollary 3.12 For t 2 !(n log n) 1. BPP and NP BPP are included in NP R CD t . 2. GI 2 NP R CND t . It follows that if R CND then the Graph isomorphism problem, GI, is in NP \ coNP. Limitations In the previous section we showed that the set R CD t is hard for MA under reductions. One might wonder whether R CD t is also hard for MA under a stronger reduction like the deterministic polynomial time Turing reduction. In this section we show that this, if true, will need a nonrelativizing proof. We will derive the following theorem. Theorem 4.1 There is a relativized world where for every polynomial t and t; . The proof of this theorem is given in Lemma 4.2 which says that the statement of Theorem 4.1 is true in any world where P NP A =poly and Theorem 4.3 which precisely shows the existence of such a world. Lemma 4.2 For any oracle A and 0 < 1 it holds that if EXP NP A NP A =poly and P t; A Proof . Suppose for a contradiction that the lemma is not true. If EXP NP NP=poly then EXP NP=poly, so EXP PH ([Yap83]). Furthermore, if EXP NP NP=poly, then certainly EXP NP EXP=poly. It then follows from [BH92] that EXP If (see [BFT97] for a denition) is in P. Then by [VV86] and so NP BPP which implies PH BPP by [Zac88]. Finally, the fact that unique-SAT is in P is equivalent to: For all x and y, C poly (xjy) CD poly (xjy) + O(1), as shown in [FK96]. We can use the proof of [FK96] to show that unique-SAT in P also implies that R CD for a particular universal machine. (Note that we need only contradict the assumption for one particular type of universal machine.) This then in its turn implies by assumption that BPP and hence EXP NP are in P NP . This however contradicts the hierarchy theorem for relativized Turing machines [HS65]. As all parts of this proof relativize, we get the result for any oracle. There's one caveat here. Though R CD t; A clearly has a meaningful in- terpretation, to talk about P R CD t; A one must of course allow P to have access to the oracle. It is not clear that P can ask any question if the machine can only ask question about the random strings. Therefore, one might argue that P R CD t; AA should actually be in the statement of the lemma. This does not aect the proof. Our universal machine, say U S , is the following. On input p; x; y, U S uses the Cook-Levin reduction to produce a formula f on jxj variables with the property that x satises f if and only if p accepts x. Then U S uses the self-reducibility of f and the assumed polynomial time algorithm for unique-SAT to make acceptance of x unique. That is rst if the number of variables is not equal jyj it rejects. Then, using the well-known substitute and reduce algorithm for SAT, it veries for assignments successively obtained from the algorithm that the algorithm for precisely accepts rejects if this algorithm accepts both Using this universal machine every program accepts at most one string and therefore R CD via an obvious predicate. As argued above, this gives us our contradiction. 2 Now we proceed to construct the oracle. Theorem 4.3 There exists an oracle A such that EXP NP A Proof . The proof parallels the construction from Beigel, Buhrman and Fortnow [BBF98], who construct an oracle such that P NP A . We will use a similar setup. Let M A be a nondeterministic linear time Turing machine such that the language L A dened by is P A complete for every A. For every oracle A, let K A be the linear time computable complete set for NP A . Let N K A be a deterministic machine that runs in time 2 n and for every A accepts a language H A that is complete for EXP NP A . We will construct A such that there exists a n 2 bounded advice function f such that for for all w (Condition (Condition 1) Condition 0 will guarantee that that EXP NP NP=poly We use the term 0-strings for the strings of the form <0; w; 1 jwj 2 > and 1-strings for the strings of the form <1; z; w; v> with other strings we immediately put in A. First we give some intuition for the proof. M is a linear time Turing machine. Therefore setting the 1-strings forces the setting of the 0 strings. Condition 0 will be automatically fullled by just describing how we set the 1-strings because they force the 0-strings as dened by Condition 0. Fullling Condition 1 requires a bit more care since N K A can query exponentially long and double exponentially many 0- and 1-strings. We consider each 1-string <1; z; w; v> as a 0-1 valued variable y <z;w;v> whose value determines whether <1; z; w; v> is in A. The construction of A wil force a correspondence between the computation of N K A (x) and a low degree polynomial over variables with values in GF (2). To encode the computation properly we use the fact that the OR function has high degree. We will assign a polynomial p z over GF[2] to all of the 0-strings and 1-strings z. We ensure that for all z 1. If p z is in A. 2. If p z is not in A. First for each 1-string z = <1; z; w; v> we let p z be the single variable polynomial y <z;w;v> . We assign polynomials to the 0-strings recursively. Note that M A (x) can only query 0-strings with jwj jxj. Consider an accepting computation path of M(x) (assuming the oracle queries are guessed correctly). Let ;m be the queries on this path and b ;m be the query answers with b the query was guessed in A and b Note that m Let P be the set of accepting computation paths of M(x). We then dene the polynomial p z for > as follows: Y (p Remember that we are working over GF[2] so addition is parity. Setting the variables y <z;w;v> (and thus the 1-strings) forces the values of z for the 0-strings. We have set things up properly so the following lemma is straightforward. Lemma 4.4 For each 0-string > we have 2 and Condition 0 can be satised. The polynomial p z has degree at most Proof: Simple proof by induction on jxj. 2 The construction will be done in stages. At stage n we will code all the strings of length n of H A into A setting some of the 1-strings and automatically the 0-strings and thus fullling both condition 0 and 1 for this stage. We will need to know the degree of the multivariate multilinear polynomials representing the OR and the AND function. Lemma 4.5 The representation of the functions OR(u um ) and the um ) as multivariate multilinear polynomials over GF[2] requires degree exactly m. Proof: Every function over GF[2] has a unique representation as a multivariate multilinear polynomial. Note that AND is just the product and by using De Morgan's laws we can write OR as Y The construction of the oracle now treats all strings of length n in lexicographic order. First, in a forcing phase in which the oracle is set so that all computations of N K A remain xed for future extensions of the oracle and next a coding phase in which rst an advice string is picked and then the computations just forced are coded in the oracle in such a way that they can be retrieved by an NP machine with this advice string. Great care has of course to be taken so that the two phases don't disturb each other and do not disturb earlier stages of the construction. We rst describe the forcing phase. Without loss of generality, we will assume that machine N only queries strings of the form q 2 K A . Note that since N runs in time 2 n it may query exponentially long strings to K A . Let x 1 be the rst string of length n. When we examine the computation of N(x 1 ) we encounter the rst query q 1 to K A . We will try to extend the oracle A to A 0 A such that q 1 2 K A 0 . If such an extension does not exist we may assume that q 1 will never be in K A no matter how we extend A in the future. We must however take care that we will not disturb previous queries that were forced to be in K A . To this end we will build a set S containing all the previously encountered queries that were forced to be in K A . We will only extend A such that for all q 2 S it holds that q 2 K A 0 We will call such an extension an S-consistent extension of A. Returning to the computation of N(x 1 ) and q 1 we ask whether there is an S-consistent extension of A such that q 1 2 K A 0 . If such an extension exists we will choose the S-consistent extension of A which adds a minimal number of strings to A and put q 1 in S. Next we continue the computation of answered yes and otherwise we continue with q 1 answered no. The next lemma shows that a minimal extension of A will never add more than 2 3n strings to A. Lemma 4.6 Let S be as above and q be any query to K A and suppose we are in stage n. If there exists an S-consistent extension of A such that q 2 K A 0 then there exists one that adds at most 2 3n strings to A. Proof . Let MK be a machine that accepts K A when given oracle A and consider the computation of machine M A l be the smallest set of strings such that adding them to A is an S-consistent extension of A such that M A 0 K (q) accepts. Consider the leftmost accepting path of M A 0 K (q) and let q be the queries (both 0 and 1-queries) on that path. Moreover let b i be 1 . Dene for q the following polynomial: Y (p After adding the strings l to A we have that P by Lemma 4.4 the degree of each p q i is at most 2 2n and hence the degree of P q is at most 2 3n . Now consider what happens when we take out any number of the strings l of A 0 resulting in A 00 . Since this was a minimal extension of A it follows that M A 00 K (q) rejects and that P computes the AND on the l strings . Since by Lemma 4.5 the degree of the unique multivariate multilinear polynomial that computes the AND over l variables over GF[2] is l it follows that l 2 3n . 2 After we have dealt with all the queries encountered on N K A continue this process with the other strings of length n in lexicographic order. Note that since we only extend A S-consistently we will never disturb any computation of N K A on lexicographic smaller strings. This follows since the queries that are forced to be yes will remain yes and the queries that could not be forced with an S-consistent extension will never be forced by any S 0 -consistent extension of A, for S S 0 . After we have nished this process we have to code all the computations of N on the strings of length n. It is easy to see that jjSjj 2 2n and that at this point by Lemma 4.6 at most 2 5n strings have been added to A at this stage. Closing the forcing phase we can now pick an advice string and proceed to the coding phase. A standard counting argument shows that there is a string z of length n 2 such that no strings of the form <1; z; w; v> have been added to A. This string z will be the advice for strings of length n. Now we have to show that we can code every string x of length n correctly in A to fulll condition 1. We will do this in lexicographic order. Suppose we have coded all strings x j (for j < i) correctly and that we want to code x i . There are two cases: In this case we put all the strings <1; z; x in A and thus set all these variables to 0. Since this does not change the oracle it is an S-consistent extension. properly extend A S-consistently adding only strings of the form <1; z; x to A. The following lemma shows that this can always be done. A proper extension of A is one that adds one or more strings to A. Lemma 4.7 Let jjSjj 2 2n be as above. Suppose that N K A There exists a proper S-consistent extension of A adding only strings of the form Proof . Suppose that no such proper S-consistent extension of A exists. Consider the following polynomial: Y q2S Where P q is dened as in Lemma 4.6, equation 2. Initially Q the degree of Q x i 2 5n . Since every extension of A with strings of the w> is not S consistent it follows that Q x i computes the OR of the variables y <z;x i ;w> . Since there are 2 n 2 many of those variables we have by Lemma 4.5 a contradiction with the degree of Q x i . Hence there exists a proper S-consistent extension of A adding only strings of the form properly coded into A. 2 Stage n ends after coding all the strings of length n. This completes the proof of Theorem 4.3 2 Theorem 4.3 together with the proof of Lemma 4.2 also gives the following corollary. Corollary 4.8 There exists a relativized world where where EXP NP is in BPP and Our oracle also extends the oracle of Ko [Ko91] to CD poly complexity as follows. Corollary 4.9 There exists an oracle such that R CD t; for any t 2 !(n log(n)) and > 0 is complete for NP under strong nondeterministic reductions and . The oracle from Theorem 4.3 is a world where coNP BPP and poly poly (xjy)+O(1), hence it follows that R CD Corollary 3.12 relativizes so by Item 1 we have that BPP NP R CD t; . 2 As a byproduct our oracle shows the following. Corollary 4.10 9A Unique-SAT A 2 P A and P NP A corollary indicates that the current proof that shows that if Unique- pcan not be improved to yield a collapse to P NP using relativizing techniques. 5 PSPACE and cR CS s In this section we further study the connection between cR CS s and interactive proofs. So far we have established that strings that have suciently high CND poly complexity can be used to derandomize an IP protocol that has two rounds in such a way that the role of both the prover and the verier can be played by an NP oracle machine. Here we will see that this is also true for IP itself provided that the random strings have high enough space bounded Kolmogorov complexity. The class of quantied boolean formulas (QBF) is dened as the closure of the set of boolean variables x i and their negations x i under the operations ^, _ , 8x i and 9x i . A QBF in which all the variables are quantied is called closed. Other QBFs are called open. We need the following denitions and theorems from [Sha92]. Denition 5.1 ([Sha92]) A QBF B is called simple if in the given syntactic representation every occurrence of each variable is separated from its point of quantication by at most one universal quantier (and arbitrarily many other symbols). For technical reasons we also assume that (simple) QBFs can contain negated variables, but no other negations. This is no loss of generality since negations can be pushed all the way down to variables. Denition 5.2 ([Sha92]) The arithmetization of a is an expression obtained from B by replacing every positive occurrence of x i by variable z i , every negated occurrence of x i by (1 z i ), every ^ by , every _ by +, every 8x i by z i 2f0;1g , and every 9x i by z i 2f0;1g . It follows that the arithmetization of a (simple) QBF in closed form has an integer value, whereas the arithmetization of an open QBF is equivalent to a (possibly multivariate) function. Denition 5.3 ([Sha92]) The functional form of a simple closed QBF is the univariate function that is obtained by removing from the arithmetization of B either z i 2f0;1g or z i 2f0;1g where i is the least index of a variable for which this is possible. be a (simple) QBF with quantiers . For be the boolean formula obtained from B by removing all its quantiers. We denote by ~ B the arithmetization of B 0 . It is well-known that the language of all true QBFs is complete for PSPACE. The restriction of true QBFs to simple QBFs remains complete. Theorem 5.4 ([Sha92]) The language of all closed simple true QBFs is complete for PSPACE (under polynomial time many-one reductions). It is straightforward that the arithmetization of a QBF takes on a positive value if and only if the QBF is true. This fact also holds relative a not too large prime. Theorem 5.5 ([Sha92]) A simple closed quantied boolean formula B is true if and only if there exists a prime number P of size polynomial in jBj such that the value of the arithmetization of B is positive modulo P . Moreover if B is false then the value of the arithmetization of B is 0 modulo any such prime. Theorem 5.6 ([Sha92]) The functional form of every simple QBF can be represented by a univariate polynomial of degree at most 3. Theorem 5.7 ([Sha92]) For every simple QBF there exists an interactive protocol with prover P and polynomial time bounded verier V such that: 1. When B is true and P is honest, V always accepts the proof. 2. When B is false, V accepts the proof with negligible probability. The proof of Theorem 5.7 essentially uses Theorem 5.6 to translate a simple QBF to a polynomial in the following way. First, the arithmetization of a simple QBF B in closed form is an integer value V which is positive if and only if B is true. Then, B's functional form F (recall: this is arithmetization of the QBF that is obtained from B by deleting the rst quantier) is a univariate polynomial p 1 of degree at most 3 which has the property that p 1 Substituting any value r 1 in p 1 gives a new integer value V 1 , which is of course the same value that we get when we substitute r 1 in F . However, F (r 1 ) can again be converted to a (low degree) polynomial by deleting its rst P or Q sign and the above game can be repeated. Thus, we obtain a sequence of polynomials. From the rst polynomial in this sequence V can be computed. The last polynomial p n has the property that p n (r things are needed: First, if any other sequence of polynomials q has the property that q 1 there has to be some i where q i (r is an intersection point of p i and q i . Second, all calculations can be done modulo some prime number of polynomial size (Theorem 5.5). We summarize this in the following observation, which is actually a skeleton of the proof of Theorem 5.7. Observation 5.8 ([Sha92],[LFKN92]) Let B be a closed simple QBF wherein the quantiers are Q if read from left to right in its syntactic representation. Let A be its arithmetization, and let V be the value of A. There exist a prime number P of size polynomial in jBj such that for any sequence r of numbers taken from [1::P ] there is a sequence of polynomials of degree at most 3 and size polynomial in jBj such that: 1. 2. 3. 4. For any sequence of univariate polynomials q (a) (b) q (c) q n (r n there is a minimal i such that p i is an intersection point of p i and q i . Where all (in)equalities hold modulo P and hold modulo any prime of polynomial size if B is false. Moreover, p i can be computed in space (jBj from B, P , r From this reformulation of Theorem 5.7 we obtain that for any sequence of univariate polynomials q and sequence of values r that items 2 and 3 in Observation 5.8 it holds that either q 1 the true value of the arithmetization of B, or there is some polynomial q i in this sequence such that r i is an intersection point of p i and q i (where p i is as in the Observation 5.8). As p i can be computed in quadratic space from B, that in the latter case r i cannot have high space bounded Kolmogorov complexity relative to B, P , Hence, if r i does have high space bounded Kolmogorov complexity, then r i is not an intersection point, so the rst case must hold (i.e., the value computed from q 1 is the true value of the arithmetization of B). The following lemma makes this precise. Lemma 5.9 Assume the following for B, P , n, 1. B is a simple false closed QBF on n variables. 2. P is a prime number 2 jBj of size polynomial in jBj. 3. is a sequence of polynomials of degree 3 with coecients in 4. r are numbers in [1::P ]. 5. 7. 8. ~ Proof: Take all calculations modulo P . Suppose q 1 It follows from Observation 5.8 that there exists a sequence items 1 through 3 of that lemma. Furthermore since B is false prime, so It follows that there must be a minimal i such that p i 6= q i and r i is an intersection point of p i and q i . However p i can be computed in space (jBj from B, P and r As both p i and q i have degree at most 3, it follows that CS n (r bounded by a constant. A contradiction. 2 This suces for the main theorem of this section. Let s be any polynomial Theorem 5.10 PSPACE NP cR CS s Proof: We prove the lemma for the proof can by padding be extended to any polynomial. There exists an NP oracle machine that accepts the language of all simple closed true quantied boolean formulas as follows. On input B rst check that B is simple. Guess a prime number P of size polynomial in B, a sequence of polynomials of degree at most 3 and with coecients in [1::P ]. Finally guess a sequence of numbers all of size jP j. Check that: 1. 2. 3. 4. nally that is at least jP j for all i n. If B is true Lemma 5.8 guarantees that these items can be guessed such that all tests are passed. If B is false and no other test fails then Lemma 5.9 guarantees that p 1 so the rst check must fail. 2 By the fact that PSPACE is closed under complement and the fact that cR CS s is also in PSPACE Theorem 5.10 gives that cR CS s is complete for PSPACE under strong nondeterministic reductions [Lon82]. Corollary 5.11 cR CS s is complete for PSPACE under strong nondeterministic reductions. Buhrman and Mayordomo [BM95] showed that for , the set R C jxjg is not hard for EXP under deterministic Turing reductions. In Theorem 5.10 we made use of the relativized Kolmogorov complexity (i.e., CS s (xjy)). Using exactly the same proof as in [BM95] one can prove that the set cR C jxjg is not hard for EXP under Turing reductions. On the other hand the proof of Theorem 5.10 also works for this set: PSPACE NP cR C t . We suspect that it is possible to extend this to show that EXP NP cR C t . So far, we have been unable to prove this. Acknowledgements We thank Paul Vitanyi for interesting discussions and providing the title of this paper. We also thank two anonymous referees who helped with a number of technical issues that cleared up much o the proofs and who pointed to us to more correct references. One of the referees also pointed out Corollary 4.8. --R Trading group theory for randomness. Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set. NP might not be as easy as detecting unique solutions. Resource bounded kolmogorov complexity revisited. Resource bounded kolmogorov complexity revisited. Six hypotheses in search of a theorem. Superpolynomial circuits An excursion to the kolmogorov random strings. Complete sets and structure in subrecursive classes. On completeness and soundness in interactive proof systems. Generalized Kolmogorov complexity and the structure of feasible computations. On the computational complexity of algorithms. Two tape simulation of multitape Turing machines. On the complexity of learning minimum time-bounded turing machines On the complexity of random strings (extended abstract). Personal communication. Strong nondeterministic polynomial-time reducibilities Completeness, the recursion theorem and e A complexity theoretic approach to randomness. NP is as easy as detecting unique solutions. Some consequences of non-uniform conditions on uniform classes Probabilistic quanti --TR

relativization;kolmogorov complexity;interactive proofs;randomness;complexity classes;arthur-merlin;merlin-arthur

586952

A Randomized Time-Work Optimal Parallel Algorithm for Finding a Minimum Spanning Forest.

We present a randomized algorithm to find a minimum spanning forest (MSF) in an undirected graph. With high probability, the algorithm runs in logarithmic time and linear work on an exclusive read exclusive write (EREW) PRAM. This result is optimal w.r. t. both work and parallel time, and is the first provably optimal parallel algorithm for this problem under both measures. We also give a simple, general processor allocation scheme for tree-like computations.

Introduction We present a randomized parallel algorithm to find a minimum spanning forest (MSF) in an edge- weighted, undirected graph. On an EREW PRAM [KR90] our algorithm runs in expected logarithmic time and linear work in the size of the input; these bounds also hold with high probability in the size of the input. This result is optimal with respect to both work and parallel time, and is the first provably optimal parallel algorithm for this problem under both measures. Here is a brief summary of related results. Following the linear-time sequential MSF algorithm of Karger, Klein and Tarjan [KKT95] (and building on it) came linear-work parallel MST algorithms for the CRCW PRAM [CKT94, CKT96] and the EREW PRAM [PR97]. The best CRCW PRAM algorithm known to date [CKT96] runs in logarithmic time and linear work, but the time bound is not known to be optimal. The best EREW PRAM algorithm known prior to our work is the result of Poon and Ramachandran which runs in O(log n log log linear work. All of these algorithms are randomized. Recently Chong, Han and Lam [CHL99] presented a deterministic EREW PRAM algorithm for MSF, which runs in logarithmic time with a linear number of processors, and hence with work O((m + n) log n), where n and m are the number of vertices and edges in the input graph. It was observed by Poon and Ramachandran [PR98] that the algorithm in [PR97] could be speeded up to run in O(log n \Delta 2 log n ) time and linear work by using the algorithm in [CHL99] as a subroutine (and by modifying the 'Contract' subroutine in [PR97]). In this paper we improve on the running time of the algorithm in [PR97, PR98] to O(log n), which is the best possible, and we improve on the algorithm in [CKT96] by achieving the logarithmic time bound on the less powerful EREW PRAM. Part of this work was supported by Texas Advanced Research Program Grant 003658-0029-1999. Seth Pettie was also supported by an MCD Fellowship. Our algorithm has a simple 2-phase structure. It makes subroutine calls to the Chong-Han- Lam algorithm [CHL99], which is fairly complex. But outside of these subroutine calls (which are made to the simplest version of the algorithm in [CHL99]), the steps in our algorithm are quite straightforward. In addition to being the first time-work optimal parallel algorithm for MSF, our algorithm can be used as a simpler alternative to several other parallel algorithms: 1. For the CRCW PRAM we can replace the calls to the CHL algorithm by calls to a simple logarithmic time, linear-processor CRCW algorithm such as the one in [AS87]. The resulting algorithm runs in logarithmic time and linear work and is considerably simpler than the MSF algorithm in [CKT96]. 2. As modified for the CRCW PRAM, our algorithm is simpler than the linear-work logarithmic-time CRCW algorithm for connected components given in [Gaz91]. 3. Our algorithm improves on the EREW connectivity and spanning tree algorithms in [HZ94, HZ96] since we compute a minimum spanning tree within the same time and work bounds. Our algorithm is simpler than the algorithms in [HZ94, HZ96]. In the following we use the notation S +T to denote union of sets S and T , and we use S + e to denote the set formed by adding the element e to the set S. We say that a result holds with high probability (or w.h.p.) in n if the probability that it fails to hold is less than 1=n c , for any constant The rest of this paper describes and analyzes our algorithm, and is organized as follows. Section 2 gives a high-level description of our algorithm, which works in two phases. Section 3 describes the details of Phase 1 of our algorithm; the main procedure of Phase 1 is Find-k-Min, which is given in section 3.4. Section 4 gives Phase 2, whose main procedure is Find-MSF. Section 5 gives the proof that our algorithm runs in expected logarithmic time and linear work, and section 6 extends this result to high-probability bounds. Section 7 addresses the issue of processor allocation in the various steps of our algorithm. Section 8 discusses the adaptability of our algorithm to realistic parallel models like the BSP [Val90] and QSM [GMR97] and the paper concludes with section 9. 2 The High-Level Algorithm Our algorithm is divided into two phases along the lines of the CRCW PRAM algorithm of [CKT96]. In Phase 1, the algorithm reduces the number of vertices in the graph from n to n=k vertices, where n is the number of vertices in the input graph, and To perform this reduction the algorithm uses the familiar recursion tree of depth log n [CKT94, CKT96, PR97], which gives rise to O(2 log n ) recursive calls, but the time needed per invocation in our algorithm is well below O(log n=2 log n ). Thus the total time for Phase 1 is O(log n). We accomplish this by requiring Phase 1 to find only a subset of the MSF. By contracting this subset of the MSF we obtain a graph with O(n=k) vertices. Phase 2 then uses an algorithm similar to the one in [PR97], but needs no recursion due to the reduced number of vertices in the graph. Thus Phase 2 is able to find the MSF of the contracted graph in O(log n) time and linear work. We assume that edge weights are unique. As always, uniqueness can be forced by ordering the vertices, then ordering identically weighted edges by their end points. Here is a high-level description of our algorithm. y We use log (r) n to denote the log function iterated r times, and log n to denote the minimum r s.t. log (r) n - 1. (Phase retain the lightest k edges in edge-list(v) G 0 :=Contract all edges in G appearing in M (Phase :=Sample edges of G 0 with prob. 1= log (2) n Theorem 2.1 With high probability, High-Level(G) returns the MSF of G in O(log n) time using processors. In the following sections we describe and analyze the algorithms for Phase 1 and Phase 2, and then present the proof of the main theorem for the expected running time. We then obtain a high probability bound for the running time and work. When analyzing the performance of the algorithms in Phase 1 and Phase 2, we use a time-work framework, assuming perfect processor allocation. This can be achieved with high probability to within a constant factor, using the load-balancing scheme in [HZ94], which requires superlinear space, or the linear-space scheme claimed in [HZ96]. We discuss processor allocation in Section 7 where we point out that a simple scheme similar to the one in [HZ94] takes only linear space on the QRQW PRAM [GMR94], which is a slightly stronger model than the EREW PRAM. The usefulness of the QRQW PRAM lies in the fact the algorithms designed on that model map on to general-purpose models such as QSM [GMR97] and BSP [Val90] just as well as the EREW PRAM. We then describe the performance of our MSF algorithm on the QSM and BSP. In Phase 1, our goal is to contract the input graph G into a graph with O(n=k) vertices. We do this by identifying certain edges in the minimum spanning forest of G and contracting the connected components formed by these edges. The challenge here is to identify these edges in logarithmic time and linear work. Phase 1 achieves the desired reduction in the number of vertices by constructing a k-Min forest (defined below). This is similar to the algorithm in [CKT96]. However, our algorithm is considerably simpler. We show that a k-Min forest satisfies certain properties, and we exploit these properties to design a procedure Bor-uvka-A, which keeps the sizes of the trees contracted in the various stages of Phase 1 to be very small so that the total time needed for contracting and processing edges in these trees is o(log n=2 log n ). Phase 1 also needs a Filter subroutine, which removes 'k-min heavy' edges. For this, we show that we can use an MSF verification algorithm on the small trees we construct to perform this step. The overall algorithm for Phase 1, Find-k-Min uses these two subroutines to achieve the stated reduction in the number of vertices within the desired time and work bounds. 3.1 k-Min Forest Phase 1 uses the familiar 'sample, contract and discard edges' framework of earlier randomized algorithms for the MSF problem [KKT95, CKT94, CKT96, PR97]. However, instead of computing a minimum spanning forest, we will construct the k-Min tree [CKT96] of each vertex (where (log (2) n) 2 ). Contracting the edges in these k-Min trees will produce a graph with O(n=k) vertices. To understand what a k-Min tree is, consider the Dijkstra-Jarnik-Prim minimum spanning tree algorithm: (choose an arbitrary starting vertex v) Repeat until T contains the MST of G Choose minimum weight edge (a; b) s.t a 2 S, b 62 S S The edge set k-Min(v) consists of the first k edges chosen by this algorithm, when started at vertex v. A forest F is a k-Min forest of G if F ' MSF(G) and for all v 2 G; k-Min(v) ' F . be the set of edges on the path from x to y in tree T , and let maxweightfAg be the maximum weight in a set of edges A. For any forest F in G, define an edge (a; b) in G to be F -heavy if weight(a; b) ? maxweightfP F (a; b)g and to be F -light otherwise. If a and b are not in the same tree in F then (a; b) is F-light. Let M be the k-Min tree of v. We define weight v (w) to be maxweightfPM (v; w)g if w appears in maxweightfk-Min(v)g. Define an edge (a; b) to be k-Min-heavy maxfweight a (b); weight b (a)g, and to be k-Min-light otherwise. 3.1 Let the measure weight v (w) be defined with respect to any k in the range [1.n]. Then weight v (w) - maxweightfPMSF (v; w)g. Proof: There are two cases, when w falls inside the k-Min tree of v, and when it falls outside. If w is inside k-Min(v), then weight v (w) is the same as maxweightfPMSF (v; w)g since k-Min(v) ' MSF . Now suppose that w falls outside k-Min(v) and weight v (w) ? maxweightfPMSF (v; w)g. There must be a path from v to w in the MSF consisting of edges lighter than maxweightfk-Min(v)g. However, at each step in the Dijkstra-Jarnik-Prim algorithm, at least one edge in PMSF is eligible to be chosen in that step. Since w 62 k-Min(v), the edge with weight maxweightfk-Min(v)g is never chosen. Contradiction. 2 Let K be a vector of n values, each in the range [1::n]. Each vertex u is associated with a value of denoted k u . Define an edge (u; v) to be K-Min-light if weight(u; v) ! maxfweight u (v); weight v (u)g, where weight u (v) and weight v (u) are defined with respect to k u and k v respectively. Lemma 3.1 Let H be a graph formed by sampling each edge in graph G with probability p. The expected number of edges in G that are K-Min-light in H is less than n=p, for any K. Proof: We show that any edge that is K-Min-light in G is also F -light where F is the MSF of H. The lemma then follows from the sampling lemma of [KKT95] which states that the expected number of F -light edges in G is less than n=p. Let us look at any K-Min-light edge (v; w). By 3.1, weight v (w) - maxweightfPMSF (v; w)g, the measure used to determine F -lightness. Thus the criterion for K-Min-lightness, maxfweight v (w); weight w (v)g, must also be less than or equal to maxweightfPMSF (v; w)g. Restating this, if (v; w) is K-Min-light, it must be F -light as well. 2 We will use the above property of a k-Min forest to develop a procedure Find-k-Min(G; l). It takes as input the graph G and a suitable positive integer l, and returns a k-Min forest of G. For runs in logarithmic time and linear work. In the next few sections we describe some basic steps and procedures used in Find-k-Min, and then present and analyze this main procedure of Phase 1. Phase 1 is concerned only with the k-Min tree of each vertex, it suffices to retain only the lightest k edges incident on each vertex. Hence as stated in the first step of Phase 1 in algorithm High-Level in Section 2 we will discard all but the lightest k edges incident on each vertex since we will not need them until Phase 2. This step can be performed in logarithmic time and linear work by a simple randomized algorithm that selects a sample of size jLj from each adjacency list L, sorts this sample, and then uses this sorted list to narrow the search for the kth smallest element to a list of size O(jLj 3=4 ). 3.2 Bor-uvka-A Steps In a basic Bor-uvka step [Bor26], each vertex chooses its minimum weight incident edge, inducing a number of disjoint trees. All such trees are then contracted into single vertices, and useless edges discarded. We will call edges connecting two vertices in the same tree internal and all others external. All internal edges are useless, and if multiple external edges join the same two trees, all but the lightest are useless. Our algorithm for Phase 1 uses a modified Bor-uvka step in order to reduce the time bound to o(log n) per step. All vertices are classified as being either live or dead. After a modified Bor-uvka step, vertex v's parent pointer is is the edge of minimum weight incident on v. In addition, each vertex has a threshold which keeps the weight of the lightest discarded edge adjacent to v. The algorithm discards edges known not to be in the k-Min tree of any vertex. The threshold variable guards against vertices choosing edges which may not be in the MSF. A dead vertex v has the useful property (shown below) that for any edge (a; b) in k-Min(v), weight(a; b) - weight(v; p(v)), thus dead vertices need not participate in any more Bor-uvka steps. It is well-known that a Bor-uvka step generates a forest of pseudo-trees, where each pseudo-tree is a tree together with one extra edge that forms a cycle of length 2. In our algorithm we will assume that a Bor-uvka step also removes one of the edges in the cycle so that it generates a collection of rooted trees. The following three claims refer to any tree resulting from a modified Bor-uvka step. Their proofs are straightforward and are omitted. 3.2 The sequence of edge weights encountered on a path from v to root(v) is monotonically decreasing. 3.3 If consists of the edges in the path from v to root(v). Furthermore, the weight of (v; p(v)) is greater than any other edge in d-Min(v). 3.4 If the minimum-weight incident edge of u is (u; v), k-Min(u) ' (k-Min(v) T be a tree induced by a Bor-uvka step, and let T 0 be a subtree of T . If e is the minimum weight incident edge on T , then the minimum weight incident edge on T 0 is either e or an edge of T . Proof: Suppose, on the contrary that the minimum weight incident edge on T 0 is e 0 62 T , and let v and v 0 be the end points of e and e 0 which are inside T . Consider the paths P (v 0 ) to the root of T . By Claim 3.2, the edge weights encountered on P and P 0 are monotonically decreasing. There are two cases. If T 0 contains some, but not all of P 0 , then e 0 must lie along P 0 . Contradiction. If T 0 contains all of P 0 , but only some of P , then some edge e 00 2 P is adjacent to The procedure Bor-uvka-A(H; l; F ) given below returns a contracted version of H with the number of live vertices reduced by a factor of l. Edges designated as parent pointers, which are guaranteed to be in the MSF of H, are returned in F . Initially Repeat log l times: (log l modified Bor-uvka steps) For each live vertex v Choose min. weight edge (v; w) (1) If weight(v; w) ? threshold(v), v becomes dead, stop else Each tree T induced by edges of F 0 is one of two types: If root of T is dead, then (2) Every vertex in T becomes dead (Claim 3.4) If T contains only live vertices (3) If depth(v) - k, v becomes dead (Claim 3.3) Contract the subtree of T made up of live vertices The resulting vertex is live, has no parent pointer, and keeps the smallest threshold of its constituent vertices Lemma 3.2 If Bor-uvka-A designates a vertex as dead, its k-Min tree has already been found. Proof: Vertices make the transition from live to dead only at the lines indicated by a number. By our assumption that we only discard edges that cannot be in the k-Min tree of any vertex, if the lightest edge adjacent to any vertex has been discarded, we know its k-Min tree has already been found. This covers line (1). The correctness of line (2) follows from Claim 3.4. Since (v; p(v)) is the lightest incident edge on v, k-Min(v) ' be called dead. Since the root of a tree is dead, vertices at depth one are dead, implying vertices at depth two are dead, and so on. The validity of line (3) follows directly from Claim 3.3. If a vertex finds itself at depth - k, its k-Min tree lies along the path from the vertex to its root. 2 Lemma 3.3 After a call to Bor-uvka-A(H; k tree of each vertex is a subset of F . Proof: By Lemma 3.2, dead vertices already satisfy the lemma. After a single modified Bor-uvka step, the set of parent pointers associated with live vertices induce a number of trees. Let T (v) be the tree containing v. We assume inductively that after dlog ie modified Bor-uvka steps, the tree of each vertex in the original graph has been found (this is clearly true for For any live vertex v let (x; y) be the minimum weight edge s.t. x 2 T (v); y 62 T (v). By the inductive hypothesis, the (i \Gamma 1)-Min trees of v and y are subsets of T (v) and T (y) respectively. By is the first external edge of T (v) chosen by the Dijkstra-Jarnik-Prim algorithm, starting at v. As every edge in (i \Gamma 1)-Min(y) is lighter than (x; y), is a subset of chosen in the (dlog ie th modified Bor-uvka step, is a subset of T (v) after dlog ie modified Bor-uvka steps. Thus after steps, the k-Min tree of each vertex has been found. 2 Lemma 3.4 After b modified Bor-uvka steps, the length of any edge list is bounded by k k b Proof: This is true for Assuming the lemma holds for modified Bor-uvka steps, the length of any edge list after that many steps is - k k . Since we only contract trees of height ! k, the length of any edge list after b steps is . 2 It is shown in the next section that our algorithm only deals with graphs that are the result of O(log modified Bor-uvka steps. Hence the maximum length edge list is k k O(log The costliest step in Bor-uvka-A is calculating the depth of each vertex. After the minimum weight edge selection process, the root of each induced tree will broadcast its depth to all depth 1 vertices, which in turn broadcast to depth 2 vertices, etc. Once a vertex knows it is at depth may stop, letting all its descendents infer that they are at depth - k. Interleaved with each round of broadcasting is a processor allocation step. We account for this cost separately in section 7. Lemma 3.5 Let G 1 have m 1 edges. Then a call to Bor-uvka-A(G 1 ; l; F ) can be executed in time O(k O(log processors. Proof: Let G 1 be the result of b modified Bor-uvka steps. By Lemma 3.4, the maximum degree of any vertex after the i th modified Bor-uvka step in the current call to Bor-uvka-A is k k b+i . Let us now look at the required time of the i th modified Bor-uvka step. Selecting the minimum cost incident edge takes time log k k b+i , while the time to determine the depth of each vertex is k \Delta log k k b+i . Summing over the log l modified Bor-uvka steps, the total time is bounded by P log l As noted above, the algorithm performs O(log modified Bor-uvka steps on any graph, hence the time is k O(log The work performed in each modified Bor-uvka step is linear in the number of edges. Summing over log l such steps and dividing by the number of processors, we arrive at the second term in the stated running time. 2 3.3 The Filtering Step The Filter Forest Concurrent with each modified Bor-uvka step, we will maintain a Filter forest, a structure that records which vertices merged together at what time, and the edge weights involved. (This structure appeared first in [King97]). If v is a vertex of the original graph, or a new vertex resulting from contracting a set of edges, there is a corresponding vertex OE(v) in the Filter for- est. During a Bor-uvka step, if a vertex v becomes dead, a new vertex w is added to the Filter forest, as well as a directed edge (OE(v); w) having the same weight as (v; p(v)). If live vertices are contracted into a live vertex v, a vertex OE(v) is added to the Filter forest in addition to directed edges having the weights of edges (v It is shown in [King97] that the heaviest weight in the path from u to v in the MSF is the same as the heaviest weight in the path from OE(u) to OE(v) in the Filter forest (if there is such a path). Hence the measures weight v (w) can be easily computed in the following way. Let P f (x; y) be the path from x to y in the Filter forest. If OE(v) and OE(w) are not in the same Filter tree, then weight weight w If v and w are in the same Filter tree, let weight 3.6 The maximum weight on the path from OE(v) to root(OE(v)) is the same as the maximum weight edge in r-Min(v), for some r. Proof: If root(OE(v)) is at height h, then it is the result of h Bor-uvka steps. Assume that the claim holds for the first i ! h Bor-uvka steps. After a number of contractions, vertex v of the original graph is now represented in the current graph by v c . Let T vc be the tree induced by the th Bor-uvka step which contains v c , and let e be the minimum weight incident edge on T vc . By the inductive hypothesis, maxweightfP f (OE(v); OE(T vc As was shown in the proof of Claim 3.5, all edges on the path from v c to edge e have weight at most weight(e)g. Each of the edges (v c ; p(v c )) and e has a corresponding edge in the Filter forest, namely (OE(v c ); p(OE(v c ))) and (OE(T vc ); p(OE(T vc ))). Since both these edges are on the path from OE(v) to p(OE(T vc )), maxweightfP f (OE(v); p(OE(T vc . Thus the claim holds after The Filter Step In a call to Filter(H; F ) in Find-k-Min, we examine each edge e from H if weight(e) ? maxfweight v (w); weight w (v)g In order to carry out this test we can use the O(log n) time, O(m) work MSF verification algorithm of [KPRS97], where we modify the algorithm for the case when x and y are not in the same tree to test the pairs (OE(x); root(OE(x)) and (OE(y); root(OE(y)), and we delete e if both of these pairs are identified to be deleted. This computation will take time O(log r) where r is the size of the largest tree formed. The procedure Filter discards edges that cannot be in the k-Min tree of any vertex. When it discards an edge (a; b), it updates the threshold variables of both a and b, so that threshold(a) is the weight of the lightest discarded edge adjacent to a. If a's minimum weight edge is ever heavier than threshold(a), k-Min(a) has already been found, and a becomes dead. be a graph formed by sampling each edge in H with probability p, and F be a k-Min forest of H 0 . The call to Filter(H; F ) returns a graph containing a k-Min forest of H, whose expected number of edges is n=p. Proof: For each vertex v, Claim 3.6 states that maxweightfP f (OE(v); Min(v) for some value k v . By building a vector K of such values, one for each vertex, we are able to check for K-Min-lightness using the Filter forest. It follows from Lemma 3.1 that the expected number of K-Min-light edges in H is less than n=p. Now we need only show that a k-Min-light edge of H is not removed in the Filter step. Suppose that edge (u; v) is in the k-Min tree of u in H, but is removed by Filter. If v is in the k u -Min tree of u (w.r.t. H 0 ), then edge (u; v) was the heaviest edge in a cycle and could not have been in the MSF, much less any k-Min tree. If v was not in the k u -Min tree of u (w.r.t. H 0 ), then weight(u; v) ? maxweightfk u -Min(u)g, meaning edge (u; v) could not have been picked in the first k steps of the Dijkstra-Jarnik-Prim algorithm. 2 3.4 Finding a k-Min Forest We are now ready to present the main procedure of Phase 1, Find-k-Min. (Recall that the initial call - given in Section 2 - is Find-k-Min(G t ; log n), where G t is the graph obtained from G by removing all but the k lightest edges on each adjacency list.) Find-k-Min(H; i) sample edges of H c with prob. 1=(log (i\Gamma1) n) 2 H is a graph with some vertices possibly marked as dead; i is a parameter that indicates the level of recursion (which determines the number of Bor-uvka steps to be performed and the sampling probability). Lemma 3.6 The call Find-k-Min(G t ; log n) returns a set of edges that includes the k-Min tree of each vertex in G t . Proof: The proof is by induction on i. Base: returns F , which by Lemma 3.3 contains the k-min tree of each vertex. Induction Step: Assume inductively that Find-k-Min(H; i\Gamma1) returns the k-min tree of H. Consider the call Find-k-Min(H; i). By the induction assumption the call to Find-k-Min(H s returns the k-min tree of each vertex in H s . By Claim 3.7 the call to Filter(H c ; F s ) returns in H f a set of edges that contains the k-Min trees of all vertices in H c . Finally, by the inductive assumption, the set of edges returned by the call to Find-k-min(H f contains the k-Min trees of all vertices in contains the (log (i\Gamma1) n)-Min tree of each vertex in H, and Find-k-Min(H; i) returns returns the edges in the k-Min tree of each vertex in H. 2 3.8 The following invariants are maintained at each call to Find-k-min. The number of live vertices in H - n=(log (i) n) 4 , and the expected number of edges in H - m=(log (i) n) 2 , where m and n are the number of edges and vertices in the original graph. Proof: These clearly hold for the initial call, when log n. By Lemma 3.3, the contracted graph H c has no more than n=(log (i\Gamma1) n) 4 live vertices. Since H s is derived by sampling edges with probability 1=(log (i\Gamma1) n) 2 , the expected number of edges in H s is - m=(log (i\Gamma1) n) 2 , maintaining the invariants for the first recursive call. By Lemma 3.1, the expected number of edges in H f - n(log (i\Gamma1) n) 2 (log (i\Gamma1) n) 4 has the same number of vertices as H c , both invariants are maintained for the second recursive call.3.5 Performance of Find-k-Min Lemma 3.7 Find-k-min(G t ; log n) runs in expected time O(log n) and work O(m n). Proof: Since recursive calls to Find-k-min proceed in a sequential fashion, the total running time is the sum of the local computation performed in each invocation. Aside from randomly sampling the edges, which takes constant time and work linear in the number of edges, the local computation consists of calls to Filter and Bor-uvka-A. In a given invocation of Find-k-min, the number of Bor-uvka steps performed on graph H is the sum of all Bor-uvka steps performed in all ancestral invocations of Find-k-min, i.e. P log n which is O(log (3) n). ?From our bound on the maximum length of edge lists (Lemma 3.4), we can infer that the size of any tree in the Filter forest is k k O(log (3) n) , thus the time needed for each modified Bor-uvka step and each Filter step is k O(log (3) n) . Summing over all such steps, the total time required is o(log n). The work required by the Filter procedure and each Bor-uvka step is linear in the number of edges. As the number of edges in any given invocation is O(m=(log (i) n) 2 ), and there are O(log (i) n) Bor-uvka steps performed in this invocation, the work required in each invocation is O(m= log (i) n) (recall that the i parameter indicates the depth of recursion). Since there are 2 log n\Gammai invocations with depth parameter i, the total work is given by P log n log n\Gammai O(m= log (i) n), which is O(m).4 Phase 2 Recall the Phase 2 portion of our overall algorithm High-Level: (the number of vertices in G s is - n=k) G s :=Sample edges of G 0 with prob. 1= log (2) n The procedure Filter(G; F ) ([KPRS97]) returns the F -light edges of G. The procedure Find- described below, finds the MSF of G 1 in time O((m 1 =m) log n log (2) n), where m 1 is the number of edges in G 1 . The graphs G s and G f each have expected m= log (2) n edges since G s is derived by sampling each edge with probability 1= k, and by the sampling lemma of [KKT95], the expected number of edges in G f is (m=k)=(1= k. Because we call Find-MSF on graphs having expected size O(m= log (2) n), each call takes O(log n) time. 4.1 The Find-MSF Procedure The procedure Find-MSF(H) is similar to previous randomized parallel algorithms, except it uses no recursion. Instead, a separate base case algorithm is used in place of recursive calls. We also use slightly different Bor-uvka steps, in order to reduce the work. These modifications are inspired by [PR97] and [PR98] respectively. As its Base-case, we use the simplest version of the algorithm of Chong et al. [CHL99], which takes time O(log n) using (m+n) log n processors. By guaranteeing that it is only called on graphs of expected size O(m= log 2 n), the running time remains O(log n) with (m processors. Find-MSF(H) H s := Sample edges of H c with prob. After the call to Bor-uvka-B, the graph H c has ! m= log 4 n vertices. Since H s is derived by sampling the edges of H c with probability 1= log 2 n, the expected number of edges to the first BaseCase call is O(m= log 2 n). By the sampling lemma of [KKT95], the expected number of edges to the second BaseCase call is ! (m= log 4 n)=(1= log 2 n), thus the total time spent in these subcalls is O(log n). Assuming the size of H conforms to its expectation of O(m= log (2) n), the calls to Filter and Bor-uvka-B also take O(log n) time, as described below. The Bor-uvka-B(H; l; F ) procedure returns a contracted version of H with O(m=l) vertices. It uses a simple growth control schedule, designating vertices as inactive if their degree exceeds l. We can determine if a vertex is inactive by performing list ranking on its edge list for log l time steps. If the computation has not stopped after this much time, then its edge list has length ? l. Bor-uvka-B(G; Repeat log l times For each vertex, let it be inactive if its edge list has more than l edges, and active otherwise. For each active vertex v choose min. weight incident edge e Using the edge-plugging technique, build a single edge list for each induced tree (O(1) time) Contract all trees of inactive vertices The last step takes O(log n) time; all other steps take O(log l) time, as they deal with edge lists of length O(l). Consequently, the total running time is O(log l). For each iteration of the main loop, the work is linear in the number of edges. Assuming the graph conforms to its expected size of O(m= log (2) n), the total work is linear. The edge-plugging technique as well as the idea of a growth control schedule were introduced by Johnson & Metaxas [JM92]. 5 Proof of Main Theorem Proof: (Of Theorem 2.1) The set of edges M returned by Find-k-Min is a subset of the MSF of G. By contracting the edges of M to produce G 0 , the MSF of G is given by the edges of M together with the MSF of G 0 . The call to Filter produces graph G f by removing from G 0 edges known not to be in the MSF. Thus the MSF of G f is the same as the MSF of G 0 . Assuming the correctness of Find-MSF, the set of edges F constitutes the MSF of G f , thus M + F is the MSF of G. Earlier we have shown that each step of High-Level requires O(log n) time and work linear in the number of edges. In the next two sections we show that w.h.p, the number of edges encountered in all graphs during the algorithm is linear in the size of the original graph. 2 6 High Probability Bounds Consider a single invocation of Find-k-min(H; i), where H has m 0 edges and n 0 vertices. We want to place likely bounds on the number of edges in each recursive call to Find-k-min, in terms of m 0 and i. For the first recursive call, the edges of H are sampled independently with probability 1=(log (i\Gamma1) n) 2 . Call the sampled graph H 1 . By applying a Chernoff bound, the probability that the size of H 1 is less than twice its expectation is Before analyzing the second recursive call, we recall the sampling lemma of [KKT95] which states that the number of F -light edges conforms to the negative binomial distribution with parameters is the sampling probability, and F is the MSF of H 1 . As we saw in the proof of Lemma 3.1, every k-Min-light edge must also be F -light. Using this observation, we will analyze the size of the second recursive call in terms of F -light edges, and conclude that any bounds we attain apply equally to k-Min-light edges. We now bound the likelihood that more than twice the expected number of edges are F -light. This is the probability that in a sequence of more than 2n 0 =p flips of a coin, with probability p of heads, the coin comes up heads less than n 0 times (since each edge selected by a coin toss of heads goes into the MSF of the sampled graph). By applying a Chernoff bound, this is exp(\Gamma\Omega\Gamma n 0 )). In this particular instance of Find-k-min, n 0 - m=(log (i\Gamma1) n) 4 and so the probability that fewer than 2m=(log (i\Gamma1) n) 2 edges are F -light is Given a single invocation of Find-k-min(H; i), we can bound the probability that H has more than 2 log n\Gammai m=(log (i) n) 2 edges by exp(\Gamma\Omega\Gamma m=(log (i) n) 4 )). This follows from applying the argument used above to each invocation of Find-k-min from the initial call to the current call at depth log Summing over all recursive calls to Find-k-min, the total number of edges (and thus the total work) is bounded by P log n The probability that Phase 2 uses O(m) work is We omit the analysis as it is similar to the analysis for Phase 1. The probability that our bounds on the time and total work performed by the algorithm fail to hold is exponentially small in the input size. However, this assumes perfect processor allocation. In the next section we show that the probability that work fails to be distributed evenly among the processors is less than 1=m !(1) . Thus the overall probability of failure is very small, and the algorithm runs in logarithmic time and linear work w.h.p. 7 Processor Allocation As stated in Section 2, the processor allocation needed for our algorithm can be performed by a fairly simple algorithm given in [HZ94] that takes logarithmic time and linear work but uses super-linear space, or by a more involved algorithm claimed in [HZ96] that runs in logarithmic time and linear work and space. We show here that a simple algorithm similar in spirit to the one in [HZ94] runs in logarithmic time and linear work and space on the QRQW PRAM [GMR94]. The QRQW PRAM is intermediate in power between the EREW and CRCW PRAM in that it allows concurrent memory accesses, but the time taken by such accesses is equal to the largest number of processors accessing any single memory location. We assume that the total size of our input is n, and that we have processors. We group the q processors into q=r groups of size r = log n and we make an initial assignment of O(r log n) elements to each group. This initial assignment is made by having each element choose a group randomly. The expected number of elements in each group is r log n and by a Chernoff bound, w.h.p. there are O(r log n) elements in each group. Vertices assigned to each group can be collected together in an array for that group in O(log n) time and O(n) work and space by using the QRQW PRAM algorithm for multiple compaction given in [GMR96], which runs in logarithmic time and linear work with high probability. (We do not need the full power of the algorithm in [GMR96] since we know ahead of time that each group has - c log 2 n elements w.h.p., for a suitable constant c. Hence it suffices to use the heavy multiple compaction algorithm in [GMR96] to achieve the bounds of logarithmic time and linear work and space.) A simple analysis using Chernoff bounds shows that on each new graph encountered during the computation each group receives either ! log n elements, or within a constant factor of its expected number of elements w.h.p. Hence in O(log log n) EREW PRAM steps each processor within a group can be assigned 1=(log n) of the elements in its group. This processor re-allocation scheme takes O(log log n) time per stage and linear space overall, and with high probability, achieves perfect balance to within a constant factor. The total number of processor re-allocation steps needed by our algorithm is O(2 log n \Delta k log log log n), hence the time needed to perform all of the processor allocation steps is O(log n) w.h.p. We note that the probability that processors are allocated optimally (to within a constant can be increased to 1 \Gamma n \Gamma!(1) by increasing the group size r. Since we perform o((log (2) n) 3 ) processor allocation steps, r can be set as high as n 1=(log (2) n) 3 without increasing the overall O(log n) running time. Thus the high probability bound on the number of items in each group being O(r log n) becomes 1\Gamman \Gamma!(1) . It is shown in [GMR96] that the heavy multiple compaction algorithm runs in time O(log n log m= log log m) time w.h.p. in m, for any m ? 0. By choosing log log n= log n , we obtain O(log n) running time for this initial step with probability which is also the overall probability bound for processor allocation. 8 Adaptations to other Practical Parallel Models Our results imply good MSF algorithms for the QSM [GMR97] and BSP [Val90] models, which are more realistic models of parallel computation than the PRAM models. Theorem 8.1 given below follows directly from results mapping EREW and QRQW computations on to QSM given in [GMR97]. Theorem 8.2 follows from the QSM to BSP emulation given in [GMR97] in conjunction with the observation that the slowdown in that emulation due to hashing does not occur for our algorithm since the assignment of vertices and edges to processors made by our processor allocation scheme achieves the same effect. Theorem 8.1 An MSF of an edge-weighted graph on n nodes and m edges can be found in O(g log n) time and O(g(m using O(m n) space on the QSM with a simple processor allocation scheme, where g is the gap parameter of the QSM. Theorem 8.2 An MSF of an edge-weighted graph on n nodes and m edges can be found on the BSP in O((L + g) log n) time w.h.p., using (m processors and O(m n) space with a simple processor allocation scheme, where g and L are the gap and periodicity parameters of the BSP. 9 Conclusion We have presented a randomized algorithm for MSF on the EREWPRAM which is provably optimal both in time and work. Our algorithm works within the stated bounds with high probability in the input size, and has good performance in other popular parallel models. An important open question that remains is to obtain a deterministic parallel MSF algorithm that is provably optimal in time and work. Recently an optimal deterministic sequential algorithm for MSF was presented in [PR00]; an intriguing aspect of this algorithm is that the function describing its running time is not known at present, although it is proven in [PR00] that the algorithm runs within a small constant factor of the best possible. Parallelizing this optimal sequential algorithm is a topic worth investigating. --R New connectivity and MSF algorithms for shuffle-exchange networks and PRAM O jist'em probl'emu minima'aln ' im. Moravsk'e P On the parallel time complexity of undirected connectivity and minimum spanning trees. A linear-work parallel algorithm for finding minimum spanning trees Finding minimum spanning trees in logarithmic time and linear work using random sampling. A note on two problems in connexion with graphs. The QRQW PRAM: Accounting for contention in parallel algorithms. Efficient low-contention parallel algorithms Can a shared-memory model serve as a bridging model for parallel computation? Theory of Computing Systems An optimal randomized logarithmic time connectivity algorithm for the EREW PRAM. Optimal randomized EREW PRAM algorithms for finding spanning forests and for other basic graph connectivity problems. Connected components in O(log 3 A simpler minimum spanning tree verification algorithm. A randomized linear-time algorithm to find minimum spanning trees An optimal EREW PRAM algorithm for minimum spanning tree verification. Parallel algorithms for shared-memory machines A randomized linear work EREW PRAM algorithm to find a minimum spanning forest. Private communication An optimal minimum spanning tree algorithm. A bridging model for parallel computation. Shortest connection networks and some generalizations. --TR --CTR Aaron Windsor, An NC algorithm for finding a maximal acyclic set in a graph, Proceedings of the sixteenth annual ACM symposium on Parallelism in algorithms and architectures, June 27-30, 2004, Barcelona, Spain Vladimir Trifonov, An O(log n log log n) space algorithm for undirected st-connectivity, Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, May 22-24, 2005, Baltimore, MD, USA David A. Bader , Guojing Cong, Fast shared-memory algorithms for computing the minimum spanning forest of sparse graphs, Journal of Parallel and Distributed Computing, v.66 n.11, p.1366-1378, November 2006 Guojing Cong , David A. Bader, Designing irregular parallel algorithms with mutual exclusion and lock-free protocols, Journal of Parallel and Distributed Computing, v.66 n.6, p.854-866, June 2006

EREW PRAM;optimal algorithm;parallel algorithm;minimum spanning tree

586956

Hardness of Approximate Hypergraph Coloring.

We introduce the notion of covering complexity of a verifier for probabilistically checkable proofs (PCPs). Such a verifier is given an input, a claimed theorem, and an oracle, representing a purported proof of the theorem. The verifier is also given a random string and decides whether to accept the proof or not, based on the given random string. We define the covering complexity of such a verifier, on a given input, to be the minimum number of proofs needed to "satisfy" the verifier on every random string; i.e., on every random string, at least one of the given proofs must be accepted by the verifier. The covering complexity of PCP verifiers offers a promising route to getting stronger inapproximability results for some minimization problems and, in particular, (hyper)graph coloring problems. We present a PCP verifier for NP statements that queries only four bits and yet has a covering complexity of one for true statements and a superconstant covering complexity for statements not in the language. Moreover, the acceptance predicate of this verifier is a simple not-all-equal check on the four bits it reads. This enables us to prove that, for any constant c, it is NP-hard to color a 2-colorable 4-uniform hypergraph using just c colors and also yields a superconstant inapproximability result under a stronger hardness assumption.

Introduction The study of probabilistically checkable proof (PCP) systems has led to major breakthroughs in theoretical computer science in the past decade. In particular this study has led to a surprisingly clear understanding of the complexity of nding approximate solutions to optimization problems. A recurring theme in this study is the association of new complexity measures to veriers of PCP systems, and construction of e-cient veriers under the new measure. The new measures are then related to some special subclass of optimization problems to gain new insight about the approximability of problems in this subclass of optimization problems. This paper presents yet another such complexity measure, the covering complexity of a verier, and relates it to a subclass of optimization problems, namely hypergraph coloring problems. Below we elaborate on some of the notions above, such as PCP, approximability, hypergraph coloring, and introduce our new complexity measure. Probabilistically checkable proofs. The centerpiece of a PCP system is the probabilistic veri- er. This verier is a randomized polynomial time algorithm whose input is a \theorem", and who is also given oracle access to a \proof". Using the traditional equivalence associated with randomized algorithms, it is convenient to think of the verier as having two inputs, the \theorem" and a \random string". Based on these two inputs the verier settles on a strategy to verify the proof | namely, it decides on a sequence of queries to ask the oracle, and prepares a predicate P . It then queries the oracle and if it receives as response bits a applies the predicate P (a and accepts i the predicate is satised. 1 The quality of the PCP system is roughly related to its ability to distinguish valid proofs (true \theorems" with correct \proofs") from invalid theorems (incorrect \theorems" from any purported \proof") | hopefully the verier accepts valid proofs with much higher probability than it does invalid theorems. To study the power of PCP systems in a complexity-theoretic setting, we quantify some of the signicant resources of the verier above, and then study the resources needed to verify proofs of membership for some hard language. Fix such a language L and consider a verier V whose goal is to verify proofs of membership in L. The above paragraph already hints at four measures we may associate with such a verier and we dene them in two steps. For functions say that a V is (r; q)-restricted if, on input x (implying the theorem x 2 L) of length n, a random string of length r(n) and makes q(n) queries to the proof oracle. We say that veries L with completeness c and soundness s, if (1) For every x 2 L, there exists an oracle such that V , on input x and oracle access to , outputs accept with probability at least c, and (2) For every x 62 L and every oracle , V outputs accept with probability at most s. The class of all languages L that have an (r; q)-restricted verier verifying it with completeness c and soundness s is denoted Covering Complexity. In the variant of PCPs that we consider here, we stick with (r; q)- restricted veriers, but alter the notion of completeness and soundness. Instead of focusing on the one proof that maximizes the probability with which the verier accepts a given input, here we allow multiple proofs to be provided to the verier. We say that a set of proofs f covers a verier V on input x if for every random string, there exists one proof i such that V accepts i on this random string. We are interested in the smallest set of proofs that satisfy this property and the cardinality of this set is said to be the covering complexity of the verier on this input. This description of a verier is somewhat restrictive. More general denitions allow the verier to be adaptive, deciding on later queries based on response to previous ones. For this paper, the restricted version su-ces. Analogous to the class PCP, we may dene the class cPCP c;s [r; q] (for covering PCP) to be the class of all languages for which there exist (r; q)-restricted veriers that satisfy the following conditions: (Completeness) If x 2 L, the covering complexity of V on x is at most 1=c. (Soundness) If x 62 L then the covering complexity of V on x is at least 1=s. Notions somewhat related to covering complexity have been considered in the literature implicitly and explicitly in the past. Typically these notions have been motivated by the approximability of minimization problems, such as graph coloring, set cover, and the closest vector problem. Our specic notion is motivated by graph and hypergraph coloring problems. We describe our motivation next. We defer the comparison with related notions to later in this section. Hypergraph coloring, approximability, and inapproximability. An l-uniform hypergraph H is given by a set of vertices V and a set of edges E where an edge e 2 E is itself a subset of V of cardinality l. A k-coloring of H is a map from V to the set kg such that no edge is monochromatic. The hypergraph coloring problem is that of nding, given H, the smallest k for which a k-coloring of H exists. When l = 2, then the hypergraph is just a graph, and the hypergraph coloring problem is the usual graph coloring problem. Graph and hypergraph coloring problems have been studied extensively in the literature from both the combinatorial and algorithmic perspective. The task of determining if a l-uniform graph is k-colorable is trivial if almost so if l 2. Every other case turns out to be NP-hard. The case of l = 2, k 3 is a classical NP-hard problem, while the case of was shown NP-hard by Lovasz [23]. Thus, even the property of a hypergraph being 2-colorable is non-trivial. This property, also called Property B, has been studied in the extremal combinatorics literature for long. Much work has been done on determining su-cient conditions under which a hypergraph family is 2-colorable and on solving the corresponding algorithmic questions [11, 6, 7, 25, 26, 29, 27]. The hardness of the coloring problem motivates the study of the approximability of the graph and hypergraph coloring problems. In context of these problems, an (l; k; k 0 )-approximation algorithm is one that produces (in polynomial time) a k 0 -coloring of every k-colorable l-uniform hypergraph for some k 0 > k, with the \approximation" being better as k 0 gets closer to k. Even this problem turns out to be non-trivial, with the best known algorithms for coloring even 3-colorable graphs requiring colors [9, 19], where n is the number of vertices. Similarly, inspired in part by the approximate graph coloring algorithms, several works [1, 10, 22] have provided approximation algorithms for coloring 2-colorable hypergraphs. The best known result for 2-colorable 4-uniform hypergraphs is a polynomial time coloring algorithm that uses ~ O(n 3=4 ) colors [1, 10]. To justify the intractability of the approximation versions of the hypergraph coloring problem, one looks for inapproximability results. Inapproximability results show that it is NP-hard to achieve the goals of an (l; k; k 0 )-approximation algorithm by producing a polynomial time computable reduction from, say, SAT to a \gap" problem related to hypergraph coloring. Here we assume a conservative denition of such a reduction, namely, the many-one reduction. The many-one version of such a reduction would reduce a formula ' to an l-uniform hypergraph H such that H is k-colorable if ' is satisable, and H is not k 0 -colorable if ' is not satisable. Since the existence of an (l; k; k 0 )-approximation algorithm now gives the power to decide if ' is satisable or not, this shows that the approximation problem is NP-hard. In the sequel, when we say that an )-approximation problem is NP-hard, we always implicitly mean that the \gap version" of the problem is NP-hard. This methodology combined with the PCP technique has been employed heavily to get hardness results of graph coloring problems. This approach started with the results of [24], and culminates with, essentially tight, the results of [12] who show that the (2; is NP-hard under randomized reductions. However, for graphs whose chromatic number is a small constant, the known hardness results are much weaker. For example, for 3-colorable graphs the best known hardness result only rules out coloring using 4 colors [20, 16]. This paper is motivated by the quest for strong (super-constant) inapproximability for coloring graphs whose chromatic number is a small constant. We do not get such results for graph coloring, but do get such inapproximability results for hypergraph coloring and in particular for coloring 4-uniform hypergraphs. Graph coloring and covering PCPs. In examining the reasons why the current techniques have been unable to show strong hardness results for inapproximability of coloring 3-colorable graphs, a natural question arises: Are PCPs really necessary to show such hardness results, or would something weaker su-ce? To date there are no reasons showing PCPs are necessary. And while the rst result showing the intractability of coloring 3-colorable graphs with 4 colors [20] did use the PCP technique, [16] show that PCPs are not needed in this result. The starting point of our work is the observation that covering PCPs are indeed necessary for showing strong hardness results for graph coloring. Specically, in Proposition 2.1, we show that if the (2; c; !(1))-approximation problem for coloring is NP-hard for c < 1, then NP cPCP results can also be derived from hardness results for coloring hypergraphs, though we don't do so here.) Previous approaches have realized this need implicitly, but relied on deriving the required results via PCPs. In particular, they use the trivial containment PCP 1;s [r; q] cPCP 1;s [r; q] and build upon the latter result to derive hardness for coloring. (Notice, that we do not have such a simple containment when the completeness parameter is not equal to 1. This special case of important in general and is referred to as perfect completeness.) For our purposes, however, this trivial containment is too weak. In particular it is known that PCP c;s [log; q] P for every c; s; q such that c > s2 q (cf. [8, Lemma 10.6]). Thus it is not possible to show NP cPCP for any constant > 0, using the trivial containment mentioned above (and such a covering PCP is essential for super-constant hardness results for coloring hypergraphs). Thus it becomes evident that a direct construction of covering PCPs may be more fruitful and we undertake such constructions in this paper. Related Notions. Typically, every approach that applies PCP to minimization problems has resulted, at least implicitly, in some new complexity measures. Two of those that are close to, and likely to be confused with, the notion of covering complexity are the notions of \multiple assignments" [2], and the \covering parameter" of [12]. Here we clarify the distinctions. In the former case, the multiple assignments of [2], the proof oracle is expected to respond to each query with an element of a large alphabet (rather than just a bit). When quantifying the \quality" of a proof, however, the oracle is allowed to respond with a subset of the alphabet, rather than just a single element, and the goal of the prover is to pick response sets of small sizes so that on every random string, the verier can pick one elements from each response set to the dierent queries so that it leads to acceptance. Once again we have a notion of covering all random strings with valid proofs, but this time the order of quantiers is dierent. The notion of multiple assignments is interesting only when the alphabet of the oracles responses are large, while our notion remains interesting even when the oracle responds with an element of a binary alphabet. The second related notion is the covering parameter of Feige and Kilian [12]. Since the names are confusingly similar (we apologize for not detecting this at an early stage), we refer to their notion as the FK-covering parameter. In a rather simplied sense, their parameter also allows multiple proofs to be presented to the verier. But their notion of coverage requires that on every random string and every possible accepting pattern of query responses for the verier, there should exist a proof which gives this accepting pattern (and is hence accepted). For any xed verier and input, the FK-covering number is always larger than ours, since we don't need every accepting pattern to be seen among the proofs. Though the notions appear close, the motivation, the application, and the technical challenges posed by the FK-covering parameter and ours are completely dierent. Both notions arise from an attempt to study graph coloring, but their focus is on general graphs (with high chromatic number), while ours is on graphs of small chromatic number. In their case, separation of the FK-covering parameter is su-cient, but not necessary, to give inapproximability of coloring. For our parameter, separation is necessary, but not su-cient to get the same. Finally, in their constructions the challenge is to take a traditional PCP and enhance it to have small FK- covering completeness and they use the PCP directly to argue that the soundness is not large. In our case, the completeness is immediate and the soundness needs further analysis. Gadgets and covering complexity. Returning to our notion of covering complexity, while it seems essential to study this to get good hardness results on coloring, the reader should also be warned that this notion is somewhat less robust that usual notions that one deals with in PCPs. Specically, prior notions were not very sensitive to the predicate applied by the verier in deciding its nal output. They could quantify the power of the verier by simple parameters such as number of bits read, or number of accepting congurations. Here we are forced to pay attention to the verier's computations and restrict these to get interesting results. It is reasonable to ask why this happens and we attempt to give some justication below. In standard PCPs, it is often possible to use \gadgets" (whenever they are available) to convert the acceptance predicate of the verier from one form to another, for only a small loss in the performance. For example suppose one has a PCP verier V 1 that reads three bits of a proof and accepts if they are not all equal (NAE). Such a verier would directly prove the hardness of the \Max NAE-3SAT" problem. But by application of a gadget the same verier can be transformed into one that proves the hardness of the \Max 3SAT" problem. The gadget notices that the function NAE(a; b; c) for three Boolean variables a; b; c is simply (a _ b _ c) ^ (:a _:b_:c), which is a conjunction of two 3SAT clauses. Thus a transformed verier V 2 which picks three bits of the proof as V 1 does, and then picks one of the two clauses implied by the check performed by V 1 and veries just this one clause, is now a verier whose acceptance predicate is a 3SAT condition. Furthermore, if the acceptance probability of V 1 on the same proof is 1 , then the acceptance probability of V 2 on some given proof is exactly 1 =2. Thus if V 1 proves inapproximability of proves inapproximability of Max 3SAT. Unfortunately, a similar transformation does not apply in the case of covering complexity. Notice that two proofs, the oracle that always responds with 0 and the one that responds with 1, always su-ces to cover any verier whose acceptance predicate is 3SAT. Yet there exist NAE 3- SAT veriers that can not be covered by any constant number of proofs. (For example, the verier that picks 3 of the n bits of the proof uniformly and independently at random and applies the NAE 3-SAT predicate to them, needs eds n) proofs to be covered.) Thus even though a gadget transforming NAE 3SAT to 3SAT does exist, it is of no use in preserving covering complexity of veriers. This non-robust behavior of cPCP veriers forces us to be careful in designing our veriers and our two results dier in mainly the predicate applied by the verier. Our results. Our rst result is a containment of NP in the class cPCP 1;" [O(log n); 4], for every " > 0. If the randomness is allowed to be slightly super-logarithmic, then the soundness can be reduced to some explicit o(1) function. Technically, this result is of interest in that it overcomes the qualitative limitation described above of passing through standard PCPs. Furthermore, the proof of this result is also of interest in that it shows how to apply the (by now) standard Fourier- analysis based techniques to the studying of covering complexity as well. Thus it lays out the hope for applying such analysis to other cPCP's as well. Unfortunately, the resulting cPCP fails to improve inapproximability of graph coloring or even hypergraph coloring. As noted earlier covering PCPs are only necessary, but not su-cient to get hardness results for hypergraph coloring. In order to get hardness results for hypergraph coloring from covering PCPs, one needs veriers whose acceptance condition is a NAE SAT (not-all-equal) predicate (though, in this case, it is also reasonable to allow the responses of the queries to be elements of a non-binary alphabet, and a result over q-ary alphabet will give a result for q-colorable hypergraphs). Keeping this objective in mind, we design a second verier (whose query complexity is also 4 bits), but whose acceptance predicate simply checks if the four queried bits are not all equal. The verier has perfect completeness and its covering soundness can be made an arbitrarily large constant (Theorem 4.2). This result immediately yields a super-constant lower bound on coloring 2-colorable 4-uniform hypergraphs: we prove that c-coloring such hypergraphs is NP-hard for any constant c (Theorem 4.4), and moreover there exists a constant c 0 > 0 such that, unless NP DTIME(n O(log log n) ), there is no polynomial time algorithm to color a 2-colorable 4-uniform hypergraph using c 0 log log n log log log n colors (Theorem 4.6). A similar hardness result also holds for coloring 2-colorable k-uniform hypergraphs for any k 5 by reduction from the case of 4-uniform hypergraphs (Theorem 4.7). Prior to our work, no non-trivial inapproximability results seem to be known for coloring 2-colorable hypergraphs, and in fact it was not known if 3-coloring a 2-colorable 4-uniform hypergraph is NP-hard. We note that we do not have analogous results for the hardness of coloring 2-colorable 3-uniform hypergraphs. The di-culty in capturing the problem stems from the di-culty of analyzing the underlying maximization problem. The natural maximization version of hypergraph 2-coloring is the following: color the vertices with two colors so that a maximum number of hyperedges are non- monochromatic. For l-uniform hypergraphs, this is problem is known as Max l-Set Splitting. For case we study here), a tight hardness result of 7=8 known [17] and this fact works its way into our analysis. For hardness result is not known for the maximization version (see [14]) and our inability to show hardness results for 3-uniform hypergraphs seems to stem from this fact. Organization. In Section 2, we go over some of the denitions more formally and relate covering complexity to approximability of hypergraph coloring. In Section 3, we analyze a simple cPCP verier that makes 4 queries and has perfect completeness and o(1) soundness. In Section 4, we analyze a more complicated cPCP verier with similar parameters whose acceptance condition is not-all-equal-sat. This yields the hardness result for coloring 2-colorable, 4-uniform hypergraphs. This is the complete version of the conference paper [15]. Preliminaries In this section we introduce covering PCPs formally, and establish a connection (in the wrong direction) between covering PCPs and inapproximability of hypergraph coloring. 2.1 Probabilistically checkable proofs (PCPs) We rst give a formal denition of a PCP. Below veriers are probabilistic oracle Turing machines whose output, on input x and random string r with oracle O, is denoted V O (x; r). The output is a bit with 1 denoting acceptance and 0 denoting rejection. Denition 1 Let c and s be real numbers such that 1 c > s 0. A probabilistic polynomial time oracle Turing machine V is a PCP verier with soundness s and completeness c for a language L For x 2 L there exists oracle such that Prob r [V (x; For x 62 L, for all Two parameters of interest in a PCP are the number of random bits used by the verier and the number of queries it makes to the proof oracle. Most of the time the symbols of are bits and whenever this is not the case, this is stated explicitly. Denition 2 For functions restricted if, on any input of length n, it uses at most r(n) random bits and makes at most q(n) queries to . We can now dene classes of languages based on PCPs. Denition 3 (PCP) A language L belongs to the class PCP c;s [r; q] if there is an (r; q)-restricted verier V for L with completeness c and soundness s. Next we have the denition of covering PCP. Denition 4 (Covering PCP) A language L belongs to the class cPCP c;s [r; q] if there is an q)-restricted verier V such that on input x: (i) if x 2 L then there is a set of proofs f 1=c such that for every random string r there exists a proof i such L, then for every set of k proofs f with k < 1=s, there is a random string r for which V rejects every i , 1 i k. One usually requires \perfect completeness" seeking PCP characterizations. It is clear from the above denitions that PCP 1;s [r; q] cPCP 1;s [r; q] and thus obtaining a PCP characterization for a language class is at least as hard as obtaining a covering PCP characterization with similar parameters. 2.2 Covering PCPs and Graph Coloring We now verify our intuition that \good" covering PCPs (i.e., those which have a large gap in covering complexity between the completeness and soundness cases) are necessary for strong lower bounds on the approximating the chromatic number. As usual, for a graph G, we denote by (G) its chromatic number, i.e., the minimum number of colors required in a proper coloring of G. Below, we use the phrase \it is NP-hard to distinguish f(n)-colorable graphs from g(n)-colorable graphs" to mean that \the (2; f; g)-approximation problem is NP-hard". As mentioned in Section 1, note that we are using a conservative denition of NP-hardness and hence this statement implies that there is a many-one reduction from SAT that maps satisable instances of SAT to f(n) colorable graphs and maps unsatisable instances to graphs that are not g(n)-colorable. Under this assumption, we show how to get nice covering PCPs. Below and throughout this paper, the function log denotes logarithms to base two. Proposition 2.1 Suppose for functions f; given a graph G on n vertices, it is NP-hard to distinguish between the cases (G) f(n) and (G) g(n). Then Proof: Let the vertex set of G be g. The covering PCP consists of proofs that correspond to \cuts" G, i.e., each i is n-bits long, with the j th bit being 1 or 0 depending on which side of the cut i contains v j . The verier simply picks two vertices and at random such that they are adjacent in G, and then check if the j th 1 and j thbits dier in any of the k proofs. The minimum number k of proofs required to satisfy the verier for all its random choices is clearly the cut cover number (G) of G, i.e., the minimum number of cuts that cover all edges of G. It is easy to see that (G)e, and therefore the claimed result follows. 2 One can get a similar result for any base q, by letting the proofs be q-ary strings and the verier read two q-ary symbols from the proof. In light of this, we get the following. Corollary 2.2 Suppose that there exists an " > 0 such that it is NP-hard, given an input graph G, to distinguish between the cases when G is 3-colorable and when (G) cPCP 1;(" log 3 n) 1 [O(log n); 2] where the covering PCP is over a ternary alphabet, and the verier's action is to simply read two ternary symbols from the proof and check that they are not equal. In light of the above Corollary, very powerful covering PCP characterizations of NP are necessary in order to get strong hardness results for coloring graphs with small chromatic number. A result similar to Proposition 2.1, with an identical proof, also holds for hypergraph coloring, and thus motivates us to look for good covering PCP characterizations of NP in order to prove hardness results for coloring 2-colorable hypergraphs. Proposition 2.3 Suppose that there exists a function f given an input r-uniform hypergraph on n vertices, it is NP-hard to distinguish between the cases when it is 2- colorable and when it is not f(n)-colorable. Then, NP cPCP log f(n) [O(log n); r]. In particu- lar, if c-coloring 2-colorable r-uniform hypergraphs is NP-hard for every constant c, then NP [O(log n); r] for every constant k 1. 3 PCP Construction I We now move on to the constructions of our proof systems. To a reader familiar with PCPs we rst give a preview of our constructions. Both our PCPs (of this section and the next) go through the standard path. We start with strong 2-prover 1-round proof systems of Raz [28], apply the composition paradigm [5], and then use the long code of [8] at the bottom level. One warning: in the literature it is common to use a variant of the long code | called the \folded long code" | we do not use the folded version. (Readers unfamiliar with the terms above may nd elaborations in Section 3.1.) As usual, the interesting aspects in the constructions are choice of the inner veriers and the analyses of their soundness. The inner veriers that we use are essentially from [17]: The inner verier in Section 4 is exactly the same as the one used by [17, Section 7] to show hardness of Max 4-Set Splitting, while the one in this section is a small variant. The goals of the analyses are dierent, since we are interested in the number of proofs required to cover all random strings. Despite the dierence, we borrow large parts of our analysis from that of [17]. In the current section our analysis essentially shows that if our verier, on some xed input, rejects every proof oracle with probability at least , then on any set of k proofs nearly k fraction of random strings end up rejecting all the proofs. Thus the standard soundness of the verier we construct is of interest and we analyze this using lemmas from [17]. The analysis of the verier in Section 4 does involve some new components and we will comment upon these in the next section. 3.1 Preliminaries: Label cover, Long codes, Proof composition Our PCP constructions (also) follow the paradigm of proof composition, by composing an \outer verier" with an \inner verier". In its most modern and easy to apply form, one starts with an outer proof system which is a 2-Prover 1-Round proof system (2P1R) construction for NP. We abstract the 2P1R by a graph-theoretic optimization problem called Label Cover. The specic version of Label Cover we refer to is the maximization version LabelCover max discussed in [3] (see [3] for related versions and the history of this problem). Label Cover. A LabelCover max instance LC consists of a bipartite graph vertex set U [ W and edge set F , \label sets" LU ; LW which represent the possible labels that can be given to vertices in U; W respectively, and projection functions for each W such that (u; w) 2 F . The optimization problem we consider is to assign a label '(u) 2 LU (resp. '(w) 2 LW ) to each u 2 U (resp. w 2 W ) such that the fraction of edges (call such an edge \satised") is maximized. The optimum value of a LabelCover max instance LC, denoted OPT(LC), is the maximum fraction of \satised" edges in any label assignment. In the language of LabelCover max , the PCP theorem [5, 4] together with the parallel repetition theorem of Raz [28] yields parts (i)-(iii) of the theorem below. Here we need an additional property that is also used in [17, Sections 6, 7]. First we need a denition: For min The denition above is quite technical (and borrowed directly from [17]) but the intuition is that projects mostly onto dierent elements of LU i the \measure" is large. Theorem 3.1 ([3, 17]) There exist d 0 ; e 0 < 1 and c > 0 and a transformation that, given a parameter - > 0, maps instances ' of Sat to instances of LabelCover max , in time n O(log - 1 ) , such that where n is the size of the Sat instance '. (iii) If ' is satisable then ' is not satisable then OPT(LC) -. (iv) For every Fg. Remark: As mentioned earlier, conditions (i)-(iii) are standard for LabelCover max . The need for Condition (iv) is inherited from some lemmas of [17] that we use (specically, Lemmas 3.3 and 3.4). This condition is shown in Lemma 6.9 of [17]. To use the hardness of label cover, we use the standard paradigm of proof composition. The use of this paradigm requires an error-correcting code, which in our case is again the long code. We dene this next. The Long Code. We rst remark on some conventions and notation we follow through the rest of this paper: We represent Boolean values by the set f1; 1g with 1 standing for False and 1 for True. This representation has the nice feature that Xor just becomes multiplication. For any domain D, denote by FD the space of all Boolean functions f 1g. For any set D, jDj denotes its cardinality. We now describe a very redundant error-correcting code, called the long code. The long code was rst used by [8], and has been very useful in most PCP constructions since. The long code of an element x in a domain D, denoted LONG(x), is simply the evaluations of all the 2 jDj Boolean functions in FD at x. If A is the long code of a, then we denote by A(f) the coordinate of A corresponding to function f , so that We note that most of the proofs used in the literature use the \folded long code" which is a code of half the length of the long code, involving evaluations of the elements x at exactly one of the functions f or f (but not both). For reasons that will become clearer later, we cannot use the folded long code here and work with the actual long code. Constructing a \Composed" PCP. Note that Theorem 3.1 implies a PCP where the proof is simply the labels of all vertices in U; W of the LabelCover max instance and the verier picks an edge at random and checks if the labels of u and w are \consistent", i.e., u;w An alternative is to choose a random neighbor w 0 of u and instead checking u;w dening '(u) to be the most common value of u;w 0 ('(w 0 )) it is easy to see that the probability of acceptance in the latter PCP (that uses w; w 0 for the check) is at most the probability of acceptance in the former PCP (that uses u; w for the check). By the properties guaranteed in Theorem 3.1, either PCP uses O(log n log - 1 ) randomness, has perfect completeness and soundness at most -. While the soundness is excellent, the number of bits it reads from the proof in total (from the two \locations" it queries) is large (namely, O(log - 1 )). In order to improve the query complexity, one \composes" this \outer" verication with an \inner" verication procedure. The inner verier is given as input a projection function has oracle access to purported encodings, via the encoding function Enc of some error-correcting code, of two labels a 2 LU and b 2 LW , and its aim is to check that (b) = a (with \good" accuracy) by making very few queries to Enc(a) and Enc(b). The inner veriers we use have a slightly dierent character: they are given input two projections 1 and 2 (specically u;w and u;w 0 ) and have oracle access to purported encodings Enc(b) and Enc(c) of two labels b; c 2 LW , and the aim is to test whether 1 This interesting feature was part of and necessary for Hastad's construction for set splitting [17], and our PCPs also inherit this feature. In our nal PCP system, the proof is expected to be the encodings of the labels '(w) of all vertices using the encoding Enc. For e-cient constructions the code used is the long code of [8], i.e., Enc =LONG. We denote the portion of the (overall) proof that corresponds to w by LP(w), and in a \correct" proof LP(w) would just be LONG('(w)) (the notation LP stands for \long proof"). The construction of a PCP now reduces to the construction of a good inner verier that given a pair of strings B; C which are purportedly long codes, and projection functions 1 and 2 , checks if these strings are the long codes of two \consistent" strings b and c whose respective projections agree (i.e., satisfy 1 Given such an inner verier IV, one can get a \composed verier" V comp using standard techniques as follows (given formula ' the verier rst computes the LabelCover max instance LC in polynomial time and then proceeds with the verication): 1. Pick u 2 U at random and w; w 0 2 N(u) at random 2. Run the inner verier with input u;w and u;w 0 and oracle access to LP(w) and LP(w 0 ). 3. Accept i the inner verier IV accepts We denote by V comp (IV) the composed verier obtained using inner verier IV. The (usual) soundness analysis of the composed PCP proceeds by saying that if there is a proof that causes the verier V comp to accept with large, say (s "), probability, where s is the soundness we are aiming for, then this proof can be \decoded" into labels for U [ W that \satisfy" more than a fraction - of the edges in the LabelCover max instance, and by Theorem 3.1 therefore the original formula ' was satisable. In our case, we would like to make a similar argument and say that if at most k proofs together satisfy all tests of V comp , then these proofs can be \decoded" into labels for U [W that satisfy more than - fraction of edges of LC. 3.2 The Inner Verier We now delve into the specication of our rst \inner verier", which we call Basic-IV4. This inner verier is essentially the same as the one for 4-set splitting in [17], but has a dierent acceptance predicate. Recall the inner verier is given input two projections functions has oracle access to two tables and aims to check that B (resp. C) is the long code of b (resp. c) which satisfy 1 Inner Verifier Basic-IV4 B;C Choose uniformly at random f 2 FLU , Choose at random g 0 ; h 0 2 FLW such that 8b 2 LW , For a technical reason, as in [17], the nal inner verier needs to run the above inner verier for the bias parameter p chosen at random from an appropriate set of values. The specic distribution we use is the one used by Hastad [17] (the constant c used in its specication is the constant from Equation (2) in the statement of Theorem 3.1). Inner Verifier IV4 B;C e, t. Choose uniformly at random. Run Basic-IV4 B;C Note that the inner verier above has perfect completeness. Indeed when B; C are long codes of b; c where 1 then for each f 2 FLU , if so these are not equal, and similarly for the case when 3.3 Covering Soundness analysis Let X( ) be the indicator random variable for the rejection of a particular proof Wg by the composed verier V comp (IV4 )). The probability that V 1 ( rejects taken over its random choices is clearly the expectation Here B; C are shorthand for LP(w) and LP(w 0 ) respectively and equal respectively in a \correct" proof. We wish to say that no k proofs can together satisfy all the tests which performs. ) is the indicator random variable for the rejection of a set of k proofs fLP by the verier V 1 ( ), then the overall probability that V 1 ( rejects all these k proofs, taken over its random choices, is exactly Y where we use the shorthand We now argue (see Lemma 3.2 below) that if this rejection probability is much smaller than there is a way to obtain labels '(u) for than - fraction of the edges (u; w) are satised by this labeling, i.e., Together with Theorem 3.1, this implies that the rejection probability (from Equation (4)) for any set of k proofs for a false claim of satisability (of '), can be made arbitrarily close to 1 , and in particular is non-zero, and thus the covering soundness of the composed verier is at most 1=k. Lemma 3.2 There exists a 0 < 1 such that for every integer k 1, every ", 0 < " < 4 k , and all a 0 Before presenting the formal proof of Lemma 3.2, we rst highlight the basic approach. The power of arithmetizing the rejection probability for a set of k proofs as in Equation (4) is that one can expand out the product and analyze the expectation of4 k where products are dened to be 1. A special term is which is the constant 1. We analyze the rest of the terms individually. We can now imagine two new proofs ~ are exclusive-ors of subsets of the k given proofs. Now one can apply existing techniques from [17] to analyze terms involving the tables ~ B and ~ C and show that ~ cannot be too negative, and similarly if the expectation of ~ much below zero, then in fact OPT(LC) is quite large. In short, at a high level, we are saying that if there exist k proofs such that the verier accepts at least one of them with good probability, then some exclusive-or of these proofs is also accepted by the verier with good probability, and we know this cannot happen by the soundness analysis of [17] for the case of a single proof. This intuition is formalized via the next two lemmas from [17]. Before stating the lemmas, we make a slight digression to point out the relevance of not employing folding here. Folded long codes are typically used as follows: Given a table supposedly giving the long code of the encoding of the label assigned to w, conceptually we assume we have a long proof A which respects the constraints A 0 such an A 0 for ourselves from A 0 by setting A 0 xed element of the concerned domain (i.e., LU or LW as the case might be). Such a table A 0 , which satises A 0 ( f) = A 0 (f) for every function f , is said to be folded. We then pretend the verier works with the long code, but carry out the soundness analysis only for folded tables. In our case also we could do the same to analyze the acceptance of a single proof. However when faced with multiple proofs, the intermediate tables we consider, such as B S above, need not be folded even if the original proofs we were given were folded | in particular, this will be the case when S has even cardinality. Thus our analysis needs to work with non \folded tables" as well. This is why we work with the long code directly. Now we go back to the technical lemmas. Lemma 3.3 ([17]) For every where the distribution of p; f; 2 is the same as the one in IV4 . This lemma is Lemma 7.9 in [17] combined with calculation in the rst half of Lemma 7.14 in the same paper. Similarly the next lemma follows from Lemma 7.12 of the same paper and a similar calculation. Lemma 3.4 ([17]) There exists a < 1 such that for every > 0 and all proof tables fBw g and fCw g, indexed by w 2 W with Bw is at leastOPT(LC) where the expectation is taken over and where the distribution of is the same as the one in IV4 . We are now ready to prove Lemma 3.2. Proof of Lemma 3.2: The proof is actually simple given Lemmas 3.3 and 3.4. We pick a that satises < ". By Equation (4), if E [X k ( ", then there exist subsets S 1 ;, such that Suppose one of S 1 , S 2 is empty, say S applied to B S 1 (which is a function mapping which together with Equation (6) above yields > ", a contradiction since "=8. Now suppose both S 1 and S 2 are non-empty. Now we apply Lemma 3.4 to B S 1 and C S 2 to get that the expectation in Equation (6) is at least 7 a. Together with Equation this yields (using " 8 a> a 0 for some absolute constant a 0 . 2 We are now ready to state and prove the main Theorem of this section. Theorem 3.5 For every constant k, NP cPCP Proof: The theorem follows from Lemma 3.2 and Theorem 3.1. Let pick - > 0 small enough so that a 0 > -. By Lemma 3.2 we have implies OPT(LC) > -. Consider the PCP with verier V comp (IV4 ). Using Theorem 3.1, we get that if the input formula ' is not satisable, the verier V comp (IV4 rejects any k proofs with probability at least 1 it clearly has perfect completeness and makes only 4 queries, the claimed result follows. 2 Remark on tightness of the analysis: In fact, Lemma 3.2 can be used to show that for any " > 0, there exists a (covering) PCP verier that makes 4 queries, has perfect completeness and which rejects any set of k proofs with probability at least 1 ". Note that this analysis is in fact tight for the verier V comp (IV4) since a random set of k proofs is accepted with probability 1 4 k . It would have been su-cient to prove that for any k proofs the set of verier coins causing the verier to reject all k proofs is nonempty. We do not know a simpler proof of this weaker statement. Construction II and Hardness of Hypergraph Coloring In the previous section we gave a PCP construction which made only 4 queries into the proof and had covering soundness smaller than any desired constant. This is already interesting in that it highlights the power of taking the covering soundness approach (since as remarked in the introduction one cannot achieve arbitrarily low soundness using classical PCPs with perfect completeness that make some xed constant number of queries). We next turn to applying this to get a strong inapproximability result for hypergraph coloring. The predicate tested by the inner verier IV4 is F (x; w), and to get a hardness result for hypergraph coloring, we require the predicate to be NAE(x; which is true unless all of x; are equal. Note that NAE(x; true, so one natural approach is to simply replace the predicate F tested by IV4 by NAE without losing perfect completeness. The challenge of course is to prove that the covering soundness does not suer in this process, and this is exactly what we accomplish. For completeness we describe the inner verier below. Inner Verifier IV-NAE4 B;C Pick p as in IV4 . Pick f; as in Basic-IV4 p . Accept i not all of B(g 1 are equal. To analyze the soundness of the resulting composed verier, we need to understand the \not- all-equal" predicate NAE. Note that NAE(x; rejects i8 and this sum equals zero otherwise. With similar notation as in the previous section this implies that for a given choice of the verier rejects all k proofs i where denotes the exclusive-or of characteristic vectors, or worded dierently, symmetric dier- ence of sets. If the verier accepts one the proofs then the right hand side of (7) must equal zero. Hence we study the expected value of this quantity. Before proceeding with the analysis we shed some insight into the analysis and explain what is new this time. Let . The terms corresponding to T being the empty set are exactly the terms that appeared in the analysis of the verier of Section 3. Let us turn our attention to terms where T 6= ;. Typically, when a sum as the above appears, we would just go ahead and analyze the individual terms. Unfortunately, it turns out that we are unable to do this in our case. To see why, consider a typical summand above, namely These are more general than the terms analyzed in Section 3, which were of the form B The rst two elements of such a product come from an identical distribution, and similarly for the last two elements of the product. This in turn enabled a certain \pairing" up of terms from which a good solution to the label cover instance could be extracted (see the analysis in Lemma 7.12 of [17] for more details). But now, since T 6= ;, the rst two tables, and are dierent, and so are the last two. Therefore, we now have to deal with individual terms which are the product of four elements each of which comes from a dierent distribution. It does not seem possible to analyze such a term by itself and extract meaningful solutions to the label cover instance. To cope with this problem, we now bunch together terms that involve the same T but dierent S 1 and S 3 . (Alternatively, one could think of this as xing T , and then picking S 1 and S 3 as random subsets of [k] and considering the expectation of the terms This makes the distribution of the rst pair as a whole identical to that of the second pair, and allows us to analyze the terms above. More formally, for each non-empty T [k], and similarly for C. Using this notation the sum in Equation (7) equals where the rst four terms correspond to the case where can be used to lower bound the expectation of the rst two sums over Lemma 3.4 can be used to lower bound the expectation of the third sum as a function of the optimum of the label cover instance. Thus we only need to study the last sum. We show that if the last term is too negative, then one can extract an assignment of labels to the provers. The intuition behind the proof is as follows. B T and C T are two functions chosen independently from the same distribution. Further, the queried pairs (g are also chosen from the same distribution, but are not independent of each other (and are related via f ). If we ignore this dependence for a moment, then we get: and this would be good enough for us. Unfortunately, (g are not independent. The intuition behind the proof of the next inequality is that if this correlation aects the expectation of then there is some correlation between the tables for B T and C T and so a reasonable strategy for assigning labels to w and w 0 can be extracted. Specically, we get the following lemma: Lemma 4.1 There exists a 0 < 1 such that the following holds: Let T "=8 be such that "; where the expectation is taken over the distribution of u; w; w as in IV-NAE4 . Then OPT(LC) a 0 As usual we postpone the proof of the lemma, and instead prove the resulting theorem. Theorem 4.2 For every constant k, NP cPCP moreover the predicate veried by the PCP upon reading bits x; Proof: We only have to analyze the soundness of the verier. Let a be the constant from Lemma 3.4 and a 0 be the constant from Lemma 4.1. Let g. Let and let - < b. To create a verier for an instance SAT, reduce the instance of SAT to an instance of Label Cover using Theorem 3.1 with parameter - and then use the verier based on using IV-NAE4 as the inner verier. To show soundness, we need to show that if this verier is covered by k proofs, then the instance of Label Cover has an optimum greater than -. Suppose we have k proofs such that the verier always accept one of the proofs. This implies that the expectation, over u; w; w of (9) is 0. This implies that at least one of summands in (9) is less than or equal to 2 (k+2) in expectation (since there are at most summands in the expression). If it is a summand in one of the rst two sums then this contradicts Lemma 3.3. If it is a summand in the third sums then by Lemma 3.4, we get that a> -. If it is a summand in the last sum, then by Lemma 4.1 we get that a 0 > -. Thus in the last two cases we get that the optimum is more than - as desired. 2 Before going on to the proof of Lemma 4.1, we discuss the consequences of Theorem 4.2 to hypergraph coloring. Before doing so, we just note that in fact one can prove a stronger claim in Theorem 4.2 that given any k proofs, the probability that the verier rejects all of them is at least8 k ", for " > 0 as small as we seek. The proof is really the same as that of Theorem 4.2, since we have argued that all terms in the expansion (9) are arbitrarily small in the case when optimum value of the label cover instance is very small. Once again this soundness analysis is tight, since a random set of k proofs will, in expectation, satisfy a fraction 1 1 8 k of the verier's checks. 4.1 Hardness results for hypergraph coloring Since the predicate used by the PCP of Theorem 4.2 is that of 4-set splitting, we get the following Corollary. Corollary 4.3 For every constant k 2, given an instance of 4-set splitting, it is NP-hard to distinguish between the case when there is a partition of the universe that splits all the 4-sets, and when for every set of k partitions there is at least one 4-set which is is not split by any of the k partitions. The above hardness can be naturally translated into a hardness result for coloring 4-uniform hy- pergraphs, and this gives us our main result: Theorem 4.4 (Main Theorem) For any constant c 2, it is NP-hard to color a 2-colorable 4-uniform hypergraph using c colors. Proof: Follows from the above Corollary since a 4-set splitting instance can be naturally identied with a 4-uniform hypergraph whose hyperedges are the 4-sets, and it is easy to see that the minimum number of partitions k needed to split all 4-sets equals dlog ce where c is the minimum number of colors to color the hypergraph such that no hyperedge is monochromatic. 2 In light of the discussion after the proof of Theorem 4.2, we in fact have the following stronger result. Theorem 4.5 For any constant c 2 and every " > 0, it is NP-hard to color a 2-colorable 4- uniform hypergraph using c colors such that at least a fraction (1 1 ") of the hyperedges are properly colored (i.e., are not monochromatic). Theorem 4.6 Assume NP 6 DTIME(n O(log log n) ). Then there exists an absolute constant c 0 > 0 such that there is no polynomial time algorithm that can color a 2-colorable 4-uniform hypergraph using c 0 log log n log log log n colors, where n is the number of vertices in the hypergraph. Proof: This follows since the covering soundness of the PCP in Theorem 4.2 can be made an explicit o(1) function. Indeed, nothing prevents from having a k that is a function of n. We need to have and to reach a contradiction - < O( . The proof size we need is . We can thus have n O(log log n) size proofs by letting log log n log log log n ). Similarly to Theorem 4.4, this implies 2 k -coloring a 2-colorable 4-uniform hypergraph is hard unless NP DTIME(n O(log log n) ). 2 We now show that a hardness result similar to Theorem 4.4 also holds for 2-colorable k-uniform hypergraphs for any k 5. Theorem 4.7 Let k 5 be an integer. For any constant ' 2, it is NP-hard to color a 2-colorable k-uniform hypergraph using ' colors. Proof: The proof works by reducing from the case of 4-uniform hypergraphs, and the claimed hardness then follows using Theorem 4.4. Let H be a 4-uniform hypergraph with vertex set V . Suppose that Construct a k-uniform hypergraph H 0 as follows. The vertex set of H 0 is where the sets V (j) are independent copies of V . On each V (j) , take a collection F (j) of 4-element subsets of V (j) that correspond to the hyperedges in H. A hyperedge of H 0 (which is a (4s element subset of now given by the union of s 4-sets belonging to s dierent F (j) 's, together with t vertices picked from a 4-set belonging to yet another F (j) . More formally, for every set of (s every choice of elements and every t-element subset f j s+1 of e j s+1 , there is a hyperedge If H is 2-colorable then clearly any 2-coloring of it induces a 2-coloring of H 0 , and hence H 0 is 2-colorable as well. Suppose H is not '-colorable and that we are given an '-coloring of H 0 . Since H is not '- colorable, each F (j) , for 1 j s' must contain a monochromatic set g j . By the pigeonhole principle, there must be a color c such that dierent g j 's have color c. The hyperedge of constructed from those (s + 1) sets is then clearly monochromatic (all its vertices have color c) and we conclude that H 0 is not '-colorable. Since the reduction runs in polynomial time when k and ' are constants the proof is complete.4.2 Discrete Fourier transforms Before going on to the proof of Lemma 4.1 we now introduce a tool that has been crucial in the analysis on inner veriers. This was hidden so far from the reader but already used in the proofs of Lemmas 3.3 and 3.4 in [17]. Now we need to introduce them explicitly. In general we consider functions mapping D to f1; 1g. For D and f 2 FD , let x2 f(x). Notice that fxg is the long code of x. For any function A mapping FD to the reals, we have the corresponding Fourier coe-cients f A(f) (f) where D. We have the Fourier inversion formula given by A (f) and Plancherel's equality that states that f In the case when A is a Boolean function the latter sum is clearly 1. We think of an arbitrary table A as being somewhat close to (or coherent with) the long code of x if there exists a small set containing x such that ^ A is non-negligibly large. Thus, when viewing the long proofs of w and w 0 , our goal is to show that the LP(w) and LP(w 0 ) have coherence with the long codes of strings x and y such that u;w (x) and u;w 0 (y) are equal. 4.3 Proof of Lemma 4.1 Fix T [k]. Throughout this section the quantities that we dene depend on T , but we don't include it as a parameter explicitly. Recall we need to show that if the expectation, over u; w; w is too negative (less than "), then we can assign labels to the label cover problem with acceptance probability more that -. Recall that dened in terms of other random variables f and g 0 and similarly h 2 in terms of f and h 0 . For brevity, we let X denote the quantity that X is a random variable depending on all the variables above. We rst analyze the expectation of X over g 1 and h 1 (for xed choice of u; w; w we calculate the expectation over f , g 0 and h 0 . In both stages we get exact expressions. Finally we make some approximations for the expectation over u; w; w 0 . (The careful reader may observe that we don't take expectations over p | in fact the lemma holds for every choice of p of the inner verier IV-NAE4 .) The crux of this proof are the functions B dened as follows: We let and Note that for a xed choice of f and g 0 we have We get a similar expression for C T and thus we get: Let us call the above quantity Y . In what follows, we rely crucially on the properties of the Fourier coe-cients of B and C . F and ^ G denote the Fourier coe-cients of B and C respectively. From the denitions and some standard manipulation, we get Using simple Fourier expansion, we can rewrite the quantity we are analyzing as: Y Y The main property about the Fourier coe-cients of B and C is that their L 1 norm is bounded. Specically, we have: A We start by dening the strategy we use to assign labels and prove that if the expectation (of Y ) is large, then the labels give an assignment to the label cover instance with objective of at least a 0 Strategy. Given w 2 W , and tables corresponding to LP(w) in k dierent proofs, compute B S for every S [k], B and its Fourier coe-cients. Pick a non-empty set LW with F j and assign as label to w, an element x 2 chosen uniformly at random. With remaining probability, since may be less than 2 k , assign no label to w. Preliminary analysis. We now give a preliminary expression for the success probability of the strategy. Consider picking u; w, and w 0 (and the associated 1 and 2 ) at random and checking for the event 1 )). The probability of this event is lower bounded by the probability that 1 () and 2 ( 0 ) intersect and we assign the elements corresponding to this intersection to w and w 0 . The probability of these events is at least: Below we show that this quantity is large if the expectation of Y is too small. We now return to the expectation of Y . An exact expression for the expectation of Y . We start with some notation. Fix u; w; w and 1 and 2 . For x 2 LU and ; 0 LW , let s x Since the argument of s x is always and the argument of t x is always 0 , we use the shorthand s x for s x () and t x for t x ( 0 ). Further for real p and non-negative integers s; t, let (p; s; t) =2 , and let (p; we show that F Y To prove the above it su-ces to show that Y Y Factors corresponding to y and z with dierent projections on LU are independent, and thus the expectation can be broken down into a product of expectations, one for each x 2 LU . Fix x 2 LU and consider the term Y Y If (or \false") the rst product equals ( 1) sx and the second equals (1 2p) t x . Similarly, If the rst product equals (1 2p) sx and the second equals ( 1) t x . The events happen with probability 1=2 each and thus giving that the expectation above (for xed x) equals2 Taking the product over all x's gives (12). Inequalities on E [Y ]. For every u, we now show how to lower bound the expectation of Y , over in terms of a sum involving only 's and 0 that intersect in their projections. This brings us much closer to the expression derived in our preliminary analysis of the success probability of our strategy for assigning labels and we lower bound a closely related quantity. Specically we now use the inequality E guaranteed by the Lemma statement) to show: x minf1; ps x g is the quantity dened in Equation (1). inequality with p ( 0 ) in the exponent follows by symmetry.) To prove the above, consider the following expression, which is closely related to the expectation of Y as given by (12). Y x Y x First we note that E 0. (Here we are using the fact that the tables B T and C T are chosen from the same distribution.) Next we note that the dierence between Y and Y 1 arises only from terms involving ; 0 such that 1 To verify this, note that if for every x, if 1 we get that terms corresponding to such pairs of ; 0 vanish in We conclude: Y x Y x Using taking absolute values we get, Y x Y x Y x Y x where the last inequality uses j(p; t)j 1 for every t 0. Next we simplify the terms of the LHS above. First we show that for every t 0, First we note both j(p; s; t)j and j(p; s)j are upper bounded by 1 If p s 1 , then we have 1 s , let us set To show (15), we need to prove that (z) 0 for z 2 [0; 1]. We have in the interval in question and we only have to check the inequality at the end points. We have Using (15) we conclude that Y x Y x Y x Substituting the above into (14) gives (13). Based on (13). we want to prove that the strategy for assigning labels is a good one. First we prove that large sets do not contribute much to the LHS of the sum in (13). Dene We have Lemma 4.8 We have "=4: Proof: By Property (iv) of Theorem 3.1 we have that the probability that p () (4k is at most "2 (2k+4) . A similar chain of inequalities as (10) shows that The sum in the lemma can hence be estimated as "=4; and the lemma follows. 2 By the same argument applied to 0 of size at least K, together with Equation (13), we get "=2: (16) We now relate to the probability of success of our strategy for assigning labels. From (11) we know this quantity is at least "2 (2k+2) where the last inequality uses (16). (It is easy to convert this randomized strategy for assigning labels to a deterministic one that does equally well.) The dominating factor in the expression is the term p O(1) (from the denition of K) which can be calculated to be O( and the proof of Lemma 4.1 is complete. 2 4.3.1 Comparison to previous proof of Theorem 4.4 We point out that the conference version of this paper [15] contained a dierent proof of Theorem 4.4. The current proof is signicantly simpler, and furthermore it is only a minor adjustment of similar proofs in [17]. The key observation to make the current proof possible is the insight that we should treat the terms of (7) in the collections given by B T (g does not seem possible to handle them one by one in an e-cient manner. The previous proof did not make this observation explicitly and ended up being signicantly more complicated. This \simplicity" in turn has already enabled some further progress on the hypergraph coloring problem | in partic- ular, using this style of analysis, Khot [21] shows a better super-constant hardness for a-colorable 4-uniform hypergraphs for a 7. 4.3.2 Subsequent related work In a very recent work, Holmerin [18] showed that the vertex cover problem considered on 4-uniform hypergraphs is NP-hard to approximate within a factor of (2 ") for arbitrary " > 0. of vertices of a hypergraph H is said to be a vertex cover if every hyperedge of H intersects S.) He proves this by modifying the soundness analysis of Hastad's 4-set splitting verier (which is also the verier we use in Section 4) to show that any proof which sets only a fraction " of bits to 1 will cause some 4-tuple tested by the verier to consist of only 1's. This in turn shows that for every constant " > 0, given a 2-colorable 4-uniform hypergraph, it is NP-hard to nd an independent set that consists of a fraction " of vertices. Note that this result is stronger as a small independent set implies a large chromatic number and it thus immediately implies the hardness of coloring such a 2-colorable 4-uniform hypergraph with 1=" colors, and hence our main result (Theorem 4.4). We stress that the verier in Holmerin's paper is the same as the one in this paper; however, the analysis in [18] obtains our result without directly referring to covering complexity. Acknowledgments We would like to thank the anonymous referees and Oded Goldreich for useful comments on the presentation of the paper. --R Coloring 2-colorable hypergraphs with a sublinear number of colors The hardness of approximate optima in lattices Hardness of Approximations. Proof veri Probabilistic checking of proofs: A new characterization of NP. An algorithmic approach to the Lov Free bits Improved approximation for graph coloring. Coloring bipartite hypergraphs. Improved approximation algorithms for maximum cut and satis Inapproximability results for set splitting and satis On the hardness of 4-coloring a 3-colorable graph Vertex cover on 4-regular hyper-graphs is hard to approximate within (2 ") Approximate graph coloring using semide On the hardness of approximating the chromatic number. Hardness results for approximate hypergraph coloring. Approximate coloring of uniform hypergraphs. On the hardness of approximating minimization problems. A random recoloring method for graphs and hypergraphs. Hypergraph coloring and the Lov Improved bounds and algorithms for hypergraph 2- coloring A parallel repetition theorem. Coloring n-sets red and blue --TR --CTR Adi Avidor , Ricky Rosen, A note on unique games, Information Processing Letters, v.99 n.3, p.87-91, August 2006 Subhash Khot, Guest column: inapproximability results via Long Code based PCPs, ACM SIGACT News, v.36 n.2, June 2005

covering PCP;set splitting;hypergraph coloring;hardness of approximations;PCP;graph coloring

586963

Simple Learning Algorithms for Decision Trees and Multivariate Polynomials.

In this paper we develop a new approach for learning decision trees and multivariate polynomials via interpolation of multivariate polynomials. This new approach yields simple learning algorithms for multivariate polynomials and decision trees over finite fields under any constant bounded product distribution. The output hypothesis is a (single) multivariate polynomial that is an $\epsilon$-approximation of the target under any constant bounded product distribution.The new approach demonstrates the learnability of many classes under any constant bounded product distribution and using membership queries, such as j-disjoint disjunctive normal forms (DNFs) and multivariate polynomials with bounded degree over any field.The technique shows how to interpolate multivariate polynomials with bounded term size from membership queries only. This, in particular, gives a learning algorithm for an O(log n)-depth decision tree from membership queries only and a new learning algorithm of any multivariate polynomial over sufficiently large fields from membership queries only. We show that our results for learning from membership queries only are the best possible.

Introduction From the start of computational learning theory, great emphasis has been put on developing algorithmic techniques for various problems. It seems that the great progress has been made in learning using membership queries, especially such functions as decision trees and multivariate polynomials. Generally speaking, three different techniques were developed for those tasks: the Fourier transform technique, the lattice based techniques and the Multiplicity Automata technique. All the techniques use membership queries (which is also called substitution queries for nonbinary fields). The Fourier transform technique is based on representing functions using a basis, where a basis function is essentially a parity of a subset of the input. Any function can be represented as a linear combination of the basis functions. Kushilevitz and Mansour [KM93] gave a general technique to recover the significant coefficients. They showed that this is sufficient for learning decision trees under the uniform distribution. Jackson [J94] extended the result to learning DNF under the uniform distribution. The output hypothesis is a majority of parities. (Also, Jackson [J95] generalizes his DNF learning algorithm from uniform distribution to any fixed constant bounded product distribution.) The lattice based techniques are, at a very high level, performing a traversal of the binary cube. Moving from one node to its neighbor, in order to reach some goal node. Angluin [A88] gave the first lattice based algorithm for learning monotone DNF. Bshouty [Bs93] developed the monotone theory, which gives a technique for learning decision trees under any distribution. (The output hypothesis in that case is depth 3 formulas.) Schapire and Sellie [SS93] gave a lattice based algorithm for learning multivariate polynomials over a finite field under any distribution. (Their algorithm depends polynomially on the size of the monotone polynomial that describes the function.) Multiplicity Automata theory is a well studied field in Automata theory. Recently, some very interesting connections where given, connecting learning such automata and learning decision trees and multivariate polynomials. Ohnishi, Seki and Kasami [OSK94] and Bergadano and gave an algorithm for learning Multiplicity Automata. Based on this work Catlan and Varricchio [BCV96] show that this algorithm learns disjoint DNF. Then Beimel et. al. [BBB+96] gave an algorithm that is based on Hankel matrices theory for learning Multiplicity Automata and show that multivariate polynomials over any field are learnable in polynomial time. (In all the above algorithms the output hypothesis is a Multiplicity Automaton.) All techniques, the Fourier Spectrum, the Lattice based and the Multiplicity Automata algorithms give also learnability of many other classes such as learning decision trees over parities (nodes contains parities) under constant bounded product distributions, learning CDNF (poly size DNF that has poly size CNF) under any distribution and learning j-disjoint DNF (DNF where the intersection of any j terms is 0). In this paper we develop a new approach for learning decision trees and multivariate polynomials via interpolation of multivariate polynomials over GF (2). This new approach leads to simple learning algorithms for decision trees over the uniform and constant bounded product distributions, where the output hypotheses is a multivariate polynomial (parity of monotone terms). The algorithm we develop gives a single hypothesis that approximate the target with respect to any constant bounded product distribution. In fact the hypothesis is a good hypothesis under any distribution that supports small terms. That is any distribution D where for a term T of size !(log n) we have PrD Previous algorithms do not achieve this property. It is also known that any DNF is learnable with membership queries under constant bounded product distribution [J95], where the output hypothesis is a majority of parities. Our contribution for j-disjoint DNF is to use an output hypothesis that is a parity of terms and to show that the output hypothesis is an ffl approximation of the target against any constant bounded distribution. We also study the learnability of multivariate polynomials from membership queries only. We give a learning algorithm for multivariate polynomials over n variables with maximal degree for each variable, where c ! 1 is constant, and with terms of size d log d) using only membership queries. This result implies learning decision trees of depth O(log n) with leaves from a field F from membership queries only. This result is a generalization of the result in [B95b] and [RB89], where the learning algorithm uses membership and equivalence queries in the former and only membership queries in the latter. The second result is a generalization of the result in [KM93] for learning boolean decision tree from membership queries. The above result also give an algorithm for learning any multivariate polynomial over fields of size log d)) from membership queries only. This result is a generalization of the results in [BT88, CDG+91, Z90] for learning multivariate polynomials under any field. Previous algorithms for learning multivariate polynomial over finite fields F require asking membership queries with assignments in some extension of the field F [CDG+91]. In [CDG+91] it is shown that an extension n of the field is sufficient to interpolate any multivariate polynomial (when membership queries with assignments from an extension field are allowed). The organization of the paper is as follows. In section 2 we define the learning model and the concept classes. In section 3 we give the algorithm for learning multivariate polynomial for the boolean domain. In section 4 we give some background for multivariate interpolation. In section 5 we show how to reduce learning multivariate polynomials to zero testing and to other problems. Then in section 6 we give the algorithm for zero testing and also give a lower bound for zero testing multivariate polynomials. 2 The Learning Model and Concept Classes 2.1 Learning Models The learning criterion we consider is exact learning [A88] and PAC-learning[Val84]. In the exact learning model there is a function f called the target function f : F n ! F which is a member of a class functions C defined over the variable set field F . The goal of the learning algorithm is to output a formula h that is equivalent to f . The learning algorithm performs a membership query (also called substitution query for the nonbinary fields) by supplying an assignment a to the variables in V as input to a membership oracle and receives in return the value of f(a). For our algorithms we will regard this oracle as a procedure MQ f (). The procedure input is an assignment a and its output is The learning algorithm performs an equivalence query by supplying any function h as input to an equivalence oracle with the oracle returning either "YES", signifying that h is equivalent to f , or a counterexample, which is an assignment b such that h(b) 6= f(b). For our algorithms we will regard this oracle as a procedure EQ f (h). We say the hypothesis class of the learning algorithm is H if the algorithm supplies the equivalence oracle functions from H. We say that a class of boolean function C is exactly learnable in polynomial time if for any there is an algorithm that runs in polynomial time, asks a polynomial number of queries (polynomial in n and in the size of the target function) and outputs a hypothesis h that is equivalent to f . The PAC learning model is as follows. There is a function f called the target function which is a member of a class of functions C defined over the variable set g. There is a distribution D defined over the domain F n . The goal of the learning algorithm is to output a formula h that is ffl-close to f with respect to some distribution D, that is, Pr D The function h is called an ffl-approximation of f with respect to the distribution D. In the PAC or example query model, the learning algorithm asks for an example from the example oracle, and receives an example (a; f(a)) where a is chosen from f0; 1g n according to the distribution D. We say that a class of boolean functions C is PAC learnable under the distribution D in polynomial time if for any f 2 C over V n there is an algorithm that runs in polynomial time, asks polynomial number of queries (polynomial in n, 1=ffl, 1=ffi and the size of the target function) and with probability at least outputs a hypothesis h that is ffl-approximation of f with respect to the distribution D. It is known from [A88] that if a class is exactly learnable in polynomial time from equivalence queries and membership queries then it is PAC learnable with membership queries in polynomial time under any distribution D. Let D be a set of distribution. We says that C is PAC learnable under D if there is a PAC-learning algorithm for C such that for any distribution D 2 D unknown to the learner and for any f 2 C the learning algorithm runs in polynomial time and outputs a hypothesis h that is an ffl-approximation of f under any distribution D 0 2 D. 2.2 The Concept Classes and Distributions A function over a field F is a function f set X. All classes considered in this paper are classes of functions where . The elements of F n are called assignments. We will consider the set of variables V describe the value of the i-projection of the assignment in the domain F n of f . For an assignment a, the i-th entry of a will be denoted by a i . A literal is a nonconstant polynomial p(x i ). A monotone literal is x r nonnegative integer r. A term (monotone term) is a product of literals (monotone literals). A multivariate polynomial is a linear combination of monotone terms. A multivariate polynomial with nonmonotone terms is a linear combination of terms. The degree of a literal p(x i ) is the degree of the polynomial p. The size of a term Let MULF (n; k; t; d) be the set of all multivariate polynomials over the field F over n variables with at most t monotone terms where each term is of size at most k and each monotone literal is of degree at most d. For the binary field B the degree is at most so we will use MUL(n; k; t). F (n; k; t; d) will be the set of all multivariate polynomial with nonmonotone terms with the above properties. We use MUL ? (n; k; t) when the field is the binary field. Throughout the paper we will assume that t - n. Since every term in MUL ? F (n; k; t; d) can be written as multivariate polynomial in MUL F (n; k; (d d) we have Proposition 1 F (n; k; t; d) ' MUL F (n; k; t(d For the boolean field (disjunctive normal form) is a disjunction of terms. A j-disjoint DNF is a DNF where the disjunction of any j terms is 0. A k-DNF is a DNF with terms of size at most k literals. A decision tree (with leaves from some field F) over V n is a binary tree whose nodes are labeled with variables from V n and whose leaves are labeled with constants from F . Each decision tree T represents a function f To compute f T (a) we start from the root of the tree the root is labeled with x i then f T TR (a) if a TR is the right subtree of the root (i.e., the subtree of the right child of the root with all its descendent). Otherwise (when a is the left subtree of the root. If T is a leaf then f T (a) is the label of this leaf. It is not hard to see that a boolean decision tree of depth k can be represented in MUL ? (n; (each leaf in the decision tree defines a term and the function is the sum of all terms), and that a j-disjoint k-DNF of size t can be represented in MUL ? (n; example [K94].) So for constant k and d = O(log n) the number of terms is polynomial. For a DNF and multivariate polynomial, f , we define size(f) to be the number of terms in f . For a decision tree the size will be the number of leaves in the tree. A product distribution is a distribution D that satisfies D(a distributions D i on F . A product distribution is fixed constant bounded if there is a constant 1=2, that is independent of the number of variables n, such that for any variable x i , distribution D supports small terms if for every term of size !(log n), we have PrD is the number of variables. 3 Simple Algorithm for the Boolean Domain In this section we give an algorithm that PAC-learns with membership queries MUL ? (n; n; t) under any distribution that supports small terms in polynomial time in n and t. We remind the reader that we assume t - n. All the algorithms in the paper run in polynomial time also when 3.1 Zero test MUL(n; k; t) We first show how to zero-test elements in MUL(n; k; t) in polynomial time in n and 2 k assuming k is known to the learner. The algorithm will run in polynomial time for Choose a term of maximal size in f . Choose any values from f0; 1g for the variables not in T . The projection will not be the zero function because the term T will stay alive in the projection. Since the projection is a nonzero function with variables there is at least one assignment for x that gives value 1 for the function. This shows that for a random and uniform assignment a, with probability at least . So to zero test a function f 2 MUL(n; k; t) randomly and uniformly choose polynomial number of assignments a i . If f(a i ) is zero for all the assignments then with high probability we have f j 0. Now from the above we have the probability that randomly chosen elements is at most ffi. This implies there is a polynomial time probabilistic zero testing algo- rithm, that succeeds with high probability. 3.2 Learning MUL(n; k; t). We now show how to reduce zero-test to learning. We first show how to find one term in f . If we know that is a term in f . If Since we can zero-test we can find the minimal This implies that f x 1 some multivariate polynomial f 1 . If f 1 we know that is a term in f . We continue recursively with f 1, in this case is a term in f . After we find a term T we define - This removes the term T from f , and thus - 1). We continue recursively with - f until we recover all the terms of f . Membership queries for - f can be simulated by memebership for f because MQ - (a). The complexity of the interpolation is performing nt calls to the zero testing procedure. This gives there is an algorithm that with probability at least learns f with nt log nt membership queries. In particular this gives, there is a polynomial time probabilistic interpolation algorithm, that succeeds with high probability to learn f from membership queries. 3.3 Learning MUL ? We now give a PAC-learning algorithm that learns MUL ? (n; n; t) under any distribution that support small terms. We first give the idea of the algorithm. The formal proof is after Theorem 1. To PAC-learn f we randomly choose an assignment a and define a). A term in f of size k will have on average k=2 monotone literals in f 0 , and terms with will have with high probability \Omega\Gamma literals. We perform a zero-restriction, i.e. for each i, with probability 1=2 we substitute x i / 0 in f 0 . Since a term of size k in f has on average k=2 monotone literals after the first shift f(x + a), in the second restriction this term will be zero with probability (about) This probability is greater than Therefore with high probability all the terms of size more than O(log t) will be removed by the second restriction. This ensures that with high probability the projection f 00 is in MUL ? (n; O(log t); t), and therefore by Proposition 1 Now we can use the algorithm in subsection 3.2 to learn f 00 . Notice that for multivariate polynomial h (with monotone terms) when we performed a zero restriction, we delete some of the monotone terms from h, therefore, the monotone terms of f 00 are monotone terms of f 0 . We continue to take zero-restrictions and collect terms of f 0 until the sum of terms that appear in at least one restriction defines a multivariate polynomial which is a good approximation of f 0 . We get a good approximation of f 0 with respect to any distribution that supports small terms since we collect all the small (i.e. O(log t)) size terms. Theorem 1 There is a polynomial time probabilistic PAC-learning algorithm with membership queries, that learns MUL ? (n; n; t) under any distribution that support small terms. We now prove that the algorithm sketched above PAC-learns with membership queries any multivariate polynomial with non-monotone terms under distributions that support small terms. For the analysis of the correctness of the algorithm we first need to formalize the notion of distributions that support small terms. The following is one way to define this notion. Definition 1. Let D c;t;ffl be the set of distributions that satisfy the following: For every c;t;ffl and any DNF f with t terms of size greater than c log(t=ffl) we have Pr Notice that all the constant bounded product distributions D where for all i are in D 1= log(1=d);t;ffl . In what follows we will assume that c - 2 and ffl ! 1=2. We will use Chernoff bound (see [ASE]). independent random variables where Pr[X Then for any a we have Pr be a multivariate polynomial where T are terms and jT 1 Our algorithm starts by choosing a random assignment a and defines f 0 All terms that are of size s (in f 0 ) will contain on average s=2 monotone literals. Therefore by Chernoff bound we have Lemma 7 With probability at least 1=2 all the terms in f 0 of size more than ffc log(t=ffl), contain at least (ff=4)c log(t=ffl) monotone literals, where ff - 4 and c - 1. Proof. Let T be any term of size ffc log(t=ffl). Let P (T ) be the number of monotone literals in T . We have Pr Since the number of terms of f 0 is t and ffl ! 1=2 the result follows.2 With probability at least 1=2 all the terms of size more than 4c log(t=ffl) will contain at least c log(t=ffl) monotone literals and all terms of size 8c log(t=ffl) will contain at least 2c log(t=ffl) monotone literals. Now we split the function f 0 into 3 functions f 1 , f 2 and f 3 . The function will contain all terms that are of size at most 4c log(t=ffl). The function will contain all terms of size between 4c log(t=ffl) and 8c log(t=ffl) and the function f all terms of size more than 8c log(t=ffl). Similarly, Our algorithm will find all the terms in f 1 , some of the terms in f 2 and none of the terms in f 3 . Therefore we will need the following claim. is a multivariate polynomial that contains some of the terms in f 2 . Then for any D 2 D c;t;ffl we have Pr Proof. The error is Pr Let ~ ~ ~ is the part of the term that contains monotone literals and ~ is the part that contains the nonmonotone literals. If - that when we change - ~ to sum of monotone terms we get Y q2S So every monotone term in f 2 will contain one of the terms - Therefore we can where f 2;i are multivariate polynomial with monotone terms. Since h is a multivariate polynomial that contains some of the terms in f 2 we have . Since j - by the definition of distribution that support small terms we have The algorithm will proceed as follows. We choose zero restrictions . Recall that a zero restriction p of f 0 is a function f 0 (p) where with probability 1=2,x i / 0 and with probability 1=2 it remains alive. We will show that with probability at least 1=2 we have the following: (A) For every term in f 1 there is a restriction p i such that f 1 (p i ) contains this term. (B) For every We will regard A and B as events. Let T 1 be the set of terms in f 1 . We know that jT 1 and every term in T 1 is of size at most 4c log(t=ffl). Let T 3 be the set of terms in f 3 . We know that the number of terms in T 3 is at most t and every term has at least 2c log(t=ffl) monotone literals. We have Pr[not and, for c - 2, Pr[not Therefore we have both events with probability at least 1=2. This shows that with probability at least 1=2 all the projections f(p i ) contains terms of size at most 8c log(t=ffl). Therefore, the algorithm proceed by learning each projection f(p i using the previous algorithm and collecting all the terms of size 2c log(t=ffl).2 The number of membership queries of the above algorithm is O((t=ffl) k n) for some constant k. For the uniform distribution k - 19. The above analysis algorithm can also be used to learn functions f of the form are terms and + is the addition of a field F . These functions can be computed as follows. For an assignment a, This gives the learnability of decision trees with leaves that contain elements from the field F . 4 Multivariate Interpolation In this section we show how to generalize the above algorithm for any multivariate polynomial over any field. Let ff2I a ff x ff 1 be a multivariate polynomial over the field F where a ff 2 F and ff are integers. We will denote the class of all multivariate polynomials over the field F and over the variables x by F [x The number of terms of f is denoted by jf j. We have jf all a ff are not zero. When d be the maximal degree of variables in f , i.e., I ' [d] n where are d constants where is the zero of the field. A univariate polynomial over the field F of degree at most d can be interpolated from membership queries as follows. Suppose where \Delta (i) (f) is the coefficient of x i in f in its polynomial representation. Then This is a linear system of equations and can be solved for \Delta (i) (f ), as follows, det is the Vandermonde matrix. If f is a multivariate polynomial then f can be written as where \Delta (i) (f) is a multivariate polynomial over the variables x . We can still use (1) to find \Delta (i) (f) by replacing each f(fl i ) with Notice that from the first equation in the system, since ?From (1) a membership query for \Delta (i) can be simulated using d queries to f . From (2), a membership query to \Delta (0) can be simulated using one membership query to f . We now extend the \Delta operators as follows: for Here \Delta always operates on the variable with the smallest index. So \Delta i 1 operates on x 1 in f to give a function f 0 that depends on x operates on x 2 in f 0 and so on. We will also write x i for the term x k . The weight of i, denoted by wt(i), is the number of nonzero entries in i. The operator \Delta i (f) gives the coefficient of x i 1 in f when represented in F [x the operator gives the coefficient of x i when f is represented in F [x Suppose I ' [d] k be such that I are the k-suffixes of all terms of f . Here the k-suffix of a term x n is x k . Since I if and only if x i is a k-suffix of some term in f , it is clear that jIj - jf j and we must have i2I We now will show how to simulate membership queries for using a polynomial number (in n and jf j) of membership queries to f . Suppose we want to find for some c 2 F n\Gammak using membership queries to f . We take r assignments - ask membership queries for (-fl i ; c) for all Now then the above linear system of equations can be solved in time The solution gives f)(c). The existence of - where the above determinant is not zero will be proven in the next section. 5 Reducing Learning to Zero-testing (for any Field) In this section we show how to use the results from the previous section to learn multivariate polynomials. Let MULF (n; k; t; d) be the set of all multivariate polynomials over the field F over n variables with t terms where each term is of size k and the maximal degree of each variable is at most d. We would like to answer the following questions. Let f 2 MUL F (n; k; t; d). 1. Is there a polynomial time algorithm that uses membership queries to f and decides whether 2. Given i - n. Is there a polynomial time algorithm that uses membership queries to f and decides whether f depends on x i ? 3. Given fi t. Is there an algorithm that runs in polynomial time and finds such that r 4. Is there a polynomial time algorithm that uses membership queries to f and identifies f? When we say polynomial time we usually mean polynomial time in n; k; t and d but all the results of this section hold for any time complexity T if we allow a blow up of poly(n; t) in the complexity. We show that 1,2 and 4 are equivalent and 1 ) 3. Obviously 2 2. We will show To prove 1 notice that f 2 MUL F (n; k; t; d) is independent of x i if and only if is the coefficient of x i in f we have g 2 MUL F (n; k; t; d). Therefore we can zero-test g in polynomial time. To prove 1 s be a zero-test for functions in MULF (n; k; t; d), that is, run the algorithm that zero-test for the input 0 and take all the membership queries in the algorithm . We now have f 2 MUL F (n; k; t; d) is 0 if and only if f(fl i Consider the s \Theta r matrix with rows [fl i 1 ]. If this matrix have rank r then we choose r linearly independent rows. If the rank is less than r then its columns are dependent and therefore there are constants c i , r such that r s: This shows that the multivariate polynomial in MUL F (n; k; t; d) we get a contradiction. Now we show that 1+2+3 ) 4. This will use results from the previous section. The algorithm first checks whether f depends on x 1 , and if yes it generates a tree with a root labeled with x 1 that has d children. The ith child is the tree for \Delta i (f ). If the function is independent of x 1 it builds a tree with one child for the root. The child is \Delta 0 (f ). We then recursively build the tree for the children. The previous section shows how to simulate membership queries at each level in polynomial time. This algorithm obviously works and it correctness follows immediately from the previous section and (1)-(3). The complexity of the algorithm is the size of the tree times the membership query simulation. The size of the tree at each level is bounded by the number terms in f , and the depth of the tree is bounded by n, therefore, the tree has at most O(nt) nonzero nodes. The total number of nodes is at most a factor of d from the nonzero nodes. Thus the algorithm have complexity the same as zero testing with a blow up of poly(n; t; d) queries and time. Now that we have reduced the problem to zero testing we will investigate in the next section the complexity of zero testing of MULF (n; k; t; d). 6 Zero-test of MULF (n; k; t; d) In this section we will study the zero testing of MUL F (n; k; ?; d) when the number of terms is unknown and might be exponentially large. The time complexity for the zero testing should be polynomial in n and d (we have k ! n so it is also polynomial in k). We will show the following Theorem 2. The class MUL F (n; k; ?; d), where d - cjF j, is zero testable in randomized polynomial time in n, d and t (here t is not the number of terms in the target) for some constant only if d The algorithm for the zero testing is simply to randomly and uniformly choose poly(n; d) points a i from F n and query f at a i , and receive f(a i ). If for all the points a i , f is zero then with high This theorem implies Theorem 3. The class MULF (n; k; t; d) where d ! cjF j for some constant c is learnable in randomized polynomial time (in n, d and t) from membership queries if Proof of Theorem 2 Upper Bound. Let OE(n; k; d) the maximal possible number of roots of a multivariate polynomial in MUL F (n; k; ?; d). We will show the following facts 1. OE(n; k; d) - jF j n\Gammak OE(k; k; d), and 2. OE(k; k; d) - jF 3. OE(1; Both facts implies that if f 6j 0, when we randomly uniformly choose an assignment a 2 F n , we have Pr a [f(a) For d - cjF j we have that this probability is bounded by 1 poly(n;d;t) . Therefore the expected running time to detect that f is not 0 is poly(n; d; t). It remain to prove conditions (1) and (2). To prove (1) let f 2 MUL F (n; k; ?; d) with maximal number of roots. Let m be a term in f with a maximal number of variables. Suppose, without loss of generality, k . For any substitution a of the variables x the term m will stay alive in the projection because it is maximal in f . Since g has at most OE(k; k; d) roots the result (1) follows. The proof of (2) is similar to the proof of Schwartz [Sch80] and Zippel [Zip79]. Let f 2 MUL F (k; k; ?; d). Write f as polynomial in F [x Let t be the number of roots of f d . Since f d d) we have For assignments a for x we have f d (a) 6= 0. For those assignments we get a polynomial in x 1 of degree d that has at most d roots for x 1 . For t assignments a for x we have f d is zero and then the possible values of x 1 (to get a root for f) is bounded by jF j. This implies The theorem follows by induction on k. 2 Proof of Theorem 2 Lower Bound Let A be a randomized algorithm that zero tests asks membership queries to f and if f 6j 0 it returns with probability at least 2=3 the answer "NO". If all the membership queries in the algorithm returns 0 the algorithm returns the answer "YES" indicating that f j 0. We run the algorithm for f j 0. Let D be the distributions that the membership assignments a a l are chosen to zero test f . Notice that if all membership queries answers are 0 while running the algorithm for f j 0 it would again choose membership queries according to the distributions D l . Now randomly and uniformly choose fl i;j 2 F , Y d Y otherwise. Note that for any input a we have that Therefore This shows that there exists f ? 6j 0 such that running algorithm A for f ? it will answer the wrong answer "YES" with probability more than 2=3. This is a contradiction. 2 --R Machine Learning The probabilistic method. A deterministic algorithm for sparse multivariate polynomial interpolation Exact learning of boolean functions via the monotone theory. A Note on Learning Multivariate Polynomials under the Uniform Distri- bution On the applications of multiplicity automata in learning. Learning sat-k-DNF formulas from membership queries Learning behaviors of automata from multiplicity and equivalence queries. On zero-testing and interpolation of k-sparse multivariate polynomials over finite fields An efficient membership-query algorithm for learning DNF with respect to the uniform distribution On Learning DNF and related circuit classes from helpfull and not-so-helpful teachers On using the Fourier transform to learn disjoint DNF. Learning decision trees using the Fourier spectrum. Randomized interpolation and approximation of sparse polynomials. A polynomial time learning algorithm for recognizable series. Interpolation and approximation of sparse multivariate polynomials over GF(2). Fast probabilistic algorithms for verification of polynomial identities. Learning sparse multivariate polynomial over a field with queries and counterexamples. A theory of the learnable. Probabilistic algorithms for sparce polynomials. Interpolating polynomials from their values. --TR --CTR Homin K. Lee , Rocco A. Servedio , Andrew Wan, DNF are teachable in the average case, Machine Learning, v.69 n.2-3, p.79-96, December 2007

decision tree learning;multivariate polynomial;learning interpolation

586974

Simple Confluently Persistent Catenable Lists.

We consider the problem of maintaining persistent lists subject to concatenation and to insertions and deletions at both ends. Updates to a persistent data structure are nondestructive---each operation produces a new list incorporating the change, while keeping intact the list or lists to which it applies. Although general techniques exist for making data structures persistent, these techniques fail for structures that are subject to operations, such as catenation, that combine two or more versions. In this paper we develop a simple implementation of persistent double-ended queues (deques) with catenation that supports all deque operations in constant amortized time. Our implementation is functional if we allow memoization.

Introduction . Over the last fteen years, there has been considerable development of persistent data structures, those in which not only the current version, but also older ones, are available for access (partial persistence) or updating (full per- sistence). In particular, Driscoll, Sarnak, Sleator, and Tarjan [5] developed e-cient general methods to make pointer-based data structures partially or fully persistent, and Dietz [3] developed an e-cient general method to make array-based structures fully persistent. These general methods support updates that apply to a single version of a structure at a time, but they do not accommodate operations that combine two dierent versions of a structure, such as set union or list catenation. Driscoll, Sleator, and Tarjan [4] coined the term con uently persistent for fully persistent structures that support such combining operations. An alternative way to obtain persistence is to use purely functional programming. We take here an extremely strict view of pure functionality: we disallow lazy evaluation, memoization, and other such techniques. For list-based data structure design, purely functional programming amounts to using only the LISP functions cons, car, cdr. Purely functional data structures are automatically persistent, and indeed con uently persistent. A simple but important problem in data structure design that makes the issue of con uent persistence concrete is that of implementing persistent double-ended queues (deques) with catenation. A series of papers [2, 4] culminated in the work of Kaplan and Tarjan [11, 10], who developed a con uently persistent implementation of deques with catenation that has a worst-case constant time and space bound for any deque operation, including catenation. The Kaplan-Tarjan data structure and its precursors obtain con uent persistence by being purely functional. Department of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. haimk@math.tau.ac.il. y Department of Computer Science, Columbia University, New York, NY 10027. Research at Carnegie Mellon University supported by the Advanced Research Projects Agency CSTO under the title \The Fox Project: Advanced Languages for Systems Software", ARPA Order No. C533, issued by ESC/ENS under Contract No. F19628-95-C-0050. cokasaki@cs.columbia.edu. z Department of Computer Science, Princeton University, Princeton, NJ 08544 and InterTrust Technologies Corporation, Sunnyvale, CA 94086. Research at Princeton University partially supported by NSF Grant No. CCR-9626862. ret@cs.princeton.edu. If all one cares about is persistence, purely functional programming is unnecessarily restrictive. In particular, Okasaki [14, 15, 16] observed that the use of lazy evaluation in combination with memoization can lead to e-cient functional (but not purely functional in our sense) data structures that are con uently persistent. In order to analyze such structures, Okasaki developed a novel kind of debit-based amortization. Using these techniques and weakening the time bound from worst-case to amortized, he was able to considerably simplify the Kaplan-Tarjan data structure, in particular to eliminate its complicated skeleton that encodes a tree extension of a redundant digital numbering system. In this paper we explore the problem of further simplifying the Kaplan-Tarjan result. We obtain a con uently persistent implementation of deques with catenation that has a constant amortized time bound per operation. Our structure is substantially simpler than the original Kaplan-Tarjan structure, and even simpler than Okasaki's catenable deques: whereas Okasaki requires e-cient persistent deques without catenation as building blocks, our structure is entirely self-contained. Furthermore our analysis uses a standard credit-based approach. We give two alternative, but closely related implementations of our method. The rst uses memoization. The second, which saves a small constant factor in time and space, uses an extension of memoization in which any expression can replace an equivalent expression. The remainder of the paper consists of ve sections. In Section 2, we introduce terminology and concepts. In Section 3, we illustrate our approach by developing a persistent implementation of deques without catenation. In Section 4, we extend our approach to handle stacks with catenation. In Section 5, we develop our solution for deques with catenation. We conclude in Section 6 with some remarks and open problems. An extended abstract of this work appeared in [9]. 2. Preliminaries. The objects of our study are lists. As in [11, 10] we allow the following operations on lists: return a new list containing the single element x. return a new list formed by adding element x to the front of list L. pop(L): return a pair whose rst component is the rst element on list L and whose second component is a list containing the second through last elements of L. return a new list formed by adding element x to the back of list L. return a pair whose rst component is a list containing all but the last element of L and whose second component is the last element of L. catenate(L; R): return a new list formed by catenating L and R, with L rst. We seek implementations of these operations (or specic subsets of them) on persistent lists: any operation is allowed on any previously constructed list or lists at any time. For discussions of various forms of persistence see [5]. A stack is a list on which only push and pop are allowed. A queue is a list on which only inject and pop are allowed. A steque (stack-ended queue) is a list on which only push, pop, and inject are allowed. Finally, a deque (double-ended queue) is a list on which all four operations push, pop, inject, and eject are allowed. For any of these four structures, we may or may not allow catenation. If catenation is allowed, push and inject become redundant, since they are special cases of catenation, but it is sometimes convenient to treat them as separate operations because they are easier to implement than general catenation. We say a data structure is purely functional if it can be built and manipulated using the LISP functions car, cons, cdr. That is, the structure consists of a set of immutable nodes, each either an atom or a node containing two pointers to other nodes, with no cycles of pointers. The nodes we use to build our structures actually contain a xed number of elds; reducing our structures to two elds per node by adding additional nodes is straightforward. Various nodes in our structure represent lists. To obtain our results, we extend pure functionality by allowing memoization, in which a function is evaluated only once on a node; the second time the same function is evaluated on the same node, the value is simply retrieved from the previous computation. In all our constructions, there are only a constant number of memoized functions (one or two). We can implement memoization by having a node point to the results of applying each memoized function to it. Initially each such pointer is undened. The rst function evaluation lls in the appropriate pointer to indicate the result. Subsequent evaluations merely follow the pointer to the result, which takes time. We also consider the use of a more substantial extension of pure functionality, in which we allow the operation of replacing a node in a structure by another node representing the same list. Such a replacement can be performed in an imperative setting by replacing all the elds in the node, for instance in LISP by using replaca and replacd. Replacement can be viewed as a generalization of memoization. In our structures, any node is replaced at most twice, which means that all our structures can be implemented in a write-once memory. (It is easy to convert an algorithm that overwrites any eld only a xed constant number of times into a write-once algorithm, with only a constant-factor loss of e-ciency.) The use of overwriting instead of memoization saves a small constant factor in running time and storage space and slightly simplies the amortized analysis. To perform amortized analysis, we use a standard potential-based framework. We assign to each conguration of the data structure (the totality of nodes currently existing) a potential. We dene the amortized cost of an operation to be its actual cost plus the net increase in potential caused by performing the operation. In our applications, the potential of an empty structure is zero and the potential is always non-negative. It follows that, for any sequence of operations starting with an empty structure, the total actual cost of the operations is bounded above by the sum of their amortized costs. See the survey paper [17] for a more complete discussion of amortized analysis. 3. Noncatenable Deques. In this section we describe an implementation of persistent noncatenable deques with a constant amortized time bound per operation. The structure is based on the analogous Kaplan-Tarjan structure [11, 10] but is much simpler. The result presented here illustrates our technique for doing amortized analysis of a persistent data structure. At the end of the section we comment on the relation between the structure proposed here and previously existing solutions. 3.1. Representation. Here and in subsequent sections we say a data structure is over a set A if it stores elements from A. Our representation is recursive. It is built from bounded-size deques called buers, each containing at most three ele- 4 H. KAPLAN, C. OKASAKI, and R. E. TARJEN ments. Buers are of two kinds: prexes and su-xes. A nonempty deque d over A is represented by an ordered triple consisting of a prex over A, denoted by pr(d); a (possibly empty) child deque of ordered pairs over A, denoted by c(d); and a su-x over A, denoted by sf(d). Each pair consists of two elements from A. The child deque c(d), if nonempty, is represented in the same way. We dene the set of descendants of a deque d in the standard way|namely, c 0 exist. The order of elements in a deque is dened recursively to be the one consistent with the order of each triple, each buer, each pair, and each child deque. Thus, the order of elements in a deque d is rst the elements of pr(d), then the elements of each pair in c(d), and nally the elements of sf(d). In general the representation of a deque is not unique|the same sequence of elements may be represented by triples that dier in the sizes of their prexes and su-xes, as well as in the contents and representations of their descendant deques. Whenever we refer to a deque d we actually mean a particular representation of d, one that will be clear from the context. The pointer representation for this representation is the obvious one: a node representing a deque d contains pointers to pr(d), c(d), and sf(d). Note that the pointer structure of d is essentially a linked list of its descendants, since c i (d) contains a pointer to c i+1 (d), for each i. 3.2. Operations. Implementing the deque operations is straightforward, except for maintaining the size bounds on buers. Specically, a push on a deque is easy unless its prex is of size three, a pop on a deque is easy unless its prex is empty, and symmetric statements hold for inject and eject. We deal with buer over ow and under ow in a proactive fashion, rst xing the buer so that the operation to be performed cannot violate its size bounds and then actually doing the operation. The details are as follows. We dene a buer to be green if it contains one or two elements, and red if it contains zero or three. We dene two memoized functions on a deque: gp, which constructs a representation of the same list but with a green prex; and gs, which constructs a representation of the same list with a green su-x. We only apply gp (gs, respectively) to a list whose prex (su-x) is red and can be made green. Specically, for gp, if the prex is empty, the child deque must be nonempty, and symmetrically for gs. Below we give implementations of push, pop, and gp; the implementations for inject, eject, and gs are symmetric. We denote a deque with prex p, child deque c, and su-x s by [p; c; s]. As mentioned in Section 2, we can implement the memoization of gp and gs by having each node point to the nodes resulting from applying gp and gs to it; initially, such pointers are undened. to If pr(d) is empty and c(d) is not, let is nonempty, let (x; return the pair (x; [p; c(e); sf(e)]). Otherwise (c(e) must be empty), let (x; return the pair (x; [;; ;; s]). y, z be the three elements in pr(d). Let p be a prex containing only x, and let c (pr(d) is empty and c(d) is not), let ((x; be a prex containing x followed by y. Return [p 3.3. Analysis. The amortized analysis of this method relies on the memoization of gp and gs. We call a node representing a deque secondary if it is returned by a call to gp or gs and primary otherwise. If a secondary node y is constructed by a call gp(x) (gs(x), respectively), the only way to access y later is via another call gp(x) (gs(x), respectively): no secondary node is returned as the result of a push, pop, inject, or eject operation. This means that gp and gs are called only on primary nodes. We devide the nodes representing deques into three states: such a node is rr if both its buers are red, gr if exactly one of its buers is red, and gg if both its buers are green. We subdivide the rr and gr states: an rr node is rr0 if neither gp nor gs has been applied to it, rr1 if exactly one of gp and gs has been applied to it, and rr2 if both gp and gs have been applied to it; a gr node is gr0 if neither gp nor gs has been applied to it, and gr1 otherwise. By the discussion above every secondary node is gr0 or gg. We dene #rr0, #rr1, and #gr0 to be the numbers of primary nodes in states rr0, rr1, and gr0, respectively. We dene the potential of a collection of nodes representing deques to be 4#rr0 A call to push is either terminal or results in a call to gp, which in turn calls push. Similarly, a call to pop is either terminal or results in a call to gp, which in turn calls pop. We charge the O(1) time spent in a call to gp (exclusive of the inner call to push or pop) to the push or pop that calls gp. A call to push results in a sequence of recursive calls to push (via calls to gp), of which the bottommost is terminal and the rest are nonterminal. A nonterminal push has one of the following eects: it converts a primary rr0 node to rr1 and creates a new primary gr0 node (the result of the push) and a new secondary gr0 node (the result of the call to gp); it converts a primary rr1 node to rr2 and creates a new primary gr0 node and a new secondary gr0 node; or, it converts a primary gr0 node to gr1 and creates a new primary gg node and a new secondary gg node. In each case the total potential drops by one, paying for the time needed for the push (excluding the recursive call). A terminal push takes O(1) time, creates O(1) new nodes, and increases the potential by O(1). We conclude that push takes O(1) amortized time. Analogous arguments apply to pop, inject, and eject, giving us the following theorem: Theorem 3.1. Each of the operations push, pop, inject, and eject dened above takes O(1) amortized time. 3.4. Implementation Using Overwriting. With the memoized implementation described above, a primary rr node can give rise to two secondary gr nodes representing the same list; a primary gr node can give rise to a secondary gg node representing the same list. These redundant representations exist simultanously. A gr representation, however, dominates an rr representation for performing deque op- erations, and a gg representation dominates a gr representation. This allows us to improve the e-ciency of the implementation by using overwriting in place of memo- ization: when gp is called on a node, it overwrites the contents of the node with the results of the gp computation, and similarly for gs. Then only one representation of a list exists at any time, and it evolves from rr to gr to gg (via one of two alternative paths, depending on whether gp or gs is called rst). Each node now needs only three elds (for prex, child deque, and su-x) instead of ve (two extra for gp and gs). Not only does the use of overwriting save a constant factor in running time and storage space, but it also simplies the amortized analysis, as follows. We dene #rr and #gr to be the number of nodes in states rr and gr, respectively. (There are now no secondary nodes.) We dene the potential of a collection of nodes to be 3#rr +#gr. A nonterminal push has one of the following eects: it converts an rr 6 H. KAPLAN, C. OKASAKI, and R. E. TARJEN node to gr and creates a new gr node, or converts a gr node to gg and creates a new gg node. In either case it reduces the potential by one, paying for the O(1) time required by the push (excluding the recursive call). A terminal push takes O(1) time and can increase the potential by O(1). We conclude that push takes O(1) amortized time. Similar arguments apply to pop, inject, and eject. 3.5. Related Work. The structure just described is based on the Kaplan-Tarjan structure of [10, Section 4], but simplies it in three ways. First, the skeleton of our structure (the sequence of descendants) is a stack; in the Kaplan-Tarjan structure, this skeleton must be partitioned into a stack of stacks in order to support worst-case constant-time operations (via a redundant binary counting mechanism). Second, the recursive changes to the structure to make its nodes green are one-sided, instead of two-sided: in the original structure, the stack-of-stacks mechanism requires coordination to keep both sides of the structure in related states. Third, the maximum buer size is reduced, from ve to three. In the special case of a steque, the maximum size of the su-x can be further reduced, to two. In the special case of a queue, both the prex and the su-x can be reduced to maximum size two. There is an alternative, much older approach that uses incremental recopying to obtain persistent deques with worst-case constant-time operations. See [7] for a discussion of this approach. The incremental recopying approach yields an arguably simpler structure than the one presented here, but our structure generalizes to allow catenation, which no one knows how to implement e-ciently using incremental recopying. Also, our structure can be extended to support access, insertion, and deletion d positions away from the end of a list in O(log d) amortized time, by applying the ideas in [12]. 4. Catenable Steques. In this section we show how to extend our ideas to support catenation. Specically, we describe a data structure for catenable steques that achieves an O(1) amortized time bound for push, pop, inject, and catenate. The data structure is based on the same recursive decomposition of lists as that in Section 5 of [10]. The pointer structure that we need here is much simpler than that in [10], and the analysis is amortized, following the framework outlined in Section 2 and used in Section 3. 4.1. Representation. Our structure is similar to the structure of Section 3, but with slightly dierent denitions of the component parts. As in Section 3, we use buers of two kinds: prexes and su-xes. Each prex contains two to six elements and each su-x contains one to three elements. A nonempty steque d over A is represented either by a su-x sf(d) only, or by an ordered triple consisting of a prex pr(d) over A, a child steque c(d) of pairs over A, and a su-x sf(d) over A. In contrast to Section 3, a pair over A is dened to be an ordered pair containing a prex and a possibly empty steque of pairs over A. Observe that this denition adds an additional kind of recursion (pairs store steques) to the structure of Section 3. This extra kind of recursion is what allows e-cient catenation. The order of elements in a steque is the one consistent with the order of each triple, each buer, each pair, each steque within a pair, and each child steque. As in Section 3, there can be many dierent representations of a steque containing a given list of elements; when speaking of a steque, we mean a particular representation of it. The pointer structure for this representation is straightforward. Each triple is represented by a node containing three pointers: to a prex, a child steque, and a su-x. Each pair is represented by a node containing two pointers: to a prex and a steque. 4.2. Operations. The implementation of the steque operations is much like the implementation of the noncatenable deque operations presented in Section 3.2. We call a prex red if it contains either two or six elements, and green otherwise. We call a su-x red if it contains three elements and green otherwise. The prex in a su-x-only steque is considered to have the same color as the su-x. We dene two memoized functions, gp, and gs, which produce green-prex and green-su-x representations of a steque, respectively. Each is called only when the corresponding buer is red and can be made green. We dene push, pop, and inject to call gp or gs when necessary to obtain a green buer. In the denitions below, we represent a steque with prex child steque c, and su-x s by [p; c; s]. Case 1: Steque d is represented by a triple. If Case 2: Steque d is represented by a su-x only. If create a prex p containing x and the rst two elements of sf(d), create a su-x s containing the last element of sf(d), and return [p; ;; s]. Otherwise, create a su-x s by pushing x onto sf(d) and return [;; ;; s]. Case 1: Steque d is represented by a triple. If Case 2: Steque d is represented by a su-x only. If create a su-x s containing x, and return [sf(d); ;; s]. Otherwise, create a su-x s by injecting x into sf(d) and return [;; ;; s]. Case 1: d 1 and d 2 are represented by triples. First, catenate the buers sf(d 1 ) and pr(d 2 ) to obtain p. Now, calculate c 0 as follows: If jpj 5 then let c 9. Create two new prexes p 0 and containing the rst four elements of p and p 00 containing the remaining elements. Let c In either case, return Case 2: d 1 or d 2 is represented by a su-x only. Push or inject the elements of the su-x-only steque one-by-one into the other steque. Note that both push and catenate produce valid steques even when their second arguments are steques with prexes of length one. Although such steques are not normally allowed, they may exist transiently during a pop. Every such steque is immediately passed to push or catenate, and then discarded, however. In order to dene the pop, gp, and gs operations, we dene a n aive-pop operation that simply pops its steque argument without making sure that the result is a valid steque. If d is represented by a triple, let (x; return the consists of a su-x only, let (x; the pair (x; ;) if Case 1: Steque d is represented by a su-x only or jpr(d)j > 2. Return n aive-pop(d). Case 2: Steque d is represented by a triple, x be the rst element on pr(d) and y the second. If jsf(d)j < 3, push y onto sf(d) to form s and 8 H. KAPLAN, C. OKASAKI, and R. E. TARJEN return (x; [;; ;; s]). Otherwise (jsf(d)j = 3), form p from y and the rst two elements on sf(d), form s from the last element on sf(d), and return (x; [p; ;; s]). Case 3: Steque d is represented by a triple, create two new prexes p and p 0 by splitting pr(d) equally in two. Let c c(d) 6= ;), proceed as follows. Inspect the rst pair (p; d 0 ) in c(d). If jpj 4 or d 0 is not empty, let ((p; d 0 Now inspect p. Case 1: p contains at least four elements. Pop the rst two elements from p to form inject these two elements into pr(d) to obtain p 0 . Let c Return Case 2: p contains at most three elements. Push the two elements in pr(d) onto p to obtain p 0 . Let c is nonempty, or c Return (Steque d is represented by a triple with contain the rst two elements of sf(d) and s the last element on sf(d). Let c Return 4.3. Analysis. The analysis of this method is similar to the analysis in Section 3.3. We dene primary and secondary nodes, node states, and the potential function exactly as in Section 3.3: the potential function, as there, is 4#rr0 where #rr0, #rr1, and #gr0 are the numbers of primary nodes in states rr0, rr1, and gr0, respectively. As in Section 3.3, we charge the O(1) cost of a call to gp or gs (excluding the cost of any recursive call to push, pop, or inject) to the push, pop, or inject that calls gp or gs. The amortized costs of push and inject are O(1) by an argument identical to that used to analyze push in Section 3.3. Operation catenate calls push and inject a constant number of times and creates a single new node, so its amortized cost is also O(1). To analyze pop, assume that a call to pop recurs to depth k (via intervening calls to gp). By an argument analogous to that for push, each of the rst k 1 calls pays for itself by decreasing the potential by one. The terminal call to pop can result in a call to either push or catenate, each of which has O(1) amortized cost. It follows that the overall amortized cost of pop is O(1), giving us the following theorem: Theorem 4.1. Each of the operations push, pop, inject, and catenate dened above takes O(1) amortized time. We can improve the time and space e-ciency of the steque data structure by constant factors by using overwriting in place of memoization, exactly as described in Section 3.4. If we do this, we can also simplify the amortized analysis, again exactly as described in Section 3.4. 4.4. Related work. The structure presented in this section is analogous to the Kaplan-Tarjan structure of [10, Section 5] and the structure of [8, Section 7], but simplies them as follows. First, the buers are of constant-bounded size, whereas the structure of [10, Section 5] uses noncatenable steques as buers, and the structure of [8, Section 7] uses noncatenable stacks as buers. These buers in turn must be represented as in Section 3 of this paper or by using one of the other methods mentioned there. In contrast, the structure of the present section is entirely self- contained. Second, the skeleton of the present structure is just a stack, instead of a stack of stacks as in [10] and [8]. Third, the changes used to make buers green are applied in a one-sided, need-driven way; in [10] and [8], repairs must be made simultaneously to both sides of the structure in carefully chosen locations. Okasaki [14] has devised a dierent and somewhat simpler implementation of con uently persistent catenable steques that also achieves an O(1) amortized bound per operation. His solution obtains its e-ciency by (implicitly) using a form of path reversal [18] in addition to lazy evaluation and memoization. Our structure extends to the double-ended case, as we shall see in the next section; whether Okasaki's structure extends to this case is an open problem. 5. Catenable Deques. In this section we show how to extend our ideas to support all ve list operations. Specically, we describe a data structure for catenable deques that achieves an O(1) amortized time bound for push, pop, inject, eject, and catenate. Our structure is based upon an analogous structure of Okasaki [16], but simplied to use constant-size buers. 5.1. Representation. We use three kinds of buers: prexes, middles, and su-xes. A nonempty deque d over A is represented either by a su-x sf(d) or by a 5-tuple that consists of a prex pr(d), a left deque of triples ld(d), a middle md(d), a right deque of triples rd(d), and a su-x sf(d). A triple consists of a rst middle buer , a deque of triples, and a last middle buer. One of the two middle buers in a triple must be nonempty, and in a triple that contains a nonempty deque both middles must be nonempty. All buers and triples are over A. A prex or su-x in a 5-tuple contains three to six elements, a su-x in a su-x-only representation contains one to eight elements, a middle in a 5-tuple contains exactly two elements, and a nonempty middle buer in a triple contains two or three elements. The order of elements in a deque is the one consistent with the order of each 5-tuple, each buer, each triple, and each recursive deque. The pointer structure is again straightforward, with the nodes representing 5-tuples or triples containing one pointer for each eld. 5.2. Operations. We call a prex or su-x in a 5-tuple red if it contains either three or six elements and green otherwise. We call a su-x in a su-x-only representation red if it contains eight elements and green otherwise. The prex of a su-x-only deque is considered to have the same color as the su-x. We introduce two memoizing functions functions gp and gs as in Sections 3.2 and 4.2, which produce green-prex and green-su-x representations of a deque, respectively, and which are called only when the corresponding buer is red but can be made green. Below we give the implementations of push, pop, gp, and catenate; the implementations of inject, eject, and gs are symmetric to those of push, pop, and gp, respectively. We denote a deque with prex p, left deque l, middle m, right deque r, and su-x s by [p; l; m; Case 1: Deque d is represented by a 5-tuple. If otherwise, let Case 2: Deque d is represented by a su-x only. If sf(d) < 8, return a su-x-only deque with su-x push(x; sf(d)). Otherwise, push x onto sf(d) to form s, with nine elements. Create a new prex p with the rst four, a middle with the next two, and a su-x s with the last three. Return [p; ;; m; ;; s]. As in Section 4.2, the implementation of pop uses n aive-pop. Case 1: Deque d is represented by a su-x only or jpr(d)j > 3. Return n aive-pop(d). Case 2: Case 3: x be the rst element on pr(d). If create a new su-x s containing all the elements in pr(d), md(d), and sf(d) except x, and return the pair consisting of x and the deque represented by s only. Otherwise, form p from pr(d) by popping x and injecting the rst element on md(d), m 0 from md(d) by popping the rst element and injecting the rst element on sf(d), form s from sf(d) by popping the rst element, and return (x; create two new prexes p and p 0 , with p containing the rst four elements of jpr(d)j and p 0 the last two; return [p; push((p proceed as follows. Case 1: ld(d) 6= ;. Inspect the rst triple t on ld(d). If either the rst nonempty middle buer in t contains 3 elements or t contains a nonempty deque, let (t; and assume that x is nonempty if t consists of only one nonempty middle buer. Apply the appropriate one of the folowing two subcases. Case 1.1: 3. Form x 0 from x and p from pr(d) by popping the rst element from x and injecting it into pr(d). Return Case 1.2: 2. Inject both elements in x into pr(d) to form p. If d 0 and y are empty, return [p; l; md(d); rd(d); sf(d)]. Otherwise (d 0 and y are nonempty) let l Case 2: Inspect the rst triple t in rd(d). If either the rst nonempty middle buer in t contains 3 elements or t contains a nonempty deque, let assume that x is nonempty if t consists of only one nonempty middle buer. Apply the appropriate one of the following two subcases. Case 2.1: x 0 from pr(d), m, and x by popping an element from m and injecting it into pr(d) to form p, popping an element from m and injecting the rst element from x to form m 0 , and popping the rst element from x to form x 0 . Return Case 2.2: 2. Inject the two elements in md(d) into pr(d) to form p. Let are empty or r Return Case 1: Both d 1 and d 2 are represented by 5-tuples. Let y be the rst element in pr(d 2 ), and let x be the last element in sf(d 1 ). Create a new middle m containing x followed by y. Partition the elements in sf(d 1 ) fxg into at most two buers s 0 1 and 1 , each containing two or three elements in order, with s 00 possibly empty. let ld 0 ld 00 otherwise, let ld 00 ld 0 1 . Similarly, partition the elements in pr(d 1 ) fyg into at most two prexes , each containing two or three elements in order, with possibly empty. Let rd 0 2 . Return [pr(d 1 ); ld 00 Case 2: d 1 or d 2 is represented by a su-x only. Push or inject the elements of the su-x-only deque one-by-one into the other deque. 5.3. Analysis. To analyze this structure, we use the same denitions and the same potential function as in Sections 3.3 and 4.3. The amortized costs of push, inject, catenate, and pop are O(1) by an argument analogous to that in Section 4.3. The amortized cost of eject is O(1) by an argument symmetric to that for pop. Thus we obtain the following theorem: Theorem 5.1. Each of the operations push, pop, inject, eject, and catenate dened above takes O(1) amortized time. Just as in Sections 3.4 and 4.3, we can improve the time and space constant factors and simplify the analysis by using overwriting in place of memoization. Overwriting is the preferred implementation, unless one is using a functional programming language that supports memoization but does not easily allow overwriting. 5.4. Related Work. The structure presented in this section is analogous to the structures of [16, Chapter 11] and [8, Section 9] but simplies them as follows. First, the buers are of constant size, whereas in [16] and [8] they are noncatenable deques. Second, the skeleton of the present structure is a binary tree, instead of a tree extension of a redundant digital numbering system as in [8]. Also, our amortized analysis uses the standard potential function method of [17] rather than the more complicated debit mechanism used in [16]. Another related structure is that of [10, Section 5], which represents purely functional, real-time deques as pairs of triples rather than 5-tuples, but otherwise is similar to (but simpler than) the structure of [8, Section 9]. It is straightforward to modify the structure presented here to use pairs of triples rather than 5-tuples. 6. Further Results and Open Questions. If the universe A of elements over which deques are constructed has a total order, we can extend the structures described here to support an additional heap order based on the order on A. Specically, we can support the additional operation of nding the minimum element in a deque (but not deleting it) while preserving a constant amortized time bound for every operation, including nding the minimum. We merely have to store with each buer, each deque, and each pair or triple the minimum element in it. For related work see [1, 2, 6, 13]. We can also support a ip operation on deques. A ip operation reverses the linear order of the elements in the deque: the ith from the front becomes the ith from the back, and vice-versa. For the noncatenable deques of Section 3, we implement ip by maintaining a reversal bit that is ipped by a ip operation. If the reversal bit is set, a push becomes an inject, a pop becomes an eject, an inject becomes a push, and an eject becomes a pop. To support catenation as well as ip we use reversal bits at all levels. We must also symmetrize the denition in Section 5 to allow a deque to be represented by a prex only, and extend the various operations to handle this possibility. The interpretation of reversal bits is cumulative. That is, if d is a deque and x is a deque inside of d, x is regarded as being reversed if an odd number of reversal bits are set to 1 along the path of actual pointers in the structure from the node for d to the node for x. Before performing catenation, if the reversal bit of either or both of the two deques is 1, we push such bits down by ipping such a bit of a deque x to 0, ipping the bits of all the deques to which x points, and swapping the appropriate buers and deques. (The prex and su-x exchange roles, as do the left deque and right deque; the order of elements in the prex and su-x is reversed as well.) We do such push-downs of reversal bits by assembling new deques, not by overwriting the old ones. We have devised an alternative implementation of catenable deques in which the sizes of the prexes and su-xes are between 3 and 5 instead of 3 and 6. We do this by memoizing the pop and eject operations and avoiding creating a new structure with a green prex (su-x, respectively) representing the original deque when performing pop (eject, respectively). Using a more complicated potential function than the ones used in earlier sections, we can show that such an implementation runs in O(1) amortized time per operation. One direction for future research is to nd a way to simplify our structures fur- ther. Specically, consider the following alternative representation of catenable de- ques, which uses a single recursive subdeque rather than two such subdeques. A nonempty deque d over A is represented by a triple that consists of a prex pr(d), a (possibly empty) child deque of triples c(d), and a su-x sf(d). A triple consists of a nonempty prex , a deque of triples, and a nonempty su-x, or just of a nonempty prex or su-x. All buers and triples are over A. The operations push, pop, inject, and eject have implementations similar to their implementations in Section 5. The major dierence is in the implementation of catenate, which for this structure requires a call to pop. Specically, let d 1 and d 2 be two deques to be catenated. catenate pops c(d 1 ) to obtain a triple (p; d and a new deque c, injects (s; c; sf(d 1 to obtain d 00 , and then pushes (p; d 00 ; pr(d 2 . The nal result has prex pr(d 1 ), child deque c 0 , and su-x sf(d 2 ). It is an open question whether this algorithm runs in constant amortized time per operation for any constant upper and lower bounds on the buer sizes. Another research direction is to design a con uently persistent representation of sorted lists such that accesses or updates d positions from either end take O(log d) time, and catenation takes O(1) time. The best structure so far developed for this problem has a doubly logarithmic catenation time [12]; it is purely functional, and the time bounds are worst-case. Acknowledgment . We thank Michael Goldwasser for a detailed reading of this paper, and Jason Hartline for discussions that led to our implementations using memoization --R Data structural bootstrapping Con uently persistant deques via data structural boot- strapping Fully persistent arrays Fully persistent lists with catenation Making data structures persistent Deques with heap order lists, PhD thesis Simple con uently persistent catenable lists (ex- tended abstract) An optimal RAM implementation of catenable min double-ended queues Amortized computational complexity Worst case analysis of set union algorithms --TR --CTR Amos Fiat , Haim Kaplan, Making data structures confluently persistent, Journal of Algorithms, v.48 n.1, p.16-58, August George Lagogiannis , Yannis Panagis , Spyros Sioutas , Athanasios Tsakalidis, A survey of persistent data structures, Proceedings of the 9th WSEAS International Conference on Computers, p.1-6, July 14-16, 2005, Athens, Greece

queue;memoization;functional programming;data structures;stack;double-ended queue deque;stack-ended queue steque;persistent data structures

586990

Taking a Walk in a Planar Arrangement.

We present a randomized algorithm for computing portions of an arrangement of n arcs in the plane, each pair of which intersect in at most t points. We use this algorithm to perform online walks inside such an arrangement (i.e., compute all the faces that a curve, given in an online manner, crosses) and to compute a level in an arrangement, both in an output-sensitive manner. The expected running time of the algorithm is $O(\lambda_{t+2}(m+n)\log n)$, where m is the number of intersections between the walk and the given arcs. No similarly efficient algorithm is known for the general case of arcs. For the case of lines and for certain restricted cases involving line segments, our algorithm improves the best known algorithm of [M. H. Overmars and J. van Leeuwen, J. Comput. System Sci., 23 (1981), pp. 166--204] by almost a logarithmic factor.

Introduction S be a set of n x-monotone arcs in the plane. Computing the whole (or parts of the) arrangement S), induced by the arcs of " S, is one of the fundamental problems in computational geometry, and has received a lot of attention in recent years [SA95]. One of the basic techniques used for such problems is based on randomized incremental construction of the vertical decomposition of the arrangement (see [BY98] for an example). If we are interested in only computing parts of the arrangement (e.g., a single face or a zone), the randomized incremental technique can still be used, but it requires non-trivial modifications Intuitively, the added complexity is caused by the need to "trim" parts of the plane as the algorithm advances, so that it will not waste energy on regions which are no longer relevant. In fact, this requirement implies that such an algorithm has to know in advance what are the regions we are interested in at any stage during the randomized incremental construction. A variation of this theme, with which the existing algorithms cannot cope efficiently, is the following online scenario: We start from a point and we find the face f of A( " that contains p(0). Now the point p starts moving and traces a connected curve fp(t)g t0 . As our walk continues, we wish to keep track of the face of A( " S) that contains the current point 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/ p(t). The collection of these faces constitutes the zone of the curve p(t). However, the function p(t) is not assumed to be known in advance, and it may change when we cross into a new face or abruptly change direction in the middle of a face (see [BDH97] for an application where such a scenario arises). The only work we are aware of that can deal with this problem efficiently is due to Overmars and van Leeuwen [OvL81], and it only applies to the case of lines (and, with some simple modifications to the case of segments as well). It can compute such a walk in (deterministic) O((n +m) log 2 n) time, inside an arrangement of n lines, where m is the number of intersections of the walk with the lines of " S. This is done by maintaining dynamically the intersection of half-planes that corresponds to the current face. The algorithm of [OvL81] is somewhat complicated and it is probably not practical for actual implementation. In this paper, we propose a new randomized algorithm that computes the zone of the walk in a general arrangement of arcs, as above, in O( t+2 (n+m) log n) expected time, where t+2 (n+m) is the maximum length of a Davenport-Schinzel sequence of order t [SA95]. The new algorithm can be interpreted as a third "online" alternative to the algorithms of [CEG dBDS95]. The algorithm is rather simple and appears to be practical. As a matter of fact, we are currently implementing and experimenting with a variant of the algorithm. As an application of the new algorithm, we present a new algorithm for computing a level in an arrangement of arcs. It computes a single level in O( t+2 (n +m) log n) expected time, where m is the complexity of the level. Both results improve by almost a logarithmic factor over the best previous result of [OvL81], for the case of lines. For the case of general arcs, we are not aware of any similarly efficient previous result. The paper is organized as follows. In Section 2 we describe the algorithm. In Section 3 we analyze its performance. In Section 4 we mention a few applications of the algorithm, including that of computing a single level. Concluding remarks are given in Section 5. 2 The Algorithm In this section, we present the algorithm for performing an online walk inside a planar arrangement Randomized Incremental Construction of the Zone Using an Oracle. Given a set " of n x-monotone arcs in the plane, so that any pair of arcs of " S intersect at most t times (for some fixed constant S) denote the arrangement of " namely, the partition of the plane into faces, edges, and vertices as induced by the arcs of " S (see [SA95] for details). We assume that " S is in general position, meaning that no three arcs of " S have a common point, and that the x-coordinates of the intersections and endpoints of of the arcs of " are pairwise distinct. The vertical decomposition of A( " S), denoted by A VD S), is the partition of the plane into vertical pseudo-trapezoids, obtained by erecting two vertical segments up and down from each vertex of S), (i.e., points of intersections between pairs of arcs and endpoints of arcs) and extending each of them until it either reaches an arc of " S, or otherwise all the way to infinity. See [BY98] for more details concerning vertical decomposition. To simplify (though slightly abuse) the notation, we refer to the cells of A VD S) as trapezoids. A selection R of " S is an ordered sequence of distinct elements of " S. By a slight abuse of notation, we also denote by R the unordered set of its elements. 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 Computing the decomposed arrangement A VD S) can be done as follows. Pick a random permutation S. Compute incrementally the decomposed arrangements A VD (S i ), inserting the i-th arc s i of S into A VD (S To do so, we compute the district D i of s i in A VD (S i\Gamma1 ), which is the set of all trapezoids in A VD (S i\Gamma1 ) that intersect s i . We split each trapezoid of D i into O(1) 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 arcs. The merging step guarantees that the resulting decomposition is A VD (S i ), independently of the insertion order of elements Let fl be the curve of the walk. For a selection R 2 S), let Z fl (R) denote the zone of fl in A(R); this is the set of all faces of A(R) that have a nonempty intersection with fl. Let A fl;VD S) denote the union of all trapezoids that cover Z S). Our goal is to compute A S). We assume for the moment that we are supplied with an oracle O(S \Delta), that can decide in constant time whether a given vertical trapezoid \Delta is in A fl;VD (S i ). Equipped with this oracle, computing A fl;VD (S) is fairly easy, using a variant of the randomized incremental construction, outlined above. The algorithm is depicted in Figure 1. We present this algorithm at a conceptual level only, because this is not the algorithm that we shall actually use. It is given to help us to describe and analyze the actual online algorithm that we shall describe later. Note that the set of trapezoids maintained by the algorithm in the i-th iteration is a superset of A fl;VD (S i ) (there might be trapezoids in C i that are no longer in Z i . However, this implies that those trpaezoids will be eliminated the first time an arc belonging to their conflict list will be handled. Moreover, the algorithm CompZoneWithOracle can be augmented to compute a history DAG (as in [SA95]), whose nodes are the trapezoids created by the algorithm and where each trapezoid destroyed during the execution of the algorithm points to the trapezoids that were created from it. Let HT fl (S i ) denote this structure after the i-th iteration of the algorithm. Definitions. A trapezoid created by the split operation of CompZoneWithOracle is called a transient trapezoid if it is later merged (in the same iteration) to form a larger trapezoid. A trapezoid generated by CompZoneWithOracle is final if it is not transient. The rank rank(\Delta) of a trapezoid \Delta is the maximum of the indices i; j of the arcs containing the bottom and top edges of \Delta in the permutation S. We denote by D(\Delta) the defining set of a final trapezoid \Delta; this is the minimal set D such that \Delta 2 A VD (D). It is easy to verify that jD(\Delta)j 4. We can also define D(\Delta) for a transient trapezoid \Delta, to be the minimal set D such that \Delta can be trnasient during an incramental construction of A VD (D). Here it is easy to verify that jD(\Delta) 6. The index index(\Delta) of a trapezoid \Delta is the minimum i such that D(\Delta) ' S i . For a trapezoid \Delta, we denote by cl(\Delta) the conflict list of \Delta; that is, the set of arcs of " S that intersect \Delta in its interior. Let next(\Delta) denote the first element of cl(\Delta), according to the ordering of S. For a trapezoid \Delta generated by CompZoneWithOracle (which was not merged into a larger trapezoid), we denote by father(\Delta) the trapezoid that \Delta was generated from. A vertical side Algorithm CompZoneWithOracle( " Input: A set " S of n arcs, a curve fl, an oracle O Output: A fl;VD begin Choose a random permutation S. for i from 1 to n do for each \Delta 2 D i such that int do split(\Delta; s) is the operation of splitting a vertical trapezoid \Delta crossed by an arc s into a constant number of vertical trapezoids, as in [dBvKOS97], such that the new trapezoids cover \Delta, and they do not intersect s in their interior. end for Merge all the adjacent trapezoids of Temp that have the same top and bottom arcs. Let Temp 1 be the resulting set of trapezoids. Let Temp 2 be the set of all trapezoids of Temp 1 that are in A fl;VD (S i ). Compute this set using jT emp 1 j calls to O. end for return C n CompZoneWithOracle Figure 1: A randomized incramental algorithm for constructing the zone of a walk in an arrage- ment of arcs, using an oracle of a vertical trapezoid \Delta is called a splitter. A splitter is transient if it is not incident to the intersection point (or endpoint) that induced the vertical edge that contains (this means that the two trapezoids adjacent to are transient, and will be merged into a larger final trapezoid). Figure 2 for an illustration of some of these definitions. It is easy to verify that a trapezoid \Delta is transient if and only if at least one of its bounding splitters is transient. Thus, one can decide whether a trapezoid is transient, by inspecting its splitters, in constant time. An Online Algorithm for Constructing the Zone. Let us assume that the random permutation S has been fixed in advance. Note that S predetermines HT (S). The key observation in the online algorithm is that in order to construct a specific leaf of HT fl (S) we do not have to maintain the entire DAG, and it suffices to compute only the parts of the DAG that lie on paths connecting the leaf with the root of HT fl (there might be several such paths, since our structure is a DAG, and not a tree). To facilitate this computation, we maintain a partial history DAG T . The nodes of T are of two types: (i) final nodes - those are nodes whose corresponding trapezoids appear in HT fl (S), l 1 l 2 l 3 l 4 l 5 Figure 2: Illustration of the defintions: (i) is a transient splitter, and thus ; 0 are both transient. We have rank( and (ii) transient nodes - these are some of the leaves of T , whose corresponding trapezoids are transient. Namely, all the internal nodes of T are copies of identical nodes of HT fl (whose corresponding trapezoids are final), while some of the leaves of T might be transient. Intuitively, T stores the portion of HT fl that we have computed explicitly so far. The transient leaves of delimit poritions of HT fl that have not been expanded yet. Inside each node of T , we also maintain the conflict list of the corresponding trapezoid. Suppose we wish to compute a leaf of HT fl which contains a given point p. We first locate the leaf of T that contains p. This is done by traversing a path in T , starting from the root of T , and going downward in each step into the child of the current trapezoid that contains p (this requires O(1) time, because the out-degree of any node of HT fl is bounded by a constant that depends on t). At the end we either reach a final leaf which is the required leaf of HT fl , or we encounter a transient leaf v. In the latter case, we need to expand T further below v, and the first step is to replace v by the coresponding node v of HT fl , obtained by merging the transient trapezoid of v with adjacent transient trapezoids, to form final trapezoid associated with v . Assume for the moment that we are supplied with a method (to be described shortly) to generate all those transient trapezoids, whose union forms the final trapezoid that is stored at v in HT fl . Then we do the following: (i) Merge all those transient trapezoids into a new (final) trapezoid \Delta; (ii) Compute the conflict list cl(\Delta) from the conflict lists of the transient trapezoids; 1 (iii) Compute the first element s \Delta in cl(\Delta) according to the permutation S; and (iv) Compute all the transient trapezoids or final generated from \Delta by splitting it by s \Delta (this generates O(1) new trapezoids). Overall, this requires O(k is the number of transient trapezoids that are merged, and l is the total length of the conflict lists of the transient trapezoids. Thus, we had upgraded a transient node v at T into a final node v . We denote this operation by Expand(v). We can now continue going down in T , passing to the child of \Delta that contains p and repeating recursively the above procedure at that child, until reaching the desired leaf of HT fl that contains p. Let be a newly created trapezoid. If is transient, then one of its splitters must be transient. Let denote this transient splitter, and let us assume that is the right edge of . This implies that either the top arc or the bottom arc of are the cause of the splitting that generated . In particular, next( f ) is either the top or bottom arc of , where f denotes the trapezoid that was generated from. To perform the merging of the conflict lists in linear time, one may use either a hash-table, or a bit-vector, or one maintains the conflict lists in a consistent ordering. See [BY98]. We compute the transient trapezoid 0 that lies to the right of , by taking te midpoint p of , and by performing a point-location query of p in T . During this point-location process, we always go down into the trapezoid \Delta that contains p in its interior or on its left edge. We stop as soon as we encounter a transient trapezoid 0 that has a left edge identical to the right edge of . This happens when and 0 have the same top and bottom edges; namely, we stop when Intuitively, if the trapezoid 0 has rank smaller than rank( ), then its left edge is longer than the right edge of ; the first time when both and 0 have indentical connecting edge is when their top and bottom edges are identical, namely, when rank( continue this process of collecting adjacent transient trapezoids using point-location queries on midpoints of transient splitters, until the two extreme splitters (to the left and to the right) are non-transient. We take the union of those trapezoids to be the new expanded trapezoid. See Figure 2. Of course, during this point-location process, we might be forced into going into parts of HT fl that do not appear yet in T . In such a case, we will compute those parts in an online manner, by performing Expand calls on the relevant transient trapezoids that we might encounter while going down T . Thus, the process of turning a transient trapezoid into a final trapezoid is a recursive process, which might be quite substantial. Let G S denote the adjacency graph of A VD (S). This is a graph having a vertex for each trapezoid in A VD (S), such that an edge connects two vertices if their corresponding trapezoids share a common vertical side. Moreover, under general position assumptions, a vertex in G S has degree at most 4. It is easy to verify that a connected component of G S corresponds to a face of A(S). By performing a point-location query, as described above, for a point p in T , we can compute the node v of G S whose trapezoid \Delta v contains p. Furthermore, by carrying out a point-location query similar to that used in the Expand operation, we can compute a node u of G S . Indeed, during such a point-location query, we traverse down T until we reach a leaf u of HT fl (i.e., the conflict list of the corresponding trapezoid is empty). The node u is a node of G S adjacent to v (i.e., uv is an edge of G S ). Repeating this process, we can perform DFS in G S , which corresponds to the entire face of A( " S) that contains p. Let fl be the curve of the online walk whose zone we wish to compute. We consider fl to be a directed curve, supplied to us by the user through a function EscapePoint fl . The function EscapePoint fl (p; \Delta) receives as input a point p 2 fl, and a trapezoid \Delta that contains p, and outputs the next intersection point of fl with @ \Delta following p. If we reach the end of fl, the function returns nil. We assume (although this is not crucial for the algorithm) that fl does not intersect itself. Thus, given a walk fl, we can compute its zone by the algorithm depicted in Figure 3. Note that by the time the algorithm terminates, the final parts of T are contained in HT fl . proper inclusion might arise; see Remark 3.6.) In analyzing the performance of the algorithm, we first bound the overall expected time required to compute HT fl , which can be done by bounding the expected running time of CompZoneWithOracle (in an appropriate model of computaiton). Next, we will bound the additional time spent by the algorithm in traversing between adjacent trapezoids (i.e., the time spent in performing the point-location queries). Remark 2.1 By skipping the expansion of the face that contains the current point p in CompZoneOnline, we get a more efficient algorithm that only computes the D of the walk. There might be cases where this will be sufficient. Algorithm CompZoneOnline( " Input: A set " S of n arcs, a starting point p of the walk, and a function EscapePoint fl that represents the walk Output: The decomposed zone of fl in A( " begin Choose a random permutation S. - a partial history DAG with a root corresponding to the whole plane where leaf(HT is the leaf of HT fl whose associate trapezoid contains p. (All the paths in HT fl from v to the root now exist in T .) Compute the face F containing \Delta v in A fl;VD (S), and add it to the output zone. Z while Compute v, the next leaf of HT fl , such that This is done by performing a point-location query in T , as described in the text, and enlarging T accordingly. Compute the face F of \Delta v in A fl;VD (S) (if it was not computed already), and add it to the output zone. Z while return Z. CompZoneOnline Figure 3: Algorithm for constructing the zone of a walk in an arragement of arcs in an online manner 2.1 Correctness In this section, we try to prove the correctness of CompZoneOnline. Observation 2.2 During the execution of CompZoneOnline, the union of trapezoids of the leaves of T form a pairwise disjoint covering of the plane by vertical trapezoids. Corollary 2.3 Each conflict list, computed for some trapezoid \Delta by the procedure CompZoneOnline is the list of all arcs of S that cross \Delta. Proof: By induction on the steps of CompZoneOnline. Observe that regions that \Delta was generated from cover \Delta, and thus the union of their conflict lists must contain the conflict list of \Delta. Corollary 2.4 For a trapezoid \Delta created by CompZoneOnline, we have that the all the curves of D(\Delta) appears in S before all the curves of K (\Delta). Lemma 2.5 Point-location query in the middle of transient splitters never fails, namely, such a query always generate a transient trapezoid whcih is adjacent to the current transient trapezoid, and they have the save top and bottom arcs. Proof: Let be the current transient trapezoid, let be a transient splitter, and let p be the point located in the middle of and assume without loss of generality that is the right edge of . The point-location query must end up in a trapezoid \Delta, which is currently a node of T that contains on its left edge. From this point on, the algorithm "refines" \Delta by going down in T , performing a sequence of splitting and expansion operations. be the sequence of trapezoids created (or visited) having p on their left side, computed during the "hunt" for a transient trapezoid adjacent to . First, note that during this process we can not perform an insertion of arc having an endpoint in the interior of . Since this will either contradict our general position assumption, or will imply that is not a transient splitter. Let \Delta i be the trapezoid having rank(s), such that i is maximl. Clearly, the left edge of \Delta must contain . Otherwise, there is an arc s l of S i\Gamma1 that intersects the interior of , but this implies that the computation of the conflict list of is incorrect, contradicting Coroallary 2.3. Thus, must have the same top and bottom arcs as . Implying, that the left side of \Delta i is . We conclude that during the point-location process we will compute \Delta i , and Definition 2.6 For a permutation S of " S, let Hist = Hist(S) denote the history-DAG generated by computing the whole vertical decomposition A VD (S), by an incremental construction that inserts the curves in their order in S. Lemma 2.7 For any final trapezoid \Delta created by the Expand procedure, during the execution of CompZoneOnline, there exists i 0, such that \Delta is a trapezoid of A VD (S i ). As a matter of fact, we have Proof: By induction on the depth of the nodes in T , where the depth of a node is defined to be the length of the longest path from the root of T to this node. Indeed, for the base of the induction, a node of depth 0 must be the root of T , which is being computed during initialization of the algorithm, and is thus the only trapezoid of A VD (S 0 ), Let \Delta be a final trapezoid of depth k in T that was generated (directly) from a trapezoid by the procedure Expand. Let the final trapezoid that was split to generate . By our induction hypthesis, f is a trapezoid of A VD (S l ), where Corollary 2.3, the conflict list of f was computed correctly. If is final, then by the above, we have is a trapezoid of A VD (S i ), where (namley, next(father( Otherwise, is transient, and during its expansion, we had computed several transient trapezoids using point-location queries. Note that those point-location queries were performed by placing points on transient splitters; namely, as soon as we encounter a non-transient splitter, we had aborted the expansion in this direction. Thus, must have the same two arcs as floor and ceiling (otherwise, either the algorithm performed a point-location in the middle of a non-trnasient splitter, or the computation of the conflict lists is incorrect). Let Clearly, \Delta is a trapezoid, and its two splitters are non-transient. We claim that the two splitters of \Delta are induced by intersections having index at most was generated from father( i ) by the splitting caused by s m . And father( i ) (and its conflict list) was computed correctly, by our induction hypothesis. Moreover, the left splitter of \Delta is either empty (i.e., \Delta left vertical side is an intersection point), or it is final. If is final, then it is adjacent to the intersection point that induces it, which by the above is defined by (at most) three arcs that must appear is Sm . Similarly, the right splitter of \Delta is final, and is defined by (at most) three arcs that appear in Sm . Thus, \Delta is a trapezoid that does not intersect any arc of Sm in its interior, its top and bottom arcs belong to Sm , and its two splitters are final and defined by arcs of Sm . This implies that \Delta is in A VD (Sm ). Lemma 2.8 All the final nodes computed by CompZoneOnline appear in HT fl . Proof: Let \Delta be a final trapezoid computed by CompZoneOnline. The trapezoid \Delta was generated during a sequence of recursive calls to Expand. Let be the set of final trapezoids created direcly by those recursive calls, such that and they are ordered according to their recursive call ordering. Let l The trapezoid \Delta 1 was created because we performed a point-location query for a point p that appear in Z(S). Since Z(S i+1 Moreover, if \Delta i+1 was computed during the computation of \Delta i , then there must be a point p i that lie inside both vertical trapezoids, because the computation of \Delta i+1 was initiated by a point-location query must also lie inside \Delta i+1 . This implies that p i 2 Z(S l i Thus, It follows, that \Delta i+1 appears in HT fl (S l i+1 By induction, it follows that \Delta k 2 HT fl (S). 3 The Analysis 3.1 Constructing the History DAG In the following, we analyze the performance of CompZoneWithOracle. We assume that it maintains for each trapezoid a conflict-lists that stores the set of arcs that cross it. Thus, the cost of each operation on a trapezoid is proportional to the size of its conflict list. We also assume that a call to the Oracle takes O(1) time. Lemma 3.1 The algorithm CompZoneWithOracle computes the zone of fl in A VD (S) in O ( t+2 (n +m) log n) expected time, and the expected number of trapezoids that it generates is O ( t+2 (n +m)). Proof: The proof is a straightforward adaptation of the proof of [CEG We omit the easy details. Observation 3.2 The trapezoids computed by CompZoneOnline are either (final) trapezoids computed by CompZoneWithOracle (and thus appear in HT fl ), or transient trapezoids that were split from trapezoids of HT fl . Lemma 3.3 The expected number of transient trapezoids generated by CompZoneOnline is O( t+2 (n +m)), and the expected total size of their conflict lists is O( t+2 (n +m) log n). Proof: Each final trapezoid generated by CompZoneOnline might be split into O(1) transient trapezoids. Each final trapezoid computed by CompZoneOnline is also computed by CompZoneWithOracle. By Lemma 3.1, the expected number of such trapezoids is O( t+2 (n+m)). The second part of the lemma follows by a similar argument. Definition 3.4 A curve fl is locally x-monotone in A( " S), if it can be decomposed inside each face of A( " into a constant number of x-monotone curves. Theorem 3.5 The algorithm CompZoneOnline computes the zone of fl in A(S) in O ( t+2 (n +m) log n) expected time, provided that fl is a locally x-monotone curve in A( " S). Proof: The time spent by CompZoneOnline is bounded by the time required to construct the history DAG, by the time spent in maintaining the conflict lists of the trapezoids, and by the time spent on performing point-location queries, as we move from one trapezoid to another in A fl;VD (S). By Lemmas 3.1 and 3.3, the expected time spent on maintaining the conflict lists of the trapezoids computed by the algorithm is O( t+2 (n m) log n), since the total time spent on handling the conflict lists is proportional to their total length. By Lemma 3.3, the expected total size of those conflict lists is O( t+2 (n +m) log n). Moreover, the depth of the history DAG constructed by the algorithm is O(log n) with a probability polynomially close to 1 [Mul94]. Thus, the expected time spent directly on performing a single point-location query (ignoring the time spent on maintaining the conflict lists) as we move from one trapezoid to the next, is O(log n). The curve fl is locally x-monotone, which implies that it intersects the splitters of each trapezoid of A fl;VD (S) at most O(1) times. Thus, the expected number of point-location queries performed by the algorithm is proportional to the expected number of transient trapezoids created plus O(m). By Lemma 3.3, we have that the expected running time is O Remark 3.6 Note that CompZoneWithOracle computes the zone of fl in A VD (S i ), for each In fact, it might compute a trapezoid \Delta 2 A fl;VD (S i ) that does not intersect the zone of fl in A fl;VD (S). In particular, such a trapezoid \Delta will not be computed by CompZoneOnline. This is a slackness in our analysis that we currently do not know whether it can be exploited to further improve the analysis of the algorithm (we suspect that it cannot). Remark 3.7 The only result of this type that we are aware of, is a classical result due to Overmars and van Leeuwen [OvL81]. It maintains dynamically the convex hull of n points in the plane in O(log 2 n) time for each insertion or deletion operation. The dual variant of this results (maintaining the intersection of halfplanes) can be used to perform walks inside line arrangements in (deterministic) O((n m) log 2 n) time, where m is the number of intersections of the walk with the lines. The algorithm of [OvL81] requires somewhat involved rebalancing of the tree that represents the current intersection of halfplanes. Our algorithm is somewhat simpler, faster, and applies to more general arrangements. As for segments and general arcs, we are not aware of any result of this type in the literature. Of course, if the curve fl is known in advance (and is simple, in the sense that one can compute quickly its intersections with any arc of " S), we can compute the single face in the modified arrangement (as in the proof of the general planar Zone Theorem [SA95, Theorem XX]) using the algorithms of [dBDS95, CEG + 93]. These algorithms are slightly simpler than the algorithm of Theorem 3.5, although they have the same expected performance. However, these algorithms are useless for online walks. Applications In this section we present several applications of the algorithm CompZoneOnline. 4.1 Computing a Level in an Arrangement of Arcs In this subsection we show how to modify the algorithm of the previous section to compute a level in an arrangement of x-monotone arcs. Definition 4.1 Let " S be a set of n x-monotone arcs in the plane, any pair of which intersect at most t times (for some fixed constant t). We assume that " S is in general position, as above. The level of a point in the plane is the number of arcs of " lying strictly below it. Consider the closure l of the set of all points on the arcs of " having level l (for 0 l ! n). E l is a x-monotone (not necessarily connected) curve (which is polygonal in the case of lines or segments), which is called the level l of the arrangement A( " S). At x-coordinates where a vertical line intersects less than l lines of S, we consider E l to be undefined. Levels are a fundamental structure in computational and combinatorial geometry, and have been subject to intensive research in recent years (see [AACS98, Dey98, TT97, TT98]). Tight bounds on the complexity of a single level, even for arrangements of lines, proved to be surprisingly hard to obtain. Currently, the best known upper bound for the case of lines is O(n(l+1) 1=3 ) [Dey98], while the lower bound is \Omega\Gamma n log (l bounds for other classes of arcs. First, note that if " S is a set of lines, then, once we know the leftmost ray that belongs to E l , then the level l is locally defined: as we move from left to right along E l , each time we encounter an intersection point (a vertex of A( " we have to change the line that we traverse. (This is also depicted in Figure 4.) In particular, we can compute the level E l in O( 3 (n using CompZoneOnline. The same procedure can be used to compute a level in an arrangement of more general arcs. The only non-local behavior we have to watch for are jump discontinuities of the level caused when an endpoint of an arc appears below the current level, or when the Figure 4: The first level in an arrangement of segments (the vertical edges show the jump discontinuities of the level, but are not part of the level). current level reaches an endpoint of an arc (see Figure 4). See below for details concerning the handling of those jumps. In the following, let l; 0 l ! n be a prescribed parameter. Let E l denote the level l in the arrangement S). The following adaption of CompZoneOnline to our setting is rather straightforward, but we include it for the sake of completeness. We sort the endpoints of the arcs of " S by their x- coordinates. Each time our walk reaches the x-coordinate of the next endpoint, we updated E l by jumping up or down to the next arc, if needed. This additional work requires O(n log n) time. During the walk, we maintain the invariant that the top edge of the current trapezoid is part of l . To compute the first trapezoid in the walk, we compute the intersection of level l with the y-axis (this can be done by sorting the arcs according to their intersections with the y-axis). Let 0 be this starting point. We perform a point-location query with p 0 in our virtual history DAG to compute the starting trapezoid \Delta 0 . Now, by walking to the right of \Delta 0 we can compute the part of E l lying to the right of the y-axis. Indeed, let \Delta be the current trapezoid maintained by the algorithm, such that its top edge is a part of E l . Let p(\Delta) denote the top right vertex of \Delta. By performing point-location queries in our partial history DAG T , we can compute all the trapezoids of A VD (S) that contain p(\Delta) (by our general position assumption, the number of such trapezoids is at most 6; this number materializes when p(\Delta) lies in the intersection of two x-monotone arcs). By inspecting this set of trapezoids, one can decide where E l continues to the right of \Delta, and determine the next trapezoid having E l as its roof. The algorithm sets \Delta to be this trapezoid. If the algorithm reaches an x-coordinate of an endpoint of an arc, we have to update E l by jumping up (if this is the right endpoint of an arc and it lies on or below the level) or down (if it is a left endpoint and lies below the level); namely, we set \Delta to be the trapezoid lying above (or below) the current \Delta. The algorithm continues in this manner, until reaching the last edge of E l . The algorithm then performs a symmetric walk to the left of the y-axis to compute the other portion of the level. Let CompLevel denote this modified algorithm. We summarize our result: Theorem 4.2 The algorithm CompLevel computes the level l in A( " S) in O ( t+2 (n expected time. Remark 4.3 Since CompLevel is online, we can use it to compute the first m 0 points of E l , in expected O( t+2 (n +m 0 ) log n) time. Remark 4.4 A straightforward extension of CompLevel allows us to compute any connected path within the union of " S (i.e., we restrict our "walk" to the arcs of " S) in an on-line manner, in randomized expected time O ( t+2 (m + n) log n), where m is the number of vertices of the path. As above, the extended version can also handle jumps between adjacent arcs during the walk. 4.2 Other Applications In this subsection, we provide some additional applications of CompZoneOnline. Theorem 4.5 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+ ")(8r randomized expected time O log n , where ff(n) is the inverse of the Ackermann function [SA95]. Proof: This follows by plugging the algorithm of Theorem 4.2 and Remark 4.3 into the algorithm described in [HP98]. For a discussion of cuttings of small asymptotic size, and their applications, see [Mat98, HP98]. Remark 4.6 Theorem 4.5 improves the previous result of [HP98] by almost a logarithmic factor. Remark 4.7 Once we have computed the level l (in an arrangement of general arcs), we can clip the arcs to their portions below the level. Supplied with those clipped arcs, we can compute the arrangement below the level l in O((m+n) log n+ r) time, where is the complexity of the level l, and r is the complexity of the first l levels of A( " Thus, we can compute the first l levels of A( " S) in O( t+2 (m+n) log n+r) expected time, using randomized incremental construction [Mul94]. This improves over the previous result of [ERvK96] that computes this portion of the arrangement in O(n log n (note that this running time is not output sensitive). A byproduct of the technique of CompZoneOnline is the ability to perform point-location queries using the partial history DAG mechanism. Definition 4.8 For a point set P , and a set of arcs " S) denote a connected polygonal set, such S), and (ii) the number of intersections between S) and the arcs of " S is minimum. Let wM S) denote the number of such intersections. The set S) can be interpreted as the minimum spanning Steiner tree of P , under the metric of intersections with the arcs of " S. Lemma 4.9 Given a set of " S of n arcs in the plane. One can answer point-location queries for a set P of m points in an online manner, such that the overall expected time to answer those queries is O( t+2 (n log n) time. Proof: We precompute a random permutation S of " S, and let T be our partial history DAG. We answer the point-location queries, by computing the relevant parts of the history DAG of A VD (S), as in CompZoneOnline. By the time the algorithm terminates, T is contained in A fl;VD (S), where S). However, the expected total weight of the trapezoids of T Z(S) is O( t+2 (n m) log(n)), by Lemma 3.1. Which bounds the overall expected query time. Remark 4.10 The result of Lemma 4.9 is somewhat disappointing, since wM [Aga91], while for the case of lines, faces can be computed in, roughly, O(n 2=3 2=3 ) [AMS98] (i.e,, j). We are not aware of any algorithm with a better running time, than the algorithm of Lemma 4.9, for the case of lines, where the query points are given in an online fashion. Currently, for the case of general arcs, no better bound than O is known, on the complexity of faces in arrangement of n arcs (see [EGP The algorithm of Lemma 4.9 is simple, and it have the favorable additional property of being adaptive. Namely, if wM S) is smaller (i.e., the query point are "close together") the overall query time improves. Furthermore, if there are alot of queries close together, the first query will be slow, and the later ones will be fast (since the later queries use parts of paths that already exists in the partial history DAG). Conclusions In this paper we have presented a new randomized algorithm for computing a zone in a planar arrangement, in an online fashion. This algorithm is the first efficient algorithm for the case of planar arcs, it performs faster (by nearly a logarithmic factor) than the algorithm of [OvL81] for the case of lines and segments, and it is considerably simpler. We also presented an efficient randomized algorithm for computing a level in an arrangement of arcs in the plane, whose expected running time is faster than any previous algorithm for this problem. The main result of this paper relies on the application of point-location queries to compute the relevant parts of an "off-line" structure (i.e., the history DAG). The author believes that this technique should have additional applications. In particular, this approach might be useful also for algorithms in higher dimensions. We leave this as an open question for further research. Acknowledgments The author wishes to thank Pankaj Agarwal, Danny Halperin and Micha Sharir for helpful discussions concerning the problems studied in this paper and related problems. --R On levels in arrangements of lines Intersection and decomposition algorithms for planar arrangements. The area bisectors of a polygon and force equilibria in programmable vector fields. Algorithmic Geometry. Computing a face in an arrangement of line segments. On lazy randomized incremental construction. Computational Geometry: Algorithms and Applications. Improved bounds for planar k-sets and related problems Algorithms in Combinatorial Geometry. Arrangements of curves in the plane: Topology An optimal algorithm for the Constructing cuttings in theory and practice. The complexity of many cells in the overlay of many arrangements. Computational Geometry: An Introduction Through Randomized Algorithms. Maintenance of configurations in the plane. A characterization of planar graphs by pseudo-line arrangements How to cut pseudo-parabolas into segments --TR --CTR Nisheeth Shrivastava , Subhash Suri , Csaba D. Tth, Detecting cuts in sensor networks, Proceedings of the 4th international symposium on Information processing in sensor networks, April 24-27, 2005, Los Angeles, California Naoki Katoh , Takeshi Tokuyama, Notes on computing peaks in k-levels and parametric spanning trees, Proceedings of the seventeenth annual symposium on Computational geometry, p.241-248, June 2001, Medford, Massachusetts, United States

planar arrangements;single face;levels;computational geometry

586992

The Density of Weakly Complete Problems under Adaptive Reductions.

Given a real number $\alpha < 1$, every language that is weakly $\leq_{n^{\alpha / 2} - {\rm T}}^{{\rm P}} $-hard for E or weakly $\leq_{n^{\alpha} - {\rm T}}^{\rm P}$-hard for E2 is shown to be exponentially dense. This simultaneously strengthens the results of Lutz and Mayordomo (1994) and Fu (1995).

Introduction In the mid-1970's, Meyer[15] proved that every - P -complete language for exponential time-in fact, every - P -hard language for exponential time-is dense. That is, linear ), DENSE is the class of all dense languages, DENSE c is the complement of DENSE, and Pm(DENSE c ) is the class of all languages that are - P -reducible to non-dense languages. language A 2 f0; 1g is dense if there is a real number ffl ? 0 such that jA -n for all sufficiently large n, where .) Since that time, a major objective of computational complexity theory has been to extend Meyer's result from - P m -reductions to - P -reductions, i.e., to prove that every - P -hard language for E is dense. That is, the objective is to prove that where PT(DENSE c ) is the class of all languages that are - P T -reducible to non-dense lan- guages. The importance of this objective derives largely from the fact (noted by Meyer[15]) that the class PT(DENSE c ) contains all languages that have subexponential circuit-size complexity. language A ' f0; 1g has subexponential circuit-size complexity if, for every real number ffl ? 0, for every sufficiently large n, there is an n-input, 1-output Boolean This research was supported in part by National Science Foundation Grant CCR-9157382, with matching funds from Rockwell International, Microware Systems Corporation, and Amoco Foundation. circuit that decides that the set and has fewer than 2 n ffl gates. Other- wise, we say that A has exponential circuit-size complexity.) Thus a proof of (2) would tell us that E contains languages with exponential circuit-size complexity, thereby answering a major open question concerning the relationship between (uniform) time complexity and circuit-size complexity. Of course (2) also implies the more modest, but more famous conjecture, that where SPARSE is the class of all sparse languages. language A ' f0; 1g is sparse if there is a polynomial q(n) such that jA -n j - q(n) for all n 2 N.) As noted by Meyer[15], the class PT(SPARSE) consists precisely of all languages that have polynomial circuit-size complexity, so (3) asserts that E contains languages that do not have polynomial circuit-size complexity. Knowing (1) and wanting to prove (2), the natural strategy has been to prove results of the form for successively larger classes P r (DENSE c ) in the range The first major step beyond (1) in this program was the proof by Watanabe[17] that i.e., that every language that is - P O(log n)\Gammatt -hard for E is dense. The next big step was the proof by Lutz and Mayordomo[10] that, for every real number ff ! 1, This improved Watanabe's result from O(log n) truth-table (i.e., nonadaptive) queries to n ff such queries for ff arbitrarily close to 1 (e.g., to n 0:99 truth-table queries). Moreover, Lutz and Mayordomo[10] proved (5) by first proving the stronger result that for all ff ! 1, which implies that every language that is weakly - P poly ) is dense. language A is weakly - P r -hard for a complexity class C if -(P r (A) j C) 6= 0, i.e., nonnegligible subset of C in the sense of the resource-bounded measure developed by Lutz[9]. A language A is weakly - P r -complete for C if A 2 C and A is weakly r -hard for C. See [12] or [2] for a survey of resource-bounded measure and weak com- pleteness.) The set of weakly - P languages for E is now known to have p-measure in the class C of all languages, while the set of all - P languages for E has measure 0 unless (which is generally conjectured to be true), almost every language is weakly - P n ff \Gammatt -hard, but not - P for E, so the result of Lutz and Mayordomo [10] is much more general than the fact that every - P n ff \Gammatt -hard language for E is dense. A word on the relationship between hardness notions for E and E 2 is in order here. It is well known that a language is - P m -hard for E if and only if it is - P m -hard for this is (E). The same equivalence holds for - P T -hardness. It is also clear that every language that is - P n ff \Gammatt -hard for E. However, it is not generally the case that Pm(P n ff \Gammatt may well be the case that a language can be - P n ff \Gammatt -hard for E, but not for E 2 . These same remarks apply to - P -hardness. The relationship between weak hardness notions for E and E 2 is somewhat different. Juedes and Lutz [8] have shown that weak - P m -hardness for E implies -hardness for proof of this fact also works for T -hardness. However, Juedes and Lutz [8] also showed that weak - P m -hardness for does not generally imply -hardness for E, and it is reasonable to conjecture (but has not been proven) that the same holds for -hardness. We further conjecture that the notions of weak - P and are incomparable, and similarly for -hardness. In any case, (6) implies that, for every ff ! 1, every language that is weakly - P for either E or E 2 is dense. Shortly after, but independently of [10], Fu[7] used very different techniques to prove that, for every ff ! 1, and That is, every language that is - P is dense. These results do not have the measure-theoretic strength of (6), but they are a major improvement over previous results on the densities of hard languages in that they hold for Turing reductions, which have adaptive queries. In the present paper, we prove results which simultaneously strengthen results of Lutz and Mayordomo[10] and the results of Fu[7]. Specifically, we prove that, for every ff ! 1, and These results imply that every language that is weakly - P weakly dense. The proof of (9) and (10) is not a simple extension of the proof in [10] or the proof in [7], but rather combines ideas from both [10] and [7] with the martingale dilation technique introduced by Ambos-Spies, Terwijn, and Zheng [3]. Our results also show that the strong hypotheses - p (NP) 6= 0 and - p 2 (NP) 6= 0 (surveyed in [12] and [2]) have consequences for the densities of adaptively hard languages for NP. Mahaney [13] proved that and Ogiwara and Watanabe [16] improved this to That is, if P 6= NP, then no sparse language can be - P btt -hard for NP. Lutz and Mayordomo [10] used (6) to obtain a stronger conclusion from a stronger hypothesis, namely, for all By and (10), we now have, for all ff ! 1, and Thus, if - p (NP) 6= 0, then every language that is - P n 0:49 \GammaT -hard for NP is dense. If - p2 (NP) 6= 0, then every language that is - P n 0:99 \GammaT -hard for NP is dense. Preliminaries The Boolean value of a condition, / is ae 0 if not /: The standard enumeration of f0; 1g is s enumeration induces a total ordering of f0; 1g which we denote by !. All languages here are subsets of f0; 1g . The Cantor space is the set C of all languages. We identify each language A 2 C with its characteristic sequence, which is the infinite binary sequence is the standard enumeration of f0; 1g . For A to indicate that w is a prefix of (the characteristic sequence of) A. The symmetric difference of the two languages A and B is A 4 The cylinder generated by a string w 2 f0; 1g is the set Note that C In this paper, a set X ' C that appears in a probability Pr(X) or a conditional probability Pr(XjCw ) is regarded as an event in the sample space C with the uniform probability measure. Thus, for example, Pr(X) is the probability that A 2 X when the language A ' f0; 1g is chosen probabilistically by using an independent toss of a fair coin to decide membership of each string in A. In particular, Pr(Cw . The complement of a set is the set X exactly t(n)-time- computable if there is an algorithm that, on input runs for at most O(t(k steps and outputs an ordered pair (a; b) 2 Z \Theta Z such that b . A function f : N d \Theta f0; 1g ! R is t(n)-time-computable if there is an exactly t(n)-time-computable function b We briefly review those aspects of martingales and resource-bounded measure that are needed for our main theorem. The reader is referred to [2], [9], [12], or [14] for more thorough discussion. A martingale is a function d : f0; 1g ! [0; 1) such that, for all w 2 f0; 1g , a t(n)-martingale is a martingale that is t(n)-time-computable, and an exact t(n)-martingale is a (rational-valued) martingale that is exactly t(n)-time-computable. A martingale d succeeds on a language A 2 C if, for every c 2 N, there exists w v A such that d(w) ? c. The success set of a martingale d is the set Cjd succeeds on Ag: The unitary success set of d is w2f0;1g The following result was proven by Juedes and Lutz [8] and independently by Mayor- domo [14]. Lemma 2.1 (Exact Computation Lemma) Let t : N ! N be nondecreasing with t(n) - n 2 . Then, for every t(n)-martingale d, there is an exact n d such that d]. A sequenceX a of series of terms a j;k 2 [0; 1) is uniformly p-convergent if there is a polynomial such that, for all j; r 2 N,X a r). The following sufficient condition for uniform p-convergence is easily verified by routine calculus. Lemma 2.2 Let a j;k 2 [0; 1) for all N. If there exist a real number ffl ? 0 and a polynomial N such that a j;k - e \Gammak ffl for all j; k 2 N with k - g(j), then the seriesX a are uniformly p-convergent. A uniform, resource-bounded generalization of the classical first Borel-Cantelli lemma was proved by Lutz [9]. Here we use the following precise variant of this result. Theorem 2.3 Let ff; e ff, and let be an exactly 2 (log n) ff -time-computable function with the following two properties. (i) For each j; k 2 N, the function d j;k defined by d j;k is a martingale. (ii) The seriesX are uniformly p-convergent. Then there is an exact 2 (log n) e ff -martingale e ff such k=t d]: Proof (sketch). Assume the hypothesis, and fix ff ff. Since ff it suffices by Lemma 2.1 to show that there is a 2 (log n) ff 0 martingale d 0 such k=t Fix a polynomial testifying that the seriesX are uniformly p-convergent, and define for all w 2 f0; 1g . Then, for each w 2 f0; 1g , so 1). It is clear by linearity that d 0 is a martingale. To see that (16) holds, assume that A 2[ k=t arbitrary. Then there exist 2c) such that A 2 S 1 [d j;k ]. Fix w v A such that d j;k (w) - 1. Then arbitrary here, it follows that A 2 S 1 [d 0 ], confirming (16). To see that d 0 is 2 (log n) ff 0 -time-computable, define dA follows, using the abbreviation 2. dA s dB s 2s s 2s For all r 2 N and w 2 f0; 1g , it is clear that and it is routine to verify the inequalities dA dB whence we have for all r 2 N and w 2 f0; 1g . Using formula (17), the time required to compute dC (r; w) exactly is no greater than and q is a polynomial. Since q(n) \Delta 2 (log n) ff it follows that exactly 2 (log n) ff 0 -time-computable. By (18), then, d 0 is a 2 (log n) ff 0 -martingale. The proof of our main theorem uses the techniques of weak stochasticity and martingale dilation, which we briefly review here. As usual, an advice function is a function h Given a function q we write ADV(q) for the set of all advice functions h such that jh(n)j - q(n) for all n 2 N. Given a language B and an advice function h, we define the language is a standard string-pairing function, e.g., ! x; y ?= 0 jxj 1xy. Given functions t; we define the advice class Definition (Lutz and Mayordomo[10], Lutz[11]) For t; language A is weakly (t; q; -stochastic if, for all B; C 2 DTIME(t)=ADV(q) such that jC =n j -(n) for all sufficiently large n, lim We write WS(t; q; -) for the set of all weakly (t; q; -stochastic languages. The following result resembles the weak stochasticity theorems proved by Lutz and Mayordomo [10] and Lutz [11], but gives a more careful upper bound on the time complexity of the martingale. Theorem 2.4 (Weak Stochasticity Theorem) Assume that ff; fi; fl; - 2 R satisfy ff - there is an exact 2 (log n) - -martingale d such that Proof. Assume the hypothesis, and assume without loss of generality that ff; fi; fl; - 2 Q . Fix ) be a language that is universal for DTIME(2 n ff ) in the following sense. For each Ng. Define a function d is not a power of 2, then where the sets Y i;j;k;y;z are defined as follows. If is the set of all A 2 C such that The definition of conditional probability immediately implies that, for each N, the function d 0 i;j;k is a martingale. Since U 2 DTIME(2 n ff 0 to compute each Pr(Y i;j;k;y;z jCw ) using binomial coefficients is at most O(2 (log(i+j+k)) - 00 steps, so the time required to compute d 0 i;j;k (w) is at most O((2 n fi steps. Thus d 0 is exactly 2 (log n) - 0 -time-computable. As in [10] and [11], the Chernoff bound tells us that, for all whence e, let 4 , and fix k 0 2 N such that for all j 2 N. Then g is a polynomial and, for all It follows by Lemma 2.2 that the seriesX i;j;k (-), for are uniformly p-convergent. Theorem 2.3 that there is an exact 2 (log n) - -martingale d such k=t Now assume that A 62 WS(2 n ff by the definition of weak stochasticity, we can fix and an infinite set J ' N such that, for all . For each n 2 J , then, there is a prefix w v A such that Cw ' Y i;j;k;h 1 (n);h2 (n), whence i;j;k ]. This argument shows that[ k=t It follows by (19) that The technique of martingale dilation was introduced by Ambos-Spies, Terwijn, and Zheng [3]. It has also been used by Juedes and Lutz[8] and generalized considerably by Breutzmann and Lutz [6]. We use the notation of [8] here. The restriction of a string to a language A ' f0; 1g is the string w-A obtained by concatenating the successive bits b i for which s i 2 A. If strictly increasing and d is a martingale, then the f -dilation of d is the function f-d : f0; 1g ! [0; 1) defined by for all w 2 f0; 1g . Lemma 2.5 (Martingale Dilation Lemma - Ambos-Spies, Terwijn, and Zheng[3]) If f : strictly increasing and d is a martingale, then f-d is also a martingale. Moreover, for every language A 2 f0; 1g , if d succeeds on f \Gamma1 (A), then f-d succeeds on A. Finally, we summarize the most basic ideas of resource-bounded measure in E and E 2 . A p-martingale is a martingale that is, for some k 2 N, an n k -martingale. A p 2 -martingale is a martingale that is, for some k 2 N, a 2 (log n) k -martingale. Definition (Lutz [9]) 1. A set X of languages has p-measure 0, and we write - there is a p- martingale d such that X ' S 1 [d]. 2. A set X of languages has p 2 -measure 0, and we write - there is a -martingale d such that X ' S 1 [d]. 3. A set X of languages has measure 0 in E, and we write 4. A set X of languages has measure 0 in E 2 , and we write -(XjE 2 5. A set X of languages has measure 1 in E, and we write In this case, we say that contains almost every element of E. 6. A set X of languages has measure 1 in E 2 , and we write -(XjE 2 In this case, we say that contains almost every element of E 2 . 7. The expression -(XjE) 6= 0 means that X does not have measure 0 in E. Note that this does not assert that "-(XjE)" has some nonzero value. Similarly, the expression means that X does not have measure 0 in E 2 . It is shown in [9] that these definitions endow E and E 2 with internal measure structure. This structure justifies the intuition that, if negligibly small subset of E (and similarly for The key to our main theorem is the following lemma, which says that languages that are -reducible to non-dense languages cannot be very stochastic. Lemma 3.1 (Main Lemma) For all real numbers ff ! 1 and fi Proof. Let assume without loss of generality that ff and fi are rational. Let A 2 P n ff \GammaT (DENSE c ). It suffices to show that A is not weakly stochastic. there exist a non-dense language S, a polynomial q(n), and a q(n)-time-bounded oracle Turing machine M such that A = L(M S ) and, for every makes exactly bjxj ff cqueries (all distinct) on input x with oracle B. Call these queries c) in the order in which M makes them. For each B 2 f0; 1g and n 2 N, define an equivalence relation -B;n on f0; 1g -q(n) by and an equivalence relation jB;n on f0; 1g n by Note that -B;n has at most 2jB -q(n) j+1 equivalence classes, so jB;n has at most (2jB -q(n) j+ equivalence classes. 2 , and let J be the set of all n 2 N for which the following three conditions hold. conditions (ii) and (iii) hold for all sufficiently large n. S is not dense, condition (i) holds for infinitely many n. Thus the set J is infinite. Define an advice function h be a maximum-cardinality equivalence class of the relation j S;n . For each Let Note that For each n 2 N, let be the set of all coded pairs such that x; y denotes the ith query of M on input w when the successive oracle answers are be the set of all such coded pairs in C n such that M accepts on input x when the successive oracle answers are b Finally, define the languages It is clear that B; C 2 DTIME(2 n ). Also, by our construction of these sets and the advice function h, for each n 2 N, we have ae and For each n 2 J , if -(n) is the number of equivalence classes of j S;n , then so It follows that j(C=h) =n 2 for all n 2 N. Finally, for all n 2 J , Since J is infinite, it follows that for all n 2 N, this shows that A is not weakly )-stochastic. We now prove our main result. Theorem 3.2 (Main Theorem) For every real number ff ! 1, Proof. Let 2. By Theorem 2.4, there is an exact 2 (log n) 2 -martingale d such that By Lemma 3.1, we then have Since d is a p 2 -martingale, this implies that - p2 (P n ff \GammaT (DENSE c Then f is strictly increasing, so f-d, the f-dilation of d, is a martingale. The time required to compute f-d(w) is steps, where w steps to compute w 0 and then O(2 (log jw 0 steps to compute d(w 0 ).) Now jw 0 j is bounded above by the number of strings x such that jxj 2 - js jwj jwj)c, so Thus the time required to compute f-d(w) is steps, so f-d is an n 2 -martingale. Now let A 2 P n ff=2 \GammaT (DENSE c ). Then f This shows that P n ff=2 \GammaT (DENSE c ) ' S 1 [f-d]. Since f-d is an We now develop a few consequences of the Main Theorem. The first is immediate. Corollary 3.3 For every real number ff ! 1, The following result on the density of weakly complete (or weakly hard) languages now follows immediately from Corollary 3.3. Corollary 3.4 For every real number ff ! 1, every language that is weakly - P for E or weakly - P Our final two corollaries concern consequences of the strong hypotheses - p (NP) 6= 0 The relative strengths of these hypotheses are indicated by the known implications E) (The leftmost implication was proven by Juedes and Lutz[8]. The remaining implications follow immediately from elementary properties of resource-bounded measure.) Corollary 3.5 Let ff ! 1. If - p (NP) 6= 0, then every language that is - P NP is dense. If - p2 (NP) 6= 0, then every language that is - P n ff \GammaT -hard for NP is dense. We conclude by considering the densities of languages to which SAT can be adaptively reduced. Definition A function g : N ! N is subradical if log It is easy to see that a function g is subradical if and only if, for all k ? 0, n). (This is the reason for the name "subradical.") Subradical functions include very slow-growing functions such as log n and (log n) 5 , as well as more rapidly growing functions such as 2 (log n) 0:99 Corollary 3.6 If - p (NP) 6= 0, g : N ! N is subradical, and SAT - P dense. Proof. Assume the hypothesis. Let A 2 NP. Then there is a - P -reduction f of A to SAT. Fix a polynomial q(n) such that, for all x 2 f0; 1g , jf(x)j - q(jxj). Composing f with the - P g(n)\GammaT -reduction of SAT to H that we have assumed to exist then gives a - P reduction of A to H. Since g is subradical, log so for all sufficiently large n, 4 . Thus A - P The above argument shows that H is - P -hard for NP. Since we have assumed Corollary 3.5 that H is dense. To put the matter differently, Corollary 3.6 tells us that if SAT is polynomial-time reducible to a non-dense language with at most 2 (log n) 0:99 adaptive queries, then NP has measure 0 in E and in E 2 . Questions As noted in the introduction, the relationships between weak hardness notions for E and under reducibilities such as - P remain to be resolved. Our main theorem also leaves open the question whether - P languages for E must be dense when2 - ff ! 1. We are in the curious situation of knowing that the classes P n 0:99 \Gammatt (DENSE c ) and P n 0:49 \GammaT (DENSE c ) have p-measure 0, but not knowing whether P n 0:50 \GammaT (DENSE c ) has p-measure 0. Indeed, at this time we cannot even prove that E 6' P n 0:50 \GammaT (SPARSE). Further progress on this matter would be illuminating. --R Theoretical Computer Science. Relative to a random oracle A On isomorphism and density of NP and other complete sets. Equivalence of measures of complexity classes EXP is not polynomial time Turing reducible to sparse sets. Weak completeness in E and Almost everywhere high nonuniform complexity the density of hard languages. Theoretical Computer Science The quantitative structure of exponential time Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis. Contributions to the Study of Resource-Bounded Measure Reported in On polynomial bounded truth-table reducibility of NP sets to sparse sets On the Structure of Intractable Complexity Classes. --TR --CTR John M. Hitchcock, The size of SPP, Theoretical Computer Science, v.320 n.2-3, p.495-503, June 14, 2004

weakly complete problems;resource-bounded measure;complexity classes;computational complexity;polynomial reductions

586993

A Generalization of Resource-Bounded Measure, with Application to the BPP vs. EXP Problem.

We introduce resource-bounded betting games and propose a generalization of Lutz's resource-bounded measure in which the choice of the next string to bet on is fully adaptive. Lutz's martingales are equivalent to betting games constrained to bet on strings in lexicographic order. We show that if strong pseudorandom number generators exist, then betting games are equivalent to martingales for measure on E and EXP. However, we construct betting games that succeed on certain classes whose Lutz measures are important open problems: the class of polynomial-time Turing-complete languages in EXP and its superclass of polynomial-time Turing-autoreducible languages. If an EXP-martingale succeeds on either of these classes, or if betting games have the "finite union property" possessed by Lutz's measure, one obtains the nonrelativizable consequence $\mbox{BPP} \neq \mbox{EXP}$. We also show that if $\mbox{EXP} \neq \mbox{MA}$, then the polynomial-time truth-table-autoreducible languages have Lutz measure zero, whereas if

Introduction Lutz's theory of measure on complexity classes is now usually defined in terms of resource-bounded martingales. A martingale can be regarded as a gambling game played on unseen languages A. Let be the standard lexicographic ordering of strings. The gambler G starts with capital places a bet A." Given a fixed particular language A, the bet's outcome depends only on whether s 1 2 A. If the bet wins, then the new capital while if the bet loses, C . The gambler then places a bet on (or against) membership of the string s 2 , then on s 3 , and so forth. The gambler succeeds if G's capital C i grows toward +1. The class C of languages A on which G succeeds (and any subclass) is said to have measure zero. One also says G covers C. Lutz and others (see [Lut97]) have developed a rich and extensive theory around this measure-zero notion, and have shown interesting connections to many other important problems in complexity theory. We propose the generalization obtained by lifting the requirement that G must bet on strings in lexicographic order. That is, G may begin by choosing any string x 1 on which to place its first bet, and after the oracle tells the result, may choose any other string x 2 for its second bet, and so forth. Note that the sequences x may be radically different for different oracle languages A-in complexity-theory parlance, G's queries are adaptive. The lone restriction is that G may not query (or bet on) the same string twice. We call G a betting game. Our betting games remedy a possible lack in the martingale theory, one best explained in the context of languages that are "random" for classes D such as E or EXP. A language L is D-random if L cannot be covered by a D-martingale. Based on one's intuition about random 0-1 sequences, the language L should likewise be D-random, where flip(x) changes every 0 in x to a 1 and vice-versa. However, this closure property is not known for E-random or EXP-random languages, because of the way martingales are tied to the fixed lex ordering of \Sigma . Betting games can adapt to easy permutations of \Sigma such as that induced by flip. Similarly, a class C that is small in the sense of being covered by a (D-) betting game remains small if the languages are so permuted. In the r.e./recursive theory of random languages, our generalization is similar to "Kolmogorov-Loveland place-selection rules" (see [Lov69]). We make this theory work for complexity classes via a novel definition of "running in time t(n)" for an infinite process. Our new angle on measure theory may be useful for attacking the problem of separating BPP from EXP, which has recently gained prominence in [IW98]. In Lutz's theory it is open whether the class of EXP-complete sets-under polynomial-time Turing reductions-has EXP-measure zero. If so (in fact if this set does not have measure one), then by results of Allender and Strauss [AS94], BPP 6= EXP. Since there are oracles A such that BPP A = EXP A [Hel86], this kind of absolute separation would be a major breakthrough. We show that the EXP-complete sets can be covered by an EXP betting game-in fact, by an E-betting game. The one technical lack in our theory as a notion of measure is also interesting here: If the "finite unions" property holds for betting games (viz. martingales do enjoy the permutation-invariance of betting games, then BPP 6= EXP. Finally, we show that if a pseudorandom number generator (PRG) of security 2 n\Omega\Gamma/1 exists, then for every EXP-betting game G one can find an EXP-martingale that succeeds on all sets covered by G. PRGs of higher security\Omega\Gamma n) likewise imply the equivalence of E-betting games and E-measure. Ambos-Spies and Lempp [ASL96] proved that the EXP-complete sets have E-measure zero under a different hypothesis, namely Measure theory and betting games help us to dig further into questions about PRGs and complexity-class separations. Our tool is the notion of an autoreducible set, whose importance in complexity theory was argued by Buhrman, Fortnow, van Melkebeek, and Torenvliet [BFvMT98] (after [BFT95]). A language L is - p -autoreducible if there is a polynomial-time oracle TM Q such that for all inputs x, Q L correctly decides whether x 2 L without ever submitting x itself as a query to L. If Q is non-adaptive (i.e., computes a polynomial-time truth-table reduction), we say L is - p tt -autoreducible. We show that the class of - p T -autoreducible sets is covered by an E-betting game. Since every EXP-complete set is - p -autoreducible [BFvMT98], this implies results given above. The subclass of - p tt -autoreducible sets provides the following tighter connection between measure statements and open problems about EXP: ffl If the - p tt -autoreducible sets do not have E-measure zero, then ffl If the - p -autoreducible sets do not have E-measure one in EXP, then EXP 6= BPP. Here MA is the "Merlin-Arthur" class of Babai [Bab85, BM88], which contains BPP and NP. Since EXP 6= MA is strongly believed, one would expect the class of - p -autoreducible sets to have E-measure zero, but proving this-or proving any of the dozen other measure statements in Corollaries 6.2 and 6.5-would yield a proof of EXP 6= BPP. In sum, the whole theory of resource-bounded measure has progressed far enough to wind the issues of (pseudo-)randomness and stochasticity within exponential time very tightly. We turn the wheels a few more notches, and seek greater understanding of complexity classes in the places where the boundary between "measure one" and "measure zero" seems tightest. Section 2 reviews the formal definitions of Lutz's measure and martingales. Section 3 introduces betting games, and shows that they are a generalization of martingales. Section 4 shows how to simulate a betting game by a martingale of perhaps-unavoidably higher time complexity. Section 5, however, demonstrates that strong PRGs (if there are any) allow one to compute the martingale in the same order of time. Section 6 presents our main results pertaining to autoreducible sets, including our main motivating example of a concrete betting game. The concluding Section 7 summarizes open problems and gives prospects for future research. A preliminary version of this paper without proofs appeared in the proceedings of STACS'98, under the title "A Generalization of Resource-Bounded Measure, With an Application." Martingales A martingale is abstractly defined as a function d from f 0; 1 g into the nonnegative reals that satisfies the following "average law": for all w The interpretation in Lutz's theory is that a string w 2 f 0; 1 g stands for an initial segment of a language over an arbitrary alphabet \Sigma as follows: Let s be the standard lexicographic ordering of \Sigma . Then for any language A ' \Sigma , write w v A if for all iff the ith bit of w is a 1. We also regard w as a function with domain and range writing w(s i ) for the ith bit of w. A martingale d succeeds on a language A if the sequence of values d(w) for w v A is unbounded. If J is a set of strings such that for any w 2 f 0; 1 g and any b 2 f 0; 1 g, d(wb) 6= d(w) implies s jwbj 2 J , we say that the martingale d is active only on J . stand for the (possibly empty, often uncountable) class of languages on which d succeeds. Definition 2.1 (cf. [Lut92, May94]). Let \Delta be a complexity class of functions. A class C of languages has \Delta-measure zero, written - \Delta there is a martingale d computable in \Delta such that C ' S 1 [d]. One also says that d covers C. Lutz defined complexity bounds in terms of the length of the argument w to d, which we denote by N . However, we also work in terms of the largest length n of a string in the domain of w. For N ? 0, n equals blog Nc; all we care about is that Because complexity bounds on languages we want to analyze will naturally be stated in terms of n, we prefer to use n for martingale complexity bounds. The following correspondence is helpful: - measure on EXP Our convention lets us simply write "- E " for E-measure (regarding \Delta as E for functions), similarly "- EXP " for EXP-measure, and generally - \Delta for any \Delta that names both a language and function class. Abusing notation similarly, we define: Definition 2.2 ([Lut92]). A class C has \Delta-measure one, written - \Delta The concept of resource bounded measure is known to be robust under several changes [May94]. The following lemma has appeared in various forms [May94, BL96]. It essentially says that we can assume a martingale grows almost monotonically (sure winnings) and not too fast (slow winnings). Lemma 2.1 ("Slow-but-Sure-Winnings" lemma for martingales) Let d be a martingale. Then there is a martingale d 0 with S 1 [d] ' S 1 [d 0 ] such that If d is computable in time t(n) , then d 0 is computable in time O(2 n t(n)). The idea is to play the strategy of d, but in a more conservative way. Say we start with an initial capital of $1. We will deposit a part c of our capital on a bank and only play the strategy underlying d on the remaining liquid part e of our capital. We start with no savings and a liquid capital of $1. When our liquid capital e would reach $2 or exceed that level, we deposit an additional $1 or $2 to our savings account c so as to keep the liquid capital in the range $[1; 2) at all times. If d succeeds, it will push the liquid capital infinitely often to $2 or above, so c grows to infinity, and d 0 succeeds too. Since we never take money out of our savings account c, and the liquid capital e is bounded by $2, once our total capital d reached a certain level, it will never go more than $2 below that level anymore, no matter how bad the strategy underlying d is. On the other hand, since we add at most $2 to c in each step, d 0 (w) cannot exceed 2(jwj We now give the formal proof. Proof. (of Lemma 2.1) Define d Checking the time and space complexity bounds for d 0 is again straightforward. We can show by induction on jwj that and that from which it follows that d 0 is a martingale. If d succeeds on !, e(w) will always remain positive for w v !, and d(wb) or more infinitely often. Consequently, lim wv!;jwj!1 that S 1 [d] ' S 1 [d 0 ]. Moreover, by (4) and the fact that c does not decrease along any sequence, we have that Since c can increase by at most 2 in every step, c(w) - 2jwj. Together with (4), this yields that One can also show that S 1 [d 0 ] ' S 1 [d] in Lemma 2.1, so the success set actually remains intact under the above transformation. As with Lebesgue measure, the property of having resource-bounded measure zero is monotone and closed under union ("finite unions property"). A resource-bounded version of closure under countable unions also holds. The property that becomes crucial in resource-bounded measure is that the whole space \Delta does not have measure zero, which Lutz calls the "measure conservation" property. With a slight abuse of meaning for "6=," this property is written - \Delta (\Delta) 6= 0. In particular, of \Delta that require substantially fewer resources, do have \Delta-measure zero. For example, P has E-measure zero. Indeed, for any fixed c ? 0, DTIME[2 cn ] has E-measure zero, and DTIME[2 n c has EXP-measure zero [Lut92]. Apart from formalizing rareness and abundance in complexity theory, resource-bounded martingales are also used to define the concept of a random set in a resource-bounded setting. Definition 2.3. A set A is \Delta-random if - \Delta (fAg) 6= 0. In other words, A is \Delta-random if no \Delta-martingale succeeds on A. Betting Games To capture intuitions that have been expressed not only for Lutz measure but also in many earlier papers on random sequences, we formalize a betting game as an infinite process, rather than as a Turing machine that has finite computations on string inputs. Definition 3.1. A betting game G is an oracle Turing machine that maintains a "capital tape" and a "bet tape," in addition to its standard query tape and worktapes, and works in stages Beginning each stage i, the capital tape holds a nonnegative rational number C i\Gamma1 . Initially, C computes a query string x i to bet on, a bet amount B i , g. The computation is legal so long as x i does not belong to the set f x of strings queried in earlier stages. G ends stage i by entering a special query state. For a given oracle language A, if x i 2 A and b i =+1, or if x 2 A and b then the new capital is given by C i := C . The query and bet tapes are blanked, and G proceeds to stage i + 1. Since we require that G spend the time to write each bet out in full, it does not matter whether we suppose that the new capital is computed by G itself or updated instantly by the oracle. In this paper, we lose no generality by not allowing G to "crash" or to loop without writing a next bet and query. Note that every oracle set A determines a unique infinite computation of G, which we denote by G A . This includes a unique infinite sequence x 1 query strings, and a unique sequence telling how the gambler fares against A . Definition 3.2. A betting machine G runs in time t(n) if for all oracles A, every query of length made by G A is made in the first t(n) steps of the computation. A similar definition can be made for space usage, taking into account standard issues such as whether the query tape counts against the space bound, or whether the query itself is preserved in read-only mode for further computation by the machine. Definition 3.3. A betting game G succeeds on a language A, written A 2 S 1 [G], if the sequence of values C i in the computation G A is unbounded. If A 2 S 1 [G], then we also say G covers A. Our main motivating example where one may wish not to bet in lexicographic order, or according to any fixed ordering of strings, is deferred to Section 6. There we will construct an E-betting game that succeeds on the class of - p -autoreducible languages, which is not known to have Lutz measure zero in E or EXP. We now want to argue that the more liberal requirement of being covered by a time t(n) betting game, still defines a smallness concept for subclasses of DTIME[t(n)] in the intuitive sense Lutz established for his measure-zero notion. The following result is a good beginning. Theorem 3.1 For every time-t(n) betting game G, we can construct a language in DTIME[t(n)] that is not covered by G. Proof. Let Q be a non-oracle Turing machine that runs as follows, on any input x. The machine simulates up to t(jxj) steps of the single computation of G on empty input. Whenever G bets on and queries a string y, Q gives the answer that causes G to lose money, rejecting in case of a zero bet. If and when G queries x, Q does likewise. If t(jxj) steps go by without x being queried, then Q rejects x. The important point is that Q's answer to a query y 6= x is the same as the answer when Q is run on input y. The condition that G cannot query a string x of length n after t(n) steps have elapsed ensures that the decision made by Q when x is not queried does not affect anything else. Hence Q defines a language on which G never does better than its initial capital C 0 , and so does not succeed. In particular, the class E cannot be covered by an E-betting game, nor EXP by an EXP-betting game. Put another way, the "measure conservation axiom" [Lut92] of Lutz's measure carries over to betting games. To really satisfy the intuition of "small," however, it should hold that the union of two small classes is small. (Moreover, "easy" countable unions of small classes should be small, as in [Lut92].) Our lack of meeting this "finite union axiom" will later be excused insofar as it has the non- relativizing consequence BPP 6= EXP. Theorem 3.1 is still good enough for the "measure-like" results in this paper. We note also that several robustness properties of Lutz's measure treated in Section 2 carry over to betting games. This is because we can apply the underlying transformations to the capital function c G of G, which is defined as follows: Definition 3.4. Let G be a betting games, and i - 0 an integer. (a) A play ff of length i is a sequence of i-many oracle answers. Note that ff determines the first i-many stages of G, together with the query and bet for the next stage. (b) c G (ff) is the capital C i that G has at the end of the play ff (before the next query). Note that the function c G is a martingale over plays ff. The proof of Lemma 2.1 works for c G . We obtain: Lemma 3.2 ("Slow-But-Sure Winnings" lemma for betting games) Let G be a betting game that runs in time t(n). Then we can construct a betting game G 0 running in time O(t(n)) such that S 1 [G] makes the same queries in the same order as G, and: 2: Proof. The proof of Lemma 2.1 carries over. The only additional observation is that c G 0 can be constructed on the fly, and this allows G 0 to run in time O(t(n)). To begin comparing betting games and martingales, we note first that the latter can be considered a direct special case of betting games. Say a betting game G is lex-limited if for all oracles A, the sequence x 1 queries made by G A is in lex order. (It need not equal the lex enumeration Theorem 3.3 Let T (n) be a collection of time bounds that is closed under multiplication by 2 n , such as 2 O(n) or 2 n O(1) . Then a class C has time-T (n) measure zero iff C is covered by a time-T (n) lex-limited betting game. Proof. From a martingale d to a betting game G, each stage i of G A bets on s i an amount B i with is the first bits of the characteristic sequence of A. This takes O(2 n ) evaluations of d to run G up through queries of length n, hence the hypothesis on the time bounds T (n). In the other direction, when G is lex-limited, one can simulate G on a finite initial segment w of its oracle up to a stage where all queries have been answered by w and G will make no further queries in the domain of w. One can then define d(w) to be the capital entering this stage. That this is a martingale and fulfills the success and run-time requirements is left to the reader. Hence in particular for measure on E and EXP, martingales are equivalent to betting games constrained to bet in lex order. Now we will see how we can transform a general betting game into an equivalent martingale. 4 From Betting Games to Martingales This section associates to every betting game G a martingale dG such that S 1 [G] ' S 1 [d G ], and begins examining the complexity of dG . Before defining dG , however, we pause to discuss some subtleties of betting games and their computations. Given a finite initial segment w of an oracle language A, one can define the partial computation G w of the betting game up to the stage i at which it first makes a query x i that is not in the domain of w. Define d(w) to be the capital C i\Gamma1 that G had entering this stage. It is tempting to think that d is a martingale and succeeds on all A for which G succeeds-but neither statement is true in general. The most important reason is that d may fail to be a martingale. To see this, suppose x i itself is the lexicographically least string not in the domain of w. That is, x i is indexed by the bit b of wb, and w1 v A iff x i 2 A. It is possible that G A makes a small (or even zero) bet on x i , and then goes back to make more bets in the domain of w, winning lots of money on them. The definitions of both d(w0) and d(w1) will then reflect these added winnings, and both values will be greater than d(w). For example, suppose G A first puts a zero bet on x then bets all of its money on x not being in A, and then proceeds with x Put another way, a finite initial segment w may carry much more "winnings potential" than the above definition of d(w) reflects. To capture this potential, one needs to consider potential plays of the betting game outside the domain of w. Happily, one can bound the length of the considered plays via the running time function t of G. Let n be the maximum length of a string indexed by (jwj)c. Then after t(n) steps, G cannot query any more strings in the domain of so w's potential is exhausted. We will define dG (w) as an average value of those plays that can happen, given the query answers fixed by w. We use the following definitions and notation: Definition 4.1. For any t(n) time-bounded betting game G and string w 2 \Sigma , define: (a) A play ff is t-maximal if G completes the first jffj stages, but not the query and bet of the next stage, within t steps. (b) A play ff is G-consistent with w, written ff -G w, if for all stages j such that the queried string x j is in the domain of w, ff That is, ff is a play that could possibly happen given the information in w. Also let m(ff; w) stand for the number of such stages j whose query is answered by w. (c) Finally, put dG ff t(n)\Gammamaximal;ff- Gw The weight 2 m(ff;w)\Gammajffj in Equation (7) has the following meaning. Suppose we extend the simulation of G w by flipping a coin for every query outside the domain of w, for exactly i stages. Then the number of coin-flips in the resulting play ff of length i is its probability. Thus dG (w) returns the suitably-weighted average of t(n)-step computations of G with w fixed. The interested reader may verify that this is the same as averaging d(wv) over all v of length 2 t(n) (or any fixed longer length), where d is the non-martingale defined at the beginning of this section. Lemma 4.1 The function dG (w) is a martingale. Proof. First we argue that Observe that when ff This is because none of the queries answered by fi can be in the domain of w, else the definition of G running in time t(n) would be violated. Likewise if ff -G w then m(ff Finally, since c G is a martingale, These facts combine to show the equality of (7) and (8). By the same argument, the right-hand side of (8) is unchanged on replacing "t(n)" by any Now consider w such that jwj + 1 is not a power of 2. Then the "n" for w0 and w1 is the same as the "n" for dG (w). Let P 0 stand for the set of ff of length t(n) that are G-consistent with w0 but not with w1, P 1 for those that are G-consistent with w1 but not w0, and P for those that are consistent with both. Then the set f ff : equals the disjoint union of P , and P 1 . Furthermore, for ff 2 P 0 we have m(ff; dG (w0)+d G c G (ff)2 m(ff;w1)\Gammat(n) c G (ff)2 m(ff;w)\Gammat(n) c G (ff)2 m(ff;w)\Gammat(n) Finally, if jwj a power of 2, then dG (w0) and dG (w1) use t 0 their length of ff. However, by the first part of this proof, we can replace t(n) by t 0 in the definition of dG (w) without changing its value, and then the second part goes through the same way for t 0 . Hence dG is a martingale. It is still the case, however, that dG may not succeed on the languages on which the betting game G succeeds. To ensure this, we first use Lemma 3.2 to place betting games G into a suitable "normal form" satisfying the sure-winnings condition (5). Lemma 4.2 If G is a betting game satisfying the sure-winnings condition (5), then S 1 [G] ' Proof. First, let A 2 S 1 [G], and fix k ? 0. Find a finite initial segment w v A long enough to answer every query made in a play ff of G such that c G (ff) long enough to make t(n) in the definition of dG (w) (Equation 7) greater than jffj. Then every ff 0 of length t(n) such that has the form ff fffi. The sure-winnings condition (5) implies that the right-hand side of defining dG (w) is an average over terms that all have size at least k. Hence dG (w) - k. Letting k grow to infinity gives A 2 S 1 [d G ]. Now we turn our attention to the complexity of dG . If G is a time-t(n) betting game, it is clear that dG can be computed deterministically in O(t(n)) space, because we need only cycle through all ff of length t(n), and all the items in (7) are computable in space O(t(n)). In particular, every E-betting game can be simulated by an ESPACE-martingale, and every EXP-betting game by an EXPSPACE-martingale. However, we show in the next section that one can estimate dG (w) well without having to cycle through all the ff, using a pseudo-random generator to "sample" only a very small fraction of them. 5 Sampling Results First we determine the accuracy to which we need to estimate the values d(w) of a hard-to-compute martingale. We state a stronger version of the result than we need in this section. We will use the strenghtening in Sections 6.2 and 6.3. Recall that Lemma 5.1 Let d be a martingale that is active only on J ' f 0; 1 g , and let [ffl(i)] 1 i=0 be a non-negative sequence such that converges to a number K. Suppose we can compute in time t(n) a function g(w) such that jg(w) \Gamma d(w)j - ffl(N ) for all w of length N . Then there is a martingale d 0 computable in time O(2 n t(n)) such that for all In this section, we will apply Lemma 5.1 with In Section 6.3 we will apply Lemma 5.1 in cases where J is finite. Proof. First note that for any w (with In case inductively define: Note that d 0 satisfies the average law (1), and that we can compute d 0 (w) in time O(2 n t(n)). By induction on jwj, we can show using the estimate (9) that It follows that and that This establishes the lemma in case . The generalization to other subsets J of is left to the reader. Next, we will specify precisely which function f G we will sample in order to estimate dG , and how we will do it. Let G be a t(n) time-bounded betting game. Consider a prefix w, and let n denote the largest length of a string in the domain of w. With any string ae of length t(n), we can associate a unique "play of the game" G defined by using w to answer queries in the domain of w, and the successive bits of ae to answer queries outside it. We can stop this play after t(n) steps-so that the stopped play is a t(n)-maximal ff-and then define f G (w; ae) to be the capital c G (ff). Note that we can compute f G (w; ae) in linear time, i.e. in time O(jwj t(n)). The proportion of strings ae of length t(n) that map to the same play ff is exactly the weight 2 m(ff;w)\Gammajffj in the equation (7) for dG (w). Letting E stand for mathematical expectation, this gives us: We will apply two techniques to obtain a good approximation g(w) to this average: ffl sampling using pseudo-random generators, and ffl approximate counting using alternation. 5.1 Sampling via Pseudo-Random Generators First, we need some relevant background on pseudo-random generators. Definition 5.1 ([NW94]). (a) The hardness HA (n) of a set A at length n is the largest integer s such that for any circuit C of size at most s with n inputs, where x is uniformly distributed over \Sigma n . (b) A pseudo-random generator (PRG) is a function D that, for each n, maps \Sigma n into \Sigma r(n) where n. The function r is called the stretching of D. We say that D is computable in C if every bit of D(y) is computable in C, given y and the index of the bit in binary. (c) The security SD (n) of D at length n is the largest integer s such that for any circuit C of size at most s with r(n) inputs s where x is uniformly distributed over \Sigma r(n) and y over \Sigma n . For our purposes, we will need a pseudo-random generator computable in E that stretches seeds super-polynomially and has super-polynomial security at infinitely many lengths. We will use the one provided by the following theorem. Theorem 5.2 If MA 6= EXP, there is a pseudo-random generator D computable in E with stretching '(log n) such that for any integer k, SD (n) - n k for infinitely many n. The proof follows directly from the next results of Babai, Fortnow, Nisan and Wigderson [BFNW93], and Nisan and Wigderson [NW94], combined with some padding. Theorem 5.3 ([BFNW93]) If MA 6= EXP, there is a set A 2 EXP such that for any integer k, for infinitely many n. Theorem 5.4 ([NW94]) Given any set A 2 EXP, there is a pseudo-random generator D computable in EXP with stretching n '(log n) such that SD n)=n). We will also make use of pseudo-random generators with exponential security and computable in exponential time. They have the interesting property that we can blow up the stretching exponentially without significantly reducing the security. Theorem 5.5 ([GGM86]) If there is a pseudo-random generator computable in EXP (respec- tively, E) with security 2 n) ), then there is such a pseudo-random generator with stretching 2 p(n) for any fixed polynomial p. The following general result shows how PRGs can be used to approximate averages. It provides the accuracy and time bounds needed for applying Lemma 5.1 to get the desired martingale. Theorem 5.6 Let D be a pseudo-random generator computable in time ffi(n) and with stretching r(n). be a linear-time computable function, and s; constructible functions such that s(N) - N and the following relations hold for any integer N - 0, Then we can approximate to within N \Gamma2 in time O(2 m(N) \Delta (s(N) Proof. For any integer N - 0, let IN be a partition of the interval [\GammaR(N ); R(N )] into subintervals of length 1 . Note that jI N I 2 IN and any string w of length The predicate underlying -(I; w) can be computed by circuits of size O(s(N )). Since SD (m(N)) 2 !(s(N )), it follows that approximates -(I; w) to within an additive error of (S D (m(N))) \Gamma1 , and we can compute it in time We define the approximation ~ h(w) for h(w) as I2IN ~ Since we can write h(w) as I2IN we can bound the approximation error as follows: I2IN I2IN Computing ~ h(w) requires jI N evaluations of ~ -, which results in the claimed upper bound for the time complexity of ~ h. Now, we would like to apply Theorem 5.6 to approximate by (7) to within N \Gamma2 , by setting However, for a general betting game G running in time t(n), we can only guarantee an upper bound of R(N) t(log N) on jf(w; ae)j. Since SD can be at most exponential, condition (10) would force m(N) to be \Omega\Gamma t(log N )). In that case, Theorem 5.6 can only yield an approximation computable in time 2 O(t(log N)) . However , we can assume wlog. that G satisfies the slow-winnings condition (6) of Lemma 3.2, in which case an upper bound of holds. Then the term s(N) in the right-hand side of (10) dominates, provided n) . Taking everything together, we obtain the following result about transforming E- and EXP- betting games into equivalent E- respectively EXP-martingales: Theorem 5.7 If there is a pseudo-random generator computable in E with security 2 n) , then for every E-betting game G, there exists an E-martingale d such that S 1 [G] ' S 1 [d]. If there is a pseudo-random generator computable in EXP with security 2 n\Omega\Gamma/1 , then for every EXP-betting game G, there exists an EXP-martingale d such that S 1 [G] ' S 1 [d]. Proof. By Lemma 3.2, we can assume that c G satisfies both the sure-winnings condition (5) as well as the slow-winnings condition (6). Because of Lemma 4.2 and Lemma 5.1 (since the series suffices to approximate the function dG (w) given by (7) to within N \Gamma2 in , where Under the given hypothesis for E, we can meet the conditions for applying Theorem 5.6 to we obtain the approximation of dG we need. The same holds in like manner for EXP, for which we have s(N) 2 2 (log N) O(1) 5.2 Approximate Counting using Alternation Instead of hypothesizing the existence of strong pseudo-random generators, we can also use the following theorem of Stockmeyer on approximate counting. Theorem 5.8 ([Sto83]) For any h 2 #P and any polynomial p, there is a function g 2 3 such that for any input w of length N , Theorem 5.9 (a) If then for every E-betting game G, there exists an E-martingale d such that S 1 [G] ' S 1 [d]. (b) If NP ' DTIME[2 (log n) O(1) then for every EXP-betting game G, there exists an EXP- martingale d such that S 1 [G] ' S 1 [d]. The proof plugs (12) into the above sampling results in a similar manner. 6 Autoreducible Sets An oracle Turing machine M is said to autoreduce a language A if L(M A and for all strings M A on input x does not query x. That is, one can learn the membership of x by querying strings other than x itself. If M runs in polynomial time, then A is P-autoreducible-we also write -autoreducible. If M is also non-adaptive, then A is - p -autoreducible. One can always code M so that for all oracles, it never queries its own input-then we call M an autoreduction. Hence we can define an effective enumeration [M i of polynomial-time autoreductions, such that a language A is autoreducible iff there exists an i such that L(M A (For a technical aside: the same M i may autoreduce different languages A, and some M i may autoreduce no languages at all.) The same goes for - p -autoreductions. Autoreducible sets were brought to the polynomial-time context by Ambos-Spies [AS84]. Their importance was further argued by Buhrman, Fortnow, Van Melkebeek, and Torenvliet [BFvMT98], who showed that all - p T -complete sets for EXP are - p -autoreducible (while some complete sets for other classes are not). Here we demonstrate that autoreducible sets are important for testing the power of resource-bounded measure. 6.1 Adaptively Autoreducible Sets As stated in the Introduction, if the - p T -autoreducible sets in EXP (or sufficiently the - p sets for EXP) are covered by an EXP-martingale, then EXP 6= BPP, a non-relativizing consequence. However, it is easy to cover them by an E-betting game. Indeed, the betting game uses its adaptive freedom only to "look ahead" at the membership of lexicographically greater strings, betting nothing on them. Theorem 6.1 There is an E-betting game G that succeeds on all - p -autoreducible sets. Proof. Let be an enumeration of - p T -autoreductions such that each M i runs in time on inputs of length n. Our betting game G regards its capital as composed of infinitely many "shares" c i , one for each M i . Initially, c Letting h\Delta; \Deltai be a standard pairing function, inductively define n During a stage makes a query of length less than n looks up the answer from its table of past queries. Whenever M i makes a query of length n s\Gamma1 or more, G places a bet of zero on that string and makes the same query. Then G bets all of the share c i on 0 n s\Gamma1 according to the answer of the simulation of M i . Finally, G "cleans up" by putting zero bets on all strings with length in [n that were not queries in the previous steps. If M i autoreduces A, then share c i doubles in value at each stage hi; ji, and makes the total capital grow to infinity. And G runs in time 2 O(n) -indeed, only the "cleanup" phase needs this much time. Corollary 6.2 Each of the following statements implies BPP 6= EXP: 1. The class of - p T -autoreducible sets has E-measure zero. 2. The class of - p -complete sets for EXP has E-measure zero. 3. E-betting games and E-martingales are equivalent. 4. E-betting games have the finite union property. The same holds if we replace E by EXP in these statements. Proof. Let C stand for the class of languages that are not - p -hard for BPP. Allender and Strauss [AS94] showed that C has E-measure zero, so trivially it is also covered by an E-betting game. Now let D stand for the class of - p -complete sets for EXP. By Theorem 6.1 and the result of [BFvMT98] cited above, D is covered by an E-betting game. contains all of EXP, and: ffl If D would have E-measure zero, so would C [ D and hence EXP, contradicting the measure conservation property of Lutz measure. ffl If E-betting games would have the finite-union property, then C [ D and EXP would be covered by an E-betting game, contradicting Theorem 3.1. Since (1) implies (2), and (3) implies (4), these observations suffice to establish the corollary for E. The proof for EXP is similar. Since there is an oracle A giving EXP A = BPP A [Hel86], this shows that relativizable techniques cannot establish the equivalence of E-martingales and E-betting games, nor of EXP-martingales and EXP-betting games. They cannot refute it either, since there are oracles relative to which strong PRGs exist-all "random" oracles, in fact. 6.2 Non-Adaptively Autoreducible Sets It is tempting to think that the non-adaptively P-autoreducible sets should have E-measure zero, or at least EXP-measure zero, insofar as betting games are the adaptive cousins of martingales. However, it is not just adaptiveness but also the freedom to bet out of the fixed lexicographic order that adds power to betting games. If one carries out the proof of Theorem 6.1 to cover the class of -autoreducible sets, using an enumeration [M i ] of - p -autoreductions, one obtains a non-adaptive E-betting game (defined formally below) that (independent of its oracle) bets on all strings in order given by a single permutation of \Sigma . The permutation itself is E-computable. It might seem that an E-martingale should be able to "un-twist" the permutation and succeed on all these sets. However, our next results, which strengthen the above corollary, close the same "non-relativizing" door on proving this with current techniques. Theorem 6.3 For any k - 1, the - p tt -complete sets for \Delta p are - p -autoreducible. Here is the proof idea, which follows techniques of [BFvMT98] for the theorem that all EXP- complete sets are - p -autoreducible. Call a closed propositional formula that has at most k blocks of like quantifiers (i.e., at most k \Gamma 1 quantifier alternations) a "QBF k formula," and let TQBF k stand for the set of true QBF formulas. Let A be a - p tt -complete set for \Delta p k . Since TQBF k is \Sigma p k -hard, there is a deterministic polynomial-time oracle Turing machine M that accepts A with oracle TQBF k . Let q(x; i) stand for the i-th oracle query made by M on input x. Whether belongs to TQBF k forms a \Delta p -question, so we can - p -reduce it to A. It is possible that this latter reduction will include x itself among its queries. Let b i denote the answer it gives to the question provided that any query to x is answered "yes," and similarly define i in case x is answered "no." i , which holds in particular if x is not queried, then we know the correct answer b i to the i-th query. If this situation occurs for all queries, we are done: We just have to run M on input x using the b i 's as answers to the oracle queries. The b i 's themselves are obtained without submitting the (possibly adaptive) queries made by M , but rather by applying the latter tt -reduction to A to the pair hx; ii, and without submitting any query on x itself. Hence this process satisfies the requirements of a - p tt -autoreduction of A for the particular input x. Now suppose that b + for some i, and let i be minimal. Then we will have two opponents play the k-round game underlying the QBF k -formula that constitutes the i-th oracle query. One player claims that b + i is the correct value for b i , which is equivalent to claiming that x 2 A, while the other claims that b \Gamma i is correct and that A. Write -A A. The players' strategies will consist of computing the game history so far, determining their optimal next move, - p tt -reducing this computation to A, and finally producing the result of this reduction under their respective assumption about -A (x). This approach will allow us to recover the game history in polynomial time with non-adaptive queries to A different from x. Moreover, it will guarantee that the opponent making the correct assumption about -A (x) plays optimally. Since this opponent is also the one claiming the correct value for b i , he will win the game. So, we output the winner's value for b i . It remains to show that we can compute the above strategies in deterministic polynomial time with a \Sigma p oracle, i.e. in FP \Sigma p k . It seems crucial that the number k of alternations be constant here. Proof. (of Theorem 6.3) Let A be a - p tt -complete set for \Delta p accepted by the polynomial-time oracle Turing machine M with oracle TQBF k . Let q(x; i) denote the i-th oracle query of M TQBF k on input x. Then q(x; i) can be written in the form (9y 1 )(8y stand for the vectors of variables quantified in each block, or in the opposite form beginning with the block (8y 1 ). By reasonable abuse of notation, we also ley y r stand for a string of 0-1 assignments to the variables in the r-th block. Without loss of generality, we may suppose every oracle query made by M has this form where each y j is a string of length jxj c , and M makes exactly jxj c queries, taking the constant c from the polynomial time bound on M . Note that the function q belongs to FP \Sigma p k . Hence the language belongs to \Delta p k+1 . Since A is - p k+1 , there is a polynomial-time nonadaptive oracle that accepts L 0 with oracle A. Now define b We define languages -reductions inductively as follows: k. The set L ' consists of all pairs hx; ji with 1 - j - jxj c , such that there is a smallest (x), and the following condition holds. For let the s-th bit of y r equal r (hx; si) otherwise. We put hx; ji into L ' iff there is a lexicographically least y ' such that and the j-th bit of y ' is set to 1. The form of this definition shows that L ' belongs to \Delta p . Hence we can take N ' to be a polynomial-time non-adaptive oracle TM that accepts L ' with oracle A. Now, we construct a - p -autoreduction for A. On input x, we compute b as well as y (b) r for b 2 f0; 1g and 1 - r - jxj c . The latter quantity y (b) r is defined as follows: for 1 - s - jxj c , the s-th bit of y (b) r equals N A[fxg r (hx; si) if r j b mod 2, and N Anfxg r (hx; si) otherwise. Note that we can compute all these values in polynomial time by making non-adaptive queries to A none of which equals x. run M on input x using b as the answer to the i-th oracle query. Since it always holds that at least one of b the correct oracle answer b i (x), we faithfully simulate M on input x, and hence compute -A (x) correctly. Otherwise, let i be the first index for which b + i, we can determine q(x; i) by simulating M on input x until it asks the i-th query. We then k )], and claim this value equals -A (x). In order to prove the claim, consider the game history y (b k . The opponent claiming the correct value for b i (x) gets to play the rounds that allow him to win the game (provided he plays well) no matter what the other player does. Since the former opponent is also the one making the correct assumption about -A (x), an inductive argument shows that he plays optimally: At his stages ', the string y ' in the above construction of L ' exists, and he plays it. The key for the induction is that at later stages ' 0 ? ', the value of y r for the value of y ' at stage '. So, the player with the correct assumption about -A (x) wins the game-that is, his guess for b i (x) (and not the other player's guess). In order to formalize the strengthening of Corollary 6.2 that results from Theorem 6.3, we call a betting game G non-adaptive if the infinite sequence x 1 of queries G A makes is the same for all oracles A. If G runs in 2 O(n) time, and this sequence hits all strings in \Sigma , then the permutation - of the standard ordering s defined by -(s computable and invertible in 2 O(n) time. It is computable in this amount of time because in order to hit all strings, G must bet on all strings in f 0; 1 g n within the first 2 O(n) steps. Hence its ith bet must be made in a number of steps that is singly-exponential in the length of s i . And to compute - need only be run for 2 O(jx i j) steps, since it cannot query x i after this time. Since - and its inverse are both E-computable, - is a reasonable candidate to replace lexicographic ordering in the definition of E-martingales, and likewise for EXP-martingales. We say a class C has -E-measure zero if C can be covered by an E-martingale that interprets its input as a characteristic string in the order given by -. Theorem 6.4 The class of - p -autoreducible languages can be covered by a non-adaptive E-betting game. Hence there is an E-computable and invertible permutation - of \Sigma such that this class has -E-measure zero. Proof. With reference to the proof of Theorem 6.1, we can let be an enumeration of tt -autoreductions such that each M i runs in time n i +i. The machine G in that proof automatically becomes non-adaptive, and since it queries all strings, it defines a permutation - of \Sigma as above with the required properties. Corollary 6.5 Each of the following statements implies BPP 6= EXP, as do the statements obtained on replacing "E" by "EXP." 1. The class of - p tt -autoreducible sets has E-measure zero. 2. The class of - p tt -complete sets for EXP has E-measure zero. 3. Non-adaptive E-betting games and E-martingales are equivalent. 4. If two classes can be covered by non-adaptive E-betting games, then their union can be covered by an E-betting game. 5. For all classes C and all E-computable and invertible orderings -, if C has -E-measure zero, then C has E-measure zero. Proof. It suffices to make the following two observations to argue that the proof of Corollary 6.2 carries over to the truth-table cases: ffl The construction of Allender and Strauss [AS94] actually shows that the class of sets that are not - p tt -hard for BPP has E-measure zero. ffl If Theorem 6.3 implies that all - p tt -complete sets for EXP are - p because BPP ' \Sigma p Theorem 6.4 and the finite-unions property of Lutz's measures on E and EXP do the rest. The last point of Corollary 6.5 asserts that Lutz's definition of measure on E is invariant under all E-computable and invertible permutations. These permutations include flip from the Introduction and (crucially) - from Theorem 6.4. Hence this robustness assertion for Lutz's measure implies BPP 6= EXP. Our "betting-game measure" (both adaptive and non-adaptive) does enjoy this permutation invariance, but asserting the finite-unions property for it also implies BPP 6= EXP. The rest of this paper explores conditions under which Lutz's martingales can cover classes of autoreducible sets, thus attempting to narrow the gap between them and betting games. 6.3 Covering Autoreducible Sets By Martingales This puts the spotlight on the question: Under what hypotheses can we show that the - p autoreducible sets have E-measure zero? Any such hypothesis must be strong enough to imply EXP 6= BPP, but we hope to find hypotheses weaker than assuming the equivalence of (E- or betting games and martingales, or assuming the finite-union property for betting games. Do we need strong PRGs to cover the - p -autoreducible sets? How close can we come to covering the - p -autoreducible sets by an E-martingale? Our final results show that the hypothesis MA 6= EXP suffices. This assumption is only known to yield PRGs of super-polynomial security (at infinitely many lengths) rather than exponential security (at almost all lengths). Recall that MA contains both BPP and NP; in fact it is sandwiched between NP BPP and BPP NP . Theorem 6.6 If MA 6= EXP, then the class of - p tt -autoreducible sets has E-measure zero. We actually obtain a stronger conclusion. Theorem 6.7 If MA 6= EXP, then the class of languages A autoreducible by polynomial-time OTMs that always make their queries in lexicographic order has E-measure zero. To better convey the essential sampling idea, we prove the weaker Theorem 6.6 before the stronger Theorem 6.7. The extra wrinkle in the latter theorem is to use the PRG twice, to construct the set of "critical strings" to bet on as well as to compute the martingale. Proof. (of Theorem 6.6) Let [M i enumerate the - p -autoreductions, with each M i running in time n i . Divide the initial capital into shares s i;m for with each s i;m valued initially at (1=m 2 )(1=2 i ). For each share s i;m , we will describe a martingale that is only active on a finite number of strings x, namely only if i - m=2dlog 2 me and m - further only if x belongs to a set constructed below. We will arrange that whenever M i autoreduces A, there are infinitely many m such that share s i;m attains a value above 1 (in fact, close to m) along A. Hence the martingale defined by all the shares succeeds on A. We will also ensure that each active share's bets on strings of length n are computable in time 2 an , where the constant a is independent of i. This is enough to make the whole martingale E-computable and complete the proof. To describe the betting strategy for s i;m , first construct a set I = I i;m starting with I and iterating as follows: Let y be the lexicographically least string of length m that does not appear among queries made by M i on inputs x 2 I. Then add y to I. Do this until I has 3dlog 2 me strings in it. This is possible because the bound 3dlog 2 mem i on the number of queries M i could possibly make on inputs in I is less than 2 m . Moreover, 2 m bounds the time needed to construct I. Thus we have arranged that for all x; y 2 I with x y, M i (x) does not query y. (13) Now let J stand for I together with all the queries M i makes on inputs in I. Adapting ideas from Definition 4.1 to this context, let us define a finite Boolean function to be consistent with M i on I, written fi - I M i , if for all x 2 I, M i run on input x with oracle answers given by agrees with the value fi(x). Given a characteristic prefix w, also write fi - w if fi(x) and w(x) agree on all x in J and the domain of w. Since I and J depend only on i and m, we obtain a "probability density" function for each share s i;m via The martingale d i;m standardly associated to this density (as in [Lut92]) is definable inductively by d i;m d i;m (In case - i;m = 0, we already have d i;m (w) = 0, and so both d i;m (w1) and d i;m (w0) are set to 0.) Note that the values - i;m (wb) for only differ from - i;m (w) if the string x indexed by b belongs to J ; i.e., d i;m is only active on J . sufficiently large m, if share s i;m could play the strategy d i;m , then on A its value would rise to (at least) m=2 i . That is, s i;m would multiply its initial value by (at least) m 3 . To see this, first note that for any w v A long enough to contain J in its domain, - i;m We want to show that for any v short enough to have domain disjoint from I, - i;m To do this, consider any fixed 0-1 assignment fi 0 to strings in J n I that agrees with v. This assignment determines the computation of M i on the lexicographically first string x 2 I, using fi 0 to answer queries, and hence forces the value of fi(x) in order to maintain consistency on I. This in turn forces the value fi(x 0 ) on the next string x 0 in I, and so on. Hence only one out of 2 jIj possible completions of fi 0 to fi is consistent with M i on I. Thus - i;m by (15), and 2 The main obstacle now is that (14), and hence d i;m (w), may not be computable in time 2 an with a independent of i. The number of assignments fi to count is on the order of 2 jJj Here is where we use the E-computable PRG D, with super-polynomial stretching and security, obtained via Theorem 5.2 from the hypothesis MA 6= EXP. For all i and sufficiently large m, D stretches a seed s of length m into at least 3dlog 2 mem i bits, which are enough to define an assignment fi s to J (agreeing with any given w). We estimate - i;m (w) by . By Theorem 5.2 there are infinitely many "good" m such that SD (m) ? m i+4 . 6.9 For all large enough good m, every estimate - Suppose not. First note that both (14) and (16) do not depend on all of w, just on the up-to- bits in w that index strings in J , and these can be hard-wired into circuits. The tests [fi - I M i ] can also be done by circuits of size o(m i+1 ), because a Turing machine computation of time r can be simulated by circuits of size O(r log r) [PF79]. Hence we get circuits of size less than SD (m) achieving a discrepancy greater than 1=SD (m), a contradiction. This proves Claim 6.9. Finally, observe that the proof of Claim 6.8 gives us not only d i;m (w) - i;m (w) \Delta m 3 , but also d i;m A. For w v A and good m, we thus obtain estimates g(w) for d i;m (w) within error bounds applying Lemma 5.1 for this g(w) and yields a martingale d 0 i;m (w) computable in time 2 an , where the constant a is independent of i. This d 0 i;m (w) is the martingale computed by the actions of share s i;m . Since actually obtain jd 0 which is stronger than what we needed to conclude that share s i;m returns enough profit. This completes the proof of Theorem 6.6. To prove Theorem 6.7, we need to construct sets I = I i;m with properties similar to (13), in the case where M i is no longer a - p -autoreduction, but makes its queries in lexicographic order. To carry out the construction of I, we use the pseudorandom generator D a second time, and actually need only that M i on input 0 m makes all queries of length ! m before making any query of length m. To play the modified strategy for share s i;m , however, appears to require that all queries observe lex order. Proof. (of Theorem 6.7). Recall that the hypothesis EXP 6= MA yields a PRG D computable in stretching m bits to r(m) bits such that for all i, all sufficiently large m give r(m) ? and infinitely many m give hardness SD be a standard enumeration of T -autoreductions that are constrained to make their queries in lexicographic order, with each running in time O(n i ). We need to define strategies for "shares" s i;m such that whenever M i autoreduces A, there are infinitely many m such that share s i;m grows its initial capital from 1=m 2 2 i to 1=2 i or more. The strategy for s i;m must still be computable in time 2 am where a is independent of i. To compute the strategy for s i;m , we note first that s i;m can be left inactive on strings of length ! m. The overall running time allowance 2 O(m) permits us to suppose that by the time s i;m becomes active and needs to be considered, the initial segment w 0 of A (where A is the language on which the share happens to be playing) that indexes strings of length up to Hence we may regard w 0 as fixed. For any ff 2 f 0; 1 let M ff stand for the computation in which w 0 is used to answer any queries of length ! m and ff is used to answer all other queries. Because of the order in which M i makes its queries, those queries y answered by w 0 are the same for all ff, so that those answers can be coded by a string u 0 of length at most m i . Now for any string y of length equal to m, define Note that given u 0 and ff, the test "M ff queries y" can be computed by circuits of size O(m i+1 ). Hence by using the PRG D at length m, we can compute uniformly in E an approximation PD (x; y) for P (x; y) such that for infinitely many m, said to be "good" m, all pairs x; y give jP D (x; Here is the algorithm for constructing I = I i;m . Start with I := ;, and while jIj ! 3 log 2 m, do the following: Take the lexicographically least string y I such that for all x 2 I, . The search for such a y will succeed within jIj \Delta m i+4 trials, since for any particular x, there are fewer than m i+4 strings y overall that will fail the test. (This is so even if m is not good, because it only involves PD , and because PD involves simulating M D(s) i over all seeds s.) There is enough room to find such a y provided which holds for all sufficiently large m. The whole construction of I can be completed within time 2 2am . It follows that for any sufficiently large good m and x; y 2 I with x ! y, Pr ff [M ff At this point we would like to define J to be "I together with the set of strings queried by M i on inputs in I " as before, but unlike the previous case where M i was non-adaptive, this is not a valid definition. We acknowledge the dependence of the strings queried by M i on the oracle A by defining JA queries y g: is, JA has the same size as J in the previous proof. This latter definition will be OK because M i makes its queries in lexicographic order. Hence the share s i;m , having already computed I without any reference to A, can determine the strings in JA on which it should be active on the fly, in lex order. Thus we can well-define a mapping fi from f 0; 1 g r to f 0; 1 g so that for any k - r, means that the query string y that happens to be kth in order in the on-the-fly construction of JA is answered "yes" by the oracle. Then we may write J fi for JA , and then write in place of Most important, given any x 2 I, every such fi well-defines a computation M fi i (x). This entitles us to carry over the two "consistency" definitions from the proof of Theorem 6.6: Finally, we may apply the latter notion to initial subsets of I, and define for 1 - 3 log m the predicate does not query x k . 6.10 For all ', Pr fi [R ' (fi)] - 1=2 ' . For the base case does not query x 1 , M i being an autoreduction, and because whether fi - x1 M i depends only on the bit of fi corresponding to x 1 . Working by induction, suppose Pr fi [R . If R '\Gamma1 (fi) holds, then taking fi 0 to be fi with the bit corresponding to x ' flipped, R holds. However, at most one of R ' (fi) and does not query x ' . Hence Pr fi [R ' (fi)] - (1=2)Pr fi [R and this proves Claim 6.10. (It is possible that neither R ' (fi) nor R ' (fi 0 ) holds, as happens when some j, but this does not hurt the claim.) Now we can rejoin the proof of Theorem 6.6 at equation (14), defining the probability density function - i;m We get a martingale d i;m from - i;m as before, and this represents an "ideal" strategy for share s i;m to play. The statement corresponding to Claim 6.8 is: autoreduces A and m is good and sufficiently large, then the ideal strategy for share s i;m multiplies its value by at least m 3 =2 along A. To see this, note that we constructed I above so that for all Pr ff [M ff It follows that d3 log me! me 2 . Hence, using Claim 6.10 with log m, we get: Since the fi defined by A satisfies fi - I M i , it follows by the same reasoning as in Claim 6.8 that d i;m profits by at least a fraction of m 3 =2 along A. This proves Claim 6.11. Finally, we (re-)use the PRG D as before to expand a seed s of length m into a string fi s of (at least) bits. Given any w, fi s well-defines a fi and a set J fi of size at most r as constructed above, by using w to answer queries in the domain of w and fi s for everything else. We again obtain the estimate - equation (16), with the same time complexity as before. Now we repeat Claim 6.9 in this new context: 6.12 For all large enough good m, every estimate - i;m (w) satisfies j- i;m (w) \Gamma - i;m (w)j - ffl. If not, then for some fixed w the estimate fails. The final key point is that because M i always makes its queries in lexicographic order, the queries in the domain of w that need to be covered are the same for every fi s . Hence the corresponding bits of w can be hard-wired by circuitry of size at most r. The test [fi s - I M i ] can thus still be carried out by circuits of size less than m i+1 , and we reach the same contradiction of the hardness value SD . Finally, we want to apply Lemma 5.1 to replace d i;m (w) by a martingale d 0 i;m (w) that yields virtually the same degree of success and is computable in time 2 O(n) . Unlike the truth-table case we cannot apply Lemma 5.1 verbatim because we no longer have a single small set J that d 0 is active on. However, along any set A, the values d 0 i;m (w) and d 0 or 1) can differ only for cases where b indexes a string in the small set J corresponding to A, and the reader may check that the argument and bounds of Lemma 5.1 go through unscathed in this case. This finishes the proof of Theorem 6.7. Conclusions The initial impetus for this work was a simple question about measure: is the pseudo-randomness of a characteristic sequence invariant under simple permutations such as that induced by flip in the Introduction ? The question for flip is tantalizingly still open. However, in Section 6.2 we showed that establishing a "yes" answer for any permutation that intuitively should preserve the same complexity-theoretic degree of pseudo-randomness, or even for a single specific such permutation as that in the simple proof of the non-adaptive version of Theorem 6.1, would have the major consequence that EXP 6= BPP. Our "betting games" in themselves are a natural extension of Lutz's measures for deterministic time classes. They preserve Lutz's original idea of ``betting'' as a means of ``predicting'' membership in a language, without being tied to a fixed order of which instances one tries to predict, or to a fixed order of how one goes about gathering information on the language. We have shown some senses in which betting games are robust and well-behaved. We also contend that some current defects in the theory of betting games, notably the lack of a finite-unions theorem pending the status of pseudo-random generators, trade off with lacks in the resource-bounded measure theory, such as being tied to the lexicographic ordering of strings. The main open problems in this paper are interesting in connection with recent work by Impagliazzo and Wigderson [IW98] on the BPP vs. EXP problem. First we remark that the main result of [IW98] implies that either or BPP has E-measure zero [vM98]. Among the many measure statements in the last section that imply BPP 6= EXP, the most constrained and easiest to attack seems to be item 4 in Corollary 6.5. Indeed, in the specific relevant case starting with the assumption one is given a non-adaptive E-betting game G and an E-martingale d, and to obtain the desired contradiction that proves BPP 6= EXP, one need only construct an EXP-betting game G 0 that covers S What we can obtain is a "randomized" betting game G 00 that flips one coin at successive intervals of input lengths to decide whether to simulate G or d on that interval. (The intervals come from the proof of Theorem 6.4.) Any hypothesis that can de-randomize this G 00 implies BPP 6= EXP. We do not know whether the hypotheses considered in [IW98], some of them shown to follow from BPP 6= EXP itself, are sufficient to do this. Stepping back from trying to prove BPP 6= EXP outright or trying to prove that these measure statements are equivalent to BPP 6= EXP, we also have the problem of narrowing the gap between BPP 6= EXP and the sufficient condition EXP 6= MA used in our results. Moreover, does EXP 6= MA suffice to make the - p T -autoreducible sets have E-measure zero? Does that suffice to simulate every betting game by a martingale of equivalent complexity? We also inquire whether there exist oracles relative to which strong PRGs still exist. Our work seems to open many opportunities to tighten the connections among PRGs, the structure of classes within EXP, and resource-bounded measure. The kind of statistical sampling used to obtain martingales in Theorems 5.6 and 5.7 was originally applied to construct martingales from "natural proofs" in [RSC95]. The derandomization technique from [BFNW93] based on EXP 6= MA that is used here is also applied in [BvM98, KL98, LSW98]. "Probabilistic martingales" that can use this sampling to simulate betting games are formalized and studied in [RS98]. This paper also starts the task of determining how well the betting-game and random-sampling ideas work for measures on classes below E. Even straightforward attempts to carry over Lutz's definitions to classes below E run into difficulties, as described in [May94] and [AS94, AS95]. We look toward further applications of our ideas in lower-level complexity classes. Acknowledgments The authors specially thank Klaus Ambos-Spies, Ron Book (pace), and Jack Lutz for organizing a special Schloss Dagstuhl workshop in July 1996, where preliminary versions of results and ideas in this paper were presented and extensively discussed. We also thank the referees for helpful comments. --R Measure on small complexity classes Measure on P: Robustness of the notion. "Algorithmic Information Theory and Randomness" Trading group theory for randomness. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Using autoreducibility to separate complexity classes. Separating complexity classes using autoreducibility. In 13th Annual Symposium on Theoretical Aspects of Computer Science Hard sets are hard to find. How to construct random functions. On relativized exponential and probabilistic complexity classes. Randomness vs. time: De-randomization under a uniform assumption On the A variant of the Kolmogorov concept of complexity. Resource bounded measure and learn- ability Almost everywhere high nonuniform complexity. The quantitative structure of exponential time. Contributions to the Study of Resource-Bounded Measure Hardness versus randomness. Relations among complexity measures. Probabilistic martingales and BPTIME classes. Pseudorandom generators The complexity of approximate counting. On the measure of BPP. --TR --CTR Klaus Ambos-Spies , Wolfgang Merkle , Jan Reimann , Sebastiaan A. Terwijn, Almost complete sets, Theoretical Computer Science, v.306 n.1-3, p.177-194, 5 September

probabilistic computation;resource-bounded measure;theory of computation;betting games;pseudorandom generators;autoreducibility;complexity classes;sampling;computational complexity;polynomial reductions

587001

Message Multicasting in Heterogeneous Networks.

In heterogeneous networks, sending messages may incur different delays on different links, and each node may have a different switching time between messages. The well-studied telephone model is obtained when all link delays and switching times are equal to one unit. We investigate the problem of finding the minimum time required to multicast a message from one source to a subset of the nodes of size k. The problem is NP-hard even in the basic telephone model. We present a polynomial-time algorithm that approximates the minimum multicast time within a factor of O(log k). Our algorithm improves on the best known approximation factor for the telephone model by a factor of $O(\frac{\log n}{\log\log k})$. No approximation algorithms were known for the general model considered in this paper.

Introduction . The task of disseminating a message from a source node to the rest of the nodes in a communication network is called broadcasting. The goal is to complete the task as fast as possible assuming all nodes in the network participate in the eort. When the message needs to be disseminated only to a subset of the nodes this task is referred to as multicasting. Broadcasting and multicasting are important and basic communication primitives in many multiprocessor systems. Current networks usually provide point-to-point communication only between some of the pairs of the nodes in the network. Yet, in many applications, a node in the network may wish to send a message to a subset of the nodes, where some of them are not connected to the sender directly. Due to the signicance of this operation, it is important to design e-cient algorithms for it. Broadcast and multicast operations are frequently used in many applications for message-passing systems (see [11]). It is also provided as a communication primitive by several collective communication libraries, such as Express by Parasoft [8] and the Message Passing Library (MPL) [1, 2] of the IBM SP2 parallel systems. This operation is also included as part of the collective communication routines in the Message-Passing Interface (MPI) standard proposal [7]. Application domains that use broadcast and multicast operations extensively include scientic computations, network management protocols, database transactions, and multimedia applications. Part of this work was done while the rst three authors visited IBM T.J. Watson Research Center. A preliminary version of this paper appeared in the Proc. of the 30th ACM Symp. on Theory of Computing, 1998. y AT&T Research Labs, 180 Park Ave., P.O. Box 971, Florham Park, NJ 07932. On leave from the Faculty of EE, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: amotz@research.att.com. z Computer Science Department, Stanford University, Stanford, CA 94305. Supported by an IBM Fellowship, ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. E-mail: sudipto@cs.stanford.edu. x Bell Laboratories, Lucent Technologies, 600 Mountain Ave., Murray Hill, NJ 07974. On leave from the Computer Science Department, Technion, Haifa 32000, Israel. E-mail: naor@research.bell-labs.com. { IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598. E-mail: sbar@watson.ibm.com. A. BAR-NOY, S. GUHA, J. NAOR AND B. SCHIEBER In most of these applications the e-ciency depends on the time it takes to complete the broadcast or multicast operations. There are two basic models in which trivial optimal solutions exist. In the rst model, all nodes are assumed to be connected, a node may send a message to at most one other node in each round, and it takes one unit of time (round) for a message to cross a link. Therefore, in each round the number of nodes receiving the message can be doubled. If the target set of nodes is of size k, then this process terminates in dlog ke rounds. In the second model the communication network is represented by an arbitrary graph, where each node is capable of sending a message to all of its neighbors in one unit of time. Here, the number of rounds required to deliver a message to a subset of the nodes is the maximum distance from the source node to any of the nodes in the subset. The model in which a node may send a message to at most one other node in each round is known as the Telephone model. It is known that for arbitrary communication graphs, the problem of nding an optimal broadcast in the Telephone model is NP-hard [12], even for 3-regular planar graphs [20]. Following the two easy cases given above, it is not hard to verify that in the Telephone model two trivial lower bounds hold for the minimum broadcast time. The rst one is dlog ne, where n denotes the number of nodes in the graph, and the second one is the maximum distance from the source node to any of the other nodes. Research in the past three decades has focused on nding optimal broadcast algorithms for various classes of graphs such as trees, grids, and hypercubes. Also, researchers have looked for graphs with minimum number of links for which a broadcast time of dlog ne can be achieved from any source node. Problems related to broadcast which were extensively investigated are the problems of broadcast multiple messages, gossiping, and computing certain functions on all n inputs in a network. See, e.g., [4, 5, 6, 9, 13, 14, 15, 19, 22, 24, 25]. An assumption central to the Telephone model is that both sender and receiver are busy during the whole sending process. That is, only after the receiver received the message, both ends may send the message to other nodes. More realistic models in this context are the Postal model [3] and the LogP model [18]. The idea there is that the sender may send another message before the current message is completely received by the receiver, and the receiver is free during the early stages of the sending process. We note that in both the Postal model and the LogP model it is assumed that the delay of a message between any pair of nodes is the same. Optimal solutions for broadcast in the Postal model are known for the case of a complete graph, and for some other classes of graphs. However, not much is known for arbitrary graphs. In the Postal model, researchers have also concentrated on other dissemination primitives and almost always assumed that the communication graph is complete. 1.1. Our results. In this paper we dene a more general model based on the Postal model and call it the heterogeneous postal model. Assume node u sends a message to node v at time 0 and the message arrives at v at time uv . The assumption is that u is free to send a new message at time s u , and v is free from time 0 to time uv r v . We call uv the delay of the link (u; v), s u the sending (or switching) time of u, and r v the receiving time of v. By denition, both s u and r v are smaller than uv . In the single message multicast problem each node receives no more than a single message. Thus, for this problem the receiving time has almost no relevance. Because of this, and to keep the presentation clearer we assume for the rest of the paper that r nodes u. Observe that when the delay, sending time, and receiving time are all equal to 1, we obtain the Telephone model. We believe that our framework may be useful to model modern communication networks, where the major components { the processors and the communication links { are not homogeneous. Some processors are faster than others, and some links have more bandwidth than others. These disparities are captured by the dierent values of the delay and the switching time. Since nding the minimum multicast time is NP-hard even in the Telephone model, we turn our focus to approximation algorithms. The main result we present is an approximation algorithm for computing a multicast scheme in the heterogeneous Postal model. The approximation factor is O (log k), where k denotes the number of processors in the target set. Previous approximation algorithms for multicasting were known only in the Telephone model. Kortsarz and Peleg [17] gave an approximation algorithm that produces a solution whose value is bounded away from the optimal solution by an O( n) additive term. This term is quite large, especially for graphs in which the broadcast (multicast) time is polylogarithmic. Later, Ravi [21], gave an algorithm that achieves a multiplicative approximation factor of O log n log k log log k We also show that it is NP-hard to approximate the minimum broadcast time within a factor of three in a model which is only slightly more complicated than the Telephone model. The rest of the paper is organized as follows. In Section 2 we dene our model. In Section 3 we describe our solution. Finally, in Section 4 we show that this problem is hard to approximate by a small constant factor. 2. The Model and the Problem. We dene our model as follows. Let (V; E) be an undirected graph representing a communication network, where V is a set of n nodes and E is the set of point to point communication links. Let U V denote a special set of terminals, and let r be a special node termed the root. Let the cardinality of the set U be k. To simplify notation assume that r 2 U . We associate with each node v 2 V a parameter s v that denotes the sending time. We sometimes refer to s v as the switching time of v to indicate that this is the time it takes node v to send a new message. In other words, 1=s v is the number of messages node v can send in one round (unit of time). We associate with each node that denotes the receiving time. We assume that r each node v. We associate with each link (u; v) 2 E a length uv that denotes the communication delay between nodes u and v. By denition, uv is greater than both s u and r v (= s v ). We can think of the delay uv as taking into account the sending time at u and the receiving time at v. Let the generalized degree of node v 2 V be the actual degree of v in the graph G multiplied by the switching time s v . Observe that the generalized degree measures the time it would have taken the node v to send a message to all of its neighbors. Our goal is to nd a minimum time multicast scheme; that is, a scheme in which the time it takes for all nodes in the set U to receive the message from the root r is minimized. Without loss of generality, we may consider only multicast schemes that are \not lazy"; i.e., schemes in which a node that has not nished sending the message to its neighbors (but has already started) is not idle. Such multicast schemes can be represented by an outward directed tree T that is rooted at r and spans all the nodes in U , together with orderings on the edges outgoing from each node in the tree. The multicast scheme corresponding to such a tree and orderings is a multicast in which each node in the tree upon receiving the message (through its single incoming edge) sends the message along each of its outgoing edges in the specied order. From now A. BAR-NOY, S. GUHA, J. NAOR AND B. SCHIEBER on, we refer to the tree in the representation of a multicast scheme as the tree \used" by the scheme. For a rooted tree T , denote by T its maximum generalized degree, and by L T the maximum distance from r to any of the nodes in U (with respect to the lengths xy associated with each link (x; y)). By denition, the multicast time of tree T is greater than T and greater than L T . Hence, Lemma 2.1. Let OPT denote the multicast time of an optimal solution using tree T , then OPT 1( T 3. The Approximation Algorithm. In this section we describe the approximation algorithm for multicasting a message to a set of terminals U from a root node The main tool used by our algorithm is a procedure ComputeCore(U 0 ) that computes for a given set of terminals U 0 , where r 2 U 1. A subset W U 0 which we call the core of U 0 , of size at most 3jU 0 j, where r 2 W . 2. A scheme to disseminate a message known to all the nodes in W to the rest of the nodes in U 0 in time proportional to the minimum multicast time from r to U 0 . The algorithm that computes the multicast scheme proceeds in ' phases. Let U Upon termination, U frg. In the ith phase, is invoked to compute: 1. The core of U denoted by U i . 2. A scheme to disseminate the message from U i to the set U i 1 in time proportional to the minimum multicast time from r to U i 1 . we have that k). The resulting multicast scheme is given by looking at the rounds of the algorithm in backward order. Namely, starting at in each round of the multicast scheme the message is disseminated from U i to U i 1 . Since U ' U ' 1 U each dissemination phase takes time proportional to the minimum multicast time from r to U . It follows that the multicast time is up to O(log times the optimal multicast time. In the rest of the section we describe the procedure ComputeCore(U 0 ). Let OPT be the minimum multicast time from r to U 0 . Lemma 2.1 implies that there exists a tree T spanning the set U 0 such that T . The procedure ComputeCore(U 0 ) has two main parts. In the rst part, we nd a set of jU 0 j paths, one for each terminal, where the ith path connects the terminal u i to another terminal called The paths have the following path properties: Length Property: The length of each path is at most 4 ( T Congestion Property: The generalized degree of the nodes in the graph induced by the paths is at most 6 ( T In the second part we design a dissemination scheme using the above paths. We do it by transforming the paths into a set of disjoint spider graphs { graphs in which at most one node has degree more than two. These spider graphs have the following spider properties: Each spider contains at least two terminals from U 0 . The set of spiders spans at least half the nodes in U 0 . The diameter of each spider is at most 4 ( T The generalized degree of the center of a spider is at most 6 ( T where the center of a spider is the unique node with degree larger than two, if such exists, or one of the endpoints of the spider, otherwise. Now, for each spider, we arbitrarily select one of the nodes from U 0 to the core of U 0 . Note that each such node can multicast the message to the rest of the terminals in its spider in O( T +L T ) time (linear in OPT ). We add all the terminals not contained in any of the spiders to the core of U 0 . We claim that the size of the core is at most4 To see this, let x denote the number of spiders and let y be the number of the terminals in all the spiders. It follows that the size of the core is jU x. By the rst spider property we have that x y=2 and by the second spider property we get that y jU 0 j=2. Thus, We now turn to describe each of the two parts of the procedure ComputeCore(U 0 ). 3.1. Finding a set of paths. We rst claim the following lemma which is analogous to the \tree pairing" lemma of Ravi [21]. Lemma 3.1. Let T be a tree that spans a set U 0 V , and suppose that jU 0 j is even. There exists a way to pair the nodes of U 0 , and nd paths (in the tree T ) connecting each pair such that: 1. the paths are edge disjoint, 2. the length of each path is bounded by 2L T , 3. the generalized degree of each node in the graph induced by the paths is at most T . Proof. The tree pairing lemma ([21]) states that there exists a pairing such that the paths in T connecting each of the pairs are edge disjoint. Consider these paths. Clearly the length of each of these paths is bounded by 2L T . The degree, and hence the generalized degree, of every node in the graph induced by the paths is no more than the (generalized) degree in T since we only use the edges of the tree T . Hence, it is bounded by T . The following corollary handles the odd case as well. Corollary 3.2. Let T be a tree that spans a set U 0 V . There exists a way to pair the nodes of U 0 , and nd paths (in the tree T ) connecting each pair such that: 1. the length of each path is bounded by 2L T , 2. the generalized degree of each node in the graph induced by the paths is at most 2 T . Proof. The corollary clearly holds if jU 0 j is even. If jU 0 j is odd, we pair jU 0 j 1 of the nodes as in Lemma 3.1, and pair the last node with any other node. The length of the path connecting the last pair is still bounded by 2L T . However, the degree of the subgraph may double up to 2 T . Recall that the tree T spans the nodes of U 0 and Our objective is to nd the set of paths as guaranteed by Corollary 3.2 with respect to T . However, we do not know T . Thus, instead, we nd a set of fractional paths satisfying similar properties. To this end, we write a linear program for nding a set of (fractional) paths that minimizes the sum of two quantities: (1) the maximum over all pairs of the average length of the paths connecting a pair, and (2) the maximum generalized degree of the subgraph induced by the paths connecting the pairs. The linear program is a variant of multicommodity ow. For each edge (u; v), we dene the directed edges (u; v) and (v; u) both of length uv . Let U g. With each node v j 2 U 0 we associate commodity j. Node v j is the source of commodity j and we create an articial sink t j with r t j We connect each of the nodes A. BAR-NOY, S. GUHA, J. NAOR AND B. SCHIEBER by a directed edge (v The objective is to minimize (L exactly one unit of ow has to be shipped from each v j to t j , such that the average length of the ow paths from v j to t j is at most 2L, and the maximum weighted congestion (generalized degree) of the induced subgraph is at most 3. More formally, let A denote the set of directed edges, and let f i (u; v) denote the ow of commodity i on directed edge (u; v). The linear programming formulation is as follows. subject to: For all 1 i h and (v;w)2A For all 1 i For all 1 i (v i ;u)2A For all 1 i For all 3 For all 1 i We now show that the set of paths guaranteed by Corollary 3.2 with respect to T can be modied so as to obtain an integral solution for the linear program as follows. If jU 0 j is even, the solution is obtained by using each path connecting a pair to ship one unit of ow from u j through u i to t j , and another unit of ow from u i through u j to t i . The length of each path is bounded by 2L T , and since we use each path twice, the generalized degree is bounded by 2 T . If jU 0 j is odd, the solution is obtained by using each of the 1(jU 0 j 1) paths connecting the rst jU nodes of twice (once in each direction), and using the path connecting the last node in U 0 to its mate to ship ow out of this node. The length of each path is still bounded by However, because of the additional path, the degree is only bounded by 3 T . It follows that the value of the objective function for this solution is T +L T , and thus the linear program is guaranteed to nd a solution whose value is upper bounded by this value. Let L T and T denote the values of the length and congestion in the optimal solution of the above linear program. The optimal solution is a \fractional" solution in the sense that the (unit) ow of each commodity is split between several ow paths. We round the fractional solution into an integral solution using an algorithm proposed by Srinivasan and Teo [23]. This algorithm builds on a theorem proved by Karp et al. [16]. For completeness and since the details are slightly dierent, we now describe the rounding of the fractional solution. Theorem 3.3. [16] Let A be a real valued r s matrix, and y be a real valued s-vector. Let b be a real valued vector such that Ay = b. Let t be a positive real number such that in every column of A, 1. the sum of all positive entries t, and 2. the sum of all negative entries t. Then, we can compute (in polynomial time) an integral vector y such that for every t. We now show how to nd an integral ow of congestion at most 6 T +4L T , where each ow path (of each commodity) has length at most 4L T . We rst decompose the ow into (polynomially many) ow paths. If any path in this decomposition is longer than 4L T , we discard it. We observe that since the average length is less than 2L T , discarding these long paths leaves at least half of a unit of ow between each pair (v We scale the ows appropriately such that the total ow to each t i is exactly 1. This can at most double the ow on an edge, and the total congestion is now at most 6 T . denote the length bounded ow paths. Denote the set of nodes in a path P i by V (P i ) and the set of edges by E(P i ). Let f(P i ) denote the amount of ow pushed on path P i . Dene the set P j as the set of all paths that carry ow of the jth commodity. Observe that each path belongs to exactly one P j . The linear system needed for Theorem 3.3 is dened by the following linear equations, where the i-th equation corresponds to the i-th row of A and the i-th element of b. for each v s v i: for all j 4L T The second set of inequalities captures the fact that the ow on all the paths corresponding to commodity j is exactly 1. Now the sum of the positive entries in a column is (length of path The second part of the inequality follows since s v vw for all v; w and s t j sum of the negative entries in a column is at most 4L T , this follows due to the fact that each P i belongs to exactly one P j . Invoking Theorem 3.3 gives us a set of paths such that, for each v s v i: for all j 4L T The second set of inequalities implies that each commodity has at least one ow path. So we have a set of ow paths such that the congestion is at most 6 T and their length is at most 4L T . Since T +L T T +L T these paths satisfy the length and congestion properties as desired. 3.2. Finding a spider decomposition. We now show how to obtain a spider decomposition satisfying the spider properties previously dened. Recall that we are now given a set of paths connecting each terminal u j with another terminal Mate(u j ), and that this set of paths satises the length and congestion properties. We nd a set of at least jU 0 j=2 trees that satisfy the following properties which are similar to the spider properties. A. BAR-NOY, S. GUHA, J. NAOR AND B. SCHIEBER Each tree spans at least two terminals from U 0 . The diameter of each tree is at most 4L T 4 ( T The generalized degree of each node in each of the trees is at most 6 T Before showing how to nd these trees, we show how to transform them into the required spiders. Repeatedly, consider the tree edges, and remove a tree edge if it separates the tree into two subtrees such that either, both subtrees contain at least two terminals, or one of them contains no terminals (in this case this subtree is removed as well). Repeat this process until no more edges can be removed. The process terminates since the number of edges is nite. Observe that upon termination, if a connected component is not a spider, then another edge could be deleted. Thus, we get the following claim. 3.4. When the process terminates each connected component is a spider. Clearly, all the terminals spanned by the trees are also spanned by the spiders. The diameter of each of these spiders is at most 4L T , since the distance between every pair of nodes in U 0 spanned by a tree is at most 4L T to begin with. Also, the generalized degree of the \center" of the spider is at most the generalized degree of its originating tree since we have not introduced any new edges in the process. We conclude that the spiders satisfy the desired spider properties. Now, we show how to nd the required trees. Dene G p to be the undirected graph induced by the paths from each terminal to its mate. Observe that a spanning forest of this graph may not satisfy the required diameter property above and hence some extra renement is necessary. For each node u in G p , nd a unique terminal in U 0 that is closest to u (with respect to the lengths xy associated with each link (x; y)). Ties are broken arbitrarily. We modify the paths starting at each terminal as follows. From each terminal u begin tracing the path connecting u to Mate(u). At some node v along this path, the closest terminal to v will not be u. We are guaranteed to encounter such a node because the closest node to Mate(u) is Mate(u) itself. From this node v trace the path to its closest terminal. This creates a path from u to another terminal denoted NewMate(u). Note that NewMate(u) may be dierent from Mate(u). However, we are guaranteed that the path from u to NewMate(u) is not longer than the path from u to Mate(u) and thus bounded by 4L T . Dene an auxiliary directed graph H on the set of terminals U 0 with the set of edges . By denition each node in H has outdegree one. Thus, each connected component of (the undirected skeleton of) H contains exactly one directed cycle. Discard one edge from each such connected component to make it a rooted tree in which all edges are oriented towards the root. (The root is unique since the outdegree of each node is one.) Note that every non-trivial strongly connected component of H is a cycle. Thus, this can be done just by discarding an arbitrary edge from each strongly connected component of H . Let H 0 be the resulting forest. Dene the level of a node in H 0 to be the number of edges in the path to it from the root of its component. (We ip the direction of the edges in H 0 for the purpose of measuring distances.) Distinguish between nodes of even level and nodes of odd level. Each edge of H 0 goes either from an odd level node to an even level node or vice-versa. Consider two collections of stars in H 0 . One collection consisting of edges from odd level nodes to even level nodes, and the other consisting of edges from even level nodes to odd level nodes. Every terminal with positive indegree and outdegree (in spanned by a star in each one of the two collections. Every terminal with either indegree or outdegree zero (in H 0 ) is spanned by a star in only one of the two collections. However, by a simple pigeon-hole argument, at least one of the collections spans at least half of the terminals. Consider such a collection. First, note that each star in this collection induces an undirected tree in the original graph when replacing each star edge by its originating path. We now claim the following, Lemma 3.5. The induced trees of any two stars belonging to the same collection are node disjoint. Proof. To obtain a contradiction assume they are not disjoint. Then, there exist two distinct terminals with the same even or odd parity, say u and v, such that NewMate(u) 6= NewMate(v), but the paths traced from u to NewMate(u) and from v to NewMate(v) have a common node x. Consider the terminal chosen by x as its closest terminal. We distinguish between two cases. Case 1: The terminal chosen by x is u. Then u must be NewMate(v), contradicting the fact that u and v are of the same parity. The case where v is the chosen terminal of x is symmetric. Case 2: The terminal chosen by x is NewMate(u). Then NewMate(v) must be the same as NewMate(u); a contradiction. The case where NewMate(v) is the chosen terminal of x is symmetric. It is easy to see that the trees induced by the stars in the collection satisfy the required properties. This concludes the construction. 4. Hardness of Approximations. In this section we show that the best possible approximation factor of the minimum broadcast time in the heterogeneous Postal model is 3 . We show this hardness result even for a restricted model in which d. Note that when s broadcast the message concurrently to all of its neighbors. The proof is by a reduction to the set cover problem. In the unweighted version of the set cover problem we are given a set U of elements and a collection S of subsets of U . The goal is to nd the smallest number of subsets from S whose union is the set U . Feige [10] proved the following hardness result. Lemma 4.1. Unless NP DT IME(n log log n ), the set cover problem cannot be approximated by a factor which is better than ln n, and hence it cannot be approximated within any constant factor. In our proof, we will only use the fact that it is NP-Hard to approximate the optimal set cover within any constant factor. Theorem 4.2. It is NP-Hard to approximate the minimum broadcast time of any graph within a factor of 3 . Proof. Assume to the contrary that there exists an algorithm that violates the claim of the theorem for some . We show how to approximate the set cover problem within a constant factor using this algorithm. To approximate the set cover problem we "guess" the size of the optimal set cover and use our approximate minimum broadcast time algorithm to check our guess. Since the size of the optimal set cover is polynomial, we need to check only a polynomial number of guesses. Consider an instance of set cover I = (U; S) where U is the set of elements, and S a collection of subsets of U . Let jU m. Let the guess on the size of the optimal set cover be k. We construct the following graph G. The graph G, consists of 1 vertices: a distinguished root vertex r, vertices e A. BAR-NOY, S. GUHA, J. NAOR AND B. SCHIEBER corresponding to the elements of U , vertices um corresponding to the subsets, and k additional vertices a r e ae1 a The root r has switching time and is connected to a by edges with delay ra ' = 1. Each vertex a ' has switching time s(a ' connected to all u j with delay a ' u j = 1. Each vertex u j has switching time connected to a vertex e i i the jth set contains the ith element. The delay of such an edge is u d, where d > 4 2 is a constant. Each vertex e i has switching time 1. Finally, to complete the instance of the multicasting problem, the target multicast set consists of all vertices e i . We rst show that if there is a set cover of size k, then there is a multicast scheme of length d 2. After time 1, all the vertices a ' receive the message. After time 2, all the vertices u j corresponding to sets which are in this cover, receive the message. This is possible since all a ' are connected to all u j . Finally, these vertices send the message to all the elements that they cover. Since s(u j that the multicast time is d 2. Suppose that the algorithm for the multicasting problem completes the multicasting at time t. By the contradiction assumption, its approximation factor is 3 . Since by our guess on the size of the set cover the optimal multicast time no more than d+2 we have t (3 )(d 2. The strict inequality follows from the choice of d. We rst claim that all the vertices u j that participate in the multicast receive the message from some a ' . Otherwise there exists a vertex e i 0 that received the message via a path of a type (r; a This means that e i 0 received the message at or after time 3d t. Our second claim is that each vertex a ' sends the message to at most 2d vertices u j . This is because the (2d 1)st vertex would receive the message at time 2d would not be able to help in the multicast eort that is completed before time 3d 2. Combining our two claims we get that the multicasting was completed with the help of 2dk vertices u j . The corresponding 2dk sets cover all the elements e i . This violates Lemma 4.1 that states that the set cover problem cannot be approximated within any constant factor. Remark:. In our proof we considered a restricted model in which the switching time may only get two possible values and the delay may get only three possible values (assuming that when an edge does not exist then the delay is innity). Observe that this hardness result does not apply to the Telephone model in which the switching time is always 1 and the delay is either 1 or innity. We have similar hardness results for other special cases. However, none of them is better than 3 and all use similar techniques. Acknowledgment . We thank David Williamson for his helpful comments. --R The IBM external user interface for scalable parallel systems CCL: a portable and tunable collective communication library for scalable parallel comput- ers Designing broadcasting algorithms in the Postal model for message-passing systems Optimal Multiple Message Broadcasting in Telephone-Like Communication Systems Optimal multi-message broadcasting in complete graphs Algebraic construction of e-cient networks Document for a standard message-passing interface Express 3.0 Introductory Guide Broadcast time in communication networks A threshold of ln n for approximating set cover Solving Problems on Concurrent Processors Computers and Intractability: A Guide to the Theory of NP-Completeness On the construction of minimal broadcast networks Tight bounds on minimum broadcast networks A survey of gossiping and broadcasting in communication networks Global wire routing in two-dimensional arrays Approximation algorithm for minimum time broadcast Optimal broadcast and summation in the LogP model Broadcast networks of bounded degree Minimum broadcast time is NP-complete for 3-regular planar graphs and deadline 2 Rapid rumor rami Generalizations of broadcasting and gossiping A constant-factor approximation algorithm for packet routing A new method for constructing minimal broadcast networks A class of solutions to the gossip problem --TR --CTR Teofilo F. Gonzalez, An Efficient Algorithm for Gossiping in the Multicasting Communication Environment, IEEE Transactions on Parallel and Distributed Systems, v.14 n.7, p.701-708, July Michael Elkin , Guy Kortsarz, Sublogarithmic approximation for telephone multicast, Journal of Computer and System Sciences, v.72 n.4, p.648-659, June 2006 David Kempe , Jon Kleinberg , Amit Kumar, Connectivity and inference problems for temporal networks, Journal of Computer and System Sciences, v.64 n.4, p.820-842, June 2002 Pierre Fraigniaud , Bernard Mans , Arnold L. Rosenberg, Efficient trigger-broadcasting in heterogeneous clusters, Journal of Parallel and Distributed Computing, v.65 n.5, p.628-642, May 2005 Eli Brosh , Asaf Levin , Yuval Shavitt, Approximation and heuristic algorithms for minimum-delay application-layer multicast trees, IEEE/ACM Transactions on Networking (TON), v.15 n.2, p.473-484, April 2007

postal model;logp model;combinatorial optimization;approximation algorithms;heterogeneous networks;multicast

587029

Beyond Competitive Analysis.

The competitive analysis of online algorithms has been criticized as being too crude and unrealistic. We propose refinements of competitive analysis in two directions: The first restricts the power of the adversary by allowing only certain input distributions, while the other allows for comparisons between information regimes for online decision-making. We illustrate the first with an application to the paging problem; as a byproduct we characterize completely the work functions of this important special case of the k-server problem. We use the second refinement to explore the power of lookahead in server and task systems.

Introduction The area of On-Line Algorithms [16, 10] shares with Complexity Theory the following characteristic: Although its importance cannot be reasonably denied (an algorithmic theory of decision-making under uncertainty is of obvious practical relevance and significance), certain aspects of its basic premises, modeling assumptions, and results have been widely criticized with respect to their realism and relation to computational practice. We think that now is a good time to revisit some of the most often-voiced criticisms of competitive analysis (the basic framework within which on-line algorithms have Computer Science Department, University of California, Los Angeles, CA 90095. Re-search supported in part by NSF grant CCR-9521606. y Computer Science and Engineering, University of California, San Diego, La Jolla, CA 92093. Research supported in part by the National Science Foundation. been heretofore studied and analyzed), and to propose and explore some better-motivated alternatives. In competitive analysis, the performance of an on-line algorithm is compared against an all-powerful adversary on a worst-case input. The competitive ratio of a problem-the analog of worst-case asymptotic complexity for this area-is defined as (1) Here A ranges over all on-line algorithms, x over all "inputs", opt denotes the optimum off-line algorithm, while A(x) is the cost of algorithm A when presented with input x. This clever definition is both the weakness and strength of competitive analysis. It is a strength because the setting is clear, the problems are crisp and sometimes deep, and the results often elegant and striking. But it is a weakness for several reasons. First, in the face of the devastating comparison against an all-powerful off-line algorithm, a wide range of on-line algorithms (good, bad, and mediocre) fare equally badly; the competitive ratio is thus not very informative, fails to discriminate and to suggest good approaches. Another aspect of the same problem is that, since a worst-case input decides the performance of the algorithm, the optimal algorithms are often unnatural and impractical, and the bounds too pessimistic to be informative in practice. Even enhancing the capabilities of the on-line algorithm in obviously desirable ways (such as a limited lookahead capability) brings no improvement to the ratio (this is discussed more extensively in a few paragraphs). The main argument for competitive analysis over the classical expectation maximization is that the distribution is usually not known. However, competitive analysis takes this argument way too far: It assumes that absolutely nothing is known about the distribution, that any distribution of the inputs is in principle possible; the worst-case "distribu- tion" prevailing in competitive analysis is, of course, a worst-case input with probability one. Such complete powerlessness seems unrealistic to both the practitioner (we always know, or can learn, something about the distribution of the inputs) and the theoretician of another persuasion (the absence of a prior distribution, or some information about it, seems very unrealistic to a probabilist or mathematical economist). The paging problem, perhaps the most simple, fundamental, and practically important on-line problem, is a good illustration of all these points. An unreasonably wide range of deterministic algorithms (both the good in practice LRU and the empirically mediocre FIFO) have the same competitive ratio-k, the amount of available memory. Even algorithms within more powerful information regimes-for example, any algorithm with lookahead pages-provably can fare no better. Admittedly, there have been several interesting variants of the framework that were at least partially successful in addressing some of these concerns. Randomized paging algorithms have more realistic performance [5, 11, 15]. Some alternative approaches to evaluating on-line algorithms were proposed in [1, 14] for the general case and in [2, 6, 7, 17] specifically for the paging problem. In this paper we propose and study two refinements of competitive analysis which seem to go a long way towards addressing the concerns expressed above. Perhaps more importantly, we show that these ideas give rise to interesting algorithmic and analytical problems (which we have only begun to solve in this paper). Our first refinement, the diffuse adversary model, removes the assumption that we know nothing about the distribution-without resorting to the equally unrealistic classical assumption that we know all about it. We assume that the actual distribution D of the inputs is a member of a known class \Delta of possible distributions. That is, we seek to determine, for a given class of distributions \Delta, the performance ratio A ED (A(x)) ED (opt(x)) (2) That is, the adversary picks a distribution D among those in \Delta, so that the expected, under D, performance of the algorithm and the off-line optimum algorithm are as far apart as possible. Notice that, if \Delta is the class of all possible distributions, (1) and (2) coincide since the worst possible distribution is the one that assigns probability one to the worst-case input and probability zero everywhere else. Hence the diffuse adversary model is indeed a refinement of competitive analysis. In the paging problem, for example, the input distribution specifies, for each page a and sequence of page requests ae, prob(ajae)-the probability that the next page fault is a, given that the sequence so far is ae. It is unlikely that an operating system knows this distribution precisely. On the other hand, it seems unrealistic to assume that any distribution at all is possible. For example, suppose that the next page request is not predictable with absolute certainty: prob(ajae) - ffl, for all a and ae, where ffl is a real number between capturing the inherent uncertainty of the request sequence. This is a simple, natural, and quite well-motivated assumption; call the class of distributions obeying this inequality \Delta ffl . An immediate question is, what is the resulting competitive ratio As it turns out, the answer is quite interesting. If k is the storage capacity, the ratio shown to coincide with the expected cost of a simple random walk on a directed graph with approximately nodes. For this value is easy to estimate: It is between 1 ffl; for larger values of k we do not have a closed-form solution for the ratio. There are two important byproducts of this analysis: First, extending the work in [8], we completely characterize the work functions of the paging special case of the k-server problem. Second, the optimum on-line algorithm is robust- that is, the same for all ffl's-and turns out to be a familiar algorithm that is also very good in practice: LRU. It is very interesting that LRU emerges from the analysis as the unique "natural" optimal algorithm, although there are other algorithms that are also optimal. The second refinement of competitive analysis that we are proposing deals with the following line of criticism: In traditional competitive analysis, the all-powerful adversary frustrates not only interesting algorithms, but also powerful information regimes. The classical example is again from paging: In paging the best competitive ratio of any on-line algorithm is k. But what if we have an on-line algorithm with a lookahead of ' steps, that is, an algorithm that knows the immediate future? It is easy to see that any such algorithm must fare equally badly as algorithms without lookahead. In proof, consider a worst case request sequence abdc \Delta \Delta \Delta and take its (' stuttered version, a '+1 b '+1 d '+1 c '+1 \Delta \Delta \Delta It is easy to see that an algorithm with lookahead ' is as powerless in the face of such a sequence as one without a lookahead. Once more, the absolute power of the adversary blurs practically important distinctions. Still, lookahead is obviously a valuable feature of paging algorithms. How can we use competitive analysis to evaluate its power? Notice that this is not a question about the effectiveness of a single algorithm, but about classes of algorithms, about the power of information regimes-ultimately, about the value of information [12]. To formulate and answer this and similar questions we introduce our second refinement of competitive analysis, which we call comparative analysis. Suppose that A and B are classes of algorithms-typically but not necessarily usually a broader class of algorithms, a more powerful information regime. The comparative ratio R(A;B) is defined as follows: min A2A This definition is best understood in terms of a game-theoretic interpre- tation: B wants to demonstrate to A that it is a much more powerful class of algorithms. To this end, B proposes an algorithm B among its own. In response, A comes up with an algorithm A. Then B chooses an input x. Finally, A pays B the ratio A(x)=B(x). The larger this ratio, the more powerful B is in comparison to A. Notice that, if we let A be the class of on-line algorithms and B the class of all algorithms-on-line or off-line-then equations (1) and (3) coincide, and Hence comparative analysis is indeed a refinement of competitive analysis. We illustrate the use of comparative analysis by attacking the question of the power of lookahead in on-line problems of the "server" type: If L ' is the class of all algorithms with lookahead ', and L 0 is the class of on-line algorithms, then we know that, in the very general context of metrical task systems [3] we have (that is, the ratio is at most 2' systems, and it is exactly while in the more restricted context of paging Diffuse Adversaries The competitive ratio for a diffuse adversary 1 is given in equation (2). In order to factor out the disadvantage of the algorithm imposed by the initial conditions, we shall allow an additive constant in the numerator. More precisely, an on-line algorithm A is c-competitive against a class \Delta of input distributions if there exists a constant d such that for all distributions D 2 \Delta: Diffuse adversaries are not related to the diffusion processes in probability theory which are continuous path Markov processes. The competitive ratio of the algorithm A is the infimum of all such c's. Fi- nally, the competitive ratio R(\Delta) of the class of distributions is the minimum competitive ratio achievable by an on-line algorithm. It is important to observe that \Delta is a class of acceptable conditional probability distributions; each is the distribution of the relevant part of the world conditioned on the currently available information. In the case of the paging problem with a set of pages M , \Delta may be any set of functions of the form where for all ae 2 M P 1. In the game-theoretic interpretation, as the sequence of requests ae develops, the adversary chooses the values of D(ajae) from those available in \Delta to maximize the ratio. Since we deal with deterministic algorithms, the adversary knows precisely the past decisions of A, but the adversary's choices may be severely constrained by \Delta. It is indicative of the power of the diffuse adversary model that most of the proposals for a more realistic competitive analysis are simply special cases of it. For example, the locality of reference in the paging problem [2, 6] is captured by the diffuse adversary model where \Delta consists of the following conditional probability distributions: there is no edge from b to a in the access graph and otherwise. Simirarly, the Markov paging model [7] and the statistical adversary model [14] are also special cases of the diffuse adversary model. In this section we apply the diffuse adversary model to the paging prob- lem. We shall focus on the class of distributions \Delta ffl , which contains all functions ffl]-that is to say, all conditional distributions with no value exceeding ffl. Since the paging problem is the k-server problem on uniform metric spaces, we shall need certain key concepts from the k-server theory (see [8, 4] for a more detailed exposition). denote the number of page slots in fast memory, and let M be the set of pages. A configuration is a k-subset of M ; we denote the set of all configurations by C. Let ae 2 M . The work function associated with ae, w ae (or w when ae is not important or understood from context) is a function defined as follows: If (X) is the optimum number of page faults when the sequence of requests is ae and the final memory configuration is X. Henceforth we use the symbols to denote set union and set difference respectively. Also, we represent unary sets with their element, e.g. we write a instead of fag. Definition 2 If w is a work function, define the support of w to be all configurations such that there is no Y 2 C, different from X, with j. Intuitively, the values of w on its support completely determine w. The following lemmata, specific for the paging problem and not true in general for the k-server problem, characterize all possible work functions by determining the structure of their support. A similar, but more complicated, characterization is implicit in the work of [11]. The first lemma states that all values in the support are the same, and hence what matters is the support itself, not the values of w on it. Lemma 1 The support of a work function w contains only configurations on which w achieves its minimum value. Proof. The proof is by contradiction. Suppose that the lemma does not hold. That is, there is a configuration A in the support of w such that Choose now a configuration B with w(B) ! w(A) that minimizes jB \Gamma Aj. By the quasiconvexity condition in [8], there is an a A such that This can hold only if either w(A) ? In the first case, we get that 1 and this contradicts the assumption that A is in the support of w. The second case also leads to a contradiction since it violates the choice of B with minimum because An immediate consequence of Lemma 1 is that the off-line cost is always equal to the minimum value of a work function. The next lemma determines the structure of the support. Lemma 2 For each work function w there is an increasing sequence of sets the most recent request, such that the support of w is precisely Note that the converse, not needed in the sequel, also holds: Any such tower of P j 's defines a reachable work function. For a work function w define its signature to be the k-tuple its type to be the k-tuple Proof. The proof is by induction on the length of the request sequence. The basis case is obvious: Let P is the initial configuration. For the induction step assume that we have the signature be the resulting work function after request r. Consider first the case that r belongs to P t and not to P kg. Since there is at least one configuration in the support of w that contains the request r, the minimum value of w 0 is the same with the minimum value of w. Therefore, the support of w 0 is a subset of the support of w. It consists of the configurations that belong to the support of w and contain the last request r. It is easy now to verify that the signature of w 0 is: If, on the other hand, the request r does not belong to P k , the minimum value of w 0 is one more than the minimum value of w. In this case, the support of w 0 consists of all configurations in the support of w where one point has been replaced by the request r, i.e. a server has been moved to service r. Consequently, the signature of w 0 is given by: The induction step now follows. We now turn to the \Delta ffl diffuse adversary. Let w be the current work function with signature We want to determine the optimal on-line and off-line strategies. A natural guess is that the best on-line strategy should prefer to have in the fast memory pages from P i than pages from P j. The reason is that pages in P i are more valuable to the off-line algorithm than pages in P a configuration from the support of w remains in the support when we replace any page from P a page from P i ; the converse does not hold in general. For the same reason the off-line strategy should prefer the next request to be in P i than in Furthermore, the adversary should try to avoid placing a request in a page already in the on-line fast memory, because although this doesn't affect the on-line cost, it may shrink the support of the current work function. This leads to the following strategy for the adversary: First, order all pages in such a way that pages in P i precede pages in P pages in P k precede pages in P k , the complement of Call such an ordering a canonical ordering with respect to the signature . The adversary now assigns probability ffl to the first 1=ffl pages of a canonical ordering that are not in the on-line fast memory. Notice that this presupposes that there are at least k + 1=ffl pages in total. Although, both strategies seem optimal, we don't have a simple proof for this. Instead, we will proceed carefully to establish that this is in fact the case. But first we analyze the competitive ratio that results from the above strategies. It is not difficult to see that the off-line cost is the expected cost of a Markov chain M k;ffl : The states of M k;ffl are the types of the work functions, i.e., all tuples (p From a state (p 1 which corresponds to a signature there are transitions to the following (not necessarily distinct) They correspond to the case of the next request r being in P t , the case of corresponds to simply repeating the last request and we can safely omit it. There is also a transition to the type that corresponds to the case r 62 P k . The cost of each transition is the associated off-line cost. All transitions have cost zero, except the last one, which has cost one, because a request r increases the minimum value of the work function if and only if r 62 P k . Finally, the transition probabilities are determined by the adversary's strategy. The total probability of the first t transitions is maxf(p (each page that is not in on-line fast memory has probability ffl). The probability of the last transition is the remaining probability also shows that there is no need to consider types with p k greater than k+1=ffl because these types are 'unreachable'. The significance of this fact is that the Markov process M k;ffl is finite (it has O((k Let c(M k;ffl ) be the expected cost of each step of the Markov process M k;ffl , which is also the expected off-line cost. If we assume that the on-line cost of each step is one, we get that the competitive ratio resulting from the above strategies is 1=M k;ffl . We now turn our attention to the optimal strategy for the adversary. It should be clear that in each step the adversary assigns probabilities to pages which are not in the on-line fast memory, based on the current work function and the configuration of the on-line algorithm. Our aim is to make the adversary's strategy independent of the configuration of the on-line algorithm and this can be achieved by allowing the on-line algorithm to swap any two pages without any extra cost immediately after servicing a request. The on-line algorithm suffers one page fault in each step, but it can move to the best configuration relative to the current work function with no extra penalty. Consider now a work function w with signature We will first argue that the optimal strategy for the adversary prefers to assign probability to pages in P i than to pages in j. The reason is that the type (1; of the resulting work function w 1 after a request in P i is no less than the type of the resulting work function w 2 after a request in P precisely, by symmetry and the assumption that the on-line algorithm is always at the best configuration, the only relevant part of a work function to the off-line strategy is its type. Therefore, instead of comparing w 2 directly to w 1 we can compare w 2 to any work function w 0 1 that has the same type as w 1 . It is easy to see that there is a work function w 0 1 with the same type as w 1 such that any configuration in the support of w 2 belongs to the support of w 0 1 . Consequently, for any request sequence ae the off-line cost to service it with initial function w 0 1 is no more than the off-line cost to service ae with initial work function w 2 . Similarly, we will argue that the optimal adversary prefers to assign probability to pages in P k than to pages in P k . However, there is a trade-off involved when the adversary assigns probability to pages in P k : The adversary suffers an immediate page fault but the support of the resulting work function is larger and therefore the adversary will pay less in the future. We want to verify that the adversary never recovers a future payoff larger than the current loss (the page fault). Again, we cannot directly compare the two work functions w 1 and w 2 that result from a request in P k and a request in P k respectively, but it suffices to compare w 1 with a work function w 0 2 that has the same type with w 2 . The next lemma shows that for any request sequence ae the cost of servicing it with initial work function w 1 differs by at most one from the cost of servicing it with initial work function w 0 . As we argued above, the worst case happens when the request in P k is actually a request in therefore we need to compare two work functions with types 1). In the next lemma we use for simplicity q Lemma 3 Let w 1 be a work function with type (1; Then there is a work function w 2 with type (1; 2; such that for any configuration in the support of w 1 there is a configuration in the support of w 2 that differs in at most one position. Proof. Let be the signature of w 1 and let a be a page in be the work function with signature Consider a configuration X in the support of w 1 . We will show that there exists a configuration Y in the support of w 2 such that jY \Gamma Xj - 1. Consider a canonical ordering with respect to the signature of w 2 and let x k be the last page of X in this order. Also, let b be the first page in this ordering not in X \Gamma x k . We claim that differs in at most one position from X, is in the support of w 2 . Notice first that Y contains the page in P 1 . It also contains the page a, since a is in the the second place in the ordering and therefore either a 2 X or a = b. It remains to show that 1. There are two cases to consider: The first case, when 1, follows from the fact that x k is in P j only if b is in P j . For the second case, when suffices to note that In summary, assuming that the on-line algorithm suffers cost one in each step but can swap pages freely, the optimum strategy for the adversary is to assign probabilities to pages which are not in the fast on-line memory and are first in a canonical ordering with respect to the current signature. On the other hand, the on-line strategy should prefer to have in its fast memory the first pages of this ordering. Therefore, if a 1 ; a is a canonical ordering then the best configuration for an on-line algorithm is g. This poses the question whether it is always possible for the on-line algorithm to be in such a configuration by a simple swap in each step. Fortunately, there is a familiar algorithm that does exactly this: the Least Recently Used (LRU) algorithm. Surprisingly, LRU does not even remember the whole signature of the work function. It simply remembers the first k pages a 1 ; a of a canonical ordering, and services the next request by swapping the a k page. It is easy to verify that after the next request, its fast memory contains the first k pages of some canonical ordering with respect to the resulting signature. Therefore, we have shown: Theorem 1 When there are at least k+1=ffl pages, algorithm LRU is optimal against a diffuse adversary with competitive ratio 1=c(M k;ffl ). It seems difficult to determine the exact competitive ratio 1=c(M k;ffl ). For the extreme values of ffl we know that 1. The first case is when the adversary has complete power and the second when the adversary suffers a page fault in almost every step. In fact, the next corollary suggests that the competitive ratio may not be expressible by a simple closed-form expression. It remains an open problem to find a simple approximation for the competitive ratio. Corollary 1 If and 1=ffl is an integer then Y Therefore, ffl. Proof. It is not difficult to see that the Markov process is identical to the following random process: in each phase repeatedly choose uniformly a number from f1; phase ends when a number is chosen twice. This random process is a generalization of the well-known birthday problem in probability theory. A phase corresponds to a cycle in the Markov chain that starts (and ends) at state with type (1; 2). The cost of the Markov chain is the expected length of a phase minus one (all transitions in the cycle have cost one except the last one). It is not hard now to verify the expression for R(ffl). In order to bound the expected length of a phase, notice that each of the first n numbers has probability at most 1= p n to end the phase. In contrast, each of the next n numbers has probability at least 1= n to end the phase. Elaborating on this observation we get that 1 ffl. Comparative Analysis On-line algorithms deal with the relations between information regimes. Formally but briefly, an information regime is the class of all functions from a domain D to a range R which are constant within a fixed partition of D. Refining this partition results in a richer regime. Traditionally, the literature on on-line algorithms has been preoccupied with comparisons between two basic information regimes: The on-line and the off-line regime (the off-line regime corresponds to the fully refined partition). As we argued in the intro- duction, this has left unexplored several intricate comparisons between other important information regimes. Comparative analysis is a generalization of competitive analysis allowing comparisons between arbitrary information regimes, via the comparative ratio defined in equation (3). Naturally, such comparisons make sense only if the corresponding regimes are rich in algorithms-single algorithms do not lend themselves to useful comparisons. As in the case of the competitive ratio for the diffuse adversary model, we usually allow an additive constant in the numerator of equation (3). We apply comparative analysis in order to evaluate the power of lookahead in task systems. An on-line algorithm for a metrical task system has lookahead if it can base its decision not only on the past, but also on the next ' requests. All on-line algorithms with lookahead ' comprise the information regime L ' . Thus, L 0 is the class of all traditional on-line algorithms. Metrical task systems [3] are defined on some metric space M; a server resides on some point of the metric space and can move from point to point. Its goal is to process on-line a sequence of tasks cost c(T j ; a j ) of processing a task T j is determined by the task T j and the position a of the server while processing the task. The total cost of processing the sequence is the sum of the distance moved by the server plus the cost of servicing each task Theorem 2 For any metrical task system, R(L there are metrical task systems for which R(L Proof. Trivially the theorem holds for Assume that ' ? 0 and fix an algorithm B in L ' . We shall define an on-line algorithm A without lookahead whose cost on any sequence of tasks is at most 2' times the cost of B. Algorithm A is a typical on-line algorithm in comparative analysis: It tries to efficiently "simulate" the more powerful algorithm B. In particular, A knows the position of B ' steps ago. In order to process the next task, A moves first to B's last known position, and then processes the task greedily, that is, with the minimum possible cost. be a sequence of tasks and let b be the points where algorithm B processes each task and a 1 ; a the corresponding points for algorithm A. For simplicity, we define also points b negative j's. Then the cost of algorithm B is and the cost of algorithm A is Recall that in order to process the j-th task, algorithm A moves to B's last known position b j \Gamma' and then processes the task greedily, that is, d(b is the smallest possible. In particular, ?From this, the fact that costs are nonnegative and the triangle inequality we get Combining these with the triangle inequalities of the form we get that the cost of algorithm A is at most The last expression is times the cost of algorithm B. For the converse, observe that when c(T all triangle inequalities above hold as equalities then a comparative ratio of can be achieved. Of course, for certain task systems the comparative ratio may be less that 1. For the paging problem, it is ' + 1. Theorem 3 For the paging problem Proof. Let be an algorithm for the paging problem in the class L ' , that is, with lookahead '. Without loss of generality we assume that B moves its servers only to service requests. Consider the following on-line algorithm A: In order to service a request r, A moves a server that has not been moved in the last n times such that the resulting configuration is as close as possible to the last known configuration of B. Fix a worst request sequence ae and let A the configurations of A and B that service ae. Without loss of generality, we assume that A moves a server in each step. By definition, A services the t-th request by moving a server not in B t\Gamman (unless A We will first show by induction that A t and B t , differ in at most n points: jB n. This is obviously true for t - n. Assume that it holds for t\Gamman then clearly the statement holds, because in each step the difference can increase by at most one. Otherwise, A services the t-th request by moving the server from some point x, x 62 B t\Gamman . Observe that A t can differ from B t in more than n points only if x 2 A However, x can belong to B only if x was requested at least once in the steps servers only to service requests. Therefore, A moved a server at x in the last n steps and it cannot move it again. Hence, A t and B t cannot differ in more than n points. The theorem now follows from the observation that for every moves of A there is a move of some server of B. The reason is this: in the same configuration then A will converge to the same configuration in at most n moves (recall that A moves a different server each time). 4 Open problems We introduced two refinements of competitive analysis, the diffuse adversary model and comparative analysis. Both restrict the power of the adversary: The first by allowing only certain input distributions and the second by restricting the refinement of the adversary's information regime. In general, we believe that the only natural way to deal with uncertainty is by designing algorithms that perform well in the worst world which is compatible with the algorithm's knowledge. There are numerous applications of these two frameworks for evaluating on-line algorithms. We simply mention here two challenging open problems. The Markov diffuse adversary. Consider again the paging problem. Suppose that the request sequence is the output sequence of an unknown Markov chain (intuitively, the program generating the page requests) with at most s states, which we can only partially observe via its output. That is, the output f(q) of a state q of the unknown Markov process is a page in M . The allowed distributions \Delta are now all output distributions of s-state Markov processes with output. We may want to restrict our on-line algorithms to ones that do not attempt to exhaustively learn the Markov process-one way to do this would be to bound the length of the request sequence to O(s). We believe that this is a useful model of paging, whose study and solution may enhance our understanding of the performance of actual paging systems. The power of vision. Consider two robots, one with vision ff (its visual sensors can detect objects in distance ff) and the other with vision fi, fi ? ff. We want to measure the disadvantage of the first robot in navigating or exploring a terrain against the second robot. Naturally, comparative analysis seems the most appropriate framework for this type of problems. Different restrictions on the terrain and the objective of the robot result in different problems but we find the following simple problem particularly challenging: On the plain, there are n opaque objects. The objective of the robot is to construct a map of the plain, i.e., to find the position of all n objects. We ask what the comparative ratio R(V ff ; V fi ) for this problem is, where V ff and V fi denote the information regimes of vision ff and fi, respectively. --R A new measure for the study of on-line algorithms Competitive paging with locality of reference. An optimal on-line algorithm for metrical task systems The server problem and on-line games Competitive paging algorithms. Strongly competitive algorithms for paging with locality of reference. Markov paging. On the k-server conjecture Beyond competitive analy- sis Competitive algorithms for on-line problems A strongly competitive randomized paging algorithm. On the value of information. Shortest paths without a map. A statistical adversary for on-line algorithms Memory versus randomization in on-line algorithms Amortized efficiency of list update and paging rules. --TR --CTR Peter Damaschke, Scheduling Search Procedures, Journal of Scheduling, v.7 n.5, p.349-364, September-October 2004 Marek Chrobak , Elias Koutsoupias , John Noga, More on randomized on-line algorithms for caching, Theoretical Computer Science, v.290 n.3, p.1997-2008, 3 January Marcin Bienkowski, Dynamic page migration with stochastic requests, Proceedings of the seventeenth annual ACM symposium on Parallelism in algorithms and architectures, July 18-20, 2005, Las Vegas, Nevada, USA Peter Damaschke, Scheduling search procedures: The wheel of fortune, Journal of Scheduling, v.9 n.6, p.545-557, December 2006 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 Marek Chrobak, SIGACT news online algorithms column 8, ACM SIGACT News, v.36 n.3, September 2005 James Aspnes , Orli Waarts, Compositional competitiveness for distributed algorithms, Journal of Algorithms, v.54 n.2, p.127-151, February 2005

paging problem;online algorithms;competitive analysis;metrical task systems

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Using redundancies to find errors.

This paper explores the idea that redundant operations, like type errors, commonly flag correctness errors. We experimentally test this idea by writing and applying four redundancy checkers to the Linux operating system, finding many errors. We then use these errors to demonstrate that redundancies, even when harmless, strongly correlate with the presence of traditional hard errors (e.g., null pointer dereferences, unreleased locks). Finally we show that how flagging redundant operations gives a way to make specifications "fail stop" bydetecting dangerous omissions.

INTRODUCTION Programming languages have long used the fact that many high-level conceptual errors map to low-level type errors. This paper demonstrates the same mapping in a diVerent direction: many high-level conceptual errors also map to low-level redundant operations. With the exception of a few stylized cases, programmers are generally attempting to perform useful work. If they perform an action, it was because they believed it served some purpose. Spurious operations violate this belief and are likely errors. For example, impossible Boolean conditions can signal mistaken expressions; critical sections without shared state can signal the use of the wrong variables written but not read can signal an unintentionally lost result. At the least, these conditions signal conceptual confusion, which we would also expect to correlate with hard errors - deadlocks, null pointer dereferences, etc. - even for harmless redundancies. We use redundancies to find errors in three ways: (1) by writing checkers that automatically flag redundancies, (2) using these errors to predict non-redundant errors (such as null pointer dereferences), and (3) using redundancies to find incomplete program specifications. We discuss each below. We wrote four checkers that flagged potentially dangerous redundancies: (1) idempotent operations, (2) assignments 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. SIGSOFT 2002/FSE-10, November 18-22, 2002, Charleston, SC, USA. that were never read, (3) dead code, and (4) conditional branches that were never taken. The errors found would largely be missed by traditional type systems and checkers. For example, as Section 2 shows, assignment of variables to themselves can signal mistakes, yet such assignments will type check in any language we know of. Of course, some legitimate actions cause redundancies. Defensive programming may introduce "unnecessary" operations for robustness; debugging code, such as assertions, can check for "impossible" conditions; and abstraction boundaries may force duplicate calculations. Thus, to eVectively find errors, our checkers must separate such redundancies from those induced by error. We wrote our redundancy checkers in the xgcc extensible compiler system [16], which makes it easy to build system-specific static analyses. Our analyses do not depend on an extensible compiler, but it does make it easier to prototype and perform focused suppression of false positive classes. We evaluated how eVective flagging redundant operations is at finding dangerous errors by applying the above four checkers to the Linux operating system. This is a good test since Linux is a large, widely-used source code base (we check roughly 1.6 million lines of it). As such, it serves as a known experimental base. Also, because it has been written by many people, it is representative of many diVerent coding styles and abilities. We expect that redundancies, even when harmless, strongly correlate with hard errors. Our relatively uncontroversial hypothesis is that confused or incompetent programmers tend to make mistakes. We experimentally test this hypothesis by taking a large database of hard Linux errors that we found in prior work [8] and measuring how well redundancies predict these errors compared to chance. In our tests, files that have redundancy errors are roughly 45% to 100% more likely to have hard errors compared to files drawn by chance. This diVerence holds across the diVerent types of redundancies. Finally, we discuss how traditional checking approaches based on annotations or specifications can use redundancy checks as a safety net to find missing annotations or incomplete specifications. Such specification mistakes commonly map to redundant operations. For example, assume we have a specification that binds shared variables to locks. A missed binding will likely lead to redundancies: a critical section with no shared state and locks that protect no variables. We can flag such omissions because we know that every lock should protect some shared variable and that every critical section should contain some shared state. This paper makes four contributions: 1. The idea that redundant operations, like type errors, commonly flag correctness errors. 2. Experimentally validating this idea by writing and applying four redundancy checkers to real code. The errors found often surprised us. 3. Demonstrating that redundancies, even when harmless, strongly correlate with the presence of traditional hard errors. 4. Showing how redundancies give a way to make specifications "fail stop" by detecting dangerous omissions. The main caveat with our approach is that the errors we count might not be errors, since we were examining code we did not write. To counter this, we only diagnosed errors that we were reasonably sure about. We have had close to two years of experience with Linux bugs, so we have reason-able confidence that our false positive rate of bugs that we diagnose, while non-zero, is probably less than 5%. Section 2 through Section 5 present our four checkers. Section 6 correlates the errors they found with traditional hard errors. Section 7 discusses how to check for completeness using redundancies. Section 8 discusses related work. Finally, Section 9 concludes. 2. IDEMPOTENT OPERATIONS System Bugs Minor False Table 1: Bugs found by the idempotent checker in Linux version 2.4.5-ac8. The checker in this section flags idempotent operations where a variable is: (1) assigned to itself by itself (x / x), (3) bitwise or'd with itself (x | x) or (4) bitwise and'd with itself (x & x). The checker is the simplest in the paper (it requires about 10 lines of code in our system). Even so, it found several interesting cases where redundancies signal high-level errors. Four of these were apparent typos in variable assignments. The clearest example was the following code, where the programmer makes a mistake while copying structure sa to structure da: /* 2.4.1/net/appletalk/aarp.c:aarp_rcv */ else { /* We need to make a copy of the entry. */ This is a good example of how redundant errors catch cases that type systems miss. This code - an assignment of a variable to itself - will type check in all languages we know of, yet clearly contains an error. Two of the other errors were caused by integer overflow (or'ing an 8-bit variable by a constant that only had bits set in the upper 16 bits). The final one was caused by an apparently missing conversion routine. The code seemed to have been tested only on a machine where the conversion was unnecessary, which prevented the tester from noticing the missing routine. The minor errors were operations that seemed to follow a nonsensical but consistent coding pattern, such as adding 0 to a variable for typographical symmetry with other non-zero additions. Curiously, each of the three false positives was annotated with a comment explaining why the redundant operation was being done. This gives evidence for our belief that programmers regard redundant operations as somewhat unusual. Macros are the main source of potential false positives. They represent logical actions that may not map to a concrete action. For example, networking code contains many calls of the form used to reorder the bytes in variable x in a canonical "network order" so that a machine receiving the data can unpack it appropriately. However, on machines on which the data is already in network order, the macro will expand to nothing, resulting in code that will simply assign x to itself. To suppress these false positives, we modified the preprocessor to note which lines contain macros - we simply ignore errors on these lines. 3. REDUNDANT ASSIGNMENTS System Bugs False Uninspected Linux 2.4.5-ac8 129 26 1840 Table 2: Bugs found by the redundant assignment checker in Linux version 2.4.5-ac8 and the xgcc system used in this paper. There were 1840 uninspected errors for variables assigned but never used in Linux - we expect a large number of these will be actual errors given the low number of false positives in our inspected results. The checker in this section flags cases where a value assigned to a variable is not subsequently used. The checker tracks the lifetime of variables using a simple global analy- sis. At each assignment it follows the variable forward on all paths. It emits an error message if the variable is read on no path before either exiting scope or being assigned another value. As we show, in many cases such lost values signal real errors, where control flow followed unexpected paths, results that were computed were not returned, etc. The checker finds thousands of redundant assignments in a system the size of Linux. Since it was so eVective, we minimized the chance of false positives by radically restricting the variables it would follow to non-global variables that were not aliased in any way. Most of the checker code deals with diVerentiating the errors into three classes, which it ranks in the following order: 1. Variables assigned values that are not read. Empirically, these errors tend to be the most serious, since they flag unintentionally lost results. 2. Variables assigned a non-constant that is then overwritten without being read. These are also commonly er- rors, but tend to be less severe. False positives in this class tend to come from assigning a return value from a function call to a dummy variable that is ignored. 3. Variables assigned a constant and then reassigned other values without being read. These are frequently due to defensive programming, where the programmer always initializes a variable to some safe value (most com- monly: NULL, 0, 0xffffffff, and -1) but does not read it before use. We track the value and emit it when reporting the error so that messages using a common defensive value can be easily suppressed. Suppressing false positives. As with many redundant checkers, macros and defensive programming cause most false positives. To minimize the impact of macros, the checker does not track variables killed or produced by macros. Its main remaining vulnerability are to values assigned and then passed to debugging macros that are turned oV: Typically there are a small number of such macros, which we manually turn back on. We use ranking to minimize the impact of defensive pro- gramming. Redundant operations that can be errors when done within the span of a few lines can be robust programming practice when separated by 20. Thus we rank errors based on (1) the line distance between the assignment and reassignment and (2) the number of conditions on the path. Close errors are most likely; farther errors become more arguably defensive programming. The errors. This checker found more errors than all the other checkers we have written combined. There were two interesting error patterns that showed up as redundant as- signments: (1) variables whose values were (unintentionally) discarded and (2) variables whose values were not used because of surprising control flow (e.g., an unexpected return). Figure 1 shows a representative example of the first pat- tern. Here, if the function signal pending returns true (a signal is pending to the current process), an error code is set and the code breaks out of the enclosing loop. The value in err must be passed back to the calling application so that it will retry the system call. However, the code always returns 0 to the caller, no matter what happens inside the loop. This will lead to an insidious error: the code usually works but, occasionally, it will abort but return a success code, causing the client to assume the operation happened. There were numerous similar errors on the caller side, where the result of a function was assigned to a variable, but then ignored rather than being checked. In both of these cases, the fact that logically the code contains errors is readily flagged by looking for variables assigned but not used. The second class of errors comes from calculations that are aborted by unexpected control flow. Figure 2 gives one ex- ample: here all paths through a loop end in a return, wrongly aborting the loop after a single iteration. This error is caught by the fact that an assignment used to walk down a linked list is never read because the loop iterator that would do so is dead code. Figure 3 gives a variation on the theme of unexpected control flow. Here an if statement has an extraneous statement terminator at its end, making the subsequent return to be always taken. In these cases, a coding mistake caused "dangling assignments" that were not used. This fact allows /* 2.4.1/net/decnet/af_decnet.c:dn_wait_run */ do { if { lost value */ break; if (scp->state != DN_RUN) return 0; Figure 1: Lost return value caught by flagging the redundant assignment to err. /* 2.4.1/net/atm/lec.c:lec_addr_delete: */ entry != NULL; { /* BUG: never reached */ if (.) { lec_arp_remove(priv->lec_arp_tables, return 0; Figure 2: A single-iteration loop caught by flagging the redundant assignment entry#next. The assignment appears to be read in the loop iteration statement (entry = next) but it is dead code, since the loop always exits after a single iteration. The logical result will be that if the entry the loop is trying to delete is not the first one in the list, it will not be deleted. us to flag such bogus structures even when we do not know how control flows in the code. The presence of these errors led us to write the dead-code checker in the next section. Reassigning values is typically harmless, but it does signal fairly confused programmers. For example: /* 2.4.5-ac8/drivers/net/wan/sdla_x25.c: alloc_and_init_skb_buf */ struct sk_buff Where new skb is assigned the value *skb but then immediately reassigned another allocated value. A diVerent case shows a potential confusion about how C's iteration works: /* 2.4.1/drivers/scsi/scsi.c: */ for (; SCpnt; SCpnt = SCnext) { Where the variable SCnext is assigned and then immediately reassigned in the loop. The logic behind this decision remains unclear. The most devious error. A few of the values reassigned before being used were suspicious lost values. One of the worst (and most interesting) was from a commercial system which had the equivalent of the following code: 2.4.5-ac8/fs/ntfs/unistr.c:ntfs_collate_names */ for { if (ic) { if (c1 < upcase_len) if (c2 < upcase_len) /* [META] stray terminator! */ return err_val; if (c1 < c2) return -1; Figure 3: Catastrophic return caught by the redundant assignment to c2. The last conditional is accidentally terminated because of a stray statement terminator (";") at the end of the line, causing the routine to always return err val. /* 2.4.1/net/ipv6/raw.c:rawv6_getsockopt */ switch (optname) { case IPV6_CHECKSUM: if (opt->checksum == else /* BUG: always falls through */ default: return -ENOPROTOOPT; Figure 4: Unintentional switch "fall through" causing the code to always return an error. This maps to the low-level redundancy that the value assigned to val is never used. System Bugs False Linux 2.4.5-ac8 66 26 Table 3: Bugs found by the dead code checker on Linux version 2.4.5-ac8. At first glance this seems like an obvious copy-and-paste error. It turned out that the redundancy flags a much more devious error. The array buf actually pointed to a "memory mapped" region of kernel memory. Unlike normal memory, reads and writes to this memory cause the CPU to issue I/O commands to a hardware device. Thus, the reads are not idempotent, and the two of them in a row rather than just one can cause very diVerent results to happen. However, the above code does have a real (but silent) error - in the variant of C that this code was written, pointers to memory mapped IO must be declared as "volatile." Otherwise the compiler is free to optimize duplicate reads away, especially since in this case there were no pointer stores that could change their values. Dangerously, in the above case buf was declared as a normal pointer rather than a volatile one, allowing the compiler to optimize as it wished. Fortunately the error had not been triggered because the GNU C compiler that was being used had a weak optimizer that conservatively did not optimize expressions that had many levels of indirection. However, the use of a more aggressive compiler or later version gcc could have caused this extremely diYcult to track down bug to surface. 4. DEAD CODE The checker in this section flags dead code. Since programmers generally write code to run it, dead code catches logical errors signaled by false beliefs that an impossible path can execute. The core of the dead code checker is a straightforward mark-and-sweep algorithm. For each routine it (1) marks all blocks reachable from the routine's entry node and (2) traverses all blocks in the routine, flagging any that are not marked. It has three modifications to this basic algorithm. First, it truncates all paths that reach functions that would not return. Examples include "panic," "abort" and "BUG" which are used by Linux to signal a terminal kernel error and reboot the system - code dominated by such calls cannot run. Second, we suppress error messages for dead code caused by constant conditions, such as printf("in foo"); since these frequently signaled code "commented out" by using a false condition. We also annotate error messages when the code they flag is a single statement that contains a break or return. These are commonly a result of defensive pro- gramming. Finally, we suppress dead code caused by macros. Despite its simplicity, dead code analysis found a high number of clearly serious errors. Three of the errors caught by the redundant assignment checker are also caught by the dead code extension: (1) the single iteration loop in Figure 2, (2) the mistaken statement terminator in Figure 3, and (3) the unintentional fall through in Figure 4. Figure 5 gives the most frequent copy-and-paste error. Here the macro "pseterr" returns, but the caller does not realize it. Thus, at all seven call sites that use the macro, there is dead code after the macro that the client intended to have executed. /* 2.4.1/drivers/char/rio/rioparam.c:RIOParam */ if (retval == RIO_FAIL) { rio_spin_unlock_irqrestore(&PortP->portSem, flags); returns */ return RIO_FAIL; Figure 5: Unexpected return: The call pseterr is a macro that returns its argument value as an error. Unfortunately, the programmer does not realize this and inserts subsequent op- erations, which are flagged by our dead code checker. There were many other similar mistaken uses of the same macro. Figure 6 gives another common error - a single-iteration loop that always terminates because it contains an if-else statement that breaks out of the loop on both paths. It is hard to believe that this code was ever tested. Figure 7 gives a variation on this, where one branch of the if statement breaks out of the loop but the other uses C's ``continue'' statement, which skips the rest of the loop body. Thus, none of the code at the end of the body can be executed. /* 2.4.1/drivers/scsi/53c7,8xx.c: return_outstanding_commands */ for (struct NCR53c7x0_cmd *) c->next) { if (c->cmd->SCp.buffer) { printk ("."); break; } else { printk ("Duh? ."); break; /* BUG: cannot be reached */ (struct scatterlist *) list; list if (free) { Figure Broken loop: the first if-else statement of the loop contains a break on both paths, causing the loop to always abort, without ever executing the subsequent code it contains. 5. REDUNDANT CONDITIONALS The checker in this section flags redundant branch conditionals branch statements (if, while, for, etc) with non-constant conditionals that always evaluate to either /* 2.4.5-ac8/net/decnet/dn_table.c: dn_fib_table_lookup */ { if (!dn_key_leq(k, f->fn_key)) break; else /* BUG: cannot be reached */ f->fn_state |= DN_S_ACCESSED; if (f->fn_state&DN_S_ZOMBIE) if (f->fn_scope < key->scope) Figure 7: Useless loop body: similarly to Figure 6 this loop has a broken if-else statement. One branch aborts the loop, the other uses C's continue statement to skip the body and begin another iteration. /* 2.4.1/drivers/net/arcnet/arc-rimi.c: arcrimi_found */ /* reserve the irq */ { if (request_irq(dev->irq, &arcnet_interrupt .)) BUGMSG(D_NORMAL, "Can't get IRQ %d!\n", dev->irq); return -ENODEV; Figure 8: Unexpected return: misplaced braces from the insertion of a debugging statement causes control to always return. true or false; (2) switch statements with impossible case's. Both cases are a result of logical inconsistency in the program and are therefore likely to be errors. The checker is based on the false-path pruning (FPP) feature in the xgcc system. FPP was originally designed to prune away false positives arising from infeasible paths. It symbolically evaluates variable assignments and comparisons, either to constants (e.g. or to other variables (e.g. using a simple congruence closure algorithm [11]. It will stop the checker from checking the current execution path as soon as it detects a logical conflict. With FPP, the checker is implemented using a simple mark-and-sweep algorithm. For each routine, it explores all feasible execution paths and marks branches (as opposed to basic blocks in Section visited along the way. Then it takes the set of unmarked branches and flags conditionals associated with them as redundant. The checker was able to find hundreds of redundant conditionals in Linux 2.4.1. The main source of false positives arises from the following two forms of macros: (1) those with embedded conditionals, and (2) constant macros that are used in conditional statements (e.g. "if (DEBUG) {.}," where DEBUG is defined to be 0). After suppressing those, we are left with three major classes of about 200 problematic cases, which we describe below. The first class of errors is the least serious of the three that we characterize as "overly cautious programming style." This includes cases where the programmer checks the same condition multiple times within very short program distances. We believe this could be an indication of a novice programmer and the conjecture is supported by the statistical analysis described in section 6. Figure 9 shows a redundant check of the above type from Linux 2.4.1. Although it is almost certainly harmless, it shows the programmer has a poor grasp of the code. One might be willing to bet on the presence of a few surrounding bugs. /* 2.4.1/drivers/media/video/cpia.c:cpia_mmap */ if (!cam || !cam->ops) return -ENODEV; /* make this _really_ smp-safe */ if (down_interruptible(&cam->busy_lock)) return -EINTR; if (!cam || !cam->ops) /* REDUNDANT! */ return -ENODEV; Figure 9: Overly cautious programming style: the second check of (!cam || !cam->ops) is redundant. Figure shows a more problematic case. As one can see, the else branch of the second if statement will never be taken, because the first if condition is weaker than the negation of the second. Interestingly, the function returns diVerent error codes for essentially the same error, indicating a possibly confused programmer. /* 2.4.1/drivers/net/wan/sbni.c:sbni_ioctl */ if(!(slave && slave->flags & IFF_UP && dev->flags & IFF_UP)) { . /* print some error message, back out */ return -EINVAL; if (slave) { . } /* BUG: !slave is impossible */ else { . /* print some error message */ return -ENOENT; Figure 10: Overly cautious programming style. The check of slave is guaranteed to be true and also notice the diVerence in return value. The second class of errors we catch are again seemingly harmless, but when we examine them carefully, we find serious errors around them. With some guesswork and cross- referencing, we assume the while loop in Figure 11 is trying to recover from hardware errors encountered when reading a network packet. But since the variable err is never up-dated in the loop body, the condition (err != SUCCESS) is always true and the loop body is never executed more than once, which is nonsensical. This could signal a possible bug where the author forgets to update err in the large chunk of recovery code in the loop. This bug, if confirmed, could be diYcult to detect dynamically, because it is in the error recovery code that is easy to miss in testing. The third class of errors are clearly serious bugs. Figure 12 shows an example detected by the redundant condi- /* 2.4.1/drivers/net/tokenring/smctr.c: smctr_rx_frame */ { . /* large chunk of apparent recovery code, with no updates to err */ if (err != SUCCESS) break; Figure Redundant conditional that suggests a serious program error. tional checker. As one can see, the second and third if statements carry out entirely diVerent actions on identical condi- tions. Apparently, the programmer cut-and-pasted the conditional without changing one of the two NODE LOGGED OUT into a fourth possibility: NODE NOT PRESENT. /* 2.4.1/drivers/fc/iph5526.c: rscn_handler */ if ((login_state == NODE_LOGGED_IN) || (login_state == NODE_PROCESS_LOGGED_IN)) { else if (login_state == NODE_LOGGED_OUT) tx_adisc(fi, ELS_ADISC, node_id, else /* BUG: redundant conditional */ if (login_state == NODE_LOGGED_OUT) tx_logi(fi, ELS_PLOGI, node_id); Figure 12: Redundant conditionals that signal errors: a conditional expression being placed in the else branch of another, identical one Figure 13 shows another serious error. One can see that the author intended to insert an element pointed to by sp into a doubly-linked list with head q->q first, but the while loop really does nothing other than setting srb p to NULL, which is nonsensical. The checker detects this error by inferring that the exit condition for the while loop conflicts with the true branch of the ensuing if statement. The obvious fix is to replace the while condition (srb p) with (srb p && srb p->next). This bug can be dangerous and hard to detect, because it quietly discards everything that was in the original list and constructs a new one with sp as the only element in it. As a matter of fact, the same bug is still present in the latest 2.4.19 release of the Linux kernel source as of this writing. 6. PREDICTINGHARDERRORSWITHRE- DUNDANCIES In this section we show the correlation between redundant errors and hard bugs that can crash a system. The redundant errors come from the previous four sections. The hard /* 2.4.1/drivers/scsi/qla1280.c: qla1280_putq_t */ while (srb_p ) if (srb_p) { /* BUG: this branch is never taken*/ if (srb_p->s_prev) else q->q_first } else { q->q_last Figure 13: A serious error in a linked list insertion imple- mentation: srb p is always null after the while loop (which appears to check the wrong Boolean condition). bugs were collected from Linux 2.4.1 with checkers described in [8]. These bugs include use of freed memory, dereferences of null pointers, potential deadlocks, unreleased locks, and security violations (e.g., the use of an untrusted value as an array index). We show that there is a strong correlation between these two error populations using a statistical technique called the contingency table method [6]. Further, we show that a file containing a redundant error is roughly 45% to 100% more likely to have a hard error than a file selected at random. These results indicate that (1) files with redundant errors are good audit candidates and (2) redundancy correlates with confused programmers who will probably make a series of mistakes. 6.1 Methodology This subsection describes the statistical methods used to measure the association between program redundancies and hard errors. Our analysis is based the 2 - 2 contingency table [6] method. It is a standard statistical tool for studying the association between two diVerent attributes of a popula- tion. In our case, the population is the set of files we have checked, and the two attributes are: (a) whether a file contains redundancies, and (b) whether it contains hard errors. In the contingency table approach, the sample population is cross-classified into four categories based on two attributes, say A and B, of the population. We obtain counts (o ij ) in each category, and tabularize the result as follows: A True False Totals True False Totals The values in the margin (n 1- , n 2- , n are row and column totals, while n - is the grand total. The null hypothesis H 0 of this test is that the A and B are mutually independent, i.e. knowing A does not give us any additional information about B. More precisely, if H 0 holds, we are expecting that: . 1 . We can then compute expected values for the four cells in the table as follows: We use a "chi-squared" test statistic [15]: to measure how far the observed values (o ij ) deviates from the expected values (e ij ). Using the T statistic, we can derive the the probability of observing the null hypothesis H 0 is true, which is called the p-value 2 . The smaller the p-value, the stronger the evidence against H 0 , thus the stronger the correlation between attributes A and B. 6.2 Data acquisition and test results In our previous work [8], we used the xgcc system to check 2055 files in Linux 2.4.1 kernel. We had focused on serious system crashing hard bug and were able to collect more than 1800 serious hard bugs in 551 files. The types of bugs we checked for included null pointer dereference, deadlocks, and missed security checks. We use these bugs to represent the class of serious hard errors, and derive correlation with program redundancies. We cross-classify the program files in the Linux kernel into the following four categories and obtain counts in 1. number of files with both redundancies and hard errors. 2. number of files with redundancies but not hard errors. 3. number of files with hard errors but not redundancies 4. number of files with neither redundancies nor hard errors. We can then carry out the test described in section 6.1 for the following three redundancy checkers: redundant assignment checker, dead code checker, and redundant conditional checker (the idempotent operation is excluded because of its small sample size). The result of the tests are given in Tables 4, 5, 6, and 7. As we can see, the correlation between redundancies and hard To see this is true, consider 100 white balls in an urn. We first randomly draw 40 of them and put a red mark on them. We put them back in the urn. Then we randomly draw of them and put a blue mark on them. Obviously, we should expect roughly 80% of the 40 balls with red marks to have blue marks, as should we expect roughly 80% of the remaining the red mark to have a blue mark. 2 Technically, under H 0 , T has a # 2 distribution with one degree of freedom. p-value can be looked up in the cumulative distribution table of the # 2 1 distribution. For example, if T is larger than 4, the p-value will go below 5%. Redundant Hard Bugs Assignments Totals 551 1504 2055 Table 4: Contingency table: Redundant Assignments vs. Hard Bugs. There are 345 files with both error types, 435 files with an assign error and no hard bugs, 206 files with a hard bug and no assignment error, and 1069 files with no bugs of either type. A T-statistic value above four gives a p-value of less than .05, which strongly suggests the two events are not independent. The observed T value of 194.37 gives a p-value of essentially 0, noticeably better than this standard threshold. Intuitively, the correlation between error types can be seen in that the ratio of 345/435 is considerably larger than the ratio if the events were independent, we expect these two ratios to be close. Hard Bugs Dead Code Yes No Totals Totals 551 1504 2055 Table 5: Contingency table: Dead code vs. Hard Bugs errors are extremely high, with p-values being approximately 0 in all four cases. It strongly suggests that redundancies often signal confused programmers, and therefore are a good predictor for hard, serious errors. 6.3 Predicting hard errors In addition to correlation, we want to know how much more likely it is that we will find a hard error in a file that has one or more redundant operations. More precisely, let E be the event that a given source file contains one or more hard errors, and R be the event that it contains one or more forms of redundant operations, we can compute a confidence interval for T which is a measure of how much more likely we are to find hard errors in a file given program redundancies. The prior probability of hard errors is computed as follows Number of files with hard errors Total number of files checked We tabularize the conditional probabilities and T # values in Table 8. (Again, we excluded the idempotent operation checker because of its small bug sample.) As shown in ta- ble, given any form of redundant operation, it is roughly more likely we will find an error in that file than otherwise. Furthermore, redundancies even predict hard errors across time: we carried out the same test between re- Redundant Hard Bugs Totals 551 1504 2055 Table Contingency table: Redundant Conditionals vs. Hard Bugs Hard Bugs Aggregate Totals 551 1504 2055 Table 7: Contingency table: Program Redundancies (Aggre- gate) vs. Hard Bugs dundancies found in Linux 2.4.5-ac8 and hard errors in 2.4.1 (roughly a year older) and found similar results. 7. FAIL-STOP SPECIFICATION This section describes how to use redundant code actions to find several types of specification errors and omissions. Often program specifications give extra information that allow code to be checked: whether return values of routines must be checked against null, which shared variables are protected by which locks, which permission checks guard which sensitive operations, etc. A vulnerability of this approach is that if a code feature is not annotated or included in the specification, it will not be checked. We can catch such omissions by flagging redundant operations. In the above cases, and in many others, at least one of the specified actions makes little sense in isolation - critical sections without shared states are pointless, as are permission checks that do not guard known sensitive actions. Thus, if code does not intend to do useless operations, then such redundancies will happen exactly when checkable actions have been missed. (At the very least we will have caught something pointless that should be deleted.) We sketch four examples below, and close with a checker that uses redundancy to find when it is missing checkable actions. Detecting omitted null annotations. Tools such as LCLint [12] let programmers annotate functions that can return a null pointer with a "null" annotation. The tool emits an error for any unchecked use of a pointer returned from a null routine. In a real system, many functions can return making it easy to forget to annotate them all. We can catch such omissions using redundancy. We know only the return value of null functions should be checked. Thus, a check on a non-annotated function means that either the function: (1) should be annotated with null or (2) the function cannot return null and the programmer has misunderstood the interface. Finding missed lock-variable bindings. Data race detection tools such as Warlock [20] let users explicitly bind locks Confidence Interval for T # Assign 353 889 0.3971 0.1289 0.0191 48.11% - 13.95% Dead Code Conditionals Aggregate 372 945 0.3937 0.1255 0.0187 46.83% - 13.65% Table 8: Program files with redundancies are roughly 50% more likely to contain hard errors to the variables they protect. The tool flags when annotated variables are accessed without their lock held. However, lock- variable bindings can easily be forgotten, causing the variable to be (silently) unchecked. We can use redundancy to catch such mistakes. Critical sections must protect some shared state: flagging those that do not will find either (1) useless locking (which should be deleted for good performance) or (2) places where a shared variable was not annotated. Missed "volatile" annotations. As described in Section 4, in C, variables with unusual read/write semantics must be annotated with the "volatile" type qualifier to prevent the compiler from doing optimizations that are safe on normal variables, but incorrect on volatile ones, such as eliminating duplicate reads or writes. A missing volatile annotation is a silent error, in that the software will usually work, but only occasionally give incorrect errors. As shown, such omissions can be detected by flagging redundant operations (reads or writes) that do not make sense for non-volatile variables. Missed permission checks. A secure system must guard sensitive operations (such as modifying a file or killing a pro- cess) with permission checks. A tool can automatically catch such mistakes given a specification of which checks protect which operations. The large number of sensitive operations makes it easy to forget a binding. As before, we can use redundancy to find such omissions: assuming programmers do not do redundant permission checks, then finding permission check that does not guard a known sensitive operation signals an incomplete specification. 7.1 Case study: Finding missed security holes In a separate paper [3] we describe a checker that found operating system security holes caused when an integer read from untrusted sources (network packets, system call param- eters) was passed to a trusting sink (array indices, memory copy lengths) without being checked against a safe upper and lower bound. A single violation can let a malicious attacker take control of the entire system. Unfortunately, the checker is vulnerable to omissions. An omitted source means the checker will not track the data produced. An omitted sink means the checker will not flag when unsanitized data reaches the sink. When implementing the checker we used the ideas in this section to detect such omissions. Given a list of known sources and sinks, the normal checking sequence is: (1) the code reads data from an unsafe source, (2) checks it, and (3) passes it to a trusting sink. Assuming programmers do not do gratuitous sanitization, then a missed sink can be detected by flagging when code does steps (1) and (2), but not (3). Reading a value from a known source and sanitizing it implies the code believes the value will reach a dangerous operation. If the value does not reach a known sink, we have likely missed one. Similarly, we could (but did not) infer missed sources by doing the converse of this analysis: flagging when the OS sanitizes data we do not think is tainted and then passes it to a trusting sink. The analysis found roughly 10 common uses of sanitized inputs in Linux 2.4.6 [3]. Nine of these uses were harmless; however one was a security hole. Unexpectedly, this was not from a specification omission. Rather, the sink was known, but our inter-procedural analysis had been overly simplistic, causing us to miss the path to it. The fact that redundancy flags errors both in the specification and in the tool itself was a nice surprise. 8. RELATED WORK Two existing types of analysis have focused on redundant operations: optimizing compilers and "anomoly detection" work. Optimizing compilers commonly do dead-code elimination and common-subexpression elimination [1] which remove redundancies to improve performance. One contribution of our work is the realization that these analyses have been silently finding errors since their invention. While our analyses are closely mirror these algorithms at their core, they have several refinements. First, we operate on a higher-level representation than a typical optimizer since a large number of redundant operations are introduced due to the compilation of source constructs to the intermediate representation. Second, in order to preserve semantics of the program, compiler optimizers have to be conservative in its analysis. In contrast, since our goal is to find possible errors, it is perfectly reasonable to flag a redundancy even if we are only 95% sure about its legitimacy. In fact, we report all suspicious cases and sort in order of a confidence heuristic (e.g. distance between redundancies, etc) in the report. Finally, the analysis tradeoVs we make diVer. For example, we use a path-sensitive algorithm to suppress false paths; most optimizers omit path- sensitive analyses because their time complexity outweighs their benefit. The second type of redundant analysis includes checking tools. Fosdick and Osterweil first applied data flow "anomaly detection" techniques in the context of software reliability. In their DAVE system [18], they used a depth first search algorithm to detect a fixed set of variable def-use type of anomalies such as uninitialized read, double definition, etc. Static approaches like this [13, 14, 18] are often path-insensitive, and therefore could report bogus errors from infeasible paths. Dynamic techniques [17, 7] instruments the program and detect anomalies that arise during execution. However, dynamic approaches are weaker in that they can only find errors on executed paths. Further the run-time overhead and diY- culty in instrumenting operating systems limits the usage of this approach. The dynamic system most similar to our work is Huang [17]. He discusses a checker similar to the assignment checker in Section 3. It tracks the lifetime of variables using a simple global analysis. At each assignment it follows the variable forward on all paths. It gives an error if the variable is read on no path before either exiting scope or being assigned another value. However, no experimental results were given. Further, because it is dynamic it seems predisposed to report large numbers of false positives in the case where a value is not read on the current executed path but would be used on some other (non-executed) path. Other tools such as lint, LCLint [12], or the GNU C compiler's -Wall option warn about unused variables and routines and ignored return values. While these have long found redundancies in real code (we use them ourselves daily), these redundancies have been commonly viewed as harmless stylistic issues. Evidence for this perception is that to the best of our knowledge the many recent error checking projects focus solely on hard errors such as null pointer dereferences or failed lock releases, rather than redundancy checking [4, 10, 5, 9, 2, 19, 21]. A main contribution of this paper is showing that redundancies signal real errors and experimentally measuring how well this holds. 9. CONCLUSION This paper explored the hypothesis that redundancies, like type errors, flag higher-level correctness mistakes. We evaluated the approach using four checkers which we applied to the Linux operating system. These simple analyses found many surprising (to us) error types. Further, they correlated well with known hard errors: redundancies seemed to flag confused or poor programmers who were prone to other error types. These indicators could be used to decide where to audit a system. 10. ACKNOWLEDGEMENT We would like to thank the anonymous reviewers for their helpful comments. This work was supported by NFS award 0086160 and by DARPA contract MDA904-98-C-A933. 11. --R Detecting races in relay ladder logic programs. Using programmer-written compiler extensions to catch security holes Automatically validating temporal safety properties of interfaces. A static analyzer for finding dynamic programming errors. Statistical Inference. An empirical study of operating systems errors. Enforcing high-level protocols in low-level software An overview of the extended static checking system. Variations on the common subexpression problem. A tool for using specifications to check code. An algebra for data flow anomaly detection. Data flow analysis in software reliability. A system and language for building system-specific Detection of data flow anomaly through program instrumentation. A dynamic data race detector for multithreaded programming. A first step towards automated detection of buVer overrun vulnerabilities. --TR Compilers: principles, techniques, and tools AIDAMYAMPERSANDmdash;a dynamic data flow anomaly detection system for Pascal programs LCLint Variations on the Common Subexpression Problem A static analyzer for finding dynamic programming errors Data Flow Analysis in Software Reliability Enforcing high-level protocols in low-level software Automatically validating temporal safety properties of interfaces An empirical study of operating systems errors A system and language for building system-specific, static analyses Detecting Races in Relay Ladder Logic Programs An algebra for data flow anomaly detection Using Programmer-Written Compiler Extensions to Catch Security Holes --CTR David Hovemeyer , Jaime Spacco , William Pugh, Evaluating and tuning a static analysis to find null pointer bugs, ACM SIGSOFT Software Engineering Notes, v.31 n.1, January 2006 Zhang , Neelam Gupta , Rajiv Gupta, Locating faults through automated predicate switching, Proceeding of the 28th international conference on Software engineering, May 20-28, 2006, Shanghai, China Zhang , Neelam Gupta , Rajiv Gupta, Pruning dynamic slices with confidence, ACM SIGPLAN Notices, v.41 n.6, June 2006 Neelam Gupta , Haifeng He , Xiangyu Zhang , Rajiv Gupta, Locating faulty code using failure-inducing chops, Proceedings of the 20th IEEE/ACM international Conference on Automated software engineering, November 07-11, 2005, Long Beach, CA, USA Yuriy Brun , Michael D. Ernst, Finding Latent Code Errors via Machine Learning over Program Executions, Proceedings of the 26th International Conference on Software Engineering, p.480-490, May 23-28, 2004 David Hovemeyer , William Pugh, Finding bugs is easy, ACM SIGPLAN Notices, v.39 n.12, December 2004 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

error detection;extensible compilation

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Model exploration with temporal logic query checking.

A temporal logic query is a temporal logic formula with placeholders. Given a model, a solution to a query is a set of assignments of propositional formulas to placeholders, such that replacing the placeholders with any of these assignments results in a temporal logic formula that holds in the model. Query checking, first introduced by William Chan \citechan00, is an automated technique for finding solutions to temporal logic queries. It allows discovery of the temporal properties of the system and as such may be a useful tool for model exploration and reverse engineering.This paper describes an implementation of a temporal logic query checker. It then suggests some applications of this tool, ranging from invariant computation to test case generation, and illustrates them using a Cruise Control System.

INTRODUCTION Temporal logic model-checking [7] allows us to decide whether a property stated in a temporal logic such as CTL [6] holds in a state-based model. Typical temporal logic formulas are AG(p # "both p and q hold in every state of the system", or AG(p # "every state in which p holds is always followed by a state in which q holds". Model checking was originally proposed as a verification tech- however, it is also extremely valuable for model understand- 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. SIGSOFT 2002/FSE-10, November 18-22, 2002, Charleston, SC, USA. r r Figure 1: A simple state machine. ing [2]. We rarely start the study of a design with a complete specification available. Instead, we begin with some key properties, and attempt to use the model-checker to validate them. When the properties do not hold, and they seldom do, what is at fault: the properties or the design? Typically, both need to be modified: the design if a bug was found, and the properties if they were too strong or incorrectly expressed. Thus, this process is aimed not only at building the correct model of the system, but also at discovering which properties it should have. Query checking was proposed by Chan [2] to speed up design understanding by discovering properties not known a priori. A temporal logic query is an expression containing a symbol ?x , referred to as the placeholder, which may be replaced by any propositional formula 1 to yield a CTL formula, e.g. AG?x , AG(?x # p). The solution to a query is a set of strongest propositional formulas that make the query true. For example, consider evaluating the query AG?x , i.e., "what are the invariants of the system", on a model in Figure 1. (p # q) # r is the strongest invariant: all others, e.g., p# q or r, are implied by it. Thus, it is the solution to this query. In turn, if we are interested in finding the strongest property that holds in all states following those in which -q holds, we form the query for the model in Figure 1, evaluates to In solving queries, we usually want to restrict the atomic propositions that are present in the answer. For example, we may not care about the value of r in the invariant computed for the model in Figure 1. We phrase our question as AG(?x{p, q}), thus explicitly restricting the propositions of interest to p and q. The answer we get is p # q. Given a fixed set of n atomic propositions of interest, the query checking problem defined above can be solved by taking propositional formulas over this set, substituting them for the placeholder, verifying the resulting temporal logic formula, tab- propositional formula is a formula built only from atomic propositions and boolean operators. ulating the results and then returning the strongest solution(s) [1]. The number n of propositions of interest provides a way to control the complexity of query checking in practice, both in terms of computation, and in terms of understanding the resulting answer. In his paper [2], Chan proposed a number of applications for query checking, mostly aimed at giving more feedback to the user during model checking, by providing a partial explanation when the property holds and diagnostic information when it does not. For example, instead of checking the invariant AG(a#b), we can evaluate the query AG?x{a, b}. Suppose the answer is a # b, that is, holds in the model. We can therefore inform the user of a stronger property and explain that a # b is invariant because a # b is. We can also use query checking to gather diagnostic information when a does not hold. For example, if is false, that is, a request is not always followed by an acknowledgment, we can ask what can guarantee an acknowledgment: AG(?x # AF ack). In his work, Chan concentrated on valid queries, that is, queries that always yield a single strongest solution. All of the queries we mentioned so far are valid. Chan showed that in general it is expensive to determine whether a CTL query is valid. Instead, he identified a syntactic class of CTL queries such that every formula in the class is valid. He also implemented a query-checker for this class of queries on top of the symbolic CTL model-checker SMV. Queries may also have multiple strongest solutions. Suppose we are interested in exploring successors of the initial state of the model in Figure 1. Forming a query EX?x , i.e., "what holds in any of the next states, starting from the initial state s0 ?", we get two incomparable solutions: p # q # r and -p # q # r. Thus, we know that state s0 has at least two successors, with different values of p in them. Furthermore, in all of the successors, q # r holds. Clearly, such queries might be useful for model exploration. Checking queries with multiple solutions can be done using the method of Bruns and Godefroid [1]. They extend Chan's work by showing that the query checking problem with a single placeholder can be solved using alternating automata [17]. In fact, the queries can be specified in temporal logics other than CTL. However, so far this solution remains purely theoretical: no implementation of such a query-checker is available. The range of applications of query checking can be expanded further if we do not limit queries to just one placeholder. In partic- ular, queries with two placeholders allow us to ask questions about pairs of states, e.g., dependencies between a current and a next state in the system. This paper describes three major contributions: 1. We enrich the language of queries to include several place- holders. The previous methods only dealt with one place- holder, referring to it as "?". In our framework, placeholders need to be named, e.g. "? x ", "? y ", "? pre ". 2. We describe the temporal logic query checking tool which we built on top of our existing multi-valued model-checker # Chek [4, 5]. The implementation not only allows one to compute solutions to the placeholders but also gives witnesses paths through the model that explain why solutions are as computed. 3. We outline a few uses of the temporal logic query checking, both in domains not requiring witness computation and in those that depend on it. The rest of this paper is organized as follows: in Section 2, we give the necessary background for this paper, briefly summarizing model-checking, query checking, and multi-valued CTL model-checking. Section 3 defines the reduction of the query checking problem to multi-valued model-checking. Section 4 describes some possible uses of query checking for model exploration. We illustrate these on an example of the Cruise Control System [16]. This section can be read without the material in Sections 2 and 3. We conclude in Section 5 with the summary of the paper and the directions for future work. Proofs of theorems that appear in this paper can be found [11]. 2. BACKGROUND In this section, we briefly outline CTL model-checking, describe the query checking problem, and give an overview of multi-valued model-checking. 2.1 Model-Checking model-checking [6] is an automatic technique for verifying properties expressed in a propositional branching-time temporal logic called Computation Tree Logic (CTL). The system is represented by a Kripke structure, and properties are evaluated on a tree of infinite computations produced by unrolling it. A Kripke structure is a tuple (S, s0 , A, R, I) where: . S is a finite set of states, and s0 is the initial state; . A is a set of propositional variables; . R # S - S is the (total) transition relation; . I : S # 2 A is a labeling function that maps each state onto the set of propositional variables which hold in it. CTL is defined as follows: 1. Constants true and false are CTL formulas. 2. Every atomic proposition a # A is a CTL formula. 3. If # and are CTL formulas, then so are -# AG#. The boolean operators -, # and # have the usual meaning. The temporal operators have two components: A and E quantify over paths, while X , F , U and G indicate "next state", "eventually (fu- ture)", "until", and "always (globally)", respectively. Hence, AX# is true in state s if # is true in the next state on all paths from s. E[#U ] is true in state s if there exists a path from s on which # is true at every step until becomes true. The formal semantics of CTL is given in Figure 2. In this figure, we use a function {true, false} to indicate the result of checking a formula # in state s. We further define the set of successors for a state s: The more familiar notation for indicating that a property # holds in a state s of a Kripke structure K (K, s |= #) can be defined as follows: We also say that a formula # holds in a Kripke structure K if # holds in K's initial state. In Figure 2, we used conjunction and disjunction in place of the more familiar universal and existential #, for # {true, false} s s Figure 2: Formal semantics of CTL operators. true false {true} {false, p, -p, true} (a) (b) (c) Figure 3: Lattices for set same as (b), represented using minimal elements. quantification. The semantics of EX and AX can be alternatively expressed as Semantics of EG and EU is as follows: Finally, the remaining CTL operators are given below: AX# -EX-# AF# A[true U #] EF# E[true U #] AG# -EF-# For example, consider the model in Figure 1, where s0 is the initial state and A = {p, q, r}. Properties AG(p# q) and AF q are true in this model, whereas AXp is not. 2.2 Query Checking Fundamentals This exposition follows the presentation in [1]. A lattice is a partial order (L, #), where every finite subset B # L has a least upper bound (called "join" and written as #B) and a greatest lower bound (called "meet" and written #B). # and # are the maximal and the minimal elements of a lattice, respectively. A lattice is distributive if join distributes over meet and vice versa. Given a set of atomic propositions P , let PF (P ) be the set of propositional formulas over P . For example, PF false, p, -p}. This set forms a distributive lattice under implication (see Figure 3(a)). Since p # true, p is under true in this lattice. Meets and joins in this lattice correspond to classical # and # op- erations, respectively. propositional formula is a solution to a query # in state s if substituting for the placeholder in # is a formula that holds in the Kripke structure K in state s. A query # is positive [2] if when 1 is a solution to # and 1 # 2 , then 2 is also a solution. For example, if p # q is a solution to #, then so is p. In other words, the set of all solutions to a positive query is a set of propositional formulas that is upward closed with respect to the implication ordering: if some propositional formula is a solution, so is every weaker formula. Alternatively, a query is positive if and only if all placeholders in it occur under an even number of negations [2]. Further, for positive queries it makes sense to look just for the strongest solutions because all other solutions can be inferred from them. These notions are formalized below. Given the ordered set (L, #) and a subset B # L, we define For example, for the ordered set (PF ({p}), #) shown in Figure subset B of L is an upset if -p} is not an upset whereas {p, -p, true} is. We write U(L, #) for the set of all upsets of L. The distributive lattice formed by elements of U(PF ({p}), #) ordered by set inclusion is shown in Figure 3(b). We refer to this as an upset lattice. Finally, we note that each upset can be uniquely represented by the set of its minimal elements. For example, for is sufficient to represent the set {p, -p, true}, and {false} is sufficient to represent {p, -p, true, false}. Figure 3(c) shows the lattice using minimal elements. In the remainder of this paper, when we say that X is a solution to a query, we mean that X # PF (P ), X is the set of minimal solutions to this query, and #X is the set of all of its solutions. 2.3 Multi-Valued Model-Checking Multi-Valued CTL model-checking [5] is a generalization of the model-checking problem. Let B refer to the classical algebra with values true and false ("classical logic"). Instead of using B , multi-valued model-checking is defined over any De Morgan algebra #) is a finite distributive lattice and - is any operation that preserves involution (-#) and De Morgan laws. Conjunction and disjunction are defined using meet and join operations of (L, #), respectively. In this algebra, we get -# and -#, but not necessarily the law of non-contradiction (#) or excluded middle (#). Properties are specified in a multiple-valued extension of CTL called has the same syntax as CTL, except that any # L is also a # CTL formula. However, its semantics is somewhat different. We modify the labeling function of the Kripke structure to be I : S - A # L, so that for each atomic proposition a # A, I(s, means that the variable a has value # in state s. Thus, The other operations are defined as their CTL counterparts (see Figure 2), where # and # are interpreted as lattice # and #, re- spectively. In fact, in the rest of this paper we often write "#" and "#" in place of "#" and "#", even if the algebra we use is different from B . The complexity of model-checking a # CTL formula # on a Kripke structure over an algebra O(|S| - h - |#|), where h is the height of the lattice (L, #), provided that meets, joins, and quantification operations take constant time [5]. We have implemented a symbolic model-checker # Chek [4] that receives a Kripke structure K and a # CTL formula # and returns an element of the algebra corresponding to the value of # in K. The exact interpretation of this value depends on the domain. For example, if the algebra is B , # Chek returns true if # holds in K and false if it does not: this is the classical model-checking. For more information about multi-valued model-checking, please consult [4, 5]. 3. TEMPORALLOGICQUERY-CHECKER In this section, we describe the computation of query-checking solutions in detail. We express the query-checking problem for one placeholder in terms of the multiple-valued model-checking frame-work described in Section 2. We then discuss how to deal with queries containing multiple placeholders, and finally what to do in the case of non-positive queries. Recall that multi-valued model-checking is an extension of model-checking to an arbitrary De Morgan algebra. In our case, the algebra is given by the upset lattice of propositional formulas (see Figure 3). In order to reduce query-checking to multi-valued model- checking, we need to translate a given query into a # CTL formula such that the element of the upset lattice corresponding to the value of the # CTL formula is the set of all solutions to the query. 3.1 Intuition Consider two simple examples of temporal logic queries, using the model in Figure 1. First, we ask ?x , meaning "what propositional are true in a state". Solving this query with respect to s0 , we notice that the formula p#-q#r holds in s0 , and all other formulas that hold in s0 are implied by it. Thus, it is the strongest solution, and the set of all solutions is given by #{p # -q # r}. Next, we look at AX?x , which means "what single formula holds in all successor states". To solve this query with respect to the state s0 , we must first identify all successors of s0 , solve the query ?x for each of them, and finally take the intersection of the results. The solution to ?x in the two successors of s0 : s1 and s2 , is #{-p # q # r} and #{p # q # r}, respectively. The intersection of these solutions is #{q # r}; thus, q # r holds in all successors of s0 , and any other solution to AX?x is implied by it. Notice that this computation is corresponds to meet in the upset lattice, this precisely matches the # CTL semantics of AX from Figure 2. Based on this observation, we show how query-checking is reduced to multi-valued model-checking 3.2 Reduction to # CTL The translation is defined top-down. All operators, constants, and propositional variables are translated one-to-one: # is mapped to the disjunction of the translation of # with the translation of #; any variable p is mapped to itself; true is mapped to the constant symbol #; and so forth. For the model in Figure 1, We now show how to translate the placeholder ?x . Consider the computation of ?x{p, q} in state s0 of the model in Figure 1. The solution #{p#-q} to the query is obtained by examining the values of p and q in s0 . We formalize this using the case-statement: case ([[p # case ([[p # and since all of the cases are disjoint, this yields the (syntactic) So, to evaluate ?x{p, q}, we use the fact that [[p # q]](s0 to get To illustrate this idea further, consider a more complex query evaluated in state s0 of Figure 1. The set of all solutions to the subquery EX?x{p, q} is #{p # q, -p # q}, and the set of all solutions to ?x is #{p # -q}. To get the set of all solutions to our query, we intersect the results to get #{p, p #= q}. THEOREM 1. Let T be the above translation from CTL queries into # CTL. Then, for any CTL query # and state s in the model, contains exactly the solutions to # in state s. As stated in Section 2.3, multiple-valued model-checking has time complexity O(|S| - h - |#|), where h is the height of the lattice. Thus, to estimate the complexity of query-checking, we need to compute the height of the upset lattice used in the reduction of query-checking to multi-valued model-checking. If the place-holder is restricted to n atomic propositions {p1 , . , pn}, then, since there are 2 n propositional formulas in n variables, the height of the upset lattice (U(PF ({p1 , . , pn}), #) is 2 2 n +1. The complexity of query-checking is O(|S| Recall that in traditional model-checking, the height of the model-checking lattice is 2, and the complexity is O(|S| - |#|). Thus, solving a query is, in the worst case, 2 2 n times slower than checking an equivalent model-checking property. However, we find that in practice the running time of the query-checker is much better than the worst case (see Section 4.4). 3.3 More Complex Queries More than one placeholder can be required to express some properties of interest. In this section, we give an extension of query- checking which allows for multiple placeholders, where each may depend on a different set of propositional variables. Furthermore, we describe how to solve non-positive queries. 3.3.1 Multiple Placeholders If a query contains multiple placeholders, it is transformed into a CTL formula by substituting a propositional formula for each placeholder. Thus, given a query on n placeholders, with L i being the lattice of propositional formulas for the ith placeholder, the set of all possible substitutions is given by the cross product . We can lift the implication order, pointwise, to the elements of L, thus forming a lattice. For two placeholders, Once again, the set of all solutions to a query is an element of the upset lattice over L. We now show how to translate queries with multiple placeholders to # CTL. Consider the query ?x # (EX?x # AX?y ). Each potential solution to this query is an element of To solve this query, we first find solutions to each subformula and then combine the results. Let B # L1 be the set of all solutions to ?x when viewed as a query with just one placeholder. However, since we have two placeholders, each solution, including the intermediate ones, must be a subset of L. The query ?x does not depend on the placeholder ?y ; therefore, any substitution for ?y (i.e., any element of L2 ) is acceptable. This results in Similarly, the set of all solutions for EX?x is C - L2 , and for AX?y - L1 -D, for some C # L1 and D # L2 . Combining these results, we get Thus, the set of solutions to this query is {(x, y) | x For example, let us compute the solution to the query ?x{p, q}# EX?y{p, q} in state s0 of the model in Figure 1. We know from the example in Section 3.2 with just one placeholder, Further, recall that Each solution to ?x{p, q} #EX?y{p, q} is an element of the lattice In this lattice, Putting these together, yields Thus, this query has two minimal solutions: (p # -q, p # q), and (p # -q, -p # q). 3.3.2 Negation Every query can be converted to its negation-normal form - the representation where negation is applied only to atomic propositions and placeholders. A query is positive if and only if all of its placeholders are non-negated when the query is put into its negation-normal form. Furthermore, we say that an occurrence of a placeholder in a query is negative if it appears negated in the negation-normal form of the query, and positive otherwise. In this section, we describe how non-positive queries can be solved by transforming them into positive form, query-checking, and post-processing the solution. Note that the solution-set for negated placeholders depend on the maximal solutions 2 , rather than the minimal ones. We consider two separate cases: (1) when all occurrences of a placeholder are either negative or positive, and (2) when a given placeholder appears in both negative and positive forms. In case (1), the query is converted to the positive form by removing all of negations that appear in front of a placeholder, and then solved as described in the previous section. Finally, if the ith placeholder occurred in a negative position, the ith formula in the solution is negated to yield the correct result. THEOREM 2. If (#1 , . , #n ) is a solution to a query Q, and query Q # is identical to Q except that the ith placeholder appears negated, then (#1 , . , -# i , . , #n ) is a solution to Q # . An element # of a solution-set X # PF (P ) is maximal if, for all We sketch the proof by giving an example for a query with a single placeholder. Consider the query AG-?x . We obtain a solution- set to AG?x and choose one formula # from it. Since AG# holds in the model, so does AG-#); therefore, -# is in the solution- set for AG-?x . In case (2), if a placeholder ?x appears both in the positive and the negative forms, we first replace each positive occurrence with ?x+ and each negative occurrence with ?x- , and then solve the resulting query. Finally, the set of all solutions to ?x is given by the intersection of solutions to ?x+ and ?x- . The complexity of using multi-valued model-checking for query- checking with multiple placeholders remains determined by the height of the lattice. We show the result for two placeholders: ?x{p1 , . , pn} and ?y{q1 , . , qm}. There are 2 2 n possible solutions to ?x , and to ?y ; therefore, there are 2 2 possible simultaneous solutions. The height of the powerset lattice of solutions 1, and so the complexity is O(|S| |#|). This result generalizes easily to any number of placehold- ers. As with the case of a single placeholder, we find that in practice query checking is more feasible than its worst case (see Section 4.4). 4. APPLICATIONS AND EXPERIENCE In this section, we show two different techniques for model exploration using temporal logic queries. The technique presented in Section 4.2 uses only the solutions to the query-checking problem and is essentially an extension of the methodology proposed by Chan in [2]. The technique presented in Section 4.3 is completely new and is based on the fact that in addition to computing the solution to a query, our model-checker can also provide a witness explaining it. The examples in this section are based on our own experience in exploring an SCR specification of a Cruise Control System [16], described in Section 4.1. Please refer to Table 3 for the running time of various queries used in this section. 4.1 The Cruise Control System (CCS) The Cruise Control System (CCS) is responsible for keeping an automobile traveling at a certain speed. The driver accelerates to the desired speed and then presses a button on the steering wheel to activate the cruise control. The cruise control then maintains the car's speed, remaining active until one of the following events occurs: (1) the driver presses the brake pedal (Brake); (2) the driver presses the gas pedal (Accel); (3) the driver turns the cruise control off the engine stops running (Running); (5) the driver turns the ignition off (Ignition); (6) the car's speed becomes uncontrollable (Toofast). If any of the first three events listed above occur, the driver can re-activate the cruise control system at the previously set speed by pressing a "resume" button The SCR method [12] is used to specify event-driven systems. System outputs, called controlled variables, are computed in terms of inputs from the environment, called monitored variables, and the system state. To represent this state, SCR uses the notion of mode- classes - sets of states, called modes, that partition the monitored environment's state space. The system changes its state as the result of events - changes in the monitored variables. For example, an event @T(a) WHEN b, formalized as -a # b # a # , indicates that a becomes true in the next state while b is true in the current state. We prime variables to refer to their values in the next state. We use the simplified version of CCS [3] which has 10 monitored variables and 4 controlled variables. One of these, Throttle, is described below. The system also has one modeclass CC, described in Table 1. Each row of the mode transition table specifies an event that activates a transition from the mode on the left to the mode on the right. The system starts in mode Off if Ignition is false, and transitions to mode Inactive when Ignition becomes true. Table 2 shows the event table for Throttle. Throttle assumes the value tAccel, indicating that the throttle is in the accelerating position, when (1) the speed becomes too slow while the system is in mode Cruise, as shown in the first row of Table 2; or (2) the system returns to the mode Cruise, indicated by @T(Inmode), and the speed has been determined to be too slow (see the second row of the table). 4.2 Applications of Queries without Witnesses Below we show how temporal logic queries can replace several questions to a CTL model-checker to help express reachability properties and discover system invariants and transition guards. Reachability analysis. A common task during model exploration is finding which states are reachable. For example, in CCS we may want to know whether all of the modes of the modeclass CC are reachable. This can be easily solved by checking a series of EF properties. For example, EF holds if and only if the mode Cruise is reachable. However, queries provide a more concise representation: the solution to the single query EF ?x{CC} corresponds to all of the reachable modes, i.e., those values p i for which EF In our example, the solutions include all of the modes; thus, all modes are reachable. Similarly, finding all possible values of Throttle when the system is in mode Cruise is accomplished by the query EF . More complex analysis can be done by combining EF queries with other CTL operators. For an exam- ple, see the queries in rows 6 and 7 of Table 3. Discovering invariants. Invariants concisely summarize complex relationships between different entities in the model, and are often useful in identifying errors. To discover all invariants, we simply need to solve the query AG?x , with the placeholder restricted to all atomic propositions in the model. Unfortunately, in all but the most trivial models, the solution to this query is too big to be used effectively [2]. However, it is easy to restrict our attention to different parts of the model. For example, the set of invariants of the mode Inactive, with respect to the variables Ignition and Running, is the solution to the query which evaluates to using multiple placeholders, we can find all invariants of each mode using a single query. For example, each solution to the query AG(?x{CC} #?y{Ignition, Running}) corresponds to invariants of each individual mode. In our example, the solution Running indicates that Ignition and Running remain true while the system is the mode Cruise. Moreover, this query can also help the analyst determine which invariants are shared between modes. From the solution Ignition we see that Ignition not only stays true throughout the mode Inactive, but it is also invariant in the modes Cruise and Override. The mode invariants for CCS that we were able to discover using query-checking are equivalent to the invariants discovered by the algorithms in [14, 15]. Notice that the strength of the invariants obtained through query-checking depends on the variables to which the placeholder is restricted. The strongest invariant is obtained by restricting the placeholder to all of the monitored variables of the system. Guard discovery. Finally, we illustrate how queries can be used to discover guards [18]. Suppose we are given a Kripke structure translation of an SCR model, i.e., events that enable transitions between modes are not explicitely represented. We can reverse-engineer the mode transition table by discovering guards in the Kripke structure. Formally, a guard is defined as the weakest propositional formula over current (pre-) and next (post-) states such that the invariant # holds, where # is the guard, and # and # are the pre- and post-conditions, respectively. Notice that since we define the guard to be the weakest solution, the guard does not directly correspond to an SCR event. Later we show that SCR events can be discovered by combining guards with mode invariants. Since guards are defined over pre- and post-states, two placeholders are required to express the query used to discover them, making the guard the weakest solution to the query AG(#?pre # AX(?post #)) We now show how this query is used to discover an event that causes CCS to switch from the mode Cruise to Inactive. In this case, we let furthermore, for practical reasons we restrict the ?pre and ?post placeholders to the set {Toofast, Running, Brake}. After solving this query, we obtain two solutions: Toofast # -Running, ?post = true Toofast Before analyzing the result, we obtain the invariant for the mode Cruise: -Toofast # Running using the invariant discovery technique presented in Section 4.2. We notice that the first solution violates the invariant, making the antecedent of the implication false; however, from the second solu- tion, it follows that AX((-Running # Toofast) # holds, yielding the guard Toofast # . Finally, combining this with the invariant for the mode Cruise, we determine that the mode transition is guarded by two independent events, @F(Running) and @T(Toofast), just as indicated in the mode transition table. 4.3 Applications of Queries with Given an existential CTL formula that holds in the model, a model-checker can produce a trace through the model showing why the formula holds. This trace is called a witness to the formula. Similarly, given an existential query, the query-checker can produce a set of traces, which we also refer to as a witness, showing why each of the minimal solutions satisfies the query. For example, consider the query EX?x{p} for the model in Figure 1. It has two minimal solutions: fore, the witness consists of two traces, one for each solution, as shown in Figure 4. The trace s0 , s2 corresponds to the solution p, and the trace s0 , s1 - to the solution -p. All of the traces comprising a witness to a query start from the Old Mode Event New Mode Off @T(Ignition) Inactive Inactive @F(Ignition) Off Ignition AND Running AND Cruise @F(Ignition) Off Inactive @F(Running) WHEN Ignition Inactive Ignition AND Running AND Ignition AND Running AND Initial Mode: Off WHEN NOT Ignition Table 1: Mode transition table for mode class CC of the cruise control system. Modes Events Cruise @T(Inmode) @T(Inmode) @T(Inmode) @F(Inmode) Throttle Table 2: Event table for the controlled variable Throttle. Figure 4: A witness for EX?x{p}, in the model in Figure 1. initial state, so they can be represented as a tree. In addition, our query-checker labels each branch in the tree with the set of solutions that are illustrated by that branch. In the example in Figure 4, the left branch is labeled with and the right - with -p. The benefit of treating a witness as a tree rather than a set of independent traces is that it becomes possible to prefer certain over others. For example, we may prefer a witness with the longest common prefix, which usually results in minimizing the total number of traces comprising the witness. We now show how witnesses can be used in several software engineering activities. Guided simulation. The easiest way to explore a model is to simulate its behavior by providing inputs and observing the system behavior through outputs. However, it is almost impossible to use simulation to guide the exploration towards a given objective. Any wrong choice in the inputs in the beginning of the simulation can result in the system evolving into an "uninteresting" behavior. For example, let our objective be the exploration of how CCS evolves into its different modes. In this case, we have to guess which set of inputs results in the system evolving into the mode Cruise, and then which set of inputs yields transition into the mode Inactive, etc. Thus, the process of exploring the system using a simulation is usually slow and error prone. An interesting alternative to a simple simulation is guided simu- lation. In a guided simulation setting, the user provides a set of ob- jectives, and then only needs to choose between the different paths through the system in cases where the objective cannot be met by a single path. Moreover, each choice is given together with the set of objectives it satisfies. Query-checking is a natural framework for implementing guided simulations. The objective is given by a query, and the witness serves as the basis for the simulation. For example, suppose we want to devise a set of simulations to illustrate how CCS evolves into all of its modes. We formalize our objective by the query EF ?x{CC} and explore the witness. Moreover, we indicate that we prefer a witnesses with the largest common prefix, which results in a single trace through the system going through modes Off, Inactive, Cruise, and finally Override. This trace corresponds to a simulation given by the sequence of events: @T(Ignition), bOff). Since our objective was achieved by a single trace, the simulation was generated completely automatically, requiring no user input. Test case generation. Although the primary goal of model-checking is to verify a model against temporal properties, it has recently been used to generate test cases [10, 9, 13, 18]. Most of the proposed techniques are based on the fact that in addition to computing expected outputs, a model-checker can produce witnesses (or counter- examples) which can be used to construct test sequences. The properties that are used to force the model-checker to generate desired test sequences are called trap properties [10]. Gargantini and Heitmeyer [10] proposed a method that uses an SCR specification of a system to identify trap properties satisfying a form of branch coverage testing criterion. Their technique uses both mode transition and condition tables to generate test se- Query Time Explanation what are all reachable modes what values of Throttle are reachable in mode Cruise 3 AGEF?x{CC} 0.787s what modes are globally reachable 4 EFEG?x{CC} 0.720s what modes have self-loops Inactive #?x{Ignition, Running}) 0.267s what are the invariants, over Ignition and Running, of mode Inactive 6 AG(?x{CC} #?x{Ignition, Running}) 0.942s find all mode invariants, restricted to Ignition and Running what modes can follow Off 8 EF (? old {CC} # EX?new{CC}) 1.204s what pairs of modes can follow each other 9 EF how do values of Toofast and Inactive change as the system EX(?y{Toofast, Running} # goes between modes Cruise and Inactive Table 3: Summary of queries used in Section 4. Query Time Table 4: Comparison between model-checking and query-checking. quences. Here, we illustrate how our technique is applicable on mode transition tables; other tables can be analyzed similarly. The method in [10] assures a form of branch coverage by satisfying the following two rules: (1) for each mode in the mode transition table, test each event at least once; (2) for each mode, test every case when the mode does not change (no-change) at least once. For example, the two test sequences need to be generated for mode Off, one testing the event @T(Ignition), and the other testing the no-change case. These can be obtained using the following trap properties: Alternatively, the two test sequences can be obtained from a witness to a single query EF . Sim- ilarly, the set of test sequences that cover the full mode transition table is obtained from the witness of the query EF (? old {CC} # EX?new{CC}). Since all of the traces comprising a witness to a query are generated at the same time, it is possible to minimize the number of different test sequences that guarantee the full coverage of the mode transition table. Moreover, whenever an EF query has more then one minimal solution, the query-checker can produce each minimal solution, and, if necessary, a witness for it, as soon as the new solution is found. Therefore, even in the cases when the complexity of the model-checking precludes obtaining the results for all of the trap properties, the query-checker can produce a solution to some of the trap properties as soon as possible. Although the method suggested above generates a set of test sequences that cover every change (and every no-change) in the mode of the system, it does not necessarily cover all of the events. For ex- ample, the change from the mode Cruise to the mode Inactive is guarded by two independent events, @T(Toofast) and @F(Running); however, the witness for our trap query contains only a single trace corresponding to this change, covering just one of the events. We can first identify the events not covered by the test sequences from the witness to the query, and then use the method from [10] to generate additional test sequences for the events not yet covered. Alternatively, if we know the variables comprising the event for a given mode transition, we can remedy the above problem by using an additional query. In our current example, the events causing the change from the mode Cruise to the mode Inactive depend on variables Toofast and Running. To cover these events, we form the query The witness to this query corresponds to two test sequences: one testing the change on the event @T(Toofast) and the other - on the event @F(Running). 4.4 Running Time Theoretical complexity of query-checking in Section 3.2 seems to indicate that query-checking is not feasible for all but very small models. However, our experience (see running times of queries used in this section in Table 3) seems to indicate otherwise. We address this issue in more detail below. Theoretically, solving a query with a single placeholder restricted to two atomic propositions is slower than model-checking an equivalent formula by a factor of . To analyze the difference between the theoretical prediction and the actual running times, we verified several CTL formulas and related queries and summarized the results in Table 4. CTL formulas are checked using # Chek, parametrized for B . The query in the second row is restricted to two atomic propositions required to encode the enumerated type for CC. However, the running time of this query is only double that of the corresponding CTL formula (row 1). A similar picture can be seen by comparing the CTL formula in row 3 with the query in row 4 of the table. Finally, increasing the number of variables that a placeholder depends on, should slow down the analysis significantly. Yet, comparing queries in rows 4 and 5 of the table, we see that the observed slowdown is only three-fold. Although we have not conducted a comprehensive set of experiments to evaluate the running time of our query-checker, we believe that our preliminary findings indicate that query-checking is in fact feasible in practice. 5. CONCLUSION In this section, we summarize the paper and suggest venues for future work. 5.1 Summary and Discussion In this paper, we have extended the temporal logic query-checking of Chan [2] and Bruns and Godefroid [1] to allow for queries with multiple placeholders, and shown the applicability of this extension on a concrete example. We have implemented a query-checker for multiple placeholders using the multi-valued model-checker # Chek. Our implementation allows us not only to generate solutions to temporal logic queries, but also to provide witnesses explaining the answers. Further, our preliminary results show that it is feasible to analyze non-trivial systems using query-checking. Please send e-mail to xchek@cs.toronto.edu for a copy of the tool. Building a query-checker on top of our model-checker has two further advantages. First, we allow query-checking over systems that have fairness assumptions. For example, we can compute invariants of CCS under the assumption that Brake is pressed infinitely often. As far as we know, Chan's system does not implement fairness. Further, the presentation in this paper used CTL as our temporal logic. However, since the underlying framework of # Chek is based on -calculus, we can easily extend our query- checker to handle -calculus queries. We are also convinced that temporal logic query-checking has many applications in addition to the ones we explored here. In par- ticular, we see immediate applications in a variety of test case generation domains and hope that practical query-checking can have the same impact as model-checking for model exploration and analysis Finally, note that query-checking is a special case of multi-valued model-checking. Multi-valued model-checking was originally designed for reasoning about models containing inconsistencies and disagreements [8]. Thus, the reasoning was done over algebras derived from the classical logic, where the # relation in #, -) indicates "more true than or equal to". Query-checking is done over lattices, and algebras over them, that have a different interpretation - sets of propositional formulas. We believe that there might be yet other useful interpretations of algebras, making # Chek the ideal tool for reasoning over them. 5.2 Future Work In this paper, we have only considered queries where the place-holders are restricted to sets of atomic propositions. However, through our experience we found that it is useful to place further restrictions on the placeholders. For example, we may want to restrict the solutions to the query EF ?x{p, q, r} only to those cases in which p and q are not true simultaneously. From the computational point of view, our framework supports it; however, expressing such queries requires an extension to the query language and some methodology to guide its use. We are currently exploring a query language inspired by SQL, in which the above query would be expressed as follows: EF ?x where ?x in PF ({p, q, r}) and not (?x # p # q) In the future, we plan to conduct further case studies to better assess the feasibility of query-checking on realistic systems. We also believe that the existence of an effective methodology is crucial to the success of query-checking in practice. We will use our case studies to guide us in the development of such a methodology. 6. ACKNOWLEDGEMENTS We gratefully acknowledge the financial support provided by NSERC and CITO. We also thank members of the UofT Formal Methods reading group for their suggestions for improving the presentation of this work. 7. --R "Temporal Logic Query-Checking" "Temporal-Logic Queries" Towards Usability of Formal Methods" Multi-Valued Model-Checker" "Model-Checking Over Multi-Valued Logics" "Automatic Verification of Finite-State Concurrent Systems Using Temporal Logic Specifications" Model Checking. "A Framework for Multi-Valued Reasoning over Inconsistent Viewpoints" "Test Generation for Intelligent Networks Using Model Checking" "Using Model Checking to Generate Tests from Requirements Specifications" "Temporal Logic Query Checking through Multi-Valued Model Checking" "Automated Consistency Checking of Requirements Specifications" " Automatic Test Generation from Statecharts Using Model Checking" "Automatic Generation of State Invariants from Requirements Specifications" "An Algorithm for Strengthening State Invariants Generated from Requirements Specifications" "Example NRL/SCR Software Requirements for an Automobile Cruise Control and Monitoring System" "An Automata-Theoretic Approach to Branching-Time Model Checking" "Coverage Based Test-Case Generation using Model Checkers" --TR Automatic verification of finite-state concurrent systems using temporal logic specifications Automated consistency checking of requirements specifications Automatic generation of state invariants from requirements specifications Using model checking to generate tests from requirements specifications Model checking An automata-theoretic approach to branching-time model checking A framework for multi-valued reasoning over inconsistent viewpoints Test Generation for Intelligent Networks Using Model Checking Model-Checking over Multi-valued Logics An Algorithm for Strengthening State Invariants Generated from Requirements Specifications Queries chi-Chek Temporal Logic Query Checking --CTR Steve Easterbrook , Marsha Chechik , Benet Devereux , Arie Gurfinkel , Albert Lai , Victor Petrovykh , Anya Tafliovich , Christopher Thompson-Walsh, Chek: a model checker for multi-valued reasoning, Proceedings of the 25th International Conference on Software Engineering, May 03-10, 2003, Portland, Oregon Dezhuang Zhang , Rance Cleaveland, Efficient temporal-logic query checking for presburger systems, Proceedings of the 20th IEEE/ACM international Conference on Automated software engineering, November 07-11, 2005, Long Beach, CA, USA

CTL;query-checking;multi-valued model-checking

587177

Large-Scale Computation of Pseudospectra Using ARPACK and Eigs.

ARPACK and its {\sc Matlab} counterpart, {\tt eigs}, are software packages that calculate some eigenvalues of a large nonsymmetric matrix by Arnoldi iteration with implicit restarts. We show that at a small additional cost, which diminishes relatively as the matrix dimension increases, good estimates of pseudospectra in addition to eigenvalues can be obtained as a by-product. Thus in large-scale eigenvalue calculations it is feasible to obtain routinely not just eigenvalue approximations, but also information as to whether or not the eigenvalues are likely to be physically significant. Examples are presented for matrices with dimension up to 200,000.

Introduction . The matrices in many eigenvalue problems are too large to allow direct computation of their full spectra, and two of the iterative tools available for computing a part of the spectrum are ARPACK [10, 11] and its Matlab counter- part, eigs. 1 For nonsymmetric matrices, the mathematical basis of these packages is the Arnoldi iteration with implicit restarting [11, 23], which works by compressing the matrix to an "interesting" Hessenberg matrix, one which contains information about the eigenvalues and eigenvectors of interest. For general information on large-scale nonsymmetric matrix eigenvalue iterations, see [2, 21, 29, 31]. For some matrices, nonnormality (nonorthogonality of the eigenvectors) may be physically important [30]. In extreme cases, nonnormality combined with the practical limits of machine precision can lead to di#culties in accurately finding the eigenvalues. Perhaps the more common and more important situation is when the nonnormality is pronounced enough to limit the physical significance of eigenvalues for applications, without rendering them uncomputable. In applications, users need to know if they are in such a situation. The prevailing practice in large-scale eigenvalue calculations is that users get no information of this kind. There is a familiar tool available for learning more about the cases in which nonnormality may be important: pseudospectra. Figure 1 shows some of the pseudospectra of the "Grcar matrix" of dimension 400 [6], the exact spectrum, and converged eigenvalue estimates (Ritz values) returned by a run of ARPACK (seeking the eigenvalues of largest modulus) for this matrix. In the original article [23] that described the algorithmic basis of ARPACK, Sorensen presented some similar plots of di#culties encountered with the Grcar matrix. This is an extreme example where the nonnormality is so pronounced that even with the convergence tolerance set to its lowest possible value, machine epsilon, the eigenvalue estimates are far from the true spectrum. From the Ritz values alone, one might not realize that anything was # Received by the editors June 6, 2000; accepted for publication (in revised form) January 15, 2001; published electronically July 10, 2001. http://www.siam.org/journals/sisc/23-2/37322.html Oxford University Computing Laboratory, Parks Road, Oxford OX1 3QD, UK (TGW@comlab. ox.ac.uk, LNT@comlab.ox.ac.uk). In Matlab version 5, eigs was an M-file adapted from the Fortran ARPACK codes. Starting with Matlab version 6, the eigs command calls the Fortran ARPACK routines themselves. G. WRIGHT AND LLOYD N. TREFETHEN Fig. 1. The #-pseudospectra for of the Grcar matrix (dimension 400), with the actual eigenvalues shown as solid stars and the converged eigenvalue estimates (for the eigenvalues of largest modulus) returned by ARPACK shown as open circles. The ARPACK estimates lie between the 10 -16 and 10 -17 pseudospectral contours. amiss. Once the pseudospectra are plotted too, it is obvious. Computing the pseudospectra of a matrix of dimension N is traditionally an expensive task, requiring an O(N 3 ) singular value decomposition at each point in a grid. For a reasonably fine mesh, this leads to an O(N 3 ) algorithm with the constant of the order of thousands. Recent developments in algorithms for computing pseudospectra have improved the constant [28], and the asymptotic complexity for large sparse matrices [3, 12], but these are still fairly costly techniques. In this paper we show that for large matrices, we can cheaply compute an approximation to the pseudospectra in a region near the interesting eigenvalues. Our method uses the upper Hessenberg matrix constructed after successive iterations of the implicitly restarted Arnoldi al- gorithm, as implemented in ARPACK. Among other things, this means that after performing an eigenvalue computation with ARPACK or eigs, a user can quickly obtain a graphical check to indicate whether the Ritz values returned are likely to be physically meaningful. Our vision is that every ARPACK or eigs user ought to plot pseudospectra estimates routinely after their eigenvalue computations as a cheap "sanity check." Some ideas related to ours have appeared in earlier papers by Nachtigal, Reichel, and Trefethen [17], Ruhe [19], Sorensen [23], Toh [24], and Toh and Trefethen [25]. For example, Sorensen plotted level curves of filter polynomials and observed that they sometimes approximated pseudospectra, and Ruhe showed that pseudospectra could be approximated by a rational Krylov method. What is new here is the explicit development of a method for approximating pseudospectra based on ARPACK. Of course, one could also consider the use of di#erent low-dimensional compressions of a matrix problem such as those constructed by the Jacobi-Davidson algorithm [22]. Preliminary experiments, not reported here, show that this kind of Jacobi-Davidson approximation of pseudospectra can also be e#ective. We start by giving an overview of pseudospectra calculations and the implicitly restarted Arnoldi iteration, followed by the practical details of our implementation along with a discussion of some of the problems we have had to deal with. After this we give some examples of the technique in practice. We also mention our Matlab graphical user interface (GUI), which automates the computation of pseudospectra LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA 593 after the eigenvalues of a matrix have been computed by eigs in Matlab. The computations presented in this paper were all performed using eigs in Matlab version 6 (which essentially is ARPACK) and our GUI rather than the Fortran ARPACK, although our initial experiments were done with the Fortran code. 2. Pseudospectra. There are several equivalent ways of defining # (A), the #-pseudospectrum of a matrix A. The two most important (see, e.g., [27]) are perhaps and E) for some E with #E#}. When the norms are taken to be the 2-norm, the definitions are equivalent to where # min (-) denotes minimum singular value. This provides the basis of many algorithms for computing pseudospectra. The most familiar technique is to use a grid over the region of the complex plane of interest and calculate the minimum singular value of zI - A at each grid point z. These values can then be passed to a contour plotter to draw the level curves. For the rest of this paper, we consider the 2-norm; other norms are discussed in [8, 28]. The reason for the cost of computation of pseudospectra is now clear: the amount of work needed to compute the minimum singular value of a general matrix of dimension N is O(N 3 ) (see, e.g., [5]). However, several techniques have been developed to reduce this cost [28]. Here are two important ones. (I) Project the matrix onto a lower dimensional invariant subspace via, e.g., a partial Schur factorization (Reddy, Schmid, and Henningson [18]). This works well when the interesting portion of the complex plane excludes a large fraction of the eigenvalues of the matrix. In this case, the e#ect of the omitted eigenvalues on the interesting portion of the pseudospectra is typically small, especially if the undesired eigenvalues are well conditioned. Projection can significantly reduce the size of the matrix whose pseudospectra we need to compute, making the singular value computation dramatically faster. In general, the additional cost of projecting the matrix is much less than the cost of repeatedly computing the smallest singular value for the shifted original matrix. Perform a single matrix reduction to Hessenberg or triangular form before doing any singular value decompositions (Lui [12]), allowing the singular value calculations to be done using a more e#cient algorithm. One way of combining these ideas is to do a complete Schur decomposition of the and then to reorder the diagonal entries of the triangular matrix to leave the "wanted" eigenvalues at the top. The reordered factorization can then be truncated leaving the required partial Schur factorization. We can now find the singular values of the matrices shifted for each grid point z using either the original matrix A or the triangular matrix This allows us to work solely with the triangular matrix T once the O(N 3 ) factorization has been completed. The minimum singular value of zI - T can be determined 594 THOMAS G. WRIGHT AND LLOYD N. TREFETHEN in O(N 2 ) operations 2 using the fact that # min (zI - T This can be calculated using either inverse iteration or inverse Lanczos iteration, which require solutions to systems of equations with the matrix (zI - T can be solved in two stages, each using triangular system solves. By combining these techniques with more subtle refinements we have an algorithm which is much more e#cient than the straightforward method. It is suggested in [28] that the speedup obtained is typically a factor of about N/4, assuming the cost of the Schur decomposition is negligible compared with that of the rest of the algorithm. This will be the case on a fine grid for a relatively small matrix (N of the order of a thousand or less), but for larger matrices the Schur decomposition is relatively expensive, and it destroys any sparsity structure. 3. Arnoldi iteration. The Arnoldi iteration for a matrix A of dimension N works by projecting A onto successive Krylov subspaces K j of dimension starting vector v 1 [1, 20]. It builds an orthonormal basis for the Krylov subspace by the Arnoldi factorization is an upper Hessenberg matrix of dimension j, the columns of V j form an orthonormal basis for the Krylov subspace, and f j is orthogonal to the columns of V j . The residual term f j e # j can be incorporated into the first term V j H j by augmenting the matrix H j with an extra row, all zeros except for the last entry, which is #f j #, and including the next basis vector, v #, in the matrix V j . The matrix H j is now rectangular, of size (j The matrix being rectangular, does not have any eigenvalues, but we can define its pseudospectra in terms of singular values by (2.3). (With the definition occasionally used that # is an eigenvalue of a rectangular matrix A if A - #I is rank-deficient, where I is the rectangular matrix of appropriate dimension with 1 on the diagonal and 0 elsewhere. In general, a rectangular matrix will have no eigenvalues, but it will have nonempty #-pseudospectra for large enough #.) It can then be shown [14, 25] that the pseudospectra of successive are nested. Theorem 3.1. Let A be an N - N matrix which is unitarily similar to a Hessenberg matrix H, and let denote the upper left (j section (in particular could be created using a restarted Arnoldi iteration). Then for any # 0, Thus as the iteration progresses, the pseudospectra of the rectangular Hessenberg matrices better approximate those of A, which gives some justification for the approximation Unfortunately, this is only the case for the rectangular matrices There do not appear to be any satisfactory theorems to justify a similar approximation for the square matrices H j , and of course for # su#ciently small the #-pseudospectra of H j must be disjoint from those of A, since they will be small sets surrounding the eigenvalues of H j , which are in general distinct from those of A. This is not the case for the rectangular matrix as there will not be points in the complex plane with infinite resolvent norm unless a Ritz value exactly matches an can also be used for Hessenberg matrices [12], but for those we do not have the advantage of projection. LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA 595 eigenvalue of the original matrix. That is, # ( typically empty for su#ciently small #. Although the property (3.2) is encouraging, theorems guaranteeing rapid convergence in all cases cannot be expected. The quality of the approximate pseudospectra depends on the information in the Krylov subspace, which in turn depends on the starting vector v 1 . Any guarantee of rapid convergence could at best be probabilistic. 3.1. Implicitly restarted Arnoldi. In its basic form, the Arnoldi process may require close to N iterations before the subspace contains good information about the eigenvalues of interest. However, the information contained within the Hessenberg matrix is very dependent on the starting vector relatively small components of the eigenvectors corresponding to the eigenvalues which are not re- quired, convergence may be quicker and the subspace size need not grow large. To avoid the size of the subspace growing too large, practical implementations of the Arnoldi iteration restart when the subspace size j reaches a certain threshold [20]. A new starting vector - v 1 is chosen which has smaller components in the directions of eigenvectors corresponding to unwanted eigenvalues, and the process is begun again. Implicit restarting [11, 23] is based upon the same idea, except that subspace is only implicitly compressed to a single starting vector - v 1 . What is explicitly formed is an Arnoldi factorization of size k based on this new starting vector, where k is the number of desired eigenvalues, and this Arnoldi factorization is obtained by carrying out implicitly shifted steps of the QR algorithm, with shifts possibly corresponding to unwanted eigenvalue estimates. The computation now proceeds in an accordion-like manner, expanding the subspace to its maximum size p, then compressing to a smaller subspace. 3 This is computationally more e#cient than simple restarting because the subspace is already of size k when the iteration restarts, and in addition, the process is numerically stable due to the use of orthogonal transformations in performing the restarting. This technique has made the Arnoldi iteration competitive for finding exterior eigenvalues of a wide range of nonsymmetric matrices. 3.2. Arnoldi for pseudospectra. In a 1996 paper, Toh and Trefethen [25] demonstrated that the Hessenberg matrix created during the Arnoldi process can sometimes provide a good approximation to the pseudospectra of the original matrix. They provided results for both the square matrix H j and the rectangular matrix We choose to build our method around the rectangular Hessenberg matrices though this makes the pseudospectral computation harder than if we worked with the square matrix. The advantage of this is that we retain the properties of Theorem 3.1, and the following in particular: For every # 0, the approximate #-pseudospectrum generated by our ARPACK algorithm is a subset of the #-pseudospectrum of the original matrix, This is completely di#erent from the familiar situation with Ritz values, which are, after all, the points in the 0-pseudospectrum of a square Hessenberg matrix. Ritz values need not be contained in the true spectrum. Simply by adding one more row to consider a rectangular matrix, we have obtained a guaranteed inclusion for every #. The results presented by Toh and Trefethen focus on trying to approximate the full pseudospectra of the matrix (i.e., around the entire spectrum) and they do not use 3 Our use of the variable p follows eigs. In ARPACK and in Sorensen's original paper [23], this would be p 596 THOMAS G. WRIGHT AND LLOYD N. TREFETHEN any kind of restarting in their implementation of the Arnoldi iteration. While this is a useful theoretical idea, we think it is of limited practical value for computing highly accurate pseudospectra since good approximations are often obtained generally only for large subspace dimensions. Our work is more local; we want a good approximation to the pseudospectra in the region around the eigenvalues requested from ARPACK or eigs. By taking advantage of ARPACK's implicit restarting, we keep the size of the subspace (and hence reasonably small, allowing us to compute (local) approximations to the pseudospectra more rapidly, extending the idea of [25] to a fully practical technique (for a more restricted problem). 4. Implementation. In deciding to use the rectangular Hessenberg matrix we have made the post-ARPACK phase of our algorithm more di#cult. While the simple algorithm of computing the minimum singular value of zI -A at each point has approximately the same cost for a rectangular matrix as a square one, the speedup techniques described in section 2 are di#cult to translate into the rectangular case. The first idea, projection to a lower dimensional invariant subspace, does not make sense for rectangular matrices because there is no such thing as an invariant subspace. The second idea, preliminary triangularization using a Schur decompo- sition, also does not extend to rectangular matrices, for although it is possible to triangularize the rectangular matrix while keeping the same singular values (by performing a QR factorization, for example), doing so destroys the vital property of shift-invariance (see (2.4)). However, our particular problem has a feature we have not yet considered: the matrix is Hessenberg. One way to exploit this property is to perform a QR factorization of the matrix obtained after shifting for each grid point. The upper triangular matrix R has the same singular values as the shifted matrix, and they are also unchanged on removing the last row of zeros, which makes the matrix square. We can now use the inverse Lanczos iteration as in section 2 to find its smallest singular value. The QR factorization can be done with an O(N 2 ) algorithm (see, e.g., [5, p. 228]), which makes the overall cost O(N 2 ). Unfortunately, the additional cost of the QR factorization at each stage makes this algorithm slightly slower for the small matrices (dimensions 50-150) output from ARPACK than for square matrices of the same size, but this appears to be the price to be paid for using matrices which have the property of (3.3). 4.1. Refinements. In some cases we have found that inverse iteration to find the minimum eigenvalue of (zI - R) # (zI - R) is more e#cient than inverse Lanczos iteration but only when used with continuation (Lui [12]). Continuation works by using the vector corresponding to the smallest singular value from the previous grid point as the starting guess for the next grid point. This sounds like a good idea; if the two shifted matrices di#er by only a small shift, their singular values (and singular vectors) will be similar. When it works, it generally means that only a single iteration is needed to satisfy the convergence crite- rion. However, as Lui indicates, there is a problem with this approach if the smallest and second smallest singular values "change places" between two values of z: the iteration may converge to the second smallest singular value instead of the smallest, since the starting vector had such a strong component in the direction of the corresponding singular vector. This leads to the convergence criterion being satisfied for the wrong singular value (even after several iterations). LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA 597 Choose max subspace size p-larger p for better pseudospectra. Choose number of eigenvalues k-larger k for better pseudospectra. Run ARPACK(A, p, k) to obtain Define a grid over a region of C enclosing converged Ritz values. For each grid point z: Perform reduced QR factorization of shifted matrix: z - I - # Get # max (z) from Lanczos iteration on (R # R) -1 , random starting vector. end. Start GUI and create contour plot of the # min values. Allow adjustment of parameters (e.g., grid size, contour levels) in GUI. Fig. 2. Pseudocode for our algorithm. In the course of our research, we have found several test matrices which su#er from this problem, including the so-called Tolosa matrix [4]. Accordingly, because of our desire to create a robust algorithm, we do not use inverse iteration. In theory it is also possible to use continuation with inverse Lanczos iteration, but our experiments indicate that the benefit is small and it again brings a risk of misconvergence. Our algorithm (the main loop of which is similar to that in [28]) is summarized in Figure 2. 5. Practical examples. While one aim of our method is to approximate the pseudospectra of the original matrix accurately, this is perhaps no more important than the more basic mission of exhibiting the degree of nonnormality the matrix has, so that the ARPACK or eigs user gets some idea of whether the Ritz values returned are likely to be physically meaningful. Even in cases where the approximations of the sets # (A) are inaccurate, a great deal may still be learned from their qualitative properties. In the following examples, ARPACK was asked to look for the eigenvalues of largest real part except where otherwise indicated. However, the choice of region of the complex plane to focus on is unimportant for our results and is determined by which eigenvalues are of interest for the particular problem at hand. The number of requested eigenvalues k was chosen rather arbitrarily to be large enough so that the approximate pseudospectra clearly indicate the true behavior in the region of the complex plane shown, and the maximum subspace size p was chosen to ensure convergence of ARPACK for the particular choice of k. Experiments show that the computed pseudospectra are not very sensitive to the choices of k and p, provided they are large enough, but we have not attempted to optimize these choices. 5.1. Two extremes. Our first example (Figure 3), from Matrix Market [15], shows a case where the approximation is extremely good. The matrix is the Jacobi matrix of dimension 800 for the reaction-di#usion Brusselator model from chemical engineering [7], and one seeks the rightmost eigenvalues. The matrix is not highly non- normal, and the pseudospectra given by the approximation almost exactly match the G. WRIGHT AND LLOYD N. TREFETHEN Pseudospectra Approximate pseudospectra Fig. 3. Pseudospectra for the matrix rdb800l (left) computed using the standard method, and pseudospectra of the upper Hessenberg matrix of dimension computed using ARPACK (right) in about 9% of the computer time (on the fine grid used here). Levels are shown for . The number of matrix-vector products needed by ARPACK (nv ) is 1,493. true pseudospectra around the converged Ritz values. This is a case where the pseudospectra computed after running ARPACK indicate that the eigenvalues returned are both accurate and physically meaningful, and that no further investigation is nec- essary. In this computation we used a maximum subspace dimension of requested eigenvalues. The second case we consider is one where the matrix has a high degree of non- normality-the Grcar matrix. As seen in Figure 1, ARPACK can converge to Ritz values which are eigenvalues of a perturbation of order machine precision of the original matrix, and the nonnormality of this particular matrix (here of dimension 400) means that the Ritz values found can lie a long way from the spectrum of the matrix. Figure 4 shows that the pseudospectra of the Hessenberg matrix (computed using asking for eigenvalues of largest modulus) in this case are not good approximations to the pseudospectra of the original one. This is typical for highly nonnormal matrices-the Hessenberg matrix cannot capture the full extent of the nonnormality, particularly when more than p eigenvalues of the original matrix lie within the region of the complex plane in which the pseudospectra are computed. In other words, the approximation is typically not so good in areas away from the Ritz values computed, and then only accurately approximates the pseudospectra of the original matrix when the Ritz values are good approximations to the eigenvalues. Despite this, a plot like that of Figure 4 will instantly indicate to the ARPACK user that the matrix at hand is strongly nonnormal and needs further investigation. 5.2. A moderately nonnormal example. While the above examples show two extreme cases, many important applications are more middle-of-the-range, where LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA 599 Pseudospectra ARPACK approximate pseudospectra Fig. 4. The pseudospectra of the Grcar matrix of dimension 400 (left) computed using the standard method, and the pseudospectra of the upper Hessenberg matrix of dimension 50 computed using ARPACK (right) in about 8% of the computer time (on this fine grid). Contours are shown -0.4 -0.3 -0.2 -0.4 -0.3 -0.2 -0.10.10.3Pseudospectra Approximate pseudospectra Fig. 5. Pseudospectra for linearized fluid flow through a circular pipe at Reynolds number 10,000 (streamwise-independent disturbances with azimuthal wave number machine precision is su#cient to accurately converge the eigenvalues, but pronounced nonnormality may nevertheless diminish the physical significance of some of them. A good example of a case in which this is important is the matrix created by linearization about the laminar solution of the Navier-Stokes equations for fluid flow in an infinite circular pipe [26]. (Our matrix is obtained by a Petrov-Galerkin spectral discretization of the Navier-Stokes problem due to Meseguer and Trefethen [16]. The axial and azimuthal wave numbers are 0 and 1, respectively, and the matrix dimension is 402.) The pseudospectra are shown in Figure 5, and although the eigenvalues all have negative real part, implying stability of the flow, the pseudospectra protrude far into the right half-plane. This implies pronounced transient growth of some perturbations of the velocity field in the pipe, which in the presence of nonlinearities in practice may lead to transition to turbulence [30]. The approximate pseudospectra also highlight G. WRIGHT AND LLOYD N. TREFETHEN -226 -0.50.5Brusellator Wave Model Crystal Fig. 6. Left: The pseudospectra for the Brusselator wave model, nv= 16,906. Right: Pseudospectral contours for a matrix of dimension 10,000 from the Crystal set at Matrix Market, this behavior. The parameters used here were 5.3. Larger examples. We now consider four larger examples. The first is the Brusselator wave model from Matrix Market (not to be confused with the very first example), which models the concentration waves for reaction and transport interaction of chemical solutions in a tubular reactor [9]. Stable periodic solutions exist for a parameter when the rightmost eigenvalues of the Jacobian are purely imaginary. For a matrix of dimension 5,000, using a subspace of dimension 100 and asking ARPACK for 20 eigenvalues, we obtained the eigenvalue estimates and approximate pseudospectra shown in Figure 6 (left). The departure from normality is evidently mild, and the conclusion from this computation is that the Ritz values returned by ARPACK are likely to be accurate and the corresponding eigenvalues physically meaningful. Figure 6 (right) shows approximate pseudospectra for a matrix of dimension 10,000, taken from the Crystal set at Matrix Market, which arises in a stability analysis of a crystal growth problem [32]. The eigenvalues of interest are the ones with largest real part. The fact that we can see the 10 -13 pseudospectrum (when the axis scale is O(1)) indicates that this matrix is significantly nonnormal, and although the matrix is too large for us to be able to compute its exact pseudospectra for compar- ison, this is certainly a case where the nonnormality could be important, making all but the rightmost few eigenvalues of dubious physical significance in an application. The ARPACK parameters we used in this case were and the computation took about one hour on our Sun Ultra 5 workstation. Although we do not have the true pseudospectra in this case, we would expect that the rightmost portion should be fairly accurate where there is a good deal of Ritz data and relatively little nonnormality. We expect that the leftmost portion is less accurate where the e#ect of the remaining eigenvalues of the matrix unknown to the approximation begins to become important. The third example, Figure 7 (left), shows the Airfoil matrix created by performing transient stability analysis of a Navier-Stokes solver [13], also from Matrix Market. In this case the matrix appears fairly close to normal, and the picture gives every reason to believe that the eigenvalues have physical meaning. Using ARPACK took about 9 hours to converge to the eigenvalues, while we were able to plot the pseudospectra in about 3 minutes (even on the fine grid used here). Our final example is a matrix which is bidiagonal plus random sparse entries LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA 601 -0.4 Airfoil Random Fig. 7. Left: The pseudospectra for the Airfoil matrix from Matrix Market of dimension 23,560, nv= 72,853. Right: The for our random matrix, with nv= 61,347. elsewhere, created in Matlab by The approximation to the pseudospectra of the matrix of dimension 200,000 is shown in Figure 7 (right), from which we can conclude that the eigenvalue estimates returned are probably accurate, but that the eigenvalues toward the left of the plot would likely be of limited physical significance in a physical application, if there were one, governed by this matrix. We used a subspace size of 50 and requested 30 eigenvalues from this example, and the whole computation took about 26 hours. 6. MATLAB GUI. We have created a Matlab GUI to automate the process of computing pseudospectra, and Figure 8 shows a snapshot after a run of ARPACK. Initially the pseudospectra are computed on a coarse grid to give a fast indication of the nonnormality of the matrix, but the GUI allows control over the number of grid points if a higher quality picture is desired. Other features include the abilities to change the contour levels shown without recomputing the underlying values, and to select parts of the complex plane to zoom in for greater detail. The GUI can also be used as a graphical front end to our other pseudospectra codes for computing pseudospectra of smaller general matrices. The codes are available on the World Wide Web from http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/. 7. Discussion. The examples of section 5 give an indication of the sort of pictures obtained from our technique. For matrices fairly close to normal, the approximation is typically a very close match to the exact pseudospectra, but for more highly nonnormal examples the agreement is not so close. This is mainly due to the e#ect of eigenvalues which the Arnoldi iteration has not found: their e#ect on the pseudospectra is typically more pronounced for nonnormal matrices. The other point to note is that if we use the Arnoldi iteration to look (for ex- G. WRIGHT AND LLOYD N. TREFETHEN Fig. 8. A snapshot of the Matlab GUI after computing the pseudospectra of a matrix. ample) for eigenvalues of largest real part, the rightmost part of the approximate pseudospectra will be a reasonably good approximation. This can clearly be seen in Figure 4 where we are looking for eigenvalues of largest modulus: the top parts of the pseudospectra are fairly good and only deteriorate lower down where the e#ect of the "unfound" eigenvalues becomes important. However, as mentioned in the introduction, creating accurate approximations of pseudospectra was only part of the motivation for this work. Equally important has been the goal of providing information which can help the user of ARPACK or eigs decide whether the computed eigenvalues are physically meaningful. For this purpose, estimating the degree of nonnormality of the matrix is more important than getting an accurate plot of the exact pseudospectra. One of the biggest advantages of our technique is that while the time spent on the computation grows as the dimension of the matrix increases, the time spent on the pseudospectra computation remains roughly constant. This is because the pseudospectra computation is based just on the final Hessenberg matrix, of dimension typically in the low hundreds at most. Figure 9 shows the proportion of time spent on the pseudospectra part of the computation for the examples we have presented here. These timings are based on the time to compute the initial output from our LARGE-SCALE COMPUTATION OF PSEUDOSPECTRA 603 Airfoil Crystal Grcar Pipe Flow R.-D. Brusselator Random Brusellator Wave Model Dimension, N Pseudospectra time Total time Fig. 9. The proportion of the total computation time spent on computing the pseudospectra for the examples presented in this paper. For large N, the pseudospectra are obtained at very little additional cost. R.-D. Brusselator Grcar Fig. 10. Lower-quality plots of pseudospectra produced by our GUI on its default coarse Such plots take just a few seconds. The matrices shown are rdb800l with Figure and the Grcar matrix with Figure 4). GUI using a coarse grid such as those illustrated in Figure 10, which is our standard resolution for day-to-day work. For the "publication quality" pseudospectra of the resolution of the other plots in this paper, the cost is about thirty times higher, but G. WRIGHT AND LLOYD N. TREFETHEN this is still much less than the cost of ARPACK for dimensions N in the thousands. Acknowledgments . We would like to thank Rich Lehoucq for his advice on beginning during his visit to Oxford in October-November 1999, Penny Anderson of The MathWorks, Inc., for her help with the beta version of the new eigs command, and Mark Embree for his comments on drafts of the paper. --R The principle of minimized iteration in the solution of the matrix eigenvalue problem Vorst, eds., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide Computing the field of values and pseudospectra using the Lanczos method with continuation Stability analysis in aeronautical industries Matrix Computations Operator Coe Theory and Applications of Hopf Bifurcation A block algorithm for matrix 1-norm estimation Waves in Distributed Chemical Systems: Experiments and Computations Package, http://www. Computation of pseudospectra by continuation Eigenvalue calculation procedure for an Euler- Navier-Stokes solver with applications to flows over airfoils On hybrid iterative methods for nonsymmetric systems of linear equations Matrix Market A Spectral Petrov-Galerkin Formulation for Pipe Flow I: Linear Stability and Transient Growth A hybrid GMRES algorithm for non-symmetric linear systems Pseudospectra of the Orr-Sommerfeld operator Rational Krylov algorithms for nonsymmetric eigenvalue problems. Variations on Arnoldi's method for computing eigenelements of large unsymmetric matrices Numerical Methods for Large Eigenvalue Problems A Jacobi-Davidson iteration method for linear eigenvalue problems Implicit application of polynomial filters in a k-step Arnoldi method Matrix Approximation Problems and Nonsymmetric Iterative Methods Calculation of pseudospectra by the Arnoldi iteration Spectra and pseudospectra for pipe Poiseuille flow Pseudospectra of matrices Computation of Pseudospectra Hydrodynamic stability without eigenvalues Computational Methods for Large Eigenvalue Problems Numerical Computation of the Linear Stability of the Di --TR --CTR S.-H. Lui, A pseudospectral mapping theorem, Mathematics of Computation, v.72 n.244, p.1841-1854, October Lorenzo Valdettaro , Michel Rieutord , Thierry Braconnier , Valrie Frayss, Convergence and round-off errors in a two-dimensional eigenvalue problem using spectral methods and Arnoldi-Chebyshev algorithm, Journal of Computational and Applied Mathematics, v.205 n.1, p.382-393, August, 2007 Kirk Green , Thomas Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, v.196 n.2, p.567-578, 15 November 2006 C. Bekas , E. Kokiopoulou , E. Gallopoulos, The design of a distributed MATLAB-based environment for computing pseudospectra, Future Generation Computer Systems, v.21 n.6, p.930-941, June 2005 Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization, Applied Numerical Mathematics, v.49 n.1, p.39-61, April 2004

implicit restarting;pseudospectra;ARPACK;arnoldi;eigenvalues

587192

A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations.

Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200. For fixed wave numbers, it displays grid-independent convergence rates and has costs proportional to the number of unknowns.

Introduction . Multigrid algorithms are effective for the numerical solution of many partial differential equations, providing a solution in time proportional to the number of unknowns. For some important classes of problems, however, standard multigrid algorithms have not been useful, and in this paper we focus on developing effective multigrid algorithms for one such class, the discrete Helmholtz equation. Our main interest lies in solving exterior boundary value problems of the form on\Omega ae R d (1.1) @\Omega (1.2) such as arise in the modeling of time-harmonic acoustic or plane-polarized electromagnetic scattering by an obstacle. The boundary \Gamma represents the scattering obstacle, and the boundary operator B can be chosen so that a Dirichlet, Neumann or Robin boundary condition is imposed. The original unbounded domain is truncated to the finite domain\Omega by introducing the artificial boundary \Gamma 1 on which the radiation boundary condition (1.3) approximates the outgoing Sommerfeld radiation condition. Depending on what type of radiation condition is chosen, M can be either a (local) differential operator or a global integral operator coupling all points on \Gamma 1 (see [14]). The data for the problem are given by the right hand side f and the boundary data g. In the most common case, f j 0 and \Gammag is the boundary data of an incident plane wave. The critical parameter is the wave number k, which is positive in the case of unattenuated wave propagation. Due to the radiation boundary condition, the solution of (1.1)-(1.3) is a complex-valued function u Discretization of (1.1)-(1.3) by finite differences or finite elements leads to a linear system of equations in which the coefficient matrix A is complex-symmetric, i.e., not Hermitian. Moreover, for large values of the wave number k, it becomes highly indefinite. It is this indefiniteness that until recently has prevented multigrid methods from being applied to the solution of the discrete equations with the same success as these methods have enjoyed for symmetric positive definite problems. As will be illustrated in Section 2, the difficulties with standard multigrid methods applied to Helmholtz problems concern both of the main multigrid components: smoothing and coarse grid correction. In particular, standard smoothers such as Jacobi or Gau-Seidel relaxation become unstable for indefinite problems since there are always error components- usually the smooth ones-which are amplified by these smoothers. The difficulties with the coarse grid correction are usually attributed to the poor approximation of the Helmholtz operator on very coarse meshes, since such meshes cannot adequately resolve waves with wavelength of which the solution primarily consists. We show, however, that although the coarse grid correction is inaccurate when coarse-grid eigenvalues do not agree well with their fine-grid counterparts, coarse meshes can still yield useful information in a multigrid cycle. In this paper, we analyze and test techniques designed to address the difficulties in both smoothing and coarse grid correction for the Helmholtz equation. For smoothing, our approach is to use a standard, damped Jacobi, relaxation when it works reasonably well (on fine enough grids), and then to replace this with a Krylov subspace iteration when it fails as a smoother. Earlier work such as in Bank [2] and Brandt and Ta'asan [11] have employed relaxation on the normal equations in this context. Krylov subspace smoothing, principally using the conjugate gradient method, has been considered by a variety of authors [3, 7, 8, 27, 29]. For coarse grid correction, we identify the type and number of eigenvalues that are handled poorly during the correction, and remedy the difficulty by introducing an outer acceleration for multigrid; that is, we use multigrid as a preconditioner for an outer, Krylov subspace, iteration. This approach has been used by many authors, e.g. [28, 31] but only for problems in which the coarse grid is restricted to be fairly fine. It has also been used in other settings [20, 23]. Any Krylov subspace method is an option for both the smoother and the outer iteration; we use GMRES [25]. In contrast to many multilevel strategies [2, 6, 9, 31], the resulting algorithm has no requirements that the coarse grid be sufficiently fine. For approaches based on preconditioning indefinite problems by preconditioners for the leading term, see [4, 5, 15, 32]. In more recent work, Brandt and Livshits [10] have developed an effective multi-grid approach for the Helmholtz equation based on representing oscillatory error components on coarse grids as the product of an oscillatory Fourier mode and a smooth amplitude-or ray-function. The standard V-cycle is augmented by so-called ray cy- cles, in which the oscillatory error components are eliminated by approximating the associated ray functions in a multigrid fashion. This wave-ray methodology has also been combined by Lee et al. [21] with a first-order system least-squares formulation for the Helmholtz equation. These approaches require construction of and bookkeeping for extra grids associated with the ray functions. An outline of the paper is as follows. In Section 2, we perform a model problem analysis, using a one-dimensional problem to identify the difficulties encountered by both smoothers and coarse grid correction, and supplementing these observations with an analysis of how dimensionality of the problem affects the computations. In Section 3, we present the refined multigrid algorithms and test their performance on a set of two-dimensional benchmark problems on a square domain. In particular, we demonstrate the effectiveness of an automated stopping criterion for use with GMRES smoothing, and we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for wave numbers as large as two hundred. In Section 4, we show the performance of the multigrid solver on an exterior scattering problem. Finally, in Section 5, we draw some conclusions. 2. Model Problem Analysis. Most of the deficiencies of standard multigrid methods for solving Helmholtz problems can be seen from a one-dimensional model problem. Therefore, we consider the Helmholtz equation on the unit interval (0; 1) with homogeneous Dirichlet boundary conditions This problem is guaranteed to be nonsingular only if k 2 is not an eigenvalue of the negative Laplacian, and we will assume here that this requirement holds. The problem is indefinite for which is the smallest eigenvalue of the negative Laplacian. Finite difference discretization of (2.1) on a uniform grid containing N interior points leads to a linear system of equations (1.4) with the N \Theta N coefficient matrix denotes the mesh width and I denotes the identity matrix. Under the assumptions on k above, it is well-known (see [26]) that for sufficiently fine discretizations, the discrete problems are also nonsingular. We also assume that all coarse grid problems are nonsingular. The eigenvalues of A are and the eigenvectors are 2h [sin ij-h] N The choice of Dirichlet boundary conditions in (2.1) allows us to perform Fourier analysis using these analytic expressions for eigenvalues and eigenvectors. In experiments described in Section 3, we will examine how our observations coincide with performance on problems with radiation conditions, which are nonsingular for all k [18]. Aspects of the algorithm that depend on the dimensionality of the problem will be considered at the end of this section. 2.1. Smoothing. For the smoothing operator, we consider damped Jacobi re- laxation, defined by the stationary iteration em um denote the residual and error vectors at step m, respectively. denotes the matrix consisting of the diagonal of A, and ! is the damping parameter. The associated error propagation matrix is and the eigenstructure of this matrix governs the behavior of the error em . Since D is a multiple of the identity matrix, S! is a polynomial in A and hence shares the same system of orthonormal eigenvectors (2.3). The eigenvalues of S! are Thus, the eigenvalue - j of S! is the damping factor for the error component corresponding to the eigenvalue - j of A. We now consider the effects of damped Jacobi smoothing on three levels of grids: fine, coarse, and intermediate. 2.1.1. Fine Grids. The fine grid mesh size is determined by accuracy requirements on the discretization, and this allows us to make certain assumptions on the size of h versus k on the fine grid. Recall that the wavelength - associated with a time-harmonic wave with wave number k ? 0 is given by 2-=k. The quantity is the number of mesh points per wavelength, and it measures the approximability of the solution on a given mesh. A commonly employed engineering rule of thumb [17] states that, for a second-order finite difference or linear finite element discretization, equivalently, kh -=5 (2.5) is required, and we will enforce (2.5) in all experiments. We also note that, for reasons of stability, a bound on the quantity h 2 k 3 is also required [18]; for high wave numbers this bound is more restrictive than the bound on kh. As a consequence of (2.5), the quantity multiplying the smoothing parameter ! in (2.4) will vary between about \Gamma1=4 and 9=4 for smoothing results in a slight amplification of the most oscillatory modes as well as of the smoothest modes. One can adjust ! so that the most oscillatory mode is damped, and this is the case as long as For S! to be an effective smoother, ! is usually chosen to maximize damping for the oscillatory half of the spectrum. This leads to the choice which is equal to the familiar optimal value of 2=3 for the Laplacian [22, p. 11] when But the smoothest mode is amplified for any positive choice of ! when the discrete problem is indefinite, and this is the case for the discrete Helmholtz operator . As can be seen from (2.4), more smooth-mode eigenvalues of S! become larger than one in magnitude as h is increased, thus making damped Jacobi-as well as other standard smoothers-increasingly more unstable as the mesh is coarsened. Figure 2.1 shows the damping factors - j for each of the eigenvalues - j of A for wave number on a grid with 31. The maximal amplification occurs for the smoothest mode, corresponding to the leftmost eigenvalue of A. When this amplification factor is approximately equal to Figure 2.2 shows how ae varies with kh. Limiting this largest amplification factor, say to ae - 1:1, would lead to the mesh size restriction kh - 0:52, somewhat stronger than (2.5). One also observes that, for kh ? 6, this mode is once again damped. In summary, the situation on the finest grids is similar to the positive definite case, except for the small number of amplified smooth modes whose number and amplification factors increase as the mesh is coarsened. l Fig. 2.1. The damping factors for the damped Jacobi relaxation plotted against the eigenvalues of A (+) for kh Fig. 2.2. The variation of the damping/amplification factor of the smoothest mode as a function of kh for 2.1.2. Very Coarse Grids. As the mesh is coarsened, the eigenvalues of A that correspond to the larger eigenvalues of the underlying differential operator disappear from the discrete problem, while the small ones-those with smooth eigenfunctions- remain. This means that, for a fixed k large enough for the differential problem to be indefinite, there is a coarsening level below which all eigenvalues are negative. For the model problem (2.1), this occurs for kh ? 2 cos(-h=2) for any fixed k ? -. In this (negative definite) case, the damped Jacobi iteration is convergent for with and the spectral radius of S! is minimized for This would permit the use of (undamped) Jacobi as a smoother on very coarse grids, but we shall not make use of this. 2.1.3. Intermediate Grids. What remains is the difficult case: values of kh for which the problem is not yet negative definite but for which a large number of smooth modes are amplified by damped Jacobi relaxation. Jacobi smoothing and other standard smoothers are therefore no longer suited, and it becomes necessary to use a different smoothing procedure. In [11] and [16] it was proposed to replace classical smoothers with the Kaczmarz iteration, which is Gau-Seidel relaxation applied to the symmetric positive-definite system AA for the auxiliary variable v defined by A This method has the advantage of not amplifying any modes, but it suffers from the drawback that the damping of the oscillatory modes is very weak. In the following section we propose using Krylov subspace methods such as GMRES for smoothing. These methods possess the advantage of reducing error components on both sides of the imaginary axis without resorting to the normal equations. 2.2. Coarse Grid Correction. The rationale behind coarse-grid correction is that smooth error components can be well represented on coarser grids, and hence a sufficiently good approximation of the error can be obtained by approximating the fine grid residual equation using the analogous system on a coarser mesh. This assumes both that the error consists mainly of smooth modes and that the solution of the coarse grid residual equation is close to its counterpart on the fine grid. In this section, we present an analysis of what goes wrong for the Helmholtz problem. 2.2.1. Amplification of Certain Modes. Assume the number of interior grid points on the fine grid is odd, and consider the next coarser mesh, with interior points. We identify R N and R n , respectively, with the spaces of grid functions on these two meshes that vanish at the endpoints, and we indicate the mesh such vectors are associated with using the superscripts h and H. Let e the fine grid error, let r denote the residual, and let denote the coarse mesh size. Let the coarse-to-fine transformation be given by the interpolation operator I h \Theta I h The following indication of what can go wrong with the (exact) coarse grid correction was given in [11]: consider a fine-grid error e consisting of only the smoothest eigenvector v h of A h with associated eigenvalue - h . The fine-grid residual is thus given by r since we are assuming that v h is smooth, its restriction - r H := I H h to the coarse grid will again be close to an eigenvector of the coarse-grid operator A H , but with respect to a slightly different eigenvalue - H . The coarse grid version of the correction is I H Hence the error on the fine grid after the correction is where we have assumed that the smooth mode v h is invariant under restriction followed by interpolation. This tells us that, under the assumption that the restrictions of smooth eigenvectors are again eigenvectors of A H , the quality of the correction depends on the ratio - h =- H . If the two are equal, then the correction is perfect, but if the relative error is large, the correction can be arbitrarily bad. This occurs whenever one of - h , - H is close to the origin and the other is not. Moreover, if - h and - H have opposite signs, then the correction is in the wrong direction. We now go beyond existing analysis and examine which eigenvalues are problematic in this sense for finite differences; a similar analysis can also be performed for linear finite elements. Consider the coarse-grid eigenfunctions v H . To understand the effects of interpolation of these grid functions to the fine grid, we must examine both the first n fine-grid eigenfunctions fv h and their complementary modes fv h are related by \Theta v h \Theta v h . As is easily verified, there holds [12] I h with c j := cos j-h=2 and s j := sin j-h=2, If full weighting is used for the restriction operator I H componentwise \Theta I H \Theta \Theta and the relation I h . The following mapping properties are easily established I H with c j and s j as defined above. If A H denotes the coarse-grid discretization matrix, then the corrected iterate ~ possesses the error propagation operator C := I \Gamma I h h A h . Denoting the eigenvalues of A h and A H by f- h respectively, we may summarize the action of C on the eigenvectors using (2.9) and as follows: Theorem 2.1. The image of the fine-grid eigenfunctions fv h under the error propagation operator C of the exact coarse grid correction is given by As a consequence, the two-dimensional spaces spanned by a smooth mode and its complementary mode are invariant under The following result shows the dependence of the matrices C j on Theorem 2.2. Using the notation defined above, there holds Moreover, lim Proof. Both (2.13) and (2.14) are simple consequences of (2.12) and the representation (2.2) of the eigenvalues - h . Application of the error propagation operator to a smooth mode v h If the entries of the first column of C j are small, then this mode is damped by the coarse grid correction. However, if the (1; 1)-entry is large then this mode is amplified, and if the (2; 1)-entry is large (somewhat less likely), then the smooth mode is corrupted by its complementary mode. As seen from (2.13), these difficulties occur whenever - H is small in magnitude. From the limits (2.14), it is evident that no such problems arise in the symmetric positive-definite case (a fact that is well-known), but they also do not occur when kh is very large, i.e., when the coarse grid Helmholtz operator is negative definite. These observations can be extended by returning to (2.8) and using (2.2), wherein it holds that That is, the coarse-grid correction strongly damps smooth error modes for either very small or very large values of kh, but it may fail to do so in the intermediate range associated with a smooth mode. We also note that in the limit the eigenvalues of C j are 0 and 1, so that C j is a projection, and in this case the projection is orthogonal with respect to the inner product induced by the symmetric and positive definite operator A h . The projection property is lost for k ? 0, since the coarse grid operator as we have defined it fails to satisfy the Galerkin condition A h A h I h H . (The Galerkin condition is, however, satisfied e.g. for finite element discretizations with interpolation by inclusion) Moreover, regardless of the type of discretization, the term A h -orthogonality ceases to makes sense once k is sufficiently large that A h is indefinite. 2.2.2. Number of Sign Changes. In this section, we discuss the number of eigenvalues that undergo a sign change during the coarsening process, and thereby inhibit the effectiveness of coarse grid correction. This is the only aspect of the algorithm that significantly depends on the dimensionality of the problem. Thus, here we are considering the Helmholtz equation (1.1) on the d-dimensional unit cube (0; 1) d , with homogeneous Dirichlet boundary conditions. We consider standard finite differences (second order three-point, five-point or seven-point discretization of the Laplacian in one, two or three dimensions, respectively), as well as the class of low order finite elements consisting of linear, bilinear or trilinear elements. We first state the issue more precisely using finite differences. In d dimensions, the eigenvalues of the discrete operator on a grid with mesh size h and N grid points in each direction are d sin For any fixed multi-index I, this eigenvalue is a well-defined function of h that converges to the corresponding eigenvalue of the differential operator as h ! 0. Our concern is the indices for which this function changes sign, for these are the troublesome eigenvalues that are not treated correctly by some coarse grid correction. As the mesh is coarsened, the oscillatory modes (j i ? N=2 for some i) are not represented on the next coarser mesh, but the smooth-mode eigenvalues f- H I g are slightly shifted to the left with respect to their fine-grid counterparts f- h I g, and some of these eigenvalues change sign at some point during the coarsening process. The following theorem gives a bound, as a function of k, on the maximal number eigenvalue sign changes occurring on all grids. Theorem 2.3. For finite difference discretization of the Helmholtz equation with Dirichlet boundary conditions on the unit cube in d dimensions 3), the number of eigenvalues that undergo a change in sign during the multigrid coarsening process is bounded above by 3: For the finite element discretizations, the number of sign changes is bounded above by pj 3: Proof. For finite differences, let fine denote the number of negative eigenvalues on some given fine grid, and let lim denote the number of negative eigenvalues of the continuous Helmholtz operator. Because eigenvalues (2.16) with the same index I shift from right to left with grid coarsening, it follows that this is an equality for all fine enough grids, as the discrete eigenvalues tend to the continuous ones. To identify lim , consider the continuous eigenvalues It is convenient to view the indices of these eigenvalues as lying in the positive orthant of a d-dimensional coordinate system. The negative eigenvalues are contained in the intersection of this orthant with a d-dimensional sphere of radius k=- centered at the origin. Let N denote this intersection, and let - N denote the d-dimensional cube enclosing N . The number of indices in - N is bk=-c d , and the number in N is aebk=-c d , where is the ratio of the volume of N to that of - N . It follows that 3: Now consider the eigenvalues of discrete problems. Again, since sign changes occur from right to left with coarsening, the mesh size that yields the maximum number of negative eigenvalues is the smallest value h for which the discrete operator is negative semidefinite. With N mesh points in each coordinate direction, this is equivalent to d sin 2 N-h 3: Thus, d=k, and d 3: Combining (2.19) with the fact that fine fine The latter difference, shown in (2.17), is then a bound on number of sign changes. For finite elements, we are concerned with the eigenvalues of the coefficient matrix A h , but it is also convenient to consider the associated operator A h defined on the finite element space V h . The eigenvalues of A h are those of the generalized matrix eigenvalue problem A h u where M h is the mass matrix. These eigenvalues tend to those of the continuous operator. Moreover, since V H is a subspace of V h , the Courant-Fischer min-max theorem implies that eigenvalues oe h and oe H with a common index shift to the right with coarsening (or to the left with refinement). In addition, since M h is symmetric positive-definite, Sylvester's inertia theorem implies that the number of negative eigenvalues of A h is is the same as that of (2.21). It follows from these observations Sign changes Sign changes Fig. 2.3. Indices of eigenvalues undergoing a sign change during coarsening of an N \Theta N finite element grid with during further coarsening of the next coarser (n \Theta n) grid with that the maximal number of negative eigenvalues of A h is bounded above by the fine grid limit lim . This is also a bound on the number of sign changes. It can be improved by examining the eigenvalues of A h more closely. Using the tensor product form of the operators, we can express these eigenvalues as 1), the indices run from 1 to N and Consider the requirement - h so that A h is negative semidefinite. This is equivalent to Since the expression - j =- j is monotonically increasing with j, the largest eigenvalue in d dimensions equals zero if -N 12d=k. For this value of h, there are 12d) d negative eigen- values, and on coarser meshes, the problem remains negative definite. Consequently, none of these quantities undergo a sign change, giving the bound j \Gamma of (2.18). Figure 2.3 gives an idea of how sign changes are distributed for bilinear elements in two dimensions. At levels where the changes take place, the indices of the eigenvalues lie in a curved strip in the two-dimensional plane of indices. Typically, there is one level where the majority of sign changes occur. As k is increased and h decreased correspondingly via (2.5), the shape of these strips remains fixed but the number of indices contained in them grows like O(h \Gammad however, that (2.5) is not needed for the analysis.) The behavior for finite differences is similar. The remedy suggested in [11] for these difficulties consists of maintaining an approximation of the eigenspace V H of the troublesome eigenvalues. A projection scheme is then used to orthogonalize the coarse grid correction against V H , and the coefficients of the solution for this problematic space are obtained separately. Since it involves an explicit separate treatment of the problematic modes, this approach is restricted to cases where there are only very a small number of these. 3. Incorporation of Krylov Subspace Methods. In view of the observations about smoothing in Section 2.1 and coarse grid correction in Section 2.2, we modify the standard multigrid method in the following way to treat Helmholtz problems: ffl To obtain smoothers that are stable and still provide a strong reduction of oscillatory components, we use Krylov subspace iteration such as GMRES as smoothers on intermediate grids. ffl To handle modes with eigenvalues that are either close to the origin on all grids-and hence belong to modes not sufficiently damped on any grid-or that cross the imaginary axis and are thus treated incorrectly by some coarse grid corrections, we add an outer iteration; that is, we use multigrid as a preconditioner for a GMRES iteration for (1.4). We will demonstrate the effectiveness of this approach with a series of numerical experiments. In all tests the outer iteration is run until the stopping criterion is satisfied, where Aum is the residual of the mth GMRES iterate and the norm is the vector Euclidean norm. The multigrid algorithm is a V-cycle in all cases; the smoothing schedules are specified below. 3.1. GMRES Accelerated Multigrid. We begin with an experiment for the one-dimensional Helmholtz equation on the unit interval with forcing term inhomogeneous Dirichlet boundary condition on the left and Sommerfeld condition on the right. We discretize using linear finite elements on a uniform grid, where the discrete right hand side f is determined by the boundary conditions. We apply both a V-cycle multigrid algorithm and a GMRES iteration preconditioned by the same V-cycle multigrid method. The smoother in these tests is one step of damped Jacobi iteration for both presmoothing and postsmoothing, using in (2.6). The initial guess was a vector with normally distributed entries of mean zero and variance one, generated by the Matlab function randn. Table 3.1 shows the iteration counts for increasing numbers of levels beginning with fine grids containing elements and for wave numbers which correspond to two and four wavelengths in the unit inter- val, respectively. We observe first that both methods display typical h\Gammaindependent multigrid behavior until the mesh size on the coarsest grid reaches kh -=2. (With 256 elements, this occurs for coarsest mesh 1=8, and for coarsest 1=16). At this point both methods require noticeably more iterations, the increase being much more pronounced in the stand-alone multigrid case. When yet coarser levels are added, multigrid diverges, whereas the # levels MG GMRES MG GMRES MG GMRES MG GMRES Table Iteration counts for multigrid V-cycle as a stand-alone iteration and as a preconditioner for GMRES applied to the one-dimensional model Helmholtz problem, with damped Jacobi smoothing. A dash denotes divergence of the iteration. 128 \Theta 128 elements 256 \Theta 256 elements # levels MG GMRES MG GMRES MG GMRES MG GMRES Table Iteration counts for the two-dimensional problem for fine grids with 128 \Theta 128 and 256 \Theta 256 meshes. A dash denotes divergence of the iteration. multigrid preconditioned GMRES method again settles down to an h-independent iteration count, which does, however, increase with k. Table 3.2 shows the same iteration counts for the two-dimensional Helmholtz problem on the unit square with a second order absorbing boundary condition (see [1, 13]) imposed on all four sides and discretized using bilinear quadrilateral finite elements on a uniform mesh. Since the problem cannot be forced with a radiation condition on the entire boundary, in this and the remaining examples of Section 3, an inhomogeneity was imposed by choosing a discrete right hand side consisting of a random vector with mean zero and variance one, generated by randn. The initial guess was identically zero. (Trends for problems with smooth right hand sides were the same.) In addition, for all two-dimensional problems, we use two Jacobi pre- and postsmoothing steps whenever Jacobi smoothing is used. The damping parameter ! is chosen to maximize damping of the oscillatory modes. For the grids on which we use damped Jacobi smoothing this optimum value was determined to be 8=9. The results show the same qualitative behavior as for the one-dimensional problem in that stand-alone multigrid begins to diverge as coarse levels are added while the GMRES- accelerated iteration converges in an h-independent number of iterations growing with # elements on coarsest grid 512 256 128 64 GMRES iterations 152 78 42 25 Table As more coarse grid information is used, the number of iterations decreases, for the one-dimensional problem with and a fine grid containing k, although with a larger number of iterations than in the one-dimensional case. A natural question is whether corrections computed on the very coarse grids, in particular those associated with mesh widths larger than 1/10 times the wavelength any contribution at all towards reducing the error. We investigate this by repeating the GMRES accelerated multigrid calculations for the one-dimensional problem with omitting all calculations-be they smoothing or direct solves-on an increasing number of coarse levels. The results are shown in Table 3.3. The leftmost entry of the table shows the iteration counts when no coarse grid information is used, i.e., for GMRES with preconditioning by two steps of iteration. Reading from left to right, subsequent entries show the counts when smoothings on a succession of coarser grids are included, but no computations are done at grid levels below that of the coarsest grid. For the rightmost entry, a direct solve was done on the coarsest mesh; this is a full V-cycle computation. The results indicate that the computations on all grids down to that at level 2, which has eight elements and only two points per wavelength, still accelerate the convergence of the outer iteration. These results show that, although multigrid by itself may diverge, it is nevertheless a powerful enough preconditioner for GMRES to converge in an h-independent number of steps. Two additional questions are whether replacing the unstable Jacobi smoother with a Krylov subspace iteration leads to a convergent stand-alone multigrid method, and how sensitive convergence behavior is as a function of the wave number k. We address the former in the following section. 3.2. GMRES as a Smoother. In this section we replace the unstable Jacobi smoother with GMRES smoothing. We use GMRES on all levels j where kh j - 1=2 and continue using damped Jacobi relaxation when kh choice is motivated by the discussion at the end of Section 2.1.1, and it ensures that the largest amplification factor for the Jacobi smoother does not become too large. The results of Section 2.1.2 show that we could switch back to Jacobi smoothing for very coarse meshes, but we have not explored this option. 3.2.1. Nonconstant Preconditioners. This introduces a slight complication with regard to the outer GMRES iteration when multigrid is used as a preconditioner. The inner GMRES smoothing steps are not linear iterations, and therefore a different preconditioner is being applied at every step of the outer iteration. A variant of GMRES able to accommodate a changing preconditioner (known as flexible GMRES is due to Saad [24]. It requires the following minor modification of the standard (right preconditioned) GMRES algorithm: if the orthonormal basis of the (m+1)st Krylov space Km+1 (AM in the case of a constant preconditioner M is denoted by then the Arnoldi relation AM Hm holds with an (m upper Hessenberg matrix ~ Hm . If the preconditioning and matrix multiplication step z is performed with a changing preconditioner results in the modified Arnoldi relation The residual vector is now minimized over the space need no longer be a Krylov space. This requires storing the vectors fz j g in addition to the orthonormal vectors fv j g, which form a basis of 3.2.2. Hand-Tuned Smoothing Schedules. Numerical experiments with a fixed number of GMRES smoothing steps at every level did not result in good perfor- mance. To get an idea of an appropriate smoothing schedule, we proceed as follows. For given k, we calculate the number o max of FGMRES iterations needed with j- level multigrid preconditioning, where we use Jacobi smoothing on all grids for which do a direct solve at the next coarser grid, making j grids in all. We then replace the direct solve on the coarsest grid of the j-level scheme with GMRES smoothing on this grid, coupled with a direct solve on the next coarser grid, and determine the smallest number m j of GMRES smoothing steps required for the outer iteration to converge in omax steps. For example, for the first line of Table 3.4, 6 outer FGMRES steps were needed for a 5-level scheme, and then m smoothing steps were needed for the outer iteration of the new 6-level preconditioner to converge in 6 steps. When the number m j has been determined, we could fix the number of GMRES smoothing steps to m j on this grid, add one coarser level, determine the optimal number of GMRES smoothing steps on the coarser grid and continue in this fashion until the maximal number of levels is reached. This approach is modified slightly by, whenever possible, trying to reduce the number of smoothings on finer levels once coarser levels have been added. This is often possible, since replacing the exact solve on the coarsest grid with several GMRES smoothing steps often has a regularizing effect, avoiding some damage possibly done by an exact coarse grid correction in modes whose eigenvalues are not well represented on the coarse grid. This hand-tuning procedure gives insight into the best possible behavior of this algorithm. In contrast to classical linear smoothers, whose damping properties for different modes is fixed, the damping properties of GMRES depend on the initial residual. In particular, since GMRES is constructed to minimize the residual, it will most damp those modes that lead to the largest residual norm reduction. For this reason, we will favor post-smoothing over pre-smoothing to prevent the unnecessary damping of smoother modes that should be handled by the coarse-grid correction. We do include two GMRES pre-smoothing steps to avoid overly large oscillatory components in the residual prior to restricting it to the next lower level, which could otherwise lead to spurious oscillatory error components being introduced by the coarse grid correction. The results are shown in Table 3.4. The entry 'D' denotes a direct solve on the corresponding level and 'J' indicates that damped Jacobi smoothing was used on this level. Looking at the smoothing schedules, we observe a 'hump' in the number of GMRES smoothing steps on the first two levels on which GMRES smoothing is used. Below this, the number decreases and is often zero for the coarsest levels. However, 256 \Theta 256, # levels Smoothing schedule MG FGMRES 9 J J J J 13 128 \Theta 128, # levels Smoothing schedule MG FGMRES 256 \Theta 256, # levels Smoothing schedule MG FGMRES 256 \Theta 256, # levels Smoothing schedule MG FGMRES 6 J J 9 J J 256 \Theta 256, # levels Smoothing schedule MG FGMRES 9 J Table Manually optimized GMRES smoothing schedule for the two-dimensional model Helmholtz prob- lem: 'J' denotes Jacobi smoothing and 'D' denotes a direct solve. The FGMRES algorithm uses the multigrid V-cycle as a preconditioner. GMRES smoothing still helps on levels which are extremely coarse with regard to resolution of the waves: in the case performing three GMRES smoothing steps on level 4 (which corresponds to 1/2 point per wavelength) still improves convergence. We remark that the number of outer iterations in all these tests, for both preconditioned FGMRES and standalone MG, is the same as for the corresponding two-grid versions of these methods, so we cannot expect faster convergence with respect to the wave number k. We also note that the number of iterations for standalone multigrid is very close to that that for FGMRES with multigrid preconditioning. We believe this is because the relatively large number of GMRES smoothing steps on intermediate levels eliminates lower frequency errors, and this mitigates the effects of axis crossings. We will return to this point in Section 3.4 3.3. A Stopping Criterion Based on L 2 -Sections. Hand tuning as in the previous section is clearly not suitable for a practical algorithm. In this section, we develop a heuristic for finite element discretizations that automatically determines a stopping criterion for the GMRES smoother. This technique is based on an idea introduced in [29]. We briefly introduce some standard terminology for multilevel methods applied to second order elliptic boundary value problems on a bounded domain\Omega ae R 2 (see [30]). We assume a nested hierarchy of finite element spaces in which the largest space V J corresponds to the grid on which the solution is sought. We require the L 2 -orthogonal projections defined by where (\Delta; \Delta) denotes the L 2 -inner product on \Omega\Gamma Let \Phi denote the basis of the finite element space V ' of dimension n ' used in defining the stiffness and mass matrices. By the nestedness property V ' ae V '+1 , there exists an n '+1 \Theta n ' matrix I '+1 whose columns contain the coefficients of the basis \Phi ' in terms of the basis \Phi '+1 , so that, writing the bases as row vectors, The stopping criterion we shall use for the GMRES smoothing iterations is based on the representation of the residual r ' of an approximate solution ~ u ' of the level-' equation as the sum of differences of L 2 -projections, which we refer to as residual sections. The following result for coercive problems, which was proven in [29], shows that the error u u ' is small if each appropriately weighted residual section is small: Theorem 3.1. Assume the underlying elliptic boundary value problem is H 1 - elliptic and H 1+ff -regular with ff ? 0. Then there exists a constant c independent of the level ' such that the H -norm of the error on level ' is bounded by The boundary value problem (1.1)-(1.3) under consideration is not H 1 -elliptic and therefore does not satisfy the assumptions of this theorem. We have found, however, that the bound (3.1) suggests a useful stopping criterion: terminate the GMRES smoothing iteration on level ' as soon as the residual section (Q has become sufficiently small. To obtain a formula for the computation of these sections, assume the residual r ' is represented by the coefficient vector r ' in terms of the dual basis of \Phi ' . The representation of Q with respect to the dual basis of \Phi '\Gamma1 is then given by the coefficient vector I Returning to the representation with respect to the basis \Phi '\Gamma1 requires multiplication , so that we obtain I If the sequence of triangulations underlying the finite element spaces V ' is quasi- uniform, then the mass matrix of level ' is uniformly equivalent to the identity scaled by h d , where d denotes the dimension of the domain. For the case consideration, this means that the Euclidean inner product on the coordinate space denoted by (\Delta; \Delta) E , when scaled by h 2 ' , is uniformly equivalent (with respect to the mesh size) to the L 2 -inner product on V ' . Therefore, the associated norms satisfy ch 2 where v ' is the coordinate vector of v ' with respect to \Phi ' . Using this norm equivalence it is easily shown that I I for some constants c and C uniformly for all levels '. As a result, the residual sections may be computed sufficiently accurately without the need for inverting mass matrices. In [29], it was suggested that the GMRES smoothing iteration for a full multigrid cycle be terminated as soon as the residual section on the given level is on the order of the discretization error on that level. For the problem under consideration here, we shall use the relative reduction of L 2 -sections as a stopping criterion, so that roughly an equal error reduction for all modes is achieved in one V-cycle. On the first level on which GMRES smoothing is used, we have the additional difficulty that many eigenvalues may be badly approximated on the next-coarser level. For this reason, it is better to also smooth the oscillatory modes belonging to the next lower level and base the stopping criterion on the residual section use this 'safer' choice on all levels. Numerical experiments with optimal smoothing schedules have shown the relative reduction of this residual section to scale like kh ' , so that we arrive at the stopping criterion I ' I I A complete description of the multigrid V-cycle algorithm starting on the finest level ' is as follows: Algorithm 3.1. ~ V-cycle with GMRES smoothing on coarse levels ~ else steps of damped Jacobi smoothing to obtain u (1) else perform 2 steps of GMRES smoothing to obtain u (1) endif steps of damped Jacobi smoothing to obtain ~ else perform GMRES smoothing until stopping criterion (3.2) is satisfied or to obtain ~ endif endif In the standalone multigrid V-cycle, Algorithm 3.1 is used recursively beginning with the finest level and iterated until the desired reduction of the relative residual is achieved on the finest level. In the FGMRES variant, Algorithm 3.1 represents the action of the inverse of a preconditioning operator being applied to the vector f ' . 3.4. Experiments with Automated Stopping Criterion. We now show how the multigrid solver and preconditioner perform with the automated stopping criterion for GMRES smoothing. Each method is applied to the two-dimensional Helmholtz problem on the unit square with second-order absorbing boundary condition and random right hand side data. In these tests, we used in (3.2), and we also imposed an upper bound mmax on the number of GMRES smoothing steps, terminating the smoothing if the stopping criterion is not satisfied after mmax steps; we tested two values, levels, where damped Jacobi smoothing is used, the number of pre-smoothings and post-smoothings was 2. Standalone MG MG-preconditioned FGMRES Nnk 2- 4- 8- 16- 32- 64- 2- 4- 8- 16- 32- 64- Table Iteration counts for standalone multigrid and multigrid-preconditioned FGMRES for various fine grid sizes and wave numbers. In all cases, GMRES smoothing is performed on levels for which kh ? 1=2 and the smoothing is terminated by the L 2 -section stopping criterion or when mmax smoothing steps are reached. This count was extrapolated from the maximum of 47 steps that memory constraints permitted. We present three sets of results. Table 3.5 shows iteration counts for a variety of wave numbers and mesh sizes. Table 3.6 examines performance in more detail by showing the automatically generated smoothing schedules for two wave numbers, Finally, to give an idea of efficiency, Table 3.7 shows an estimate for the operation counts required for the problems treated in Table 3.6. Grid # levels Smoothing schedule Iterations 128 \Theta 128 7 J J 128 \Theta 128 7 J J 17 17 11 2 D 13 Grid # levels Smoothing schedule Iterations 128 \Theta 128 7 J J 19 17 11 2 D 9 128 \Theta 128 7 J J 17 17 11 2 D 13 Grid # levels Smoothing schedule Iterations 512 \Theta 512 9 J J 33 37 512 \Theta 512 9 J J 31 37 Grid # levels Smoothing schedule Iterations 512 \Theta 512 9 J J 20 20 512 \Theta 512 9 J J 20 20 Table Smoothing schedules with automated stopping criterion, for selected parameters. We make the following observations on these results: ffl For low wave numbers, the number of iterations of standalone multigrid is close to that for FGMRES. The difference increases as the wave number in- creases, especially for the case 20. For large enough k, multigrid fails to converge whereas MG-preconditioned FGMRES is robust. This behavior is explained by the results of Section 2.2.2. For large wave numbers, the increased number of amplified modes eventually causes standalone multigrid to fail; a larger number of smoothing steps mitigates this difficulty, presumably by eliminating some smooth errors. The (outer) FGMRES iteration handles this situation in a robust manner. ffl The automated stopping criterion leads to smoothing schedules close to those obtained by hand tuning (see Table 3.4), and correspondingly similar outer iteration counts. ffl The operation counts shown in Table 3.7 suggest that MG-preconditioned FGMRES is more efficient than standalone multigrid even when the latter Grid MG FGMRES MG FGMRES 64 \Theta 64 13.2 13.3 128 \Theta 128 24.0 22.1 256 \Theta 256 61.2 43.2 1091.2 971.1 512 \Theta 512 196.6 148.1 1418.1 1377.8 Table Operation counts (in millions) for selected parameters, with method is effective. ffl For fixed wave number, outer iteration counts are mesh independent, so that standard "multigrid-like" behavior is observed. Moreover, because Jacobi smoothing is less expensive than GMRES smoothing, during the initial stages of mesh refinement the costs per unknown are increasing at less than a linear rate. ffl The growth in outer iteration counts with increasing wave number is slower than linear in k. The operation counts increase more rapidly, however, because of the increased number of smoothing steps required for larger wave numbers. 4. Application to an Exterior Problem. As a final example we apply the algorithm to an exterior scattering problem for the Helmholtz equation as given in (1.1)-(1.3). The domain\Omega consists of the exterior of an ellipse bounded externally by a circular artificial boundary \Gamma 1 on which we impose the exact nonlocal Dirichlet-to- Neumann (DtN) boundary condition (see [19]). The source function is forcing is due to the boundary condition on the boundary \Gamma of the scatterer, given by with data g(x; representing a plane wave incident at angle ff to the positive x-axis. The solution u represents the scattered field associated with the obstacle and incident field g; the resulting total field u+g then satisfies a homogeneous Dirichlet or Neumann boundary condition on \Gamma, respectively. An angle of incidence was chosen to avoid a symmetric solution. The problems were discretized using linear finite elements beginning with a very coarse mesh which is successively refined uniformly to obtain a hierarchy of nested finite element spaces. The finest mesh, obtained after five refinement steps, contains 32768 degrees of freedom. Several combinations of k and h were tested, where in each case kh ! 0:5 on the finest mesh. Figure 4.1 shows a contour plot of the solution u of the Dirichlet problem for 8-. The computations make use of the PDE Toolbox of the Matlab 5.3 computing environment. The problems were solved using both the standalone and FGMRES-accelerated versions of multigrid, with GMRES smoothing using the residual section stopping criterion with stopping criterion requiring residual reduction by a \Gamma6 as in Section 3, and zero initial guess. In all examples, we used the maximal number of levels with the exception of the Dirichlet problem for where we also varied the number of levels from six down to two. The results are shown in Table 4.1. The table gives the wave number k and the length of the ellipse 2-=k. The third column gives the maximum value of kh on Dirichlet problem 28 4 100 26 Neumann problem 28 Table Iteration counts for the exterior scattering problem with Dirichlet or Neumann plane wave data on the boundary of an ellipse for various wave numbers, grid sizes and numbers of levels. the finest mesh and the fourth column indicates the number of levels used in each computation. The last two columns list the iteration counts. We observe that the preconditioned iteration performs well in all cases, with a growth in number of iterations slower than linear in k. The standalone multigrid variant performs less well in comparison, requiring more than 100 steps to converge in several cases and even diverging in one case. This is particularly the case for the Neumann problem, where the superiority of the preconditioned variant is even more pronounced. For the Neumann problems we also notice a slight growth in iteration counts for fixed k and decreasing h. 5. Conclusions. The results of this paper show that the addition of Krylov subspace iteration to multigrid, both as a smoother and as an outer accelerating procedure, enables the construction of a robust multigrid algorithm for the Helmholtz equation. GMRES is an effective smoother for grids of intermediate coarseness, in that it appears not to amplify any error modes and in addition tends to have a regularizing effect on the contribution to the coarse grid correction coming from smoothing on a given level. The combination of our multigrid algorithm as a preconditioner with FGMRES is effective in handling the deficiencies of standard multigrid methods for the Helmholtz equation, and the outer FGMRES acceleration is necessary particularly for high wave numbers. In addition, results in the paper indicate that grids too coarse to result in a meaningful discretization of the Helmholtz equation may still provide some useful information for coarse-grid corrections. Using an automated stopping criterion based on L 2 -sections of the residual leads to smoothing cycles that are close to hand-tuned optimal smoothing schedules. An important aspect of our algorithm is that it consists of familiar building blocks and is thus easily implemented. For very large wave numbers for which the discretiza- Fig. 4.1. Contour plot of the solution of the Dirichlet problem with wave number tion must not only keep kh but also k 3 h 2 small, the grid hierarchy will contain more grids fine enough to use Jacobi smoothing, thus making the algorithm more efficient. The result is a multigrid method that appears to converge with a rate independent of the mesh size h and with only moderate dependence on the wave number k. Finally, the numerical results show that we are able to effectively solve Helmholtz problems with wave numbers of practical relevance. --R A comparison of two multilevel iterative methods for nonsymmetric and indefinite elliptic finite element equations Sharp estimates for multigrid rates of convergence with general smoothing and acceleration An iterative method for the Helmholtz equation The cascadic multigrid method for elliptic problems On the combination of the multigrid method and conjugate gradients The analysis of multigrid algorithms for nonsymmetric and indefinite elliptic problems Multigrid methods for nearly singular and slightly indefinite Absorbing boundary conditions for the numerical simulation of waves Multigrid preconditioners applied to the iterative solution of singularly perturbed elliptic boundary value problems and scattering problems Finite element method for the Helmholtz equation in an exterior domain Exact non-reflecting boundary conditions Analysis and comparison of relaxation schemes in robust multigrid and conjugate gradient methods A Multigrid Preconditioner for Stabilised Discretisations of Advection-Diffusion Problems Iterative Methods for Sparse linear Systems GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems An observation concerning Ritz-Galerkin methods with indefinite bilinear forms Some estimates of the rate of convergence for the cascadic conjugate-gradient method Multigrid techniques for highly indefinite equations On the performance of Krylov subspace iterations as smoothers in multigrid meth- ods A new class of iterative methods for nonselfadjoint or indefinite problems On the multilevel spltting of finite element spaces for indefinite elliptic boundary value problems --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 Peng Li , L. T. Pileggi, Efficient harmonic balance simulation using multi-level frequency decomposition, Proceedings of the 2004 IEEE/ACM International conference on Computer-aided design, p.677-682, November 07-11, 2004

multigrid;helmholtz equation;krylov subspace methods

587201

On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization.

We consider the application of the conjugate gradient method to the solution of large equality constrained quadratic programs arising in nonlinear optimization. Our approach is based implicitly on a reduced linear system and generates iterates in the null space of the constraints. Instead of computing a basis for this null space, we choose to work directly with the matrix of constraint gradients, computing projections into the null space by either a normal equations or an augmented system approach. Unfortunately, in practice such projections can result in significant rounding errors. We propose iterative refinement techniques, as well as an adaptive reformulation of the quadratic problem, that can greatly reduce these errors without incurring high computational overheads. Numerical results illustrating the efficacy of the proposed approaches are presented.

Introduction A variety of algorithms for nonlinearly constrained optimization [7, 8, 12, 29, 31] use the conjugate gradient (CG) method [25] to solve subproblems of the form minimize x subject to In nonlinear optimization, the n-vector c usually represents the gradient rf of the objective function or the gradient of the Lagrangian, the n \Theta n symmetric matrix H stands for either the Hessian of the Lagrangian or an approximation to it, and the solution x represents a search direction. The equality constraints (1.2) are obtained by linearizing the constraints of the optimization problem at the current iterate. We will assume here that A is an m \Theta n that A has full row rank so that the constraints (1.2) constitute linearly independent equations. We also assume for convenience that H is positive definite in the null space of the constraints, as this guarantees that (1.1)-(1.2) has a unique solution. This positive definiteness assumption is not needed in trust region methods, but our discussion will also be valid in that context because trust region methods normally terminate the CG iteration as soon as negative curvature is encountered (see [36, 38], and, by contrast, [23]). The use of an iterative method such as CG is attractive in large scale optimization because, when the number of variables is large, it can be cost effective to solve (1.1)- (1.2) approximately, and only increase the accuracy of the solution as the iterates of the optimization algorithm approach the minimizer. In addition, the properties of the CG method merge very well with the requirements of globally convergent optimization methods (see e.g. [36]). In this paper we study how to apply the preconditioned CG method to (1.1)- (1.2) so as to keep the computational cost at a reasonable level while ensuring that rounding errors do not degrade the performance of the optimization algorithm. The quadratic program (1.1)-(1.2) can be solved by computing a basis Z for the null space of A, using this basis to eliminate the constraints, and then applying the CG method to the reduced problem. We will argue, however, that due to the form of the preconditioners used in practice, the explicit use of Z will cause the iteration to be very expensive, and that significant savings can be achieved by means of approaches that bypass the computation of Z altogether. The price to pay for these alternatives is that they can give rise to excessive roundoff errors that can slow the optimization iteration and may even prevent it from converging. As we shall see, these errors cause the constraints (1.2) not to be satisfied to the desired accuracy. We describe iterative refinement techniques that can improve the accuracy of the solution in highly ill-conditioned problems. We also propose a mechanism for redefining the vector c adaptively that does not change the solution of the quadratic problem but that has more favorable numerical properties. Notation. Throughout the paper k \Delta k stands for the ' 2 matrix or vector norm, while the G-norm of the vector x is defined to be x T Gx, where G is a given positive-definite matrix. We will denote the floating-point unit roundoff (or machine precision) by ffl m . We let -(A) denote the condition number of A, i.e. are the nonzero singular values of A. 2. The CG method and linear constraints A common approach for solving linearly constrained problems is to eliminate the constraints and solve a reduced problem (c.f. [17, 20]). More specifically, suppose that Z is an n \Theta (n \Gamma m) matrix spanning the null space of A. Then the columns of A T together with the columns of Z span R n , and any solution x of the linear equations (1.2) can be written as for some vectors x A . The constraints (1.2) yield which determines the vector x A . Substituting (2.1) into (1.1), and omitting constant terms is a constant now) we see that x Z solves the reduced problem minimize x Z2 where As we have assumed that the reduced Hessian H ZZ is positive definite, (2.3) is equivalent to the linear system We can now apply the conjugate gradient method to compute an approximate solution of the problem (2.3), or equivalently the system (2.4), and substitute this into (2.1) to obtain an approximate solution of the quadratic program (1.1)-(1.2). This strategy of computing the normal component A T x A exactly and the tangential component Zx Z inexactly is compatible with the requirements of many nonlinear optimization algorithms which need to ensure that, once linear constraints are satisfied, they remain so throughout the remainder of the optimization calculation (cf. [20]). Let us now consider the practical application of the CG method to the reduced system (2.4). It is well known that preconditioning can improve the rate of convergence of the CG iteration (c.f. [1]), and we therefore assume that a preconditioner W ZZ is given. W ZZ is a symmetric, positive definite matrix of dimension which might be chosen to reduce the span of, and to cluster, the eigenvalues of W \Gamma1 or could be the result of an automatic scaling of the variables [7, 29]. Regardless of how W ZZ is defined, the preconditioned conjugate gradient method applied to (2.4) is as follows (see, e.g. [20]). Algorithm I. Preconditioned CG for Reduced Systems. Choose an initial point x Z , compute r ZZ r Z and p \Gammag Z . Repeat the following steps, until a termination test is satisfied: r Z Z ZZ r Z \Gammag Z Z / g Z and r Z / r Z This iteration may be terminated, for example, when r Z ZZ r Z is sufficiently small. Once an approximate solution is obtained, it must be multiplied by Z and substituted in (2.1) to give the approximate solution of the quadratic program (1.1)-(1.2). Alternatively, we may rewrite Algorithm I so that the multiplication by Z and the addition of the term A T x A is performed explicitly in the CG iteration. To do so, we introduce, in the following algorithm, the n-vectors x; Algorithm II Preconditioned CG (in Expanded Form) for Reduced Systems. Choose an initial point x satisfying (1.2), compute \Gammag. Repeat the following steps, until a convergence test is satisfied: x This will be the main algorithm studied in this paper. Several types of stopping tests can be used, but since their choice depends on the requirements of the optimization method, we shall not discuss them here. In the numerical tests reported in this paper we will use the quantity r T ZZ r Z to terminate the CG iteration. Note that the vector g, which we call the preconditioned residual, has been explicitly defined to be in the range of Z. As a result, in exact arithmetic, all the search directions generated by Algorithm II will also lie in the range of Z, and thus the iterates x will all satisfy (1.2). Rounding errors when computing (2.17) may cause p to have a component outside the range of Z, but this component will normally be too small to cause difficulties. 3. Implementation of the Projected CG Method Algorithm II constitutes an effective method for computing the solution to (1.1)-(1.2) and has been successfully used in various algorithms for large scale optimization (cf. [16, 28, 39]). The main drawback is the need for a null-space basis matrix Z, whose computation and manipulation can be costly, and which can sometimes give rise to unnecessary ill-conditioning [9, 10, 18, 24, 33, 37]. These difficulties will become apparent when we describe practical procedures for computing Z and when we consider the types of preconditioners W ZZ used in practice. Let us begin with the first issue. 3.1. Computing a basis for the null space There are many possible choices for the null-space matrix Z. Possibly the best strategy is to choose Z so as to have orthonormal columns, for this provides a well conditioned representation of the null space of A. However computing such a null-space matrix can be very expensive when the number of variables is large; it essentially requires the computation of a sparse LQ factorization of A and the implicit or explicit generation of Q, which has always been believed to be rather expensive when compared with the alternatives described in [24]. Recent research [30, 35] has suggested that it is in fact possible to generate Q as a product of sparse Householder matrices, and that the cost of this may, after all, be reasonable. We have not experimented with this approach, however, because, to our knowledge, general purpose software implementing it is not yet available. Another possibility is to try to compute a basis of the null-space which involves as few nonzeros as possible. Although this problem is computationally hard [9], sub-optimal heuristics are possible but still rather expensive [10, 18, 33, 37]. A more economical alternative is based on simple elimination of variables [17, 20]. To define Z we first group the components of x into m basic or dependent variables (which for simplicity are assumed to be the first m variables) and and partition A as where the m \Theta m basis matrix B is assumed to be nonsingular. Then we define \GammaB I which clearly satisfies and has linearly independent columns. In practice Z is not formed explicitly; instead we compute and store sparse LU factors [13] of B, and compute products of the form Zv and Z T v by means of solves using these LU factors. Ideally we would like to choose a basis B that is as sparse as possible and whose condition number is not significantly worse than that of A, but these requirements can be difficult to achieve. In simply ensuring that B is well conditioned can be difficult when the task of choosing a basis is delegated to a sparse LU factorization algorithm such as MA48 [15]. Some recent codes (see, e.g., [19]) have been designed to compute a well-conditioned basis, but it is not known to us to what extent they reach their objective. 3.2. Preconditioning These potential drawbacks of the null-space basis (3.1) are not sufficiently serious to prevent its effective use in Algorithm II. However, when considering practical choices for the preconditioning matrix W ZZ , one exposes the weaknesses of this approach. Ideally, one would like to choose W ZZ so that W \Gamma1 thus ZZ is the perfect preconditioner. However, it is unlikely that Z T HZ or its inverse are sparse matrices, and even if Z T HZ is of small dimension, forming it can be quite costly. Therefore operating with this ideal preconditioner is normally out of the question. In this paper we consider preconditioners of the form ZZ where G is a symmetric matrix such that Z T GZ is positive definite. Some suggestions on how to choose G have been made in [32]. Two particularly simple choices are The first choice is appropriate when H is dominated by its diagonal. This is the case, for example, in barrier methods for constrained optimization that handle bound constraints l - x - u by adding terms of the form \Gamma- to the objective function, for some positive barrier parameter -. The choice I arises in several trust region methods for constrained optimization [7, 12, 29], where the preconditioner (which derives from a change of variables) is thus given by ZZ Regardless of the choice of G, the preconditioner (3.3) requires operations with the inverse of the matrix Z T GZ. In some applications [16, 39] Z, defined by (3.1), has a simple enough structure that forming and factorizing the (n \Gamma m) \Theta (n \Gamma m) matrix Z T GZ is not expensive when G has a simple form. But if the LU factors of B are not very sparse and the number of constraints m is large, forming Z T GZ may be rather costly, even if as it requires the solution of 2m triangular systems with these LU factors. In this case it is preferable not to form Z T GZ, but rather compute products of the form (Z solving (Z T using the CG method. This inner CG iteration has been employed in [29] with I, and can be effective on some problems-particularly if the number of degrees of freedom, very small. But it can fail when Z is badly conditioned and tends to be expensive. Moreover, since the matrix Z T GZ is not known explicitly, it is difficult to construct effective preconditioners for accelerating this inner CG iteration. In summary when the preconditioner has the form (3.3), and when Z is defined by means of (3.1), the computation (2.15) of the preconditioned residual g is often so expensive as to dominate the cost of the optimization algorithm. The goal of this paper is to consider alternative implementations of Algorithm II whose computational cost is more moderate and predictable. Our approach is to avoid the use of the null-space basis Z altogether. 3.3. Computing Projections To see how to bypass the computation of Z, let us begin by considering the simple case when so that the preconditioner W ZZ is given by (3.4). If P Z denotes the orthogonal projection operator onto the null space of A, then the preconditioned residual (2.15) can be written as This projection can be performed in two alternative ways. The first is to replace P Z by the equivalent formula and thus to replace (3.6) with We can express this as is the solution of Noting that (3.10) are the normal equations, it follows that v + is the solution of the least squares problem minimize and that the desired projection g + is the corresponding residual. This approach can be implemented using a Cholesky factorization of AA T . The second possibility is to express the projection (3.6) as the solution of the augmented system / I A T !/ r +! This system can be solved by means of a symmetric indefinite factorization that uses 1 \Theta 1 and 2 \Theta 2 pivots [21]. Let us suppose now that the preconditioner has the more general form (3.3). The preconditioned residual (2.15) now requires the computation This may be expressed as if G is non-singular, and can be found as the solution of G A T !/ r +! whenever Z T GZ is non-singular (see, e.g., [20, Section 5.4.1]). While (3.14) is far from appealing when G \Gamma1 does not have a simple form, (3.15) is a useful generalization of (3.12). Clearly the system (3.12) may be obtained from (3.15) by setting I, and the perfect preconditioner results if other choices for G are also possible; all that is required is that Z T GZ be positive definite. The idea of using the projection (3.7) in the CG method dates back to at least [34]; the alternative (3.15), and its special case (3.12), are proposed in [8], although [8] unnecessarily requires that G be positive definite. A more recent study on preconditioning the projected CG method is [11]. Hereafter we shall write (2.15) as where P is any of the projection operators we have mentioned above. Note that (3.8), (3.12) and (3.15) do not make use of the null space matrix Z and only require factorization of matrices involving A. Unfortunately they can give rise to significant round-off errors, particularly as the CG iterates approach the solution. The difficulties are caused by the fact that as the iterations proceed, the projected vector increasingly small while r does not. Indeed, the optimality conditions of the quadratic program (1.1)-(1.2) state that the solution x satisfies for some Lagrange multiplier vector -. The vector Hx + c, which is denoted by r in Algorithm II, will generally stay bounded away from zero, but as indicated by (3.16), it will become increasingly closer to the range of A T . In other words r will tend to become orthogonal to Z, and hence, from (3.13), the preconditioned residual g will converge to zero so long as the smallest eigenvalue of Z T GZ is bounded away from zero. That this discrepancy in the magnitudes of will cause numerical difficulties is apparent from (3.9), which shows that significant cancellation of digits will usually take place. The generation of harmful roundoff errors is also apparent from (3.12)/(3.15) because will be small while the remaining components v + remain large. Since the magnitude of the errors generated in the solution of (3.12)/(3.15) is governed by the size of the large component v + , the vector g + will contain large relative errors. These arguments will be made more precise in the next section. Example 1. We applied Algorithm II to solve problem CVXEQP3 from the CUTE collection [4], with In this and all subsequent experiments, we use the simple preconditioner (3.4) corresponding to the choice used both the normal equations (3.8) and augmented system (3.12) approaches to compute the projection. The results are given in Figure 1, which plots the residual r T g as a function of the iteration number. In both cases the CG iteration was terminated when r T g became negative, which indicates that severe errors have occurred since r T must be positive-continuing the iteration past this point resulted in oscillations in the norm of the gradient without any significant improvement. At iteration 50 of both runs, r is of order 10 5 whereas its projection g is of Figure also plots the cosine of the angle between the preconditioned residual g and the rows of A. More precisely, we define A T where A i is the i-th row of A. Note that this cosine, which should be zero in exact arithmetic, increases indicating that the CG iterates leave the constraint manifold Severe errors such as these are not uncommon in optimization calculations; see x7 and [27]. This is of grave concern as it may cause the underlying optimization algorithms to behave erratically or fail. In this paper we propose several remedies. One of them is based on an adaptive redefinition of r that attempts to minimize the differences in magnitudes between . We also describe several forms of iterative refinement for the projection operation. All these techniques are motivated by the roundoff error analysis given next. 4. Analysis of the Errors We now present error bounds that support the arguments made in the previous section, particularly the claim that the most problematic situation occurs in the latter stages of the PCG Augmented System Iteration resid cos PCG Normal Equations Iteration resid cos Figure 1: Conjugate gradient method with two options for the projection iteration when g + is converging to zero, but r + is not. For simplicity, we shall assume henceforth that A has been scaled so that shall only consider the simplest possible preconditioner, as opposed to exact, quantity will be denoted by a subscript c. Let us first consider the normal equations approach. Here is given by (3.9) where (3.10) is solved by means of the Cholesky factorization of AA T . In finite precision, instead of the exact solution v + of the normal equations we obtain v the error \Deltav with . Recall that ffl m denotes unit roundoff and -(A) the condition number of A. We can now study the total error in the projection vector g + . To simplify the analysis, we will ignore the errors that arise in the computation of the matrix-vector product A T v and in the subtraction given in (3.9), because these errors will be dominated by the error in v + whose magnitude is estimated by (4.1). Under these assumptions, we have from (3.9) that the computed projection and the exact projection 1 The bound (4.1) assumes that there are no errors in the formation of AA T and Ar + , or in the backsolves using the Cholesky factors; this is a reasonable assumption in our context. We should also note that (4.1) can be sharpened by replacing the term possible diagonal scalings D. and thus the error in the projection lies entirely in the range of A T . We then have from (4.1) that the relative error in the projection satisfies This error can be significant when -(A) is large or when is large. Let us consider the ratio (4.4) in the case when kr much larger than its projection We have from (3.9) that kr and by the assumption that Suppose that the inequality above is achieved. Then (4.4) gives which is simpler to interpret than (4.4). We can thus conclude that the error in the projection (4.3) will be large when either -(A) or the ratio kr large. When the condition number -(A) is moderate, the contribution of the ratio (4.4) to the relative error (4.3) is normally not large enough to cause failure of the optimization calculation. But as the condition number -(A) grows, the loss of significant digits becomes severe, especially since -(A) appears squared in (4.3). In Example 1, and we have mentioned that the ratio (4.4) is of order O(10 6 ) at iteration 50. The bound (4.3) indicates that there could be no correct digits in g + , at this stage of the CG iteration. This is in agreement with our test, for at this point the CG iteration could make no further progress. Let us now consider the augmented system approach (3.15). Again we will focus on the choice I, for which the preconditioned residual is computed by solving I A T !/ r +! using a direct method. There are a number of such methods, the strategies of Bunch and Kaufman [5] and Duff and Reid [14] being the best known examples for dense and sparse matrices, respectively. Both form the LDL T factorization of the augmented matrix (i.e. the matrix appearing on the left hand side of (4.5)), where L is unit lower triangular and D is block diagonal with 1 \Theta 1 or 2 \Theta 2 blocks. This approach is usually (but not always) more stable than the normal equations ap- proach. To improve the stability of the method, Bj-orck [2] suggests introducing a parameter ff and solving the equivalent system !/ r +! An error analysis [3] shows that where j depends on n and m and in the growth factor during the factorization, and oe 1 - are the nonzero singular values of A. It is important to notice that now -(A)-and not - 2 (A)-enters in the bound. If ff - oe m (A), this method will give a solution that is never much worse than that obtained by a tight perturbation analysis, and therefore can be considered stable for practical purposes. But approximating oe m (A) can be difficult, and it is common to simply use In the case which concerns us most, when kg converges to zero while kv the term inside the last square brackets in (4.7) is approximately kv + k, and we obtain where we have assumed that ff = 1. It is interesting to compare this bound with (4.3). We see that the ratio (4.4) again plays a crucial role in the analysis, and that the augmented system approach is likely to give a more accurate solution than the method of normal equations in this case. This cannot be stated categorically, however, since the size of the factor j is difficult to predict. The residual update strategy described in x6 aims at minimizing the contribution of the ratio (4.4), and as we will see, has a highly beneficial effect in Algorithm II. Before presenting it, we discuss various iterative refinement techniques designed to improve the accuracy of the projection operation. 5. Iterative Refinement Iterative refinement is known as an effective procedure for improving the accuracy of a solution obtained by a method that is not backwards stable. We will now consider how to use it in the context of our normal equations and augmented system approaches. 5.1. Normal Equations Approach Let us suppose that we choose I and that we compute the projection P A r the normal equations approach (3.9)-(3.10). An appealing idea for trying to improve the accuracy of this computation is to apply the projection repeatedly. Therefore rather than computing in (2.15), we let where the projection is applied as many times as necessary to keep the errors small. The motivation for this multiple projections technique stems from the fact that the computed projection have only a small component, consisting entirely of rounding errors, outside of the null space of A, as described by (4.2). Therefore applying the projection P A to the first projection c will give an improved estimate because the ratio (4.4) will now be much smaller. By repeating this process we may hope to obtain further improvement of accuracy. The multiple projection technique may simply be described as setting g + performing the following steps: solve L(L set where L is the Cholesky factor of AA T . We note that this method is only appropriate when although a simple variant is possible when G is diagonal. Example 2. We solved the problem given in Example 1 using multiple projections. At every CG iteration we measure the cosine (3.17) of the angle between g and the columns of A. If this cosine is greater than 10 \Gamma12 , then multiple projections are applied until the cosine is less than this value. The results are given in Figure 2, and show that the residual r T g was reduced much more than in the plane CG iteration (Figure 1). Indeed the ratio between the final and initial values of r T g is 10 \Gamma16 , which is very satisfactory. It is straightforward to analyze the multiple projections strategy (5.1)-(5.2) provided that, as before, we make the simplifying assumption that the only rounding errors we make are in forming L and solving (5.1). We obtain the following result which can be proved by induction. For where as in (4.1) A simple consequence of (5.3)-(5.4) and the assumption that A has norm one is that and thus that the error converges R-linearly to zero with rate Of course, this rate can not be sustained indefinitely as the other errors we have ignored in (5.1)-(5.2) become important. Nonetheless, one would expect (5.5) to reflect the true behaviour until k(g small multiple of the unit roundoff ffl m . It should Iteration residual Figure 2: CG method using multiple projections in the normal equations approach. be stressed, however, that this approach is still limited by the fact that the condition number of A appears squared in (5.5); improvement can be guaranteed only if We should also note that multiple projections are almost identical in their form and numerical properties to fixed precision iterative refinement to the least squares problem [3, p.125]. Fixed precision iterative refinement is appropriate because the approach we have chosen to compute projections is not stable. To see this, compare (4.3) with a perturbation analysis of the least squares problem [3, Theorem 1.4.6]), which gives Here the dependence on the condition number is linear-not quadratic. Moreover, since is multiplied by kg is small the effect of the condition number of A is much smaller in (5.7) than in (4.3). We should mention two other iterative refinement techniques that one might consider, but that are either not effective or not practical in our context. The first is to use fixed-precision iterative refinement [3, Section 2.9] to attempt to improve the solution v + of the normal equations (3.10). This, however, will generally be unsuccessful because fixed-precision iterative refinement only improves a measure of backward stability [21, p.126], and the Cholesky factorization is already a backward stable method. We have performed numerical tests and found no improvement from this strategy. However, as is well known, iterative refinement will often succeed if extended-precision is used to evaluate the residuals. We could therefore consider using extended precision iterative refinement to improve the solution v + of the normal equations (3.10). So long as and the residuals of (3.10) are smaller than one in norm, we can expect that the error in the solution of (3.10) will decrease by a factor ffl m-(A) 2 until it reaches O(ffl m ). But since optimization algorithms normally use double precision arithmetic for all their computations, extending the precision may not be simple or efficient, and this strategy is not suitable for general purpose software. For the same reason we will not consider the use of extended precision in (5.1)-(5.2) or in the iterative refinement of the least squares problem. 5.2. Augmented System Approach We can apply fixed precision iterative refinement to the solution obtained from the augmented system (3.15). This gives the following iteration. Compute solve G A T !/ \Deltag ae g ae v and update Note that this method is applicable for general preconditioners G. When an appropriate value of ff is in hand, we should incorporate it in this iteration, as described in (4.6). The general analysis of Higham [26, Theorem 3.2] indicates that, if the condition number of A is not too large, we can expect high accuracy in v + and good accuracy in g + in most cases. Example 3. We solved the problem given in Example 1 using this iterative refinement technique. As in the case of multiple projections discussed in Example 2, we measure the angle between g and the columns of A at every CG iteration. Iterative refinement is applied as long as the cosine of this angle is greater than 10 \Gamma12 . The results are given in Figure 3. We observe that the residual r T g is decreased almost as much as with the multiple projections approach, and attains an acceptably small value. We should point out, however, that the residual increases after it reaches the value 10 \Gamma10 , and if the CG iteration is continued for a few hundred more iterations, the residual exhibits large oscillations. We will return to this in x6.1. In our experience 1 iterative refinement step is normally enough to provide good accu- racy, but we have encountered cases in which 2 or 3 steps are beneficial. 6. Residual Update Strategy We have seen that significant roundoff errors occur in the computation of the projected residual vector is much smaller than the residual r + . We now describe a procedure Iteration residual Figure 3: CG method using iterative refinement in the augmented system approach. for redefining r + so that its norm is closer to that of g + . This will dramatically reduce the roundoff errors in the projection operation. We begin by noting that Algorithm II is theoretically unaffected if, immediately after computing r + in (2.14), we redefine it as for some y This equivalence is due to the condition and the fact that r is only used in (2.15) and (2.16). It follows that we can redefine r + by means of (6.1) in either the normal equations approach (3.8)/(3.13) or in the augmented system approach (3.12)/(3.15) and the results would, in theory, be unaffected. Having this freedom to redefine r + , we seek the value of y that minimizes where G is any symmetric matrix for which Z T GZ is positive definite, and G \Gamma1 is the generalized inverse of G. The vector y that solves (6.2) is obtained as This gives rise to the following modification of the CG iteration. Algorithm III Preconditioned CG with Residual Update. Choose an initial point x satisfying (1.2), compute find the vector y that minimizes kr\GammaA T y, compute and set \Gammag. Repeat the following steps, until a convergence test is satisfied: x This procedure works well in practice, and can be improved by adding iterative refinement of the projection operation. In this case, at most 1 or 2 iterative refinement steps should be used. Notice that there is a simple interpretation of Steps (6.6) and (6.7). We first obtain y by solving (6.2), and as we have indicated the required value is (3.15). But (3.15) may be rewritten as G A T !/ and thus when we obtain g + in Step (6.7), it is as if we had instead found it by solving (6.11). The advantage of using (6.11) compared to (3.15) is that the solution in the latter may be dominated by the large components v + , while in the former g + are the large componentsof course, in floating point arithmetic, the zero component in the solution of (6.11) will instead be tiny rounded values provided (6.11) is solved in a stable fashion. Viewed in this way, we see that Steps (6.6) and (6.7) are actually a limited form of iterative refinement in which the computed v + , but not the computed g + which is discarded, is used to refine the solution. This "iterative semi-refinement" has been used in other contexts [6, 22]. There is another interesting interpretation of the reset r / r \Gamma A T y performed at the start of Algorithm III. In the parlance of optimization, c is the gradient of the objective function (1.1) and r \Gamma A T y is the gradient of the Lagrangian for the problem (1.1)-(1.2). The vector y computed from (6.2) is called the least squares Lagrange multiplier estimate. (It is common, but not always the case, for optimization algorithms to set in (6.2) to compute these multipliers.) Thus in Algorithm III we propose that the initial residual be set to the current value of the gradient of the Lagrangian, as opposed to the gradient of the objective function. One could ask whether it is sufficient to do this resetting of r at the beginning of Algorithm III, and omit step (6.6) in subsequent iterations. Our computational experience shows that, even though this initial resetting of r reduces its magnitude sufficiently to avoid errors in the first few CG iteration, subsequent values of r can grow, and rounding errors may reappear. The strategy proposed in Algorithm III is safe in that it ensures that r is small at every iteration, but one can think of various alternatives. One of them is to monitor the norm of r and only apply the residual update when it seems to be growing. 6.1. The Case There is a particularly efficient implementation of the residual update strategy when I. Note that (6.2) is precisely the objective of the least squares problem (3.11) that occurs when computing via the normal equations approach, and therefore the desired value of y is nothing other than the vector v + in (3.10) or (3.12). Furthermore, the first block of equations in (3.12) shows that r . Therefore, in this case (6.6) can be replaced by r and (6.7) is In other words we have applied the projection operation twice, and this is a special case of the multiple projections approach described in the previous section. Based on these observations we propose the following variation of Algorithm III that requires only one projection per iteration. We have noted that (6.6) can be written as . Rather than performing this projection, we will define r where g is the projected residual computed at the previous iteration. The resulting iteration is given by Algorithm III with the following two changes: Omit Replace (6.10) by g / g + and r / This strategy has performed well in our numerical experiments and avoids the extra storage and computation required by Algorithm III. We now show that it is mathematically equivalent to Algorithm III - which in turn is mathematically equivalent to Algorithm II. The arguments that follow make use of the fact that, when we have that The first iteration is clearly the same as that of Algorithm III, except that the value we store in r in the last step is not r us consider the effect that this has on the next iteration. The numerator in the definition (6.3) of ff now becomes T g which equals r T P g. Thus the formula of ff is theoretically unchanged, but the has the advantage that it can never be negative, as is the case with (6.3) when rounding errors dominate the projection operation. Next, the step which is different from the value calculated in Algorithm III. Step (6.6) is omitted in the new variant of Algorithm III. The projected residual calculated in (6.7) is now P (P r +ffHp) which is mathematically equivalent to the value PP (r +ffHp) calculated in Algorithm III (recall that (6.6) can be written as that the new strategy applies the double projection only to r. Finally let us consider the numerator in (6.8). In the new variant, it is given by whereas in Algorithm III it is given by By expanding these expressions we see that the formula for fi is mathematically equivalent in both cases, but that in the new variant the projection is applied selectively. Example 4. We solved the problem given in Example 1 using this residual update strategy with I. The results are given in Figure 4 and show that the normal equations and augmented system approaches are equally effective in this case. We do not plot the cosine (3.17) of the angle between the preconditioned residual and the columns of A because it was very small in both approaches, and did not tend to grow as the iteration progressed. For the normal equations approach this cosine was of order 10 \Gamma14 throughout the CG iteration; for the augmented system approach it was of order 10 \Gamma15 . Note that we have obtained higher accuracy than with the iterative refinement strategies described in the previous section; compare with Figures 2 and 3. Augmented System Iteration residual Normal Equations Iteration residual Figure 4: Conjugate gradient method with the residual update strategy. To obtain a highly reliable algorithm for the case when I we can combine the residual update strategy just described with iterative refinement of the projection operation. This gives rise to the following iteration which will be used in the numerical tests reported in x7. Algorithm IV Residual Update and Iterative Refinement for Choose an initial point x satisfying (1.2), compute where the projection is computed by the normal equations (3.8) or augmented system (3.12) approaches, and set \Gammag. Choose a tolerance ' max . Repeat the following steps, until a convergence test is satisfied: x Apply iterative refinement to P r until (3.17) is less than ' We conclude this discussion by elaborating on the point made before Example 4 concerning the computation of the steplength parameter ff. We have noted that the formula is preferable to (6.12) since it cannot give rise to cancellation. Similarly the stopping test should be based on g T g rather than on g T r. The residual update implemented in Algorithm IV does this change automatically, but we believe that these expressions are to be recommended in other implementations of the CG iteration, provided the preconditioner is based on To test this, we repeated the computation reported in Example I using the augmented system approach; see Figure 1. The only change is that Algorithm II now used the new for ff and for the stopping test. The CG iteration was now able to continue past iteration 70 and was able to reach the value We also repeated the calculation made in Example 3. Now the residual reached the level and the large oscillations in the residual mentioned in Example 3 no longer took place. Thus in both cases these alternative expressions for ff and for the stopping test were beneficial. 6.2. General G We can also improve upon the efficiency of Algorithm III for general G, using slightly outdated information. The idea is simply to use the obtained when computing g + in (6.7) as a suitable y rather than waiting until after the following step (6.5) to obtain a slightly more up-to-date version. The resulting iteration is given by Algorithm III, with the following two changes: Omit Replace (6.10) by g / g + and r / r obtained as a bi-product from (6.7). Notice, however, that for general G, the extra matrix-vector product A T v + will be required, since we no longer have the relationship that we exploited when Although we have not experimented on this idea here, it has proved to be beneficial in other, similar circumstances [22]. 7. Numerical Results We now test the efficacy of the techniques proposed in this paper on a collection of quadratic programs of the form (1.1)-(1.2). The problems were generated during the last iteration of the interior point method for nonlinear programming described in [7], when this method was applied to a set of test problems from the CUTE [4] collection. We apply the CG method with preconditioner (3.4) (i.e. with to solve these quadratic programs. We use the augmented system and normal equations approaches to compute projections, and for each we compare the standard CG iteration (stand) with the iterative refinement (ir) techniques described in x5 and the residual update strategy combined with iterative refinement (update) as given in Algorithm IV. The results are given in Table 1. The first column gives the problem name, and the second, the dimension of the quadratic program. To test the reliability of the techniques proposed in this paper we used a very demanding stopping test: the CG iteration was terminated when In these experiments we included several other stopping tests in the CG iteration, that are typically used by trust region methods for optimization. We terminate if the number of iterations exceeds 2(n \Gamma m) where denotes the dimension of the reduced system (2.4); a superscript 1 in Table 1 indicates that this limit was reached. The CG iteration was also stopped if the length of the solution vector is greater than a "trust region radius" that is set by the optimization method (see [7]). We us a superscript 2 to indicate that this safeguard was activated, and note that in these problems only excessive rounding errors can trigger it. Finally we terminate if p T Hp ! 0, indicated by 3 or if r T g ! 0, indicated by 4 . Note that the standard CG iteration was not able to meet the stopping test for any of the problems in Table 1, but that iterative refinement and update residual were successful in most cases. Table 2 reports the CPU time for the problems in Table 1. Note that the times for the standard CG approach (stand) should be interpreted with caution, since in some of these problems it terminated prematurely. We include the times for this standard CG iteration only to show that the iterative refinement and residual update strategies do not greatly increase the cost of the CG iteration. Next we report on 3 problems for which the stopping test could not be met by any of the variants. For these three problems, Table 3 provides the least residual norm attained for each strategy. As a final, but indirect test of the techniques proposed in this paper, we report the results obtained with the interior point nonlinear optimization code described in [7] on 29 nonlinear programming problems from the CUTE collection. This code applies the CG method to solve a quadratic program at each iteration. We used the augmented system Augmented System Normal Equations Problem dim stand ir update stand ir update CORKSCRW 147 COSHFUN OPTCTRL6 Table 1: Number of CG iterations for the different approaches. A 1 indicates that the iteration limit was reached, 2 indicates termination from trust region bound, 3 indicates negative curvature was detected and 4 indicates that r T Augmented System Normal Equations Problem dim stand ir update stand ir update COSHFUN OPTCTRL6 Table 2: CPU time in seconds. 1 indicates that the iteration limit was reached, 2 indicates termination from trust region bound, 3 indicates negative curvature was detected and 4 indicated that r T Augmented System Normal Equations Problem dim stand ir update stand ir update OBSTCLAE 900 2.3D-07 1.5D-07 5.5D-08 2.3D-07 9.9D-08 4.2D-08 Table 3: The least residual norm: r T g attained by each option. and normal equations approaches to compute projections, and for each of these strategies we tried the standard CG iteration (stand) and the residual update strategy (update) with iterative refinement described in Algorithm IV. The results are given in Table 4, where "fevals" denotes the total number of evaluations of the objective function of the nonlinear problem, and "projections" represents the total number of times that a projection operation was performed during the optimization. A * indicates that the optimization algorithm was unable to locate the solution. Note that the total number of function evaluations is roughly the same for all strategies, but there are a few cases where the differences in the CG iteration cause the algorithm to follow a different path to the solution. This is to be expected when solving nonlinear problems. Note that for the augmented system approach, the residual update strategy changes the number of projections significantly only in a few problems, but when it does the improvements are very substantial. On the other hand, we observe that for the normal equations approach (which is more sensitive to the condition number -(A)) the residual update strategy gives a substantial reduction in the number of projections in about half of the problems. It is interesting that with the residual update, the performance of the augmented system and normal equations approaches is very similar. 8. Conclusions We have studied the properties of the projected CG method for solving quadratic programming problems of the form (1.1)-(1.2). Due to the form of the preconditioners used by some nonlinear programming algorithms we opted for not computing a basis Z for the null space of the constraints, but instead projecting the CG iterates using a normal equations or augmented system approach. We have given examples showing that in either case significant roundoff errors can occur, and have presented an explanation for this. We proposed several remedies. One is to use iterative refinement of the augmented system or normal equations approaches. An alternative is to update the residual at every iteration of the CG iteration, as described in x6. The latter can be implemented particularly efficiently when the preconditioner is given by I in (3.3). Our numerical experience indicates that updating the residual almost always suffices to keep the errors to a tolerable level. Iterative refinement techniques are not as effective by themselves as the update of the residual, but can be used in conjunction with it, and the numerical results reported in this paper indicate that this combined strategy is both economical and accurate. 9. Acknowledgements The authors would like to thank Andy Conn and Philippe Toint for their helpful input during the early stages of this research. Augmented System Normal Equations f evals projections f evals projections Problem n m stand update stand update stand update stand update CORKSCRW 456 350 64 61 458 422 COSHFUN GAUSSELM 14 11 25 26 92 93 28 41 85 97 HAGER4 2001 1000 OBSTCLAE 1024 0 26 26 6233 6068 26 26 6236 6080 OPTCNTRL OPTCTRL6 122 Table 4: Number of function evaluations and projections required by the optimization method for the different implementations of the CG iteration. --R Iterative solution methods. CUTE: Constrained and unconstrained testing environment. Some stable methods for calculating inertia and solving symmetric linear equations. Linear least squares solutions by Housholder transfor- mations Primal and primal-dual methods for nonlinear programming Linearly constrained optimization and projected preconditioned conjugate gradients. The null space problem I: Complexity. The null space problem II: Algorithms. A preconditioned conjugate gradient approach to linear equality constrained minimization. A global convergence theory for general trust-region based algorithms for equality constrained optimization Direct methods for sparse matrices. The multifrontal solution of indefinite sparse symmetric linear equations. The design of MA48 Practical Methods of Optimization. Computing a sparse basis for the null-space SNOPT: an SQP algorithm for large-scale constrained optimization Practical Optimization. Matrix Computations. Iterative methods for ill-conditioned linear systems from optimiza- tion Solving the trust-region subproblem using the Lanczos method Sparse orthogonal schemes for structural optimization using the force method. Methods of conjugate gradients for solving linear systems. Iterative refinement and LAPACK. Implicit nullspace iterative methods for constrained least squares problems. On the implementation of an algorithm for large-scale equality constrained optimization Multifrontal computation with the orthogonal factors of sparse matrices. Indefinitely preconditioned inexact newton method for large sparse equality constrained nonlinear programming problems. Preconditioning reduced matrices. Substructuring methods for computing the null space of equilibrium matrices. The conjugate gradient method in extremal problems. QR Factorization of Large Sparse Overdetermined and Square Matrices with the Multifrontal Method in a Multiprocessing Environment. The conjugate gradient method and trust regions in large scale optimiza- tion Nested dissection for sparse nullspace bases. Towards an efficient sparsity exploiting Newton method for minimization. On large-scale nonlinear network optimization --TR --CTR 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 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. 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 Nicholas I. M. Gould , Dominique Orban , Philippe L. Toint, GALAHAD, a library of thread-safe Fortran 90 packages for large-scale nonlinear optimization, ACM Transactions on Mathematical Software (TOMS), v.29 n.4, p.353-372, December 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 Nicholas I. M. Gould , Philippe L. Toint, An iterative working-set method for large-scale nonconvex quadratic programming, Applied Numerical Mathematics, v.43 n.1-2, p.109-128, October 2002 Meizhong Dai , David P. Schmidt, Adaptive tetrahedral meshing in free-surface flow, Journal of Computational Physics, v.208 n.1, p.228-252, 1 September 2005 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

conjugate gradient method;nonlinear optimization;quadratic programming;iterative refinement;preconditioning

587203

Practical Construction of Modified Hamiltonians.

One of the most fruitful ways to analyze the effects of discretization error in the numerical solution of a system of differential equations is to examine the "modified equations," which are equations that are exactly satisfied by the (approximate) discrete solution. These do not actually exist in general but rather are defined by an asymptotic expansion in powers of the discretization parameter. Nonetheless, if the expansion is suitably truncated, the resulting modified equations have a solution which is remarkably close to the discrete solution. In the case of a Hamiltonian system of ordinary differential equations, the modified equations are also Hamiltonian if and only if the integrator is symplectic. Evidence for the existence of a Hamiltonian for a particular calculation is obtained by calculating modified Hamiltonians and monitoring how well they are conserved. Also, energy drifts caused by numerical instability are better revealed by evaluating modified Hamiltonians. Doing this calculation would normally be complicated and highly dependent on the details of the method, even if differences are used to approximate derivatives. A relatively simple procedure is presented here, nearly independent of the internal structure of the integrator, for obtaining highly accurate estimates for modified Hamiltonians. As a bonus of the method of construction, the modified Hamiltonians are exactly conserved by a numerical solution in the case of a quadratic Hamiltonian.

Introduction One of the most fruitful ways to analyze the effects of discretization error in the numerical solution of differential equations is to examine the "modified equations," which are the equations that are exactly satisfied by the (approximate) discrete solution. These do not actually exist (in general), but rather are defined by an asymptotic expansion in powers of the discretization parameter. Nonetheless, if the expansion is suitably truncated, the resulting modified equations have a solution which is remarkably close to the discrete solution [9]. In the case of a Hamiltonian system of The work of R. D. Skeel was supported in part by NSF Grants DMS-9971830, DBI-9974555 and NIH Grant P41RR05969 and completed while visiting the Mathematics Department, University of California, San Diego. y z ordinary differential equations, the modified equations are also Hamiltonian if and only if the integrator is symplectic. The existence of a modified, or "shadow" [4], Hamiltonian is an indicator of the validity of statistical estimates calculated from long time integration of chaotic Hamiltonian systems [18]. In addition, the modified Hamiltonian is a more sensitive indicator than is the original Hamiltonian of drift in the energy (caused by instability). Evidence for the existence of a Hamiltonian for a particular calculation can be obtained by calculating modified Hamiltonians and monitoring how well they are conserved. Doing this calculation would normally be complicated and highly dependent on the details of the method, even if differences are used to approximate higher derivatives. Presented here is a relatively simple procedure, nearly independent of the internal structure of the integrator, for obtaining highly accurate estimates for modified Hamiltonians. Consider a step by step numerical integrator x evolves an approximate solution x n - x(nh) for a system of ordinary differential equations - f(x). For such discrete solutions there exists modified equations - defined by an asymptotic expansion such that formally the numerical solution x (nh). The modified right-hand-side function f h is defined uniquely by postulating an asymptotic expansion f in powers of h, substituting this into the equations for the numerical solution, expanding in powers of h, and equating coefficients [26, 6, 22]. The asymptotic expansion does not generally converge except for (reasonable integrators applied to) linear differential equations. A Hamiltonian system is of the form I for some Hamiltonian H(x), . The modified equation for an integrator \Phi h applied to this system is Hamiltonian, i.e., f modified Hamiltonian H h (x), if and only if the integrator is symplectic [23, 20]. The integrator is symplectic if \Phi h;x (x)J T \Phi h;x (x) j J . There is theoretical [2, 8, 18] and empirical evidence that very small error for a very long time where x h is the solution for a suitably truncated Hamiltonian H h . In what follows we assume that H h is such a Hamiltonian and we neglect the very small error. If we plot total energy as a function of time for a numerical integrator such as leapfrog/St-ormer/ Verlet applied to a molecular dynamics simulation, we get a graph like Fig. 3. What we observe are large fluctuations in the original Hamiltonian, as the trajectory moves on a hypersurface of constant modified Hamiltonian. A small drift or jump in the energy would be obscured by the fluctuations. A plot of a modified Hamiltonian might be more revealing. As an example, the plots of modified Hamiltonians in Fig. 4 show a clear rise in energy already in a 400-step simulation. This indicates that plots of suitable modified Hamiltonians can make it easier to test integration algorithms for instability and programming bugs. Details of this and other numerical tests are given in section 2. Before continuing, it is worth emphasizing that the concern of this paper is stability monitoring-not the monitoring and enhancement of accuracy, as in [4] and [15]. The goal is to construct an approximate modified Hamiltonian that can be conveniently assembled from quantities, such as forces and energies, already available from the numerical integration. We consider the special separable Hamiltonian H(q; which the system is of the form A "brute force" approach would be to determine an asymptotic expansion for H h and of the quantities available for making an approximation and then to form a suitable linear combination of the latter. By such a matching of asymptotic expansions one could derive the following modified Hamiltonians for the leapfrog method: Here a superscript n denotes evaluation at q n , the centered difference operator is defined by ffi w the averaging operator is defined by -w are defined in terms of q n , p n by the leapfrog method. An easier and more elegant construction is presented in Secs. 3-5. The technique is developed only for splitting methods. It is likely that a similar construction is also possible for symplectic implicit Runge-Kutta methods. The idea is to add a new position and conjugate momentum variable to get an extended Hamiltonian - H h (y) which is homogeneous of order 2. For such a Hamiltonian Jy h (t). Thus the problem is reduced to that of forming an approximation for using the numerical solution of an extended Hamiltonian system. It is plausible that such a construction might be useful theoretically due to the existence of robust approximation techniques. Eq. (1) for H [2] contains an h 2 term which is not needed for achieving 2nd order accuracy. It is present because the truncations H [2k] are designed to exactly conserve energy for the numerical solution when H is quadratic. (See Secs. 4.1 and 5.1.) This is a very useful property because typical applications, including molecular dynamics, are dominated by harmonic motion. The existence of a modified Hamiltonian that is exactly conserved for a quadratic Hamiltonian is noted in [17, Eq. (4.7b)], and the search for similar methods having this property was central to the results of this paper. For a quadratic Hamiltonian the modified Hamiltonian H h actually exists (if h is not too large), but H [2k] 6= H h . simple derivation of H h for the one-dimensional case is given in [22].) Also, it should be noted that the Hamiltonians H [2k] will not detect numerical instability in the case of quadratic Hamiltonians H . The modified Hamiltonians H [2k] (x), defined by Eqs. (15), (8), (14), (7) are computed and plotted as functions of time for numerical solutions generated by the leapfrog method, given energy time (fs) 8th order 6th order 4th order energy 2nd order Figure 1: Energy and various truncations of modified Hamiltonians for decalanine. by (9). The unmodified Hamiltonians are those of classical molecular dynamics. The testing was done with a molecular dynamics program written by the second author, which is compatible with NAMD [11, 16] but limited in features to facilitate algorithm testing. The first couple of experiments demonstrate the quality of the modified Hamiltonians. The test problem is a 66-atom peptide, decalanine, in a vacuum [1]. The force field parameters are those of CHARMM 22 for proteins [13, 14] without cutoffs for nonbonded forces. Figure 1 shows a plot of the Hamiltonian and 2nd, 4th, 6th, and 8th order modified Hamiltonian approximations vs. time for 100000 fs (femtoseconds) for a step size with the energy sampled every 8th step. The level graph at the top is the 8th order truncation, the one just barely beneath it is 6th order, and the one under that is 4th order. The greatly fluctuating graph is the energy itself and the undulating one well below it is the 2nd order truncation. Note how well the asymptotic theory holds for the higher order truncations-one could not obtain such flat plots by simply smoothing the original Hamiltonian. Figure 2 expands the vertical scale to show fluctuations in the 8th, 6th, and 4th order truncations of modified Hamiltonians. An explanation is in order concerning the initial drop in energy. Because a symplectic method preserves volume in phase space and because there is less phase space volume at lower energies, it can be inferred that the first part of the trajectory is simply the second half of a very unusual fluctuation. In other words the initial conditions are atypical, i.e., not properly equilibrated (with respect to the original Hamiltonian). This is particularly well revealed by the plot of the 2nd order truncation. The remaining experiments demonstrate the ability of modified Hamiltonians to detect instabil- ity. The test problem is a set of 125 water molecules harmonically restrained to a 10 - A-radius sphere. The water is based on the TIP3P model [25] without cutoffs and with flexibility incorporated by adding bond stretching and angle bending harmonic terms (cf. Ref. [12]). energy time (fs) 8th order 6th order 4th order Figure 2: Closer look at higher order truncations for decalanine. energy time (fs) energy Figure 3: Energy for flexible water with step size 2.5 fs. Figure 3 shows a plot of the energy vs. time for 1 000 fs for a step size with the energy sampled every step. Note that the large fluctuations make it is difficult to determine whether or not there is energy drift. Figure 4 shows a plot of the 6th and 8th order modified Hamiltonians for the same step size energy time (fs) 8th order 6th order Figure 4: 6th and 8th order truncations with step size 2.5 fs. fs. An upward energy drift is now obvious. The 2nd and 4th order approximations are not shown because neither of them were as flat. Normal mode analysis for this system [10] shows that the 250 fastest frequencies have periods in the range 9.8-10.2 fs and use of the formula in [22, p. 131] shows that a 2.5 fs step size is 30% of the effective period for discrete leapfrog dynamics. It is remarkable that the 8th order approximation is the flattest, even for such a large step size. Figure 5, shows a plot of the 6th and 8th order modified Hamiltonians for step size There is no apparent upward drift of the energy. Theoretically instability due to 4:1 resonance [21] should occur for the leapfrog method at h \Delta angular frequency = 2, which is in the range 2.2-2.3 fs for flexible water. 3 Augmenting the Integrator We assume that one step of size h for the given method applied to a system with Hamiltonian H is the composition of exact h-flows for Hamiltonian systems with Hamiltonians H 1 , H 2 , . , HL . Each H l (x) is assumed to be sufficiently smooth on some domain containing the infinite time trajectory. For example, 1. the leapfrog method for separable Hamiltonian systems H(q; 2. the Rowlands method [19] for special separable Hamiltonian systems uses H 1 3. double time-stepping [7, 24] uses 4 U fast (q), H 2 fast (q), energy time (fs) 8th order 6th order Figure 5: 6th and 8th order truncations with step size 2.15 fs. 4. Molly [5] does the same as double time-stepping except for the substitution of U slow (A(q)) for U slow (q) where A(q) is a local temporal averaging of q over vibrational motion. We defined the homogeneous extension of a Hamiltonian by If H is quadratic, then - H is homogeneous of order 2: . The extended Hamiltonian yields the augmented system With initial condition and the system simplifies to For p+U(q), the extended Hamiltonian is - and the simplified augmented system is Remark. The association of ff with q rather than fi is of practical importance in that we want to get values of - p is calculated. The following proposition shows that the value of the extended Hamiltonian can be calculated knowing just the solution: H(y) be the homogeneous extension of a given Hamiltonian H(x), and let y(t) be a solution of the extended Hamiltonian system with ff initially 1. Then J is the matrix I of augmented dimension. Proof. Differentiating Eq. (4) with respect to oe gives - Because - H is a homogeneous extension of H , the solution of - H "includes" that of H and we have Of course, the goal is not to calculate the original Hamiltonian, for which we know a formula but not the solution; rather, it is to calculate a modified Hamiltonian, for which we know the solution (at grid points) but not a formula. Therefore, we must augment the integrator so that its solution at grid points is that of the homogeneous extension of the modified Hamiltonian. For an integrator that is a composition of Hamiltonian flows, this is accomplished by using the homogeneous extension of each of the constituent Hamiltonians. More specifically, we define the augmented method y H to be the composition of exact flows for systems with Hamiltonians - HL where H l (q; ff; Lemma 1 (Commutativity) The method \Psi h defined above has a modified Hamiltonian where H h (q; p) is the modified Hamiltonian of the original method \Phi h , i.e., the following diagram commutes: homogeneous extension discretization # discretization homogeneous extension Proof. The modified Hamiltonian - H h for method \Psi h can be expressed as an asymptotic expansion using the Baker-Campbell-Hausdorf formula [20]. This formula combines Hamiltonians using the operations of scalar multiplication, addition, and the Poisson bracket fH; x JN x . It is thus sufficient to show that each of these three commute with the operation of forming the homogeneous extension. We show this only for the last of these. The homogeneous extension of the Poisson bracket is This is exactly the same as the (extended) Poisson bracket of ff Remark. The aim is to discretize the extended Hamiltonian so that this commutativity property holds. Extension of this technique to implicit Runge-Kutta methods would require an augmentation of the method so that commutativity holds. The following corollary allows the value of the Hamiltonian to be approximated from known values of y h (t) at grid points: Proposition 2 Let x h (t) and y h (t) be the solutions for modified Hamiltonians H h and - tively. Then Jy h (t): Proof. Similar to that of Prop. 1. 2 4 Using Full Step Values This section presents the construction of H [2k] for even values of k. Let y h (t) be the solution of the modified extended Hamiltonian system with initial condition y. It has values y h (t) be the degree k polynomial interpolant of these values. (For large k it may be preferable, instead, to use trigonometric interpolation suitably modified [3].) From Prop. 2, 1- jh Z jh=2 \Gammaj h=22 - The interpolant - k where the error (t) and with the brackets denoting a 1)th divided difference. Noting that - jh Z jh=2 \Gammaj h=22 - Je(t)dt \Gammajh Z jh=2 \Gammaj h=22 y h (t) T - Z jh=2 \Gammaj h=22 - Je(t)dt +O(h 2k+2 ) Z jh=2 \Gammaj h=2 where the second equation is obtained by integrating by parts and where fl(t) def Jy [k+1] (t). This can be expressed as an expansion By forming a suitable linear combination of the values H k;j , it is expected that one can get - H h with the first k=2 \Gamma 1 leading error terms eliminated: linear combination of the H Note. The value - contains a leading term that is only O(h k ), so it is not useful for eliminating error terms. The case is the 4th order accurate formula For the case \Gammah and Z 2h \Gamma2h and hence, Below are given formulas for H [8] and for H [4] in terms of values of y h (t) at grid points. Let a j be the jth centered difference of y h (t) at where the centered difference operator is defined by ffi and the averaging operator is defined by The 4th degree interpolant in divided difference form is Hence, 6 a 3 s(s and 6 a 4 s(s and we have2 powers of s: Averaging over and averaging over \Gamma2 - s - 2 yields 90 A 14 \Gamma 107 Therefore, For a 2nd degree interpolant it follows from Eq. (5) that An implementation of these formulas might calculate H [2k] consecutive values of n in terms of quantities A n defined in terms of centered differences of y n which can be obtained from the x n . (Only 1st and higher differences of fi n are needed.) Example 1. To make this concrete, we calculate H [4] (x) for the leapfrog method, as given by Eq. (2). The leapfrog method advances one step by We have Suppressing the n in the superscript, y whence not needed7 7 From so Therefore, 4.1 The case of a quadratic Hamiltonian The following result implies that, in the case where H(x) is quadratic, the numerical solution exactly conserves an approximate modified Hamiltonian which is a linear functional of 1- where -(t) is a linear combination of numerical solution values. Proposition 3 Assume that \Phi h is the composition of flows for systems with quadratic Hamiltonians and that \Psi h is constructed as in Proposition 1. Then the quantity a i;j \Psi i where the sum is taken over a finite set of pairs of integers, is exactly conserved by method \Psi h . Proof. The mapping \Psi h is the composition of flows for systems with homogeneous quadratic Hamiltonians. Then a i;j (S i Sy) T - a a i;j \Psi i (y):5 Using Intermediate Values This section presents the construction of H [2k] for odd values of k. For most numerical integrators one can define "sensible" mid-step values, and these can be used instead of full step values to get an estimate accurate up to O(h 2k ). We assume that \Psi \Gammah=2 \Psi h=2 is a composition of exact flows of homogeneously extended Hamiltonians. Remark. It is not necessary that the mid-step values be approximations to y(t) at midpoints nor that \Psi h be time symmetric (\Psi \Gamma1 we need is that \Psi where each of \Psi 1;h , \Psi 2;h is a composition of exact flows of homogeneously extended Hamiltonians. For example, the leapfrog method separates into half steps - \Gammah=2 \Psi h=2 with - \Psi h=2 as follows: Remark. For the leapfrog method the estimate over an interval from (n\Gamma 1 2 k)h to (n+ 1 k is odd actually uses only values of energy and forces from the shorter interval from (n \Gamma 1k to (n The mid-step values are values at midpoints of some function z h (t) which can be used to construct the Hamiltonian: Proposition 4 Let z h z h (t) T - Jz h (t): Proof. For any real s we define \Psi s sh-flow for - . Because - h is symplectic, z h Hamiltonian system with Hamiltonian - . Also, - h is the composition of flows of Hamiltonians that are 2nd order homogeneous, and hence - is homogeneous of order 2. z h (t) T - (t) be the degree k polynomial interpolant of the values z h (y), As before, let jh Z jh=2 \Gammaj h=22 - The interpolant - k where the error h (t) and z [k+1] Similar to before, we get Z jh=2 \Gammaj h=2 z h (t) T - Jz [k+1] h (t). This can be expressed as an expansion Again, it is expected that a suitable linear combination of the different values of H k;j yields - H h with the first (k leading error terms eliminated: linear combination of the H Note. It seems not to be possible to combine values obtained from full steps with those from half steps to further increase the order of accuracy, because the error expansions for the two kinds of averages do not have terms in common that can cancel. For we have the 2nd order formula For Z h=2 \Gammah=2 and Z 3h=2 \Gamma3h=2 e Let b j be the jth centered difference of z h (t) at using mid-step values: The 3rd degree interpolant is Hence, and and we powers of s: Averaging and averaging over \Gamma 3 yields Therefore, For a 1st degree interpolant it follows from Eq. (13) that Example 2. We calculate H [2] (x) for the leapfrog method, as given by Eq. (1). We have J(-y n Suppressing the n in the superscript, y e Therefore, Example 3. We calculate H [6] (x) for the leapfrog method, as given by Eq. (3). From Eqs. (16) we have We have h-ffiF from Eq. (11b), and ffi 2 (q T F ) is given by Eq. (12). Then and, therefore, 5.1 The case of a quadratic Hamiltonian Proposition 5 Assume that \Phi h is the composition of flows for systems with quadratic Hamiltoni- ans, that \Psi h is constructed as in Proposition 1, and that - \Psi h=2 is as assumed at the beginning of this section. Then the quantity a i;j where the sum is taken over a finite set of pairs of integers, is exactly conserved by method \Psi h . Proof. Because - are the compositions of flows for systems with homogeneous quadratic Hamiltonians, the mappings - a a a i;j Acknowledgment The authors are grateful for the assistance of Justin Wozniak, who did preliminary tests of the second order truncation for the H'enon-Heiles Hamiltonian and for decalanine. --R http://www. On the Hamiltonian interpolation of near to the identity symplectic mappings with application to symplectic integration algorithms. Residual acceleration as a measure of the accuracy of molecular dynamics simulations. Shadow mass and the relationship between velocity and momentum in symplectic numerical integration. On the scope of the method of modified equations. Generalized Verlet algorithm for efficient molecular dynamics simulations with long-range interactions The life-span of backward error analysis for numerical integrators Asymptotic expansions and backward analysis for numerical inte- grators Longer time steps for molecular dynamics. NAMD2: Greater scalability for parallel molecular dynamics. Molecular Modelling: Principles and Applications. Common molecular dynamics algorithms revisited: Accuracy and optimal time steps of St-ormer-leapfrog An analysis of the accuracy of Langevin and molecular dynamics algorithm. Backward error analysis for numerical integrators. A numerical algorithm for Hamiltonian systems. Numerical Hamiltonian Problems. Nonlinear resonance artifacts in molecular dynamics simulations. Integration schemes for molecular dynamics and related applications. Some Geometric and Numerical Methods for Perturbed Integrable Systems. Reversible multiple time scale molecular dynamics. The modified equation approach to the stability and accuracy analysis of finite difference methods. --TR --CTR Robert D. Engle , Robert D. Skeel , Matthew Drees, Monitoring energy drift with shadow Hamiltonians, Journal of Computational Physics, v.206 n.2, p.432-452, 1 July 2005 Jess A. Izaguirre , Scott S. Hampton, Shadow hybrid Monte Carlo: an efficient propagator in phase space of macromolecules, Journal of Computational Physics, v.200 n.2, p.581-604, November 2004

integrator;symplectic;modified equation;backward error;hamiltonian;numerical

587230

Asynchronous Parallel Pattern Search for Nonlinear Optimization.

We introduce a new asynchronous parallel pattern search (APPS). Parallel pattern search can be quite useful for engineering optimization problems characterized by a small number of variables (say, fifty or less) and by objective functions that are expensive to evaluate, such as those defined by complex simulations that can take anywhere from a few seconds to many hours to run. The target platforms for APPS are the loosely coupled parallel systems now widely available. We exploit the algorithmic characteristics of pattern search to design variants that dynamically initiate actions solely in response to messages, rather than routinely cycling through a fixed set of steps. This gives a versatile concurrent strategy that allows us to effectively balance the computational load across all available processors. Further, it allows us to incorporate a high degree of fault tolerance with almost no additional overhead. We demonstrate the effectiveness of a preliminary implementation of APPS on both standard test problems as well as some engineering optimization problems.

Introduction We are interested in solving the unconstrained nonlinear optimization problem: We introduce a family of asynchronous parallel pattern search (APPS) methods. Pattern search [15] is a class of direct search methods which admits a wide range of algorithmic possibilities. Because of the exibility aorded by the denition of pattern search [23, 16], we can adapt it to the design of nonlinear optimization methods that are intended to be eective on a variety of parallel and distributed computing platforms. Our motivations are several. First, the optimization problems of interest to us are typically dened by computationally expensive computer simulations of complex physical processes. Such a simulation may take anywhere from a few seconds to many hours of computation on a single processor. As we discuss further in x2, the dominant computational cost for pattern search methods lies in these objective function evalu- ations. Even when the objective function is inexpensive to compute, the relative cost of the additional work required within a single iteration of pattern search is negligible. Given these considerations, one feature of pattern search we exploit is that it can compute multiple, independent function evaluations simultaneously in an eort both to accelerate the search process and to improve the quality of the result ob- tained. Thus, our approach can take advantage of parallel and distributed computing platforms. We also have a practical reason, independent of the computational environment, for using pattern search methods for the problems of interest. Simply put, for problems dened by expensive computer simulations of complex physical processes, we often cannot rely on the gradient of f to conduct the search. Typically, this is because no procedure exists for the evaluation of the gradient and the creation of such a procedure has been deemed untenable. Further, approximations to the gradient may prove unreliable. For instance, if the accuracy of the function can only be trusted to a few signicant decimal digits, it is di-cult to construct reliable nite-dierence approximations to the gradient. Finally, while the theory for pattern search assumes that f is continuously dierentiable, pattern search methods can be eective on nondierentiable (and even discontinuous) problems precisely because they do not explicitly rely on derivative information to drive the search. Thus we focus on pattern search for both practical and computational reasons. However, both the nature of the problems of interest and the features of the current distributed computing environments raise a second issue we address in this work. The original investigation into parallel pattern search (PPS) methods 1 [7, 22] made two 1 The original investigations focused on parallel direct search (PDS), a precursor to the more general PPS methods discussed here. fundamental assumptions about the parallel computation environment: 1) that the processors were both homogeneous and tightly coupled and 2) that the amount of time needed to complete a single evaluation of the objective was eectively constant. It is time to reexamine these two assumptions. Clearly, given the current variety of computing platforms including distributed systems comprising loosely-coupled, often heterogeneous, commercial o-the-shelf components [21], the rst assumption is no longer valid. The second assumption is equally suspect. The standard test problems used to assess the eectiveness of a nonlinear optimization algorithm typically are closed-form, algebraic expressions of some function. Thus, the standard assumption that, for a xed choice of n, evaluations complete in constant time is valid. However, given our interest in optimizing problems dened by the simulations of complex physical processes, which often use iterative numerical techniques themselves, the assumption that evaluations complete in constant computational time often does not hold. In fact, the behavior of the simulation for any given input is di-cult to assess in advance since the behavior of the simulation can vary substantially depending on a variety of factors. For both the problems and computing environments of interest, we can no longer assume that the computation proceeds in lockstep. A single synchronization step at the end of every iteration, such as the global reduction used in [22], is neither appropriate nor eective when any of the following factors holds: function evaluations complete in varying amounts of time (even on equivalent processors), the processors employed in the computation possess dierent performance characteristics, or the processors have varying loads. Again our goal is to introduce a class of APPS methods that make more eective use of a variety of computing environments, as well as to devise strategies that accommodate the variation in completion time for function evaluations. Our approach is outlined in x3. The third, and nal, consideration we address in this paper is incorporating fault tolerant strategies into the APPS methods since one intent is to use this software on large-scale heterogeneous systems. The combination of commodity parts and shared resources raises a growing concern about the reliability of the individual processors participating in a computation. If we embark on a lengthy computation, we want reasonable assurance of producing a nal result, even if a subset of processors fail. Thus, our goal is to design methods that anticipate such failures and respond to protect the solution process. Rather than simply checkpointing intermediate computations to disk and then restarting in the event of a failure, we are instead considering methods with heuristics that adaptively modify the search strategy. We discuss the technical issues in further detail in x4. In x5 we provide numerical results comparing APPS and PPS on both standard and engineering optimization test problems; and nally, in x6 we outline additional questions to pursue. Although we are not the rst to embark on the design of asynchronous parallel optimization algorithms, we are aware of little other work, particularly in the area of nonlinear programming. Approaches to developing asynchronous parallel Newton or quasi-Newton methods are proposed in [4, 8], though the assumptions underlying these approaches dier markedly from those we address. Specically, both assume that solving a linear system of equations each iteration is the dominant computational cost of the optimization algorithm because the dimensions of the problems of interest are relatively large. A dierent line of inquiry [20] considers the use of quasi-Newton methods, but in the context of developing asynchronous stochastic global optimization algorithms. For now, we focus on nding local minimizers. Parallel Pattern Search Before proceeding to a discussion of our APPS methods, let us rst review the features of direct search, in general, and pattern search, in particular. Direct search methods are characterized by neither requiring nor explicitly approximating derivative information. In the engineering literature, direct search methods are often called zero-order methods, as opposed to rst-order methods (such as the method of steepest descent) or second-order methods (such as Newton's method) to indicate the highest order term being used in the local Taylor series approximation to f . This characterization of direct search is perhaps the most useful in that it emphasizes that in higher-order methods, derivatives are used to form a local approximation to the function, which is then used to derive a search direction and predict the length of the step necessary to realize decrease. Instead of working with a local approximation of f , direct search methods work directly with f . Pattern search methods comprise a subset of direct search methods. While there are rigorous formal denitions of pattern search [16, 23], a primary characteristic of pattern search methods is that they sample the function over a predened pattern of points, all of which lie on a rational lattice. By enforcing structure on the form of the points in the pattern, as well as simple rules on both the outcome of the search and the subsequent updates, standard global convergence results can be obtained. For our purposes, the feature of pattern search that is amenable to parallelism is that once the candidates in the pattern have been dened, the function values at these points can be computed independently and, thus, concurrently. To make this more concrete, consider the following particularly simple version of a pattern search algorithm. At iteration k, we have an iterate x k 2 R n and a step-length parameter k > 0. The pattern of p points is denoted by g. For the purposes of our simple example, we choose D fe represents the jth unit vector. As we discuss at the end of this section, other choices for D are possible. We now have several algorithmic options open to us. One possibility is to look successively at the pattern points x k either we nd a point x+ for which f(x+ ) < f(x k ) or we exhaust all 2n possibilities. At the other extreme, we could determine x+ 2 fx k 2ng such that (which requires us to compute f(x k for all 2n vectors in the set D). Fig. 1 illustrates the pattern of points among which we search for x+ when 2. r z }| { Figure 1: A simple instance of pattern search In either variant of pattern search, if none of the pattern points reduces the ob- jective, then we set x reduce by setting otherwise, we set We repeat this process until some suitable stopping criterion, such as k < tol, is satised. There are several things to note about the two search strategies we have just outlined. First, even though we have the same pattern in both instances, we have two dierent algorithms with dierent search strategies that could conceivably produce dierent sequences of iterates and even dierent local minimums. Second, the design of the search strategies re ects some intrinsic assumptions about the nature of both the function and the computing environment in which the search is to be executed. Clearly the rst strategy, which evaluates only one function value at a time, was conceived for execution on a single processor. Further it is a cautious strategy that computes function values only as needed, which suggests a frugality with respect to the number of function evaluations to be allowed. The second strategy could certainly be executed on a single processor, and one could make an argument as to why there could be algorithmic advantages in doing so, but it is also clearly a strategy that can easily make use of multiple processors. It is straightforward to then derive PPS from this second strategy, as illustrated in Fig. 2. Before proceeding to a description of APPS, however, we need to make one more remark about the pattern. As we have already seen, we can easily derive two dierent search strategies using the same basic pattern. Our requirements on the outcome of the search are mild. If we fail to nd a point that reduces the value of f at x k , then we must try again with a smaller value of k . Otherwise, we accept as our new iterate any point from the pattern that produces decrease. In the latter case, we may choose to modify k . In either case, we are free to make changes to the pattern to be used in the next iteration, though we left the pattern unchanged in the examples given above. However, changes to either the step length parameter or the pattern are subject to certain algebraic conditions, outlined fully in [16]. 2 The reduction parameter is usually 1but can be any number in the set (0; 1). Initialization: Select a pattern g. Select a step-length parameter 0 . Select a stopping tolerance tol. Select a starting point x 0 and evaluate f(x 0 ). Iteration: 1. Evaluate concurrently. 2. Determine x+ such that f(x+ (synchronization point). 3. If f(x+ ) < f(x k ), then set x Else set x 4. If k+1 < tol, exit. Else, repeat. Figure 2: The PPS Algorithm There still remains the question of what constitutes an acceptable pattern. We borrow the following technical denition from [6, 16]: a pattern must be a positive spanning set for R n . In addition, we add the condition that the spanning set be composed of rational vectors. Denition 1 A set of vectors fd positively spans R n if any vector z 2 R n can be written as a nonnegative linear combination of the vectors in the set; i.e., for any z 2 R n there exists A positive spanning set contains at least n+1 vectors [6]. It is trivial to verify that the set of vectors (used to dene the pattern for our examples above) is a positive spanning set. 3 3 The terminology \positive" spanning set is a misnomer; a more proper name would be \non- negative" spanning set. Asynchronous Parallel Pattern Search Ine-ciencies in processor utilization for the PPS algorithm shown in Fig. 2 arise when the objective function evaluations do not complete in approximately the same amount of time. This can happen for several reasons. First, the objective function evaluations may be complex simulations that require dierent amounts of work depending on the input parameters. Second, the load on the individual processors may vary. Last, groups of processors participating in the calculation may possess dierent computational characteristics. When the objective function evaluations take varying amounts of time those processors that can complete their share of the computation more quickly wait for the remaining processors to contribute their results. Thus, adding more processors (and correspondingly more search directions) can actually slow down the PPS method given in Fig. 2 because of an increased synchronization penalty. The limiting case of a slow objective function evaluation is when one never com- pletes. This could happen if some processor fails during the course of the calculations. In that situation, the entire program would hang at the next synchronization point. Designing an algorithm that can handle failures plays some role in the discussion in this section and is given detailed coverage in the next. The design of APPS addresses the limitations of slow and failing objective function evaluations and is based on a peer-to-peer approach rather than master-slave. Although the master-slave approach has advantages, the critical disadvantage is that, although recovery for the failure of slave processes is easy, we cannot automatically recover from failure of the master process. In the peer-to-peer scenario, all processes have equal knowledge, and each process is in charge of a single direction in the search pattern D. In order to fully understand APPS, let us rst consider the single processor's algorithm for synchronous PPS in a peer-to-peer mode, as shown in Fig. 3. Here subscripts have been dropped to illustrate how the process handles the data. The set of directions from all the processes forms a positive spanning set. With the exception of initialization and nalization, the only communication a process has with its peers is in the global reduction in Step 2. To terminate, all processors detect convergence at the same time since they all have identical, albeit independent, values for trial . 4 In an asynchronous peer-to-peer version of PPS (see Fig. 4), we allow each process to maintain its own versions of x best , x+ , trial , etc. Unlike synchronous PPS, these values may not always agree with the values on the other processes. Each process decides what to do next based only on the current information available to it. If it nds a point along its search direction that improves upon the best point it knows so far, then it broadcasts a message to the other processors letting them know. It also checks for messages from other processors, and replaces its best point with the 4 In a heterogeneous environment, there is some danger that the processors may not all have the same value for trial because of slight dierences in arithmetic and the way values are stored; see [2]. Iteration: 1. Compute x trial x best is \my" direction). 2. Determine f + (and the associated x+ ) via a global reduction minimizing the f trial values computed in Step 1. 3. If f best , then fx best ; f best g fx+ g. Else trial 1 4. If trial > tol, go to Step 1. Else, exit. Figure 3: Peer-to-peer version of (synchronous) PPS Iteration: Consider each incoming triplet fx+ received from another processor. best , then fx best ; f best ; best g fx+ trial best . 1. Compute x trial x best is \my" direction). 2. g. 3. If f best , then fx best ; f best ; best g fx+ best , and broadcast the new minimum triplet fx best ; f best ; best g to all other processors. Else trial 1 4. If trial > tol, goto Step 0. Else broadcast a local convergence message for the pair fx best ; f best g. 5. Wait until either (a) enough of processes have converged for this point or (b) a better point is received. In case (a), exit. In case (b), goto Figure 4: Peer-to-peer version of APPS incoming one if it is an improvement. If neither its own trial point nor any incoming messages are better, it performs a contraction and continues. Convergence is a trickier issue than in the synchronous version because the processors do not reach trial < tol at the same time. Instead, each processor converges in the direction that it owns, and then waits for the other processes to either converge to the same point or produce a better point. Since every good point is broadcast to all the other process, every process eventually agrees on the best point. The nal APPS algorithm is slightly dierent from the version in Fig. 4 because we spawn the objective function evaluation as a separate process. Our motivation is that we may sometimes want to stop an objective function evaluation before it completes in the event that a good point is received from another processor. We create a group of APPS daemon processes that follow the basic APPS procedure outlined in Fig. 4 except that each objective function evaluation will be executed as a separate process. The result is APPS daemons working in peer-to-peer mode, each owning a single slave objective function evaluation process. The APPS daemon (see Fig. 5) works primarily as a message processing center. It receives three types of messages: a \return" from its spawned objective function evaluation and \new minimum" and \convergence" messages from APPS daemons. When the daemon receives a \return" message, it determines if its current trial point is a new minimum and, if so, broadcasts the point to all other processors. The trial that is used to generate the new minimum is saved and can then be used to determine how far to step along the search direction. The alternative would be to reset trial = 0 every time a switch is made to a new point, but then scaling information is lost which may lead to unnecessary additional function evaluations. In the comparison of the trial and best f-values, we encounter an important caveat of heterogeneous computing [2]. The comparison of values (f 's, 's, etc.) controls the ow of the APPS method, and we depend on these comparisons to give consistent results across processors. Therefore, we must ensure that values are only compared to a level of precision available on all processors. In other words, a \safe" comparison declares mach where mach is the maximum of all mach 's. A \new minimum" message means that another processor has found a point it thinks is best, and the receiving daemon must decide if it agrees. In this case, we must decide how to handle tie-breaking in a consistent manner. If f best , then we need to be able to say which point is \best" or if indeed the points we are comparing are equal (i.e., x best = x+ ). The tie breaking scheme is the following. If f best , then compare + and best and select the larger value of . If the values are also equal, check next to see if indeed the two points are the same, but rather than comparing x best and x+ directly by measuring some norm of the dierence, use a unique identier included with each point. Thus, two points are equal if and only if Return from Objective Function Evaluation. Receive f trial . 1. Update x best and/or trial . (a) If f trial < f best , then i. fx best ; f best ; best g fx trial ; f trial ; trial g. ii. Broadcast new minimum message with the triplet best ; f best ; best g to all other processors (b) Else if x best is not the point used to generate x trial , then trial best . (c) Else trial 1 2. Check for convergence and spawn next objective function evaluation. (a) If trial > tol, compute x trial x best trial d and spawn a new objective function evaluation. (b) Else broadcast convergence message with best ; f best ; best g to all processors including myself. New Minimum Message. Receive the triplet fx+ g. 1. If f best , then best or I am locally converged, then ag TRUE, else ag FALSE. (b) Set fx best ; f best ; best g fx+ g. (c) If ag is TRUE, then break current objective function evaluation spawn, compute x trial x best trial d, and spawn a new objective function evaluation. Convergence Message. Receive the triplet triplet fx+ g. 1. Go though steps for new minimum to be sure that this point is x best . 2. Then, if I am the temporary master consider all the processes that have so far converged to x best . If enough other processes have converged so that their associated directions form a positive spanning set, then output the solution, shutdown the remaining APPS daemon processes, and exit. Figure 5: APPS Daemon Message Types and Actions their f-values, -values, and unique identiers match. 5 In certain cases, the current objective function evaluation is terminated in favor of starting one based on a new best point. Imagine the following scenario. Suppose three processes, A, B, and C start o with the same value for x best , generate their own x trial 's, and spawn their objective function evaluations. Each objective function evaluation takes several hours. Process A nishes its objective function evaluation before any other process and does not nd improvement, so it contracts and spawns a new objective function evaluation. A few minutes later, Process B nishes its objective function evaluation and nds improvement. It broadcasts its new minimum to the other processes. Process A receives this message and terminates its current objective function evaluation process in order to move to the better point. This may save several hours of wasted computing time. However, Process C, which is still working on its rst objective function evaluation, waits for that to complete before considering moving to the new x best . When the daemon receives a \convergence" message, it records the converged direction, and possibly checks for convergence. The design of the method requires that a daemon cannot locally converge to a point until it has evaluated at least one trial point generated from that best point along its search direction. Each point has an associated boolean convergence table which is sent in every message. When a process locally converges, it adds a TRUE entry to its spot in the convergence table before it sends a convergence message. In order to actually check for convergence of a su-cient number of processes, it is useful to have a temporary master to avoid redundant computation. We dene the temporary master to be the process with the lowest process id. While this is usually process 0, it is not always the case if we consider faults, which are discussed in the next section. The temporary master checks to see if the converged directions form a positive spanning set, and if so outputs the result and terminate the entire computation. Checking for a positive spanning set is done as follows. Let V D be the candidate for a positive basis. We solve nonnegative least squares problems according to the following theorem. Theorem 3.1 A set is a positive spanning set if the set is in its positive span (where 1 is the vector of all 1's). Alternatively, we can check the positive basis by rst verifying that V is a spanning set using, say, a QR factorization with pivoting, and then solving a linear program. Theorem 3.2 (Wright [24]) A spanning set is a positive spanning set if the maximum of the following LP is 1. 5 This system will miss two points that are equal but generated via dierent paths. In the rst case, we can use software for the nonnegative least squares problem from Netlib due to Lawson and Hanson [14]. In the second case, the software implementation is more complicated since we need both a QR factorization and a linear program solver, the latter of which is particularly hard to come by in both a freely available, portable, and easy-to-use format. 4 Fault Tolerance in APPS The move toward a variety of computing environments, including heterogeneous distributed computing platforms, brings with it an increased concern for fault tolerance in parallel algorithms. The large size, diversity of components, and complex architecture of such systems create numerous opportunities for hardware failures. Our computational experience conrms that it is reasonable to expect frequent failures. In addition, the size and complexity of current simulation codes call into question the robustness of the function evaluations. In fact, application developers themselves will testify that it is possible to generate input parameters for which their simulation codes fail to complete successfully. Thus, we must contend with software failures as well as hardware failures. A great deal of work has been done in the computer science community with regard to fault tolerance; however, much of that work has focused on making fault tolerance as transparent to the user as possible. This often entails checkpointing the entire state of an application to disk or replicating processes. Fault tolerance has traditionally been used with loosely-coupled distributed applications that do not depend on each other to complete, such as business database applications. This lack of interdependence is atypical of most scientic applications. While checkpointing and replication are adequate techniques for scientic applications, they incur a substantial amount of unwanted overhead; however, certain scientic applications have characteristics that can be exploited for more e-cient and elegant fault tolerance. This algorithm-dependent variety of fault tolerance has already received a considerable amount of attention in the scientic computing community; see, e.g., [11, 12]. These approaches rely primarily on the use of diskless checkpointing, a signicant improvement over traditional approaches. The nature of APPS is such that we can even further reduce the overhead for fault tolerance and dispense with checkpointing altogether. There are three scenarios that we consider when addressing fault tolerance in APPS: 1) the failure of a function evaluation, 2) the failure of an APPS daemon, and the failure of a host. These scenarios are shown in Figure 6. The approaches for handling daemon and host failures are very similar to one another, but the function evaluation failure is treated in a somewhat dierent manner. When a function evaluation fails, it is respawned by its parent APPS daemon. If the failure occurs more than a specied number of times at the same trial point, then the daemon itself fails. 6 If an APPS daemon fails, the rst thing the temporary master does is check for convergence since the now defunct daemon may have been in the process of that check when it died. Next it checks whether or not the directions owned by the remaining daemons form a positive basis. If so, convergence is still guaranteed, so nothing is done. Otherwise, all dead daemons are restarted. If a host fails, then the APPS daemons that were running on that host are restarted on a dierent host according to the rules stated for daemon failures. The faulty host is then removed from the list of viable hosts and is no longer used. Exit from Function Evaluation. 1. If the number of tries at this point is less than the maximum allowed number, respawn the function evaluation. 2. Else shutdown this daemon. An APPS Daemon Failed. 1. Record failure. 2. If I am the (temporary) master, then (a) Check for convergence and, if converged, output the result and terminate the computation. (b) If the directions corresponding to the remaining daemons do not form positive spanning set, respawn all failed daemons. A Host Failed. 1. Remove host from list of available hosts. Figure Tolerance Messages and Actions Two important points should be made regarding fault tolerance in APPS. First, there are no single points of failure in the APPS algorithm itself. While there are scenarios requiring a master to coordinate eorts, this master is not xed If it should fail while performing its tasks, another master steps up to take over. This means the degree of fault tolerance in APPS is constrained only by the underlying communication architecture. The current implementation of APPS uses PVM, which has a single point of failure at the master PVM daemon [9]. We expect Harness [1], the successor to PVM, to eliminate this disadvantage. The second point of interest is that 6 This situation can be handled in dierent ways for dierent applications; attempts to evaluate a certain point could be abandoned without terminating the daemon. no checkpointing or replication of processes is necessary. The algorithm recongures on the y, and new APPS daemons require only a small packet of information from an existing process in order to take over where a failed daemon left o. Therefore, we have been able take advantage of characteristics of APPS in order to elegantly incorporate a high degree of fault tolerance with very little overhead. Despite the growing concern for fault tolerance in the parallel computing world, we are aware of only one other parallel optimization algorithm that incorporates fault tolerance, FATCOP [3]. FATCOP is a parallel mixed integer program solver that has been implemented using a Condor-PVM hybrid as the communication substrate. FATCOP is implemented in a master-slave fashion which means that there is a single point of failure at the master process. This is addressed by having the master checkpoint information to disk (via Condor), but recovery requires user intervention to restart the program in the event of a failure. In contrast, APPS can recover from the failure of any type of process, including the failure of a temporary master, on its own and has no checkpointing whatsoever. 5 Numerical Results We compare PPS 7 and APPS on several test problems as well as two engineering problems, a thermal design problem and a circuit simulation problem. The tests were performed on the CPlant supercomputer at Sandia National Labs in Livermore, California. CPlant is a cluster of DEC Alpha Miata 433 MHz Processors. For our tests, we used 50 nodes dedicated to our sole use. 5.1 Standard Test Problems We compare APPS and PPS with 8, 16, 24,and 32 processors on six four dimensional test problems: broyden2a, broyden2b, chebyquad, epowell, toint trig, and vardim [18, 5]. Since the function evaluations are extremely fast, we added extra \busy work" in order to slow them down to better simulate the types of objective functions we are interested in. 8 The parameters for APPS and PPS were set as follows. Let be the problem dimension, and let p be the number of processors. The rst 2n search directions are g. The remaining p 2n directions are vectors that are randomly generated (with a dierent seed for every run) and normalized to unit length. This set of search directions is a positive spanning set. We initialize and 7 We are using our own implementation of a positive basis PPS, as outlined in Fig. 3, rather than the well-known parallel direct search (PDS) [22]. PDS is not based on the positive basis framework and is quite dierent from the method described in Fig. 3, making comparisons di-cult. 8 More precisely, the \busy work" was the solution of a 100 101 nonnegative least squares problem. We added two additional twists to the way is updated for all tests. First, if the same search direction yields the best point two times in a row, is doubled before the broadcast. Second, the smallest allowable for a \new minimum" is such that at least three contractions will be required before local convergence. That way, we are guaranteed to have several evaluations along each search direction for each point. Method Process Function Function Init Idle Total ID Evals Breaks Time Time Time Summary 272.5 70.6 0.04 0.07 24.72 Summary 235 N/A 0.22 6.10 30.63 Table 1: Detailed results for epowell on eight processors. Before considering the summary results, we examine detailed results from two sample runs given in Table 1. Each process reports its own counts and timings. All times are reported in seconds and are wall clock times. Because APPS is asynchronous, the number of function evaluations varies for each process, in this case by as much as 25%. Furthermore, APPS sometimes \breaks" functions midway through execution. On the other hand, every process in PPS executes the same number of function eval- uations, and there are no breaks. For both APPS and PPS, the initialization time is longer for the rst process since it is in charge of spawning all the remaining tasks. The idle time varies from task to task but is overall much lower for APPS than PPS. An APPS process is only idle when it is locally converged, but a PPS process may potentially have some idle time every iteration while it waits for the completion of the global reduction. The total wall clock time varies from process to process since each starts and stops at slightly dierent times. The summary information is the average over all processes except in the case of total time, in which case the maximum over all times is reported. Because some of the search directions are generated randomly, every run of PPS and APPS generates a dierent path to the solution and possibly dierent solutions in the case of multiple minima. 9 Because of the nondeterministic nature of APPS, it gets dierent results every run even when the search directions are identical. Therefore, for each problem we report average summary results from 25 runs. Problem Procs Function Evals APPS Idle Time Total Time Name APPS PPS Breaks APPS PPS APPS PPS broyden2a 8 40:59 37:00 8:14 0:07 0:95 3:88 4:88 chebyquad 8 73:06 62:00 16:74 0:05 1:61 6:86 8:11 toint trig 8 53:83 41:00 10:97 0:04 1:11 4:99 5:60 Table 2: Results on a collection of four dimensional test problems. The test results are summarized in Table 2. These tests were run in a fairly favorable environment for PPS|a cluster of homogeneous, dedicated processors. The 9 The exception is PPS with 8. Because there are no \extra" search directions, the solution to the path is the same for every run|only the timings dier. primary di-culty for PPS is the cost of synchronization in the global reduction. In terms of average function evaluations per processor, both APPS and PPS required about the same number. In general for both APPS and PPS, the number of function evaluations per processor decreased as the number of processes increased. We expect the idle time for APPS to be less than that for PPS; and, indeed, the idle time is two orders of magnitude less. Furthermore, the idle time for PPS increases as the number of processors goes up. APPS was faster (on average) than PPS in 22 of 24 cases. The total time for APPS either stayed steady or reduced as the number of processors increased. In contrast, the total PPS time increased as the number of processors increased due to the synchronization penalty. Comparing APPS and PPS on simple problems is not necessarily indicative of results for typical engineering problems. The next two subsections yield more meaningful comparisons, given the types of problems for which pattern search is best suited. 5.2 TWAFER: A Thermal Design Problem This engineering application concerns the simulation of a thermal deposition furnace for silicon wafers. The furnace contains a vertical stack of 50 wafers and several heater zones. The goal is to achieve a specied constant temperature across each wafer and throughout the stack. The simulation code, TWAFER [10], yields measurements at a discrete collection of points on the wafers. The objective function f is dened by a least squares t of the N discrete wafer temperatures T j to a prescribed ideal T as where x i is the unknown power parameters for the heater in zone i. We consider the four and seven zone problems. For this problem, we used the following settings for APPS and PPS. The rst n+1 search directions are the points of a regular simplex centered about the origin. The remaining are generated randomly and normalized to unit length. We set There are some di-culties from the implementation point of view that are quite common when dealing with simulation codes. Because TWAFER is a legacy code, it expects an input le with a specic name and produces an output le with a specic name. The names of these les cannot be changed, and TWAFER cannot be hooked directly to PVM. As a consequence, we must write a \wrapper" program that runs an input lter, executes TWAFER via a system call, and runs an output lter. The input le for TWAFER must contain an entire description of the furnace and the wafers. We are only changing a few values within that le, so our input lter generates the input le for TWAFER by using a \template" input le. This template le contains tokens that are replaced by our optimization variables. The output le from TWAFER contains the heat measurements at discrete points. Our output lter reads in these values and computes the least squares dierence between these and the ideal temperature in order to determine the value of the objective function. An additional caveat is that TWAFER must be executed in a uniquely named subdirectory so that its input and output les are not confused with those of any other TWAFER process that may be accessing the same disk. Lastly, because TWAFER is executed via a system call, APPS has no way of terminating its execution prematurely. (APPS can terminate the wrapper program, but TWAFER itself will continue to run, consuming system resources.) Therefore, we allow all function evaluations to run to completion, that is, we do not allow any breaks. Another feature of TWAFER is that it has nonnegativity constraints on the power settings. We use a simple barrier function that returns a large value (e.g., Problem Method Procs f(x) Function Idle Total Evals Time Time 4 Zone APPS 20 0:67 334:6 0:17 395:94 4 Zone PPS 20 0:66 379:9 44:77 503:88 Table 3: Results on the four and seven zone TWAFER problems. Results for the TWAFER problem are given in Table 3. The four zone results are the averages over ten runs, and the seven zones results are averages over nine runs. (The tenth PPS run failed due to a node fault. The tenth APPS run had several faults, and although it did get the nal solution, the summary data was incomplete.) Here we also list the value of the objective function at the solution. Observe that PPS yields slightly better function values (compared to the original value of more than 1000) on average but at a cost of more function evaluations and more time. The average function evaluation execution time for the four zone problem is 1.3 seconds and for the seven zone problem is 10.4 seconds; however, the number of function evaluations includes instances where the bounds were violated in which case the TWAFER code was not executed and the execution time is essentially zero since we simply return Once again PPS has a substantial amount of idle time. The relatively high APPS idle time in the seven zone problem was due to a single run in which the idle time was particularly high for some nodes (634 seconds on average). 5.3 SPICE: A Circuit Simulation Problem The problem is to match simulation data to experimental data for a particular circuit in order to determine its characteristics. In our case, we have 17 variables representing inductances, capacitances, diode saturation currents, transistor gains, leakage inductances, and transformer core parameters. The objective function is dened as where N is the number of time steps, V SIM j (x) is the simulation voltage at time step for input x, and V EXP j is the experimental voltage at time step j. The SPICE3 [19] package is used for the simulation. Like TWAFER, SPICE3 communicates via le input and output, and so we again use a wrapper program. The input lter for SPICE is more complicated than that for TWAFER because the variables for the problem are on dierent scales. Since APPS has no mechanism for scaling, we handled this within the input lter by computing an a-ne transformation of the APPS variables. Additionally, all the variables have upper and lower bounds. Once again, we use a simple barrier function. The output lter for SPICE is also more complicated than that for TWAFER. The SPICE output les consists of voltages that are to be matched to the experimental data. The experimental data is two cycles of output voltage measured at approximately Fig. 7). The simulation data contains approximately 10 or more cycles, but only the last few complete cycles are used because the early cycles are not stable. The cycles must be automatically identied so that the data can be aligned with the experimental data. Furthermore, the time steps from the simulation may dier from the time steps in the experiment, and so the simulation data is interpolated (piecewise constant) to match the experimental data. The function value at the initial point is 465. The APPS parameters were set as follows. The search directions were generated in the same way as those for the test problems. We set tolerance corresponds to a less than 1% change in the circuit parameter). Once again, we do not allow \breaks" since the function evaluation is called from a wrapper program via a system call. The results from APPS and PPS on the SPICE problem are reported in Table 4. In this case, we are reporting the results of single runs, and we give results for 34 and 50 processors. The average SPICE run time is approximately 20 seconds; however, we once again do not dierentiate between times when the boundary conditions are violated and when the SPICE code is actually executed. Increasing the number of processors by 47% results in a 39% reduction in execution time for APPS but only 4% for PPS. For both 34 and 50 processors, APPS is faster than PPS, and even produces a slightly better objective value (compared to the starting value of more than 400). At the solution, two constraints are binding. -55Time Voltage Figure 7: Spice results. The solid line represents the experimental output. The dashed line represents the simulation output after optimization. The dotted line represents the starting point for the optimization. Method Procs f(x) Function Idle Total Evals Time Time APPS 34 26.3 57:5 111.92 1330.55 APPS 50 26.9 50:6 63.22 807.29 PPS 34 28.8 53:0 521.48 1712.24 PPS 50 34.9 47:0 905.48 1646.53 Table 4: Results for the 17 variable SPICE problem. Initial Final f(x) Total Procs Procs Time 34 34 27.8 1618.46 50 Table 5: APPS results for the 17 variable SPICE with a failure approximately every seconds. Table 5 shows the results of running APPS with faults. In this case, we used a program that automatically killed one PVM process every seconds. The PVM processes are the APPS daemons and the wrapper programs. The SPICE3 simulation is executed via a system call, and so continues to execute even if its wrapper terminates; regardless, the SPICE3 program can no longer communicate with APPS and is eectively dead. The results are quite good. In the case of 34 processors, every APPS task that fails must be restarted in order to maintain a positive basis. So, the nal number of APPS processes is 34. The total time is only increased by 21% despite approximately 50 failures; furthermore, this time is still faster than PPS. In the case of 50 processors, the nal number of processors is 32. (Recall that tasks are only restarted if there are not enough remaining to form a positive basis.) In the case of 50 processors, the solution time is only increased by 29%, and is once again still faster than PPS. In this case, however, the quality of the solution is degraded. This is likely due to the fact that the solution lies on the boundary and some of the search directions that failed were needed for convergence (see Lewis and Torczon [17]). 6 Conclusions The newly-introduced APPS method is superior to PPS in terms of overall computing time on a homogeneous cluster environment for both generic test problems and engineering applications. We expect the dierence to be even more pronounced for larger problems (both in terms of execution time and number of variables) and for heterogenous cluster environments. Unlike PPS, APPS does not have any required synchronizations and, thus, gains most of its advantage by reducing idle time. APPS is fault tolerant and, as we see in the results on the SPICE problem for 34 processors, does not suer much slow-down in the case of faults. In forthcoming work, Kolda and Torczon [13] will show that in the unconstrained case the APPS method converges (even in the case of faults) under the same assumptions as pattern search [23]. Although the engineering examples used in this work have bound constraints, the APPS method was not fully designed for this purpose, as evidenced in the poor results on the SPICE problem with faults on 50 processors. Future work will explore the algorithm, implementation, and theory in the constrained cases. In the implementation described here, the daemons and function evaluations are in pairs; however, for multi-processor (MPP) compute nodes, this means there will be several daemon/function evaluation pairs per node. An alternative implementation of APPS is being developed in which there is exactly one daemon per node regardless of how many function evaluations are assigned to it. As part of this alternative implementation, the ability to dynamically add new hosts as they become available (or to re-add previously failed hosts) will be incorporated. Another improvement to the implementation will be the addition of a function value cache in order to avoid reevaluating the same point more than once. The challenge is deciding when two points are actually equal; this is especially di-cult without knowing the sensitivity of the function to changes in each variable. The importance of positive bases in the pattern raises several interesting research questions. First, we might consider the best way to generate the starting basis. We desire a pattern that maximizes the probability of maintaining a positive basis in the event of failures. Another research area is the aect that \conditioning" of the positive basis has on convergence. Our numerical studies have indicated that the quality of the positive basis may be an issue. Last, supposing that enough failures have occurred so that there is no longer a positive basis, we may ask if we can easily determine the fewest number of vectors to add to once again have a positive basis. Our current implementation simply restarts all failed processes. A Acknowledgments Thanks to Jim Kohl, Ken Marx, Juan Meza for helpful comments and advice in the implemenation of APPS and the test problems. --R Harness: A next generation distributed virtual machine. Practical experience in the numerical dangers of heterogeneous computing FATCOP: A fault tolerant Condor-PVM mixed integer program solver Convergence and numerical results for a parallel asynchronous quasi-Newton method Testing a class of methods for solving minimization problems with simple bounds on the variables Theory of positive linear dependence An asynchronous parallel Newton method PVM: Parallel Virtual Machine: A Users' Guide and Tutorial for Network Parallel Computing A model for low pressure chemical vapor deposition in a hot-wall tubular reactor On the convergence of asynchronous parallel direct search. Solving Least Squares Problems Why pattern search works Rank ordering and positive bases in pattern search algorithms How to Build a Beowulf: A Guide to the Implementation and Application of PC Clusters PDS: Direct search methods for unconstrained optimization on either sequential or parallel machines A note on positively spanning sets. --TR --CTR Genetha A. Gray , Tamara G. Kolda, Algorithm 856: APPSPACK 4.0: asynchronous parallel pattern search for derivative-free optimization, ACM Transactions on Mathematical Software (TOMS), v.32 n.3, p.485-507, September 2006 A. Ismael Vaz , Lus N. Vicente, A particle swarm pattern search method for bound constrained global optimization, Journal of Global Optimization, v.39 n.2, p.197-219, October 2007 Steven Benson , Manojkumar Krishnan , Lois Mcinnes , Jarek Nieplocha , Jason Sarich, Using the GA and TAO toolkits for solving large-scale optimization problems on parallel computers, ACM Transactions on Mathematical Software (TOMS), v.33 n.2, p.11-es, June 2007 Genetha Anne Gray , Tamara G. Kolda , Ken Sale , Malin M. Young, Optimizing an Empirical Scoring Function for Transmembrane Protein Structure Determination, INFORMS Journal on Computing, v.16 n.4, p.406-418, Fall 2004 Jack Dongarra , Ian Foster , Geoffrey Fox , William Gropp , Ken Kennedy , Linda Torczon , Andy White, References, Sourcebook of parallel computing, Morgan Kaufmann Publishers Inc., San Francisco, CA,

distributed computing;asynchronous parallel optimization;pattern search;direct search;fault tolerance;cluster computing

587249

Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature.

Integration lattices are one of the main types of low discrepancy sets used in quasi-Monte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first bm of which forms a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

Introduction . Multidimensional integrals appear in a wide variety of applications in finance [4, 48, 50], physics and engineering [29, 42, 49, 58], and statistics [8, 12, 13]. The integration domain may often be assumed, after some appropriate transformation, to be the unit cube, in which case the integral takes the form: Z for some known integrand, f . Adaptive methods, such as [2], have been developed for approximating multidimensional integrals, but their performance deteriorates as the dimension increases. For finance problems the dimension can be in the hundreds or even thousands. An alternative to adaptive quadrature is Monte Carlo methods, where the integral is approximated by the sample mean of the integrand evaluated on a set, P , of N independent random points drawn from a uniform distribution on [0; The quadrature error for Monte Carlo methods is typically O(N \Gamma1=2 ). One reason for this relatively low accuracy is that the points in P are chosen independently of each other. Thus, some parts of the integration domain contain clumps of points while other parts are empty of points. To obtain greater accuracy one may replace the random set P by a carefully chosen deterministic set that is more uniformly distributed on [0; 1) s . As is explained in Section 3, one may define a discrepancy that measures how much the empirical distribution function of P differs from the continuous uniform distribution. Then y Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR, China. This research was supported by a HKBU FRG grant 96-97/II-67. z fred@hkbu.edu.hk, http://www.math.hkbu.edu.hk/~fred x D'epartement d'Informatique et de Recherche Op'erationnelle, Universit'e de Montr'eal C.P. 6128, Succ. Centre-Ville, Montr'eal (Qu'ebec), Canada, H3C 3J7 F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. one chooses P in quadrature rule (1.1) to have as small a discrepancy as possible. The quadrature methods based on low discrepancy sets are called quasi-Monte Carlo methods. They are discussed in several review articles [3, 16, 40, 59] and monographs [27, 44, 53]. Two important families of low discrepancy sets are: i. integration lattices [44, Chap. 5] and [53], and ii. digital nets and sequences [44, Chap. 4]. These two families are introduced in Section 2. One advantage of the second family is that any number of consecutive points from a good digital sequence has low dis- crepancy. If one needs more points, one may use additional terms from the digital sequence without discarding the original ones. On the other hand, until now, the number of points in an integration lattice has had to be specified in advance. So far, there has been no systematic way of adding points to an integration lattice while still retaining its lattice structure. The purpose of this article is to provide a method for constructing infinite lattice sequences, thereby eliminating the need to know N , the number of points, in advance. Although the emphasis is on rank-1 lattices, the method may be applied to integration lattices of arbitrary rank. Given an infinite lattice sequence one may approximate a multidimensional integrand with a quadrature rule of the form (1.1) for moderate number of points N 0 . If the error estimate is unacceptably high, then one may choose an additional points from the lattice sequence to obtain a quadrature rule with N 1 points, and so on. The following section describes the new method for obtaining infinite lattice se- quences. Section 3 briefly reviews some results on discrepancy and quadrature error analysis for quasi-Monte Carlo methods. These are used to find the generating vectors for the new lattice sequences in Section 4. The issue of error estimation is addressed in Section 5. Two practical examples are explored in Section 6, where the new lattice sequences are compared with existing quadrature methods. The last section contains some concluding remarks. 2. Integration Lattices and Digital Sequences. This section begins by introducing integration lattices. Next, digital sequences and (t; s)-sequences are de- scribed. Finally, the idea underlying digital sequences is used to produce infinite lattice sequences. 2.1. Integration Lattices. Rank-1 lattices, also known as good lattice point (glp) sets, were introduced by Korobov [31] and have been widely studied since then (see [27, 44, 53] and the references therein). The formula for a shifted rank-1 lattice set is simply where N is the number of points, h is an s-dimensional generating vector of integers (a good lattice point) that depends on N , \Delta is an s-dimensional shift vector in [0; 1) s , and fxg denotes the fractional part of a vector x, i.e. later generalized glp sets by introducing more than one generating vector. A shifted integration lattice with points based on generating vectors h is: Integration lattices and their use for quadrature are discussed in the monograph [53]. Extensible Lattice Sequences 3 For a given N there is the problem of choosing good generating vectors. Although theoretical constructions exist for in higher dimensions one typically finds generating vectors by minimizing a discrepancy or measure of non-uniformity of the lattice. Several examples of discrepancies are given in Section 3. 2.2. Digital Nets and Sequences. Digital nets and sequences are another method of constructing low discrepancy sets (see [32] and [44, Chapter 4]). Let b denote a positive integer greater than one. For any non-negative integer one may extract the digits of its base b representation, finitely many of which are nonzero: The i th term of a digital net or sequence is given by z z (i) s: (2.2d) If the generating matrices C are m \Theta m, then this construction yields a digital net fz points. If the generating matrices are 1 \Theta 1, i.e. each C j has entries c jkl defined for k; then one has a digital sequence g. The prototype digital sequence is the one-dimensional Van der Corput sequence, g. This is defined by taking equal to the identity In essence, the Van der Corput sequence, takes the b-ary representation of an integer and reflects it about the decimal point. 2.3. (t; m; s)-Nets and (t; s)-Sequences. Similarly to integration lattices one has the problem of how to choose the generating matrices C j in (2.2). Usually this is done to optimize the quality factor of the net or sequence. For any non-negative s-vector k, and for a base b consider the following set of disjoint boxes, whose union is the unit cube: Each such box in B k has volume b \Gammak 1 \Gamma\Delta\Delta\Delta\Gammak s . A (t; m; s)-net in base b is a set of points in [0; 1) s , such that every box in B k contains b m\Gammak 1 \Gamma\Delta\Delta\Delta\Gammak s of these points for any k satisfying t. Thus, any function that is piecewise 4 F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. constant on the boxes in B k will be integrated exactly according to quadrature rule The integer parameter t is called the quality parameter of the net and it takes values between 0 and m. A smaller value of t means a better net. A (t; s)-sequence is an infinite sequence of points in [0; 1) s such that the b m points numbered lb m to (l +1)b non-negative integer l. By using a (t; s)-sequence to do quasi-Monte Carlo calculations, one need not know the number of points required in advance. If the first b m1 points do not give sufficient accuracy, then one may add the next b m2 \Gamma b m1 points in the sequence to get a net with b m2 points, without throwing away the first b m1 points. There is a connection between digital nets and sequences as defined above and (t; m; s)-nets and (t; s)-sequences [32]. Let c (m) jiffl denote the vector containing the first m elements of the i th row of the the generating matrix C j . Given a non-negative integer s-vector k, let C(m; k) be the following set of the first k j rows of the j th generating matrix for s: c (m) Furthermore, let is a linearly independent set of vectors for all k with The following theorem (see [32]) gives the condition for which a digital net is a (t; m; s)- net and a digital sequence is a (t; s)-sequence. Theorem 2.1. For prime bases b the digital net defined above in (2.2) is a s)-net. If, in addition, T (s) is finite, then the digital sequence defined in (2.2) is a (T (s); s)-sequence. Finding good generating matrices for digital nets and sequences is an active area of research. Virtually all generators found so far have been based on number theoretic arguments. Early sequences include those of Sobol' [56], Faure [9] and Niederreiter [43]. Algorithms for these sequences can be found in the ACM Transactions on Mathematical Software collection. The FINDER software developed at Columbia University by Traub and Papageorgiou implements generalized Sobol' and generalized Faure se- quences. New constructions with smaller t values are given by Niederreiter and Xing [47]. 2.4. Infinite Lattice Sequences. The idea underlying digital sequences may be extended to integration lattices to obtain infinite lattice sequences. The i th term of a rank-1 lattice, which is fih=Ng, depends inherently on the number of points, N . Thus, the formula for a lattice must be rewritten in a way that does not involve N explicitly. A way to do this was first suggested in [24]. Suppose that the number of points, N , is some integer power of a base b - 2, that is, . This is the same assumption as for a digital or (t; m; s)-net. The first N values of the Van der Corput sequence, defined in (2.3) areb Extensible Lattice Sequences 5 although in a different order. Therefore, the term appears in the definition of the rank-1 lattice may be replaced by OE b (i), a term that does not depend on N . The s-dimensional generating vector h in (2.1) typically depends on N also. It may be expressed in b-ary form as: are digits. For k ? m the digits h jk do not affect the definition of the rank-1 lattice set with points since they only contribute integers to the product OE b (i)h. Therefore, each component of h may be written (in principle) as an infinite string of digits: This single "infinite" generating vector may serve for all possible values of m. The preceding paragraphs provide the basis for defining an infinite rank-1 lattice sequence. Altering the original definition in (2.1) leads to the following: Definition 2.2. An infinite rank-1 lattice sequence in base b with generating vector h of the form (2.6) and shift \Delta is defined as: The first b m terms of the infinite rank-1 lattice sequence (2.7) are a rank-1 lattice. Moreover, just as certain subsets of a (t; s)-sequence are (t; m; s)-nets, so subsets of an infinite rank-1 lattice sequence are shifted rank-1 lattices. Theorem 2.3. Suppose that P is the set consisting of the l +1 st run of b m terms of the infinite lattice rank-1 sequence defined in (2.7): Then, P is a rank-1 lattice with shift OE b (l)b \Gammam h + \Delta, that is, Proof. The proof follows directly from the definition of the Van der Corput sequence. For all note that Substituting the right hand side into the definition of P completes the proof. The definition of an infinite rank-1 lattice sequence may be extended to integration lattices of arbitrary rank. Definition 2.4. An infinite lattice sequence (of arbitrary rank) with bases generating vectors h of the form (2.6) is defined as: A practical complication for an integration lattice of rank greater than 1 is that there are multiple indices, i k , each of which may or may not tend to infinity, and each at its own rate. Because of this complication we will focus on rank-1 lattices in the sections that follow. Theorem 2.3 also has a natural extension to infinite lattice sequences of arbitrary rank, and its proof is similar. 6 F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. 3. Discrepancy. Unlike (t; s)-nets for which there exist explicit constructions of the generating matrices C j , there are no such explicit constructions of generating vectors h for rank-1 lattices for arbitrary s. Tables of generating vectors for lattices that do exist (see [8, 14, 27]) are usually obtained by minimizing some measure of non-uniformity, or discrepancy, of the lattice. This section describes several useful discrepancy measures. the quadrature error for a rule of the form (1.1) for an arbitrary set P . Worst case error analysis of the quadrature error leads to a Koksma- Hlawka-type inequality of the form [23]: where D(P ) is the discrepancy or measure of nonuniformity of the point set defining the quadrature rule, and V (f) is the variation or fluctuation of the integrand, f . The precise definitions of the discrepancy and the variation depend on the particular space of integrands. In the traditional Koksma-Hlawka inequality (see [26] and [44, Theorem 2.11]), the variation is the variation in the sense of Hardy and Krause, and the discrepancy is the L1 -star discrepancy: Here F unif is the uniform distribution on the unit cube, FP is the empirical distribution function for the sample P , and j\Deltaj denotes the number of points in a set. The notation denotes the L p -norm or the ' p -norm, depending on the context. Error bounds of the form (3.1) involving the L p -star discrepancy have been derived by [57, 62]. Error bounds involving generalizations of the star discrepancy appear in [21, 22, 23, 55]. When the integrands belong to a reproducing kernel Hilbert space, the error bound (3.1) may be easily obtained [21, 23]. The discrepancy may be written in terms of the reproducing kernel Z K(x; y) dx dy \GammaN Z For example, the L 2 -star discrepancy, whose formula was originally derived in [60], is a special case of the above formula with K(x; An advantage of considering reproducing kernel Hilbert spaces of integrands is that the computational complexity of the discrepancy is relatively small (at worst O(N 2 ) operations). By contrast the L1 -star discrepancy requires O(N s ) operations to evaluate. The discrepancy of type (3.3) can also be interpreted as an average-case quadrature error [25, 41, 61]. Suppose that the integrand is a random function lying in the sample space A, and suppose that the integrand has zero mean and covariance kernel, K(x; y), that is, Extensible Lattice Sequences 7 Then the root mean square quadrature error over A is the discrepancy as defined in If P is a simple random sample, then the mean square discrepancy is [25]: Z This formula serves as a benchmark for other (presumably superior) low discrepancy sets. Since the mean square discrepancy is O(N \Gamma1 ), the discrepancy itself is typically O(N \Gamma1=2 ) for a simple random sample. The variance of a function, f , may be defined as Z f dx The mean value of the variance over the space of average-case integrands can be shown to be [25]: Z Z which is just the term in braces in (3.5). It may seem odd at first that the discrepancy can serve both as an average-case and worst-case quadrature error. The explanation is that the space of integrands, A, in the average-case analysis is much larger than the space of integrands, W , in the worst-case analysis. See [21, 25] and the references therein for the proofs of the above results as well as further details. There are some known asymptotic results for the discrepancies of (t; m; s)-nets. The L1 -star discrepancy of any (t; m; s)-net is O(N \Gamma1 [log N Theorem 4.10]. Moreover, the typical (in the sense of an average taken over all possible nets) L 2 - star discrepancy of (0; m; s)-nets is O(N \Gamma1 [log N the best possible for any set. For discrepancies of the form (3.3) with sufficiently smooth kernels, typical (0; m; s)-nets have O(N \Gamma3=2 [log N Lattice rules, the topic of this article, are known to be particularly effective for integrating periodic functions. Suppose that the integrand has an absolutely convergent Fourier series with Fourier coefficients - Z Here k 0 x denotes the dot product of the s-dimensional wavenumber vector k with x. The quadrature error for a particular integrand with an absolutely convergent Fourier series is simply the sum of the quadrature errors of each term: "N 8 F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. where the term corresponding to does not enter because constants are integrated exactly. One can multiply and divide by arbitrary weights w(k) inside this sum. Then by applying H-older's inequality one has the following error bound of the form (3.1) [23]: "N Here 1 f\Deltag denotes the indicator function. In order to insure that the discrepancy is finite we assume that the weights increase sufficiently fast as k tends to infinity: If P is the node set of an integration lattice, then it is known that trigonometric polynomials are integrated exactly for all nonzero wavenumbers not in the dual lattice, The dual lattice is the set of all k satisfying k 0 . Thus, for node sets of lattices the definition of discrepancy above may be simplified to Certain explicit choices of w(k) have appeared in the literature. For example, one may choose where the over-bar notation is defined as are arbitrary positive weights, and ff is a measure of the assumed smoothness of the integrand. If fi is the node set of a lattice, then is a traditional figure of merit for lattices [53]. Furthermore, for the discrepancy is DF;1 (P ae(L) is the Zaremba figure of merit for lattice rules [44, Def. 5.31]. The more general case of P not a lattice is considered in [20], and the case of unequal weights discussed in [22]. Extensible Lattice Sequences 9 If the weight function w(k) takes the form (3.9) for positive integer ff, then for the infinite sum defining the discrepancy in (3.8b) may be written as a finite sum: s Y for general sets P , where B 2ff denotes the Bernoulli polynomial of degree 2ff [1, Chap. When P is the node set of an integration lattice, the double sum can be simplified to a single sum: s Y Another choice for w(k) is a weighted ' r -norm of the vector k to some power: again for arbitrary positive weights these weights are unity, P is the node set of a lattice, and kkk \Gammaff ae min kkk r oe \Gammaff For discrepancy is equivalent to the spectral test, commonly employed to measure the quality of linear congruential pseudo-random number generators [30, 33]. The spectral test has been used to select lattices for quasi-Monte Carlo quadrature in [7, 34, 35, 36]. The case which one might call an ' 1 -spectral test is also interesting. We will return to these two cases in the next section. 4. Good Generating Vectors for Lattice Sequences. As mentioned at the beginning of the previous section, finding good generating vectors for lattices typically requires optimizing some discrepancy measure. In this subsection we propose some loss functions and optimization algorithms for choosing good generating vectors for extensible rank-1 lattice sequences. In principle one would like to have an s \Theta 1 array of digits h jk , according to (2.6). However, in practice it is only necessary to have an s max \Theta mmax array of digits h jk , is the maximum number of points and s max is the maximum dimension to be considered. In finance calculations, for example, the necessity of timely forecasts may constrain one to a budget of private communication). For simplicity we consider generating vectors h that are of the form originally proposed by Korobov, that is, This means that only the digits need to be chosen, for which there are b choices. The generating vector is tested for dimensions up to s max , but in fact it can be extended to any dimension. F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. The number j defining the generating vector is chosen by minimizing a loss func- tion, G, of the form ~ Here the function ~ is related to one of the measures of discrepancy introduced in Section 3, and the maximum over some range of values of m and s insures that the resulting generating vector is good for a range of numbers of points and dimensions. However, since the discrepancy itself depends significantly on the number of points and the dimension, it must be appropriately scaled to arrive at the function ~ G. The details of this scaling are given below. 4.1. Generating Vectors Based on Minimizing P ff . The discrepancy defined in (3.11), which is a generalization of the P ff figure of merit for lattice rules, has the advantage of requiring only O(sN) operations to evaluate for lattices. To remove some of the dimension dependence of this discrepancy it is divided by the square root of right hand side of (3.6). The root mean square of this scaled discrepancy for a random sample is then N \Gamma1=2 , independent of s. The formula for the scaled discrepancy of the node set of a lattice with s Y The specific choice of the value of fi j here is not crucial, but seems to give good results. For one-dimensional lattices, i.e. evenly spaced points on the interval [0; 1), this discrepancy is N \Gamma1 , and one would expect that as the dimension increases this scaled discrepancy would tend to (or at least do no worse than) N \Gamma1=2 . Therefore, to remove this remaining dimension dependence the above scaled discrepancy is divided by the For a fixed s, D asy (m,s) is asymptotically O(b \Gammam m (s\Gamma1)=2 as N tends to infinity. This is the asymptotic order for (0; m; s)-nets [25], and what we hope to achieve for lattice sequences. Furthermore, N and s. In summary, the resulting loss function is ~ \Theta s Y The optimal values of j found by minimizing G 1 (j) for different ranges of m and s are given in Table 4.1. The algorithm for optimizing G 1 (j) may be described as an intelligent exhaustive search. One need not compute G 1 (j) for all possible values of j. Suppose at any stage of the optimization j is the best known value of j, and one finds that ~ j. Since ~ depends only on the first digits of j, one can immediately eliminate from consideration all j that have the same first m \Gamma 1 digits as ~ j. This same search strategy is also used for the other loss functions described below. Extensible Lattice Sequences 11 4.2. Generating Vectors Based on the Spectral Test. The use of the spectral test to analyze the lattice structure of linear congruential generators is described in [30] and tables of good integration lattices are given in [34]. The difference here is that nearly all smaller lattices imbedded in the largest lattice considered must have low discrepancy. In [34], only the full lattice was examined. The length of the shortest non-zero vector in the dual lattice L ? is which is related to the discrepancy (3.12) with 2. This length has the absolute upper bound d where the constants fl s and ae s depend only on s (see [34] and the references therein). The bound for s - 8 is the least upper bound for a general s-dimensional lattice with real-valued coordinates, and with b \Gammam points per unit of volume. The bound for s ? 8 is not the least upper bound, but it is still reasonably tight, as our numerical results will show. We define the normalized ' 2 -spectral test discrepancy as ~ which is larger than 1 and is the inverse of the quantity S t defined in [34]. (The different notation here is to be consistent with the rest of this article). The loss function to be minimized is of the form (4.2) with ~ We note that 1=d 2 (j; m; s) can be interpreted as the (Euclidean) distance between the successive hyperplanes that contain all the points of the primal lattice L, for the family of hyperplanes for which this distance is the largest. The problem of computing a shortest vector in (4.4) can be formulated as a quadratic optimization problem with s integer decision variables, because k can be written as a linear combination of the s vectors of a basis of the dual lattice, with integer coefficients. The decision variables are these coefficients. (See [30] for details.) We solved this problem by using the branch-and-bound algorithm of Fincke and Pohst [10], with a few heuristic modifications to improve the speed. The worst-case time complexity of this algorithm is exponential in d 2 (j; m; s), and polynomial in s for d 2 (j; m; s) fixed [10]. In practice, it (typically) works nicely even when d 2 (j; m; s) is large. For example, one can compute 2, an arbitrary j and in less than 1 second on a Pentium-II computer. 4.3. Generating Vectors Based on the ' 1 -spectral Test. With the ' 1 norm, the length of the shortest non-zero vector in L ? is which is related to the discrepancy (3.12) with 1. One has the upper bound which was established by Marsaglia [37] by applying the general convex body theorem of Minkowski. This suggests the normalized ' 1 -spectral test quantity: ~ 12 F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. Here, we want to minimize the loss function (4.2) with ~ G 3 . One can interpret d 1 (j; m; s) (or d 1 (j; m; s) \Gamma 1 in certain cases; see [30]) as the minimal number of hyperplanes that cover all the points of P . We computed via the algorithm of Dieter [6], which works fine for s up to about 10 (independently of m), but becomes very slow for larger s (the time is exponential in s). The test quantity in (4.8) has an interpretation similar that the quality parameter t for (t; m; s)-nets. Define s Since ~ (s!)]=s. Thus, the rank-1 lattice defined by j integrates exactly all trigonometric polynomials of wavenumber k when log(jk s If one considers T (j; m; s) as the quality parameter of the lattice, then this condition is similar to that in (2.5). There, t determines the resolution at which piecewise constant functions that are exactly integrated by a net. Here, T (j; m; s) determines the resolution at which trigonometric polynomials are integrated exactly by a lattice. The discrepancy for the node set of this lattice as defined in (3.12) with If one can construct an infinite sequence of digits, j, for which then the above discrepancy decays like N \Gammaff=s . Again, the parameter ff indicates the assumed smoothness of the integrands. 4.4. Tables of Coefficients. We made computer searches to find the best j's, based on minimizing the worst-case loss function ~ selected values of m . The bounds m define a range of number of points in the lattice, and maximal number of dimensions, that we are interested in. Selecting different parameters j for different ranges of values of m and s is a convenient compromise between the extreme cases of choosing a different j for each pair (m; s), and choosing the same j for all pairs (m; s). In practice one typically has a general idea of how many points one is going to take. By selecting an j specialized for that range, one can obtain a lattice with a better figure of merit for this particular range. Table 4.1 gives the optimal j's and the corresponding figures of merit (4.9) for Because of computational efficiency constraints, for the searches, we limited ourselves to m 1 - 20 for Then, for the best j that we found, we verified the performance when m 0 was reduced Extensible Lattice Sequences 13 or when m 1 or s 1 was increased, and retained the smallest m 0 and largest m 1 and s 1 for which G i was unchanged. The table also gives the value of G i;min (j; m defined by replacing max by min in (4.9). This best-case figure tells us the range of values taken by ~ over the given region. We also made exhaustive searches with all the entries where in the table, and obtained the same values of j and G cases. Table Values of j defining generating vectors of the form (4.1) for rank-1 lattice sequences with base minimizing (4.9). 5. Estimating Quadrature Error. The advantage of an extensible lattice sequence is that N need not be specified in advance. Therefore, in practice one would estimate the quadrature error for an initial lattice, and continue to increase the lattice size until the quadrature error meets the desired tolerance. Although the discrepancy, D(P ), is a good measure for the quality of a set P , it cannot be used directly for error estimation. The worst case error bounds (3.1) are often quite conservative, and there is no easy way to estimate the variation of the integrand. The average case error given, (3.4), is sensitive to how one defines the kernel - multiplying the kernel by a factor of c 2 changes the average case error by a factor of c. Quadrature error estimates for lattice rules have been investigated by [5, 28, 53]. Two different kinds of quadrature rules and error estimates have been proposed. Both involve estimating the error for terms of Q(f , where the P l are node sets of lattices and The method proposed in [53] takes P to be composed of 2 s shifted copies of a lattice. Rather than considering copy rules, we take P l to be the node sets of shifted lattices of size b m1 imbedded in the extensible lattice described in Section 2.4: 14 F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. Another quadrature rule and error estimate takes the P l to be independent random shifts of a lattice. This can be done by taking: where the \Delta l are independent, uniformly distributed random vectors in [0; 1] s . Note that for both (5.1) and (5.2) the set P can be extended in size as necessary by increasing m 1 . The theory behind the error estimates for cases (5.1) and (5.2) are given in the following theorem. Theorem 5.1. Suppose that a quadrature rule Q(\Delta; P ) based on some arbitrary P , as given in (1.1), is the average value of the quadrature rules based on the sets First, consider the case where the integrands are random functions from a sample space A as described in the paragraph preceding (3.4). Then it follows that where D is the discrepancy based on the covariance kernel for A. Secondly, consider the case of a fixed integrand, f , but where the P l are random shifts of a set P 0 , that is, for independent, uniformly distributed Proof. Assuming that the quadrature rules satisfy (5.3), it follows that the mean square deviation of the Q(f ; P l ) from Q(f ; P ) may be written asM If the integrand is a random function from a sample space A with covariance kernel K, then average case error analysis in (3.4) plus (5.7) leads to the following equations: Extensible Lattice Sequences 15 The equations above may be rearranged to give (5.4). The quadrature error for rule Q(f ; P ) may also be written as: Substituting the sum of the by equation (5.7) gives: If the P l are random shifts as in (5.5), then the expected value of the term [I(f) \Gamma vanishes for all k 6= l and taking the expected value of (5.8) yields (5.6). Some remarks are in order to explain the assumptions and conclusions of the above theorem. These are given below. Assumption (5.3), and thus conclusion (5.4), holds for both the cases (5.1) and (5.2) above. In fact, this part of the theorem holds for any imbedded rule or any rule where the P l all contain the same number of points, and their union is P or multiple copies of P . For example, (5.4) would apply to the case where P is a (t; m; s)-net made up of a union of subnets P l . Assumption (5.5), and therefore conclusion (5.6) holds for (5.2). There is a difficulty if one tries to derive a result like (5.6) for an imbedded rule of the form (5.1), where \Delta is a random shift. The argument leading to (5.6) assumes that the points in different P l are uncorrelated, which is not true for (5.1). However, if the extensible lattice is a good one, it is expected that the terms in (5.8) are on average negative. Under this assumption one may then conclude that the right hand side of (5.6) is a conservative (too large) upper bound on the expected square quadrature error. The factor in (5.4) above involving the discrepancies of P and the P l does not depend strongly on the particular choice of discrepancy, but only on the asymptotic rate of decay. If, for example, D(P ) - CN \Gammaff for some unknown C, but known ff, where N is the number of points in P , then Although conclusions (5.6) and (5.9) are derived under different assumptions, they both suggest error estimates of the form F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. indicates the rate of decay of the discrepancy for (5.9). The factor c ? 1 depends on how conservative one wishes to be. The Chebyshev inequality implies that the above inequality will hold "at least" 100(1 \Gamma c \Gamma2 )% of the time. Error estimate (5.10) leads to the stopping criteria: where ffl is the absolute error tolerance. Note that if the stopping criteria is not met, one would normally increase the size of the P l by increasing m 1 , rather than increasing the number of the P l by increasing M . For some high dimensional problems the discrepancy of a lattice (or other low discrepancy set) may decay as slowly as the Monte Carlo rate of O(N \Gamma1=2 N (see [18, 41]). Therefore, even when using (5.9), it may be advisable to make a conservative choice of ff = 1=2. This choice makes the approach of error estimate (5.9), based on random integrands, equivalent to that of (5.6), based on randomly shifted P l . Sloan and Joe [53, Section 10.3] suggest an error estimate of the form s s where P is formed from 2 s copies of a rank-1 lattice, and each Q(f ; P l ) is an imbedded rule based on half of the points in P . Although this case does not exactly fit Theorem 5.1, the arguments in the proof can be modified to obtain a result similar to to (5.4): s s s This would suggest that the error estimation formula of Sloan and Joe is reasonable when 2D(P ) on average. The disadvantage of 2 s -copy rules is that they require at least 2 s points, which may be unmanageable for large s. To summarize, both imbedded lattice rules, (5.1), and independent random shifts of lattices, (5.2), have similar error estimates, (5.10), and stopping criteria, (5.11). The advantage of the independent random shifts approach is that, the theory holds for any integrand, not the average over a space of integrands. One advantage of the imbedded rules approach is that one need only generate a single extensible lattice. Furthermore, the set P for the imbedded rule is the node set of a lattice (if M is a power of b), which is never the case for the independent random shifts approach. Thus, the accuracy of the imbedded rule approach is likely to be better. In the examples in the next section, the imbedded lattice rules based on (5.1) are used. 6. Examples of Multidimensional Quadrature. Two example problems are chosen to demonstrate the performance of the new rank-1 lattice sequences proposed in Section 2.4. The first example is the computation of multivariate normal probabilities and the second is the evaluation of a multidimensional integral arising in physics problems. Extensible Lattice Sequences 17 6.1. Multivariate Normal Probabilities. Consider the following multivariate normal Z b1 Z bs as where, a and b are known s-dimensional vectors, and \Sigma is a given s \Theta s positive definite covariance matrix. Some a j and/or b j may be infinite. Unfortunately, the original form is not well-suited for numerical quadrature. Therefore, Alan Genz [12] proposed a transformation of variables that results in an integral over an s \Gamma 1-dimensional unit cube. See [12, 13] for the details of the transformation, and see [13, 15] for comparisons of different methods for calculating multivariate normal probabilities. The particular test problem is one considered by [13, 24] and may be described as follows: i.i.d. uniformly on [0; generated randomly according to [13, 38]: (6.1d) Numerical comparisons were made using three types of algorithms: i. the adaptive algorithm DCHURE [2], ii. an older Korobov rank-1 lattice rule with a different generating vector for each N and s - this algorithm is a part of NAG and is used in [5, 12], and iii. the new rank-1 lattice sequences proposed in Section 2.4 with generating vectors given in Table 4.1 For the second and third algorithms we applied the periodizing transformation x 0 1j to the integrand over the unit cube. This appears to increase the accuracy of the lattice rule methods. The computations were carried out in FORTRAN on a Unix work station in double precision. The absolute error tolerance was chosen to be , and this was compared with the actual error E. Since the true value of the integral is unknown for this test problem, the value given by the Korobov algorithm with a tolerance of used as the "exact" value for computing the error. For the new rank-1 lattice sequences the stopping criterion (5.11) was used with M between 4 and 7 and For each dimension 50 random test problems were generated and solved by the various quadrature methods. The scaled absolute errors E=ffl and the computation times in seconds are given in the box and whisker plots of Fig. 6.1. The boxes contain the middle half of the values and the whiskers give the range of most values except the outliers (denoted by ). Ideally, the scaled error should nearly always be less than one, otherwise the error estimate is not conservative enough. On the other hand if the scaled error is too small, then the error estimate is too conservative. Fig. 6.1 shows that the adaptive rule performs well for smaller dimensions, but underestimates the error and is quite slow in higher dimensions. The lattice rules do well even in higher dimensions, and the new rank-1 lattice sequences appear to be faster than the older Korobov-type rule. This is likely due to the fact that the lattice sequences proposed here can re-use the old points when N must be increased. F. J. Hickernell, H. S. Hong, P. L' ' Ecuyer and C. Lemieux. 6.2. A Multidimensional Integral from Physics. Keister [29] considered the following multidimensional integral that has applications in physics: Z R s Z cos@ A dy; (6.2) where \Phi denotes the standard Gaussian distribution function. Keister gave an exact formula for the answer and compared the quadrature methods of McNamee and Stenger [39] and Genz and Patterson [11, 51] for evaluating this integral. Later, Papageorgiou and Traub [49] applied the generalized Faure sequence from FINDER to this problem. The results of numerical experiments for the above integral for dimension 25 are shown in Figure 6.2. The exact value of the integral is reported in [49]. To be consistent with the numerical results reported in [29, 49], we did not perform error estimation, but just computed the actual error for each kind of numerical method as a function of N , the number of points. Because \Phi there is a technical difficulty with using an unshifted lattice rule, so when performing the numerical experiments the lattice sequences were given random shifts (modulo 1). Box and whisker plots show how well the new rank-1 lattice sequences perform for 50 random shifts. According to Figure 6.2 the generalized Faure sequence (in base 29) and the lattice sequence perform much better than the other two rules. In some cases the lattice sequences perform better than the generalized Faure sequence. 7. Conclusion. Lattice rules are simpler to code than digital nets. Given the construction in Section 2.4, it is now possible to have extensible lattice sequences in the same way that one has (t; s)-sequences. Good generating vectors for these lattice sequences may be found by using the spectral test or minimizing other discrepancy measures, as shown in Section 4. The performance of these lattice rules is in many cases comparable to other multidimensional quadrature rules and in some cases superior Acknowledgments . Thanks to Alan Genz for making available his software for computing multivariate normal distributions. Also thanks to Joe Traub and Anargy- ros Papageorgiou for making available the FINDER software. --R Handbook of Mathematical Functions with Formulas An adaptive algorithm for the approximate calculation of multiple integrals Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension Randomization of number theoretic methods for multiple integration How to calculate shortest vectors in a lattice Discr'epance de suites associ'ees 'a un syst'eme de num'eration (en dimension s) Improved methods for calculating vectors of short length in a lattice A Lagrange extrpolation algorithm for sequences of approximations to multiple inte- grals Parameters for integrating periodic functions of several variables Simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results On the assessment of random and quasi-random point sets Random and Quasi-Random Point Sets A comparison of random and quasirandom points for multidimensional quadrature Computing multivariate normal probabilities using rank-1 lattice sequences Funktionen von beschr-ankter Variation in der Theorie der Gleichverteilung Applications of Number Theory to Numerical Analysis Randomization of lattice rules for numerical multiple integration Multidimensional quadrature algorithms The Art of Computer Programming The approximate computation of multiple integrals On the distribution of digital sequences Random numbers fall mainly in the planes Construction of fully symmetric numerical integration formu- las Generating quas-random paths for stochastic processes Points and sequences with small discrepancy Shiue eds Quasirandom points and global function fields Faster valuation of financial derivatives The optimum addition of points to quadrature formulae Lattice methods for multiple integration Lattice Methods for Multiple Integration Lattice methods for multiple integration: Theory An intractability result for multiple integration The distribution of points in a cube and the approximate evaluation of integrals "Nauka" Computational investigations of low discrepancy point sets Average case complexity of multivariate integration Some applications of multidimensional integration by parts --TR --CTR Fred J. Hickernell , Harald Niederreiter, The existence of good extensible rank-1 lattices, Journal of Complexity, v.19 n.3, p.286-300, June Hee Sun Hong , Fred J. Hickernell , Gang Wei, The distribution of the discrepancy of scrambled digital (t, m, s)-nets, Mathematics and Computers in Simulation, v.62 n.3-6, p.335-345, 3 March Pierre L'Ecuyer, Quasi-monte carlo methods in practice: quasi-monte carlo methods for simulation, Proceedings of the 35th conference on Winter simulation: driving innovation, December 07-10, 2003, New Orleans, Louisiana Fred J. Hickernell, My dream quadrature rule, Journal of Complexity, v.19 n.3, p.420-427, June Hee Sun Hong , Fred J. Hickernell, Algorithm 823: Implementing scrambled digital sequences, ACM Transactions on Mathematical Software (TOMS), v.29 n.2, p.95-109, June

good lattice point sets;multidimensional;discrepancy;spectral test

587257

Residual Replacement Strategies for Krylov Subspace Iterative Methods for the Convergence of True Residuals.

In this paper, a strategy is proposed for alternative computations of the residual vectors in Krylov subspace methods, which improves the agreement of the computed residuals and the true residuals to the level of O(u) ||A|| ||x||. Building on earlier ideas on residual replacement and on insights in the finite precision behavior of the Krylov subspace methods, computable error bounds are derived for iterations that involve occasionally replacing the computed residuals by the true residuals, and they are used to monitor the deviation of the two residuals and hence to select residual replacement steps, so that the recurrence relations for the computed residuals, which control the convergence of the method, are perturbed within safe bounds. Numerical examples are presented to demonstrate the effectiveness of this new residual replacement scheme.

Introduction Krylov subspace iterative methods for solving a large linear system typically consist of iterations that recursively update approximate solutions x n and the corresponding residual vectors They can be written in a general form as follows. Algorithm 1. Template for Krylov subspace Method: Input: an initial approximation x For convergence Generate a correction vector q n by the method; (the vector x n does not occur in other statements) End for Department of Mathematics, Utrecht University, P.O. Box 80010, NL-3508 Utrecht, The Netherlands E-mail: vorst@math.uu.nl y Department of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2. E-mail: ye@gauss.amath.umanitoba.ca Research supported by grants from University of Manitoba Research Development Fund and from Natural Sciences and Engineering Research Council of Canada Most Krylov subspace iterative methods, including the conjugate gradient method (CG) [12], the bi-conjugate gradient method (Bi-CG) [4, 13], CGS [19], and BiCGSTAB [22], fit in this framework (see [2, 11, 16] for other methods). In exact arithmetic, the recursively defined r n in Algorithm 1 is exactly the residual for the approximate solution x In a floating point arithmetic, however, the round-off patterns for x n and r n will be different. It is important to note that any error made in the computation of x n is not reflected by a corresponding error in r n , or in other words, computational errors to x n do not force the method to correct, since x n has no influence on the iteration process. This leads to the well known situation that b \Gamma Ax n and r n may differ significantly. This phenomenon has been extensively discussed in the literature, see [10, 11, 18] and the references cited there. Indeed, if we denote the computed results of x respectively (but we still use q n to denote the computed update vector of the algorithm), then we have where f l(z) denotes the computed result of z in finite arithmetic, the absolute value and inequalities on vectors are componentwise, and u is the machine roundoff unit. The vectors / n and rounding error terms, and they can be bounded by a straightforward error analysis (see Section 3 for details). In particular, the relations (1) and (2) show that / n and j n depend only on the iteration vectors - r n , and q n . We will call b \Gamma A-x n the true residual for the approximation - x n and call - r n , as obtained by recurrence formula (2), the computed residual (or the updated residual). Then the difference between the two satisfies (using the finite precision recurrences (1) and (2)) where we assume for now that b \Gamma Ax Hence, the difference between the true and the updated residuals is a result of accumulated rounding errors. In particular, a significant deviation of r n may be expected, if there is a - r i with large norm during the iteration (a not uncommon situation for Bi-CG and CGS). On the other hand, even when all / i or j i are small (as is common for CG), but if it takes a relatively large number of iterations for convergence, the sheer accumulation of / i and j i could also lead to a nontrivial deviation. What makes all this so important is that, in a finite precision implementation, the sequence - r n satisfies, almost to machine precision u, its defining recurrence relation, and as was observed for many Krylov subspace methods, this is the driving force behind convergence of - r n [10, 15, 18, 20]. Indeed, residual bounds have been obtained in [20] for CG and Bi-CG, which show that even a significantly perturbed recurrence relation (with perturbations much larger than the machine precision) usually still leads to eventual convergence of the computed residuals. This theoretical insight has been our motivation and justification for the residual replacement scheme to be presented in Section 2.1. On the other hand, the true residual b \Gamma A-x n itself has no self-correcting mechanism for convergence, mainly because any perturbation made to - x n does not have an effect on the iteration parameters, whereas errors in - immediately lead to other iteration parameters. Thus, in a typical convergent iteration process, - r n converges to a level much smaller than u eventually, but the true residual b \Gamma A-x n can only converge to the level dictated by \Sigma n since Usually, when - r n is still bigger than the accumulated error \Sigma n agrees well with - r n in magnitude, but when - r n has converged to a level that is smaller than the accumulated error, then just the accumulated error and has no agreement at all with - r n . In summary, a straightforward implementation would reduce the true residuals at best to bound for this has been obtained in [10] and it is called the attainable accuracy. We note that this term could be significant even if only one of / i or j i is large, or if n is large. The above problems become most serious in methods such as CGS and Bi-CG where intermediate x n and - r n can have very large norm, and this may result in a large / n or j n . Several popular methods, such as BiCGSTAB [22], BiCGSTAB(') [17], QMR [7], TFQMR [5], and composite step BiCG [1], have been developed to reduce the norm of - r n (see [6] for details). We note that controlling the size of k-r n k only does not solve the deviation problem in all situations, as, for instance, the accumulation of tiny errors over a long iteration may still result in a nontrivial deviation. A simple approach for solving the deviation problem is to replace the computed residuals by the true residuals at some iteration step to restore the agreement. Then the deviation at subsequent steps will be the error accumulation after that iteration only. This includes a complete replacement strategy that simply computes r n by b \Gamma Ax n at every iteration, and a periodic replacement strategy that updates r n by b \Gamma Ax n only at intervals of the iteration count. While such a strategy maintains agreement of the two kinds of residuals, it turns out that the convergence of the r n may deteriorate (as we will see, it may result in unacceptably large perturbations to the lanczos recurrence relation for the residual vectors that steers the convergence, see Section 2.3). Recently, Sleijpen and van der Vorst [18], motivated by suggestions made by Neumaier (see [11, 18]), introduced a very sophisticated replacement scheme that includes a so-called flying-restart procedure. It was demonstrated that this new residual replacement strategy can be very effective in the sense that it can improve the convergence of the true residuals by several orders of magnitude. For practical implementations, such a strategy is very useful because it leads to meaningful residuals and this is important for stopping the iteration process at the right point. Of course, one could, after termination of the classical process, simply test the true residual, but the risk is that the true residual stagnated already long before termination, so that much work has been done in vain. The present paper will follow the very same idea of replacing the computed residual by the true residual at selected steps, in order to maintain close agreement between the two residuals, but we propose a simpler strategy so that the replacement is done only when it is necessary and at phases in the iteration where it is harmless, that is that convergence mechanism for - r n is not destroyed. Specifically, we shall present a rigorous error analysis for iterations with residual replacement and we will propose computable bounds for the deviation between the computed and true residuals. This will be used to select the replacement phases in the iteration in such a way that the Lanczos recurrence among - r n is sufficiently well maintained. For the resulting strategy, it will be shown that, provided that the computed residuals converge, the true residual will converge to the level O(u)kAkkxk, the smallest level that one can expect for an approximation. The paper has been organized as follows. In Section 2, we develop a refined residual replacement strategy and we discuss some strategies that have been reported by others. We give an error analysis in Section 3, and we derive some bounds for the deviation to be used in the replacement condition. We present a complete implementation in Section 4. It turns out that the residual replacement strategy can easily be incorporated in existing codes. Some numerical examples are reported in Section 5, and we finish with remarks in Section 6. The vector norm used in this paper is one of the 1, 2, or 1-norm. Residual Replacement Strategy In this section, we develop a replacement strategy that maintains the convergence of the true residuals. A formal analysis is postponed to the next section. The specific iterative method can be any of those that fit in the general framework of Algorithm 1. Throughout this paper, we shall consider only iteration processes for which the computed residual - r n converges to a sufficiently small level. As mentioned in Section 1, we follow the basic idea to replace the computed residual - r m by the true residual f selected steps We will refer to such an iteration step as one where residual replacement occurs. Hence, the residual generated at an arbitrary step n could be either the usual updated residual - r or the true residual depending on whether replacement has taken place or not at step n. In order to distinguish the two possible formulations, we denote by r n the residual obtained at step n of the process with the replacement strategy, that is With the residual replacement at step m residual deviation is immediately reduced to and it can be shown (see Lemma 1 of Section 2.2) that For the subsequent iterations n ? m, but before the next replacement step, we clearly have that Therefore, the accumulated deviation before step m has no effect to the deviation after updating (n ? m). However, in order for such a strategy to succeed, two conditions must be met, namely, ffl the computed residual r n should preserve the convergence mechanism of the original process that has been steered by the - ffl from the last updating step m to the termination step K, the accumulated error \Sigma K should be small relative to u(jr which is the upperbound for j- m j. We discuss in the next two subsections how to satisfy these two objectives. 2.1 Maintaining convergence of computed residuals In order that r n maintains the convergence mechanism of the original updated residuals, it should preserve the property that gives rise to the convergence of the original - r n . We therefore need to identify the properties that lead to convergence of the iterative method in finite precision arithmetic. While this may be different for each individual method, it has been observed for several Krylov subspace methods (including CG [10, 20], Bi-CG [20], CGS, BiCGSTAB, and BiCGSTAB(') [18]), that the recurrence r and a similar one for q n is satisfied almost to machine precision and this small local error is one of the properties behind the convergence of the computed residuals. Furthermore, the analysis of [20] suggests that convergence is well maintained even when the recurrence equations are perturbed with perturbations that are significantly greater than the machine precision. This latter property is the basis for our residual replacement strategy. Therefore, we briefly discuss this perturbation phenomenon for Bi-CG (or CG), as presented in [20]. Consider the Bi-CG iteration which contains r In finite which denote the computed results of r n and p n , respectively, satisfy the perturbed recurrence are rounding error terms that can be bounded in terms of u. Combining these two equations, we obtain the following perturbed matrix equation in a normalized form r n+1 where T n is an invertible tridiagonal matrix 1 , ff 0 with ff We note that (4) is just an equation satisfied by an exact Bi-CG iteration under a perturbation F n . In particular, detailed bounds on - n and j n will, under some mild assumptions, lead to F n - O(u). The main result of [20] states that if a sequence - r n satisfies (4) and Z n+1 has full rank, then we have where n . The case F reduces to the known theoretical bound for the exact BiCG residuals [1]. Therefore, even when - r n and its exact counterpart are completely different, their norms are bounded by similar quantities and are usually comparable. Of course, in both cases, the bounds depend on the quality of the constructed basis. More importantly, a closer examination of the bound reveals that even if the perturbation F n is in magnitude much larger than u, the quantities in the bound, and thus k-r n+1 k, may not be significantly affected. Indeed, in [20] numerical experiments were presented, where relatively large artificial random perturbations had been injected to the recurrence for r n ; yet it did not significantly affect the convergence mechanism. An implication of this analysis is that, regardless of how - r n is generated but as long as it satisfies (4), its norm can be bounded by (6). Hence, we can replace - r n by r We assume that no breakdowns of the iteration process have occurred when are not too large relative to kr n k and kr (see (5)), and we may still expect it to converge in a similar fashion. Indeed, this criterion explains why the residual replacement strategies like r but do not work always (see Section 2.3). Here, it will be used to determine when it is safe to replace - r n by r note that the above discussion is for Bi-CG, but the phenomenon it reveals seems to be valid for many other methods, especially for those methods that are based on Bi-CG (CGS, BiCGSTAB, and others). Now we consider the case that residual replacement is carried out at step m, that is r It follows from the definition of ffi m and - r m that . So, the updated residual r m satisfies Thus depending on the magnitude of kj 0 k relative to kr m k and kr m\Gamma1 k, the use of r may result in large perturbations to the recurrence relation. Therefore, a residual replacement strategy should ensure that the replacement is only done when kj 0 kg is not too large. In a typical iteration, as the iteration proceeds, kffi n k, and hence kj 0 increases while k-r n k decreases. Replacement will reduce ffi n but, in order to maintain the recurrence relation, it should be carried out before kj 0 becomes too large relative to k-r n k. For this reason, we propose to set a threshold ffl and carry out a replacement when kj 0 reaches the threshold. To be precise, we replace the residual at step n by r We note that, in principle, residual replacement can be carried out for all steps up to where reaches certain point. However, from the stability point of view, it is preferred to generate the residual by the recurrence as much as possible, since kj 0 n k is generally bigger than the recurrence rounding error kj n k (of order u). 2.2 Groupwise solution updating to reduce error accumulations From the discussions of Section 2.1, we learn that residual replacement should only be carried out up to certain point. In this subsection, we will discuss how to maintain, after the last replacement, the deviation at the order of ujAjjx n j, in which case x n is a backward stable solution. Note that, for any x n , ukAkkx n k is the lowest value one can expect for its residual. This is simply because even with the exact solution x, both is the last updating step, which menas that we are in the final phase of the iteration process, then, because of (3), the deviation at step n ? m is From our updating condition, we have that kr n k - kj 0 is chosen not too close to u, kr n k is small and - m. We now discuss the three different parts of ffi n . The discussion here is only to motivate the groupwise updating strategy; a more rigorous analysis will be given in the next section. we have that \Sigma n ffl For the / i part, j/ m)ukAkkxk. If large, the accumulation of errors over steps can be significant. We note that this is the same type of error accumulation in evaluating a sum of small numbers by direct recursive additions, which can fortunately be corrected through appropriately grouping the arithmetic operations as with terms of similar order of magnitude in the same group S i \Delta. In this way, the rounding errors associated with a large number of additions inside a group S i is of the magnitude of uS i , which can be much smaller than uS. The same technique can be adopted for computing x n as Specifically, the recurrence for x n can be carried out in the following equivalent form Groupwise Solution Update: For convergence End for Such a groupwise update scheme has been suggested by Neumaier, and it has been worked out by Sleijpen and van der Vorst (see [18] for both references). By doing so, the error in the local recurrence is reduced. Indeed, for (instead of ujx i j). Hence, \Sigma n In summary, with groupwise updating of the approximated solution, all three parts of ffi n can be maintained at the level of ujAjjxj. We mention that groupwise updating can also be used to obtain better performance of a code for modern architectures, because it allows for level-3 BLAS operations. This has been suggested in [21, page 52, note 5]. 2.3 Some other residual replacement strategies We briefly comment on some other residual replacement strategies. For the naive strategy of "replacing always" (the residuals are computed always as b\GammaAx n ) or for "periodic replacement" (update periodically at every ' steps), replacement is carried out throughout the iteration, even when kr n k is very small. This, as we know, may result in large perturbations to the recurrence equations relative to kr n k, since jj 0 j is at least j- n j - ujAjjx n j, see (7). In that case, as kr n k decreases, the recurrence relation may be perturbed too much and hence the convergence property deteriorates. This is the typical behaviour observed in such implementations. We note that if - n can be made to decrease as kr n k does, then replacement can be carried out at later stages of the iterations. This leads to the strategy of "flying-restart" of Sleijpen and van der Vorst [18], which significantly reduces - n , and hence j 0 n , at a replacement step. In the flying-restart strategy b is replaced by f at some but not all of the residual replacement steps (say addition to the residual replacement r The advantage of this is that, at the flying-restart step n i+1 , the residual is updated by r n i+1 (noting that b / r n i . Then which decreases as r n i and - decrease. This is the term that determines the perturbation to the recurrence and can be kept small relative to r n . However, the deviation satisfies (assuming x n i+1 ). Namely, the deviation at each flying-restart step carries forward to the later steps. Therefore flying-restart should only be used at carefully selected steps where However, it is not easy to identify a condition to monitor this. It is also necessary to have two different conditions for the residual replacement and flying-restart. Fortunately, our discussion in the last two subsections shows that carrying out replacement carefully at some selected steps, in combination with groupwise update, is usually sufficient. We shall not pursue the flying-restart idea further in this paper. Analysis of the Residual Replacement Scheme In this section, we formally analyze the residual replacement strategy as developed in Section 2.1 (and presented in Algorithm 2 below). In particular, we develop a computable bound for kffi n k and n k, that can be used for the implementation of the residual replacement condition. We first summarize residual replacement strategy in the following algorithm, written in a form that identifies relevant rounding errors for later theoretical analysis. Algorithm 2: Iterative Method with Residual Replacement: Given an initial approximation floating point vector); set - For convergence Generate a correction vector q n by the method; if residual replacement condition (8) holds else (denote but not compute x End for Note that x n and ffi n are theoretical quantities as defined by the formulas and are not to be computed. The vectors / due to finite precision arithmetic At step n of the iterative method, q n is computed in finite precision arithmetic by the algorithm. However, the rounding errors involved in the computation of q n are irrelevant for the deviation of the two residuals, which solely depends on the different treatment of q n in the recurrences for r n and x n . Throughout this paper, we assume that A is a floating point matrix. Our error analysis is based on the following standard model for roundoff errors in basic matrix computations [8, p.66] (all inequalities are componentwise). where are floating point vectors, N is a constant associated with the matrix-vector multiplication (for instance, the maximal number of nonzero entries per row of A). It is easy to show that Using this, the following lemma, which includes (1) and (2), is obtained. Lemma 1 The error terms in the computed recurrence of Algorithm 2 are bounded as follows: For a step at which a residual replacement is carried out: Proof From (9), we have that j/ n j - uj-x This leads to the bound for j/ n j: For a residual replacement step, the updated z is x n by definition, that is x Therefore, The bounds for j n and - n follow similarly. be the number of step at which a residual replacement is carried out and let later step, but still before the next replacement step. Then, we have that and Proof The first bound follows directly from Lemma 1. For we have that q Noting that - Similarly, We now consider the deviation of the two residuals. be the number of an iteration step at which residual replacement is carried out and let n ? m denote a later iteration step, still before the next replacement step. Then, we have that Proof At step m, by the definition of xm in Algorithm 2, z with z being the updated z-vector and - Therefore . Hence, for the range of n ? m, and before the next residual replacement step: Lemma 2, we obtain the following computable bound on ffi n . Lemma be the number of an iteration step at which residual replacement is carried out and let n ? m denote a later iteration step, still before the next replacement step. Then, we have Proof The bound for kffi m k follows from that for - m , see (14). From Lemma 2 and Lemma 3, it follows that which leads to the bound for ffi n in terms of norms. We note that it is possible to obtain a sharper bound by accumulating the vectors in the bound for jffi n j. Our experiments do not show any significant advantage of such an approach. We next consider the perturbation to the recurrence. Theorem 1 Consider step n of the iteration and let m ! n be the last step before n, at which a residual replacement is carried out. If replacement is also done at step n, then let x 0 be the computed approximate solution and r 0 the residual. Then the residual r 0 n satisfies the following approximate recurrence Proof First, in the notation of Alg. 2, x 0 where we have used that b \Gamma Ax . Furthermore, by Lemma 3, Also, kAi Combining these three, and using that r 0 O(u), the bound on kj 0 n k is obtained as in Lemma 4. Note that bound (16) is computable at each iteration step. Therefore, we can implement the residual replacement criterion (8) with this bound instead of kj 0 k. We note that the factor 2 in the bound comes from the bound for q i in Lemma 2, which is pessimistic since q i - x i . Therefore, we can use the following d n as an estimate for kj 0 Hence, we shall use the following residual replacement criterion, that is residual replacement is done if With this strategy, the replaced residual vector r n satisfies the recurrence equation (15) with k. With this property, we consider situations where r n converges. We now discuss convergence of the true residual. Theorem 2 Consider Algorithm 2 with the residual replacement criterion (18), and assume that the algorithm terminates at step be the number of the last residual replacement iteration step before termination. If then Proof From (17), we have dK ? k. Furthermore, at the termination step, we have kr is the last updating step, we have for n - m, d n ? fflkr n k as otherwise there would be another residual replacement after m. That implies kr ~ which is an upper bound for kffi n k (Lemma 4) and ~ where ~ which implies ~ Thus the bound follows from We add two remarks with respect to this theorem. Remark 1: If the main condition (19) is satisfied, then the deviation, and hence the true residual, will remain at the level of uNkAkkxK k at termination. Such an approximate solution is backward stable and it is best one can expect. The condition suggests that ffl should not be chosen too small. Otherwise, the replacement strategy will be terminated too early so that the accumulation after the last replacement might become significant. As can be expected, however, the theoretical condition is more restrictive than practically necessary and our numerical experience suggests that ffl can be much smaller than what (19) dictates, without destroying the conclusion of the theorem. Remark 2: On the other hand, in Section 2.1 we have seen that ffl controls perturbations to the recurrence of r n , and for this reason it is desirable to choose it as small as possible. In our experience, there is a large range of ffl around p u that balances the two needs. Reliable Implementation of Iterative Methods In this section, we summarize the main results of the previous sections into a complete implemen- tation. We also address some implementation issues. It is easy to see from the definition of d n (see (17)) that it increases except at the residual replacement steps when it is reset to u(NkAkkxm k Our residual replacement strategy is to reduce d n whenever necessary (as determined by the replacement criterion) so as to keep it at the level of uNkAkkxK k at termination. With the use of criterion (18), however, there are situations where the residual replacement is carried out in consecutive steps while d n remains virtually unchanged, namely when kr n k stays around d n =ffl - uNkAkkx n k=ffl. ?From the stability point of view, it is preferred not to replace the residuals in such situations. To avoid unnecessary replacement in such cases, we impose as an additional condition that residual replacement is carried out only when d n has a nontrivial increase from the dm of the previous replacement step m. Therefore, we propose d n ? 1:1d m as a condition in addition to (18) for the residual replacement. The following scheme sketches a complete implementation. Algorithm 3: Reliable Implementation of Algorithm 1. Input an initial approximation residual replacement threshold ffl; an estimate of NkAk; For convergence Generate a correction vector q n by the Iterative Method; if d End for Remark: In this reliable implementation, we need estimates for N (the maximal number of nonzero entries per row of A) and kAk. In our experience with sparse matrices, the simple choice still leads to a practical estimate d n for kffi n k. In any case, we note that precise estimates are not essential, because the replacement threshold ffl can be adjusted. We also need to choose this ffl. Our extensive numerical testing (see section 5) suggests that ffl - p u is a practical criterion. However, there are examples where this choice leads to stagnating residuals at some unacceptable level. In such cases, choosing a smaller ffl will regain the convergence to O(u). The presented implementation requires one extra matrix-vector multiplication when an replacement is carried out. Since only a few steps with replacement are required, this extra cost is marginal relative to the other costs. However, some savings can be made by selecting a slightly smaller ffl and carrying out residual replacement at the step next to the one for which the residual replacement criterion is satisfied (cf [18]). It also requires one extra vector storage for the groupwise solution up-date (for z) and computation of a vector norm k-x n k for the update of d n (kr n k is usually computed in the algorithm for stopping criteria). 5 Numerical Examples In this section, we present some numerical examples to show how Algorithm 3 works and to demonstrate its effectiveness. We present our testing results for CG, Bi-CG and CGS. All tests are carried out in MATLAB on a SUN Sparc-20 workstation, with In all examples, unless otherwise specified, the replacement threshold ffl is chosen to be 10 \Gamma8 . kAk1 is explicitly computed and N is set to 1. In Examples 1 and 2, we also compare d n and the deviation kffi n k. Example 1: The matrix is a finite-difference discretization on a 64 \Theta 64 grid for with a homogeneous Dirichlet boundary condition. a(x; y. We apply CG and Reliable CG (i.e. Alg. 3) to solve this linear system and the convergence results are given in Figure 1. In Figure (and similarly in Figures 2 and 3 for the next example), we give in (a) the convergence history of the (normalized) computed residual for CG (solid line), the (normalized) true residuals for CG (dashed line) and for reliable CG (dotted line). In (b), we also give the (normalized) deviations of the two residuals kffi (dash-dotted line) and for reliable CG (dotted line) and the bound d n for reliable CG (in x-mark). Example 2: The matrix is a finite-difference discretization on a 64 \Theta 64 grid for the following convection diffusion equation with a homogeneous Dirichlet boundary condition. The function f is a constant. We consider Bi- CG and CGS for solving the linear systems with The results are shown in Figure 2 for Bi-CG, and in Figure 3 for CGS. In the above examples, we have observed the following typical convergence behaviour. For the original implementations, the deviation increases and finally stagnates at some level, which is exactly where the true residual stagnates, while the computed residual continues to converge. With the reliable implementations, when the deviation increases to a certain level relative to r n , a residual replacement is carried out and this reduces the error level. Eventually, the deviation and hence the true residual arrive at the level of ukAkkxk. We also note that the bound d n captures the behaviour of kffi n k very closely, although it may be an overestimate for ffi n by a few orders of magnitude. In all three cases, the final residual norms for the reliable implementation are smaller than the ones as obtained by the MATLAB function Anb. Example 3: In this case, we have tested the algorithm for Bi-CG (or CG if symmetric definite) and CGS, on the Harwell-Boeing collection of sparse matrices [3]. We compare the original imple- mentations, the reliable implementations and the implementations of Sleijpen and van der Vorst [18] (based on their replacement criteria (16) and (18)). In Table 1, we give the results for those matrices for which the computed residuals converge to a level smaller than ukAkkxk so that there is a deviation of the two residuals. For those cases where b is not given, we choose it such that a Figure 1: Example 1 (CG) (a): solid - computed residual of CG; dashed - true residual of CG; dotted true residual of reliable CG; (b): dash-dotted - of CG, dotted - nk of reliable CG; x - dn of reliable CG (a) Convergence History iteration number normalized residual norm (b) Residual Deviation and Bound iteration number deviation given random vector is the solution. We note that for some matrices, it may take 10n iterations to achieve that, which is not practical. However, we have included these results in order to show that even with excessive numbers of iterations, we still arrive at small true residuals eventually. We list the normalized residuals res attained by the three implementations and by Gaussian elimination with partial pivoting (MATLAB Anb). We also list the number of residual replacements (n r ) for our reliable implementations and the number of flying-restart (n f ) and the number of residual replacements (n r ) for the implementations of Sleijpen and van der Vorst (SvdV). There are two cases for which the computed residuals do not converge to O(u)kbk with the choice of 8. For those cases, a slightly smaller ffl will recover the stability and the results are listed in the last row of the table. We see that in all cases, the reliable implementation reduces the normalized residual to O(u) and res2 is the smallest among the three implementations, even smaller than MATLAB Anb. The improvement on the true residual is more apparent in CGS than in Bi-CG (or CG). Except in a few cases, both the reliable implementation presented here and the implementation of Sleijpen and van der Vorst work well and are comparable. So the main advantage of the new approach is its simplicity and an occasional improvement in accuracy. Figure 2: Example 2 (Bi-CG) (a): solid - computed residual of Bi-CG; dashed - true residual of Bi- CG; dotted - true residual of reliable Bi-CG; (b): dashed - of Bi-CG, dotted - of reliable Bi-CG; x - dn of reliable Bi-CG (a) Convergence History iteration number normalized residual norm (b) Residual Deviation and Bound iteration number deviation 6 Concluding Remarks We have presented a new residual replacement scheme for improving the convergence of the true residuals in finite precision implementations of Krylog subspace iterative methods. By carefully monitoring the deviation of the computed residual and the true residual and incorporating the earlier ideas on residual replacement, we obtain a reliable implementation that preserves the convergence mechanism of the computed residuals, as well as sufficiently small deviations. An error analysis shows that this approach works under certain conditions, and numerical tests demonstrate its effectiveness. Comparison with an earlier approach shows that the new scheme is simpler and easier to implement as an add-on to existing implementations for iterative methods. We point out that the basis for the present work is the understanding that the convergence behaviour (of computed residuals) in finite precision arithmetic is preserved under small perturbations to the recurrence relations. Such a supporting analysis is available for Bi-CG (and CG) but it is still an empirical observation for most other Krylov subspace methods. It would be interesting to derive a theoretical analysis confirming this phenomenon for those methods as well. Acknowledgements We would like to thank Ms. Lorrita McKnight for assistance in carrying out the tests on Harwell-Boeing matrices. Figure 3: Example 2 (CGS) (a): solid - computed residual of CGS; dashed - true residual of CGS; dotted - true residual of reliable CGS; (b): dashed - of CGS, dotted - nk of reliable CGS; x - dn of reliable CGS (a) Convergence History iteration number normalized residual norm (b) Residual Deviation and Bound iteration number deviation --R An Analysis of the Composite Step Biconjugate Gradient Algorithm for Solving nonsymmetric Systems Templates for the solution of linear systems: Building blocks for iterative methods Sparse Matrix Test Problems Conjugate Gradient Methods for Indefinite Systems A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems SIAM J Iterative solutions of linear systems Acta Numerica Matrix Computations Behavior of Slightly Perturbed Lanczos and Conjugate-Gradient Recurrences Estimating the attainable accuracy of recursively computed residual methods Methods of Conjugate Gradients for solving linear systems Solution of Systems of Linear Equations by Minimized Iterations On the convergence rate of the conjugate gradients in presence of rounding errors Accuracy and effectiveness of the Lanczos algorithm for the Symmetric eigenproblem Iterative Methods for Sparse Linear Systems PWS Publishing BICGSTAB(L) for linear equations involving unsymmetric matrices with complex spectrum Electronic Trans. Reliable updated residuals in hybrid Bi-CG methods Computing <Volume>56</Volume>:<Pages>144-163</Pages> (<Year>1996</Year>) Analysis of the Finite Precision Bi-Conjugate Gradient algorithm for Nonsymmetric Linear Systems The performance of FORTRAN implementations for preconditioned conjugate gradients on vector computers --TR --CTR Stefan Rllin , Martin H. Gutknecht, Variations of Zhang's Lanczos-type product method, Applied Numerical Mathematics, v.41 n.1, p.119-133, April 2002

residuals;finite precision;residual replacement;krylov subspace methods

587269

Efficient Nonparametric Density Estimation on the Sphere with Applications in Fluid Mechanics.

The application of nonparametric probability density function estimation for the purpose of data analysis is well established. More recently, such methods have been applied to fluid flow calculations since the density of the fluid plays a crucial role in determining the flow. Furthermore, when the calculations involve directional or axial data, the domain of interest falls on the surface of the sphere. Accurate and fast estimation of probability density functions is crucial for these calculations since the density estimation is performed at each iteration during the computation. In particular the values fn (X1 ), fn (X2), ... , fn (Xn) of the density estimate at the sampled points Xi are needed to evolve the system. Usual nonparametric estimators make use of kernel functions to construct fn. We propose a special sequence of weight functions for nonparametric density estimation that is especially suitable for such applications. The resulting method has a computational advantage over kernel methods in certain situations and also parallelizes easily. Conditions for convergence turn out to be similar to those required for kernel-based methods. We also discuss experiments on different distributions and compare the computational efficiency of our method with kernel based estimators.

Introduction . esti mati oni s the problem of the esti mati on of the values of a li ty ven samples from the associ ated di stri buti on. No made about the type of the di stri buti on from whi ch the samples are drawn. Thi si si n contrast to esti mati on whi ch the assumed to come from a ven fami ly, and the parameters are then esti mated by ous stati sti cal methods. Early contri butors to the theory of nonparametri c esti mati oni nclude Smi rnov [21], Rosenblatt [16], Parzen [15], and Chentsov [3]. ve descri pti ons of ous approaches to nonparametri c on along wi th a ve bi bli ography can be books by lverman [23] and Nadaraya [14]. More recent developments are presentedi n books by Scott [18] and Wand and Jones [27]. Results of the experi mental compari son of some dely used methods In addi ti on to data analysi s, ani mportant appli cati on of nonparametri c onal flui d mechani cs. When the flow calculati ons are per- Lagrangi an framework, a set of space are evolved through me usi ng the ng ons. In poi nts that werei ni ti ally close can move apart, leadi ng to mesh di storti on and cal di #culti es. Problems th mesh di storti on can be eli mi nated to a extent by the use of smoothed cle hydrodynami cs ques [2, 13, 9, 12]. SPH treats the nts bei ng tracked as samples comi ng from an unknown li ty di stri buti on. These calculati ons often requi re the computati on of the values of not only the unknown densi ty, buti ts gradi ent as well. # Received by the editors August 16, 1995; accepted for publication (in revised form) August 3, 1999; published electronically June 13, 2000. http://www.siam.org/journals/sisc/22-1/29046.html Department of Computer Science, University of California, Santa Barbara, CA 93106 (omer@cs. ucsb.edu). # Department of Mathematics, Indian Institute of Technology, Bombay, India (ashok@math. iitb.ernet.in). In contrast to appli cati ons concerned wi th the di splay of the densi ty, ci ent to esti mate the densi ty on some gri d,i n these flui d flow calculati ons the requi red at each sample nt. Another di #erencei n these two types of appli cati onsi s that when deali ng th data analysi s, usually concerned th the opti mal accuracy one can get for a ven sample si ze. In flui d flow calculati ons, where addi ti onal "data" can be ned wi thi ncreased di screti zati on, usually more concerned wi th the opti mal vari ati on of the onal e#ort as a on of error. In some appli cati ons, for example,i n ng di recti onal data [24], the samples li e on the ci rcle S 1 or along the surface of the al case of di recti onal al whi ch the c about the center of the ci rcle or the sphere, ous methods have been proposed for nonparametri c cal stati sti cs, such as the kernel [15, 1, 28] and the orthogonal seri es methods [17, 11]. The kernel method has been extensi vely studi ed, probably the most popular appli cati ons such as SPH. In thi s method, the value of the densi ty at the nt xi s esti mated as #, (1) ni s the esti mate of the ven a sample of are the posi ti ons of the samples drawn from a li ty di stri buti on wi th an unknown on Ki s a kernel hi s the ndow wi dth, and A hi s a normali zati on factor to make f ni nto a li ty ty. One of the drawbacks of the kernel method s the onal costi nvolved. Even possi ble to reduce the the one-di mensi onal case usi ng the expansi on of a polynomi al kernel and an ng strategy [19], thi s strategy cannot be ly extended to hi gher di mensi ons [5]. ng methods [5] can be usedi n any di mensi on. However, si nce the evaluated on a gri d, methodi s not sui table for the flui d flow calculati ons whi ch we arei nterested, where an requi red at each sample nt. We propose a cosi ne-based wei ght on nonparametri c esti mati on, whi chi s a al case of the class of esti mators that form a # sequence [26, 28]. Thi s mi lar to the kernel esti mator but has the ease of evaluati on of a seri es expansi on. The role of the ndow wi dth parameter h of the kernel methodi s replaced by a smoothi ng parameter our method, and f ni s now of the form (2) Our choi ce of c cularly sui table for appli cati onsi n flui d flow calculati ons where the values f n (X 1 at the sampled di recti ons themselves are requi red at each each me stepi n the flow We show that th esti mator the red n values can be computed e#ci ently usi ng only O(m 1+d n) operati ons for di recti onal data and O(m d n) operati ons for al di mensi ons, where m need not be large as long wi thout bound th n. Thi si si n contrast to the O(n 2 ) ons red by the kernel method for computati oni n the worst case and an expected complexi ty of O(h d th ng bounded support. However,i n the al case of d = 1, the of the kernel method can be reduced to li near after ng step. GL U AND ASHOK SRINIVASAN We deri ve ons under whi ch the sequence of esti mated ons f n fashi on converge to the unknown experi mentally veri fy the accuracy and the e#ci ency of our methodi n practi cal test cases. Experi ments and cal analyses alsoi ndi cate how m should vary th n for opti mal accuracy. The paperi s organi zed as follows. In secti on 2 we define the wei ght on ve the ons for the convergence of the ntegrated square error (MISE) when the sample spacei s S 1 (Theorem 3). The ons guarantee that E(f as n #. We also present correspondi ng results for S 2 . In secti on 3 schemes for e#ci ent computati on of these esti mates on S 1 and S 2 are presented. In secti ons 4 and 5, we descri be experi mental results th our esti mator and wi th the kernel method for some di stri buti ons practi ce. Our experi mentsi mply a net savi ngs on the number of ons performed over kernel methodsi n ons and also veri fy the formula found for the opti mal choi ce of m. The results show that the kernel method and our esti mator perform di #erent setti ngs, and thus complement each other. The conclusi ons are presentedi n secti on 6. The ns addi ti onal test results. 2. The cosine estimator and the convergence of MISE. In secti on, we first menti on some related work done on spheri cal data; then we define our esti mator and deri ve ons ts convergence for di recti onal data on the ci rcle, and ve correspondi ng results for di recti onal and al data on the sphere and al data on the ci rcle. The kernel method for nonparametri c esti mati on for di recti onal and al di scussedi n [6, 8]. Whi le deali ng th di recti onal data, Fi sher, Lewi s, and Embleton [6] recommend usi ng the ng kernel: exp For al data they recommend the kernel #, (4) where normali zes W to a li ty on, and C ni s the reci procal of h usedi n the defini ti on of kernel esti mators. x and X i are the Cartesi an represen- tati on of nts P and P i , respecti vely, and x T X ii s thei nner product of these two vectors. W n the role of K(x - X i ) of (1). Hall, Watson, and Cabrera [8] analyze esti mators for di recti onal data wi th the x - X i replaced by . Observe that the term x T X ii s the cosi ne of the angle between the nts ii s a measure of the di stance along the surface of the sphere between nts P and P i . Inner product plays a cruci al rolei n these esti mators. We consi der an esti mator that can be terms of powers of the nner product, the power playi ng the role of the smoothi ng parameter. Thi s enables us to expand the esti matori n a seri es and li tates fast computati on. c (x- c (x- OOc (x)32 (a) (b) -x Fig. 1. The ns c 32 (x) and c 2.1. The case of S 1 . We first define our esti mator on S 1 . Assume X 1, 2, . sequence ndependently andi denti cally di stri buted (i .i .d.) random ables (observati ons) for di recti onal data on th li ty [-#]. Wei mpose the addi ti onal condi ti on that si nce the random ables X j are defined on the ci rcle S 1 . As an esti mator of the densi ty of di recti onal data f(x), x # [-#], we der a nonparametri c esti mator of the form ven by (2) th cos 2m on [-#]. The normali zati on factor Am ven below makes c m (x)i ntegrate to 1 on [-#]: -# cos 2m 2# dx. Maki ng use of a table ofi ntegrals such as Gradshteyn and Ryzhi k [7] and by usi ng can be shown that As examples, the ons c m (x) and c m 4 are shown on S Fi gure 1(a) and on thei nterval Fi gure 1(b). We wi sh to find su#ci ent ons under whi ch the sequence of esti mators f n converges to fi n the MISE sense. In order to do thi s, we first show the convergence of the bi as and then deri ve the ons under whi ch the ance converges to 0. We shall then use these results to prove convergence of MISE on S 1 . Fi rst we show that as m #, the expected value of the esti mate f n (x) approaches the actual uni formly for any ven n. Lemma 1. Suppose f # C 2 andl et f n (x) be as given in (2). Then uniforml y, independentl y of n. Proof. -# s)f(s)ds, (7) GL U AND ASHOK SRINIVASAN as showni n lverman [20] and Whi ttle [29]. By a change of able -# x-# x-# usi ng the peri odi ci ty of c m and f , along th (7), (8), and the mean value theorem, -# -# /2)dy, where # x nt between x and y. Therefore -# -# -# From (6), the ntegral evaluates to 1, and nce yc m (y)i s an odd on, the secondi ntegral evaluates to 0. Let 2M We then have the ng esti mate for the as: -# -# -#Am cos 2m (y/2)y 2 dy. For any # such that 0 < #, |y|<#Am cos 2m (y/2)y 2 dy |y|#Am cos 2m (y/2)y 2 dy |y|<#Am cos 2m (y/2)dy Am cos 2m (#/2) nce cos y decreases as |y|i ncreases on thei nterval under consi derati on. Furthermore, bounded above by 1. Therefore, Am Am Am In order to get a bound, we wi ll choose # as a functi on of m. If we take # 0 as m #, then # as m #, then thi s term decays exponenti ally. The second (10)i s the product of thi s term and m), and thus the product approaches 0 si nce the exponenti al decay domi nates. In order to get a good bound on the first term of (10), we wi sh to choose # ng the condi ti on that # such that the # 2i s as small as possi ble. We can choose arbi trari ly small. Thus M # /mi s an asymptoti c bound on the as. Furthermore, the ndependent of x; hence, the uni form. Lemma2 . Suppose f # C 2 andl et f n (x) be as given in (2). Then uniforml y as n #, provided m # as n #, and Proof. -# -# as showni n Whi ttle [29]. As a consequence of Lemma 1, the secondi ntegral approaches asymptoti cally, and hence, the second term approach 0 nce bounded. It thus su#ces to show the convergence of the firsti ntegral to 0. x#] |f(x)|. maki ng a change of able usi ng -# n, where the expressi on on the ri ght-hand a consequence of the pressi on for nce m/n 2 #, the abovei ntegral converges to nce mi si ndependent of x, the uni form. Therefore the vari ance of uni formly to 0 under the ons of the lemma. Note that the bound on the bi as for the cosi ne method ven by Lemma 1i s of the form and the bound for the ance ven by Lemma Therefore, the role played by m for the cosi ne methodi s the same as h -2 of the kernel based methods, where hi s the ndow wi dth of the kernel esti mator. In other words the bounds on the bi as and the vari ance of the cosi ne esti mator arei n accordance wi th the behavi or of the kernel method lverman [23]. Such si mi lari ty of rates of convergencei s to be expected si nce the cosi ne essenti ally li ke the kernel esti mator, though the forms of the ons di #er. It wi ll be shown later that the advantage of the cosi ne esti mator li esi ni ts onal e#ci ency. Theorem 3. Suppose f # C 2 [-#] and f n (x) is as given in (2). If m # as n #, and -# E(f as n #. Proof. E(f as showni n Whi ttle [29]. From Lemmas 1 and 2, each of thei ntegrals approaches 0. Hence, the MISE converges to 0. GL U AND ASHOK SRINIVASAN In fact, the MISEi s of the form where the c bounds on the constants are as as we shall explai n later, the exact asymptoti c constants are not all mportant for practi cal ons. Condi ti ons for the convergence of the esti mates ts deri vati ves on the real li nei nstead of S 1 can be consi der the case when the di recti onal data li e along the surface of the 2.2. The case of S 2 . Let X 1, 2, . ., n, be a sequence ables th values on the surface of the centered at the Suppose that the li ty on f(x) of the X j has bounded second deri vati ves. We consi der a nonparametri c esti mator of the form some m to be determi ned as a functi on of n. The c m are thi s case as follows. If # xX denotes the angle between nts x and X, then cos 2m The normali zi ng factor ven below: Through a deri vati on along the li nes of the case of the ci rcle, the ng theorem can be proved for the convergence of the esti mators. Theorem 4. Suppose f # C andl et f n (x) be as given in (14). If m # as n # and E(f Analogous to (13), the form of the MISEi s found to be From expressi on for MISE we see that asi n the case of S hi s the ndow wi dth of the kernel esti mator. When deali ng th al data, we can consi der the ng al esti mator for spheri cal data: cos 2m (# xX ). We can also define a correspondi ng esti mator on the ci rcle, where we take the cosi ne of the arc length between two poi nts,i nstead of the cosi ne of half the arc length the case of di recti onal data. The between the s the same for the cases of the ci rcle and the sphere, vely. 3. E#cient evaluation of the density estimator. In secti on, we shall descri be an e#ci ent algori thm for the computati on of the esti mates f n (x) evaluated at a set of n observed nts X 2, . on the ci rcle S 1 case) and on the We also show f the value of f n at some arbi trary xi s desi red, then ly accompli shed once computed. The e#ci ency of our methodi s based on the fact that f n terms of the ons c m (x) as or (15). Suppose we represent the posi ti ons of the observed nts X 2, . thei r Cartesi an coordi nates. We show that for any x, f n (x) can be expressed as a polynomi al of total degree mi n the coordi nates of x. The coe#ci ents of polynomi al can be determi nedi n turn from the coordi nates of the X i . Moreover, the coe#ci ents are the sums of the contri buti ons due to each X ii ndependently. Fi rst we consi der the case of di recti onal data on S 1 . From (2), (5), and the half-angle formula for cosi ne we get Denote the poi nts on S 1 correspondi ng to the angles x and X i for 1, 2, . ., n, by Cartesi an coordi nates and let # represent the standardi nner product on R 2 . Then cos(x . ng thi si nto (17) we get The expressi on (18)i s a polynomi al of degree . For a fixed m, we can compute the coe#ci ents by addi ng the contri buti on of each X i as follows. Usi ng the al theorem and (18) r+s#m s x r where thei nner th Changi ng the order of summati on r+s#m s where M(r, s) =n i2, and Ami s as ven by (6). If we use the expressi on for onal ease, then (20) fies to r+s#m s GL U AND ASHOK SRINIVASAN Table Co mputatio nal co mplexityo f the co sine estimato r. Circle Axial Directional for large m. Now we consi der the number of ons requi red for the evaluati on of f n (x) ven the ons X 2, . . The powers X r i1 and X r i2 for a fixed 1, 2, . ., m can be computed th O(m) multi pli cati ons. ng thi s for 1, 2, . res O(mn) multi pli cati ons. After the conclusi on of thi s step, each of the O(m 2 ) averages M(r, s) for a ven r and s can be computed wi th an addi ti onal O(n) ons. Si nce there are a total of O(m 2 ) correspondi ng to the rs s th 0 # thi s means that the coe#ci ents of the polynomi ali n or (21) can be computed wi th a total of O(m 2 n) ons. Once the coe#ci ents of f n (x) have been computed, to evaluate f n (x) th calculate the powers x r 1 and x r 1, 2, . ., mi n O(m) ons. Si nce the coe#ci ents are already avai lable, the remai ni ng res only an multi pli cati ons and addi ti ons. The results for di #erent cases are summari zedi n Table 1. Remark. For our MISE convergence condi ti on for S 1 (Theorem 3) to be musti ncrease wi thout bound th n. Theoreti cally, we can take m to be as ng as we li ke. Then the above resulti mpli es that the computati on of the densi ty at all of the sample poi nts can be accompli shed usi ng only about O(n) the magni tude of m ves acceptable accuracy for f n (x). The problem th ng m to be too slowly that the magni tude of m controls the error our convergence proofs. An e#ci ent algori thm for the evaluati on of f n (#x) for di recti onal data on S constructed larly. When # 2, . are observati ons on S 2 drawn from an unknown can be shown [4] that r+s+t#m where the th M(r, s, t) =n i3 . Thi s ti me the coe#ci ents of the polynomi ali n (22) can be computed wi th a total of O(m 3 n) ons. After preprocessi ng, each evaluati on of f n (x) for res only an ons. ng results can be deri ved for al data as summari zedi n Table 1. It should also be noted that we needed the Cartesi an representati on of the data. If the data are cal coordi nates, then there wi ll be an addi ti onal overhead for ng the Cartesi an representati on. However, thi s overhead takes only li near ti me and so wi ll be negli gi ble for su#ci ently large data. Furthermore,i t has been shown [24] that for ani mportant class of appli cati ons, Cartesi an coordi nates are preferable to cal coordi nates, as the latter systemi s not cally stable for solvi ng the di #erenti al equati ons that se. In the subsequent parts of secti on we shall compare the onal e#- ci ency of our scheme wi th that of the kernel method. 3.1. Parallelization. One of the advantages of the onal strategy de- bed abovei s the ease of paralleli zati on. Paralleli zati oni s redi n many flui d flow calculati ons due to the large si zes of the systems. The kernel methodi s some- what di #cult to paralleli ze. If we use an e#ci ent kerneli mplementati on that performs kernel evaluati ons only for those nts whi ch are di stance h of the ven sample, then an e#ci enti mplementati on of the paralleli zati on res load ng and decomposi ti on so that poi nts that are close by remai n on the same processor, and so that each processor has roughly the same loadi n terms of the onal e#ort. Also, the communi cati on pattern for the kernel method i s not very regular. In contrast, paralleli zati on for the cosi ne esti mator can ly be accompli shed by a global reducti on operati on, for whi ch e#ci ons are usually avai lable. Thi s method requi res the same onal e#ort for each poi nt, and so the loadi s ly balanced by havi ng the same number of each processor. decomposi ti on does not play ani mportant si nce the poi nts can be on any processor. 3.2. Theoretical comparison of the kernel and the cosine estimators. Now we analyze the onal e#ci ency of the kernel and the cosi ne esti mati on methods. Ani mportant measure of the e#ci ency of the algori thmsi s not just the convergence rate of the error th sample ze n, but the of the onal e#ort C red as a functi on of the error E. For the kernel esti mator, we can te the #, (23) where hi s the smoothi ng parameter, the di mensi on, and ni s the sample ze. The onal e#ort requi red for nonparametri c esti mati on can be expressed as dependi ng on the detai ls of the algori thm used. For a ven sample ves the mal h as h # n -1/(d+4) . However, si nce the equati on for the onal also depends on h, we need to der the possi bi li ty that a value of h smaller than opti mal value may actually result lower onal e#ort. Let us consi der a vari ati on of h th n of the form From (25), (24), and (23) we # . For mi ni mum error, the exponent of both the terms on the ri ght should be the same, otherwi se the error due to the hi gher term wi ll domi nate. Thi s leads to whi chi s the same as the value of mal # for a ven n. If we let h optn represent the opti mal h mi ni mi zi ng the MISE for a ven n, and h optC represent the opti mal GL U AND ASHOK SRINIVASAN mi ni mi zi ng the onal e#ort as a functi on of the error, then the expressi on deri ved above for # does not necessari lyi mply that h si nce the relati on would sti ll sfy the expressi on for # for some constant k. If k < 1, then we can choose a subopti mal value of hi n order toi mprove the speed of the algori thm. The opti mal vari ati on of error th onal usi ng thi s value of ven by 4 . Let us now consi der the cosi ne esti mator. We can te the asymptoti c MISE as follows: where Ei s the MISE, mi s the smoothi ng parameter, di s the di mensi on the ci rcle and for the sphere), and ni s the sample ze. The onal e#ort requi red for esti mator can be expressed as n), where # can be determi ned from Table 1. The expressi on for C abovei s the same as (24) th recalli ng that m behaves as h -2 . By an mi lar to the previ ous case we can show that the opti mal vari ati on of error th onal ven by As examples, for the cosi ne esti mator on the ci rcle th al data and di recti onal data 2), the onal complexi ty and the error are related by vely. The complexi ty of the kernel esti matori s the same for al and di recti onal data. However, several di #erent possi bi li ti es st dependi ng on how e#ci ent thei mplemen- tati on of the esti matori s. If we der esti mators of the form ven by (3) and (4), then we have However,i f we consi der a kernel th bounded support, and use an e#ci mentati on of the algori thm that computes the kernel only for those poi nts that have a nonzero contri buti on, then the expected value of for data on the ci rcle. Note that the worst case remai ns (26). For the case, we can consi der an e#ci ent algori thm usi ng polynomi al kernels and ng [19], whi ch uses a li near me after ani ni ti al O(n log n) ng step. In thi s case whi ch means that the kernel method has a better than the cosi ne kernel. However there appears to be no natural generali zati on of thi s update strategy to hi gher di mensi ons [5]. Results for the di #erent cases can be determi nedi n the manner demonstrated above and are presentedi n Table 2. We wi sh to menti on that the exact constantsi n Theorem 3 are not qui te mportant (compared wi th the exponent on E), nce asymptoti cally the slowdowni ncurred by the cache domi nates the overall runni ng me. We can expect that the mpler memory access pattern of our esti mator wi ll makei t advantageous over the kernel methodi n the asymptoti c case. Table theo ptimal co mputatio nal e#o rt versus MISE. The numbers in the table represent #, where the relatio nship between the co mputatio nal es no t take into acco unt an initial so rting step. Estimator Circle Cosine, axial data 1.75 Cosine, directional data 2.25 Kernel, worst case 1.25 * Kernel, expected case 1.25 * Si nce the worst case es of the kernel method and the cosi ne esti mator for di recti onal data on the sphere have the same order, the ve e#ci enci es of the methods can be tested only through experi ments. larly, si nce the worst case complexi ty of the cosi ne esti mator for al data on the spherei s the same as the expected case for an e#ci enti mplementati on of the kernel esti mator, we need to perform experi ments to test the ve meri ts of the two esti mators. 4. Experimental results. We performed cal experi ments for al and di recti onal data on the ci rcle and spherei n order to test the e#ecti veness of our esti mator. We first plot esti mates for known di stri buti ons and then demonstrate that the MISE follows expected trends for di stri buti ons. We finally compare the onal e#ci ency of our esti mator wi th that of kernel methods. More ri cal results are presentedi n the appendi x. We consi der the on normali zes the functi on to be a densi ty on the surface of a sphere and Si s a known functi on of U . The angles # and # are the azi muth and the cal coordi nates. Thi s s the soluti on to a cular flui d mechani cs. In Fi gure 2(a) we present a cal esti mate for the versi on of the above on where # was taken to be In thi s figure, we take the data to be di recti onal. However, nce th respect to the center of the ci rcle, we can consi der the data as al and use the al esti mator. We can see from Fi gure 2(b) that res a much smaller value of m. In Fi gure 3(a) the MISEi s compared versus m and n for the one-di mensi onal # di stri buti on usi ng the al cosi ne esti mator. We also compare th one case of the di recti onal esti matori n order to show the benefit of usi ng the al esti mator. In Fi gure 3(b) the MISEi s compared versus m and n for the two-di mensi onal versi on of the # di stri buti on on the surface of a sphere usi ng the di recti onal esti mator. We next present results for experi ments compari ng the speed of the cosi ne and the kernel esti mators. We consi der the opti mal vari ati on of the onal e#ort th MISE. In order to get the opti mal onal e#ort for a ven MISE, we allow for the possi bi li ty that we may re di #erent sample si zes for the kernel and the cosi ne esti mators. Thi si s fied ve calculati ons one can ly change the "sample" ze by changi ng the di screti zati on of the system. We have performed these sons only for spheri cal data. The case of data on the ci rcle was not dered because of the asymptoti c analyses of the previ ous secti on whi ch clearly cate that the li near kernel algori thmi n the one-di mensi onal case wi ll outperform the cosi ne esti mator. However,i n a paralleli mplementati on, the ng step for the li near kernel algori thm may be slow, and then one may wi sh to consi der the cosi ne esti mator. GL U AND ASHOK SRINIVASAN (a) (b) Co sine estimates theo ne-dimensio nal caseo f the # defined abo ve. The so lid line represents the true density. (a) The dashed line represents the directio nal estimate (b) The dashed line represent the axial estimate The ng kernel was chosen for the sons: [1, 2], 0 otherwi se, (a) (b) n=1000 n=2000 Fig. 3. MISE versusm and n fo r the # density. (a) One-dimensio nal (o n the circle): exp (US cos 2 (x))/A; the so lid line sho ws the results fo r the axial co sine estimato r and the dashed line fo r the directio nal estimato r (with Two -dimensio nal (o n the exp (US cos 2 (#))/A as defined abo ve; MISE fo r the directio nal co sine estimato r. where Ai s the normali zati on constant, and the rati of the di stance between two nts along the surface of the sphere to hi s ven as the argument to the kernel on. The use of thi s kernel for compari sons can be fied ts popular use GL U AND ASHOK SRINIVASAN Time Fig. 4. Co mpariso no f time (in seco nds) versus the MISE fo r the co sine and kernel data sampled defined abo ve. The po ints marked in represent the kernel estimate. The po ints marked in x represent the co sine estimate. flui d mechani cs calculati ons [12 . Furthermore, we cannot expect any other kernel to ve a ficantly better performance, for the ng reasons: (i Iti s well known that most kernels are equally good [23, Table 3.1 th respect to "e#ci ency," as Gi ven that the e#ci encyi s about the same, the only other consi derati oni s the onal e#orti nvolved. Our kernel takes between 6 and 10 ng nt operati ons for a nonzero evaluati on (i ncludi ng the cost of computi ng the square of the di stance). Any other reasonable kernel would requi re at least 6 ng nt ons. Apart from thi s, the memory access ti mes and zero-evaluati ons would add the same constant to all kernels. Fi gures 4 and 5 compare the onal e#ort requi red for the cosi ne wei ght on esti mator and the kernel esti mator for es. We obtai ned the data usi ng the ng procedure. We performed esti mates for ous values of n, m, and h and obtai ned the MISE and the ti me for the calculati ons. For the cosi ne and the kernel esti mates, we separately plotted the data for the me requi red for the calculati ons versus the error. We chose the lower envelope of the data as the curve for that cular esti mator si nce the values of m, n for the data on the lower envelope ve the best nable speed for a ven MISE. In thei mplementati on of the kernel esti mator, we di vi ded the spherei nto cells such that the si des of the cells had length at least 2h nce the kernel defined above has ndow dth 2h, rather than h). We placed each samplei n the ate cell. When computi ng the densi ty for a cular cell, we need to search over only a few cells. The expected complexi ty of We first der data esti mated usi ng the al cosi ne esti mator and an al vari ant of the kernel esti mator. Fi gure 4 shows the results for the two-di mensi onal di stri buti on. Thi si s an example of a hi ghly nonuni form di stri buti on. We can see that the kernel and the cosi ne esti mators are about equally fast. We next consi der a more di stri buti on ven by where #i s the azi muth and #i s the on. The results presentedi n Fi gure 5(a) show that the cosi ne esti mator outperforms the kernel esti mator by more than an order of magni tude when the al forms of both the esti mators are used. After thi s we consi dered the two es menti oned above but treated the data as di recti onal and esti mated them usi ng the di recti onal vari ants of the kernel and the cosi ne esti mators. For the # di stri buti on, the cosi ne esti mator performed very poorly onal e#ci ency, and we do not present results for thi s case. Fi gure 5(b) presents the results for the di stri buti on ven by cos(#)+1/(8#). We can see that the cosi ne esti mator sti ll outperforms the kernel esti mator, though only sli ghtly. 5. Discussion. The compari son of the esti mates wi th the true densi tyi ndi cate that the cosi ne esti mator produces accurate results for the di stri buti ons tested. Plots of MISE versus m and n follow the expected trends. As the sample zei ncreases, the error for the decreases. Besi des, the opti mal value of mi ncreases as the sample zei ncreases. It can also be seen that as the number of poi ntsi ncreases, the range of m over whi ch the esti mate performs well alsoi ncreases. We can use thi s to our advantage by choosi ng a subopti mal value of m whi ch decreases the onal e#ort ficantly buti ncreases the error only sli ghtly. The experi ments compari ng the onal e#ci enci es show that the cosi ne esti mator outperforms the kernel esti mator for al data when the di stri buti oni s moderately uni form. If the di stri buti oni s hi ghly nonuni form, then the two esti ma- tors ve comparable performance for al data. The cosi ne esti mator outperforms the kernel esti mator sli ghtly for di recti onal data when the di stri buti oni s moderately uni form. However, the ng results for the cosi ne esti mator are poor for hi ghly nonuni form di recti onal data. In general, when the datai s not very nonuni form, smoother wei ght ons are used. Thi s ves a low value of m whi chi mpli es a fast evaluati on usi ng the cosi ne esti mator. However, thi s leads to a hi gher h for the kernel esti mator, whi chi mpli es that more samples contri bute to the kernel evaluati on of each sample nt and, hence, thi s leads to more onal e#ort. Conversely, when the di stri buti oni s hi ghly nonuni form, especi ally for di recti onal data, the ker- nel methodi s to be preferred. The ri cal test results presentedi n the further demonstrate nt. We also analyzed our experi mental data to esti mate an opti mum vari ati on of m th n. Usi ng the results of our experi ments, we can perform a least squares and mate m as kn 1/2.5 for one-di mensi onal esti mati on whi chi s the same as that expected based on the expressi on for the MISE. ng appears to ve reasonable esti mates for densi ty on the surface of a sphere. Thi s resulti s also stent wi th the cal predi cti ons. Here, the magni tude of k depends on the complexi ty of the functi on. It es between 1 and 10 for the di stri buti ons dered here. We also noted the values of m, n, and h, whi ch gave the opti mal onal e#ort for a ven MISE, and compared the results for the kernel and the cosi ne esti mators. We observed that the values of h were close to the values whi ch gave the mi ni mum MISE for the ven sample si ze. However, the values of m were cantly lower than the values whi ch gave the mi ni mum MISE for the ven sample ze, though the errori nvolvedi tself was not much hi gher than the mi ni mum MISE. appears that we can choose a subopti mal smoothi ng parameteri n order to ncrease the speedi n the case of the cosi ne esti mator. GL U AND ASHOK SRINIVASAN (b) (a) Time Time Fig. 5. Plo to f time (in seco nds) versus the MISE fo r the co sine and the kernel estimatio no f data sampled 1/(8#). The po ints marked in o represent the kernel estimate. The po ints marked in x represent the co sine estimate. (a) Data treated as axial. (b) Data treated as directio nal. Elevation Density Fig. 6. Plo to f density ns g(#; di#erent the elevatio n. The so lid line the dashed line the dash-do tted line and the do tted line 6. Conclu ion . In thi s paper, we have descri bed a wei ght on esti mator for nonparametri c esti mati on of li ty ons based on cosi nes, and we provi ded ons under whi ch the esti mate ts deri vati ves converge to the actual ons. We have developed a scheme for the e#ci ent computati on of the densi ty and presented experi mental results to check the performance of the esti mator for practi cal problems. These results are cularly relevant to flui d mechan- calculati ons and,i n general, to ons where the sample si ze can be controlled, for example, though refinement of the di screti zati on. We have also ven an empi r- cal formula for choosi ng the wei ght on exponent parameter of the esti mator. Our experi mental results suggest that the cosi ne esti mator outperforms the kernel esti mator for both di recti onal and al data that are moderately uni form. It ves performance comparable to the kernel esti mator for hi ghly nonuni form al data, whi le the kernel methodi s preferable for hi ghly nonuni form di recti onal data. There potenti al for further theoreti cal study of our esti mator. Appendix . Further test results. We present more test resultsi n secti on to study the ve e#ci enci es of the cosi ne and the kernel techni ques, as the ed systemati cally from bei ng vely uni form to bei ng sharply peaked on the For these tests, we chose ons g(#; si s a constant that governs the sharpness of the #i s the elevati on, and normali zes thi s to a li ty on. Fi gure 6 shows the densi ty as a functi on of the elevati on alone for di #erent values of the parameter s. Thi s c about the center of the sphere, and thus we can use the al esti mators, on 4. We can alsoi gnore our knowledge of and use the general di recti onal esti mators. GL U AND ASHOK SRINIVASAN GL U AND ASHOK SRINIVASAN GL U AND ASHOK SRINIVASAN Time (a) (b) We present the results of the experi ments as plots of me versus MISE for the kernel esti mator versus the cosi ne esti mator, for both al and di recti onal data. The kernel esti matori s the one usedi n the ri cal on 4. The tests were performed on an Intel Celeron 300MHz processor th 64 MB memory. The C code was led wi th the gcc compi ler at opti mi zati on level -O3. We can see from Fi gures 7, 8, 9, 10, and 11 that when the on i s not very sharp, the cosi ne esti mator outperforms the kernel esti mator for both al and di recti onal data. As the densi ty becomes sharper, the kernel method starts outperformi ng the cosi ne esti mator for di recti onal data, though the latteri s sti ll better for al data. When the densi ty becomes extremely sharp, the kernel method becomes better for both types of data, though for al data the two methods are sti ll comparable to a extenti n terms of speed. These results follow the cally predi cted trends and demonstrate that these two methods complement each other for di #erent types of data. Appendix . We thank the referees for thei r detai led comments and advi ce, espe- ci ally for di recti ng our attenti on to the current li terature. --R On so me glo bal measureso f the deviatio nso f density functio n estimates Estimatio no f unkno wn pro bability density basedo no bservatio ns Fast implementatio nso f no nparametric curve estimato rs Statistical Analysiso f Spherical Data Kernel density estimatio n with spherical data TREESPH: A unificatio no f SPH with the hierarchical tree metho d The estimatio no f pro bability densities and cumulatives by Fo urier series metho ds thed particle hydro dynamics On estimatio no f a pro bability density functio n and mo de Remarkso n so me no n-parametric estimateso f a density functio n Estimatio no f pro bability density by ano rtho go nal series Multivariate Density Estimatio n Fast algo rithms fo r no nparametric curve estimatio n Kernel density estimatio n using the fast Fo urier transfo rm On the appro ximatio no f pro bability densitieso f rando m variables Numerical So lutio no f Partial Di Density Estimatio n fo r Statistics and Data Analysis A new co mputatio nal metho d fo r the so lutio no f flo w pro blemso f micro structured fluids. Pro bability density estimatio n in astro no my Pro bability density estimatio n using delta sequences Kernel Smo thing On the estimatio no f the pro bability density On the smo ns --TR --CTR Jeff Racine, Parallel distributed kernel estimation, Computational Statistics & Data Analysis, v.40 n.2, p.293-302, 28 August 2002

kernel method;convergence;efficient algorithm;nonparametric estimation;fluid mechanics;probability density

587292

A Multiresolution Tensor Spline Method for Fitting Functions on the Sphere.

We present the details of a multiresolution method which we proposed at the Taormina Wavelet Conference in 1993 (see "L-spline wavelets" in Wavelets: Theory, Algorithms, and Applications, C. Chui, L. Montefusco, and L. Puccio, eds., Academic Press, New York, pp. 197--212) which is suitable for fitting functions or data on the sphere. The method is based on tensor products of polynomial splines and trigonometric splines and can be adapted to produce surfaces that are either tangent plane continuous or almost tangent plane continuous. The result is a convenient compression algorithm for dealing with large amounts of data on the sphere. We give full details of a computer implementation that is highly efficient with respect to both storage and computational cost. We also demonstrate the performance of the method on several test examples.

Introduction In many applications (e.g., in geophysics, meteorology, medical modelling, etc.), one needs to construct smooth functions defined on the unit sphere S which approximate or interpolate data. As shown in [11], one way to do this is to work with tensor-product functions of the form e defined on the rectangle where the ' i are quadratic polynomial B-splines on [\Gamma-=2; -=2], and the ~ are periodic trigonometric splines of order three on [0; 2-]. With some care in the choice of the coefficients (see Sect. 2), the associated surface 1) Institutt for Informatikk, University of Oslo P.O.Box 1080, Blindern 0316 Oslo, Norway tom@ifi.uio.no. Supported by NATO Grant CRG951291. Part of the work was completed during a stay at "Institut National des Sciences Appliqu'ees'' and "Laboratoire Approximation et Optimisation" of "Universit'e Paul Sabatier'', Toulouse, France. Department of Mathematics, Vanderbilt University, Nashville, TN 37240, s@mars.cas.vanderbilt.edu. Supported by the National Science Foundation under grant DMS-9803340 and by NATO Grant CRG951291. with will be tangent plane continuous In practice we often encounter very large data sets, and to get good fits using tensor product splines (1.1), a large numbers of knots are required, resulting in many basis functions and many coefficients. Since two spline spaces are nested if their knot sequences are nested, one way to achieve a more efficient fit without sacrificing quality is to look for a multiresolution representation of (1.1), i.e., to recursively decompose it into splines on coarser meshes and corresponding correction (wavelet) terms. Then compression can be achieved in the standard way by thresholding out small coefficients. The paper is organized as follows. In Sect. 2 we introduce notation and give details on the tensor product splines to be used here. In Sect. 3 we describe the general decomposition and reconstruction algorithm in matrix form, while in Sect. 4 we present a tensor version of the algorithms. The required matrices corresponding to the polynomial and trigonometric spline spaces, respectively, are derived in Sections 5 and 6. Sect. 7 is devoted to details of implementing the algorithm. In Sect. 8 we present test examples, and in Sect. 9 several concluding remarks. x2. Tangent plane continuous tensor splines 'm be the standard normalized quadratic B-splines associated with the knot sequence Recall that ' i is supported on the interval [x and that the B-splines form a partition of unity on [\Gamma-=2; -=2]. Let T m be the classical trigonometric B-splines of order 3 defined on the knot sequence ~ m+3 , where and ~ x ~ see Sect. 6. Recall that T j is supported on the interval ~ m, be the associated 2-periodic trigonometric B-splines, see [10]. These splines can be normalized so that for OE 2 [0; 2-] e cos x e cos x e sin x ~ Since the left and right boundaries of H map to the north and south poles, respectively, a function f of the form (1.1) will be well-defined on S if and only if x and where fS and fN are the values at the poles. Now since f is 2-periodic in the OE variable and is C 1 continuous in both variables, we might expect that the corresponding surface S f has a continuous tangent plane at nonpolar points. However, since we are working in a parametric setting, more is needed. The following theorem shows that under mild conditions on f which are normally satisfied in practice, we do get tangent plane continuity except at the poles. Theorem 2.1. Suppose f is a spline as in (1.1) which satisfies the conditions and (2.5), and that in addition f('; OE) ? 0 for all ('; OE) 2 H. Then the corresponding surface S f is tangent plane continuous at all nonpolar points of S. Proof: Since f is a C 1 spline, the partial deriviatives f ' and f OE are continuous on H. Now are two tangents to the surface S f at the point f('; OE)vv v v v v v vv ('; OE). The normal vector to the surface at this point is given by the cross product nn n n n n n nn := t 1 \Theta t 2 . By the hypotheses, continuous, and thus to assure a continuous tangent plane, it suffices to show that nn n n n n n nn has positive length (which insures that the surface does not have singular points or cusps). Using Mathematica, it is easy to see that \Theta cos(') 2 f('; which is clearly positive for all values of ('; OE) 2 H with ' 6= \Sigma-=2. With some additional side conditions on the coefficients of f , we can make the surface S f also be tangent plane continuous at the poles. The required conditions (cf. [3,11]) are that AS cos x sin and AN cos x +BN sin m, where AS ,BS ,AN , and BN are constants. x3. Basic decomposition and reconstruction formulae Suppose are a nested sequence of finite-dimensional linear subspaces of an inner-product space X, i.e. Let be the corresponding orthogonal decompositions. For our application, it is convenient to express decomposition and reconstruction in matrix form, cf. [12]. Let ' k;mk be a basis for V k , and let be a basis for W Then by the nestedness, there exists an m k \Theta m k\Gamma1 matrix P k such that where The equation (3.1) is the usual refinement relation. Similarly, there exists an m k \Theta such that where Let be the Gram matrices of size m k \Theta m k and n k\Gamma1 \Theta n k\Gamma1 , respectively. It is easy to see that Clearly, the Gram matrices G k and H are symmetric. The linear independence of the basis functions OE k;i and of / k;i implies that both G k and H are positive definite, and thus nonsingular. The following lemma shows how to decompose and reconstruct functions in V k in terms of functions in V k\Gamma1 and W k\Gamma1 . Lemma 3.1. Let f k a k be a function in V k associated with a coefficient vector a k 2 IR mk , and let be its orthogonal decomposition, where Then a Moreover, a Proof: To find a k\Gamma1 , we take the inner-product of both sides of (3.5) with ' k\Gamma1;i . Using the refinement relation (3.1) and the orthogonality of the ' k\Gamma1;i with / k\Gamma1;j , we get which gives the formula for a k\Gamma1 . If we instead take the inner-products with / we get the formula for b k\Gamma1 . In view of the linear independence of the functions , the reconstruction formula (3.6) follows immediately from (3.5) and the refinement relations. x4. Tensor-product decomposition and reconstruction In this section we discuss decomposition and reconstruction of functions in tensor product spaces V k \Theta e are as in the previous section, and where e are similar subspaces of an inner-product space e X. In particular, suppose e and that e be as in the previous section, and let e be the analogous matrices associated with the spaces e Theorem 4.1. Let f k A k;' ~ ' ' be a function in V k \Theta e associated with a coefficient matrix A k;' . Then f k;' has the orthogonal decomposition with f ~ (2) ~ where the matrices A are computed from the system of equations G e G e e e e G e e e with e Moreover, e e e e Proof: To find the formula for A k\Gamma1;'\Gamma1 , we take the inner-product of both sides of (4.1) with ' k\Gamma1;i for ' '\Gamma1;j for . The formulae for the B (i) are obtained in a similar way. The reconstruction formula follows directly from (4.1) after inserting the refinement relations and using the linear independence of the components of the vectors ' k and in ~ Note that computing the matrices A k\Gamma1;'\Gamma1 and B (i) in a decomposition step can be done quite efficiently since several matrix products occur more than once, and we need only solve linear systems of equations involving the four matrices G G H . As we shall see below, in our setting the first two of these are banded matrices, and the second two are periodic versions of banded matrices. All of them can be precomputed and stored in compact form. x5. The decomposition matrices for the polynomial splines In this section we construct the matrices P k , Q k , and G k needed for the decomposition and reconstruction of quadratic polynomial splines on the closed interval [\Gamma-=2; -=2]. Consider the nested sequence of knots where with be the associated normalized quadratic B-splines with supports on the intervals [x k . For each k, the span V k of ' k;mk is the m k dimensional linear space of C 1 quadratic splines with knots at the x k i . These spaces are clearly nested. In addition to the well-known refinement relations a simple computation shows that Equations (5.2), (5.3) provide the entries for the matrix P k . In particular, the first two and last two columns are determined by (5.3), while for any 3 - 2, the i-th column of P k contains all zeros except for the four rows which contain the numbers 1=4, 3=4, 3=4, and 1=4. For example, In general, P k has at most two nonzero entries in each row and and at most four nonzero entries in each column. In order to construct the matrices Q k , we now give a basis for the wavelet space W k\Gamma1 . Here we work with the usual L 2 inner-product on L 2 [\Gamma-=2; -=2]. Let Theorem 5.1. Given k - 1, let and for k - 2, let \Gamma6864\Gamma4967\Gamma4061 In addition, for k - 2, let form a basis for W k\Gamma1 . Proof: The wavelets in (5.5) are just the well-known quadratic spline wavelets, see e.g., [1]. As described in [5], the coefficients of the remaining wavelets can be computed by forcing orthogonality to V k\Gamma1 . In view of (3.2), the wavelets are linearly independent if and only if the matrix Q k is of full rank. This follows since the submatrix of Q k obtained by taking rows easily be seen to be diagonally dominant. For an alternate proof of linear independence, see Lemma 11 of [5]. In view of properties of B-splines, it is easy to see that 2: We now describe the matrices Q k . By Theorem 5.1, and For general k - 2, the nonzero elements in the third column of Q k are repeated in in each successive column they are shifted down by two rows. The first two and last two columns of Q k contain the same nonzero elements as Q 2 . Clearly, Q k has at most 4 nonzero entries in each row and at most 8 nonzero entries in each column. We now describe the Gram matrices G k , which in general are symmetric and five-banded. To get G k , we start with the matrix with 66h k =120 on the diagonal, 26h k =120 on the first subdiagonal, and h k =120 on the second diagonal. Then replace the entries in the 3 \Theta 3 submatrices in the upper-left and lower-right corners by @ For example, and x6. The decomposition matrices for the trigonometric splines In this section we present the matrices e needed for the decomposition and reconstruction of periodic trigonometric splines of order 3. Suppose ' - 1, and that ~ is a nested sequence of knots, where ~ h where 0; otherwise. is the usual trigonometric B-spline of order three associated with uniformly spaced knots (0; h; 2h; 3h). Set ~ ae M ';i (OE); For later use we define ~ ';em ' ' ';i for The span e ';em ' is the space of periodic trigonometric splines of order three. Clearly, these spaces are nested, and in fact we have the following refinement relation: Theorem 6.1. For all ~ where Proof: By nestedness and the nature of the support of T h , for some numbers u; v; w; z. By symmetry, it is enough to compute u and v. To find u, we note that on [0; h], Then using (6.2) we can solve for u. To find v we note that and then solve for v using (6.2). Theorem 6.1 can now be used to find the entries in the matrix e needed in Sect. 2. In particular, each column has exactly the four nonzero elements starting in the first row in column one, and shifted down by two rows each time we move one column to the right (where in the last column the last two elements are moved to the top of the column). For example, e Next we describe a basis for the wavelet space f W '\Gamma1 which has dimension ~ 1. In this case we work with the usual L 2 inner-product on Theorem 6.2. Given ' - 1, let ~ ~ where and ~ with Then ~ is a basis for the space f Proof: To construct wavelets in f we apply Theorem 5.1 of [6] which gives explicit formulae for the ~ q i in terms of inner-products of ~ ' ';i with ~ ' '\Gamma1;j . To show that ~ are linearly independent, it suffices to show that e is of full rank. To see this, we construct a ~ n '\Gamma1 \Theta ~ by moving the last column of e in front of the first column, and then selecting rows 2; We now show that this matrix is strictly diagonally dominant, and thus of full rank. First, we note that in each row of B ' the element on the diagonal is ~ while the sum of the absolute values of the off diagonal elements is j~q 1 ( ~ h ' )j simple computation shows that each of the functions D(h) and r i (h) := ~ has a Taylor expansion which is an alternating series. In particular, using the first two terms of each series, we get Now it is easy to see that \Theta j~q 3 also has an alternating series expansion, and we get for the same range of h. This shows that B ' is strictly diagonally dominant, and the proof is complete. The formulae for the ~ q i in Theorem 6.2 are not appropriate for small values of ~ . In this case we can use the following Taylor expansions: ~ ~ ~ ~ Rather than computing them each time we need them, we can precompute and store the necessary values of ~ see Table 1 in Sect. 7. We can now describe the matrix e needed in Sect. 2 for decomposing and reconstructing with trigonometric splines. For e ~ ~ ~ ~ ~ ~ where all ~ are evaluated at ~ h 1 . For ' - 2, each column of e contains the 8 entries ~ evaluated at ~ h ' . In particular, these entries start in row 1 in column 1, and are shifted down by two each time we move one column to the right (where in the last three columns, entries falling below the last row are moved to the top). Clearly, e Q ' has exactly four nonzero entries in each row. For example, e ~ ~ ~ ~ ~ ~ ~ ~ where all ~ are evaluated at ~ h 2 . Finally, we describe the Gram matrices. Theorem 6.3. For ' - 1, the 3 associated with the ~ ' ';i is given by e I 00 I 01 I I 01 I 00 I 01 I I 02 I 01 I 00 I 01 I I 01 I 00 I 01 I 02 I I 01 I 00 I 01 I 01 I where I Z ~ x ' ~ ~ I 01 := Z ~ x ' ~ ~ I 00 := Z ~ x ' ~ ~ with Moreover, e I I Proof: Using (6.2), the necessary integrals can be computed directly. The formulae in Theorem 6.3 are clearly not appropriate for small values of ~ , in which case the following formulae can be used: I I ~ I We can precompute and store the values of I 00 , I 01 , and I 02 for various levels see Table 2 in Sect. 7 for the values up to x7. Implementation 7.1. Decomposition The decomposition procedure begins with a tensor spline of the form (1.1) based on polynomial splines ' k;i (') at a given level k - 1 and periodic trigonometric splines ~ ';j (OE) at a given level ' - 1 with coefficient matrix C := A k;' of size m k \Theta e To carry out one step of the decomposition, we solve the systems (4.2) for A , and set To continue the decomposition, we now carry out the same procedure on the matrix A k\Gamma1;'\Gamma1 . This process can be repeated at most min(k; times, where at each step the new spline coefficients and wavelet coefficients are stored in C. Thus, the entire decomposition process requires no additional storage beyond the original coefficient matrix. Because of the banded nature of the matrices appearing in (4.2), with careful programming and the use of appropriate band matrix solvers, the j-th step of the decomposition can be carried out with O(m k\Gammaj+1 e To help keep the number of operations as small as possible, we precompute and store the entries of the matrices G . appearing in (4.2). Table 1 gives the values of ~ needed for the e Table gives the values of I h ' and I needed for the e G ' . The matrices H k are symmetric positive definite and seven-banded, while the e H ' are symmetric positive definite periodic seven-banded matrices. To check the robustness of the decomposition process, we computed the exact condition numbers of the matrices G k , H k , e H ' for up to eight levels. None of the condition numbers exceeded 10, and we can conclude that the algorithm is highly robust. 6 -28.996175484404513950 146.95303891951439472 -302.86111072242944246 9 -28.999940238853933503 146.99926613409589417 -302.99782947935722381 Tab. 1. Trigonometric spline wavelet coefficients for various '. 7.2. Thresholding Typically, in the j-th step of the decomposition, many of the entries in the matrices k\Gammaj;'\Gammaj of wavelet coefficients will be quite small. Thus, to achieve compression, these can be removed by a thresholding process. In view of (5.6), tangent plane continuity will be maintained at the poles if we retain all coefficients in the first two and last two rows of these matrices. Given ffl, at the j-th level we remove all other 9 0.5500021215280431720 0.21666764608909212581 0.008333384794548569195 Tab. 2. Inner products of Trigonometric B-splines for various '. wavelet coefficients in B (1) k\Gammaj;'\Gammaj whose absolute values are smaller than ffl=2 j . We do the same for B (3) k\Gammaj;'\Gammaj using a threshold value of ffl=(300 smaller threshold is applied because of the scaling of the wavelets. 7.3. Reconstruction In view of (4.3), to carry out one reconstruction step simply involves matrix multiplication using our stored matrices. Because of the band nature of these matrices, the computation of A k\Gammaj;'\Gammaj requires O(m k\Gammaj e operations. At each step of the reconstruction we can store these coefficients in the same matrix C where the decomposition was carried out. x8. Examples To test the general performance of the algorithms, we begin with the following simple example. Example 1. Let s be the tensor spline with coefficients Discussion: Since the normalized quadratic B-splines form a partition of unity, it follows from (2.1) that with these coefficients, s j 1 for all ('; OE) 2 H, i.e., the corresponding surface is exactly the unit sphere. In this case the coefficient matrix is of size 770 \Theta 1536, and involves 1,182,720 coefficients. To test the algorithms, we performed decomposition with various values of ffl, including zero. In all cases, after reconstruction we got coefficients which were correct to machine accuracy (working in double precision). The run time on a typical workstation is just a few seconds for a full 7 levels of decomposition and reconstruction. The illustrate the ability of our multiresolution approach to achieve high levels of compression while retaining important features of a surface, we now create a tensor spline fit to a smooth surface with a number of bumps. Example 2. Let B be the surface shown in the upper left-hand corner of Figure 1. Discussion: The surface B was created by fitting a spline f 8;8 to data created by choosing 10 random sized subrectangles at random positions in H, and adding tensor product quadratic B-splines of maximum height 3/4 with support on each such rectangle to the constant values corresponding to the unit sphere. For the coefficient matrix is of size 770 \Theta 768 and involves 591,360 coefficients. To test the algorithms, we performed decomposition with the thresholding values 9. Table 3 shows the results of a typical run with step nco 6 9746 Tab. 3. Reduction in coefficients in Example 2 with Almost 3=4 of the coefficients are removed in the first step of decomposition, and after 7 steps we end up with only 9745 coefficients (which amounts to a 60:1 compression ratio). Table 4 shows the differences between the original coefficients and the coefficients obtained after reconstruction. The table lists both the maximum norm and the average ' 1 norm mem are the original coefficients, and ~ c ij are the reconstructed ones. Due to the scaling of the wavelets these numbers are somewhat larger than the corresponding ffl. The surfaces corresponding to the values are shown in Figure 1. At near perfect looking reconstruction, while at the major features are reproduced with only small wiggles in the surface. At Fig. 1. Compressed surfaces for Example 2. we have larger oscillations in the surface. This example shows that there is a critical value of ffl beyond which the surface exhibits increasing oscillations with very little additional compression. x9. Remarks Remark 9.1. The approach discussed in this paper was first presented at the Taormina Wavelet Conference in October of 1993, and as far as we know was the first spherical multiresolution method to be proposed. The corresponding proceedings paper [6] focuses on the general theory of L-spline wavelets, and due to space limitations, a full description of the method could not be included. In the meantime Tab. 4. Coefficient errors in Example 2 for selected ffl. we have become aware of the recent work [2,4,8,9,13]. In [2] the authors use tensor splines based on exponential splines in the OE variable. The method in [4] uses discretizations of certain continuous wavelet transforms based on singular integral operators, while the method in [8] uses tensor functions based on polynomials and trigonometric polynomials. Finally, the method in [9] utilizes C 0 piecewise linear functions defined on spherical triangulations. Except for the last method, we are not aware of implementations of the other methods. Remark 9.2. In our original paper [6], an alternative way of making sure that tangent plane continuity is maintained at the poles was proposed. The idea is to decompose the original tensor product function s into two parts s H and s P , where mk \Gamma2 e and reconstruction can be performed on s H . After adding s P , the reconstructed spline possesses tangent plane continuity at the poles. Our implementation of this method exhibits essentially the same performance in terms of compression and accuracy as the method described here, but for higher compression ratios produces surfaces which are not quite as visually pleasing near the poles. Remark 9.3. The method described here can be extended to the case of nonuniform knots in both the ' and OE variables. In this case the computational effort increases considerably since the various matrices can no longer be precomputed and stored. Remark 9.4. In Sect. 4 we have presented the details of the tensor-product decomposition and reconstruction algorithms assuming that the initial function f k;' lies in the space V k \Phi e not necessarily the same. Since these spaces can always be reindexed, this is not strictly necessary in the abstract setting, but was convenient for our application where there is a natural indexing for our spaces. Remark 9.5. In computing the coefficients needed in Sections 5 and 6, we found it convenient to use Mathematica. Remark 9.6. There are several methods for computing approximations of the form (1.1). An explicit quasi-interpolation method using data on a regular grid (along with derivatives at the north and south poles) can be found in [11]. The same paper also describes a two-stage method which can be used to interpolate scattered data, and a least squares method which can be used to fit noisy data. A general theory of quasi-interpolation operators based on trigonometric splines can be found in [7]. Remark 9.7. A closed, bounded, connected set U in IR 3 which is topologically equivalent to a sphere is called a sphere-like surface. This means that there exists a one-to-one mapping of U onto the unit sphere S. Moreover, there exists a point O inside the volume surrounded by U , such that every point on the surface U can be seen from O. Such surfaces are also called starlike. For applications, we can focus on the class of sphere-like surfaces of the form where ae is a smooth function defined on S. Then each function f defined on U is just the composition ae of a function g defined on S. Remark 9.8. As indicated in [9], compression methods on the sphere can be adapted to the problem of creating multiresolution representations of bidirectional reflection distribution functions (BRDF's), althought the basic domain for such functions is actually a hemisphere. We will explore the use of our method for this purpose in a later paper. Remark 9.9. It is well-known that the polynomial B-splines are stable. In particular for quadratic B-splines (' i ) with general knots3 kck1 - k for all coefficient vectors c. The same bounds hold for trigonometric splines since the linear functionals introduced in [11] are dual to the ~ Analogous stability results hold for general p-norms. --R An Introduction to Wavelets Multiresolution analysis and wavelets on S 2 and S 3 Algorithms for smoothing data on the sphere with tensor product splines Spherical wavelet transform and its discretiza- tion in Wavelets: Theory efficiently representing functions on the sphere Basic Theory Fitting scattered data on spherelike surfaces using tensor products of trigonometric and polynomial splines Wavelets for Computer Graphics Biorthogonale Wavelets auf der Sph-are --TR --CTR Thomas W. Sederberg , David L. Cardon , G. Thomas Finnigan , Nicholas S. North , Jianmin Zheng , Tom Lyche, T-spline simplification and local refinement, ACM Transactions on Graphics (TOG), v.23 n.3, August 2004 El Bachir Ameur , Driss Sbibih, Quadratic spline wavelets with arbitrary simple knots on the sphere, Journal of Computational and Applied Mathematics, v.162 n.1, p.273-286, 1 January 2004 John E. Lavery, Shape-preserving interpolation of irregular data by bivariate curvature-based cubic L

spherical data compression;tensor splines;multiresolution

587299

Explicit Algorithms for a New Time Dependent Model Based on Level Set Motion for Nonlinear Deblurring and Noise Removal.

In this paper we formulate a time dependent model to approximate the solution to the nonlinear total variation optimization problem for deblurring and noise removal introduced by Rudin and Osher [ Total variation based image restoration with free local constraints, in Proceedings IEEE Internat. Conf. Imag. Proc., IEEE Press, Piscataway, NJ, (1994), pp. 31--35] and Rudin, Osher, and Fatemi [ Phys. D, 60 (1992), pp. 259--268], respectively. Our model is based on level set motion whose steady state is quickly reached by means of an explicit procedure based on Roe's scheme [ J. Comput. Phys., 43 (1981), pp. 357--372], used in fluid dynamics. We show numerical evidence of the speed of resolution and stability of this simple explicit procedure in some representative 1D and 2D numerical examples.

Introduction . The classical algorithms for image deblurring and/or denoising have been mainly based on least squares, Fourier series and other L 2 -norm approxi- mations, and, consequently, their outputs may be contaminated by Gibbs' phenomena and do not approximate well images containing edges. Their computational advantage comes from the fact that they are linear, thus fast solvers are widely available. How- ever, the effect of the restoration is not local in spatial scale. Other bases of orthogonal functions have been introduced in order to get rid of those problems, e.g., compactly supported wavelets. However, Gibbs' phenomenon, (ringing), is still present for these norms. The Total Variation (TV) deblurring and denoising models are based on a variational problem with constraints using the total variation norm as a nonlinear non-differentiable functional. The formulation of these models was first given by Rudin, Osher and Fatemi in ([19]) for the denoising model and Rudin and Osher in ([18]) for the denoising and deblurring case. The main advantage is that their solutions preserve edges very well, but there are computational difficulties. Indeed, in spite of the fact that the variational problem is convex, the Euler-Lagrange equations are nonlinear and ill-conditioned. Linear semi-implicit fixed-point procedures devised by Vogel and Oman, (see [26]), and interior-point primal-dual implicit quadratic methods by Chan, Golub and Mulet, (see [6]), were introduced to solve the models. Those methods give good results when treating pure denoising problems, but the methods become highly ill-conditioned for the deblurring and denoising case where the computational cost is very high and parameter dependent. Furthermore, those methods also suffer from the undesirable staircase effect, namely the transformation of smooth regions (ramps) into piecewise constant regions (stairs). In this paper we present a very simple time dependent model constructed by evolving the Euler-Lagrange equation of the Rudin-Osher optimization problem, multiplied by the magnitude of the gradient of the solution. The two main analytic features of y Department of Mathematics, University of California, Los Angeles, 405 Hilgard Av- enue, Los Angeles, CA 90095-1555 and Departament de Matem'atica Aplicada, Universitat de Dr. Moliner, 50, 46100 Burjassot, Spain. E-mail addresses: marquina@uv.es, URL: http://gata.uv.es/~marquina. Supported by NSF Grant INT9602089 and DGICYT Grant PB97- 1402. Department of Mathematics, University of California, Los Angeles, 405 Hilgard Avenue, Los Angeles, CA 90095-1555. E-mail address: sjo@math.ucla.edu. Supported by NSF Grant DMS this formulation are the following: 1) the level contours of the image move quickly to the steady solution and 2) the presence of the gradient numerically regularizes the mean curvature term in a way that preserves and enhances edges and kills noise through the nonlinear diffusion acting on small scales. We use the entropy-violating Roe scheme, ([16]) for the convective term and central differencing for the regularized mean curvature diffusion term. This makes a very simple, stable, explicit procedure, computationally competitive compared with other semi-implicit or implicit procedures. We show numerical evidence of the power of resolution and stability of this explicit procedure in some representative 1D and 2D numerical examples, consisting of noisy and blurred signals and images, (we use Gaussian white noise and Gausssian blur). We have observed in our experiments that our algorithm shows a substantially reduced staircase effect. 2. Deblurring and Denoising. A recording device or a camera would record a signal or image so that 1) the recorded intensity of a small region is related to the true intensities of a neighborhood of the pixel, through a degradation process usually called blurring and 2) the recorded intensities are contaminated by random noise. To fix our ideas we restrict the discussion to R 2 . An image can be interpreted as either a real function defined on \Omega\Gamma a bounded and open domain of R 2 , (for simplicity we will assume\Omega to be the unit square henceforth) or as a suitable discretization of this continuous image. Our interest is to restore an image which is contaminated with noise and blur in such a way that the process should recover the edges of the image. Let us denote by u 0 the observed image and u the real image. A model of blurring comes from the degradation of u through some kind of averaging. Indeed, u may be blurred through the application of a kernel: k(x; s; r) by means of Z \Omega u(s; r) k(x; s; ds dr (2.1) and, we denote this operation by v u. The model of degradation we assume is where n is Gaussian white noise, i.e., the values n i of n at the pixels i are independent random variables, each with a Gaussian distribution of zero mean and variance oe 2 . If the kernel k is translation invariant, i.e., there is a function j(x; y), (also called a kernel), such that k(x; s; and the blurring is defined as a 'superposition' of j 0 s: Z \Omega u(s; r) ds dr (2.3) and this isotropic blurring is called convolution. Otherwise, if the kernel k is not translation-invariant we call this blurring anisotropic. For the sake of simplicity, we suppose that the blurring is coming from a convolution, through a kernel function j such that j u is a selfadjoint compact integral operator. Typically, j has the following properties, goes to 1 and R For any ff ? 0 the so-called heat kernel, defined as is an important example that we will use in our numerical experiments. The main advantage of the convolution is that if we take the Fourier transform of (2.3) we get then, to solve the model (2.2) with we take Fourier transform and we arrive at To recover u(x; y), we need to deconvolve, i.e., this means that we have to divide the equation (2.6) by - j(k; l) and to apply the inverse Fourier transform. This procedure is generally very ill-posed. Indeed, j is usually smooth and j(x; y) ! 0 rapidly as goes to 1, thus large frequencies in u 0 get amplified considerably. The function u 0 is generally piecewise smooth with jumps in the function values and derivatives; thus the Fourier method approximation gives global error estimates of order O(h), (see ([11])) and suffers from Gibbs' phenomenon. Discrete direct methods dealing with the linear integral equation (2.6) have been designed by different authors, (see ([13] and references therein). One way to make life easier is to consider a variational formulation of the model that regularizes the problem. Our objective is to estimate u from statistics of the noise, blur and some a priori knowledge of the image (smoothness, existence of edges). This knowledge is incorporated into the formulation by using a functional R that measures the quality of the image u, in the sense that smaller values of R(u) correspond to better images. The process, in other words, consists in the choice of the best quality image among those matching the constraints imposed by the statistics of the noise together with the blur induced by j. The usual approach consists in solving the following constrained optimization problem: min subject to jjj since E( denotes the expectation of the random variable X) imply that jjj R\Omega (j Examples of regularization functionals that can be found in the literature are, r is the gradient and \Delta is the Laplacian, see Refs. [22, 8]. The main disadvantage of using these functionals is that they do not allow discontinuities in the solution, therefore the edges can not be satisfactorily recovered. In [19], the Total Variation norm or TV-norm is proposed as a regularization functional for the image restoration problem: Z \Omega Z \Omega y dx: (2.8) The norm does not penalize discontinuities in u, and thus allows us to recover the edges of the original image. There are other functionals with similar properties introduced in the literature for different purposes, (see for instance, [7, 5, 25, 2]). The restoration problem can be thus written as min Z \Omega subject to 1i Z \Omega (j Its Lagrangian is Z \Omega \Omega (j and its Euler-Lagrange equations, with homogeneous Neumann boundary conditions for u, are: \Omega (j There are known techniques, (see [3]), for solving the constrained optimization problem (2.9) by exploiting solvers for the corresponding unconstrained problem, whose Euler-Lagrange equations are (2.11) for - fixed. Therefore, for the sake of clarity, we will assume the Lagrange multiplier - to be known throughout the exposi- tion. For , we can then write the equivalent unconstrained problem as min Z \Omega (j and its Euler-Lagrange equation in the more usual form: We call (2.14) the nonlinear deconvolution model. The linear deconvolution model would be that comes from the Euler-Lagrange equation of the corresponding unconstrained problem with the norm Since the equation (2.14) is not well defined at points where due to the presence of the term 1=jruj, it is common to slightly perturb the Total Variation functional to become: Z \Omega where fi is a small positive parameter, or, Z \Omega with the notation 3. The time dependent model. Vogel and Oman and Chan, Golub and Mulet devised direct methods to approximate the solution to the Euler-Lagrange equation with an a priori estimate of the Lagrange multiplier and homogeneous Neumann boundary conditions. Those methods work well for denoising problems but the removal of blur becomes very ill-conditioned with user-dependent choice of parame- ters. However, stable explicit schemes are preferable when the steady state is quickly reached because the choice of parameters is almost user-independent. Moreover, the programming for our algorithm is quite simple compared to the implicit inversions needed in the above mentioned methods. Usually, time dependent approximations to the ill-conditioned Euler-Lagrange equation are inefficient because the steady state is reached with a very small time step, when an explicit scheme is used. This is the case with the following formulation due to Rudin, Osher and Fatemi (see [19]) and Rudin and Osher (see [18]): with u(x; given as initial data, (we have used as initial guess the original blurry and noisy image u 0 ) and homogeneous Neumann boundary conditions, i.e., @u @n the boundary of the domain. As t increases, we approach to a restored version of our image, and the effect of the evolution should be edge detection and enhancement and smoothing at small scales to remove the noise. This solution procedure is a parabolic equation with time as an evolution parameter and resembles the gradient-projection method of Rosen (see [17]). In this formulation we assume an a priori estimate of the Lagrange multiplier, in contrast with the dynamic change of - supposed in the Rosen method, (see section 6 for details). The equation (3.1) moves each level curve of u normal to itself with normal velocity equal to the curvature of the level surface divided by the magnitude of the gradient of u, (see ([23]), ([15]) and ([20])). The constraints are included in the -term and they are needed to prevent distortion and to obtain a nontrivial steady state. However, this evolution procedure is slow to reach steady state and is also stiff since the parabolic term is quite singular for small gradients. In fact, an ad hoc rule of thumb would indicate that the timestep \Deltat and the space stepsize \Deltax need to be related by \Deltat for fixed c ? 0, for stability. This CFL restriction is what we shall relax. These issues were seen in numerous experiments. In order to avoid these difficulties, we propose a new time dependent model that accelerates the movement of level curves of u and regularizes the parabolic term in a nonlinear way. In order to regularize the parabolic term we multiply the whole Euler-Lagrange equation (2.14) by the magnitude of the gradient and our time evolution model reads as follows: We use as initial guess the original blurry and noisy image u 0 and homogeneous Neumann boundary conditions as above, with an a priori estimate of the Lagrange multiplier. From the analytical point of view this solution procedure approaches the same steady state as the solution of whenever u has nonzero gradient. The effect of this reformulation, (i.e. preconditioning) is positive in various aspects: 1. The effect of the regularizing term means that the movement of level curves of u is pure mean curvature motion, (see [15]). 2. The total movement of level curves goes in the direction of the zeros of j u\Gammau 0 regularized by the anisotropic diffusion introduced by the curvature term. 3. The problem for the denoising case is well-posed in the sense that there exists a maximum principle that determines the solution, (see ([15])). 4. There are simple explicit schemes, such as Roe's scheme, that behave stably with a reasonable CFL restriction for this evolution equation. Let us remark that explicit schemes could also be applied for the 'anisotropic blurring' case. 5. This procedure is more morphological, (see [1]), in the pure denoising case, i.e., it operates mainly on the level sets of u and u 0 . This is easily seen if we replace u by h(u) and u 0 by h(u 0 ) with equation (3.3) is invariant, except that replaced by (h(u) \Gamma h(u 0 ))=h 0 (u). The anisotropic diffusion introduced in this model is a nonlinear way to discriminate scales of computation. This never occurs with a linear model, (e.g. the linear deconvolution model), because in this case we would have the linear heat equation with constant diffusion. Thus, our model (3.3) can be seen as a convection-diffusion equation with morphological convection and anisotropic diffusion. 4. Explicit numerical schemes for the 1D model. The 2D model described before is more regular than the corresponding 1D model, because the 1D original optimization problem is barely convex. For the sake of understanding the numerical behavior of our schemes, we also discuss the 1D model. The Euler-Lagrange equation in the 1D case reads as follows: x This equation can be written either as x using the small regularizing parameter fi ? 0 introduced at the end of the previous section or using the ffi -function. The Rudin-Osher-Fatemi model, (ROF model), in terms of the ffi -function will read as follows Our model in 1D will be x regularizing parameter. The parameter fi ? 0 plays a more relevant role in this case than in the 2D model. We can also state our model in terms of the ffi function as where a convolution of the ffi function must be used in practice. The intensity of this kind of convolution decides which scale acts on the diffusion term. In this paper, we always approximate ffi by A radical way to make the coefficient of u xx nonsingular is to solve the evolution model: This model works in such a manner that away from extrema we have a large multiplier of \Gammaj (j and at extrema it is just the heat equation. These evolution models are initialized with the blurry and noisy signal u 0 and homogeneous Neumann boundary conditions, and with a prescribed Lagrange multi- plier. We estimated - ? 0 near the maximum value such that the explicit scheme is stable under appropriate CFL restrictions, (see below). In order to convince the reader about the speed and programming simplicity of our model, we shall give the details of the first order scheme for the 1D pure denoising model, i.e., x Let u n j be the approximation to the value u(x Then, the scheme for the problem (4.9) will be where and ug j is the upwind gradient, i.e., \Deltax if \Deltax if Our general explicit scheme has the following features: 1. We use central differencing for u xx , 2. The convolution operator j is computed by evolving the heat equation u with the explicit Euler method in time and central differencing in space with corresponding to a oe of the 1D heat kernel: -oe 3. We use upwind Roe differencing, (see [16], [10]), checking the direction of propagation by computing the sign of the derivative of the coefficient of j (j respect to u x times the sign of this term. Indeed, for our evolution model (4.5) it is enough to check the sign of u x For the model (4.8) we get the same direction of propagation as before. We note that there is no notion of "entropy condition satisfying" discontinuities in image processing; thus we omit the usual "entropy-fix" applied to the Roe solver in this work. 4. The CFL condition depends on - and fi. Indeed, the parabolic term in our model (4.5) gives a CFL restriction \Deltat x and the convection term gives \Deltat s x for fixed c. These restrictions are reasonable at local extrema and near edges, compared with the parabolic CFL restriction that corresponds to the reaction-diffusion ROF model, (4.4): \Deltat which is too stiff along flat regions or at local extrema. The CFL restriction coming from the convection term in the radical model (4.8) is better but also unfortunate \Deltat Thus, our model is more convenient from this point of view. 5. Explicit numerical schemes for the 2D model. We can express our 2D model in terms of explicit partial derivatives as: x y using u 0 as initial guess and homogeneous Neumann boundary conditions, (i.e., absorbing boundary). The denominator, u 2 y , appearing in the diffusion term may vanish or be small along flat regions or at local extrema, when it is computed. Then, we can use either the regularizing parameter fi ? 0, (small enough to perform floating point division), or make the diffusion term equal to zero when gradient is smaller than a tolerance, (we can also use parameter fi small as tolerance cut-off). Our choice in this paper was the cut-off option, following a suggestion by Barry Merriman. Thus, concerning stability and resolution the role of parameter fi is almost irrelevant in 2D calculations. Let u n ik be the approximation to the value u(x and \Deltay and \Deltat are the spatial stepsizes and the time stepsize, respectively. We denote by v ik ). We point out that we used for j, the convolution with the 2D heat kernel, (2.4), in our experiments, aproximated by evolving the 2D heat equation u by means of the explicit Euler method in time and central differencing in space. Then our first order scheme reads as follows: ik ug x ik ik- (w n where the second order term is defined by if g x ik ik! fi and ik ik otherwise, where x y ik 2\Deltay xx ik ik yy \Deltay 2 xy ik 2\Deltax\Deltay ug x ik is the upwind gradient in the x-direction, i.e., ug x \Deltax (5.10) if g x ik (w n ug x ik \Deltax (5.11) if g x ik (w n ik is the upwind gradient in the y-direction, i.e., ug y \Deltay (5.12) if g y ik (w n ug y ik \Deltay if g y ik (w n A very simple way to extend this scheme to get high order accuracy is to follow Shu-Osher prescription, (see [21]). Thus, we consider a method of lines, using an explicit high order Runge-Kutta method in time and using a method of spatial ENO reconstruction, (see [24], [9], [21] and [12]), of the same order, for the convection term, applied on every time substep. We have tested the Van Leer second order MUSCL spatial reconstruction using the minmod function as slope-limiter together with classical second order Runge-Kutta method and the third order PHM spatial reconstruction as in [12], using as slope- limiter the harmod function, consisting of the harmonic mean of the lateral slopes when they have the same sign and zero when they have different sign, together with the third order Shu-Osher Runge-Kutta method of [21]. We have found that these explicit methods are stable and give high accuracy under the same CFL restrictions as the first order scheme. As a sample we shall describe the second order MUSCL method. Since the Runge-Kutta methods used here are linear combination of first order explicit Euler timesteps, it is enough to formulate one Euler step, (in fact, in this case it is Heun's method which is the arithmetic mean of two Euler timesteps). Following the notation used above we have: ik ik ik- (w n where the reconstructed upwind gradients rug x ik and rug y ik are computed in the following way. We reconstruct the left x-gradient in from the linear function: \Deltax (5.15) where computed in x i , i.e. gl x where the minmod function is defined as being sgn the sign function. Analogously, we have the reconstructed right x-gradient, gr x gr x where \Deltax (5.20) where Then the reconstructed upwind gradient in the x-direction is defined from the mean value as if gm x ik if gm x The procedure in the y-direction is similar. -50501500 50 100 150 200 250 300 -5050150Fig. 6.1. Left, original vs. noisy 1D image; right original vs. recovered 1D image 6. Numerical Experiments. In this section, we perform some numerical experiments in 1D and 2D. We have used 1D signals with values in the range [0; 255]. The signal of (6.1, left) represents the original signal versus the noisy signal with SNR - 5. The signal of (6.1, right) represents the original signal versus the recovered signal after 80 iterations with first order scheme with CFL 0:25. The estimated computed as the maximum value allowed for stability, using the explicit Euler method in time. We have used this experiment in order to achieve the appropiate amount of difusion at small scales. In pure denoising 1D problems the choice of the value of fi in our model depends on the SNR. Let us observe the very reduced staircase effect, compared with the usual one obtained with either fixed-point iterative methods or nonlinear primal-dual methods, (see [4]). Now, we present a pure deblurring problem in 1D. The signal of (6.2, left) represents the original signal versus the blurred signal with (as in 4.11. The signal of (6.2, right) represents the original signal versus the recovered signal after 40 iterations with first order scheme with CFL 0:1. The estimated computed as the maximum value allowed for stability, using the explicit Euler method in time. We use 0:01 in this experiment. The signal of (6.3, left) represents the original signal versus the blurred and noisy signal with (as in 4.11), and SNR - 5. The signal of (6.2, right) represents the original signal versus the recovered signal after 80 iterations with first order scheme Fig. 6.2. Left, original vs. blur 1D image; right original vs. recovered 1D image -50501500 50 100 150 200 250 300 -5050150Fig. 6.3. Left,original vs. noisy and blurred 1D signal ; right, original vs. recovered 1D signal with CFL 0:25. The estimated computed as the maximum value allowed for stability, using explicit Euler method in time. The - used for the current denoising and deblurring problem is smaller than the one used in the above pure deblurring problem, as we expected. We use this experiment to get the correct degree of difusion at small scales. This shows that the 1D problem is quite sensitive to the choice of fi, in contrast with the 2D case where the size of this parameter becomes irrelevant. Let us also observe a very reduced staircase effect. We performed many other experiments with 1D signals, obtaining similar results. All our 2D numerical experiments were performed on the original image (Fig 6.4, left) with 256 \Theta 256 pixels and dynamic range in [0; 255]. The third order scheme we used in our 2D experiments was based on the third order Runge-Kutta introduced by Shu and Osher, (see [21]), to evolve in time with a third order spatial approximation based on the PHM reconstruction introduced in ([12]). Our first 2D experiment was made on the noisy image, (6.4, right), with a SNR which is approximately 3. Details of the approximate solutions using the Chan-Golub- Mulet primal-dual method and our time dependent model using the third order Roe's scheme, (described above), are shown in Fig. 6.5. We used - 0:0713 and we perform 50 iterations with CFL number 0:1. We used the same estimated - as the one used for the primal-dual method, and we observed that this value correponds to the largest we 50 100 150 200 250100200Fig. 6.4. Left: original image, right: noisy image, SNR- 3. Chan-Golub-Mulet Primal-Dual Resolution 256x256, SNR \approx 3, Estimated =0.0713 50 100 150 200 250100200ROE-ORDER3-RK3 50 100 150 200 250100200Fig. 6.5. Left: image obtained by the Chan-Golub-Mulet primal-dual method, right: image obtained by our time evolution model,with 50 timesteps and CFL-0.1 allowed for stability with this CFL restriction. We also remark that the third order Runge-Kutta method used enhances the diffusion at small scales. The contour plots are shown in Fig 6.6. We can infer from these contours that the edges obtained by the new model are sharper than the ones obtained by the primal-dual method. This might seem surprising, since the steady state satisfies the same equation (2.14) on the analytic level. Numerically they are quite different because the approximation of the convection term involves hyperbolic upwind ideas. Our second 2D experiment is a pure deblurring problem. Fig (6.7, left), corresponds to the original image blurred with Gaussian blur where as in (2.4). We remark that we computed the convolution operator j by evolving the 2D heat equation with explicit Euler method in time and central differencing in space with a CFL number of 0.125, in order to test our model in practical conditions. In Fig (6.7, right), we represent the approximation using our third order Roe's scheme where we perform 50 iterations with CFL number 0:1. We have used (the maximum value that allows stability for the above CFL restriction), and We observe that the scheme is not sensitive to the choice of fi provided the value be small enough, (smaller than 0:1). This behavior is justified from the fact that the 2D problem is more regular. Fig. 6.6. Left: isointensity contours of part of the image obtained by the primal-dual method, right: isointensity contours of part of the image obtained by our time evolution model. 50 100 150 200 25010020050 100 150 200 250100200Fig. 6.7. Left: image blurred with Gaussian blur with image restored with our model, using third order Roe's scheme with 50 timesteps and CFL-0.1. The isointensity contours showed in (6.8) make clear the edge enhancement obtained through our algorithm. Our 2D critical experiment was performed on the blurry and noisy image represented in Fig (6.9, left), with Gaussian blur where as in (2.4) and SNR - 5. We have used the 0:01. We performed 50 iterations with a CFL number of 0:1, using our third order Roe's scheme, obtaining the approximation represented in figure (6.9, right). Let us observe the denoising and deblurring effect in the isointensity contours picture represented in figure (6.10). Finally, we shall include the convergence history of the two 1D experiments corresponding to the pure denoising problem and a denoising and deblurring problem presented above. In Figs 6.11 and 6.12 we represent the semilog plot of the L 2 -norm of the differences between consecutive iterates versus the number of iterations and the plot of the evolution of the total variation of the solution, respectively. We observe 'superlinear' convergence along the first third part of the evolution and linear convergence along the remainder. We pointed out that all our experiments were performed with a constant timestep and thus, the computational cost is very low compared with the semi-implicit methods. These usually require one third of the number of iterations we needed, but every step of the semi-implicit method requires about five iterations Fig. 6.8. Left: isointensity contours of part of the blurred image, right: isointensity contours of part of the image restored by using our time evolution model. 50 100 150 200 25010020050 100 150 200 250100200Fig. 6.9. Left: image blurred with Gaussian blur with noisy with SNR - 10, right: image restored with our model, using third order Roe's scheme with 50 timesteps and CFL-0.1. of the preconditioned conjugate gradient method to invert. 7. Concluding remarks. We have presented a new time dependent model to solve the nonlinear TV model for noise removal and deblurring together with a very simple explicit algorithm based on Roe's scheme of fluid dymamics. The numerical algorithm is stable with a reasonable CFL restriction, it is easy to program and it converges quickly to the steady state solution, even for deblurring and denoising prob- lems. The algorithm is fast and efficient since no inversions are needed for deblurring problems with noise. Our time dependent model is based on level set motion that makes the procedure morphological and appears to satisfy a maximum principle in the pure denoising case, using as initial guess the noisy image. We also have numerical evidence, (through our numerical tests), of this stability in the deblurring case, using the noisy and blurred image as initial guess. --R A variational method in image recovery Modular solvers for constrained image restoration problems Extensions to total variation denoising Image recovery via total variation minimization and related problems A nonlinear primal-dual method for total variation-based image restoration Constrained restoration and the recovery of discontinuities The theory of Tikhonov regularization for Fredholm integral equations of the first kind Uniformly high order accurate essentially non-oscillatory schemes III Numerical methods for conservation laws The Fourier method for nonsmooth data reconstructions for nonlinear scalar conservation laws Restoring images degraded by spatially variant blur Fronts propagating with curvature dependent speed: algorithms based on a Hamilton-Jacobi formulation The gradient-projection method for nonlinear programming: Part II Total variation based image restoration with free local constraints Nonlinear total variation based noise removal algorithms Cambridge University Press Efficient implementation of essentially non-oscillatory shock capturing schemes II Solutions of ill-posed problems On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature Towards the ultimate conservative difference scheme V. Variational problems and PDE's for image analysis and curve evolution Iterative methods for total variation denoising --TR --CTR John Steinhoff , Meng Fan , Lesong Wang , William Dietz, Convection of Concentrated Vortices and Passive Scalars as Solitary Waves, Journal of Scientific Computing, v.19 n.1-3, p.457-478, December Youngjoon Cha , Seongjai Kim, Edge-Forming Methods for Image Zooming, Journal of Mathematical Imaging and Vision, v.25 n.3, p.353-364, October 2006 Ronald P. Fedkiw , Guillermo Sapiro , Chi-Wang Shu, Shock capturing, level sets, and PDE based methods in computer vision and image processing: a review of Osher's contributions, Journal of Computational Physics, v.185 n.2, p.309-341, March

upwind schemes;total variation norm;image restoration;nonlinear diffusion;level set motion

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Schur Complement Systems in the Mixed-Hybrid Finite Element Approximation of the Potential Fluid Flow Problem.

The mixed-hybrid finite element discretization of Darcy's law and continuity equation describing the potential fluid flow problem in porous media leads to a symmetric indefinite linear system for the pressure and velocity vector components. As a method of solution the reduction to three Schur complement systems based on successive block elimination is considered. The first and second Schur complement matrices are formed eliminating the velocity and pressure variables, respectively, and the third Schur complement matrix is obtained by elimination of a part of Lagrange multipliers that come from the hybridization of a mixed method. The structural properties of these consecutive Schur complement matrices in terms of the discretization parameters are studied in detail. Based on these results the computational complexity of a direct solution method is estimated and compared to the computational cost of the iterative conjugate gradient method applied to Schur complement systems. It is shown that due to special block structure the spectral properties of successive Schur complement matrices do not deteriorate and the approach based on the block elimination and subsequent iterative solution is well justified. Theoretical results are illustrated by numerical experiments.

Introduction . Let be a bounded domain in R 3 with a Lipschitz continuous boundary @ The potential uid ow in saturated porous media can be described by the velocity u using Darcy's law and by the continuity equation for incompressible ow where p is the piezometric potential ( uid pressure), A is a symmetric and uniformly positive denite second rank tensor of the hydraulic resistance of medium with for all represents the density of potential sources in the medium. The boundary conditions are given by @ @ where @ @ @ N are such that @ @ @ is the outward normal vector dened (almost everywhere) on the boundary @ Assume that the domain is a polyhedron and it is divided into a collection of subdomains such that every subdomain is a trilateral prism with vertical faces and general nonparallel bases (see, e.g., [11], [14] or [15]). We will denote the discretization of the domain by E h and assume an uniform regular mesh with the discretization parameter h. Denote also the collection of all faces of elements which are not adjacent This work was supported by the Grant Agency of the Czech Republic under grant 201/98/P108 and by the grant AS CR A2030706. Revised version October 1999. y Seminar for Applied Mathematics, Swiss Federal Institute of Technology (ETH) Zurich, ETH- Zentrum, CH-8092 Zurich, Switzerland. (miro@sam.math.ethz.ch) z Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodarenskou vez 2, 182 07 Prague 8, Czech Republic. (maryska@uivt.cas.cz, miro@uivt.cas.cz, tuma@uivt.cas.cz) to the boundary @ D by @ D and introduce the set of interior faces @ We consider the following low order nite element approximation. Let be the space spanned by the linearly independent basis functions v e dened on the element e 2 E h in the form and such that they are orthonormal with respect to the set of functionals Z 5: Here f e k denotes the k-th face of the element e k;3 ) is the outward normal vector with respect to the face f e k . The velocity function u will be then approximated by vector functions linear on every element e 2 E h from the Raviart- Thomas space denotes the restriction of a function v h onto the element e 2 E h . Further denote the space of constant functions on each element e 2 E h by M 0 (e) and denote the space of constant functions on each face f 2 h by M 0 (f ). The piezometric potential p will be approximated by the space which consists of elementwise constant functions where h j e is the restriction of a function h onto element e . The Lagrange multipliers coming from the hybridization of a method will be approximated by the space of all functions constant on every face from h Here h j f denotes the restriction of a function h onto the face f 2 h . Analogously we introduce the spaces M 0(@ N ) as the spaces of functions constant on every face from [ e2E h @e \ @ D and h \ @ respectively. The detailed description of the spaces that we use can be found in [14] (see also [11] or [15]). The Raviart-Thomas approximation of the mixed-hybrid formulation for the problem (1.1) and (1.2) reads as follows (see [4]): Find @e\@ (1. @e\@ where p D;h and uN;h are approximations to the functions p D and uN on the spaces where the function q is approximated by For other details we refer to [14] or [11]. Further denote by the number of elements, by the number of interior inter-element faces and the number of faces with the prescribed Neumann boundary conditions in the discretization by j@ j. Let e be some numbered ordering of the set of prismatic elements and f k , NNC, be the ordering of the set of non-Dirichlet faces from h . For every element we denote by NIF e i the number of interior inter-element faces and by NNC e i the number of faces with Neumann boundary conditions imposed on the element e i . Let the nite-dimensional space RT 0 spanned by linearly independent basis functions v NA from the denition (1.6); let the space M 0 spanned by NE linearly independent basis functions nally the space M 0 spanned by NIF linearly independent basis functions k , +NNC. From this Raviart-Thomas approximation we obtain the system of linear algebraic equations in the formB @ C A =B @ where are unknowns, the symmetric positive denite matrix block A 2 R NA;NA is given by the terms (Av the outdiagonal block B 2 R NA;NE by (r and the block C 2 R NA;NIF+NNC by < . Here n k is the outward normal vector to with respect to the face (see [11] and [14]). Let us denote the system matrix in (1.12) by A. The symmetric matrix A is indenite due to the zero diagonal block of dimension NNC. The structure of nonzero elements in the matrix from a small model problem can be seen in Figure 1. Partition the submatrix C in A as (C 1 corresponds to the interior inter-element faces in the discretized domain and C 2 2 R NA;NNC is the face- condition incidence matrix corresponding to the element faces with Neumann boundary conditions. Note that every column of C 1 contains only two nonzero entries equal to 1. The singular values of C 1 are all equal to 2 and the matrix block C 2 has orthogonal columns. Moreover, the whole matrix block C has also singular values equal to 2 or 1. The matrix B has a special structure. The nonzero elements correspond to the face- element incidence matrix with values equal to -1. Thus all singular values of the matrix are equal to 5 (the matrix B is, up to the normalization coe-cients, orthogonal). It is easy to see from the denition of approximation spaces (see [14] or [15]) that the symmetric positive denite block is 5 5 block diagonal and it was shown in [15] Fig. 1. Structural pattern of the matrix obtained from mixed hybrid nite element approximation of a model problem with (to be discussed in Section 5). that the spectrum of the matrix block A satises are positive constants independent of the discretization parameters and dependent on the domain and the tensor A. It is also easy to see that the system matrix A in (1.12) is non-singular if and only if the block (B C) has a full column rank. Clearly, if the condition @ holds (all boundary conditions are Neumann conditions), then the matrix block (B C) is singular, due to the fact that all sums of row elements are zero. In other words, the function p is unique up to a constant function in the case @ Assuming @ it follows from the analysis presented in [15] that there exist positive constants c 3 and c 4 such that for the singular values of the matrix block (B C) we have Moreover, for eigenvalues of the whole symmetric indenite matrix A it follows asymptotically are positive constants independent of system parameters. In this paper, for solving the symmetric indenite systems (1.12), the successive reduction to Schur complement systems is proposed. We consider three successive Schur complement systems arising during the block elimination of unknowns which correspond to matrix blocks A, B and C 2 respectively, or in other words, which correspond to the elimination of the velocity variables u, the pressure variables p and of a part of the Lagrange multipliers . While the concept of reduction to the rst and second Schur complement systems is well known as a static condensation (described e.g. in [4], Section V. or in [11])), the proposed reduction to the third Schur complement system seems to be new. The main contribution of the paper consists in a detailed investigation of the structure of nonzero entries and the spectral properties of the Schur complement ma- trices. This enables thorough complexity analysis of the direct or iterative solution of corresponding Schur complement systems. A brief analysis of the structure of the rst Schur complement matrix can be found in [4] as well as a straightforward observation that its principal leading block is a diagonal. Here we extend this analysis and discuss the mutual relation between the number of nonzero entries in the rst Schur complement matrix and the number of nonzeros in the system matrix (1.12). We show further that no ll-in occurs during the process of reduction to the second and third Schur complement system. Moreover, we prove that the number of nonzeros in both these two Schur complements is always less than the number of nonzeros in (1.12). It is shown also that the spectral properties of matrices in such Schur complement systems do not deteriorate during the successive elimination. Thus an approach based on the block reduction and subsequent iterative solution is well justied. The outline of the paper is as follows. In Section 2, we examine the structural pattern of resulting Schur complement matrices and give estimates for their number of nonzero elements in terms of the discretization parameters listed above. Section 3 is devoted to the solution of the whole indenite system (1.12) via three Schur complement reductions and subsequent direct solution. Using the graph theoretical results we give the asymptotic bound of the computational complexity for the Cholesky decomposition method applied to the third Schur complement system. In Section 4, we concentrate on the spectral properties of the Schur complement system matrices. The theoretical convergence rate of the iterative conjugate gradient-type method in terms of the discretization parameters is estimated. The asymptotic bounds for the computational work of the iterative solution are given. Section 5 contains some numerical experiments illustrating the previously developed theoretical results. Finally, we give some concluding remarks and mention some open questions for our future work. 2. Structural properties of the Schur complement matrices. In this section we take a closer look to the discretized indenite system and corresponding Schur complements and we extend the brief analysis from [4]. There are several possibilities for the choice of a block ordering in the consecutive elimination. We shall concentrate on the block ordering which seems to be the most natural and e-cient from the point of view of solving the nal Schur complement system by a direct solver or by a conjugate gradient-type method. The same ordering for the elimination of the rst two blocks was used also in [4], p. 178-181 or in [11]. Note that the static condensation is not the only way to form the successive Schur complements. E.g., in [17] the case of the Raviart- Thomas discretization for the closely related nodal methods was studied and reduction to a dierent second Schur complement system was discussed. The following simple result gives the number of nonzero elements in the triangular part of the matrix A. By the triangular part of a matrix M we mean its upper (lower) strict triangle diagonal. We will deal only with the structural nonzero elements we do not take into account accidental cancellations and possible initial zero values of the Fig. 2. Structural pattern of the Schur complement matrix A=A for A from Figure 1. tensor of hydraulic permeability. By the structure of a matrix M we mean 0g. Lemma 2.1. The number of nonzeros in the triangular part of A is given by Proof. The triangular part of A has 15NE nonzeros, the block B contributes by 5NE nonzeros, C 1 has 2NIF nonzeros and C 2 contains NNC nonzeros. The symmetric positive denite matrix block A in (1.12) is block-diagonal, each 5 5 block corresponds to certain element in the discretization of the domain. Therefore it is straightforward to eliminate the velocity variables u and to obtain the rst Schur complement system with the matrix A=A =B @ A 11 A 12 A 13 A T A T The structure of the matrix A=A for our example problem is shown in Figure 2. For details we also refer to [4], p. 180-181 or [11]. For the number of nonzeros in the matrix A=A we can show the following result. Lemma 2.2. The number of nonzeros in the triangular part of the Schur complement matrix A=A is equal to Proof. Clearly, . Note that the ll-in for A 1 C 1 is considerably higher (it is equal to 10NIF ). Further, jA 13 and The number of nonzeros in A 23 is equal to Finally, note that Observe that the directed graph of the matrix B T C 1 has the set of arcs is an interior face of ig: The undirected graph of C T adjacency relation based on the connectivity through the interior faces inside the domain. It follows that where e(f) and e(f) are the two elements from E h such that considering the relation NIF: Putting all the partial sums together we get the desired result. Consider now the second Schur complement matrix A A 22 A 23 A T The structure of ( A=A)=A 11 for our example matrix is shown in Figure 3. The matrix block A 11 in the rst Schur complement matrix A=A is diagonal [4], [11]. The following result shows that it is worth to form the Schur complement matrix ( A=A)=A 11 from the matrix A=A since no further ll-in appears during the elimination of the block A 11 corresponding to pressure variables p and so we can further reduce the dimension of the system. Theorem 2.1. A 22 A 23 A T Proof. We have the following structural equivalences: Fig. 3. Structural pattern of the Schur complement matrix ( A=A)=A11 for A from Figure 1. From previous Theorem it is also easy to see that right lower block B 22 is block-diagonal with blocks of varying size (depending on number of faces with Neumann conditions in each element) each corresponding to a certain element in the discretization. So in the following we will consider the third Schur complement matrix induced by the block B 22 in the matrix ( A=A)=A 11 : We can prove a similar result to the one given in Theorem 2.1. Therefore, the Schur complement system with the matrix (A=A)=A 11 can be reduced to the Schur complement matrix (( A=A)=A 11 )=B 22 of dimension equal to NIF , without inducing any additional ll-in. Moreover, this can be done using incomplete factorization procedures. Theorem 2.2. Proof. Using Theorem 2.1 we get only in the trivial singular case with jE h we get the desired result Struct(B 11 The following simple corollary gives the number of nonzero elements in the second and third Schur complement matrices ( A=A)=A 11 and (( A=A)=A 11 )=B 22 . We shall use these results later. Corollary 2.1. The number of nonzeros in the triangular part of ( A=A)=A 11 is given by and the number of nonzeros in the triangular part of ( A=A)=A 11 )=B 22 is given by Apart from explicit assembly of the Schur complement matrices or using them implicitly there is another possibility which may be considered { keeping the Schur complements in factorized form. Consider the following decomposition: In contrast to the previous case, where the local numbering of the faces corresponding to the individual elements did not play a role, this is not the case now. Theorem 2.3. Assume that all the elements within the diagonal blocks of the matrix A are nonzero. The ll-in in if the faces with Dirichlet boundary conditions are numbered rst in the local ordering of each nite element. Proof. Because of the block structure of A we can consider the individual nite elements independently. The minimum value of the nonzero count of ^ subsequent rows which correspond to the same nite element is 1P it is easily checked to be minimal in this case. Therefore, from now we assume that within each element we have rst numbered the faces corresponding to Dirichlet boundary conditions, then the interior inter-element faces and nally the faces with Neumann boundary conditions. The matrix (2.19) can be written in the form It is clear that it is more advantageous to keep most of the blocks of (2.20) in the explicit form multiplying the factors directly. A typical example is the block ^ B, which is a diagonal matrix. The main question here is whether we can reduce the system further as in the previous case and at the same time keep the matrix blocks in a factorized form. Unfortunately, there is one basic obstacle. Whereas we are able to embed the structure of A T into the structure of A 22 we cannot in general express in the factorized form as is factor which can be easily computed. We have considered the partially factorized structure (2.20) since it is important from a computational point of view. Using a structural prediction based on such factors is exactly the way how to obtain the sparsity structure of explicit Schur complement matrices 22 in an e-cient way. In our implementations we used a procedure similar to the one from [16] to get these structures. 3. Direct solution of the Schur complement systems. In the following we will discuss the direct solution of the Schur complement systems. Namely, we will concentrate on the system with the matrix (( A=A)=A 11 )=B 22 2 R NIF;NIF : The following theorem gives a bound on the asymptotic work necessary to solve the linear system (1.12), which is dominated by the decomposition of the matrix (( A=A)=A 11 )=B 22 : Theorem 3.1. The number of arithmetic operations to solve the symmetric inde- nite system (1.12) directly via three consecutive block eliminations and using the Cholesky decomposition is O(NIF 2 Proof. We will only give a sketch of the proof here. The work is dominated by the decomposition of B 12 , which has the same nonzero structure as A 22 : Our uniform regular nite element mesh is a well-shaped mesh in a suitable sense (see [19]). The proof of Lemma 2.2 and the statements of Theorem 2.1 and 2.2 imply that the graph G of (( A=A)=A 11 )=B 22 is also the graph of a well-shaped mesh. Namely, it is a bounded-degree subgraph of some overlap graph (see [18], [19]). It was shown in [25] that the upper bound on the second-smallest eigenvalue of the Laplacian matrix of G (the Fiedler value) is of the order O(1=NIF 2=3 using the techniques from [25] we obtain that there exists a O(NIF 2=3 )-size bisector of G. Therefore, G satises the so-called NIF 2=3 -separator theorem: there exist constants such that the vertices of G can be partitioned into sets GA ; GB and the vertex separator GC such that jGA j; jGB j NIF and jGC j NIF 2=3 : Moreover, any subgraph of G satises the NIF 2=3 -separator theorem. The technique of recursive partitioning of G called generalized nested dissection and used to reorder the considered Schur complement matrix provides an elimination ordering with an O(NIF 2 )-bound on the arithmetic work of Cholesky decomposition (see Theorem 6 in [12]). Note that the explicit computation of the matrix (( A=A)=A 11 )=B 22 is necessary in the framework of direct methods. Theorem 3.1 provides a theoretical result which is based on spectral partitioning methods. The reordering algorithms based on the separators obtained by the spectral partitioning techniques and applied recursively within the nested dissection need not necessarily be the best practical approach to get a reasonable matrix reordering. Nevertheless, experimental results with various partitioning schemes show that high quality reorderings can be e-ciently computed in this way (see [7]). Also some other reorderings which combine global procedures (partitioning of large meshes) and local algorithms (like MMD) can provide reasonable strategies to nd a ll-in minimizing permutation. 4. The conjugate gradient method applied to the Schur complement sys- tems. In this section we concentrate on the iterative solution of the Schur complement systems discussed in Section 3. We consider the conjugate gradient method applied to the symmetric positive denite systems with matrices A=A, (( A=A)=A 11 ) and 22 . It is well known that the convergence rate of the conjugate gradient method can be bounded in terms of the condition number of the corresponding Schur complement matrix [9], [6], [26]. We show that the condition number of the matrix A=A is asymptotically the same as the conditioning of the negative part of spectrum of the whole indenite matrix A. Moreover, we prove that condition numbers of the matrices grow like 1=h 2 with respect to the discretization parameter h and they do not deteriorate during the successive eliminations. Based on these results we estimate the number of iteration steps necessary to achieve the prescribed tolerance in error norm reduction. We show that the number of iteration steps necessary to reduce the error norm by the factor of " grows asymptotically like 1=h for all three Schur complement systems. Therefore, the total number of ops in the iterative algorithm can be signicantly reduced due to decrease of the matrix order during the elimination. First, we consider the following theorem. Theorem 4.1. Let 1 be the eigenvalues of the positive denite block A 2 R NA;NA , 1 be the singular values of the matrix block (B C) 2 R NA;NBC . Then for the eigenvalues of the Schur complement Moreover, for the eigenvalues of the positive denite matrix blocks A 22 A 23 A T A 22 A 23 A T The condition number of the Schur complement system matrix A=A then can be bounded by the expression (4. Proof. The positive denite matrix A 1 has the spectrum 0 < 1= 1 1= 1=NA . The rst inclusion in the theorem follows from the following two inequalities 1 NA ((B C)x; (B C)x); Similarly, from the inequalities 1 we obtain the second inclusion. The third part of the proof is completely analogous to the second part. Corollary 4.1. There exist positive constants c 9 and c 10 such that for the spectrum of the Schur complement matrix A=A we have The condition number of the matrix A=A can be bounded as The Schur complement system with positive denite matrix A=A can be solved iteratively by the conjugate gradient method [9] or the conjugate residual method [6]. It is well known that the conjugate gradient method generates the approximate solutions which minimize the energy norm of the error at each iteration step [26], [6]. The closely related conjugate residual method that dier only in the denition of innerproduct, on the other hand, generates the approximate solutions which minimize their residual norm at every iteration [6]. It is also well known fact that there exists so-called peak/plateau connection between these methods [5] showing that there is no signicant dierence in the convergence rate of these methods when measured by the residual norm of an approximate solution. In our paper we use the conjugate gradient method together with the minimal residual smoothing procedure applied on its top to get monotonic residual norms [28]. Applying such technique allows better monitoring of the convergence by residual norm and it is mathematically equivalent to the residual minimizing conjugate residual method [6]. The computational cost of this technique is minimal and it costs only two inner products and one vector update per iteration. In the framework of iterative methods the number of operations in matrix-vector products is what is usually the most important. These products, performed repeatedly in each iteration loop, contribute in a substantial way to the nal e-ciency of iterative solver. When solving the system with Schur complement matrix A=A the number of ops per iteration for an unpreconditioned method is dominated by the matrix vector multiplication with the matrix A=A. Its number of nonzeros was given by Lemma 2.2. Moreover, using the estimates (1.13) and (1.14), the condition number of the Schur complement matrix A=A can be bounded by the term O( 3 Consequently, the number of ops for conjugate gradients necessary to achieve a reduction by " is of order Assuming overestimates we obtain the asymptotic estimate of order O(NE 3 The previous considerations did not take into account the Schur complement systems with matrices (( A=A)=A 11 ) and ((( A=A)=A 11 )=B 22 ). The convergence rate of the iterative conjugate gradient method applied to the second and third Schur complement systems depend analogously on the condition number of the Schur complement matrices [9], [6], [26]. The analysis of the spectrum of the matrix (( A=A)=A 11 ) is given in the following theorem. Theorem 4.2. Let 1 be the eigenvalues of the positive denite block A 2 R NA;NA , 1 be the singular values of the matrix block (B C) 2 R NA;NBC . Then for the spectrum of the Schur complement matrix A=A)=A 11 we have Consequently, the condition number of the matrix ( A=A)=A 11 can be bounded as follows Proof. From the denition of the Schur complement matrix ( A=A)=A 11 and the statement of Theorem 4.1 we have A 22 A 23 A T The bound for the minimal eigenvalue can be obtained considering the following result (see [20], p.201): A 11 A 12 A 13 A T A T A A Then from the interlacing property of the eigenvalue set of symmetric matrix A=A (see e.g. [8]) it follows Considering the previous inequalities we get the lower bound for the minimal eigenvalue of the matrix ( A=A)=A 11 , which completes the proof. We have shown that the condition number of the Schur complement system matrix A=A)=A 11 is bounded by a multiple of the condition number of the matrix A=A. Therefore the number of iteration steps for the conjugate gradient method necessary to reduce the error norm(or after smoothing the residual norm) by some factor is asymptotically the same as before. The complexity of the matrix-vector multiplication is lower and according to Corollary 2.1 is of the order Assuming again the overestimates (NIF we obtain the asymptotic estimate O(NE). The total number of ops for the conjugate gradients or the conjugate residual method necessary to achieve a reduction by the factor " is then again of order O(NE 3 NE). From the statements of Theorem 4.1 and Theorem 4.2 it is clear that the reduction to the Schur complement systems does not aect the asymptotic conditioning of the positive denite matrices A=A and ( A=A)=A 11 : The same is true for the spectral properties of the third Schur complement system with the matrix (( A=A)=A 11 )=B 22 : Since the proof is completely analogous to the proof of Theorem 4.2 we shall present only the following statement (cf. [10], p. 256). Theorem 4.3. The condition number of (( A=A)=A 11 )=B 22 is bounded by the condition number of the matrix ( A=A)=A 11 In the following we present two additional results concerning the the matrix-vector multiplications with Schur complement matrices. Theorem 4.4 compares the number of nonzeros in the Schur complement matrices ( A=A)=A 11 and (( A=A)=A 11 )=B 22 to the number of nonzeros in the original matrix A: Theorem 4.4. The number of nonzero entries in the matrix (( A=A)=A 11 ) or the matrix ((( A=A)=A 11 )=B 22 ) is smaller than the number of nonzeros in the matrix A: Proof. Using the fact that it follows from Lemma 2.1 and Theorem 2.1 that Clearly, the number of nonzeros in the matrix ((( A=A)=A 11 )=B 22 ) is even smaller. Note that the number of nonzeros in the original matrix A can be smaller or larger than the corresponding number of nonzeros in the matrix A=A: Consider now the factorized Schur complement in the form (2.20). It can be shown also that there is no clear winner between the number of oating-point operations to multiply a dense vector by the or the number of operations to get a product of a matrix (( A=A)=A 11 )=B 22 with a dense vector of appropriate dimension, respectively. The result depends on the shape of the domain and its boundary conditions. Never- theless, the following Theorem 4.5 shows that if we do not form the Schur complement explicitly it is worth to use the factorized form (2.19) and the reordering of the Schur complement from Theorem 2.3 instead of its implicit form. Theorem 4.5. Let v be a dense vector. The number of oating-point operations to compute smaller than the number of oating-point operations to computeB @ Proof. Taking into account the local ordering from Theorem 2.3 the dierence between these two quantities can be bounded by 2NIF NNC 0: 5. Numerical experiments. In the following we present numerical experiments which illustrate the results developed in the theoretical part of the paper. Two model potential ow problems (1.1) and (1.2) in a rectangular domain with Neumann conditions prescribed on the bottom and on the top of the domain have been considered. Dirichlet conditions that preserve the nonsingularity of the whole system matrix A were imposed on the rest of the boundary. The choice of boundary conditions in these examples is motivated by our application and it comes from a modelling of a conned aquifer (see [3]) between two impermeable layers. In order to verify the theoretical results derived in previous sections we will restrict our attention rst to the simplest geometrical shape - cubic domain and report the results obtained from a uniformly regular mesh renement. In practical situations, however, relatively thin aquifers with possible cracks in the rock are frequently modelled, and so the number of Neumann conditions may represent a big portion of the whole boundary. As our second model example, we consider a rectangular domain discretized by 6 layers of Model potential uid ow problem - cubic domain Discretization parameters Matrix dimensions h, NE NIF NNC A A=A ( A=A)=A 11 (( A=A)=A 11 )=B 22 1/5, 250 525 100 2125 875 625 525 1/10, 2000 4600 400 17000 7000 5000 4600 1/15, 6750 15975 900 57375 23625 16875 15975 1/20, 16000 38400 1600 136000 56000 40000 38400 1/30, 54000 131400 3600 459000 189000 135000 131400 1/35, 87750 209475 4900 728875 300125 214375 209475 1/40, 128000 313600 6400 1088000 448000 320000 313600 Table Model potential uid ow problem - realistic domain Discretization parameter Matrix dimension 95x95x6 251560 36100 937460 395960 287660 251560 elements in the mesh. As we will see later, the reduction to the third Schur complement proposed in this paper can become even more signicant than for the cubic domain. Prismatic discretizations of domains with NE elements were used [14], [11]. For the cubic domain we have then Discretization parameters h, NE, NIF , NNC, dimension N of the resulting indenite system matrix A and the dimensions of the corresponding Schur complement matrices A=A, ( A=A)=A 11 and (( A=A)=A 11 )=B 22 are given in Table 1 for a cubic domain and in Table 2 for a more realistic domain. We note again that the dierence between dimensions of the second and third Schur complement matrix is signicantly larger in the case of modelling of thin layers that arise regularly in our application. For the example of a cubic domain the spectral properties of the matrix blocks A and (B C) as well as of the whole symmetric indenite matrix A have been investigated. The extremal positive and negative eigenvalues of the matrix A and the extremal singular values of the block (B C) (squared roots of the extremal eigenvalues of the matrix were approximated by a reduction to the symmetric tridiagonal form of the matrix using 1500 steps of the symmetric Lanczos algorithm [8] and by a subsequent Spectral properties of the system matrix and its blocks - problem with a cubic domain matrix blocks spectral properties eigenvalues of the matrix A NE spectrum of A sing. values of (BjC) negative part positive part 2000 [0.33e-2, 0.2e-1] [0.927e-1, 2.64] [-2.64, -0.898e-1] [0.335e-2, 2.64] 16000 [0.66e-2, 0.4e-1] [0.467e-1, 2.64] [-2.64, -0.413e-1] [0.679e-2, 2.65] 54000 [0.99e-2, 0.6e-1] [0.312e-1, 2.65] [-2.64, -0.241e-1] [0.104e-1, 2.65] 128000 [0.13e-1, 0.8e-1] [0.234e-1, 2.65] [-2.64, -0.152e-1] [0.136e-1, 2.65] eigenvalue computation of the resulting tridiagonal matrix using the LAPACK double precision subroutine DSYEV [1]. Extremal eigenvalues of the diagonal matrix block A were computed directly by the LAPACK eigenvalue solver element by element. It can be seen that the computed extremal eigenvalues of the block A are in perfect agreement with the theory (see Table 3). Similarly, we can observe approximately a linear decrease of the computed minimal singular value of the matrix block (B C) with respect to the mesh discretization parameter h. From the computed extremal eigenvalues of the whole indenite system A we can conclude that even if our mesh size parameters h are rather small and give rise to very large system dimensions (see Table 1), they are outside of the asymptotic inclusion set (1.15). Indeed for our example and our mesh size interval we have c 1 =h c 4 , c 2 =h c 4 and with the exception of using Lemma 2.1 in [22], pp. 3-4 (see also [15]) we obtain the inclusion set in the form which is in good agreement with the results in Table 3. Using the same technique we have approximated the extremal eigenvalues of the Schur complement matrices A=A, ( A=A)=A 11 and (( A=A)=A 11 )=B 22 coming from a problem on a cubic domain. From Table 4 it can be seen that the inclusion set for the extremal eigenvalues of the rst Schur complement matrix A=A coincides with the bounds given in Theorem 4.1. We can see that the extremal eigenvalues of the second Schur complement matrix ( A=A)=A 11 are bounded by the extremal eigenvalues of the matrix A=A. Similarly, the extremal eigenvalues of the third Schur complement matrix are bounded by the extremal eigenvalues of the matrix ( A=A)=A 11 . This behaviour is in accordance with the asymptotic bounds given in Theorem 4.2 and Theorem 4.3. The smoothed conjugate gradient method has been applied to the resulting three Schur complement systems (see also the discussion in previous section). Unpreconditioned and also preconditioned versions with the IC(0) preconditioner [23], [24] for the solution of these symmetric positive denite systems have been used. For the solution of Spectral properties of Schur complement matrices - problem with a cubic domain spectral properties of Schur complement matrices 2000 [0.182e1, 0.173e4] [0.251e1, 0.596e3] [0.272e1, 0.596e3] 54000 [0.693e-1, 0.579e3] [0.966e-1, 0.199e3] [0.992e-1, 0.199e3] 128000 [0.293e-1, 0.434e3] [0.409e-1, 0.149e3] [0.417e-1, 0.149e3] the whole indenite system the minimal residual method has been used. For the preconditioned version the positive denite block-diagonal preconditioning with ILUT(0,20) for the decomposition of the block corresponding to constraints (see e.g. [22], [21]) was used. The choice of ILUT(0,20) was motivated by our eort to obtain rather precise factorization with restricted memory requirements which should be close to the full decomposition of the block (B C) T (B C). This preconditioner was found generally better than the in- denite block-diagonal preconditioning with the same ILUT(0,20) decomposition or than the indenite preconditioner discussed in [13] or [21]. The initial approximation x 0 was set to zero, the relative residual norm krnk used as the stopping criterion. For the implementation details of iterative solvers we refer to [6]. Our experiments were performed on an SGI Origin 200 with processor R10000. In Table 5 and Table 6 we consider iteration counts and CPU times in the minimal residual method (unprecondi- tioned/preconditioned) applied to the whole system (1.12) and in the conjugate gradient method (unpreconditioned/preconditioned) applied to the Schur complement systems with the matrices A=A, ( A=A)=A 11 and (( A=A)=A 11 )=B 22 for a model problem with a cubic and more realistic domain, respectively. The dependence of the iteration counts presented in all columns of Table 5 corresponds surprisingly well to the theoretical order O( 3 NE). The convergence behaviour of the smoothed conjugate gradient method applied to the third Schur complement system with the matrix (( A=A)=A 11 )=B 22 for this case is presented in Figure 4. From the results in Table 5 and Table 6 it follows that while the gain from the solution of the third Schur complement system is rather moderate in the case of a cubic domain and in the case of the realistic at domain it becomes more signicant. 6. Conclusions. Successive block Schur complement reduction for the solution of symmetric indenite systems has been considered in the paper. It was shown that due to the particular structure of matrices which arise from mixed-hybrid nite element discretization of the potential uid ow problem, the resulting Schur complement matrices remain sparse. Moreover, their spectral properties do not deteriorate and the iterative conjugate gradient method can be successfully applied. Theoretical bounds for the Number of iterations of the conjugate gradient method - problem with a cubic domain unpreconditioned/preconditioned CG applied to matrix NE A A=A ( A=A)=A 11 (( A=A)=A 11 )=B 22 2000 608/76 154/35 87/32 80/32 6.43/1.56 0.76/0.25 0.30/0.16 0.25/0.14 48.17/11.91 3.86/1.50 1.51/0.92 1.30/1.01 16000 1031/138 288/67 164/63 155/63 54000 1358/188 418/95 234/93 228/93 926.98/218.78 104.76/36.92 39.88/24.04 37.94/23.13 128000 1637/229 546/122 303/122 298/122 Table Number of iterations of the conjugate gradient method - realistic model example unpreconditioned/preconditioned CG applied to matrix NE A A=A ( A=A)=A 11 (( A=A)=A 11 )=B 22 50700 2053/336 810/172 448/166 421/166 86700 2959/403 1042/222 578/214 543/214 132300 3420/447 1272/271 706/262 663/262 iteration number relative residual norms unpreconditioned and smoothed conjugate gradient method applied to ((-A/A)/A 11 )/B 22 Fig. 4. Convergence of the smoothed conjugate gradient method applied to the third Schur complement system convergence rate of this method in terms of the discretization parameters have been developed and tested on a model problem example. Numerical experiments indicate that the given theoretical bounds on the eigenvalue set are realistic not only for the system matrix and its blocks, but also for the Schur complement matrices. The iteration counts for the conjugate gradient method are also in a good agreement with the theoretical predictions. Direct solution of the third Schur complement system is also a possible al- ternative. Nevertheless, its comparison with iterative solvers is outside the scope of this paper. In case of structured grids, a geometric multigrid solver and/or preconditioner for solving the nal Schur complement system can be used. Namely, the stencil from the rst Schur complement which expresses element-element connectivity in the domain (see proof of Lemma 2.2) remains unchanged after the subsequent two reduction and an appropriate method could be based on that. Another approach for the solution of symmetric indenite systems seems to be promising. As was pointed out in [2], the classical null-space algorithm can be imple- mented. QR factorization of the o-diagonal block (B C) is considered and the solution of the indenite system is transformed to the solution of a block lower triangular sys- tem, where the subproblem corresponding to the diagonal block can be solved using the factorization or an iterative conjugate gradient-type algorithm. This approach has the advantage of performing the matrix-vector multiplication by the Q factor using elementary Householder transformations. Although the Q factor may be structurally full, the elementary Householder vectors may be quite sparse. Moreover, a roundo error analysis of the algorithm can be carried out. 7. Acknowledgment . Authors would like to thank Michele Benzi for careful reading of manuscript and anonymous referees for their many useful comments which significantly improved the presentation of the paper. We are indebted to Jir Muzak from the Department of Mathematical Modelling in DIAMO, s.e., Straz pod Ralskem for providing us with a model numerical example for the experimental part of this paper and to Jorg Liesen for giving us the reference [20]. This work was supported by the Grant Agency of the Czech Republic under grant 201/98/P108 and by the grant AS CR A2030706. --R LAPACK User's Guide SIAM The use of QR factorization in sparse quadratic programming. Dynamics of Fluids in Porous Media. Mixed and Hybrid Finite Element Methods. Relations between Galerkin and norm-minimizing iterative methods for solving linear systems Iterative methods for large Geometric mesh partitioning: implementation and exper- iments Matrix Computations. Method of conjugate gradients for solving linear systems. Accuracy and Stability of Numerical Algorithms. Generalized nested dissection Sparse QR factorization with applications to linear least squares problems. Approximate Schur complement preconditioning of the lowest-order nodal discretizations Automatic mesh partitioning Geometric separators for Schur Complement and Statistics. A preconditioned iterative method for saddle point problems. Iterative Methods for Sparse Linear Systems. ILUT: A dual threshold incomplete ILU factorization. Spectral partitioning works: Planar graphs and Parallel iterative solution methods for linear systems arising from discretized PDE's. --TR

indefinite linear systems;preconditioned conjugate residuals;potential fluid flow problem;sparse linear systems;finite element matrices

587322

Sparse Serial Tests of Uniformity for Random Number Generators.

Different versions of the serial test for testing the uniformity and independence of vectors of successive values produced by a (pseudo)random number generator are studied. These tests partition the t-dimensional unit hypercube into k cubic cells of equal volume, generate n points (vectors) in this hypercube, count how many points fall in each cell, and compute a test statistic defined as the sum of values of some univariate function f applied to these k individual counters. Both overlapping and nonoverlapping vectors are considered. For different families of generators, such as linear congruential, Tausworthe, nonlinear inversive, etc., different ways of choosing these functions and of choosing k are compared, and formulas are obtained for the (estimated) sample size required to reject the null hypothesis of independent uniform random variables, as a function of the period length of the generator. For the classes of alternatives that correspond to linear generators, the most efficient tests turn out to have $k \gg n$ (in contrast to what is usually done or recommended in simulation books) and to use overlapping vectors.

Introduction . The aim of this paper is to examine certain types of serial tests for testing the uniformity and independence of the output sequence of general-purpose uniform random number generators (RNGs) such as those found in software libraries. These RNGs are supposed to produce \imitations" of mutually independent random variables uniformly distributed over the interval [0; 1) (i.i.d. U(0; 1), for short). Testing an RNG whose output sequence is U amounts to testing the null hypothesis are i.i.d. U(0; 1)." To approximate this multidimensional uniformity, good RNGs are usually designed (theoretically) so that the multiset t of all vectors rst successive output values, from all possible initial seeds, covers the t-dimensional unit hypercube [0; 1) t very evenly, at least for t up to some t 0 , where t 0 is chosen somewhere between 5 and 50 or so. When the initial seed is chosen randomly, this t can be viewed in some sense as the sample space from which points are chosen at random to approximate the uniform distribution over [0; 1) t . For more background on the construction of RNGs, see, for example, [13, 17, 21, 35]. For large t, the structure of t is typically hard to analyze theoretically. Moreover, even for a small t, one would often generate several successive t-dimensional vectors of the form (u statistical testing then comes into play because the dependence structure of these vectors is hard to analyze theoretically. An excessive regularity of t implies that statistical tests should fail when their sample P. L'Ecuyer and R. Simard, Departement d'Informatique et de Recherche Operationnelle, Universite de Montreal, C.P. 6128, Succ. Centre-Ville, Montreal, H3C 3J7, Canada. e- mail: lecuyer@iro.umontreal.ca and simardr@iro.umontreal.ca. S. Wegenkittl, Institute of Mathematics, University of Salzburg, Hellbrunnerstrasse 34, A-5020 Salzburg, Austria, e-mail: ste@random.mat.sbg.ac.at This work has been supported by the National Science and Engineering Research Council of Canada grants # ODGP0110050 and SMF0169893, by FCAR-Quebec grant # 93ER1654, and by the Austrian Science Fund FWF, project no. P11143-MAT. Most of it was performed while the rst author was visiting Salzburg University and North Carolina State University, in 1997-98 (thanks to Peter Hellekalek and James R. Wilson). sizes approach the period length of the generator. But how close to the period length can one get before trouble begins? Several goodness-of-t tests for H 0 have been proposed and studied in the past (see, e.g., [13, 9, 26, 41] and references therein). Statistical tests can never certify for good an RNG. Dierent types of tests detect dierent types of deciencies and the more diversied is the available battery of tests, the better. A simple and widely used test for RNGs is the serial test [1, 6, 8, 13], which operates as follows. Partition the interval [0; 1) into d equal segments. This determines a partition of [0; 1) t into cubic cells of equal size. Generate nt random numbers U construct the points V and let X j be the number of these points falling into cell j, for has the multinomial distribution with parameters 1=k). The usual version of the test, as described for example in [6, 13, 14] among other places, is based on Pearson's chi-square statistic is the average number of points per cell, and the distribution of X 2 under H 0 is approximated by the chi-square distribution with k 1 degrees of freedom when 5 (say). In this paper, we consider test statistics of the general form where f n;k is a real-valued function which may depend on n and k. We are interested for instance in the power divergence statistic real-valued parameter (by we understand the limit as could also consider - seems unnecessary in the context of this paper. Note that D . The power divergence statistic is studied in [39] and other references given there. A more general class is the '- divergence family, where f n;k (X Other forms of f n;k that we consider are f n;k (where I denotes the indicator function), which the corresponding Y is the number of cells with at least b points, the number of empty cells, and the number of collisions, respectively. We are interested not only in the dense case, where > 1, but also in the sparse case, where is small, sometimes much smaller than 1. We also consider (circular) overlapping versions of these statistics, where U replaced by V i . In a slightly modied setup, the constant n is replaced by a Poisson random variable with mean n. Then, (X a vector of i.i.d. Poisson random variables with mean instead of a multinomial vector, and the distribution of Y becomes easier to analyze because of this i.i.d. property. For large k and n, however, the dierence between the two setups is practically negligible, and our experiments are with A rst-order test observes the value of Y , say y, and rejects H 0 if the p-value is much too close to either 0 or 1. The function f is usually chosen so that p too close to 0 means that the points tend to concentrate in certain cells and avoid the others, whereas p close to 1 means that they are distributed in the cells with excessive uniformity. So p can be viewed as a measure of uniformity, and is approximately a random variable under H 0 if the distribution of Y is approximately continuous. A second-order (or two-level) test would obtain N \independent" copies of Y , say is the theoretical distribution of Y under H 0 , and compare their empirical distribution to the uniform. Such a two-level procedure is widely applied when testing RNGs (see [6, 13, 16, 29, 30]). Its main supporting arguments are that it tests the RNG sequence not only at the global level but also at a local level (i.e., there could be bad behavior over short subsequences which \cancels out" over larger subsequences), and that it permits one to apply certain tests with a larger total sample size (for example, the memory size of the computer limits the values of n and/or k in the serial test, but the total sample size can exceed n by taking N > 1). Our extensive empirical investigations indicate that for a xed total sample size Nn, when testing RNGs, a test with typically more e-cient than the corresponding test with N > 1. This means that for typical RNGs, the type of structure found in one (reasonably long) subsequence is usually found in (practically) all subsequences of the same length. In other words, when an RNG started from a given seed fails spectacularly a certain test, it usually fails that test for most admissible seeds. The common way of applying serial tests to RNGs is to select a few specic generators and some arbitrarily chosen test parameters, run the tests, and check if H 0 is rejected or not. Our aim in this paper is to examine in a more systematic way the interaction between the serial tests and certain families of RNGs. From each family, we take an RNG with period length near 2 e , chosen on the basis of the usual theoretical criteria, for integers e ranging from 10 to 40 or so. We then examine, for dierent ways of choosing k and constructing the points V i , how the p-value of the test evolves as a function of the sample size n. The typical behavior is that takes \reasonable" values for a while, say for n up to some threshold n 0 , then converges to 0 or 1 exponentially fast with n. Our main interest is to examine the relationship between n 0 and e. We adjust (crudely) a regression model of the form log e++ where and are two constants and is a small noise. The result gives an idea of what size (or period length) of RNG is required, within a given family, to be safe with respect to these serial tests for the sample sizes that are practically feasible on current computers. It turns out that for popular families of RNGs such as the linear congruential, multiple recursive, and shift-register, the most sensitive tests choose k proportional to 2 e and yield which means that n 0 is a few times the square root of the RNG's period length. The results depend of course on the choice of f in (1.2) and on how d and t are chosen. For example, for linear congruential generators (LCGs) selected on the basis of the spectral test [6, 13, 24], the serial test is most sensitive when k 2 e , in which case k). These \most e-cient" tests are very sparse ( 1). Such large values of k yield more sensitive tests than the usual ones (for which k 2 e and 5 or so) because the excessive regularity of LCGs really shows up at that level of partitioning. For k 2 e , the partition eventually becomes so ne that each cell contains either 0 or 1 point, and the test loses all of its sensitivity. For xed n, the non-overlapping test is typically slightly more e-cient than the overlapping one, because it relies on a larger amount of independent information. However, the dierence is typically almost negligible (see Section 5.3) and the non-overlapping test requires t times more random numbers. If we x the total number of U i 's that are used, so the non-overlapping test is based on n points whereas the overlapping one is based on nt points, for example, then the overlapping test is typically more e-cient. It is also more costly to compute and its distribution is generally more complicated. If we compare the two tests for a xed computing budget, the overlapping one has an advantage when t is large and when the time to generate the random numbers is an important fraction of the total CPU time to apply the test. In Section 2, we collect some results on the asymptotic distribution of Y for the dense case where k is xed and n ! 1, the sparse case where both and the very sparse case where n=k ! 0. In Section 3 we do the same for the overlapping setup. In Section 4 we brie y discuss the e-ciency of these statistics for certain classes of alternatives. Systematic experiments with these tests and certain families of RNGs are reported in Section 5. In Section 6, we apply the tests to a short list of RNGs proposed in the literature or available in software libraries and widely used. Most of these generators fail miserably. However, several recently proposed RNGs are robust enough to pass all these tests, at least for practically feasible sample sizes. 2. Power Divergence Test Statistics for Non-Overlapping Vectors. We brie y discuss some choices of f n;k in (1.2) which correspond to previously introduced tests. We then provide formulas for the exact mean and variance, and limit theorems for the dense and sparse cases. 2.1. Choices of f n;k . Some choices of f n;k are given in Table 2.1. In each case, Y is a measure of clustering: It tends to increase when the points are less evenly distributed between the cells. The well-known Pearson and loglikelihood statistics, are both special cases of the power divergence, with respectively [39]. H is related to G 2 via the relation 2). The statistics N b , W b , and C count the number of cells that contain exactly b points (for b 0), the number of cells that contain at least b points (for b 1), and the number of collisions (i.e., the number of times a point falls in a cell that already has a point in respectively. They are related by N and 2.2. Mean and Variance. Before looking at the distribution of Y , we give expressions for computing its exact mean and variance under H 0 . If the number of points is xed at n, Denoting one obtains after some algebraic manipulations: x Table Some choices of f n;k and the corresponding statistics. divergence loglikelihood negative entropy number of cells with exactly b points number of cells with at least b points number of empty cells number of collisions x x x min(n x y (f(x) )(f(y) Although containing a lot of summands, these formulas are practical in the sparse case since for the Y 's dened in Table 2.1, when n and k are large and small, only the terms for small x and y in the above sums are non-negligible. These terms converge to 0 exponentially fast as a function of x y, when x rst two moments of Y are then easy to compute by truncating the sums after a small number of terms. For example, with 1000, the relative errors on E[H ] and are less than the sums are stopped at of 1000, and if the sums are stopped at similar behavior is observed for the other statistics. The expressions (2.1) and (2.2) are still valid in the dense case, but for larger , more terms need to be considered. Approximations for the mean and variance of D - when 1, with error terms in o(1=n), are provided in [39], Chapter 5, page 65. In the Poisson setup, where n is the mean of a Poisson random variable, the X j are i.i.d. Poisson() and the expressions become 2.3. Limit Theorems. The limiting distribution of D - is a chi-square in the dense case and a normal in the sparse case. Two-moment-corrected versions of these 6 PIERRE L'ECUYER, RICHARD SIMARD, AND STEFAN WEGENKITTL results are stated in the next proposition. This means that D (C) - and D (N) - in the proposition have exactly the same mean and variance as their asymptotic distribution (e.g., 0 and 1 in the normal case). Read and Cressie [39] recommend this type of standardization, which tends to be closer to the asymptotic distribution than a standardization by the asymptotic mean and variance. The two-moment corrections become increasingly important when - gets away from around 1. The mean and variance of D - can be computed as explained in the previous subsection. Another possibility would be to correct the distribution itself, e.g., using Edgeworth-type expansions [39], page 68. This gives extremely complicated expressions, due in part to the discrete nature of the multinomial distribution, and the gain is small. Proposition 2.1. For - > 1, the following holds under H 0 . (i) [Dense case] If k is xed and n !1, in the multinomial setup convergence in distribution, and is the chi-square distribution with k 1 degrees of freedom. In the Poisson setup, D (C) (ii) [Sparse case] For both the multinomial and Poisson setups, if is the standard normal distribution. Proof. For the multinomial setup, part (i) can be found in [39], page 46, whereas part (ii) follows from Theorem 1 of [11], by noting that all the X j 's here have the same distribution. The proofs simplify for the Poisson setup, due to the independence. The p n=k are i.i.d. and asymptotically N(0; 1) in the dense case, so their sum of squares, which is X 2 , is asymptotically 2 (k). We now turn to the counting random variables N b , W b , and C. These are not approximately chi-square in the dense case. In fact, if xed k, it is clear that xed b. This implies that W b ! k and random variables are all degenerate. For the Poisson setup, each X i is Poisson(), so p b b 0 and N b is BN(k; p b ), a binomial with parameters k and p b . If k is large and p b is small, N b is thus approximately Poisson with (exact) mean for b 0: The next result covers other cases as well. Proposition 2.2. For the Poisson or the multinomial setup, under H 0 , suppose that k !1 and n !1, and let 1 , positive constants. one also has C (ii) For Proof. In (i), since 0, one has for the Poisson case E[N b+1 ]=E[N b 0. The relative contribution of W b+1 to the sum W (a sum of correlated Poisson random variables) is then negligible compared with that of N b , so N b and W b have the same asymptotic distribution (this follows from Lemma 6.2.2 of [2]). Likewise, under these conditions with 2, C has the same asymptotic distribution as N 2 , because e 1. For the multinomial setup, it has been shown (see [2], Section 6.2) that N b and W b , for b 2, are asymptotically Poisson(kp b ) when ! 0, the same as for the Poisson setup. The same argument as for W 2 applies for C, using again their Lemma 6.2.2, and this proves (i). For for the Poisson setup, we saw already that N 0 is asymptotically For the multinomial case, the same result follows from Theorem 6.D of [2], and this proves (ii). Part (iii) is obtained by applying Theorem 1 of [11]. The exact distributions of C and N 0 under H 0 , for the multinomial setup, are given by where the are the Stirling numbers of the second kind (see [13], page 71, where an algorithm is also given to compute all the non-negligible probabilities in time O(n log n)). In our implementation of the test based on C, we used the Poisson approximation for 1=32, the normal approximation for > 1=32 and n > 2 15 , and the exact distribution otherwise. 3. Overlapping vectors. For the overlapping case, let X (o) t;j be the number of overlapping vectors j. Now, the formulas (2.1) and (2.2) for the mean and variance, and the limit theorems in Propositions 2.1 and 2.2, no longer stand. The analysis is more di-cult than for the disjoint case because in general P [X (o) depends on i and P [X (o) depends on the pair (i; in a non-trivial way. Theoretical results have been available in the overlapping multinomial setup, for the Pearson statistic in the dense case. Let and let X 2 be the equivalent of X 2 for the overlapping vectors of dimension t 1: Consider the statistic ~ Good [8] has shown that E[X 2 exactly (see his Eq. (5) and top of page 280) and that when page 284). This setup, usually with n=k 5 or so, is called the overlapping serial test or the m-tuple test in the literature and has been used previously to test RNGs (e.g., [1, 29, 30]). The next proposition generalizes the result of Good to the power divergence statistic in the dense case. Further generalization is given by Theorem 4.2 of [43]. Proposition 3.1. Let the power divergence statistic for the t-dimensional overlapping vectors, and dene ~ in the multinomial setup, if - > 1, k is xed, and n !1, ~ Proof. The result is well-known for 1. Moreover, a Taylor series expansion of D -;(t) in powers of X (o) easily shows that D probability as Theorem A6.1). Therefore, ~ D -;(t) has the same asymptotic distribution as ~ D 1;(t) and this completes the proof. For the sparse case, where k; our simulation experiments support the conjecture that ~ ~ The overlapping empty-cells-count test has been discussed in a heuristic way in a few papers. For calls it the overlapping pairs sparse occupancy (OPSO) and suggests a few specic parameters, without providing the underlying the- ory. Marsaglia and Zaman [32] speculate that N 0 should be approximately normally distributed with mean ke and variance ke (1 3e ). This make sense only if is not too large or not too close to zero. We studied empirically this approximation and found it reasonably accurate only for 2 5 (approximately). The approximation could certainly be improved by rening the variance formula. Proposition 2.2 (i) and (ii) should hold in the overlapping case as well. Our simulation experiments indicate that the Poisson approximation for C is very accurate for (say) < 1=32, and already quite good for 1, when n is large. 4. Which Test Statistic and What to Expect?. The LFSR, LCG, and MRG generators in our lists are constructed so that their point sets t over the entire period are superuniformly distributed. Thus, we may be afraid, if k is large enough, that very few cells (if any) contain more than 1 point and that D - , C, N 0 , N b and W b for b 2 are smaller than expected. In the extreme case where assuming that the distribution of C under H 0 is approximately Poisson with mean n 2 =(2k), the left p-value of the collision test is . For a xed number of cells, this p-value approaches 0 exponentially fast in the square of the sample size n. For example, k, and 16 k, respectively. Assuming that k is near the RNG's period length, i.e., means that the test starts to fail abruptly when the sample size exceeds approximately 4 times the square root of the period length. As we shall see, this is precisely what happens for certain popular classes of generators. If we use the statistic b instead of C, in the same situation, we have and the sample size required to obtain a p-value less than a xed (small) constant is 2. In this setup, C and N 2 are equivalent to W 2 , and choosing e-cient test. Suppose now that we have the opposite: Too many collisions. One simple model of this situation is the alternative are i.i.d. uniformly distributed over boxes, the other k k 1 boxes being always empty." Under H 1 , W b is approximately Poisson with mean is large and 1 is small) instead of Therefore, for a given 0 , and x 0 such that the power of the test at level 0 is where x 0 depends on b. When b increases, for a xed 0 , x 0 decreases and 1 decreases as well if n=k 1 maximizes the power unless n=k 1 is large. In fact the test can have signicant power only if 1 exceeds a few units (otherwise, with large probability, one has W not rejected). This means which can be approximated by O(k (b 1)=b is reasonably large. Then, is the best choice. If k 1 is small, 1 is maximized (approximately) by taking The alternative H 1 just discussed can be generalized as follows: Suppose that the have a probability larger than 1=k, while the other k k 1 cells have a smaller probability. H 1 is called a hole (resp., peak , split) alternative if k 1 =k is near 1 (resp., near 0, near 1/2). We made extensive numerical experiments regarding the power of the tests under these alternatives and found the following. Hole alternatives can be detected only when n=k is reasonably large (dense case), because in the sparse case one expects several empty cells anyway. The best test statistics to detect them are those based on the number of empty cells N 0 , and D - with - as small as possible (e.g., 0). For a peak alternative, the power of D - increases with - as a concave function, with a rate of increase that typically becomes very small for - larger than 3 or 4 (or higher, if the peak is very narrow). The other test statistics in Table 2.1 are usually not competitive with D 4 (say) under this alternative, except for W b which comes close when b n=k 1 (however it is hard to choose the right b because k 1 is generally unknown). The split alternative with the probability of the k k 1 low-probability cells equal to 0 is easy to detect and the collision test (using C or W 2 ) is our recommendation. The power of D - is essentially the same as that of C and W 2 , for most -, because E[W 3 ] has a negligible value, which implies that there is almost a one-to-one correspondence between C, W 2 , and D - . However, with the small n that su-ces for detection in this situation, E[W 2 ] is small and the distribution of D - is concentrated on a small number of values, so neither the normal nor the chi-square is a good approximation of its distribution. Of course, the power of the test would improve if the high-probability cells were aggregated into a smaller number of cells, and similarly for the low-probability cells. But to do this, one needs to know where these cells are a priori . These observations extend (and agree with) those made previously by several authors (see [39] and references therein), who already noted that for D - , the power decreases with - for a hole alternative and increases with - for a peak alternative. This implies in particular that G 2 and H are better [worse] test statistics than X 2 to detect a hole [a peak]. In the case of a split alternative for which the cell probabilities are only slightly perturbed, X 2 is optimal in terms of Pitman's asymptotic e-ciency is optimal in terms of Bahadur's e-ciency (see [39] for details). 5. Empirical Evaluation for RNG Families. 5.1. Selected Families of RNGs. We now report systematic experiments to assess the eectiveness of serial tests for detecting the regularities in specic families of small RNGs. The RNG families that we consider are named LFSR3, GoodLCG, BadLCG2, MRG2, CombL2, InvExpl. Within each family, we constructed a list of specic RNG instances, with period lengths near 2 e for (integer) values of e ranging from 10 to 40. These RNGs are too small to be considered for serious general purpose softwares, but their study gives good indication about the behavior of larger instances from the same families. At step n, a generator outputs a number un 2 [0; 1). The LFSR3s are combined linear feedback shift register (LFSR) (or Tausworthe) generators with three components, of the form where means bitwise exclusive-or, and are constant parameters selected so that the k j are reasonably close to each other, and the sequence fung has period length (2 k1 1)(2 k2 1)(2 k3 1) and is maximally equidistributed (see [19] for the denition and further details about these generators). The GoodLCGs are linear congruential generators (LCGs), of the form where m is a prime near 2 e and a is selected so that the period length is m 1 and so that the LCG has an excellent behavior with respect to the spectral test (i.e., an excellent lattice structure) in up to at least 8 dimensions. The BadLCG2s have the same structure, except that their a is chosen so that they have a mediocre lattice structure in 2 dimensions. More details and the values of a and m can be found in [24, 26]. The MRG2 are multiple recursive generators of order 2, of the form period length m 2 1, and excellent lattice structure as for the GoodLCGs [17, 21]. The CombL2s combine two LCGs as proposed in [15]: so that the combined generator has period length (m 1 1)(m 2 1)=2 and an excellent lattice structure (see [28] for details about that lattice structure). InvExpl denotes a family of explicit inversive nonlinear generators of period length m, dened by where m is prime and (an) 1 mod 5.2. The Log-p-values. For a given test statistic Y taking value y, let the log-p-value of the test as For example, means that the right p-value is between 0.01 and 0.001. For a given class of RNGs, given Y , t, and a way of choosing k, we apply the test for dierent values of e and with sample size e+ , for where the constant is chosen so that the test starts to fail at approximately the same value of for all (or most) e. More specically, we dene ~ (resp. ) as the smallest values of for which the absolute log-p-value satises j'j 2 (resp. j'j 14) for a majority of values of e. These thresholds are arbitrary. 5.3. Test Results: Examples and Summary. Tables 5.1 and 5.2 give the log- p-values for the collision test applied to the GoodLCGs and BadLCG2s, respectively, in dimensions, with . Only the log- p-values ' outside of the set f1; 0; 1g, which correspond to p-values less than 0:01, are displayed. The symbols and ! mean ' 14 and ' 14, respectively. The columns not shown are mostly blank on the left of the table and lled with arrows on the right of the table. The small p-values appear with striking regularity, at about the same for all e, in each of these tables. This is also true for other values of e not shown in the table. One has ~ Table 5.1, while ~ in Table 5.2. The GoodLCGs fail because their structure is too regular (the left p-values are too small because there are too few collisions), whereas the BadLCG2s have the opposite behavior (the right p-values are too small because there are too many collisions; their behavior correspond to the split alternative described in Section 4). Table 5.3 gives the values of ~ and for the selected RNG families, for the collision test in 2 and 4 dimensions. All families, except InvExpl, fail at a sample size proportional to the square root of the period length . At 1=2 , the left or right p-value is less than 10 14 most of the time. The BadLCG2s in 2 dimensions are the rst to fail: They were chosen to be particularly mediocre in 2 dimensions and the test detects it. Apart from the BadLCG2s, the generators always fail the tests due to excessive regularity. For the GoodLCGs and LFSR3s, for example, there was never a cell with more than 2 points in it. For the LFSR3s, we distinguish two cases: One where d was chosen always odd and one where it was always the smallest power of 2 such that . In the latter case, the number of collisions is always 0, since no cell contains more than a single point over the entire period of the generator, as a consequence of the \maximal equidistribution" property of these generators [19]. The left p-values then behave as described at the beginning of Section 4. The InvExpl resist the tests until after their period length is exhausted. These generators have their point set t \random-looking" instead of very evenly distributed. However, they are much slower than the linear ones. We applied the power divergence tests with 4, and in most cases the p-values were very close to those of the collision test. In fact, when no cell count which we have observed frequently), there is a one-to-one correspondence between the values of C and of D - for all - > 1. Therefore, all these statistics should have similar p-values if both E[W 3 ] and the observed value of W 3 are small (the very sparse situation). For the overlapping versions of the tests, the values of , and are exactly the same as those given in Table 5.3. This means that the Table The log-p-values ' for the GoodLCGs with period length 2 e , for the collision test (based on C), in cells, and sample size . The table entries give the values of '. The symbols and ! mean ' 14 and ' 14, respectively. Here, we have ~ and 22 3 11 26 2 6 28 2 overlapping tests are more e-cient than the non-overlapping ones, because they call the RNG t times less. We applied the same tests with smaller and larger numbers of cells, such as found that ~ and increase when moves away from 2 e . A typical example: For the GoodLCGs with 6, 5, and 7 for the four choices of k given above, respectively, whereas . The classical way of applying the serial test for RNG testing uses a large average number of points per cell (dense case). We applied the test based on X 2 to the GoodLCGs, with k n=8, and found empirically This means that the required sample size now increases as O( 2=3 ) instead of O( 1=2 ) as before; i.e., the dense setup with the chi-square approximation is much less e-cient than the sparse setup. We observed the same for D - with other values of - and other values of t, and a similar behavior for other RNG families. For the results just described, t was xed and d varied with e. We now x (i.e., we take the rst two bits of each number) and vary the dimension as Table 5.4 gives the results of the collision test in this setup. Note the change in for the GoodLCGs and BadLCG2s: The tests are less sensitive for these large values of t. We also experimented with two-level tests, where a test of sample size n is replicated times independently. For the collision test, we use the test statistic C T , the total number of collisions over the N replications, which is approximately Poisson with mean Nn 2 e n=k =(2k) under H 0 . For the power divergence tests, we use as test statistics the sum of values of D (N) - and of D (C) - , which are approximately N(0; N) and 2 (N(k 1)) under H 0 , respectively. We observed the following: The power Table The log-p-values ' for the collision test, with the same setup as in Table 5.1, but for the BadLCG2 generators. Here, ~ 22 26 28 Table Collision tests for RNG families, in t dimensions, with k 2 e . Recall that ~ (resp. ) is the smallest integer for which j'j 2 (resp. j'j 14) for a majority of values of e, in tests with sample e+ . RNG family LFSR3, d power of of a test with (N; n) is typically roughly the same as that of the same test at level one with sample size n N . Single-level tests thus need a smaller total sample size than the two-level tests to achieve the same power. On the other hand, two-level tests are justied when the sample size n is limited by the memory size of the computer at hand. (For n k, the counters X j are implemented via a hashing 14 PIERRE L'ECUYER, RICHARD SIMARD, AND STEFAN WEGENKITTL Table Collision tests with divisions in each dimension and dimensions. Generators ~ CombL2 technique, for which the required memory is proportional to n instead of k). Another way of doing a two-level test with D - is to compute the p-values for the N replicates and compare their distribution with the uniform via (say) a Kolmogorov-Smirnov or Anderson-Darling goodness-of-t test. We experimented extensively with this as well and found no advantage in terms of e-ciency, for all the RNG families that we tried. 6. What about real-life LCGs?. From the results of the preceding section one can easily predict, conservatively, at which sample size a specic RNG from a given family will start to fail. We verify this with a few commonly used RNGs, listed in Table 6.1. (Of course, this list is far from exhaustive). Table List of selected generators. LCG1. LCG with LCG2. LCG with LCG3. LCG with LCG4. LCG with LCG5. LCG with LCG6. LCG with LCG7. LCG with LCG8. LCG with LCG9. LCG with RLUX. RANLUX with WEY1. Nested Weyl with (see [10]). WEY2. Shued nested Weyl with (see [10]). CLCG4. Combined LCG of [25]. CMRG96. Combined MRG in Fig. 1 of [18]. CMRG99. Combined MRG in Fig. 1 of [23]. Generators LCG1 to LCG9 are well-known LCGs, based on the recurrence x at step i. LCG1 and LCG2 are recommended by Fishman [7] and a FORTRAN implementation of LCG1 is given by Fishman [6]. LCG3 is recommended in [14], among others, and is used in the SIMSCRIPT II.5 and INSIGHT simulation languages. LCG4 is in numerous software systems, including the IBM and Macintosh operating systems, the Arena and SLAM II simulation languages (note: the Arena RNG has been replaced by CMRG99 after we wrote this paper), MATLAB, the IMSL library (which also provides LCG1 and Table The log-p-values for the collision test in cells, and sample size m. Generator Table The log-p-values for the two-level collision test (based on C T ) in cells, sample size for each replication, and replications. Generator LCG5), the Numerical Recipes [38], etc., and is suggested in several books and papers (e.g., [3, 36, 40]). LCG6 is used in the VAX/VMS operating system and on Convex computers. LCG5 and LCG9 are the rand and rand48 functions in the standard libraries of the C programming language [37]. LCG7 is taken from [6] and LCG8 is used in the CRAY system library. LCG1 to LCG4 have period length 2 31 2, LCG5, LCG6, AND LCG9 have period length m, and LCG7 and LCG8 have period length RLUX is the RANLUX generator implemented by James [12], with luxury level 24. At this luxury level, RANLUX is equivalent to the subtract-with-borrow generator with modulus 43 and proposed in [31] and used, for example, in MATHEMATICA (according to its documentation). WEY1 is a generator based on the nested Weyl sequence dened by (see [10]). WEY2 implements the shued nested Weyl sequence proposed in dened by CLCG4, CMRG96, and CMRG99 are the combined LCG of [25], the combined MRG given in Figure 1 of [18], and the combined MRG given in Figure 1 of [23]. Table 6.2 gives the log-p-values for the collision test in two dimensions, for LCG1 to LCG6, with k m and m. As expected, suspect values start to appear at sample size n 4 these LCGs are denitely rejected with n m. LCG4 has too many collisions whereas the others have too few. By extrapolation, LCG7 to LCG9 are expected to start failing with n around 2 26 , which is just a bit more than what the memory size of our current computer allowed when we wrote this paper. However, we applied the two-level collision test with . Here, the total number of collisions C T is approximately Poisson with mean . The log-p-values are in Table 6.3. With a total sample size of 32 2 24 , LCG7 and LCG8 fail decisively; they have too few collisions. We also tried 4, and the collision test with overlapping, and the results were similar. We tested the other RNGs (the last 5 in the table) for several values of t ranging from 2 to 25. RLUX passed all the tests for t 24 but failed spectacularly in 25 dimensions. With the log-p-value for the collision test is are 239 collisions, while E[CjH 0 ] 166). For a two-level test with the total number of collisions was C much more than This result is not surprising, because for this generator all the points V i in 25 dimensions or more lie in a family of equidistant hyperplanes that are 1= 3 apart (see [20, 42]). Note that RANLUX with a larger value of L passes these tests, at least for t 25. WEY1 passed the tests in 2 dimensions, but failed spectacularly for all t 3: The points are concentrated in a small number of boxes. For example, with a sample size as small as (' 14). WEY2, CLCG4, CMRG96, and CMRG99 passed all the tests that we tried. 7. Conclusion. We compared several variants of serial tests to detect regularities in RNGs. We found that the sparse tests perform better than the usual (dense) ones in this context. The choice of the function f n;k does not seem to matter much. In particular, collisions count, Pearson, loglikelihood ratio, and other statistics from the power divergence family perform approximately the same in the sparse case. The overlapping tests require about the same sample size n as the non-overlapping ones to reject a generator. They are more e-cient in terms of the quantity of random numbers that need to be generated. It is not the purpose of this paper to recommend specic RNGs. For that, we refer the reader to [22, 23, 27, 33], for example. However, our test results certainly eliminate many contenders. All LCGs and LFSRs fail these simple serial tests as soon as the sample size exceeds a few times the square root of their period length, regardless of the choice of their parameters. Thus, when their period length is less than 2 50 or so, which is the case for the LCGs still encountered in many popular software products, they are easy to crack with these tests. These small generators should no longer be used. Among the generators listed in Table 6.1, only the last four pass the tests described in this paper, with the sample sizes that we have tried. All others should certainly be discarded. --R Oxford Science Publica- tions A Guide to Simulation Inversive congruential pseudorandom numbers: A tutorial The serial test for sampling numbers and other tests for randomness A Guide to Chi-Squared Testing Pseudorandom number generator for massively parallel molecular-dynamics simulations Asymptotic normality and e-ciency for certain goodness-of- t tests RANLUX: A Fortran implementation of the high-quality pseudorandom number generator of Luscher The Art of Computer Programming Simulation Modeling and Analysis A random number generator based on the combination of four LCGs Selection criteria and testing An object-oriented random-number package with many long streams and substreams Structural properties for two classes of combined random number generators Inversive and linear congruential pseudorandom number generators in empirical tests A current view of random number generators A new class of random number generators Asymptotic divergence of estimates of discrete distri- butions Good ones are hard to The Standard C Library Portable random number generators Thoughts on pseudorandom number generators Tests for the uniform distribution On the add-with-carry and subtract-with-borrow random number generators --TR --CTR Makoto Matsumoto , Takuji Nishimura, Sum-discrepancy test on pseudorandom number generators, Mathematics and Computers in Simulation, v.62 n.3-6, p.431-442, 3 March Peter Hellekalek , Stefan Wegenkittl, Empirical evidence concerning AES, ACM Transactions on Modeling and Computer Simulation (TOMACS), v.13 n.4, p.322-333, October Pierre L'Ecuyer , Jacinthe Granger-Pich, Combined generators with components from different families, Mathematics and Computers in Simulation, v.62 n.3-6, p.395-404, 3 March Pierre L'Ecuyer, Software for uniform random number generation: distinguishing the good and the bad, Proceedings of the 33nd conference on Winter simulation, December 09-12, 2001, Arlington, Virginia

random number generation;collision test;goodness-of-fit;m-tuple test;multinomial distribution;OPSO;serial test

587368

Preconditioners for Ill-Conditioned Toeplitz Systems Constructed from Positive Kernels.

In this paper, we are interested in the iterative solution of ill-conditioned Toeplitz systems generated by continuous nonnegative real-valued functions f with a finite number of zeros. We construct new w-circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f * KN with a suitable positive reproducing kernel KN. By the restriction to positive kernels we obtain positive definite preconditioners. Moreover, if f has only zeros of even order $\le 2s$, then we can prove that the property $ \int_{-\pi}^{\pi} t^{2k} K_N (t) \, \mbox{d} t \le C N^{-2k} $ $(k=0,\hspace*{1.5pt}\ldots,s)$ of the kernel is necessary and sufficient to ensure the convergence of the PCG method in a number of iteration steps independent of the dimension N of the system. Our theoretical results were confirmed by numerical tests.

Introduction In this paper, we are concerned with the iterative solution of sequences of \mildly" ill{ conditioned Toeplitz systems are positive denite Hermitian Toeplitz matrices generated by a continuous non{negative function f which has only a nite number of zeros. Often these systems are obtained by discretization of continuous problems (partial dierential equation, integral equation with weakly singular kernel) and the dimension N is related to the grid parameter of the discretization. For further applications see [12] and the references therein. Iterative solution methods for Toeplitz systems, in particular the conjugate gradient method (CG{method), have attained much attention during the last years. The reason for this is that the essential computational eort per iteration step, namely the multiplication of a vector with the Toeplitz matrix AN , can be reduced to O(N log N) arithmetical operations by fast Fourier transforms (FFT). However, the number of iteration steps depends on the distribution of the eigenvalues of AN . If we allow the generating function f to have isolated zeros, then the condition numbers of the related Toeplitz matrices grow polynomial with N and the CG{method converges very slow [8, 28, 45]. Therefore, the really task consists in the construction of suitable preconditioners M N of AN so that the number of iteration steps of the corresponding preconditioned CG{method (PCG{method) becomes independent of N . Here it is useful to recall a result of O. Axelsson [1, p. 573] relating the spectrum of the coe-cient matrix to the number of iteration steps to achieve a prescribed precision: Theorem 1.1. Let A be a positive denite Hermitian (N; N){matrix which has p and q isolated large and small eigenvalues, respectively: Let dxe denote the smallest integer x. Then the CG{method for the solution of requires at most iteration steps to achieve precision , i.e. where jjxjj A := denotes the numerical solution after n iteration steps. In literature two kinds of preconditioners were mainly exploited, namely banded Toeplitz matrices and matrices arising from a matrix algebra AON := f O 0 where ON denotes a unitary matrix. For another approach by multigrid methods see for example [23]. Various banded Toeplitz preconditioners were examined [10, 5, 40, 36, 41]. It was proved that the corresponding PCG{methods converge in a number of iteration steps independent of N . However, there is the signicant constraint that the cost per iteration of the proposed procedure should be upper-bounded by O(N log N ). This implies some conditions on the growth of the bandwidth of the banded Toeplitz preconditioners [41]. The above constraint is trivially fullled if we chose preconditioners from matrix algebras, where the unitary matrix ON has to allow an e-cient multiplication with a vector in O(N log N) arithmetical operations. Up to now, the only preconditioners of the matrix algebra class which ensure the desired convergence of the corresponding PCG{method are the preconditioners proposed in [31, 25]. Unfortunately, the construction of these preconditioners requires the explicit knowledge of the generating function f . Extensive examinations were done with natural and optimal Tau preconditioners [6, 3]. Only for su-ciently smooth functions, where the necessary smoothness depends on the order of the zeros of f , the natural Tau preconditioners become positive denite and lead to the desired location of the eigenvalues of the preconditioned matrices. The optimal Tau preconditioner is in general a bad choice if f has zeros of order > 2. The reason for this will become clear in the following sections. In this paper, we combine our approach in [31] with the approximation of f by its convolution with a reproducing kernel KN . The kernel approach was given in [15] for positive generating functions. Interesting tests with B{spline kernels were performed by R. Chan et al. in [14]. The advantage of the kernel approach is that it does not require the explicit knowledge of the generating function. However, only for our theoretical proofs we need some knowledge about the location of the zeros of the generating function f . See remarks at the end of this section. We restrict our attention to positive kernels. This ensures that our preconditioners are positive denite. Suppose that f has only zeros of even order 2s. Then we prove that under the \moment condition" Z on the kernels KN , the eigenvalues of M 1 N AN are contained in some interval [a; b] (0 < a b < 1) except for a xed number (independent of N) of eigenvalues falling into [b; 1) so that PCG converges in O(1) steps. Note that the above kernel property with su-ciently smooth f the Jackson result denotes the modulus of continuity. On the other hand, the classical saturation result of P. P. Korovkin [29, 21] states that we cannot expect a convergence speed of jjf KN f jj 1 better than N 2 even in the presence of very regular functions f . This paper is organized as follows: In Section 2, we introduce our w{circulant positive denite preconditioners. We show how the corresponding PCG{method can be implemented with only O(N) arithmetical operations per step more than the original CG{method. Section 3 is concerned with the location of the eigenvalues of the preconditioned matrices. We will see that under some assumptions on the kernel the number of CG{iterations is independent of N . Special kernels as Jackson kernels and B{spline kernels are considered in Section 5. In Section 6, we sketch how our ideas can be extended to (real) symmetric Toeplitz matrices with trigonometric preconditioners and to doubly symmetric block Toeplitz matrices with Toeplitz blocks. Finally, Section 7 contains numerical results. After sending our manuscript to SIAM J. Sci. Comput., R. H. Chan informed us that his group has got similar results as in our preprint. See [16] and for a rened version [17]. The construction of circulant preconditioners of R. H. Chan et al. is based on Jackson kernels and the proofs are dierent from ours. In [16], the authors prove convergence of the corresponding PCG{method in O(log N) iteration steps. By a trick (see [16, Theorem 4.2]), which can also be applied to our w{circulant preconditioners, the R. H. Chan et al. need no knowledge about the location of the zeros of f . 2 Preconditioners from kernels Let C 2 denote the Banach space of 2{periodic real{valued continuous functions with norm We are interested in the solution of Hermitian Toeplitz systems a generated by a non{negative function f 2 C 2 which has only a nite number of zeros. By [10], the matrices AN (f) are positive denite such that (2.1) can be solved by the CG{method. Unfortunately, since the generating function f 2 C 2 has zeros, the related Toeplitz matrices are asymptotically ill{conditioned and the CG{method converges very slow. To accelerate the convergence of the CG{method, we are looking for suitable preconditioners of AN , where we do not suppose the explicit knowledge of the generating function f . To reach our aim, we use reproducing kernels. This method was originally proposed for Toeplitz matrices arising from positive functions f 2 C 2 in [15]. In [14], R. Chan et al. showed by numerical tests that preconditioners from special kernels related to B{splines can improve the convergence of the CG{method also if f 0 has zeros of various order. A theoretical proof of R. Chan's results was open up to now. In this paper, we restrict our attention to even trigonometric polynomials KN c N;k cos and KN 0; then KN is called a positive (trigonometric) kernel. As main examples of such kernels we consider generalized Jackson polynomials and B{spline kernels in Section 4. For denote the convolution of f with KN , i.e. or equivalently in the Fourier domain a k (f)c N;k e ikx : (2.5) We consider so{called reproducing kernels KN (N 2 N) with the property that lim kf f N k for all f 2 C 2 . We chose grids GN (N 2 N) consisting of equispaced nodes xN;l such that f(x N;l 1. Note that the choice of the grids requires some preliminary information about the location of the zeros of f . By a trick (cf. [16]) this restriction can be neglected if we accept some more outlyers. We consider matrices of the with e 2ijk=N diag (f(x N;l Obviously, the matrices M N can be written as ~ a 0 ~ aN 1 e iNwN ~ a 1 e iNwN ~ a 1 ~ a 0 with ~ a a k (f) := 1 These are ( e iNwN ){circulant matrices (see [20]). In particular, we obtain circulant matrices for skew{circulant matrices for N . As preconditioners for (2.1), we suggest matrices of the form with suitable positive reproducing kernels KN . By (2.5), the construction of these preconditioners requires only the knowledge of the Toeplitz matrices AN . It is not necessary to know the generating function f explicitly. However, for the theoretical results in this paper, we must have some information about the location of the zeros of f . Note that by a trick in [16] this information is also super uous. Here we point out that the auxiliary nontrivial problem of nding some crucial analytic properties of the generating function f has been treated and partially solved in [40]. Moreover, our preconditioners have the following desirable properties: 1. Since f 0 with a nite number of zeros and KN is a positive kernel, it follows by (2.4) that f N > 0. Thus, the matrices M N (f N ) are positive denite. 2. In the following section, we will prove that under certain conditions on the kernels KN the eigenvalues of M 1 N AN are bounded from below by a positive constant independent of N and that the number of isolated eigenvalues of M 1 N AN is independent of N . Then, by Theorem 1.1, the number of PCG{steps to achieve a xed precision is independent of N . 3. By construction (2.8), the multiplication of M N with a vector requires only O(N log N) arithmetical operations by using FFT{techniques. By a technique presented in [26] it is possible to implement a PCG{method with preconditioner M N which takes only O(N) instead of O(N log N) arithmetical operations per iteration step more than the original CG{method with respect to AN . 3 Eigenvalues of M 1 In this section, we prove that under certain assumptions on the kernels KN the eigenvalues of M 1 N AN are bounded from below by a positive constant independent of N and that the number of isolated eigenvalues of M 1 N AN is independent of N . For the proof of our main result, we need some preliminary lemmata. Lemma 3.1 Let p 2 C 2 be a non{negative function which has only a nite number of zeros. be a positive function with Then, for f := ph and any N 2 N, the eigenvalues of A 1 lie in the interval The proof can be found for example in [5, 10, 31]. A more sophisticated version for f; g 2 L 1 was proved in [38, 37]. Lemma 3.2 Let p be a real{valued non{negative trigonometric polynomial of degree s. Let N 2s: Then at most 2s eigenvalues of M N (p) 1 AN (p) dier from 1. Proof: For arbitrary f 2 C 2 with pointwise convergent Fourier series, we obtain by replacing by the Fourier series of f at xN;l ~ a a a e 2ilk=N e 2ilj=N r2Znf0g a j+rN e iw N k e iw N (j+rN) e 2ilk=N e 2ilj=N r2Znf0g a k+rN e iw This is well{known as aliasing eect. Then it follows that where BN (f) := (b j k (f)) N 1 r2Znf0g a k+rN (f) e iw We consider is of degree smaller than s N 2 , we have that b k jkj N 1 s. Consequently, BN (p) is of rank 2s. Now the assertion follows by (3.1). In the sequel, we restrict our attention to Toeplitz matrices having a non{negative generating function f 2 C 2 with a zero of even order 2s (s 2 N) at We use the trigonometric polynomial s of degree s which has also a zero of order 2s at The convergence of our PCG{method is related to the behavior of the grid functions precisely, for the proof of our main theorem, we need that fq s;N (x)g N2N is bounded for all x 2 GN from above and below by positive constants independent of N . This will be the content of the following lemmata. First, we see that the above property follows immediately for all grid points x 2 GN having some distance independent of N from the zero of f : Lemma 3.3 Let GN be dened by (2.7) with wN 6= 0. Let fKN g N2N be a sequence of positive even reproducing kernels and let q s;N be given by (3.3). Then, for xN 2 GN \ [a; b] and for every " > 0 there exists N(") such that for all N N("). Proof: Since xN 2 [a; b] (N 2 N) for some a > 0; b < 2, we have that Further, we obtain by (2.6) that for every " > 0 there exists N(") such that for all N N("). By rewriting (3.3) in the form we obtain the assertion. By Lemma 3.3, it remains to consider the sequences fq s;N for N !1 or with xN ! 2 for N !1. Since both cases require the same ideas, we consider lim The existence of a lower bound of fq s;N (x N )g N2N does also not require additional properties of the kernel KN : Lemma 3.4 Let GN be dened by (2.7) with wN 6= 0. Let fKN g N2N be a sequence of positive even reproducing kernels and let q s;N be given by (3.3). Then, for xN 2 GN with lim there exists a constant > 0 independent of N such that q s;N Proof: By denition of q s;N and p s;N , we have that and since p s 0 and KN 0, we obtain for xN < that Z xN The polynomial p s is monotonely increasing on [0; ]. Thus Z xN KN Since KN is even and fullls (2.3), we get for any sequence xN 2 GN 0 that xN It remains to examine if for any xN 2 GN with lim Here the \moment property" comes into the play. Lemma 3.5 Let GN (n 2 N) be dened by (2.7) with Let fKN g N2N be a sequence of positive even kernels and let q s;N (s 1) be given by (3.3). Then there exists a constant < 1 independent of N such that for all xN 2 GN with lim only if KN fullls the \moment property" Z Note that the restriction (3.4) on the grids GN means that we have for any xN 2 GN that w=N xN . Proof: Since sin 2 x x 2 for all x 2 R, we obtain by (3.2) that Similarly, we have for any xed 0 y =2 that sin 2 x and hence 2s 2s x 2s Using (3.6), we conclude by KN 0 that Z Z 2s 2s Z and since KN is even s 2s x 2s 2k Z Let KN satisfy (3.5). Then s 2s By (3.4), we have for any grid sequence xN 2 GN that xN w=N . Consequently, By (3.7) this implies that there exists < 1 independent of N so that q s;N On the other hand, we see by (3.7) with y := =4 that Z 2s x 2s 2k Z By denition of GN , there exists a grid sequence fxN g N2N so that xN approaches zero as N 1 1). Assume that KN does not fulll (3.5). Then we obtain for the above sequence that p s;N while we have by (3.6) that p s cannot be bounded from above. This completes the proof. By Lemma 3.3 { Lemma 3.5, we obtain that for grids GN dened by (2.7) and (3.4) and for even positive reproducing kernels with (3.5) there exist Now we can prove our main theorem. Theorem 3.6 Let fAN (f)g N2N be a sequence of Toeplitz matrices generated by a non{ negative function f 2 C 2 which has only a zero of order 2s (s 2 N) at the grids GN be dened by (2.7) and (3.4). Assume that fKN g N2N is a sequence of even positive reproducing kernels satisfying (3.5). Finally, let M N (f N ) be dened by (2.10). Then we have: i) The eigenvalues of M 1 N (f N )AN (f) are bounded from below by a positive constant independent of N . ii) For N 2s, at most 2s eigenvalues of M N (f N are not contained in the interval Here , are given by (3.8) and h min ; h max are dened as in Lemma 3.1, where h := f=p s . Proof: 1. To show ii), we consider the Rayleigh quotient By Lemma 3.1, we have that and thus, since the second factor on the right{hand side of (3.9) is positive By Lemma 3.2, we know that with a matrix RN (2s) of rank 2s and consequently and Since KN and p s are non{negative, we obtain by (2.4) and by denition of h that This implies by denition of M N (f N ) that and further by (3.3), (3.8) and since 0 < < 1 that for all u 6= oN . Assume that RN (2s) has s 1 positive eigenvalues. Then, by properties of the Rayleigh quotient and by Weyl's theorem [24, p. 184] at most s 1 eigenvalues of are larger than hmax . Similarly, we obtain by consideration of the left{ hand inequality of (3.10) that at most 2s s 1 eigenvalues of M N (f N are smaller than h min hmax . 2. To show i), we rewrite (3.9) as As in the rst part of the proof, we see that this implies Consequently, it remains to show that there exists a constant 0 < c < 1 such that c By (3.1), this is equivalent to By the special structure of BN (p s ) and AN (p s ), assertion i) follows as in the proof of Theorem 4.3 in [3]. This completes the proof. By the following theorem, the \moment property" (3.5) of the kernel is also necessary to obtain good preconditioners. Theorem 3.7 Let fAN (f)g N2N be a sequence of Toeplitz matrices generated by a non{ negative function f 2 C 2 which has only a zero of order 2s (s 2 N) at Let the grids GN be dened by (2.7) and (3.4). Assume that fKN g N2N is a sequence of even positive reproducing kernels which do not fulll (3.5). Finally, let M N (f N ) be dened by (2.10). Then, for arbitrary " > 0 and arbitrary c 2 N, there exist N("; c) such that for all N N("; c) at least c eigenvalues of M N (f N are contained in (0; "). The proof follows again the lines of the fundamental paper of F. Di Benedetto [3, Theorem 5.4]. We include the short proof with respect to our background. Proof: By the proof of Theorem 3.6, we have for all u 6= o that Hence it remains to show that M N (p s;N has an arbitrary number of eigenvalues in (0; ") for N su-ciently large. By (3.2) and [32, Theorem 3.1], we have that diag 2s I diag s I j;k=0 is an orthogonal matrix and where stoep a 0 and shank a 0 denote the symmetric Toeplitz matrix and the persymmetric Hankel matrix with rst row a 0 , respectively. Deleting the rst s 1 and the last s 1 rows and columns of we obtain AN (p s ). Thus, we have by Courants minimax theorem for the eigenvalues 2s 2s The later result is due to a technique of D. Bini et al. [7, Proposition 4.2]. Consider AN (p s ) this matrix has positive eigenvalues, while we have for arbitrary " > 0 that 2s Since KN does not fulll (3.5), we have by Lemma 3.5 that lim Thus, for j c independent of N and for su-ciently large N N("; c) the values j (AN (p s ) negative. The eigenvalues of AN (p s are continuous functions of t. Since the smallest c eigenvalues pass from a positive value for to a negative value for ") such that AN (p s zero. This is equivalent to the fact that M N (p s;N has an eigenvalue " we are done. The generalization of the above results for generating functions with dierent zeros of even order is straightforward (see [18]). By applying the polynomial Y instead of p s and following the above lines, we can show that for grids GN of the form (2.7) with xN;l 6= y m) and for kernels KN fullling (3.5) with there exist constants 0 < < 1 such that for all x 2 GN (p KN )(x) 4 Jackson polynomials and B{spline kernels In this section, we consider concrete positive reproducing kernels KN with property (3.5). The generalized Jackson polynomials of degree N 1 are dened by sin(nx=2) sin x=2 2m determined by (2.3) [22, p. 203]. It is well{known [22, p. 204], that the generalized Jackson polynomials J m;N are even positive reproducing which satisfy property (3.5) for In particular, J 1;N is the Fejer kernel which is related to the optimal circulant preconditioner [19, 15]. However, the Fejer kernel does not fulll (3.5) for s 1 such that we cannot expect a fast convergence of our PCG{method if f has a zero of order 2. Our numerical tests conrm this result. By Theorem 3.6, the generalized Jackson polynomials can be used for the construction of preconditioners. Note that preconditioners related to Jackson kernels were also suggested in [39]. However, the construction of the Fourier coe-cients of J m;N seems to be rather complicated. See also [10]. Therefore we prefer the following B{spline kernels. The \B{spline kernels" were introduced by R. Chan et al. in [14]. The authors showed by numerical tests that preconditioners from B{spline kernels of certain order seem to be good candidates for the PCG{method. Applying the results of the previous section, we are able to show the theoretical reasons for these results, at least for the positive B{spline kernels. Let [0;1) denote the characteristic function of [0; 1). The cardinal B{splines Nm (m 1) of order m are dened by and their centered version by Note that Mm is an even function with supp where sinc x := sin x Let the B{spline kernels B m;N be dened by [14] mk Note that B 1;N again coincides with the Fejer kernel. For the construction of the preconditioner, it is important, that the Fourier coe-cient c can be computed in a simple way for example by applying a simplied version of de Boor's algorithm [9, p. 54]. By (4.1), it is easy to check that B m;N is a dilated, 2{periodized version of (sinc x sinc Thus Moreover, we obtain similar to the generalized Jackson polynomials: Lemma 4.1 The B{spline kernels B m;N satisfy (3.5) if and only if m s + 1. Proof: By (4.2), we obtain that Z Z sinc dt sin N dt 2k sin u 2m du c N 2kZu 2k 2m du C N 2k for 1. Thus, for m s On the other hand, we have that Z Z sin N dt 2k sin u 2m du If m k, then the last integral is not bounded for N !1. Thus, for m s, the kernel B m;N does not fulll property (3.5). By Theorem 3.6, the B{spline kernels preconditioners 5 Generalizations of the preconditioning technique In this section, we sketch how our preconditioners can be generalized to (real) symmetric Toeplitz matrices and to doubly symmetric block Toeplitz matrices with Toeplitz blocks. We will do this in a very short way since both cases do not require new ideas. However, we have to introduce some notation to understand the numerical tests in Section 7. Symmetric Toeplitz matrices First, we suppose in addition to Section 2 that the Toeplitz matrices AN 2 R N;N are symmet- ric, i.e. the generating function f 2 C 2 is even. Note that in this case, the multiplication of a vector with AN can be realized using fast trigonometric transforms instead of fast Fourier transforms (see [32]). In this way, complex arithmetic can be completely avoided in the iterative solution of (2.1). This is one of the reasons to look for preconditioners of type (2.8), where the Fourier matrix F N is replaced by trigonometric matrices corresponding to fast trigonometric transforms. In practice, four discrete sine transforms (DST I { IV) and four discrete cosine transforms (DCT I { IV) were applied (see [46]). Any of these eight trigonometric transforms can be realized with O(N log N) arithmetical operations (see for example [2, 44]). Likewise, we can dene preconditioners with respect to any of these transforms. In this paper, we restrict our attention to the DST{II and DCT{II, which are determined by the following transform matrices: 1). Similar to (2.10), (2.8), we introduce the preconditioners (see [31]) diag l diag l We recall, that for the construction these preconditioners no explicit knowledge of the generating function is required. Since f is even, the grids GN are simply chosen as GN := fx N;l := l for the DCT{II and the DST{II preconditioners, respectively. If f(x N;l then we can prove Theorem 3.6 with respect to the preconditioners (5.1) in a completely similar way. We have only to replace the decomposition (3.1) by for the DCT{II and for the DST{II, respectively. See also [31]. Remark: Let O 0 denote the matrix algebra with respect to the unitary matrix ON . Then the optimal preconditioner of AN in AON is dened by denotes the Frobenius norm. As mentioned in the previous section, the optimal preconditioner in A FN coincides with our preconditioner (2.10) dened with respect to the Fejer kernel B 1;N and with in (2.7). It is easy to check (see [33]) that the optimal preconditioner in AON , where ON 2 fC IV N g is equal to our preconditioner M N (f N ; ON ) in (5.1) dened with respect to ON and with respect to the Fejer kernel. Unfortunately, the Fejer kernel preconditioners do not lead to a fast convergence of the PCG{method if the generating function f of AN has a zero of order 2s 2. In contrast to these results, the optimal preconditioners in AON with ON dened by the DCT I { III or by the DST I { III do not coincide with the corresponding Fejer kernel preconditioner In literature [6, 3], so{called optimal Tau preconditioners were of special interest. Using our notation, optimal Tau preconditioners are the optimal preconditioners with respect to the DST{I as unitary transform. The optimal Tau preconditioner realizes a fast convergence of the PCG{method if the generating function f of AN has only zeros of order 2s 2 [6]. Block Toeplitz matrices with Toeplitz blocks Next we are interested in the solution of doubly symmetric block Toeplitz systems with Toeplitz blocks. The construction of preconditioners with the help of reproducing kernels was applied to well{conditioned block Toeplitz systems in [27]. Following these lines, we generalize our univariate construction to ill{conditioned block Toeplitz systems with Toeplitz blocks. In the next Section we will present good numerical results also for the block case. However, in general, it is not possible to prove the convergence of PCG in a number of iteration steps independent of N . Here we refer to [34]. Note that as in the univariate case there exist banded block Toeplitz preconditioners with banded Toeplitz blocks which ensure a fast convergence of the corresponding PCG{method [35]. See also [4, 30]. We consider systems of linear equations where AM;N denotes a positive denite doubly symmetric block Toeplitz matrix with Toeplitz blocks (BTTB matrix), i.e. r;s=0 with A r := (a r;j k and a r;j = a jrj;jjj . We assume that the matrices AM;N are generated by a real{valued 2{ periodic continuous even function in two variables, i.e. a j;k := 1 Z'(s; ds dt : Note that the multiplication of a vector with a BTTB matrix requires only O(MN log(MN)) arithmetical operations (see [33]). We dene our so{called \level{2" preconditioners by N;M (S II with by r=0 with x kN+j := x 6 Numerical Examples In this section, we conrm our theoretical results by various numerical examples. The fast computation of the preconditioners and the PCG{method were implemented in MATLAB, where the C{programs for the fast trigonometric transforms were included by cmex. The algorithms were tested on a Sun SPARCstation 20. As transform length we choose and as right{hand side b of (2.1) the vector consisting of entries \1". The PCG{method started with the zero vector and stopped if kr (j) k 2 =kr (0) k 2 < denotes the residual vector after j iterations. We restrict our attention to preconditioners (2.10) and (5.1) constructed from B{spline kernels . The following tables show the number of iterations of the corresponding PCG{method to achieve a xed precision. The rst row of each table contains the exponent n of the transform length in the univariate case and the block length N in the block Toeplitz case. The kernels are listed in the rst column and the applied unitary transform in the second column of each table. Here F w N := W N F N with W N := diag( e ik=N wN := =N in (2.7). For comparison, the second row of each table contains the number of PCG{steps with preconditioner M N (f) dened by (2.8). These preconditioners, which can be constructed only if the generating function f is known, were examined in [31]. We begin with symmetric ill{conditioned Toeplitz matrices AN (f) arising from the generating functions ii) (see [3, 10, 11, 14, 31, 36]): f(x) := x 4 The Tables present the number of iteration steps with dierent preconditioners. As expected, for it is not su-cient to choose a preconditioner based on the Fejer it is not su-cient to choose a preconditioner based on the cubic B{spline kernel in order to keep the number of iterations independent of N . On the other hand, we have a similar convergence behavior for the dierent unitary transforms. This is no surprise for F w N and for S II N . However, for F N and for C II N , the corresponding grids GN contain the zero of f , namely This was excluded in Theorem 3.6. In our numerical tests it seems to play no rule that a grid point meets the zero of f . Our next example in Table 3 conrms our theoretical results for the function with zeros of order 2 at Finally, let us turn to BTTB matrices AN;N . In our examples, the matrices AN;N are generated by the functions iv) (see [4]): '(s; v) (see [30, 31]): '(s; vi) (see [30, 31]): '(s; These matrices are ill{conditioned and the CG{method without preconditioning, with Strang{ type{preconditioning or with optimal trigonometric preconditioning converges very slow (see [30, 33, 4]). Our preconditioning (5.2) leads to the number of iterations in the Tables 4 { 6. In [34], we proved that the number of iteration steps of PCG is independent of N in Example iv) and we explained the convergence behavior of PCG for the other examples. To our knowledge, there does not exist a faster PCG{method if the generating function ' is unknown up to now. Note that by [42, 43] any multilevel preconditioner is not optimal in the sense that a cluster cannot be proper [45]. Summary . We suggested new positive denite w{circulant preconditioners for sequences of Toeplitz systems with polynomial increasing condition numbers. The construction of our preconditioners is based on the convolution of the generating function with positive reproducing kernels and, by working in the Fourier domain, does not require the explicit knowledge of the generating function. As main result we proved that the quality of the preconditioner depends on a \moment property" of the corresponding kernel which is related to the order of the zeros of the generating function. This explains, e.g. why optimal circulant preconditioners arising from convolutions with the Fejer kernels fail to be good preconditioners if the generating function has zeros of order 2. --R Iterative Solution Methods. Fast polynomial multiplication and convolution related to the discrete cosine transform. Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices Preconditioning of block Toeplitz matrices by sine transforms. preconditioning for Toeplitz matrices. Capizzano. A unifying approach to abstract matrix algebra preconditioning. Spectral and computational properties of band symmetric Toeplitz matrices. Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions. Toeplitz preconditioners for Hermitian Toeplitz systems. Conjugate gradient methods of Toeplitz systems. Sine transform based preconditioners for symmetric Toeplitz systems. Circulant preconditioners from B-splines Circulant preconditioners constructed from kernels. Circulant preconditioners for ill-conditioned Hermitian Toeplitz matrices The best circulant preconditioners for Hermitian Toeplitz systems. The best circulant preconditioners for Hermitian Toeplitz systems II: The multiple-zero case An optimal circulant preconditioner for Toeplitz systems. Circulant Matrices. The Approximation of Continuous Functions by Positive Linear Operators. Constructive Approximation. Multigrid methods for Toeplitz matrices. Matrix Analysis. Iterative methods for ill-conditioned Toeplitz matrices Iterative methods for Toeplitz-like matrices A note on construction of circulant preconditioners from kernels. Linear Operators and Approximation Theory. Band preconditioners for block-Toeplitz -Toeplitz-block-systems Preconditioners for ill-conditioned Toeplitz matrices Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems. Trigonometric preconditioners for block Toeplitz systems. Preconditioning of Hermitian block-Toeplitz-Toeplitz-block matrices by level-1 preconditioners Preconditioning strategies for asymptotically ill-conditioned block Toeplitz systems An ergodic theorem for classes of preconditiones matrices On the extreme eigenvalues of Hermitian (block) Toeplitz matrices Capizzano. Korovkin theorems and linear positive gram matrix algebra approximations of toeplitz matrices. Capizzano. How to choose the best iterative strategy for symmetric Toeplitz systems Capizzano. Toeplitz preconditioners constructed from linear approximation pro- cesses Any circulant-like preconditioner for multi-level matrices is not optimal How to prove that a preconditioner can not be optimal. A polynomial approach to fast algorithms for discrete Fourier- cosine and Fourier-sine transforms Circulant preconditioners with unbounded inverses. Fast algorithms for the discrete W transform and for the discrete Fourier transform. --TR --CTR D. Noutsos , S. Serra Capizzano , P. Vassalos, Spectral equivalence and matrix algebra preconditioners for multilevel Toeplitz systems: a negative result, Contemporary mathematics: theory and applications, American Mathematical Society, Boston, MA, 2001 D. Noutsos , S. Serra Capizzano , P. Vassalos, Matrix algebra preconditioners for multilevel Toeplitz systems do not insure optimal convergence rate, Theoretical Computer Science, v.315 n.2-3, p.557-579, 6 May 2004

preconditioners;reproducing kernels;ill-conditioned Toeplitz matrices;CG method

587369

Symplectic Balancing of Hamiltonian Matrices.

We discuss the balancing of Hamiltonian matrices by structure preserving similarity transformations. The method is closely related to balancing nonsymmetric matrices for eigenvalue computations as proposed by Osborne [J. ACM, 7 (1960), pp. 338--345]and Parlett and Reinsch [Numer. Math., 13 (1969), pp. 296--304] and implemented in most linear algebra software packages. It is shown that isolated eigenvalues can be deflated using similarity transformations with symplectic permutation matrices. Balancing is then based on equilibrating row and column norms of the Hamiltonian matrix using symplectic scaling matrices. Due to the given structure, it is sufficient to deal with the leading half rows and columns of the matrix. Numerical examples show that the method improves eigenvalue calculations of Hamiltonian matrices as well as numerical methods for solving continuous-time algebraic Riccati equations.

Introduction . The eigenvalue problem for Hamiltonian matrices A G where A; G; Q 2 R n\Thetan and G, Q are symmetric, plays a fundamental role in many algorithms of control theory and other areas of applied mathematics as well as computational physics and chemistry. Computing the eigenvalues of Hamiltonian matrices is required, e.g., when computing the H1 -norm of transfer matrices (see, e.g., [9, 10]), calculating the stability radius of a matrix ([13, 29]), computing response functions [22], and many more. Hamiltonian matrices are also closely related to continuous-time algebraic Riccati equations (CARE) of the form with A; G; Q as in (1.1) and X 2 R n\Thetan is a symmetric solution matrix. Many numerical methods for solving (1.2) are based on computing certain invariant subspaces of the related Hamiltonian matrices; see, e.g., [19, 21, 26, 28]. For a detailed discussion of the relations of Hamiltonian matrices and continuous-time algebraic Riccati equations (1.2) we refer to [18]. In eigenvalue computations, matrices and matrix pencils are often preprocessed using a balancing procedure as described in [23, 25] for a general matrix A 2 R n\Thetan . First, A is permuted via similarity transformations in order to isolate eigenvalues, i.e., a permutation matrix P 2 R n\Thetan is computed such that are upper triangular matrices. Then, a diagonal matrix Universitat Bremen, Fachbereich 3 - Mathematik und Informatik, Zentrum fur Technomathe- matik, 28334 Bremen, Germany. E-mail: benner@math.uni-bremen.de is computed such that rows and columns of D \Gamma1 Z ZDZ are as close in norm as possible. That is, balancing consists of a permutation step and a scaling step. In the scaling step, the rows and columns of a matrix are scaled, which usually leads to a decrease of the matrix norm. This preprocessing step often improves the accuracy of computed eigenvalues significantly; isolated eigenvalues (i.e., those contained in T 1 and T 2 ) are even computed without roundoff error. Unfortunately, applying this balancing strategy to a Hamiltonian matrix H as given in (1.1) will in general destroy the Hamiltonian structure. This is no problem if the subsequent eigenvalue algorithm does not preserve or use the Hamiltonian struc- ture. But during the past fifteen years, several structure preserving methods for the Hamiltonian eigenproblem have been suggested. In particular, the square-reduced method [31], the Hamiltonian QR algorithm (if in (1.1), rank [12], the recently proposed algorithm based on a symplectic URV-like decomposition [7], or the implicitly restarted symplectic Lanczos method of [5] for large sparse Hamiltonian eigenproblems are appropriate choices for developing subroutines for library usage and raise the need for a symplectic balancing routine. Similarity transformations by symplectic matrices preserve the Hamiltonian structure. Thus, in order to balance a Hamiltonian matrix and to preserve its structure, the required permutation matrix and the diagonal scaling matrix should be symplectic. In Section 2 we will give some necessary background. Isolating eigenvalues of Hamiltonian matrices without destroying the structure can be achieved using symplectic permutation matrices. This will be the topic of Section 3. How to equilibrate rows and norms of Hamiltonian matrices in a similar way as proposed in [25] using symplectic diagonal scaling matrices will be presented in Section 4. When invariant subspaces, eigenvectors, or solutions of algebraic Riccati equations are the target of the computations, some post-processing steps are required. These and some other applications of the proposed symplectic balancing method are discussed in Section 5. Some numerical examples on the use of the proposed balancing strategy for eigenvalue computation and numerical solution of algebraic Riccati equations are given in Section 6. 2. Preliminaries. The following classes of matrices will be employed in the sequel. Definition 2.1. Let where I n is the n \Theta n identity matrix. a) A matrix H 2 R 2n\Theta2n is Hamiltonian if (HJ) . The Lie Algebra of Hamiltonian matrices in R 2n\Theta2n is denoted by H 2n . b) A matrix H 2 R 2n\Theta2n is skew-Hamiltonian if (HJ) . The Jordan algebra of skew-Hamiltonian matrices in R 2n\Theta2n is denoted by SH 2n . c) A matrix S 2 R 2n\Theta2n is symplectic if SJS The Lie group of symplectic matrices in R 2n\Theta2n is denoted by S 2n . d) A matrix U 2 R 2n\Theta2n is unitary symplectic if U 2 S 2n and UU . The compact Lie group of unitary symplectic matrices in R 2n\Theta2n is denoted by US 2n . Observe that every H 2 H 2n must have the block representation given in (1.1). In [11], an important relation between symplectic and Hamiltonian matrices is proved. Proposition 2.2. Let S 2 R 2n\Theta2n be nonsingular. Then S \Gamma1 HS is Hamiltonian for all H 2 H 2n if and only if S T This result shows that in general, similarity transformations that preserve the Hamiltonian structure have to be symplectic up to scaling with a real scalar. The following proposition shows that the structure of 2n \Theta 2n orthogonal symplectic matrices permits them to be represented as a pair of n \Theta n matrices. Hence, the arithmetic cost and storage for accumulating orthogonal symplectic matrices can be halved. Proposition 2.3. [24] An orthogonal matrix U 2 R 2n\Theta2n is symplectic if and only if it takes the form 2. We have the following well-known property of the spectra of Hamiltonian matrices (see, e.g., [18, 21], and the references given therein). Proposition 2.4. The spectrum of a real Hamiltonian matrix, denoted by oe (H) is symmetric with respect to the imaginary axis, i.e., if 2 oe (H), then \Gamma 2 oe (H). The spectrum of Hamiltonian matrices can therefore be partitioned as oe When solving Hamiltonian eigenproblems one would like to compute a Schur form for Hamiltonian matrices analogous to the real Schur form for non-symmetric matrices. This should be done in a structure-preserving way. Definition 2.5. a) Let " H has the form G A T A 2 R n\Thetan is in real Schur form (quasi-upper triangular) and " is real Hamiltonian quasi-triangular. b) If H 2 H 2n and there exists U 2 US 2n such that " real Hamiltonian quasi-triangular, then " H is in real Hamiltonian Schur form and U T HU is called a Hamiltonian Schur decomposition. If a Hamiltonian Schur decomposition exists such that " H is as is (2.2), then U can be chosen such that oe ( " Most of the structure-preserving methods for the Hamiltonian eigenproblem, i.e., those using symplectic (similarity) transformations, rely on the following result. For Hamiltonian matrices with no purely imaginary eigenvalues this result was first stated in [24] while in its full generality as given below it has been proved in [20]. Theorem 2.6. Let H 2 H 2n and let iff its pairwise distinct nonzero purely imaginary eigenvalues. Furthermore, let the associated H-invariant subspaces be spanned by the columns of U k , Then the following are equivalent. There exists S 2 S 2n such that S \Gamma1 HS is real Hamiltonian quasi-triangular. ii) There exists U 2 US 2n such that U T HU is in real Hamiltonian Schur form. is congruent to J for all is always of the appropriate dimension. Note that from Theorem 2.6 it follows that purely imaginary eigenvalues of H 2 must have even algebraic multiplicity in order for the Hamiltonian Schur form of H to exist. 3. Isolating Eigenvalues by Symplectic Permutations. Let P denote any n \Theta n permutation matrix. It is easy to see that symplectic permutation matrices have the form With matrices of type (3.1) it is possible to transform a Hamiltonian matrix to the ~ 22 0 -z -z -z -z -z -z where A 11 , A 33 are upper triangular and either Q The existence of such a P s will be proved in a constructive way later by Algorithm 3.4 which transforms a Hamiltonian matrix to the form given in (3.2). From a Hamiltonian matrix having the form (3.2) a total of 2(p of H can be read off directly as seen by the following result. Lemma 3.1. Let H 2 H 2n and is of the form (3.2) and either G there exists a permutation matrix 2n\Theta2n such that are upper triangular with and A 22 G 22 is a 2r \Theta 2r Hamiltonian submatrix of H. Proof. Let H be as in (3.2) and I Then 13 A 11 A 12 G 12 G 11 A 13 22 \GammaA T diag (\Pi q ; I I I r 0 I r I q 07 7 7 7 7 7 5 Thus, has the desired form. The eigenvalue relation (3.4) follows from \GammaI q 0 \GammaI q 0 13 A 11 Lemma 3.1 is merely of theoretical interest and demonstrates that in order to solve the Hamiltonian eigenvalue problem, we can proceed by working only with H 22 . But the transformations we have used in the proof are in general non-symplectic. If we want to compute invariant subspaces, eigenvectors, and/or the Hamiltonian Schur form given in Theorem 2.6, we can transform the Hamiltonian matrix in (3.2) such that it has Hamiltonian Schur form in rows and columns But this can not be accomplished using only symplectic permutation matrices of the form (3.1). Therefore we need another class of transformation matrices. Definition 3.2. A matrix P J 2 R 2n\Theta2n is called a J-permutation matrix if a) it is symplectic, i.e., P T c) each row and column have exactly one nonzero entry. As P J 2 US 2n , it is clear that a similarity transformation by a J-permutation matrix preserves the Hamiltonian structure. In analogy to standard permutations, similarity transformations with P J can be performed without floating point opera- tions. Moreover, they can be represented by a signed integer vector IP of length n, rows and columns k; j are to be interchanged while the sign indicates if the corresponding entry in P J is +1 or \Gamma1. The entries of P J in rows to 2n can be deduced from IP and Proposition 2.3. Furthermore, symplectic permutation matrices as given in (3.1) are J-permutation matrices. Lemma 3.3. For any H 2 H 2n having the form (3.2), there exists a J-permutation matrix P J such that A 11 A 12 G 11 A 22 G T" G 22 A T A T A T3 A 11 is upper triangular and with the notation in (3.2), Proof. Let a Hamiltonian matrix H be given as in (3.2). We need a J-permutation matrix only in the first step. Let I Obviously, P 1 is a J-permutation matrix and A 11 A 12 \GammaG 13 G 11 G 12 A 13 22 Now assume G ~ I q Then, 13 A T0 A 11 A 12 A 13 G 11 G 12 We thus obtain the form (3.9) by another similarity transformation with where \Pi q is defined in (3.8). For the other case, i.e., ~ I q and In both cases, " J-permutation matrix and " P is a Hamiltonian matrix having the desired form (3.9). In order to isolate eigenvalues, it is sufficient to restrict ourselves to symplectic permutations. But having computed the form (3.2), it is possible that there are still isolated eigenvalues in H 22 . Applying the same procedure used to isolate eigenvalues in H to H 22 , we can transform H 22 to the form (3.2). This process can then be repeated until no more isolated eigenvalues are found. Accumulating all permutations in a symplectic permutation matrix P s of the form (3.1), this results in a similarity transformation ~ . A . 0 . 0 gps gps Here, A j;j t, are upper triangular and for either and the Hamiltonian submatrix A t;t G t;t has no isolated eigenvalues. If we now define p := then we can partition ~ H in (3.10) as in (3.2). Then the first step in the proof of Lemma 3.1 can be performed to obtain the form (3.7). Just the block-structure of the upper left and lower right diagonal blocks in (3.7) are more complicated. But it is still possible to bring them to upper triangular form using repeatedly the same sequence of permutations used in the proof of Lemma 3.1. This shows that 2(p + q) eigenvalues of the Hamiltonian matrix can be read off directly from (3.10) and that ~ H is permutationally similar to4 ~ ~ Y ~ Z Further, we have H t;t s s If only eigenvalues are required, we can continue working only with H t;t . If also eigen-vectors and/or invariant subspaces are required, the similarity transformations used to solve the reduced-order eigenproblem for H t;t have to be applied to the whole matrix ~ H . In that case, ~ H should be transformed to the form given in (3.9). Partitioning ~ H from (3.10) as in (3.2), we can perform the first step of the proof of Lemma 3.3 with the J-permutation matrix P 1 . The subsequent steps to achieve upper triangular form in the first p + q rows and columns have then to be performed for each of the first distinguishing the cases Q A procedure to transform a Hamiltonian matrix H to the form in (3.10) is given in the following algorithm. Note that in the given algorithm, t. Algorithm 3.4. Input: Matrices A; G; Q 2 R n\Thetan , where defining a Hamiltonian Output: A symplectic permutation matrix P s ; matrices A; G; Q with defining a Hamiltonian matrix having the form ~ END IF END WHILE ~ END IF END WHILE END WHILE END In each execution of the outer WHILE-loop, we first search a row isolating an eigenvalue. If such a row is found, we look for a column isolating an eigenvalue. In this fashion it can be guaranteed that at the end, there are no more isolated eigenvalues although we always only touch the first n rows and columns of the Hamiltonian matrix. In an actual implementation one would of course never form the permutation matrices explicitly but store the relevant information in an integer vector. Multiplications by permutation matrices are realized by swapping the data contained in the rows or columns to be permuted; for details, see, e.g., [3]. It is rather difficult to give a complete account of the cost of Algorithm 3.4. If there are no isolated eigenvalues, the algorithm requires 4n floating point additions and 2n comparisons as opposed to 8n additions and 4n comparisons for the unstructured permutation procedure from [25] as implemented in the LAPACK subroutine xGEBAL [3] when applied to H 2 R 2n\Theta2n . The worst case for Algorithm 3.4 would be that in each execution of the outer WHILE-loop, an isolated eigenvalue is found in the last execution of the second inner WHILE-loop. In that case, the cost consists of 4n 3 =3 floating point additions, moving floating point numbers. But in this worst-case analysis, all eigenvalues are isolated such that after permuting, there is nothing left to do, and the Hamiltonian matrix is in Hamiltonian Schur form. A worst-case study for xGEBAL shows that the permutation part requires 8n 3 and moving floating point numbers. We can therefore conclude that Algorithm 3.4 is about half as expensive as the procedure proposed in [25] applied to a Hamiltonian matrix. 4. Symplectic Scaling. Suppose now that we have transformed the Hamiltonian matrix to the form (3.10). Since all subsequent transformations are determined from H t;t , the scaling parameters to balance H t;t have now to be chosen such that the rows and columns of H t;t (instead of ~ are as close in norm as possible. In order to simplify notation we will in the sequel call the Hamiltonian matrix again H . Let H off be the off-diagonal part of H , i.e., We may without loss of generality assume that none of the rows and columns of H off vanishes identically. Otherwise, we could isolate another pair of eigenvalues. Now we want to scale H such that the norms of its rows and columns are close in norm. As noted before, employing the technique of Parlett and Reinsch [25] destroys the Hamiltonian structure. Diagonal scaling has thus to be performed using a symplectic diagonal matrix D s . Such a matrix must have the form, where D 2 R n\Thetan is a nonsingular diagonal matrix. Let us at first note an obvious result for Hamiltonian matrices. Here and in the sequel we will use the colon notation (see, e.g., [15]) H(:; k), H(j; :) to indicate the kth column and jth row, respectively, of a matrix H . Lemma 4.1. Let H 2 R 2n\Theta2n be a Hamiltonian matrix. Then for all p 1 and for all i.e., the p-norms of the ith column equals the norm of the (n + i)th row and the norm of the ith row equals the norm of the (n Proof. The result is obvious by noting kxk for x 2 R 2n and observing that from the structure of Hamiltonian matrices, we have and furthermore, Equation (4.3) follows analogously by noting We can now conclude that it is sufficient to equilibrate the norms of the first rows and columns of a 2n \Theta 2n Hamiltonian matrix by using a consequence of Lemma 4.1. Corollary 4.2. Let H 2 R 2n\Theta2n be a Hamiltonian matrix. Then for all p 1 and for all Since a similarity transformation with any diagonal matrix does not affect the diagonal elements of the transformed matrix, it is in the following sufficient to consider H off . We will employ the notation ith column of H off ; transpose of the ith row of H off : In the sequel, we will for convenience use 1. The results also hold for any other p-norm. From a computational point of view it is also reasonable to use the 1-norm, since its computation does not involve any floating point multiplications and furthermore, reducing the norm of H in one norm usually implies also a reduction in the other norms. Equilibrating can now be achieved in a similar way as in the Parlett/Reinsch method. If fi denotes the base of the floating point arithmetic and oe i is any signed integer, then they compute fi oe i closest to the real scalar Thus, with D H is in general no longer Hamiltonian. Unfortunately, using the symplectic diagonal matrix where i and computing ~ (D (i) s we obtain and thus in general, k ~ Nevertheless, equilibrating the 1-norms of h i and h i can be achieved by requiring solving the resulting quartic equation jg ii j: It remains to show that equation (4.6) has a positive solution. Theorem 4.3. Let H 2 R 2n\Theta2n be a Hamiltonian matrix and denote its off-diagonal part by H off . Assume that none of the rows and columns of H off vanishes identically. Then there exists a unique real number such that for ~ H as in (4.4) we have Proof. Solutions of Equation (4.6) are non-zero solutions of k=0 a k t k . Recalling that g ii = the coefficients of the polynomial p satisfy a a Since there is at most one change of sign in the coefficients of the polynomial p, Descartes' rule of signs shows that there is at most one positive zero of p. So if there exists a positive solution of (4.6), it is unique. By assumption, kh Therefore, either a ii is part of h i ) and either a 3 ? 0 or a 4 ? 0 (as q ii is part of h i ). Thus, we know that p is a polynomial of degree at least 3 with positive leading coefficient and hence, lim t!1 On the other hand, if using the mean value theorem it follows that there exists a positive zero of p. If g ii = 0, then is a zero of p and The third order polynomial has a positive zero because of the mean value theorem and by has again at least one positive zero and hence equation (4.6) has at least one positive real solution regardless of the value of g ii . On the other hand, it was already observed that there is at most one such solution and we can conclude that there exists a unique solving equation (4.6) whence k ~ h The other equalities follow immediately from Corollary 4.2. Computing the exact value equilibrating the ith and (n+i)th rows and columns would require the solution of the fourth-order equation (4.6). Since the diagonal similarity transformations are to be chosen from the set of machine numbers, it is sufficient to find the machine number fi oe i closest to ffi i . This can be done similarly to the computation in the general case as proposed in [25] and implemented in the Fortran 77 subroutines BALANC from EISPACK [14] or its successor xGEBAL from LAPACK [3]. That is, starting with the quantities in (4.5) are evaluated and compared. If k ~ h then this is repeated for then we use . This is achieved by the following algorithm. Algorithm 4.4. Input: Hamiltonian matrix H 2 H 2n having no isolated eigenvalues, base of floating point arithmetic. Output: Diagonal matrix D s 2 S 2n , H is overwritten by D \Gamma1 row and column norms equilibrated as far as possible. r END WHILE r END WHILE END IF diag diag diag END One execution of the outer FOR-loop of Algorithm 4.4 can be considered as a sweep. The algorithm is terminated if for a whole sweep, all D . Usually, the row and column norms are approximately equal after very few sweeps. Afterwards, the iteration makes only very limited progress. Therefore, Parlett and Reinsch propose in [25] a modification, which, translated to our problem, becomes: Let ffi i be determined by the two inner WHILE-loops of Algorithm 4.4 and compute jg ii j: If (where fl is a given positive constant), then compute D i as in Algorithm 4.4. Otherwise, set D For the behavior is essentially the same as for Algorithm 4.4 (in a few cases, Algorithm 4.4 increases kh which can not happen if 1). For slightly smaller than one, a step is skipped if it would produce an insubstantial reduction of kh In an actual computation, the similarity transformations with the D i 's can be applied directly to the blocks A, G, and Q of the Hamiltonian matrix without forming the Hamiltonian matrix itself. Thus, each similarity transformation can be performed using only 4n \Gamma 4 multiplications. When the standard (not structure preserving) scaling procedure from [25] is applied to H , each similarity transformation requires multiplications. (Recall that in Algorithm 4.4, two rows and columns are equilibrated at a time while only one row and column is treated in each step of the inner FOR-loop of the standard procedure.) The number of sweeps required to converge is similar to those for the general case since the theory derived in [25] only requires the assumption of similarity transformations with diagonal matrices and that in step i of each sweep, the ith rows and columns are equilibrated as far as possible with . But this is accomplished by Algorithm 4.4. Moreover, if ffi i is taken as the exact solution of (4.6), the convergence of the sequence of similarity transformations to a stationary point can be proved as in [23, 16]. That is, if i is the solution of (4.6) in sweep k, then lim k!1 ffi (k) for all hence in the limit, H is a balanced Hamiltonian matrix. Note that here, each sweep has length n while in the standard balancing algorithm, one has to go through each row/column pair of the matrix and thus, each sweep has length 2n. Thus, the computational cost for scaling a 2n \Theta 2n Hamiltonian matrix by Algorithm 4.4, assuming k 1 sweeps are required, is 4n n) as opposed to 8n 2 n) for the standard scaling procedure as given in [25] with assumed sweeps required for convergence. In general, k 1 k 2 such that the structure-preserving scaling strategy is about half as expensive as the standard procedure. These flop counts are based on the assumption that the cost for determining the can be considered as small (O(1)) compared to the similarity transformations. Remark 4.5. In [17] it is proposed to solve the matrix balancing problem using a convex programming approach. To compare the complexity of this approach to that of Algorithm 4.4, suppose that Algorithm 4.4 terminates after k 1 sweeps with . For the matrix H to be balanced, let ng \Theta ng Eg. Assume that and ! e . (Here, Theorem 5 in [17] states that the complexity of computing a diagonal matrix Y with positive diagonal entries such that the rows and columns of Y \Gamma1 HY are balanced with the same accuracy as achieved by Algorithm 4.4 is O ne hmin . From numerical experience, it can be assumed that with respect to n. Hence, Algorithm 4.4 can be considered to be of complexity O(n 2 ). This complexity is clearly superior to that of the convex programming approach which is still the case if k Algorithm 4.4 requires a careful implementation to guard against over- and underflow due to a very large/small ffi i . Here, we can use the bounds discussed in [25] and implemented in LAPACK subroutine xGEBAL [3]; we just have to take into account that in each step we scale by fi \Sigma2 rather than fi as in xGEBAL. 5. Backtransformation, Ordering of Eigenvalues, and Applications. So far we have only considered the problem of computing the eigenvalues of a Hamiltonian matrix. In order to compute eigenvectors, invariant subspaces, and the solutions of algebraic Riccati equations, we have to transform the Hamiltonian matrix to real Schur form. As we are considering structure-preserving methods, the goal is to transform the Hamiltonian matrix to real Hamiltonian Schur form as given in Theorem 2.6 a) - if it exists. Assume that we have applied Algorithm 3.4 to the Hamiltonian matrix and obtained a symplectic permutation matrix P s such that P T s HP s has the form given in (3.10). Then, we have applied a J-permutation P J to the permuted Hamiltonian matrix such that its rows and columns numbered are in Hamiltonian Schur form, i.e., P T has the form given in (3.9). (From Lemma 3.3 we know that such a P J exists.) Next, we have applied Algorithm 4.4 to the Hamiltonian submatrix H t;t 2 H 2r from (3.11) and obtained a diagonal matrix diag I Then A 22 G T" G 22 A T A T A T3 A 11 2 R (p+q)\Theta(p+q) is upper triangular and the Hamiltonian submatrix " H 22 := A22 Q22 G22 A has no isolated eigenvalues and its rows and columns are equilibrated by Algorithm 4.4. Now assume the Hamiltonian Schur form of " H 22 exists and we have computed U U22 V22 V22 U22 US 2r that transforms " H 22 into real Hamiltonian Schur form. Set I 22 real Hamiltonian quasi-triangular and . The first n columns of S span a Lagrangian H-invariant subspace. In most applications, the c-stable H-invariant subspace is desired. Let us assume the method used to transform " H 22 to Hamiltonian Schur form chooses U 22 such that the first r columns of U 22 , i.e., the columns of U22 V22 , span the " H 22 -invariant subspace of choice. But there is no guarantee that the isolated eigenvalues in " A 11 are the desired ones. In that case, we have to reorder the Hamiltonian Schur form in order to move the undesired eigenvalues to the lower right block of " H and the desired ones to the upper left block. Assume that we want to compute the Lagrangian H-invariant subspace corresponding to a set ae oe (H) which is closed under complex conjugation. (Note that this is a necessary condition in order to obtain a Lagrangian invariant subspace [2]). Using the standard reordering algorithm for the real Schur form of an n \Theta n unsymmetric matrix as given in [15, 30], we can find an orthogonal matrix ~ U such that with the orthogonal symplectic matrix U diag U ; ~ U , we have that ~ A 11 ~ A 12 ~ G 11 ~ A 22 ~ G T~ G 22 A T A T A T3 where A 11 ; A 22 are quasi-upper triangular, and A T 22 A T Therefore, we have to swap the eigenvalues in ~ A 22 and \Gamma ~ A T 22 . Note that the eigenvalues to be re-ordered are among the isolated eigenvalues and hence are real. This implies that ~ A 22 is upper triangular. The re-ordering can be achieved analogously to the re-ordering of eigenvalues in the real Schur form as given in [15, 30]. The following procedure uses this standard re-ordering in order to swap eigenvalues within ~ A 22 A T 22 ) and requires rotations working exclusively in rows and columns n and 2n in order to exchange eigenvalues from ~ A 22 with eigenvalues from \Gamma ~ A T 22 . Assume A 22 A T 22 sn \Gammas n cn be a Givens rotation matrix that annihilates the second component of gnn where ~ Then U n is a symplectic Givens rotation matrix acting in planes n and 2n and ~ A 11 ~ A 12 a 1;n ~ G 11 ~ A 22 . ~ G T~ G 22 gnn A T A T A T 22 0 Here, the bar indicates elements changed by the similarity transformation. The next step is now to move n up in the upper diagonal block using again the standard ordering subroutine such that we obtain again the form given in (5.1), just A 11 2 R (n\Gammak+1)\Theta(n\Gammak+1) and again, the relations (5.2) hold. This procedure has now to be repeated until A 11 ). Remark 5.1. If the Hamiltonian matrix has the form h A which corresponds to a linear system stabilizable and (C; detectable, then each isolated eigenvalue in (5.1) given by the diagonal elements of ~ A 11 has negative real part. Otherwise, these eigenvalues are unstable or undetectable and can not be stabilized/detected. Therefore, if we have not mixed up blocks by the J-permutation matrix P J (i.e., in Algorithm 3.4, i n) and the c-stable H-invariant subspace is required, no re-ordering is necessary. Remark 5.2. When solving algebraic Riccati equations using any approach based on the Hamiltonian eigenproblem, the symplectic balancing strategy proposed here is often not enough to minimize errors caused by ill-scaling. This is due to the effect that for a balanced Hamiltonian matrix h A G we still may have kQk AE kGk which may cause large errors when computing invariant subspaces [27]. Therefore, another symplectic scaling using a similarity transformation with diag ae I n ae 2 R, should be applied to H in order to achieve kAk kGk kQk as far as possible; see [4] for details and a discussion of several heuristic strategies to achieve this. Remark 5.3. Everything derived so far for Hamiltonian matrices can be applied in the same way to skew-Hamiltonian matrices. If N 2 SH 2n , then h A G A T with . The skew-Hamiltonian structure is again preserved under symplectic similarity transformations. Hence, isolating eigenvalues, re-ordering, etc., can be achieved in the same way as for Hamiltonian matrices as all considered transformations do not depend on the signs in the matrix blocks A, G, Q, but only on the distinction zero/non-zero when isolating eigenvalues and on the absolute values of the entries when equilibrating rows and norms. Note that Algorithm 4.4 even simplifies quite a lot for real skew-Hamiltonian matrices. As q can be computed as in the general balancing algorithm for non-symmetric matrices because in (4.5) we obtain k ~ Eigenvalues of skew-Hamiltonian matrices as well as a skew-Hamiltonian Schur form can be computed in a numerically strong backward stable way by Van Loan's method [31]. It is advisable to balance skew-Hamiltonian matrices using the proposed strategies prior to applying this algorithm. Remark 5.4. We have considered so far only real Hamiltonian and skew- Hamiltonian matrices. Isolating eigenvalues and equilibrating rows and columns for complex (skew-)Hamiltonian matrices can be achieved in exactly the same way. A structure-preserving, numerically backward stable (and hence numerically strong backward stable) method for solving the complex (skew-)Hamiltonian eigenproblem has recently been proposed [8]. The proposed symplectic balancing method can (and should) also be used prior to applying this algorithm. 6. Numerical Examples. We have tested the symplectic balancing strategy for eigenvalue computations. The computations were done in Matlab 1 Version 5.2 with machine precision " 2:2204 \Theta 10 \Gamma16 . Algorithms 3.4 and 4.4 were implemented as Matlab functions. We used the modified algorithm as suggested by (4.8) where we set suggested in [25] and implemented in the LAPACK subroutine xGEBAL [3]. The eigenvalues of the balanced and the unbalanced Hamiltonian matrix 1 Matlab is a trademark of The MathWorks, Inc. were computed by the square-reduced method using a Matlab function sqred which implements the explicit version of the square-reduced method (see [31]). We also tested the effects of symplectic balancing for the numerically backward stable, structure-preserving method for the Hamiltonian eigenvalue problem presented in [7]. Like the square-reduced method, this algorithm uses the square of the Hamiltonian matrix. But it avoids forming the square explicitly using a symplectic URV-type decomposition of the Hamiltonian matrix. As reference values we used the eigenvalues computed by the unsymmetric QR algorithm with Parlett/Reinsch balancing as implemented in the LAPACK expert driver routine DGEEVX [3], applied to the Hamiltonian matrix and using quadruple precision. Moreover, we tested the effects of balancing when solving algebraic Riccati equations with the structure-preserving multishift method presented in [1] for the examples from the benchmark collection [6]. We only present some of the most intriguing results Example 6.1. [6, Example 6] The system data come from an optimal control problem for a J-100 jet engine as a special case of a multivariable servomechanism problem. The resulting Hamiltonian matrix H 2 R 60\Theta60 has 8 isolated eigenvalues: triple eigenvalues at \Sigma20:0 and simple eigenvalues at \Sigma33:3. Algorithm 3.4 returns and for the permuted Hamiltonian matrix we have Hn+1:n+i l \Gamma1;n+1:n+i l Next, the Hamiltonian submatrix is scaled using Algorithm 4.4. After six sweeps, we obtain the balanced Hamiltonian We have decreased the 2-norm of the matrix used in the subsequent eigenvalue computation by more than five orders of magnitude. If the eigenvalues are computed by the square- reduced method applied to the unbalanced Hamiltonian matrix, the triplet of isolated eigenvalues is returned as a pair of conjugate complex eigenvalues with relative errors \Gamma11 and a simple eigenvalue with relative error 3:96 \Theta 10 \Gamma11 . For the simple eigenvalue at 33:3, the relative error is 7:7 \Theta 10 \Gamma15 . For the balanced version, these eigenvalues are returned with full accuracy since they are not affected by roundoff errors. The relative errors for the other (not isolated) eigenvalues are given in Figure 6.1 where we use the relative distance of the computed eigenvalues to those computed by DGEEVX as an estimate of the real relative error. Figure 6.1 only contains the relative errors for the eigenvalues with positive real parts as sqred returns the eigenvalues as exact plus-minus pairs. The '+' for the 26th eigenvalue is missing as the computed relative error for the balanced version is zero with respect to machine precision. The eigenvalues are ordered by increasing absolute values. From Figure 6.1, the increasing accuracy for decreasing ratio kHk 2 =jj is obvious - with or without balancing. All computed eigenvalues of the balanced matrix are more accurate than for the unbalanced one. The increase in accuracy is more significant for the eigenvalues of smaller magnitude. This reflects the decrease of the ratios kHk 2 =jj which more or less determines the accuracy of the computed eigenvalues; see [31]. The decrease factor for kHk 2 is about 5 \Theta 10 \Gamma6 . The accuracy for the eigenvalues of smaller magnitude increases by almost the same factor. From Figure 6.2 we see that symplectic balancing also improves the eigenvalues computed by the method proposed in [7]. As the method does not suffer from the perturbation, the accuracy for all computed eigenvalues is similar. Also note that in the unbalanced version, the isolated eigenvalues are computed with a relative accuracy ranging from 7:0 \Theta 10 \Gamma14 to 1:2 \Theta 10 \Gamma15 . eigenvalue number relative errors '+' - with symplectic balancing, 'o' - without balancing Fig. 6.1. square-reduced method. eigenvalue number relative errors '+' - with symplectic balancing, 'o' - without balancing Fig. 6.2. symplectic URV Using the balanced matrix in order to solve algebraic Riccati equations by the multishift method as described in [1], we obtain the following results: if the multishift method is applied to the unbalanced data, the computed solution yields a residual of size 1:5 \Theta 10 \Gamma6 while using the balanced Hamiltonian matrix we get r This shows that numerical methods for solving algebraic Riccati equations can substantially be improved employing balancing. Example 6.2. [6, Example 13] The Hamiltonian matrix is defined as in (1.1) with diag (1; 0; denotes the fourth unit vector. After four sweeps of Algorithm 4.4, kHk 2 is reduced from 10 12 to 1:5 \Theta 10 6 . The accuracy of the computed eigenvalues did not improve significantly, but for the stabilizing solution of the algebraic Riccati equation, the Frobenius norm of the residual as defined in (6.1) dropped from r 7. Concluding Remarks. We have seen that isolated eigenvalues of a real Hamiltonian matrix can be deflated using similarity transformations with symplectic permutation matrices, the deflated problem can be scaled in order to reduce the norm of the deflated Hamiltonian matrix and to equilibrate its row and column norms, and the remaining (not isolated) eigenvalues can then be determined by computing the eigenvalues of the deflated, balanced Hamiltonian submatrix. If invariant subspaces are required, then we can use J-permutation matrices and a symplectic re-ordering strategy in order to obtain the desired invariant subspaces. The same method can be applied in order to balance skew-Hamiltonian and complex (skew-)Hamiltonian matrices. Numerical examples demonstrate that symplectic balancing can significantly improve the accuracy of eigenvalues of Hamiltonian matrices as well as the accuracy of solutions of the associated algebraic Riccati equations computed by structure-preserving methods. Final Remark and Acknowledgments. The work presented in this article continues preliminary results derived in [4]. The author would like to thank Ralph Byers, Heike Fabender, and Volker Mehrmann for helpful suggestions. --R A multishift algorithm for the numerical solution of algebraic Riccati equations Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case structure preserving method for computing the eigenvalues of real Hamiltonian or symplectic pencils A bisection method for computing the H1 norm of a transfer matrix and related problems A fast algorithm to compute the H1-norm of a transfer function matrix Matrix factorization for symplectic QR-like methods Matrix Eigensystem Routines- EISPACK Guide Extension Matrix Computations Matrix balancing On the complexity of matrix balancing The Algebraic Riccati Equation Invariant subspace methods for the numerical solution of Riccati equations Canonical forms for Hamiltonian and symplectic matrices and pencils The Autonomous Linear Quadratic Control Problem Solution of large matrix equations which occur in response theory A Schur decomposition for Hamiltonian matrices Balancing a matrix for calculation of eigenvalues and eigenvec- tors Computational Methods for Linear Control Systems Solving continuous-time matrix algebraic Riccati equations with condition and accuracy estimates Algorithms for Linear-Quadratic Optimization A fast algorithm to compute the real structured stability radius Algorithm 506-HQR3 and EXCHNG: Fortran subroutines for calculating and ordering the eigenvalues of a real upper Hessenberg matrix A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix --TR --CTR Peter Benner , Daniel Kressner , Volker Mehrmann, Structure preservation: a challenge in computational control, Future Generation Computer Systems, v.19 n.7, p.1243-1252, October Pierluigi Amodio, On the computation of few eigenvalues of positive definite Hamiltonian matrices, Future Generation Computer Systems, v.22 n.4, p.403-411, March 2006 Peter Benner , Daniel Kressner, Algorithm 854: Fortran 77 subroutines for computing the eigenvalues of Hamiltonian matrices II, ACM Transactions on Mathematical Software (TOMS), v.32 n.2, p.352-373, June 2006

hamiltonian matrix;symplectic method;balancing;eigenvalues

587379

An Inverse Free Preconditioned Krylov Subspace Method for Symmetric Generalized Eigenvalue Problems.

In this paper, we present an inverse free Krylov subspace method for finding some extreme eigenvalues of the symmetric definite generalized eigenvalue problem x$. The basic method takes a form of inner-outer iterations and involves no inversion of B or any shift-and-invert matrix $A-\lambda_0 B$. A convergence analysis is presented that leads to a preconditioning scheme for accelerating convergence through some equivalent transformations of the eigenvalue problem. Numerical examples are given to illustrate the convergence properties and to demonstrate the competitiveness of the method.

Introduction Iterative methods such as the Lanczos algorithm and the Arnoldi algorithm are widely used for solving large matrix eigenvalue problems (see [21, 22]). Eective applications of these algorithms typically use a shift-and-invert transformation, which is sometimes called preconditioning [22] and requires solving a linear system of equations of the original size at each iteration of the process. For truly large problems, solving the shift-and-invert equations by a direct method such as the LU factorization is often infeasible or ine-cient. In those cases, one can employ an iterative method to solve them approximately, resulting in two levels of iterations called inner-outer iterations. However, methods like the Lanczos algorithm and the Arnoldi algorithm are very sensitive to perturbations in the iterations and therefore require highly accurate solutions of these linear systems (see [8]). Therefore, the inner-outer iterations may not oer an e-cient approach for these methods. There has recently been great interest in other iterative methods that are also based on shift- and-invert equations but tolerate low accuracy solutions. One simple example of such methods is the inexact inverse iteration where the linear convergence property is preserved even when the inversion is solved to very low accuracy (see [7, 13, 14, 27]). Several more sophisticated and competitive methods have been developed that also possess such a property. They include the Jacobi-Davidson Scientic Computing and Computational Mathematics Program, Department of Computer Science, Stanford University, Stanford, CA 94305. E-mail : golub@sccm.stanford.edu. Research supported in part by National Science Foundation Grant DMS-9403899. y Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027. E-mail: qye@ms.uky.edu. Part of this Research was supported by NSERC of Canada while this author was with University of Manitoba. method [5, 24, 25], truncated RQ iterations [26, 32] and others [14, 29, 30, 31]. One di-culty with these methods is that it is not easy to determine to what accuracy the shift-and-invert equations should be solved. On the other hand, there have been several works that aim at generalizing the concept of preconditioning for linear systems to the eigenvalue problem [1, 2, 4, 11, 12, 17, 18, 15, 20, 28]. This is mostly done, however, by directly adapting a preconditioner used for inverting a certain matrix into an eigenvalue iteration and in these situations, the role of preconditioners is usually not clear, although some of them can be regarded as using inexact shift-and-invert [19]. Overall, while all these new methods have been demonstrated to work successfully in some problems, there is in general a lack of understanding of how and why they work. Furthermore, optimal eigenvalue projection methods such as the Lanczos algorithm have mostly not been incorporated in these developments. In this paper, we shall present a variation of the Krylov subspace projection methods for computing some extreme eigenvalues of the generalized eigenvalue problem called the pencil problem for (A; B), are symmetric matrices and B > 0. The method iteratively improves an approximate eigenpair, each step of which uses either the Lanczos or the Arnoldi iteration to produce a new approximation through the Rayleigh-Ritz projection on a Krylov subspace, resulting in a form of inner-outer iterations. We shall present our theoretical and numerical ndings concerning convergence properties of this method and derive bounds on asymptotic linear convergence rates. Furthermore, we shall develop from the convergence analysis some equivalent transformations of the eigenvalue problem to accelerate the convergence, which will be called preconditioning. In particular, such transformations will be based on incomplete factorization and thus generalize the preconditioning for linear systems. To the best of our knowledge, this is the rst preconditioning scheme for the eigenvalue problem that is based on and can be justied by a convergence theory. The paper is organized as follows. We present the basic algorithm in Section 2 and analyze its convergence properties in section 3. We then give some numerical examples in Section 4 to illustrate the convergence properties. We then present a preconditioned version of the algorithm in Section 5, followed by some numerical examples on the preconditioning in Section 6. We nish with some concluding remarks in Section 7. Basic Inverse Free Krylov Subspace Method In this section, we present our basic algorithm for nding the smallest eigenvalue and a corresponding eigenvector (; x) of a pencil (A; B) where A; B are symmetric with B > 0. We note that the method to be developed can be modied in a trivial way for nding the largest eigenvalue (or simply considering ( A; B)). Given an initial approximation we aim at improving it through the Rayleigh-Ritz orthogonal projection on a certain subspace, i.e. by minimizing the Rayleigh quotient on that subspace. Noting that the gradient of the Rayleigh quotient at x 0 is r the well-known steepest descent method chooses a new approximate eigenvector x by minimizing Clearly, this can be considered as the Rayleigh-Ritz projection method on the subspace K 1 spanfx g. On the other hand, the inverse iteration constructs a new approximation by x . If the inversion is solved inexactly by an iterative solver (i.e. in an inexact inverse iteration [7]), then x 1 is indeed chosen from a Krylov subspace as generated by A 0 B. Since x 1 is extracted from the Krylov subspace to solve the linear system, it may not be a good choice for approximating the eigenvector. We consider here a natural extension of these two approaches that nds a new approximate eigenvector x 1 from the Krylov subspace Km spanfx (for some xed m) by using the Rayleigh-Ritz projection method. The projection is carried out by constructing a basis for Km and then forming and solving the projection problem for the pencil B). Repeating the process, we arrive at the following iteration, which we call inverse free Krylov method for (A; B). Algorithm 1: Inverse Free Krylov Subspace Method. Input m 1 and an initial approximation x 0 with For Construct a basis Find the smallest eigenpair End In the algorithm, we apply the projection to the shifted pencil update the approximation accordingly, which is theoretically equivalent to using the projection of (A; B) directly. This formulation, however, may improve the stability while saving matrix-vector multiplications by utilizing which need to be computed in the construction of the basis. In constructing a basis z there are many possible choices and, theoretically, they are all equivalent in that the new approximate eigenpair ( obtained will be the same, which is dened by (1) However, numerically, we will consider a basis that is orthonormal on a certain inner product. Such a basis of the Krylov subspace is typically constructed through an iterative method itself, which will be called the inner iteration. The original iteration of Algorithm 1 will be called the outer iteration. We shall discuss in the subsections later three methods for constructing the basis. In our presentation of Algorithm 1, we have assumed for convenience that that we construct a full basis z Generally, if only. Then, the Rayleigh-Ritz projection is simply carried out by replacing Zm with and (1) is still valid. Numerically, however, early termination at step p of the inner iteration is not likely and a full basis is usually constructed even when p < m in theory, but this causes no problem as the larger space spanned by more vectors would yield a better approximation. Given that the basic ingredients of Algorithm 1 are the projection and the Krylov subspaces, it is not surprising that some similar methods have been considered before. In [10, 11], Knyazev discussed and analyzed some very general theoretical methods and suggested several special cases, among which is the use of Km and (1). Morgan and Scott's preconditioned Lanczos algorithm [18] takes a similar iteration but uses the smallest Ritz value of the matrix A k B rather than that of the pencil to update the eigenvalue. With an m varied with each iteration, it has a quadratic convergence property. We point out however that the quadratic convergence is not a desirable property because it is achieved at the cost of increasingly larger m and it prevents improvement of convergence by preconditioning (there is hardly any need to accelerate a quadratic convergent algorithm). Our study will be somewhat of dierent nature in that we consider accelerating convergence by changing certain conditions of the problem through equivalent transformations (see section 5) as opposed to increasing m. Also related to ours are methods based on inverting a shifted matrix A k B or its projection, which include the inverse iteration and the Jacobi-Davidson method [24]. When the inversion is solved approximately by an iterative method, the solution is extracted from a Krylov subspace generated by A k B (or its projection). In these cases, it is chosen to satisfy the related linear system. We note that the Jacobi-Davidson method also uses the Rayleigh-Ritz projection in the outer iteration, the cost of which increases with the iteration. By xing the size of subspaces for projection, the cost of Algorithm 1 is xed per outer iteration. I, it is easy to see that Algorithm 1 is just the standard restarted Lanczos algorithm for A. In this regard, our investigation is on the version with a xed m and on how m aects the convergence. Furthermore, our development will lead to a preconditioning strategy that transforms I) to the pencil problem (L 1 AL suitably chosen L), to which Algorithm 1 will be applied. This transformation to a more complicated problem may seem counter intuitive but an important feature of Algorithm 1 is that the case I oers no advantage than a more general B: We now discuss in details the construction of a basis for Km in Algorithm 1. 2.1 Orthonormal basis by the Lanczos algorithm One obvious choice of the basis of the Krylov subspace Km is the orthonormal one as constructed by applying the Lanczos algorithm to C Simultaneously with the Lanczos process, we produce a tridiagonal matrix . The Lanczos process requires m+1 matrix-vector multiplications by C Once the basis has been constructed, we multiplications by B. Note that Here we state the Lanczos algorithm. Algorithm 2: Orthonormal basis by Lanczos. B, an approximate For End With the orthonormal basis, Bm is in general a full matrix and we need to solve a generalized eigenvalue problem for (Am ; Bm ). While in the exact arithmetic, it may not be valid in a nite precision arithmetic for larger m when there could be severe loss of orthogonality among z i . This can be corrected by either computing explicitly or using reorthogonalization [6] in the Lanczos process. We note that C k Zm has been computed in the Lanczos algorithm and can be stored for forming Am . 2.2 B-orthonormal basis by the Arnoldi algorithm We can also construct a B-orthonormal basis for Km by the modied Gram-Schmidt process in the B-inner product, which is essentially the Arnoldi algorithm. The advantage of this approach is a simpler projection problem with but it is at the cost of a longer recurrence. We also need to compute . We state the algorithm here. Algorithm 3: B-orthonormal basis by Arnoldi. Input For For End Each step of the Arnoldi algorithm requires 2 matrix-vector multiplications with one by C k and one by B. In addition, we need to store Bz i from each iteration in order to save matrix-vector multiplications, resulting in storage cost of m vectors. We note again that, for larger m, the B- orthogonality among the columns of Zm may gradually be lost. This leads to deterioration of the equation In that case, we need either reorthogonalization in the Arnoldi algorithm, or explicit computations of In comparing the two constructions, the computational costs associated with them are very comparable. They both require 2(m multiplications. The Arnoldi recurrence is more expensive in both ops and storage than the Lanczos recurrence while it produces a more compact projection matrix than the Lanczos algorithm. Clearly, these dierences are very minor when m is not too large, which is the case of interest in practical implementations. In terms of numerical stability of these two theoretically equivalent processes, our testing suggests that there is very little dierence. However, for the preconditioned version of Algorithm 1 that we will discuss in Section 5, the approach by the Arnoldi algorithm seems to have some advantage, see section 5. 2.3 C k -orthogonal basis by a variation of the Lanczos algorithm It is also possible to construct Zm that is C k -orthogonal by a variation of the Lanczos algorithm with a three term recurrence. Then the projection will have a compact form, leading to a computationally more eective approach than the previous two. However, it is less stable owing to the indeniteness of C k . For the theoretical interest, we outline this variation of the Lanczos algorithm for in the form of a full matrix tridiagonalization. be the standard tridiagonalization of the Lanczos algorithm for C where T is tridiagonal and Q is orthogonal with x k =kx k k as its rst column. For the sake of simplicity in presentation, we assume here that k is between the rst and the second eigenvalue, which implies that C has exactly one negative eigenvalue. Noting that the (1; 1) entry of T is x T be the block LDL T decomposition of T , where I and Write It is easy to check that and Now a Lanczos three term recurrence can be easily derived from (2) to construct the columns of Z, which still form a basis for the Krylov subspace and is essentially C-orthogonal. However, our tests show that this is numerically less stable. Therefore, we shall not consider this further and omit a detailed algorithm here. 3 Convergence analysis. In this section, we study convergence properties of Algorithm 1 that include a global convergence result and a local one on the rate of linear convergence. In particular, we identify the factors that aect the speed of the convergence so as to develop preconditioning strategy to improve the convergence. We rst prove that Algorithm 1 always converges to an eigenpair. For that, we need the following proposition, the proof of which is straightforward. Proposition 1 Let 1 be the smallest eigenvalue of (A; B) and ( k ; x k ) be the eigenpair approximation obtained by Algorithm 1 at step k. Then and Theorem 1 Let ( k ; x k ) denote the eigenpair approximation obtained by Algorithm 1 at step k. Then k converges to some eigenvalue ^ of (A; B) and converges in direction to a corresponding eigenvector). Proof From Proposition 1, we obtain that k is convergent. Since x k is bounded, there is a convergent subsequence x n k . Let x: B)^x. Then it follows from (3) that, Suppose now ^ r 6= 0. We consider the projection of (A; B) onto rg by dening r]: Noting that f^x; ^ rg is orthogonal, we have ^ B)^r is indenite. Thus the smallest eigenvalue of B), denoted by ~ , is less than ^ , i.e. ~ Furthermore, at step k, dene r Let ~ k+1 be the smallest eigenvalue of B. Hence by the continuity property of the eigenvalue, we have On the other hand, k+1 is the smallest eigenvalue of the projection of (A; B) on which implies Finally, combining the above together, we have obtained which is a contradiction to (4). Therefore, ^ is an eigenvalue and Now, to show suppose there is a subsequence m k such that > 0. From the subsequence m k , there is a subsequence n k for which x n k is convergent. Hence by virtue of the above proof, which is a contradiction. This completes the proof. Next, we study the speed of convergence through a local analysis. In particular, we show that k converges at least linearly. be the smallest eigenvalue of (A; B), x be a corresponding unit eigenvector and be the eigenpair approximation obtained by Algorithm 1 at step k. Let 1 be the smallest eigenvalue of A k B and u 1 be a corresponding unit eigenvector. Then Asymptotically, if k ! 1 , we have Proof First, from the denition, we have Furthermore, A 1 I k B 0 and A 0I is the smallest eigenpair of is the smallest eigenpair of (A; B). Clearly A k B is indenite and hence 1 0: Now using Theorem 3 of Appendix, we have which leads to the bound (5). To prove the asymptotic expansion, let 1 (t) be the smallest eigenvalue of A tB. Then . Using the analytic perturbation theory, we obtain 0 and hence Choosing from which the expansion follows. We now present our main convergence result. We assume that k is already between the rst and the second smallest eigenvalues. Then by Theorem 1, it converges to the smallest eigenvalue. Theorem 2 Let 1 < 2 n be the eigenvalues of (A; B) and ( k+1 ; x k+1 ) be the approximate eigenpair obtained by Algorithm 1 from ( k ; x k ). Let 1 2 n be the eigenvalues of A k B and u 1 be a unit eigenvector corresponding to 1 . Assume 1 < k < 2 . Then kBk where and with Pm denoting the set of all polynomials of degree not greater than m: Proof First, write g. At step k of the algorithm, we have Let A k be the eigenvalue decomposition of A k B, where orthogonal and g. Let q be the minimizing polynomial in m with and it follows from x T and Using Proposition 1, we have y T and hence On the other hand, we also have and where we note that q( 1 Thus kBk where we have used (8) and (9). Finally, combining (7), (10) and Lemma 1, we have kBk which leads to the theorem. It is well known that m in the theorem can be bounded in terms of i as (see [16, Theorem 1.64] for example) Then the speed of convergence depends on the distribution of the eigenvalues i of A k B but not those of (A; B). This dierence is of fundamental importance as it allows acceleration of convergence by equivalent transformations that change the eigenvalues of A k B but leave those of (A; B) unchanged (see the discussion on preconditioning in Section 5). On the other hand, the bound shows accelerated convergence when m is increased. In this regard, our numerical tests suggests that the convergence rate decreases very rapidly as m increases (see Section 4). For the special case of just the steepest descent method for A. It is easy to check in this case that Using this in Theorem 2, we recover the classical convergence bound for the steepest descent method [9, p.617]. We note that there is a stronger global convergence result in this case, i.e. k is guaranteed to converge to the smallest eigenvalue if the initial vector has a nontrivial component in the smallest eigenvector (see [9, p.613]). There is no such result known for the case B 6= I. Asymptotically we can also express the bound in terms of the eigenvalues of A 1 B instead of i which is dependent of k. We state it as the following corollary; but point out that the bound of Theorem 2 is more informative. n be the eigenvalues of A 1 B. Then, we have asymptot- ically p! 2m The proof follows from combining (11) with i 4 Numerical Examples - I In this section, we present numerical examples to illustrate the convergence behavior of Algorithm 1. Here, we demonstrate the linear convergence property and the eect of m on the convergence rate. Example 1: Consider the Laplace eigenvalue problem with the Dirichlet boundary condition on an L-shape region. A nite element discretization on a triangular mesh with 7585 interior nodes (using PDE toolbox of MATLAB) leads to a pencil eigenvalue problem apply Algorithm 1 to nd the smallest eigenvalue with a random initial vector and the stopping criterion is set as kr k k=kr 0 We give the convergence history of the residual kr k k for (from top down resp.) in Figure 1. We present in Figure 2 (a) the number of outer iterations required to achieve convergence for each m in the range of correspondingly in Figure 2 (b), the total number of inner iterations. We observe that the residual converges linearly with the rate decreased as m increases. Fur- thermore, from Figure 2 (a), the number of outer iterations decreases very rapidly (quadratically or even exponentially) as m increases and it almost reaches its stationery limit for m around 70. Because of this peculiar property, we see from Figure 2 (b) that the total number of inner iterations is near minimal for a large range of m (40 < m < 80 in this case). Example 2: We consider a standard eigenvalue problem which A is a ve point nite dierence discretization of the Laplace operator on the mesh of the unit square. Again, we apply Algorithm 1 to nd the smallest eigenvalue with a random initial vector. Figure Example 1: convergence of r k for outer iterations 2-norm of residuals In this case, it is simply a restarted Lanczos algorithm and we shall consider its comparison with the Lanczos algorithm without restart. In Figure 3, we present the convergence history of k 1 where k is the approximate eigenvalue obtained at each inner iteration. They are plotted in the dot lines from top down for respectively. The corresponding plot for the Lanczos algorithm (without restart) is given in the solid line. We have also considered the number of outer iterations and the total number of inner iterations as a function of m and observed the same behavior as in Example 1. We omit a similar gure here. In particular, the nearly exponential decrease of the outer iteration count implies that the convergence history with a moderate m (in this case even 16) will be very close to the one with very large m (i.e. Lanczos without restart in the solid line ). These examples conrm the linear convergence property of Algorithm 1. Furthermore, our numerical testing has consistently shown that the number of outer iterations decreases nearly exponentially as m increases. This implies that near optimal performance of the algorithm can be achieved with a moderate m, which is very attractive in implementations. Unfortunately we have not been able to explain this interesting behavior with our convergence results. Even for the restarted Lanczos algorithm, it seems to be a phenomenon not observed before and can not be explained by the convergence theory of the Lanczos algorithm either. 5 Preconditioning In this section, we discuss how to accelerate the convergence of Algorithm 1 through some equivalent transformations, which we call preconditioning, and we shall present the preconditioned version of Algorithm 1. Figure 2: Example 1: Outer and total inner iterations vs. m outer iterations (a) parameter m (inner iterations) total inner iterations (b) From our convergence result (Theorem 2), the rate of convergence depends on the spectral distribution of C i.e. the separation of 1 from the rest of eigenvalues 2 ; ; n of C k . With an approximate eigenpair ( k ; x k ), we consider for some matrix L k the transformed pencil which has the same eigenvalues as (A; B). Thus, applying one step of Algorithm 1 to have the bound (6) of Theorem 2 with the rate of convergence 2 determined by ^ the eigenvalues of We can now suitably choose L k to obtain a favorable distribution of ^ i and hence a smaller m . We shall call (13) a preconditioning transformation. One preconditioning transformation can be constructed using the LDL T factorization of a symmetric matrix [6]. For example, if is the LDL T factorization with D k being a diagonal matrix of 1, choosing this L k results in ^ . Then, at the convergence stage with 1 < k < 2 , we have ^ which implies and thus by Theorem 2, Figure 3: Example 2: Eigenvalue convergence history against each inner iteration for restarted dot lines from top down and for Lanczos without restart in solid line total inner iterations error of Ritz value We conclude that Algorithm 1, when applied to k ) at step k using the exact LDL T factor- ization, converges quadratically. This is even true with (i.e. the steepest descent method). Similarly, in light of Corollary 1, if we use a constant L obtained from the LDL T factorization (assuming 1 is known) with D k being a diagonal matrix of 0 and 1, Algorithm 2 also converges quadratically. What we have described above is the ideal situations of fast quadratic convergence property achieved by using an exact LDL T factorization. In practice, we can use an incomplete LDL T factorization A k B L k D k L T (through incomplete LU factorization, see [23, Chapter 10]). Then we will have a nonzero but small m and hence fast linear convergence. Indeed, to be e-cient, we can consider a constant L as obtained from an incomplete LDL T factorization of where 0 is a su-ciently good approximation of 1 and apply Algorithm 1 to (13). Then, the preconditioned algorithm converges linearly with the rate determined by the eigenvalues of which has a better spectral distribution as long as ( 0 k )L 1 BL T is small relative to ^ We note that 0 k need not be very small if L 1 BL T is small (e.g. for the discretization of dierential operators). It may work even when 0 < 1 , for which A and the incomplete LDL T factorization becomes incomplete Cholesky factorization. It is also possible to construct L based on other factorization, such as approximate eigenvalue decomposition. As in the preconditioned iterative methods for linear systems, the preconditioned iteration of Algorithm 1 can be implemented implicitly, i.e. without explicitly forming the transformed problem C k . We derive a preconditioned version of the algorithm in the rest of this section. be the approximate eigenpair that has been obtained at step k for the pencil (A; B). is the corresponding approximate eigenpair for the transformed pencil (13). By applying one step of iteration to the transformed pencil, the new approximation is obtained by constructing a basis ^ z z m for the Krylov subspace and form the projection problem is the smallest eigenpair of the above projection problem, then v) is the new approximate eigenpair for (A; B). z m ]: Then, the new approximate eigenpair can be written as ( k+1 and the projection problem is Therefore, to complete the k-th iteration, we only need to construct basis for the subspace L T Km . The actual construction of z i depends on which method we use and will be given in details in the subsections later. Here, we summarize the preconditioned algorithm as follows. Algorithm 4: Preconditioned Inverse Free Krylov Subspace Method. Input m and an initial approximation x 0 with For convergence Construct a preconditioner L k ; Construct a preconditioned basis Km Find the smallest eigenvalue 1 and a eigenvector v for (Am ; Bm ); End Remark: As in the linear system case, the above algorithm takes the same form as the original one except using a preconditioned search space L T Km . In the following subsections, we discuss the construction of a preconditioned basis by the Arnoldi algorithm and the Lanczos algorithm corresponding to the construction in Sections 2.1 and 2.2. Our numerical testing suggests that the Arnoldi algorithm might be more stable than the Lanczos algorithm in some cases. 5.1 Preconditioned basis by the Arnoldi method In the Arnoldi method, we construct ^ B-orthonormal basis for Km . Correspond- ingly, z is a B-orthonormal basis for L T Km . Starting from ^ B , the recurrence for z i is where h z T z and with is B-orthonormal, and h j;i above ensures this condition. From this, we arrive at the following algorithm. Algorithm 5: Preconditioned B-orthonormal basis by Arnoldi Input and a preconditioner L k . For For End We see from the algorithm that only L T k is needed in our construction. If we use 0 < 1 and is an incomplete Cholesky factor, i.e. A 0 B LL T , then we can use any matrix approximating explicitly forming L k . For example, for dierential operators, we can use the multigrid or domain decomposition preconditioners for A 0 B directly. 5.2 Preconditioned basis by the Lanczos method In the Lanczos method, we construct ^ z z m as an orthonormal basis for Km . Then the corresponding basis z k . Starting from ^ the recurrence is z z z T z i and . The resulting tridiagonal matrix T as constructed from 's and 's satises Zm . Thus, using ^ and with Clearly, the formulas for i and i+1 ensures z j is M-orthonormal. Thus, we have the alternative formulas From this, we can derive a recurrence to construct the basis. We note that this construction normalizes z i in the M-norm. In practice, M could be nearly singular. Therefore, it is more appropriate to normalize it in the 2-norm. The following algorithm is one of several possible formulations here. Algorithm Preconditioned basis by Lanczos Input and a preconditioner L k . For End 6 Numerical Examples - II In this section, we present some numerical examples to demonstrate the eectiveness and competitiveness of the preconditioned inverse free Krylov subspace method (Algorithm 4). Example 3: A and B are the same as in Example 1. We apply Algorithm 4 to nd the smallest eigenvalue (closest to We use a constant L k as obtained by the threshold incomplete LDL T factorization of A A with the drop tolerance 10 2 . We compare our algorithm with the Jacobi-Davidson algorithm that uses the same number of inner iterations (m) and with the same kind of preconditioner. We give in Figure 4 the convergence history of the residual kr k k of Algorithm 4 in solid lines and that of the Jacobi-Davidson algorithm in dot lines from top down respectively. In Figure 5, we also present the number of outer iterations and the total number of inner iterations required to reduce kr k k=kr 0 k to 10 7 for each m in (+) mark for Algorithm 4 and in (o) mark for the Jacobi-Davidson algorithm. Comparing it with Example 1, the results clearly demonstrate the acceleration eect of pre-conditioning by signicantly reducing the number of outer iterations (Fig. 4 and Fig. 5(a)). Furthermore, the values of m at which the total number of inner iterations is near its minimum are signicantly smaller with preconditioning (around Fig. 5). Although J-D algorithm has smaller number of total inner iteration for very small m, the corresponding outer iteration count is larger, which increases its cost. We also considered for this example the ideal preconditioning with L k chosen as the exact LDL T factorization of C k . In this case, we use an initial vector with kAx 0 so that 0 is su-ciently close to 1 . We present the residual convergence history in Figure 6 for steepest descent method), 2 and 4. The result conrms the quadratic convergence property for all Figure 4: Example 3 residual convergence history for lines - Algorithm 4; dot lines - Jacobi-Davidson) outer iterations 2-norm of residuals m. We have also tested the case that uses L L from the exact factorization of A 1 B and in this case it converges in just one iteration, conrming Corollary 1. The next example is for the standard eigenvalue problem and the preconditioned algorithm implicitly transforms it to a pencil problem. Example 4: A is the same matrix as in Example 2 We use a constant L k as obtained by the incomplete LDL T decompositions of A A with no ll-in. We compare it with the Jacobi-Davidson algorithm with the same kind of preconditioner. We also consider the shift-and- spectral transformed Lanczos algorithm (i.e. applying the Lanczos to A 1 ). We give in Figure 7 the convergence history of the residual kr k k of Algorithm 4 in solid lines from top down for and that of the Jacobi-Davidson algorithm in dot lines for with the corresponding marks. The residual for the spectral transformed Lanczos is given in dash-dot (with +) line. Figure 8 is the number of outer iterations and the total number of inner iterations vs. m. Again, preconditioning signicantly accelerates convergence and our result compares very favorably with the Jacobi-Davidson method. An interesting point here is that Algorithm 4 with based on incomplete factorization outperforms the shift-and-invert Lanczos algorithm. Although we do not suggest this is the case in general, it does underline the eectiveness of the preconditioned algorithm. Figure 5: Example 3 Outer and total inner iterations vs. m (+ - Algorithm 4; outer iterations (a) total inner iterations (b) 7 Concluding Remarks We have presented an inverse free Krylov subspace method that is based on the classical Krylov subspace projection methods but incorporates preconditioning for e-cient implementations. A convergence theory has been developed for the method and our preliminary tests of the preconditioned version demonstrate its competitiveness. Comparing with the existing methods, it has a relatively well understood theory and simple numerical behavior. We point out that the algorithm has a xed cost per outer iteration, which makes it easy to implement. For the future work, we will consider generalizations in three directions, namely, an e-cient block version for computing several eigenvalues simultaneously, a strategy to compute interior eigenvalues and an algorithm for the general nonsymmetric problem. A Appendix . Perturbation Bounds for Generalized Eigenvalue Problems We present a perturbation theorem that is used in the proof of Lemma 1 but might be of general interest as well. In the following, A; B; and E are all symmetric. Theorem 3 Let 1 be the smallest eigenvalue of (A; B) with x a corresponding unit eigenvector and let 1 be the smallest eigenvalue of with u a corresponding unit eigenvector, where min (E) Figure convergence with ideal preconditioning where min (E) and max (E) denote the smallest and the largest eigenvalues of E respectively. Proof Using the minimax characterization, we have z 6=0 z T z T Bz Similarly, min (E) which completes the proof. We note that 1=x T Bx (or 1=u T Bu ) is the Wilkinson condition number for 1 (or 1 ). These bounds therefore agree with the rst order analytic expansion and will be sharper than traditional bounds based on kBk. Figure 7: Example 4 residual convergence history for lines - Algorithm 4; dot lines - Jacobi-Davidson, dash-dot - shift-and-invert Lanczos) --R A subspace preconditioning algorithm for eigenvec- tor/eigenvalue computation The Davidson method Applied Numerical Linear Algebra Minimization of the computational labor in determining the Matrix Computations Inexact inverse iterations for the eigenvalue problems Large sparse symmetric eigenvalue problems with homogeneous linear constraints: the Lanczos process with inner-outer iterations Functional Analysis in Normed Spaces Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem. Preconditioned Eigensolvers - An oxymoron? Elec Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient method An inexact inverse iteration for large sparse eigenvalue problems The inexact rational Krylov sequence method The restarted Arnoldi method applied to iterative linear solvers for the computation of rightmost eigenvalues Computer Solution of Large Linear Systems Generalizations of Davidson's method for computing eigenvalues of sparse symmetric matrices Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems A geometric theory for preconditioned inverse iteration a new method for the generalized eigenvalue problem and convergence estimate Preprint The Symmetric Eigenvalue Problem Numerical Methods for Large Eigenvalue Problems Iterative Methods for Sparse Linear Systems A Jacobi-Davidson iteration method for linear eigenvalue problems A truncated RQ iteration for large scale eigenvalue calculations Robust preconditioning of large sparse symmetric eigenvalue problems Restarting techniques for the (Jacobi-)Davidson symmetric eigenvalue method Elec Dynamic thick restarting of the Davidson and implicitly restarted Arnoldi methods Inexact Newton preconditioning techniques for large symmetric eigenvalue problems Elec. Convergence analysis of an inexact truncated RQ iterations Elec. --TR --CTR James H. Money , Qiang Ye, Algorithm 845: EIGIFP: a MATLAB program for solving large symmetric generalized eigenvalue problems, ACM Transactions on Mathematical Software (TOMS), v.31 n.2, p.270-279, June 2005 P.-A. Absil , C. G. Baker , K. A. Gallivan, A truncated-CG style method for symmetric generalized eigenvalue problems, Journal of Computational and Applied Mathematics, v.189 n.1, p.274-285, 1 May 2006

preconditioning;eigenvalue problems;krylov subspace

587382

Nonlinearly Preconditioned Inexact Newton Algorithms.

Inexact Newton algorithms are commonly used for solving large sparse nonlinear system of equations $F(u^{\ast})=0$ arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of $\|F\|$, especially for problems with unbalanced nonlinearities, because the methods do not have built-in machinery to deal with the unbalanced nonlinearities. To find the same solution $u^{\ast}$, one may want to solve instead an equivalent nonlinearly preconditioned system ${\cal F}(u^{\ast})=0$ whose nonlinearities are more balanced. In this paper, we propose and study a nonlinear additive Schwarz-based parallel nonlinear preconditioner and show numerically that the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails.

Introduction . Many computational engineering problems require the numerical solution of large sparse nonlinear system of equations, i.e., for a given nonlinear vector u # R n , such that starting from an initial guess u (0) and Inexact Newton algorithms (IN) [7, 8, 11, 17] are commonly used for solving such systems and can briefly be described here. Suppose u (k) is the current approximate solution; a new approximate solution u (k+1) can be computed through the following steps: Algorithm 1.1 (IN). 1: Find the inexact Newton direction p (k) such that Step 2: Compute the new approximate solution Here # k is a scalar that determines how accurately the Jacobian system needs to be solved using, for example, Krylov subspace methods [2, 3, 11, 12]. # (k) is another scalar that determines how far one should go in the selected inexact Newton direction [7]. IN has two well-known features, namely, (a) if the initial guess is close enough to the desired solution then the convergence is very fast, and (b) such a good initial # Department of Computer Science, University of Colorado, Boulder, CO 80309-0430 (cai@cs.colorado.edu). The work was supported in part by the NSF grants ASC-9457534, ECS- and ACI-0072089, and by Lawrence Livermore National Laboratory under subcontract B509471. Department of Mathematics & Statistics, Old Dominion University, Norfolk, VA 23529-0077; ISCR, Lawrence Livermore National Laboratory, Livermore, CA 94551-9989; and ICASE, NASA Langley Research Center, Hampton, VA 23681-2199 (keyes@icase.edu). This work was supported in part by NASA under contract NAS1-19480 and by Lawrence Livermore National Laboratory under subcontract B347882. guess is generally very di#cult to obtain, especially for nonlinear equations that have unbalanced nonlinearities [19]. The step length # (k) is often determined by the components with the worst nonlinearities, and this may lead to an extended period of stagnation in the nonlinear residual curve; see Fig 5.2 for a typical picture and more in the references [4, 14, 16, 23, 27, 28]. In this paper, we develop some nonlinearly preconditioned inexact Newton algorithms Find the solution u # R n of (1.1) by solving a preconditioned system Here the preconditioner G : R n 1. If 2. G # F -1 in some sense. 3. G(F (w)) is easily computable for w # R n . 4. If a Newton-Krylov type method is used for solving (1.4), then the matrix-vector product (G(F (w))) # v should also be easily computable for As in the linear equation case [13], the definition of a preconditioner can not be given precisely, nor is it necessary. Also as in the linear equation case, preconditioning can greatly improve the robustness of the iterative methods, since the preconditioner is designed so that the new system (1.4) has more uniform nonlinearities. PIN takes the following Algorithm 1.2 (PIN). 1: Find the inexact Newton direction p (k) such that Step 2: Compute the new approximate solution Note that the Jacobian of the preconditioned function can be computed, at least in theory, using the chain rule, i.e., #F If G is close to F -1 in the sense that G(F (u)) # u, then #G #F I . In this case, Algorithm 1.2 converges in one iteration, or few iterations, depending on how close is G to F -1 . In fact, the same thing happens as long as G(F (u)) # Au, where A is constant matrix independent of u. On the other hand, if G is a linear function, then #G would be a constant matrix independent of u. In this case the Newton equation of the preconditioned system reduces to the Newton equation of the original system and G does not a#ect the nonlinear convergence of the method, except for the stopping conditions. However, G does change the conditioning of the linear Jacobian system, and this forms the basis for the matrix-free Newton-Krylov methods. Most of the current research has been on the case of linear G; see, for example, [4, 24]. In this paper, we shall focus on the case when G is the single-level nonlinear additive Schwarz method. As an example, we show the nonlinear iteration history, in Figure 5.2, for solving a two-dimensional flow problem with various Reynolds numbers using the standard IN (top) and PIN (bottom). It can be seen clearly that PIN is much less sensitive to the change of the Reynolds number than IN. Details of the numerical experiment will be given later in the paper. Nonlinear Schwarz algorithms have been studied extensively as iterative methods [5, 9, 20, 21, 22, 25, 26], and are known, at least experimentally, to be not very robust, in general, unless the problem is monotone. However, we show in the paper that nonlinear Schwarz can be an excellent nonlinear preconditioner. We remark that nonlinear methods can also be used as linear preconditioners as described in [6], but we will not look into this issue in this paper. Nested linear and nonlinear solvers are often needed in the implementation of PIN, and as a consequence, the software is much harder to develop than for the regular IN. Our target applications are these problems that are di#cult to solve using traditional Newton type methods. Those include (1) problems whose solutions have local singularities such as shocks or nonsmooth fronts; and (2) multi-physics problems with drastically di#erent sti#ness that require di#erent nonlinear solvers based on a single physics submodel, such as coupled fluid-structure interaction problems. The rest of the paper is organized as follows. In section 2, we introduce the nonlinear additive Schwarz preconditioned system and prove that under certain assumptions it is equivalent to the original unpreconditioned system. In section 3, we derive a formula for the Jacobian of the nonlinearly preconditioned system. The details of the full algorithm is presented in section 4, together with some comments about every step of the algorithm. Numerical experiments are given in section 5. In section 6, we make some further comments and discuss some future research topics along the line of nonlinear preconditioning. Several concluding remarks are given in section 7. 2. A nonlinear additive Schwarz preconditioner. In this section, we describe a nonlinear preconditioner based on the additive Schwarz method [5, 9]. Let be an index set; i.e., one integer for each unknown u i and F i . We assume that is a partition of S in the sense that Here we allow the subsets to have overlap. Let n i be the dimension of S i ; then, in general, Using the partition of S, we introduce subspaces of R n and the corresponding restriction and extension matrices. For each S i we define V i # R n as and a n-n restriction (also extension) matrix I S i whose kth column is either the kth column of the n - n identity matrix I n-n if k # S i or zero if k # S i . Similarly, let s be a subset of S; we denote by I s the restriction on s. Note that the matrix I s is always symmetric and the same matrix can be used as both restriction and extension operator. Many other forms of restriction/extension are available in the literature; however, we only consider the simplest form in this paper. Using the restriction operator, we define the subdomain nonlinear function as F. We next define the major component of the algorithm, namely the nonlinearly preconditioned function. For any given v # R n , define T as the solution of the following subspace nonlinear system We introduce a new function which we will refer to as the nonlinearly preconditioned F (u). The main contribution of this paper is the following algorithm. Algorithm 2.1. Find the solution u # of (1.1) by solving the nonlinearly preconditioned system with u (0) as the initial guess. Remark 2.1. In the linear case, this algorithm is the same as the additive Schwarz algorithm. Using the usual notation, if then where A is the subspace inverse of A in V i . Remark 2.2. The evaluation of the function F(v), for a given v, involves the calculation of the T i , which in turn involves the solution of nonlinear systems on S i . Remark 2.3. If the overlap is zero, then this is simply a block nonlinear Jacobi preconditioner. Remark 2.4. If (2.2) is solved with Picard iteration, or Richardson's method, then the algorithm is simply the nonlinear additive Schwarz method, which is not a robust algorithm, as is known from experience with linear and nonlinear problems. Assumption 2.1 (local unique solvability). For any s # S, we assume that is uniquely solvable on s. Remark 2.5. The assumption means that, for a given subset s, if both u and are solutions on s, i.e., and then if u| This assumption maybe a little too strong. A weaker assumption could be for s to be the subsets in the partition. In this case, the proof of the following theorem needs to be modified. Theorem 2.1. Under the local unique solvability assumption, the nonlinear systems and (2.2) are equivalent in the sense that they have the same solution. Proof. Let us first assume that u # is the solution of (1.1), i.e., F immediately implies that By definition, T i satisfies Comparing (2.3) and (2.4), and using the local unique solvability assumption, we must have Therefore, u # is a solution of (2.2). Next, we assume that u # is a solution of (2.2) which means that We prove that T i in two steps. First we show that T i equals zero in the nonoverlapping part of S, then we show that T i must equal each other in the overlapping part of S. be the nonoverlapping part of S, i.e., there exists one and only one i such that k # S i }, Obviously for any 1 # j # N . Taking ASSUMPTION 2.1, we have for any 1 # j # N . Due to the uniqueness, we must have for any 1 # i, j # N . Since the sum of T i (u # )| s is zero, and they all equal to each other, they must all be zero. Thus, . This is equivalent to saying that u # is a solution of (1.1). 3. Basic properties of the Jacobian. If (2.2) is solved using a Newton type algorithm, then the Jacobian is needed in one form or another. We here provide a computable form of it, and discuss some of its basic properties. Let J be the Jacobian of the original nonlinear system, i.e., and JS i , the Jacobian of the subdomain nonlinear system, i.e., N. Note that if F (-) is sparse nonlinear function, then J is a sparse matrix and so are the JS i . Unfortunately, the same thing cannot be said about the preconditioned function F(-). Its Jacobian, generally speaking, is a dense matrix, and is very expensive to compute and store as one may imagine. In the following discussion, we denote by and JS i the Jacobian of the preconditioned whole system, and the subsystems, respectively. Because of the definition of T i , JS i is a n-n matrix. components in S i , n independent variables u 1 , . , un , and its other n-n i components are zeros. Suppose we want to compute the Jacobian J at a given point u # R n . Consider one subdomain S i . Let S c be the complement of S i in S, we can write which is correct up to an re-ordering of the independent variables u and uS c u. Using the definition of T i (u), we have that Taking the derivative of the above function with respect to uS i , we obtain # I S i - which implies that assuming the subsystem Jacobian matrix #FS i is nonsingular in the subspace V i . Next, we take the derivative of (3.1) with respect to uS c which is equivalent to Note that since the sets S i and S c do not overlap each other. Combining (3.2) and (3.3), we obtain J. Summing up (3.6) for all subdomains, we have a formula for the Jacobian of the preconditioned nonlinear system in the form of J. (3.7) is an extremely interesting formula since it corresponds exactly to the additive Schwarz preconditioned linear Jacobian system of the original un-preconditioned equation. This fact implies that, first of all, we know how to solve the Jacobian system of the preconditioned nonlinear system, and second, the Jacobian itself is already well-conditioned. In other words, nonlinear preconditioning automatically o#ers a linear preconditioning for the corresponding Jacobian system. 4. Additive Schwarz preconditioned inexact Newton algorithm. We describe a nonlinear additive Schwarz preconditioned inexact Newton algorithm (AS- PIN). Suppose u (0) is a given initial guess, and u (k) is the current approximate so- lution; a new approximate solution u (k+1) can be computed through the following steps: Algorithm 4.1 (ASPIN). 1: Compute the nonlinear residual g through the following two steps: a) Find g (k) by solving the local subdomain nonlinear systems with a starting point g (k) b) Form the global residual c) Check stopping conditions on g (k) . Step 2: Form elements of the Jacobian of the preconditioned system Step 3: Find the inexact Newton direction p (k) by solving the Jacobian system approximately Step 4: Compute the new approximate solution where # (k) is a damping parameter. ASPIN may look a bit complicated, but as a matter of fact, the required user input is the same as that for the regular IN Algorithm 1.1, i.e., the user needs to supply only two routines for each subdomain: (1) the evaluation of the original function FS i (w). This is needed in both Step 1 a) and Step 2 if the Jacobian is to be computed using finite-di#erence methods. It is also needed in Step 4 in the line search steps. (2) the Jacobian of the original function JS i in terms of a matrix-vector multipli- cation. This is needed in both Step 1 a) and Step 3. We now briefly discuss the basic steps of the algorithm. In Step 1 a) of Algorithm 4.1, N subdomain nonlinear systems have to be solved in order to evaluate the preconditioned function F at a given point. More explicitly, we solve which has n i equations and n i unknowns, using Algorithm 1.1 with a starting value . Note that the vector u (k) is needed to evaluate GS i (#), for this requires the ghost points in a mesh-based software implementation. In a parallel implementation, the ghost values often belong to several neighboring processors and communication is required to obtain their current values. We note, however, that the ghost values do not change during the solution of the subdomain nonlinear system. In Step 2, pieces of the Jacobian matrix are computed. The full Jacobian matrix J never needs to be formed. In a distributed memory parallel implementation, the submatrices JS i are formed, and saved. The multiplication of J with a given vector is carried out using the submatrices JS i . Therefore the global J matrix is never needed. Several techniques are available for computing the JS i , for example, using an analytic multi-colored finite di#erencing, or automatic di#erentiation. A triangular factorization of JS i is also performed at this step and the resulting matrices are stored. In Step 3, the matrix should not considered as a linear preconditioner since it does not appear on the right-hand side of the linear system. However, using the additive Schwarz preconditioning theory, we know that for many applications the matrix J is well-conditioned, under certain conditions. We also note that if an inexact solver is used to compute w in Step 3, the Newton search direction would be changed and, as a result, the algorithm becomes an inexact Newton algorithm. As noted above the Jacobian system, in Step 3, does not have the standard form of a preconditioned sparse linear system However, standard linear solver software packages can still be used with some slight modification, such as removing the line that performs Since the explicit sparse format of J is often not available, further preconditioning of the Jacobian system using some of the sparse matrix based techniques, such as ILU, is di#cult. A particular interesting case is when the overlap is zero; then the diagonal blocks of J are all identities, therefore, do not involve any computations when multiplied with vectors. Let us take a two-subdomain case for example, and JS i 22 J 21 I # . The same thing can also be done for the overlapping case. This is a small saving when there many small subspaces. However, the saving can be big if there are relatively few subspaces, but the sizes are large. For example, in the case of a coupled fluid-structure interaction simulation, there could be only two subdomains; one for the fluid flow and one for the structure. In Step 4, the step length # (k) is determined using a standard line search technique [7] based on the function More precisely, we first compute the initial reduction Jp (k) . Then, # (k) is picked such that Here # is a pre-selected parameter (use The standard cubic backtracking algorithm [7] is used in our computations. 5. Numerical experiments. We show a few numerical experiments in this section using ASPIN, and compare with the results obtained using a standard inexact Newton's algorithm. We are mostly interested in the kind of problems on which the regular inexact Newton type algorithm does not work well. We shall focus our discussion on the following two-dimensional driven cavity flow problem [15], using the velocity-vorticity formulation, in terms of the velocity u, v, and the vorticity #, x y Fig. 5.1. A 9 - 9 fine mesh with 3 - 3 subdomain partition. The 'o' are the mesh points. The dashed lines indicate a 3 - nonoverlapping partitioning. The solid lines indicate the "overlapping = 1" subdomains. defined on the unit -#u- #y #x Re #x #y Here Re is Reynolds number. The boundary conditions are: . bottom, left and right: . top: We vary the Reynolds number in the experiments. The boundary condition on # is given by its definition: #y #x The usual uniform mesh finite di#erence approximation with the 5-point stencil is used to discretize the boundary value problem. Upwinding is used for the divergence (convective) terms and central di#erencing for the gradient (source) terms. To obtain a nonlinear algebraic system of equations F , we use natural ordering for the mesh points, and at each mesh point, we arrange the knowns in the order of u, v, and #. The partitioning of F is through the partitioning of the mesh points. In other words, the partition is neither physics-based nor element-based. Figure 5.1 shows a typical mesh, together with an overlapping partition. The subdomains may have di#erent sizes depending on whether they touch the boundary of # The size of the overlap is as indicated in Figure 5.1. Note that since this is mesh-point based partition, the zero overlap case in fact corresponds to the 1/2 overlap case of the element-based partition, which is used more often in the literature on domain decomposition methods for finite element problems [10]. The subdomain Jacobian matrices JS i are formed using a multi-colored finite di#erence scheme. The implementation is done using PETSc [1], and the results are obtained on a cluster of DEC workstations. Double precision is used throughout the computations. We report here only the machine independent properties of the algorithms. 5.1. Parameter definitions. We stop the global PIN iterations if used for all the tests. The global linear iteration for solving the global Jacobian system is stopped if the relative tolerance or the absolute tolerance is satisfied. In fact we pick # independent of k, throughout the nonlinear iterations. Several di#erent values of # global-linear-rtol are used as given in the tables below. At the kth global nonlinear iteration, nonlinear subsystems defined in Step 1 a) of Algorithm 4.1, have to be solved. We use the standard IN with a cubic line search for such systems with initial guess g (k) 0. The local nonlinear iteration in subdomain S i is stopped if one of the following two conditions is satisfied: )# local-nonlinear-rtol #FS i or )# local-nonlinear-atol . The overall cost of the algorithm depends heavily on the choice of # local-nonlinear-rtol . We report computation results using a few di#erent values for it. 5.2. Comparison with a Newton-Krylov-Schwarz algorithm. We compare the newly developed algorithm ASPIN with a well-understood inexact Newton algorithm using a cubic backtracking line search as the global strategy, as described in [7]. Since we would like to concentrate our study on the nonlinear behavior of the algorithm, not how the linear Jacobian systems are solved, the Jacobian systems are solved almost exactly at all Newton iterations. More precisely at each IN iteration, the Newton direction p (k) satisfies with GMRES with an one-level additive Schwarz preconditioner is used as the linear solver with the same partition and overlap as in the corresponding ASPIN algorithm. The history of nonlinear residuals is shown in Figure 5.2 (top) with several di#erent Reynolds numbers on a fixed fine mesh of size 128 - 128. 5.3. Test results and observations. As the Reynolds number increases, the nonlinear system becomes more and more di#cult to solve. The Newton-Krylov- Schwarz algorithm fails to converge once the Reynolds number passes the value 770.0 on this 128 - 128 mesh, no matter how accurately we solve the Jacobian system. Standard techniques for going further would employ pseudo time stepping [18] or nonlinear continuation in h or Re [28]. However, our proposed PIN algorithm converges for a much larger range of Reynolds numbers as shown in Figure 5.2. Furthermore, the number of PIN iterations does not change much as we increase the Reynolds number. A key to the success of the method is that the subdomain nonlinear problems are well solved. In Table 5.1, we present the numbers of global nonlinear PIN iterations and the numbers of global GMRES iterations per PIN iteration for various Reynolds numbers and overlapping sizes. Two key stopping parameters are # global-linear-rtol for the global linear Jacobian systems and # local-nonlinear-rtol for the local nonlinear systems. We test several combinations of two values 10 -6 and 10 -3 . As shown in the table, the total number of PIN iteration does not change much as we change # global-linear-rtol and # local-nonlinear-rtol ; however, it does increase from 2 or 3 to 6 or 9 when the Reynolds number increases from 1 to 10 4 . The bottom part of Table 5.1 shows the corresponding numbers of GMRES iterations per PIN iteration. These linear iteration numbers change drastically as we switch to di#erent stopping parameters. Solving the global Jacobian too accurately will cost a lot of GMRES iterations and not result in much savings in the total number of PIN iterations. Table 5.1 also compares the results with two sizes of overlap. A small number PIN iterations can be saved as one increases the overlapping size from 0 to 1, or more, as shown also in Table 5.3. The corresponding number of global linear iterations decreases a lot. We should mention that the size of subdomain nonlinear systems increases as one increases the overlap, especially for three dimensional problems. The communication cost in a distributed parallel implementation also increases as we increase the overlap. Recent experiments seem to indicate that small overlap, such as overlap=1, is preferred balancing the saving of the computational cost and the increase of the communication cost, see for example [10, 14]. Of course, the observation is highly machine and network dependent. In Table 5.2, we look at the number of Newton iterations for solving the subdomain nonlinear systems. In this test case, we partition the domain into 16 subdomains, 4 in each direction, and number them naturally from the bottom to top, and left to right. Four touch the moving lid. The solution of the problem is less smooth near the lid, especially when the Reynolds number is large. As expected, the subdomains near the lid need more iterations; two to three times more than what is needed in the smooth subdomains for the large Reynolds number cases. We next show how the iteration numbers change as we change the number of subdomains with a fixed 128 - 128 fine mesh. The results are displayed in Table 5.4. As we increase the number of subdomains from 4 to 16 the number of global PIN iterations does not change much; up or down by 1 is most likely due to the last bits of the stopping conditions rather than the change of the algorithm. Note that when we change the number of subdomains, the inexact Newton direction changes, and as a result, the algorithm changes. As a matter of fact, we are comparing two mathematically di#erent algorithms. The bottom part of Table 5.4 shows that the number of GMRES iterations per PIN increases quite a bit as we increase the number Global PIN iterations. Fine mesh 128 - 128, 4 - 4 subdomain partition on 16 processors. Subdomain linear systems are solved exactly. # global-linear-rtol is the stopping condition for the global GMRES iterations. # local-nonlinear-rtol is the stopping condition for the local nonlinear iterations. The absolute tolerances are # . The finite di#erence step size is 10 -8 . number of PIN iterations number of GMRES iterations per PIN of subdomains. 6. Some further comments. We comment on a few important issues about the newly proposed algorithm including parallel scalabilities and load balancing in parallel implementations. Parallel scalability is a very important issue when using linear or nonlinear iterative methods for solving problems with a large number of unknowns on machines with a large number of processors. It usually involves two separate questions, namely how the iteration numbers change with the number of processors and with the number of unknowns. It is a little bit surprising that, from our limited experience, the number of ASPIN iterations is not sensitive at all to either the number of processors or the number of unknowns. In other words, the number of nonlinear PIN iterations is completely scalable. However, this can not be carried over to the linear solver. To Total number of subdomain nonlinear iterations. Fine mesh 128 - 128, 4 - 4 subdomain partition on processors. Subdomains are naturally ordered. Subdomain linear systems are solved exactly. # is the stopping condition for the global GMRES iterations. is the stopping condition for the local nonlinear iterations. The absolute tolerances are # 1. The finite di#erence step size is 10 -8 . subdomain # Table Varying the overlapping size. Fine mesh 128 - 128, 4 - 4 subdomain partition on 16 pro- cessors. Subdomain linear systems are solved exactly. # is the stopping condition for the global GMRES iterations. # is the stopping condition for the local nonlinear iterations. The absolute tolerances are # . The finite di#erence step size is 10 -8 . GMRES/PIN make the linear solver scalable, a coarse grid space is definitely needed. Our current software implementation is not capable of dealing with the coarse space, therefore no further discussion of this issue can be o#ered at this point. Load balancing is another important issue for parallel performance that we do not address in the paper. As shown in Table 5.2, the computational cost is much higher in the subdomains near the lid than the other subdomains, in particular for the large Reynolds number cases. To balance the computational load, idealy, one should partition the domain such that these subdomains that require more linear/nonlinear iterations contain less mesh points. However, the solution dependent cost information is not available until after a few iterations, and therefore the ideal partition has to obtained dynamically as the computation is being carried out. Di#erent subdomain partitions with the same fine mesh 128 - 128. Subdomain linear systems are solved exactly. # is the stopping condition for the global GMRES iterations. # is the stopping condition for the local nonlinear iterations. The absolute tolerances are # 1. The finite di#erence step size is 10 -8 . number of PIN iterations subdomain partition number of GMRES iterations per PIN Table Di#erent fine meshes on 16 processors. Subdomain linear systems are solved ex- actly. # is the stopping condition for the global GMRES iterations. is the stopping condition for the local nonlinear iterations. The absolute tolerances are # 1. The finite di#erence step size is 10 -8 . number of PIN iterations fine mesh number of GMRES iterations per PIN We only discussed one partitioning strategy based on the geometry of the mesh and the number of processors available in our computing system. Many other partitioning strategies need to be investigated. For example, physics-based partitions: all the velocity unknowns as# 1 and the vorticity unknowns . In this case, the number of subdomains may have nothing to do with the number of processors. Further partitions may be needed on and# 2 for the purpose of parallel process- ing. One possible advantage of this physics-based partition is that the nonlinearities between di#erent physical quantities can be balanced. An extreme case of a mesh-based partition would be that each subdomain contains only one grid point. Then, the dimension of the subdomain nonlinear system is the same as the number of variables associated with a grid point, 3 for our test case. In this situation, ASPIN becomes a pointwise nonlinear scaling algorithm. As noted in linear scaling does not change the nonlinear convergence of Newton's method, but nonlinear scaling does. Further investigation should be of great interest. --R The Portable Hybrid Krylov methods for nonlinear systems of equations Convergence theory of nonlinear Newton-Krylov algorithms Parallel Newton- Krylov-Schwarz algorithms for the transonic full potential equation Domain decomposition methods for monotone nonlinear elliptic problems Nonlinearly preconditioned Krylov subspace methods for discrete Newton algorithms Numerical Methods for Unconstrained Optimization and Nonlinear Equations On the nonlinear domain decomposition method Domain decomposition algorithms with small overlap Globally convergent inexact Newton methods Choosing the forcing terms in an inexact Newton method Matrix Computations Globalized Newton-Krylov- Schwarz algorithms and software for parallel implicit CFD Numerical Computation of Internal and External Flows Robust linear and nonlinear strategies for solution of the transonic Euler equations Iterative Methods for Linear and Nonlinear Equations Convergence analysis of pseudo-transient continuation An analysis of approximate nonlinear elim- ination On the Schwarz alternating method. On the Schwarz alternating method. On Schwarz alternating methods for incompressible Navier-Stokes equations in n dimensions NITSOL: A Newton iterative solver for nonlinear systems Parallel Multilevel Methods for Elliptic Partial Di Rate of convergence of some space decomposition methods for linear and nonlinear problems Global convergence of inexact Newton methods for transonic flow A locally refined rectangular grid finite element method: Application to computational fluid dynamics and computational physics --TR --CTR Feng-Nan Hwang , Xiao-Chuan Cai, A parallel nonlinear additive Schwarz preconditioned inexact Newton algorithm for incompressible Navier-Stokes equations, Journal of Computational Physics, v.204 Heng-Bin An , Ze-Yao Mo , Xing-Ping Liu, A choice of forcing terms in inexact Newton method, Journal of Computational and Applied Mathematics, v.200 n.1, p.47-60, March, 2007 S.-H. Lui, On monotone iteration and Schwarz methods for nonlinear parabolic PDEs, Journal of Computational and Applied Mathematics, v.161 n.2, p.449-468, 15 December D. A. Knoll , D. E. Keyes, Jacobian-free Newton-Krylov methods: a survey of approaches and applications, Journal of Computational Physics, v.193 n.2, p.357-397, 20 January 2004

incompressible flows;nonlinear preconditioning;nonlinear additive Schwarz;inexact Newton methods;krylov subspace methods;domain decomposition;nonlinear equations;parallel computing

587383

On Two Variants of an Algebraic Wavelet Preconditioner.

A recursive method of constructing preconditioning matrices for the nonsymmetric stiffness matrix in a wavelet basis is proposed for solving a class of integral and differential equations. It is based on a level-by-level application of the wavelet scales decoupling the different wavelet levels in a matrix form just as in the well-known nonstandard form. The result is a powerful iterative method with built-in preconditioning leading to two specific algebraic multilevel iteration algorithms: one with an exact Schur preconditioning and the other with an approximate Schur preconditioning. Numerical examples are presented to illustrate the efficiency of the new algorithms.

Introduction The discovery of wavelets is usually described as one of the most important advances in mathematics in the twentieth century as a result of joint eorts of pure and applied mathematicians. Through the powerful compression property, wavelets have satisfactorily solved many important problems in applied mathematics e.g. signal and image processing; see [23, 20, 34, 38] for a summary. There remain many mathematical problems to be tackled before wavelets can be used for solution of dierential and integral equations in a general setting. The traditional wavelets were designed mainly for regular domains and uniform meshes. This was one of the reasons why wavelets may not be immediately applicable to arbitrary problems. The introduction of the lifting idea, interpolatory wavelets [35, 25, 1] and adaptivity [17] provides a useful way of constructing wavelets functions in non-regular domains and in high dimensions. However, the algebraic (sparse) structure of the matrix generated by a wavelet method is usually a nger-like one that is a di-cult sparse pattern to deal with; refer to [10, 13, 15, 18]. Firstly direct solution of a linear system with such a matrix is either not feasible or ine-cient. Secondly iterative solution requires a suitable preconditioner and this choice of preconditioner is usually dependent of the smoothness of the underlying operator (in addition to the assumptions for wavelets compression). Often a diagonal preconditioner is not su-cient. For some particular problems, several preconditioning techniques have been suggested. For instance, the nger matrix from wavelets representation of a Calderon-Zygmund operator plus a non-constant diagonal matrix cannot be preconditioned eectively by a diagonal matrix. In this case, one can use the idea of two-stage preconditioning as proposed in [13] or to use other modied wavelets methods such as a centring algorithm [15]; see also [38] for another modied algorithm and [19] for using approximate inverses. there exists a gap in realizing the full e-ciency oered by wavelet bases for model prob- lems. That is to say, a generally applicable iterative algorithm is still lacking. For a recent and general survey of iterative methods, refer to [31]. This paper proposes two related and e-cient iterative algorithms based on the wavelet formulation for solving an operator equation with conventional arithmetic. Both algorithms use the Schur complements recursively but dier in how to use coarse levels to solve Schur complements equations. In the rst algorithm, we precondition a Schur complement by using coarse levels while in the second we use approximate Schur complements to construct a preconditioner. We believe that our algorithms can be adapted to higher dimensional problems more easily than previous work in the subject. The motivation of this work follows from the observation that any 1-scale compressed results (matrices) can be conveniently processed before applying the next scale. In this way, regular patterns created by past wavelet scales are not destroyed by the new scales like in the non-standard (NS) form [10] and unlike in the standard wavelet bases; we dene the notation and give further details in Section 2. This radical but simple idea will be combined in Section 3 with the Schur complement method and Richardson iterations in a multi-level iterative algorithm. Moreover the Richardson iterations can be replaced by a recursive generalized minimal residuals (GMRES) method [30]. The essential assumption for this new algorithm to work is the invertibility of an approximate band matrix; in the Appendix we show that for a class of Calderon-Zygmund and pseudo-dierential operators such an invertibility is ensured. In practice we found that our method works equally well for certain operators outside the type for which we can provide proofs. In Section 4, we present an alternative way of constructing the preconditioner by using approximate Schur complements. Section 5 discusses the complexity issues while Section 6 presents several numerical experiments to illustrate the eectiveness of the two new algorithms. We remark that our rst algorithm is similar to the framework of a NS form reformulation of the standard wavelets bases (based on the pyramid algorithm) but does not make use of the NS form itself, although our algorithm avoids a nger matrix (just like a NS form method) that could arise from overlapping dierent wavelet scales. The NS form work was by Beylkin, Coifman and Rokhlin [10] often known as the BCR paper. As a by-product, the NS form reduces the ops from O(n log n) to O(n). However, the NS form does not work with conventional arithmetic although operations with the underlying matrix (that has a regular sparse pattern) can be specially designed; in fact the NS form matrix itself is simply singular in conventional arithmetic. The recent work in [22] has attempted to develop a direct solution method based on the NS form that requires a careful choice of a threshold; here our method is iterative. In the context of designing recursive sparse preconditioners, it is similar to the ILUM type preconditioner to a certain extent [33]. Our second algorithm is similar to the algebraic multi-level iteration methods (AMLI) that were developed for nite elements [4, 2, 3, 36]; here our method uses wavelets and does not require estimating eigenvalues. Wavelets splitting of an operator This section will set up the notation to be used later and motivate the methods in the next sections. We rst introduce the standard wavelet method. For simplicity, we shall concentrate on the Daubechies' order m orthogonal wavelets with low pass lters c lters (such that d In fact, the ideas and expositions in this paper apply immediately to the more general bi-orthogonal wavelets [20]. Following the usual setting of [10, 20, 22, 11, 34], the lter coe-cients c j 's and d j 's dene the scaling function (x) and the wavelet function (x). Further, dilations and translations of (x) and (x) dene a multi-resolution analysis for L 2 in d-dimensions, in particular, (R d where the subspaces satisfy the relations In numerical realisations, we select a nite dimension space V 0 (in the nest scale) as our approximation space to the innite decomposition of L 2 in (1) i.e. eectively use to approximate L 2 (R d ). Consequently for a given operator its innite and exact operator representation in wavelet bases is approximated in space V 0 by where are both projection operators. For brevity, dene operators Then one can observe that A further observation based on T is that the wavelet coe-cients of T j 1 will be equivalently generated by the block operator Now we change the notation and consider the discretization of all continuous operators. Dene A as the representation of operator T 0 in space V 0 . Assume that A on the nest level is of dimension the dimension of matrices on a coarse level j is '. The operator splitting in (3) for the case of dimensions can be discussed similarly [10, 22]) corresponds to the two-dimensional wavelet transform e where the one level transform from j 1 to j (for any with rectangular matrices P j and Q j (corresponding to operators P j and Q j ) dened respectively as For a class of useful and strongly elliptic operators i.e. Calderon-Zygmund and pseudo-dierential operators, it was shown in BCR [10] that matrices A k;i are indeed 'sparse' satisfying the decaying property c m;j m;j is a generic constant depending on m and j only. To observe a relationship between the above level-by-level form and the standard wavelet rep- resentation, dene a square matrix of size 0 I j . Then the standard wavelet transform can be written as that transforms matrix A into e Figure 1: The level-by-level form (left) versus the non-standard wavelet form [10] (right) 22464160208Thus the diagonal blocks of e A are the same as A j 's of a level-by-level form. However the o- diagonal blocks of the former are dierent from B j and C j of the latter. To gain some insight into the structure of the o-diagonal blocks of matrix e A with the standard wavelet transform, we consider the following case of wavelets. Firstly after level 1 transform, we obtain e nn Secondly after level 2 transform, we get e e nn Finally after level 3 transform, we arrive at e e I 3n=4 nn Clearly the o-diagonal blocks of e A 3 are perturbations of that of the level-by-level form o-diagonal blocks in fact the one-sided transforms for the o-diagonal blocks are responsible for the resulting (complicated) sparsity structure. This can be observed more clearly for a typical example with Fig.1 where the left plot shows the level-by-level representation set-up that will be used in this paper and in Fig.2 where the left plot shows the standard wavelet representation as in (8). Figure 2: The standard wavelet form representation (left) versus an alternative centering form [15] (right) for the example in Figure 1 12864112Motivated by the exposition in (8) of the standard form, we shall propose a preconditioning and iterative scheme that operates on recursive 1-level transforms. Thus it will have the advantage of making full use of the NS form idea and its theory while avoiding the problem of a non-operational NS form matrix. Remark 1 Starting from the level-by-level set-up, taking T ' and the collection of all triplets 1j' as a sparse approximation for T 0 is the idea of the NS form [10, 22]. By way of comparison, in Fig.1, the NS form representation versus the level-by-level form are shown. It turns out that this work uses the identical set-up to the NS form without using the NS form formulation itself because we shall not use the proposed sparse approximation. Note that the centering algorithm [15] (see the right plot in Fig.2) is designed as a permutation of the standard wavelet form (see the left plot of Fig.2) and is only applicable to a special class of problems where its performance is better. 3 An exact Schur preconditioner with level-by-level wavelets We now present our rst and new recursive method for solving the linear system dened on the nest scale V 0 i.e. A is of size 0 discussed in the previous section, and x Instead of considering a representation of T 0 in the decomposition space (2) and then the resulting linear system, we propose to follow the space decomposition and the intermediate linear system in a level-by-level manner. A sketch of this method is given in Fig.3 (the left plot) where we try to show a relationship between the multi-resolution (MR for wavelet representation) and the multi-level (ML for preconditioning via Schur) ideas from the nest level (top) to the coarsest level (bottom). Figure 3: Illustration of Algorithms 1 (left) and 2 (right). Here we take the nest level and 3 the coarsest level), use '2' to indicate a DWT step (one level of wavelets) and ' to denote the direct solution process on the coarsest level. The arrows denote the sequence of operations (written on the arrowed lines) with each algorithm interacting the two states (two columns on the plots) of multi-resolution wavelets and multi-level Schur decomposition. The left plot shows that for Algorithm 1, a Richardson step (or GMRES) takes the results of a DWT step to the next level via the Schur decomposition while the right plot shows that the Schur decomposition takes the results of a DWT step to the next level via a Schur approximation.2 Richardson Schur LU Schur LU Schur LU Direct Solution MR/Wavelets ML/Schur2 Approximation Schur LUApproximation Schur LUSchur LU Direct Solution MR/Wavelets ML/Schur Firstly at level 0, we consider V and the wavelet transform (4) yields e where e x e nn following the general result in (5), it is appropriate to consider the approximation of A band matrices. To be more precise, let B (D) denote a banded matrix of D with semi-bandwidth where integer 0. Dene A suitable (to be specied later). Then matrix e can be approximated by nn Or equivalently matrix nn is expected to be small in some norm (refer to the Appendix). Write equation (10) as Consequently we propose to use M our preconditioner to equation (14). This preconditioner can be used to accelerate iterative solution; we shall consider two such methods: the Richardson method and the GMRES method [30]. The most important step in an iterative method is to solve the preconditioning equation: or in a decomposed form A 1 y (1)y (2)! r (1)r (2)! Using the Schur complement method we obtain y (2)= z 2 y (1) Here the third equation of (17), unless its dimension is small (i.e. V 1 is the coarsest scale), has to be solved by an iterative method with the preconditioner T 1 ; we shall denote the preconditioning step by where T 1 is of size 1 1 . This sets up the sequence of a multilevel method where the main characteristic is that each Schur complements equation in its exact form is solved iteratively with a preconditioner that involves coarse level solutions. At any level j (1 j < '), the solution of the following linear system with T j of size j j and through solving e can be similarly reduced to that of with T j+1 of size j+1 j+1 . The solution procedure from the nest level to the coarsest level can be illustrated by the following diagram (for Transform Schur Precondition j+1 j+1 A j+1 z y (2) y (1) where as with (12) and (13) A j+1 B j+1 nn nn The coarsest level is set up in the previous section, where a system like (20) is solved by a direct elimination method. As with conventional multi-level methods, each ne level iteration leads to many coarse level iteration cycles. This can be illustrated in Fig. 4 where (bottom plot) are assumed and at the coarsest level ( direct solution is used. In practice, a variable may be used to achieve certain accuracy for the preconditioning step i.e. convergence up to a tolerance is pursued whilst by way of comparison smoothing rather convergence is desired in an usual multilevel method. Our experiments have shown that are often su-cient to ensure the overall convergence. We now summarise the formulation as an algorithm. The iterative solver for (19) at level i can be the Richardson method or the GMRES method [30] for solving e (or actually a combination of the two). For simplicity and generality, we shall use the word \SOLVE" to denote such an iterative solver (either Richardson or GMRES). Algorithm 1 (Recursive I) 1. and start on the nest level. 2. Apply one level DWT to T j x to obtain ~ 3. Use j steps of SOLVE for ~ 4. In each step, implement the preconditioner Restrict to the coarse level: A j+1 z y (2) y (1) Figure 4: Iteration cycling patterns of Algorithm 1 with levels: top for 3. In each case, one solves a ne level equation (starting from the nest level 0) by iteratively solving coarser level equations times; on the coarsest level (here level 3) a direct solution is carried out.2020 5. Use SOLVE for the above third equation with the preconditioner T j+1 i.e. solve T j+1 x b j+1 . 7. If (on the coarsest level), apply a direct solver to T j x and proceed with Step 8; otherwise return to Step 2. 8. 9. Interpolate the coarse level solution to the ne level j: x (2) x (1) x (2) x (1) 10. Apply one level inverse DWT to e y j to obtain y j . 11. If (on the nest level), check the residual error | if small enough accept the solution x 0 and stop the algorithm. If j > 0, check if j steps (cycles) have been carried out; if not, return to Step 2 otherwise continue with Step 8 on level j. The rate of convergence of this algorithm depends on how well the matrix T j approximates j and this approximation is known to be accurate for a suitable and for a class of Calderon- Zygmund and pseudo-dierential operators [10]. For this class of problems, it remains to discuss the invertibility of matrix A j which is done in the Appendix; a detailed analysis on T j T j may be done along the same lines as Lemma 1. For other problem classes, the algorithm may not work at all for the simple reason that A j may be singular e.g. the diagonal of matrix may have zero entries. Some extensions based on the idea of [13] may be applied as discussed in Section 6. Remark 2 We remark that for a class of general sparse linear systems, Saad, Zhang, Botta, Wubs et al [28, 12, 32, 33] have proposed a recursive multi-level preconditioner (named as ILUM) similar to this Algorithm 1. The rst dierence is that we need to apply one level of wavelets to achieve a nearly sparse matrix while these works start from a sparse matrix and permute it to obtain a desirable pattern suitable for Schur decomposition. The second dierence is that we propose an iterative step before calling for the Schur decomposition while these works try to compute the exact Schur decomposition approximately. Therefore it is feasible to rene our Algorithm 1 to adopt the ILUM idea (using independent sets) for other problem types. However one needs to be careful in selecting the dimensions of the leading Schur block if a DWT is required for compression purpose. 4 An approximate Schur preconditioner with level-by-level wavelets In the previous algorithm, we use coarse level equations to precondition the ne level Schur complement equation. We now propose an alternative way of constructing a preconditioner for a ne level equation. Namely we approximate and compute the ne level Schur complement before employing coarse levels to solve the approximated Schur complement equation. A sketch of this method is shown in Fig.3 showing the natural coupling of wavelet representation (level-by-level form) and Schur complement. symbol '2'. To dierentiate from Algorithm 1, we change the notation for all matrices. At any level k for consider the solution (compare to (19)) A Applying 1-level of DWT, we obtain A A (k) Note that we have the block LU decomposition A (k) A (k) I #" I A (k)1 A (k)0 S (k) 22 A (k) A (k) 12 is the true Schur complement. To approximate this Schur complement, we must consider approximating the second term in the above S (k) . We propose to form band matrix approximations For level these approximations are possible for a small bandwidth ; see Appendix. Seeking a band approximation to the inverse of A (k) makes sense because A (k) is expected to have a decaying property (refer to (5)). Let S denote the set of all matrices that have the sparsity pattern of a band matrix B ). The formation of a sparse approximate inverse (SPAI) is to nd a band matrix B 11 2 S such that min Refer to [8, 24, 16]. Brie y as with most SPAI methods, the use of F-norm decouples the minimisation into least squares (LS) problems for individual columns c j of B 11 . More precisely, owning to the j-th LS problem is to solve A (k) which is not expensive since c j is sparse. Once B 11 is found, dene an approximation to the true Schur complement S (k) as 22 A (k) and set A This generates a sequence of matrices A (k) . Now comes the most important step about the new preconditioner. Setting M the ne level preconditioner M (0) is dened recursively by I #" I B (k) is an approximation to the true Schur complement S (k) of A (k) . Here Observe that this preconditioner is dened through the V-cycling pattern recursively using the coarse levels. To go beyond the V-cycling, we propose a simple residual correction idea. We view the solution y [j] k of the preconditioning equation (compare to (15) and (9)) as an approximate solution to the equation Then the residual vector is k . This calls for a repeated solution M r k and gives the correction and a new approximate solution to (25): In practice we take see the top plot in Fig.4) as our experiments suggest that 2 is su-cient to ensure the overall convergence. Thus an essential feature of this method dierent from Algorithm 1 is that every approximated Schur complement matrix needs to be transformed to the next wavelet level in order to admit the matrix splitting (23) while inverse transforms are needed to pass coarse level information back to a ne level as illustrated in Fig.3. Ideally we may wish to use A 22 A (k) the approximate Schur complement but taking A (k) 21 and A (k) 12 as full matrices would jeopardize the e-ciency of the overall iterative method. We proposed to use band matrices (or thresholding) to approximate these two quantities just as in (13) with Algorithm 1. To summarise, the solution of the preconditioning equation from the nest level to the coarsest level and back up is illustrated in Fig.5 for means the same entry and exit point. The general algorithm for solving M (0) y can be stated as follows: Algorithm 2 (Recursive II) 1. Apply one level DWT to T k to obtain A (k) (k)2. Find the approximate inverse B (k) 3. Generate the matrix T 22 A (k) 12 . Solution Stage 1. and start on the nest level. 2. Apply one level DWT to r k and consider ~ 3. Solve the preconditioning equation M (k) ~ y by Restricting to the coarse level: r (2) 4. Solve for the above third equation at the next level T k+1 y 5. (on the coarsest level), apply a direct solver to T k y and proceed with Step 8; otherwise return to Step 2. 7. Set k := k 1. 8. Interpolate the coarse level k solution to the ne level k: e y (2) y (1) i:e: e y (1) e y (2) e y (1) 9. Apply one level inverse DWT to e y k to obtain y k . 10. When (on the nest level), check the residual error | if small enough accept the solution y 0 and stop the algorithm. When k > 1, check if cycles have been carried out; if not, nd the residual vector and return to Step 2 otherwise continue with Step 7 on level k. Remark 3 It turns out that this algorithm is similar to the algebraic multi-level iteration methods (AMLI) that was developed for a class of symmetric positive denite nite element equations in a hierarchical basis [4, 2, 3, 36]. In fact, let and then S (k) is implicitly dened by the following I P A (k+1) where P is a degree polynomial satisfying As with AMLI, for the valid choice P 1 gives rise to the V-cycling pattern and For > 1, the polynomial P (t) is chosen to improve the preconditioner; ideally A (k) O(1) asymptotically. Refer to these original papers about how to work out the coe-cients of P (t) based on eigenvalue estimates. However we do not use any eigenvalue estimates to construct P . We also remark that an alternative denition of a recursive preconditioner dierent from AMLI is the ILUM method as mentioned in Remark 2, where in a purely algebraic way (using independent sets of the underlying matrix graph) A (k) 11 is dened as a block diagonal form after a suitable permutation. This would give rise to another way of approximating the true Schur complement A (k). However the sparsity structures of blocks A (k)and A (k)will aect the density of nonzeros in matrix S (k) and an incomplete LU decomposition has to be pursued as in [28, 12, 33]. 5 Complexity analysis Here we mainly compare the complexity of Algorithms 1-2 in a full cycle. Note that both algorithms can be used by a main driving iterative solver where each step of iteration will require n 2 ops (one op refers to 1 multiplication and 1 addition) unless some fast matrix vector multiplication methods are used. One way to reduce this op count is to use a small threshold so that only a sparse form of e A is stored. Also common to both algorithms are the DWT steps which are not counted here. To work out op counts for diering steps of the two algorithms, we list the main steps as follows Algorithm 1 Algorithm 2 Main 3 band solves: 3n i 2 2 band-band multiplications: 8n i 2 step 4 band-vector multiplications: 8n i 4 band-vector multiplications: 8n i 3 o-band-vector multiplications (R i Therefore an -cycling (iteration) across all levels would require these ops (assuming and after ignoring the low order terms. Therefore we obtain F II =F I 9 for a typical situation with I 10. Thus we expect Algorithm I to be cheaper than II if the same number of iteration steps are recorded. Here by Algorithm I we meant the use of a Richardson iteration (in SOLVE of Algorithm 1); however if a GMRES iteration is used for preconditioning then the op count will increase. Of course as is well known, op count is not always a reliable indicator for execution speed; especially if parallel computing is desired a lot of other factors have to be considered. Remark 4 For sparse matrices, all op counts will be much less as DWT matrices are also sparse. The above complexity analysis is done for a dense matrix case. Even in this case, setting up preconditioners only adds a few equivalent iteration steps to a conventional iteration solver. One can usually observe overall speed-up. For integral operators, the proper implementation is to use the biorthorgonal wavelets as trial functions to yield sparse matrices A directly (for a suitable threshold); in this case a dierent complexity analysis is needed as all matrices are sparse and experiments have shown that although this approach is optimal a much larger complexity constant (independent of n) is involved. For structured dense matrices where FFT is eective, wavelets may have to be applied implicitly to preserve FFT representation. Further work is in progress. 6 Numerical experiments Here we shall compare the new algorithms with two previous methods: the WSPAI method by Chan-Tang-Wan [14] (CTW) and the two stage method by Chan-Chen [13] (CC). Further comparisons with other methods such as SPAI and ILU type can be found in [14] and [39]. Note that WSPAI, where applicable, is faster that SPAI and ILU. We now present numerical experiments from two sets of problems solved by these 4 methods: M1 | Algorithm 1 with the SOLVE step replaced by a Richardson iteration method for steps on each level. M2 | Algorithm 1 with the SOLVE step replaced by a GMRES iteration method on each level; in particular GMRES(25) for the main nest level iteration and GMRES() for coarser levels. M3 | Algorithm 1 with the SOLVE step replaced by a GMRES(25) iteration method on the nest level (outer iteration method) and a Richardson iteration method for steps on coarser levels. M4 | Algorithm 2 with the main iteration method being a GMRES(25)iteration method on the nest level and steps of residual correction on all levels for preconditioning. The test problems are the following: { Example 1. Symmetric case [10]: A { Example 2. Symmetric case log ji Ljlog jj Lj 6 otherwise Set 2: { Example 3. Unsymmetric case: A { Example 4. An anisotropic PDE problem in both x and y directions: where the coe-cients are dened as ([39, 13, Ch.5]) 100 (x; y) 2 [0; 0:5] [0; 0:5] or [0:5; 1] [0:5; 1] 100 (x; y) 2 [0; 0:5] [0:5; 1] or [0:5; 1] [0; 0:5] Table 1: Number of iteration steps to reach 1). Here is for cycling pattern, n is the problem size and ' denotes the number of levels used. Method Size Levels Case of Case of used n ' Steps Steps { Example 5. A discontinuous coe-cient PDE problem as tested in [14, 39, 13]: where the coe-cients are dened as ([39, Ch.3]) Here set 1 problems fall into the class for which our algorithms are expected to work; we mainly test the dependence of the variants of Algorithms 1-2 on the parameter choices. This is because for symmetric positive denite (SPD) matrices, their principal submatrices are also SPD and invertible so the assumptions of Lemma 1 (Appendix) are satised. Set 2 problems, although outside the class of problems that we have provided an analysis, are solvable by previously studied wavelets preconditioners so we include these for comparison purpose. For all cases, we stop an iterative method when the residual in the 2-norm is reduced by . The coarsest level is xed at Tables 1-2 display the numerical results for dierent problem sizes from solving set 1 problems, where 'Steps' means the number of iteration steps required. Clearly one can observe that 'Steps' is approximately independent of the problem size which is usually expected of a good preconditioner. For set 2 problems, we do not expect M1 { M4 to work. To extend these methods, we propose to combine one step of Stage-1 preconditioning as proposed in [13] to smooth out the given matrix (w.r.t. level 0 to 1). Specically we select a diagonal matrix such that the sum of diagonal entries of B 1 and C 1 in is minimised. Then our algorithms can be applied to the new linear system D 1 points to one way of possible further work. Other approaches similar to stage 1 preconditioning Table 2: Number of iteration steps to reach 1). Here is for cycling pattern, n is the problem size and ' denotes the number of levels used. Method Size Levels Case of Case of used n ' Steps Steps 512 43 may be considered e.g. for some indenite problems we have found that the permutation idea by Du [21] can be combined with our algorithms here; the idea was recently used in [9] to extend the applicability of sparse approximate inverse preconditioners. In Table 3, we use 'Stage-1' to indicate if such a step has been carried out; whenever such an one-o stage 1 diagonal preconditioning is used, we put \Yes" in this column. The symbol '*' denotes a case where no convergence has been achieved after 100 steps. Clearly the simple idea of Stage-1 preconditioning does improve the performance of M1{M4 except for M1 (although M2, M4 appear to be more robust than M3). Therefore we shall only compare M2{M4 with the work of CTW [14] and CC [13] next. Finally in Table 4, we show results from solving Examples 4 and 5 where the CPU seconds are obtained from a Sun Ultra-2 workstation (using Matlab 5) and the other notation is the same as in Table 3. Here 'diag' refers to the diagonal preconditioner which does not work for these two examples and data with a little 'f' in column 5 are used in constructing Figs.6 and 7. Table 4 demonstrates that when combined with stage-1 preconditioning, the new algorithms can out perform the previous methods. In particular, it appears that M3 is the fastest method in the table. To compare the residuals and CPU time of M3 (the best case), CTW [14] and CC [13] in solving Examples 4 and 5, we plot respectively in Fig.6 and Fig.7 the convergence history and CPU time (all data are taken from Table 4), where one can observe that M3 (New I) out performs the others. This again conrms that the new algorithm M3 converges the fastest. As remarked already, comparisons with other non-wavelets preconditioners (e.g. SPAI and ILU) can be found in [14] and [39] where it was concluded that WSPAI is faster. As also remarked earlier, the proposed multi-level algorithms can be potentially developed much further by incorporating the ideas from [6, 13, 9, 21]. More importantly as they are closely based on wavelets compression theory, generalizations to multi-dimensions appear to be more straightforward than similar and known wavelet preconditioners; these aspects are currently being investigated. Table 3: Number of iteration steps to reach Note that this example is outside the scope of M1 M4. Here is for cycling pattern, n is the problem size and ' denotes the number of levels used. Method With Size Levels Case of Case of used Stage-1 n ' Steps Steps Table 4: Comparison of the new algorithms with previous work for Examples 4 5 (set 2). Data indicated by 'f' in column 5 are used in Figs.6-7. Problem Stage-1 Method Convergence Steps CPU Diag Diag CC [13] 54 f 302 Conclusions This paper has presented two related algorithms implementing an algebraic wavelet preconditioner. The rst one is similar to the set up of a NS form representation of a wavelet basis while the second resembles the AMLI preconditioner designed for nite elements. Both algorithms are observed to give excellent performance for a class of symmetric positive denite Calderon-Zygmund and pseudo- dierential operators. Combined with a minor stage 1 preconditioning step, they are immediately applicable to problems outside this class of problems. We note that there are several methods that are designed to deal with anisotropic and highly indenite elliptic problems [6, 9, 21] and alternative methods of constructing a sparse preconditioner [33, 12]; these should be investigated in the near future to further extend the multi-level preconditioner to an even wider class of problems. Acknowledgements This work is partially supported by grants NSF ACR 97-20257, NASA Ames NAG2-1238, Sandia Lab LG-4440 and UK EPSRC GR/R22315. The second author wishes to thank the Department of Mathematics, UCLA for its hospitality during his visits conducting part of this work. --R A wavelet-based approach for the compression of kernel data in large scale simulations of 3D integral problems Algebraic multilevel preconditioning methods I Algebraic multilevel preconditioning methods II Algebraic multilevel iteration method for Stieljes matrices The algebraic multilevel iteration methods On the additive version of the algebraic multilevel iteration method for anisotropic elliptic problems Hierarchical bases and the Preconditioning highly inde Fast wavelet transforms and numerical algorithms I Fast wavelet transforms for matrices arising from boundary element methods Matrix renumbering ILU: An e Wavelet sparse approximate inverse preconditioners Discrete wavelet transforms accelerated sparse preconditioners for dense boundary element systems An analysis of sparse approximate inverse preconditioners for boundary integral equa- tions Adaptive wavelet methods for elliptic operator equations - convergence rates Wavelet methods for second-order elliptic problems Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition Wavelet and multiscale methods for operator equations LU factorization of non-standard forms and direct multiresolution solvers Multiresolution representation and numerical algorithms: a brief review On a family of two-level preconditionings of the incomplete block factorization type Algebraic multilevel iteration preconditioning technique Preconditioning of inde ILUM: a multi-elimination ILU preconditioner for general sparse matrices Iterative Methods for Sparse Linear Systems GMRES: a generalized minimal residual algorithm for solving unsymmetric linear systems Iterative solution of linear systems in the 20th century BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems Wavelets and Filter Banks The lifting scheme: a construction of second generation of wavelets On two ways of stabilizing the hierarchical basis multilevel methods Nearly optimal iterative methods for solving Wavelet Transforms and PDE Techniques in Image Compression Scalable and multilevel iterative methods On the multi-level splitting of nite element spaces --TR

schur complements;multiresolution;sparse approximate inverse;multilevel preconditioner;wavelets;level-by-level transforms

587394

An Algebraic Multilevel Multigraph Algorithm.

We describe an algebraic multilevel multigraph algorithm. Many of the multilevel components are generalizations of algorithms originally applied to general sparse Gaussian elimination. Indeed, general sparse Gaussian elimination with minimum degree ordering is a limiting case of our algorithm. Our goal is to develop a procedure which has the robustness and simplicity of use of sparse direct methods, yet offers the opportunity to obtain the optimal or near-optimal complexity typical of classical multigrid methods.

Introduction . In this work, we develop a multilevel multigraph algorithm. Algebraic multigrid methods are currently a topic of intense research interest [17, 18, 20, 46, 12, 48, 38, 11, 44, 3, 4, 1, 2, 5, 16, 7, 29, 28, 27, 42, 41, 21]. An excellent recent survey is given in Wagner [49]. In many \real world" calculations, direct methods are still widely used [6]. The robustness of direct elimination methods and their simplicity of use often outweigh the apparent benets of fast iterative solvers. Our goal here is to try to develop an iterative solver that can compete with sparse Gaussian elimination in terms of simplicity of use and robustness and to provide the potential of solving a wide range of linear systems more e-ciently. While we are not yet satised that our method has achieved this goal, we believe that it is a reasonable rst step. In particular, the method of general sparse Gaussian elimination with minimum degree ordering is a point in the parameter space of our method. This implies that in the worst case, our method defaults to this well-known and widely used method, among the most computationally e-cient of general sparse direct methods [26]. In the best case, however, our method can exhibit the near optimal order complexity of the classical multigrid method. Our plan is to take well studied, robust, and widely used procedures and data structures developed for sparse Gaussian elimination, generalize them as necessary, and use them as the basic components of our multilevel solver. The overall iteration follows the classical multigrid V-cycle in form, in contrast to the algebraic hierarchical basis multigraph algorithm developed in [11]. In this work we focus on the class of matrices which are structurally symmetric; that is, the pattern of nonzeros in the matrix is symmetric, although the numerical values of the matrix elements may render it nonsymmetric. Such structurally symmetric matrices arise in the discretizations of partial dierential equations, say, by the nite element method. For certain problems, the matrices are symmetric and positive denite, but for others the linear systems are highly nonsymmetric and/or indenite. Thus in practice this represents a very broad class of behavior. While our main interest is in scalar elliptic equations, as in the nite element code PLTMG [8], our algorithms can formally be applied to any structurally symmetric, nonsingular, sparse matrix. Sparse direct methods typically have two phases. In the rst (initialization) phase, Department of Mathematics, University of California at San Diego, La Jolla, CA 92093. The work of this author was supported by the National Science Foundation under contract DMS-9706090. y Bell Laboratories, Lucent Technologies, Murray Hill, NJ 07974. equations are ordered, and symbolic and numerical factorizations are computed. In the second (solution) phase, the solution of the linear system is computed using the factorization. Our procedure, as well as other algebraic multilevel methods, also breaks naturally into two phases. The initialization consists of ordering, incomplete symbolic and numeric factorizations, and the computation of the transfer matrices between levels. In the solution phase, the preconditioner computed in the initialization phase is used to compute solution using the preconditioned composite step conjugate gradient (CSCG) or the composite step biconjugate gradient (CSBCG) method [9]. Iterative solvers often have tuning parameters and switches which require a certain level of a priori knowledge or some empirical experimentation to set in any particular instance. Our solver is not immune to this, although we have tried to keep the number of such parameters to a minimum. In particular, in the initialization phase, there are only three such parameters: , the drop tolerance used in the incomplete factorization (called dtol in our code). maxf il, an integer which controls to overall ll-in (storage) allowed in a given incomplete factorization. maxlvl, an integer specifying the maximum number of levels. (The case corresponds to sparse Gaussian elimina- tion.) In the solution phase, there are only two additional parameters: tol, the tolerance used in the convergence test. maxcg, an integer specifying the maximum number of iterations. Within our code, all matrices are generally treated within a single, unied frame- work; e.g., symmetric positive denite, nonsymmetric, and indenite problems generally do not have specialized options. Besides the control parameters mentioned above, all information about the matrix is generated from the sparsity pattern and the values of the nonzeros, as provided in our sparse matrix data structure, a variant of the data structure introduced in the Yale sparse matrix package [23, 10]. For certain block matrices, the user may optionally provide a small array containing information about the block structure. This input limits the complexity of the code, as well as eliminates parameters which might be needed to further classify a given matrix. On the other hand, it seems clear that a specialized solver directed at a specic problem or class of problems, and making use of this additional knowledge, is likely to outperform our algorithm on that particular class of problems. Although we do not think our method is provably \best" for any particular problem, we believe its generality and robustness, coupled with reasonable computational e-ciency, make it a valuable addition to our collection of sparse solvers. The rest of this paper is organized as follows. In section 2, we provide a general description of our multilevel approach. In section 3, we dene the sparse matrix data structures used in our code. Our incomplete factorization algorithm is a standard drop tolerance approach with a few modications for the present application. These are described in section 4. Our ordering procedure is the minimum degree algorithm. Once again, our implementation is basically standard with several modications to the input graph relevant to our application. These are described in section 5. In section 6, we describe the construction of the transfer matrices used in the construction of the coarse grid correction. Information about the block structure of the matrix, if any is provided, is used only in the coarsening procedure. This is described in section 7. Finally, in section 8, we give some numerical illustrations of our method on a variety of (partial dierential equation) matrices. 2. Matrix formulation. Let A be a large sparse, nonsingular N N matrix. We assume that the sparsity pattern of A is symmetric, although the numerical values need not be. We will begin by describing the basic two-level method for solving Let B be an N N nonsingular matrix, called the smoother, which gives rise to the basic iterative method used in the multilevel preconditioner. In our case, B is an approximate factorization of A, i.e., where L is (strict) lower triangular, U is (strict) upper triangular with the same sparsity pattern as L t , D is diagonal, and P is a permutation matrix. Given an initial guess x steps of the smoothing procedure produce iterates The second component of the two-level preconditioner is the coarse grid correction. Here we assume that the matrix A can be partitioned as A ff A fc A cf A cc where the subscripts f and c denote ne and coarse, respectively. Similar to the smoother, the partition of A in ne and coarse blocks involves a permutation matrix P . The ^ coarse grid matrix ^ A is given by A ff A fc A cf A cc I cc The matrices V cf and W t N) matrices with identical sparsity patterns; thus A has a symmetric sparsity pattern. If A A. Let I cc In standard multigrid terminology, the matrices ^ W are called restriction and prolongation, respectively. Given an approximate solution xm to (2.1), the coarse grid correction produces an iterate xm+1 as follows. (b Axm ); As is typical of multilevel methods, we dene the two-level preconditioner M implicitly in terms of the smoother and coarse grid correction. A single cycle takes an initial guess x 0 to a nal guess x 2m+1 as follows: Two-Level Preconditioner are dened using (2.3). (ii) xm+1 is dened using (2.7). are dened using (2.3). The generalization from two-level to multilevel consists of applying recursion to the solution of the equation ^ r in (2.7). Let ' denote the number of levels in the recursion. M(') denote the preconditioner for A. Then (2.7) is generalized to (b Axm ); The general ' level preconditioner M is then dened as follows: '-Level Preconditioner directly. starting from initial guess x 0 , compute x 2m+1 using (iii){(v): are dened using (2.3). (iv) xm+1 is dened by (2.8), using iterations of the ' 1 level scheme for r to dene ^ M , and with initial guess ^ are dened using (2.3). The case corresponds to the symmetric V-cycle, while the case corresponds to the symmetric W-cycle. We note that there are other variants of both the V-cycle and the W-cycle, as well as other types of multilevel cycling strategies [30]. However, in this work (and in our code) we restrict attention to just the symmetric V-cycle with presmoothing and postsmoothing iterations. For the coarse mesh solution our procedure is somewhat nontraditional. Instead of a direct solution of (2.1), we compute an approximate solution using one smoothing iteration. We illustrate the practical consequences of this decision in section 8. If A is symmetric, then so is M , and the '-level preconditioner could be used as a preconditioner for a symmetric Krylov space method. If A is also positive denite, so is M , and the standard conjugate gradient method could be used; otherwise the CSCG method [9], SYMLQ [43], or a similar method could be used. In the nonsymmetric case, the '-level preconditioner could be used in conjunction with the CSBCG method [9], GMRES [22], or a similar method. To complete the denition of the method, we must provide algorithms to compute the permutation matrix P in (2.2); compute the incomplete factorization matrix B in (2.2); compute the ne-coarse partitioning compute the sparsity patterns and numerical values in the prolongation and restriction matrices in (2.6). 3. Data structures. Let A be an N N matrix with elements A ij and a symmetric sparsity structure; that is, both A ij and A ji are treated as nonzero elements (i.e. stored and processed) if jA diagonal entries A ii are treated as nonzero regardless of their numerical values. Our data structure is a modied and generalized version of the data structure introduced in the (symmetric) Yale sparse matrix package [23]. It is a rowwise version of the data structure described in [10]. In our scheme, the nonzero entries of A are stored in a linear array a and accessed through an integer array ja. Let i be the number of nonzeros in the strict upper triangular part of row i and set The array ja is of length N+1+, and the array a is of length N+1+ if A A. If A t 6= A, then the array a is of length 2. The entries of ja(i), 1 i N are pointers dened as follows: The locations ja(i) to ja(i contain the i column indices corresponding to the row i in the strictly upper triangular matrix. In a similar manner, the array a is dened as follows: If A t 6= A, then In words, the diagonal is stored rst, followed by the strict upper triangle stored row- wise. If A t 6= A, then this is followed by the strict lower triangle stored columnwise. Since A is structurally symmetric, the column indexes for the upper triangle are identical to the row indexes for the lower triangle, and hence they need not be duplicated in storage. As an example, let A 11 A 12 A 13 0 0 A 21 A 22 0 A 24 0 A 43 A 44 0 Then a A11 A22 A33 A44 A55 A12 A13 A24 A34 A35 A21 A31 A42 A43 A53 Diagonal Upper triangle Lower triangle Although the YSMP data structure was originally devised for sparse direct methods based on Gaussian elimination, it is also quite natural for iterative methods based on incomplete triangular decomposition. Because we assume that A has a symmetric sparsity structure, for many matrix calculations a single indirect address computation in ja can be used to process both a lower and a upper triangular element in A. For example, the following procedure computes procedure mult(N, ja, a, x, y) end for for k ja(i) to ja(i end for end for For symmetric matrices, set lmtx 0; umtx 0. Also, may be readily computed by setting lmtx 0; umtx ja(N The data structure for storing quite analogous to that for A. It consists of two arrays, ju and u, corresponding to ja and a, respectively. The rst entries of ju are pointers as in ja, while entries ju(i) to ju(i contain column indices of the nonzeros of row i in of U . In the u array, the diagonal entries of D are stored in the rst N entries. Entry arbitrary. Next, the nonzero entries of U are stored in correspondence to the column indices in ju. If the nonzero entries of L follow, stored columnwise. The data structure we use for the N ^ W and the ^ are similar. It consists of an integer array jv and a real array v. The nonzero entries of are stored rowwise, including the rows of the block I cc . As usual, the rst entries of jv are pointers; entries jv(i) to jv(i contain column indices for row W . In the v array, the nonzero entries of ^ are stored rowwise in correspondence with jv but shifted by N since there is no diagonal part. If ^ W , this is followed by the nonzeros of ^ stored columnwise. 4. ILU factorization. Our incomplete (L+D)D 1 (D+U) factorization is similar to the row elimination scheme developed for the symmetric YSMP codes [23, 26]. For simplicity, we begin by discussing a complete factorization and then describe the modications necessary for the incomplete factorization. Without loss of generality, assume that the permutation matrix After k steps of elimination, we have the block factorization A 11 A 12 A 21 A 22 I where A 11 is k k and A 22 is N k N k. We assume that at this stage, all the blocks on the right-hand side of (4.1) have been computed except for the Schur complement S, given by Our goal for step k + 1 is to compute the rst row and column of S, given by Because A and (L +D)D 1 (D + U) have symmetric sparsity patterns, and our data structures take advantage of this symmetry, it is clear that the algorithms for computing are the same and in practice dier only in the assignments of shifts for the u and a arrays, analogous to lmtx and umtx in procedure mult. Thus we will focus on the computation of just . At this point, we also assume that the array ju has been computed in a so-called symbolic factorization step. The major substeps are as follows: 1. Copy the rst column of A 22 (stored in the data structures ja and a) into an expanded work vector z of size N . 2. Find the multipliers given by nonzeros of D 1 3. For each multiplier using column k of L 21 (i.e., 4. Copy the nonzeros in z into the data structures ju and u. In step 1, we need to know the nonzeros of the rst column of A 22 , which is precisely the information easily accessible in the ja and a data structures. In step 3, we need to know the nonzeros in columns of L 21 , which again is precisely the information easily available in our data structure. In step 4, we copy a column of information into the lower triangular portion of the ju and u data structures. Indeed, the only di-cult aspect of the algorithm is step 2, in which we need to know the sparsity structure of the rst column of U 12 , information that is not readily available in the data structure. This is handled in a standard fashion using a dynamic linked list structure and will not be discussed in detail here. To generalize this to the incomplete factorization case, we rst observe that the ju array can be computed concurrently with the numeric factorization simply by creating a list of the entries of the expanded array z that are updated in step 3. Next, we note that one may choose which nonzero entries from z to include in the factorization by choosing which entries to copy to the ju and u data structures in step 4. We do this through a standard approach using a drop tolerance . In particular, we neglect a pair of o-diagonal elements if j. Note D ii has not yet been computed. It is well known that the ll-in generated through the application of a criterion such as (4.4) is a highly nonlinear and matrix dependent function of . This is especially problematic in the present context, since control of the ll-in is necessary in order to control the work per iteration in the multilevel iteration. Several authors have explored possibilities of controlling the maximum number of ll-in elements allowed in each row of the incomplete decomposition [35, 47, 31]. However, for many cases of interest, and in particular for matrices arising from discretizations of partial dierential equations ordered by the minimum degree algorithm, most of the ll-in in a complete factorization occurs in the later stages, even if all the rows initially have about the same number of nonzeros. Thus while it seems advisable to try to control the total ll-in, one should adaptively decide how to allocate the ll-in among the rows of the matrix. In our algorithm, in addition to the drop tolerance , the user provides a parameter maxf il, which species that the total number of nonzeros in U is not larger than maxf il N . Our overall strategy is to compute the incomplete decomposition using the given drop tolerance. If it fails to meet the given storage bound, we increase the drop tolerance and begin a new incomplete factorization. We continue in this fashion until we complete a factorization within the given storage bound. Of course, such repeated factorizations are computationally expensive, so we developed some heuristics which allow us to predict a drop tolerance which will satisfy the storage bound. As the factorization is computed, we make a histogram of the approximate sizes of all elements that exceed the drop tolerance and are accepted for the factorization. Let m denote the number of bins in the histogram; our code. Then for each pair of accepted o-diagonal elements, we nd the largest k 2 [1; m] such that Here > 1 our code). The histogram is realized as an integer array h of size m, where h ' is the number of accepted elements that exceeded the drop tolerance by factors between ' 1 and ' for 1 ' m 1; hm contains the number of accepted elements exceeding the drop tolerance by m 1 . If the factorization reaches the storage bound, we continue the factorization but allow no further ll-in. However, we continue to compute the histogram based on (4.5), proling the elements we would have accepted had space been available. Then using the histogram, we predict a new value of such that the total number of elements accepted for U is no larger than maxf il N=. Such a prediction of course cannot be guaranteed, since the sizes and numbers of ll-in elements depend in a complicated fashion on the specic history of the incomplete factorization process; indeed, the histogram cannot even completely prole the remainder of the factorization with the existing drop tolerance, since elements that would have been accepted could introduce additional ll-in at later stages of the calculation as well as in uence the sizes of elements computed at later stages of the factorization. In our implementation, the factor varies between depending on how severely the storage bound was exceeded. Its purpose is to introduce some conservative bias into the prediction with the goal that the actual ll-in accepted should not exceed maxf il N . Finally, we note that there is no comprehensive theory regarding the stability of incomplete triangular decompositions. For certain classes of matrices (e.g., M-matrices and H-matrices), the existence of certain incomplete factorizations has been proved [39, 25, 24, 40, 51]. However, in the general case, with potentially indenite and/or highly nonsymmetric matrices, one must contend in a practical way with the possibility of failure or near failure of the factorization. A common approach is to add a diagonal matrix, often a multiple of the identity, to A and compute an incomplete factorization of the shifted matrix. One might also try to incorporate some form of diagonal pivoting; partial or complete pivoting could potentially destroy the symmetric sparsity pattern of the matrix. However, any sort of pivoting greatly increases the complexity of the implementation, since the simple but essentially static data structures ja, a, ju, and u are not appropriate for such an environment. Our philosophy here is to simply accept occasional failures and continue with the factorization. Our ordering procedure contains some heuristics directed towards avoiding or at least minimizing the possibility of failures. And when they do occur, failures often corrupt only a low dimensional subspace, so a Krylov space method such as conjugate gradients can compensate for such corruption with only a few extra iterations. In our implementation, a failure is revealed by some diagonal entries in D becoming close to zero. O-diagonal elements L ji and U ij are multiplied by D 1 ii , and the solution of (L multiplication by D 1 ii . For purposes of calculating the factorization and solution, the value of D 1 ii is modied near zero as follows: 1=D ii for jD ii j > , Here is a small constant; in our implementation, is the machine epsilon. Although many failures could render the preconditioner well-dened but essentially useless, in practice we have noted that D 1 ii is rarely modied for the large class of nite element matrices which are the main target of our procedure. 5. Ordering. To compute the permutation matrix P in (2.2), we use the well-known minimum degree algorithm [45, 26]. Intuitively, if one is computing an incomplete factorization, an ordering which tends to minimize the ll-in in a complete factorization should tend to minimize the error For particular classes of matrices, specialized ordering schemes have been developed [34, 15, 37, 36]. For example, for matrices arising from convection dominated prob- lems, ordering along the ow direction has been used with great success. However, in this general setting, we prefer to use just one strategy for all matrices. This reduces the complexity of the implementation and avoids the problem of developing heuristics to decide among various ordering possibilities. We remark that for convection dominated problems, minimum degree orderings perform comparably well to the specialized ones, provided some (modest) ll-in is allowed in the incomplete factorization. For us, this seems to be a reasonable compromise. Our minimum degree ordering is a standard implementation, using the quotient graph model [26] and other standard enhancements. A description of the graph of the matrix is the main required input. Without going into detail, this is essentially a small variant of the basic ja data structure used to store the matrix A. We will denote this modied data structure as jc. Instead of storing only column indices for the strict upper triangle as in ja, entries jc(i) to jc(i of the jc data structure contain column indices for all o-diagonal entries for row i of the matrix A. We have implemented two small enhancements to the minimum degree ordering; as a practical matter, both involve changes to the input graph data structure jc that is provided to the minimum degree code. First, we have implemented a drop tolerance similar to that used in the the factorization. In particular the edge in the graph corresponding to o-diagonal entries A ij and A ji is not included in the jc data structure if jA jj A ii j: (5.1) This excludes many entries which are likely to be dropped in the subsequent incomplete factorization and hopefully will result in an ordering that tends to minimize the ll-in created by the edges that are kept. The second modication involves some modest a priori diagonal pivoting designed to minimize the number failures (near zero diagonal elements) in the subsequent factorization. We rst remark that pivoting or other procedures based on the values of the matrix elements (which can be viewed as weights on graph edges and nodes) would destroy many of the enhancements which allow the minimum degree algorithm to run in almost linear time. Our modication is best explained in the context of a simple 2 2 example. Let b a with a; b; c 6= 0. Clearly, A is nonsingular, but the complete triangular factorization of A does not exist. However, a b a 0 c bc=a a b Now suppose that A ii 0, A jj these four elements form a submatrix of the form described above, and it seems an incomplete factorization of A is less likely to fail if the P is chosen such that vertex j is ordered before vertex i. This is done as follows: for each i such that A ii 0, we determine a corresponding j such that A jj there is more than one choice, we choose the one for which jA ij A ji =A jj j is maximized. To ensure that vertex i is ordered after vertex j, we replace the sparsity pattern for the o-diagonal entries for row (column) i with the union of those for rows (columns) i and j. If we denote the set of column indices for row i in the jc array as adj(i), then Although the sets adj(i) and adj(j) are modied at various stages, it is well known that (5.3) is maintained throughout the minimum degree ordering process [26], so that at every step of the ordering process deg(j) deg(i), where deg(i) is the degree of vertex i. As long as deg(j) < deg(i), vertex j will be ordered before vertex i by the minimum degree algorithm. On the other hand, if at some stage of the ordering process, it remains so thereafter, and (5.3) becomes In words, i and j become so-called equivalent vertices and will be eliminated at the same time by the minimum degree algorithm (see [26] for details). Since the minimum degree algorithm sees these vertices as equivalent, they will be ordered in an arbitrary fashion when eliminated from the graph. Thus, as a simple postprocessing step, we must scan the ordering provided by the minimum degree algorithm and exchange the order of rows i and j if i was ordered rst. Any such exchanges result in a new minimum degree ordering which is completely equivalent, in terms of ll-in, to the the original. For many types of nite element matrices (e.g., the indenite matrices arising from Helmholtz equations), this a priori scheme is useless because none of the diagonal entries of A is close to zero. However, this type of problem is likely to produce only isolated small diagonal entries in the factorization process, if it produces any at all. On the other hand, other classes of nite element matrices, notably those arising in from mixed methods, Stokes equations, and other saddle-point-like formulations, have many diagonal entries that are small or zero. In such cases, the a priori diagonal pivoting strategy can make a substantial dierence and greatly reduce the numbers of failures in the incomplete triangular decomposition. 6. Computing the transfer matrices. There are three major tasks in computing the prolongation and restriction matrices ^ W of (2.6). First, one must determine the sparsity structure of these matrices; this involves choosing which unknowns are coarse and which are ne. This reduces to determining the permutation P of (2.4). Second, one must determine how coarse and ne unknowns are related, the so-called parent-child relations [49]. This involves computing the sparsity patterns for the matrices V cf and W fc . Third, one must compute the numerical values for these matrices, the so-called interpolation coe-cients [50]. There are many existing algorithms for coarsening graphs. For matrices arising from discretizations of partial dierential equations, often the sparsity of the matrix A is related in some way to the underlying grid, and the problem of coarsening the graph of the matrix A can be formulated in terms of coarsening the grid. Some examples are given in [14, 13, 17, 18, 46, 12, 49]. In this case, one has the geometry of the grid to serve as an aid in developing and analyzing the coarsening procedure. There are also more general graph coarsening algorithms [32, 33, 19], often used to partition problems for parallel computation. Here our coarsening scheme is based upon another well-known sparse matrix ordering technique, the reverse Cuthill{McKee algorithm. This ordering tends to yield reordered matrices with minimal bandwidth and is widely used with generalized band elimination algorithms [26]. We now assume that the graph has been ordered in this fashion and that a jc data structure representing the graph in this ordering is available. Our coarsening procedure is just a simple postprocessing step of the basic ordering routine, in which the N vertices of graph are marked as COARSE or F INE. procedure coarsen(N, jc, type) end for for j jc(i) to jc(i end for end for This postprocessing step, coupled with the the reverse Cuthill{McKee algorithm, is quite similar to a greedy algorithm for computing maximal independent sets using breadth-rst search. Under this procedure, all coarse vertices are surrounded only by ne vertices. This implies that the matrix A cc in (2.4) is a diagonal matrix. For the sparsity patterns of matrices arising from discretizations of scalar partial dierential equations in two space dimensions, the number of coarse unknowns ^ N is typically on the order of N=4 to N=5. Matrices with more nonzeros per row tend to have smaller values of ^ N . To dene the parents of a coarse vertex, we take all the connections of the vertex to other ne vertices; that is, the sparsity structure of V cf in (2.5) is the same as that of the block A cf . In our present code, we pick V cf and W fc according to the formulae ff A fc ; ff ~ Here D ff is a diagonal matrix with diagonal entries equal to those of A ff . In this sense, the nonzero entries in V cf and W fc are chosen as multipliers in Gaussian elimi- nation. The nonnegative diagonal matrices R ff and ~ R ff are chosen such that nonzero rows of W fc and columns of V cf , respectively, have unit norms in ' 1 . Finally, the coarsened matrix ^ A of (2.5) is \sparsied" using the drop tolerance and a criterion like (5.1) to remove small o-diagonal elements. Empirically, applying a drop tolerance to ^ A at the end of the coarsening procedure has proved more e-cient, and more eective, than trying to independently sparsify its constituent matrices. If the number of o-diagonal elements in the upper triangle exceeds maxf il ^ N , the drop tolerance is modied in a fashion similar to the incomplete factorization. The o-diagonal elements are proled by a procedure similar to that for the incomplete fac- torization, but in this case the resulting histogram is exact. Based on this histogram, a new drop tolerance is computed, and (5.1) is applied to produce a coarsened matrix satisfying the storage bound. 7. Block matrices. Our algorithm provides a simple but limited functionality for handling block matrices. Suppose that the N N matrix A has the K K block structure A =B @ where subscripts for A ij are block indices and the diagonal blocks A jj are square matrices. Suppose A jj is of order The matrix A is stored in the usual ja and a data structures as described in section 3 with no reference to the block structure. A small additional integer array ib of size K + 1 is used to dene the block boundaries as follows: In words, integers in the range ib(j) to ib(j inclusive, comprise the index set associated with block A jj . Note that ib(K This block information plays a role only in the coarsening algorithm. First, the reverse Cuthill{McKee algorithm described in section 6 is applied to the block diagonal matrix A =B @ A 11 AKKC A (7.2) rather than A. As a practical matter, this involves discarding graph edges connecting vertices of dierent blocks in the construction of the graph array jc used as input. Such edges are straightforward to determine from the information provided in the ib array. The coarsening algorithm applied to the graph of A produces output equivalent to the application of the procedure independently to each diagonal block of A. As a consequence, the restriction and prolongation matrices automatically inherit the block structure A. In particular, VKKC A and ^ W jj are are rectangular matrices having the structure of (2.6) that would have resulted from the application of the algorithm independently to A jj . However, like the matrix A, are stored in the standard jv and v data structures described in section 3 without reference to their block structures. The complete matrix A is used in the construction of the coarsened matrix ^ A of (2.5). However, because of (7.1) and (7.3) A 1K A also automatically inherits the K K block structure of A. It is not necessary for the procedure forming ^ A to have any knowledge of its block structure, as this block structure can be computed a priori by the graph coarsening procedure. Like A is stored in standard ja and a data structures without reference to its block structure. Since the blocks of A have arbitrary order, and are essentially coarsened independently, it is likely that eventually some of the ^ That is, certain blocks may cease to exist on coarse levels. Since the block information is used only to discard certain edges in the construction of the graph array jc, \00" diagonal blocks present no di-culty. 8. Numerical experiments. In this section, we present a few numerical illus- trations. In our rst sequence of experiments, we consider several matrices loosely based on the classical case of 5-point centered nite dierence approximations to u on a uniform square mesh. Dirichlet boundary conditions are imposed. This leads to the n n block tridiagonal system I I T I I T I I with T the n n tridiagonal matrix This is a simple test problem easily solved by standard multigrid methods. In contrast to this example we also consider the block tridiagonal system Both A and A have the same eigenvectors and the same eigenvalues, although the association of eigenvectors and eigenvalues are reversed in the case of A. That is, the so-called smooth eigenvectors are associated with large eigenvalues, while rough eigenvectors are associated with smaller eigenvalues. Although A does not arise naturally in the context of numerical discretizations of partial dierential equations, it is of interest because it dees much of the conventional wisdom for multigrid methods. Third, we consider block 3 3 systems of the form where A is the discrete Laplacian and D is a symmetric positive denite \stabilization" matrix with a sparsity pattern similar to A. However, the nonzeros in D are of size compared to size O(1) nonzero elements in A. C x and C y also have sparsity patterns similar to that of A, but these matrices are nonsymmetric and their nonzero entries are of size O(h). Such matrices arise in stabilized discretizations of the Stokes equations. One third of the eigenvalues of S are negative, so S is quite indenite. In addition to the ja and a arrays, for the matrix S we also provided an ib array as described in section 7 to dene its 33 block structure. We emphasize again that this block information is used only in the computation of the graph input to the coarsening procedure and is not involved in any aspect of the incomplete factorization smoothing procedure. With many small diagonal elements, this class of matrices provides a good test of the a priori pivoting strategy used in conjunction with the minimum degree ordering. In Table 8.1, Levels refers to the number of levels used in the calculation. In our implementation the parameter maxlvl, which limits the number of levels allowed, was set su-ciently large that it had no eect on the computation. The drop tolerance was set to matrices. The ll-in control parameter maxf il was set su-ciently large that it had no eect on the computation. The initial guess for all problems was x In Table 8.1, the parameter Digits refers to In these experiments, we asked for six digits of accuracy. The column labeled Cycles indicates the number of multigrid cycles (accelerated by CSCG) that were used to achieve the indicated number of digits. Finally, the last two columns, labeled Init. and Solve, record the CPU time, measured in seconds, for the initialization and solution phases of the algorithm, respectively. Initialization includes all the orderings, incomplete factorizations, and computation of transfer matrices used in the multigraph preconditioner. Solution includes the time to solve (2.1) to at least six digits given the preconditioner. These experiments were run on an SGI Octane R10000 250mhz, using double precision arithmetic and the f90 compiler. In analyzing these results, it is clear that our procedure does reasonably well on all three classes of matrices. Although it appears that the rate of convergence is not independent of N , it seems apparent that the work is growing no faster than logarithmically. CPU times for larger vales of N are aected by cache performance as well as the slightly larger number of cycles. For the highly indenite Stokes matrices S, it is important to also note the ro- bustness, that the procedure solved all of the problems. With more nonzeros per row on average, the incomplete factorization was more expensive to compute than for the other cases. This is re ected in relatively larger initialization and solve times. In our next experiment, we illustrate the eect of the parameters maxlvl and . For the matrix A with we solved the problem for and 1 maxlvl 7. We terminated the iteration when the solution had six digits, Performance comparison. Digits Cycles Init. Solve Discrete Laplacian A, Stokes matrix S, as measured by (8.1). We also provide the total storage for the ja and ju arrays for all matrices, measured in thousands of entries. Since the matrices are symmetric, this is also the total ( oating point) storage for all matrices A and approximate LDU factorizations. Here we see that our method behaves in a very predictable way. In particular, decreasing the drop tolerance or increasing the number of levels improves the convergence behavior of the method. On the other hand, the timings do not always follow the same trend. For example, for the case increasing the number of levels from decreases the number of cycles but increases the time. This is because for our method defaults to the standard conjugate gradient iteration with the incomplete factorization preconditioner. When maxlvl > 1, one presmoothing and one postsmoothing step are used for the largest matrix. With the additional cost of the recursion, the overall cost of the preconditioner is more than double the cost for the case We also note that, unlike the classical multigrid method, where the coarsest matrix is solved exactly, in our code we have chosen to approximately solve the coarsest system using just one smoothing iteration using the incomplete factorization. When the maximum number of levels are used, as in Table 8.1, the smallest system is typically 1 1 or 2 2, and this is an irrelevant remark. However, in the case of Table 8.2, the fact that the smallest system is not solved exactly signicantly in uences the overall rate of convergence. This is why, unlike methods where the coarsest system is solved exactly, increasing the number of levels tends to improve the rate of convergence. In the case the coarsest matrix had an exact LDU factorization for the case (because the matrix itself was nearly diagonal), and setting maxlvl > 5 did not increase the number of levels. The cases Dependence of convergence of and maxlvl, discrete Laplacian A, maxlvl Digits Cycles Init. Solve 3 6.1 96 13.2 116.9 1077 1119 6 { { { { { { 7 { { { { { { 6.1 56 12.1 64.9 878 2106 6.1 22 16.6 31.7 878 3649 and used a maximum of 10 and 9 levels, respectively, but the results did not change signicantly from the case 7. We also include in Table 8.2 the case ination. (In fact, our code uses jjAjj as the drop tolerance when the user species to avoid dividing by zero.) Here we see that Gaussian elimination is reasonably competitive on this problem. However, we generally expect the initialization cost for to grow like O(N 3=2 ). For we expect the solution times to grow like O(N p ), p > 1. For the best multilevel choices, we expect both initialization and solution times to behave like O(N) O(N log N ). In our nal series of tests, we study the convergence of the method for a suite of test problems generated from the nite element code PLTMG [8]. These example problems were presented in our earlier work [11], where a more complete description of the problems, as well as numerical results for our hierarchical basis multigraph method and the classical AMG algorithm of Ruge and Stuben [46], can be found. As a group, the problems feature highly nonuniform, adaptively generated meshes, relatively complicated geometry, and a variety of dierential operators. For each test case, both the sparse matrix and the right-hand side were saved in a le to serve as input for the iterative solvers. A short description of each test problem is given below. Problem Superior. This problem is a simple Poisson equation with homogeneous Dirichlet boundary conditions on a domain in the shape of Lake Superior. This is the classical problem on a fairly complicated domain. The solution is generally very smooth but has some boundary singularities. Problem Hole. This problem features discontinuous, anisotropic coe-cients. The overall domain is the region between two concentric circles, but this domain is divided into three subregions. On the inner region, the problem is with In the middle region, the equation is and in the outer region the equation is Homogeneous Dirichlet boundary conditions are imposed on the inner (hole) bound- ary, homogeneous Neumann conditions on the outer boundary, and the natural continuity conditions on the internal interfaces. While the solution is also relatively smooth, singularities exist at the internal interfaces. Problem Texas. This is an indenite Helmholtz equation posed in a region shaped like the state of Texas. Homogeneous Dirichlet boundary conditions are imposed. The length scales of this domain are roughly 16 16, so this problem is fairly indenite. Problem UCSD. This is a simple constant coe-cient convection-diusion equation r (ru posed on a domain in the shape of the UCSD logo. Homogeneous Dirichlet boundary conditions are imposed. Boundary layers are formed at the bottom of the region and the top of various obstacles. Problems Jcn 0 and Jcn 180. The next two problems are solutions of the current continuity equation taken from semiconductor device modeling. This equation is a convection-diusion equation of the form r (ru in most of the rectangular domain. However, in a curved band in the interior of the domain, jj 10 4 and is directed radially. Dirichlet boundary conditions and are imposed along the bottom boundary and along a short segment on the upper left boundary, respectively. Homogeneous Neumann boundary conditions are specied elsewhere. The solutions vary exponentially across the domain which is typical of semiconductor problems. In the rst problem, Jcn 0, the convective term is chosen so the device is forward biased. In this case, a sharp internal layer develops along the top interface boundary. In the second problem, Jcn 180, the sign of the convective term is reversed, resulting in two sharp internal layers along both interface boundaries. We summarize the results in Table 8.3. As before, perhaps the most important point is that the method solved all of the problems. While convergence rates are not independent of h, once again the growth appears to be at worst logarithmic. Below we make some additional remarks. Table Performance comparison. N Levels Digits Cycles Init. Solve 20k 9 7.3 5 1.4e 0 9.4e-1 Hole, Jcn 0, Jcn For all problems, decreasing the drop tolerance will tend to increase the effectiveness of the preconditioner, although it generally will also make the preconditioner more costly to apply. Thus one might optimize the selection of the drop tolerance to minimize the decreasing number of cycles against the increasing cost per cycle. In these experiments, we did not try such systematic optimization, but we did adjust the drop tolerance in a crude way such that more di-cult problems performed in a fashion similar to the easy ones. Problem Texas is by far the most di-cult in this test suite. While we set the problem with order 80k was the only one which came close to achieving this storage limit. Most were well below this limit, and many averaged less than 10 nonzeros per row in L and U factors. For the nonsymmetric problems the CSBCG method is used for acceleration. Since the CSBCG requires the solution of a conjugate system with A t , two matrix multiplies and two preconditioning steps are required for each itera- tion. As noted in section 3, with our data structures, applying a transposed matrix and preconditioner costs the same as applying the original matrix or preconditioner. Since these are the dominant costs in the CSBCG methods, the cost per cycle is approximately double that for an equivalent symmetric system. --R On eigenvalue estimates for block incomplete factorization methods The algebraic multilevel iteration methods - theory and applications Stabilization of algebraic multilevel iteration methods Algebraic multilevel preconditioning methods I A class of hybrid algebraic multilevel preconditioning methods PLTMG: A Software Package for Solving Elliptic Partial Di An analysis of the composite step biconjugate gradient method General sparse elimination requires no permanent integer storage The incomplete factorization multigraph algorithm The hierarchical basis multigrid method and incomplete LU decomposi- tion Orderings for incomplete factorization preconditioning of nonsymmetric problems Towards algebraic multigrid for elliptic problems of second order Boundary treatments for multilevel methods on unstructured meshes Black box multigrid Variational iterative methods for non-symmetric systems of linear equations Algorithms and data structures for sparse symmetric Gaussian elimination A stability analysis of incomplete LU factorizations Algebraic analysis of the hierarchical basis preconditioner Computer Solution of Large Sparse Positive De Incomplete block factorization preconditioning for linear systems arising in the numerical solution of the helmholtz equation An algebraic hierarchical basis preconditioner Incomplete Decompositions - Theory Analysis of multilevel graph partitioning Ordering techniques for convection dominated problems on unstructured three dimensional grids Ordering strategies for modi Energy optimization of algebraic multigrid bases An analysis of the robustness of some incomplete factorizations On the stability of the incomplete LU-factorizations and characterizations of H-matrices Using approximate inverses in algebraic multilevel methods Solution of sparse inde A multigrid method based on incomplete Gaussian elimination A graph theoretic study of the numeric solution of sparse positive de ILUT: a dual threshold incomplete LU factorization Convergence of algebraic multigrid based on smoothed aggregation Introduction to algebraic multigrid An energy-minimizing interpolation for robust multi-grid methods On the robustness of ILU smoothing --TR --CTR Randolph E. Bank, Compatible coarsening in the multigraph algorithm, Advances in Engineering Software, v.38 n.5, p.287-294, May, 2007 Gh. Juncu , E. Mosekilde , C. Popa, Numerical experiments with MG continuation algorithms, Applied Numerical Mathematics, v.56 n.6, p.844-861, June 2006 J. S. Ovall, Hierarchical matrix techniques for a domain decomposition algorithm, Computing, v.80 n.4, p.287-297, September 2007 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

algebraic multigrid;multigraph methods;incomplete LU factorization

587404

A Scalable Parallel Algorithm for Incomplete Factor Preconditioning.

We describe a parallel algorithm for computing incomplete factor (ILU) preconditioners. The algorithm attains a high degree of parallelism through graph partitioning and a two-level ordering strategy. Both the subdomains and the nodes within each subdomain are ordered to preserve concurrency. We show through an algorithmic analysis and through computational results that this algorithm is scalable. Experimental results include timings on three parallel platforms for problems with up to 20 million unknowns running on up to 216 processors. The resulting preconditioned Krylov solvers have the desirable property that the number of iterations required for convergence is insensitive to the number of processors.

Introduction . Incomplete factorization (ILU) preconditioning is currently among the most robust techniques employed to improve the convergence of Krylov space solvers for linear systems of equations. (ILU stands for incomplete LU fac- torization, where L and U are the lower and upper triangular (incomplete) factors of the coe#cient matrix.) However, scalable parallel algorithms for computing ILU preconditioners have not been available despite the fact that they have been used for more than twenty years [12]. We report the design, analysis, implementation, and computational evaluation of a parallel algorithm for computing ILU preconditioners. Our parallel algorithm assumes that three requirements are satisfied. . The adjacency graph of the coe#cient matrix (or the underlying finite element or finite di#erence mesh) must have good edge separators, i.e., it must be possible to remove a small set of edges to divide the problem into a collection of subproblems that have roughly equal computational work requirements. . The size of the problem must be su#ciently large relative to the number of processors so that the work required by the subgraph on each processor is suitably large to dominate the work and communications needed for the boundary nodes. . The subdomain intersection graph (to be defined later) should have a small chromatic number. This requirement will ensure that the dependencies in factoring the boundary rows do not result in undue losses in concurrency. An outline of the paper is as follows. In section 2, we describe the steps in the parallel algorithm for computing the ILU preconditioner in detail and provide theoretical justification. The algorithm is based on an incomplete fill path theorem; the proof and discussion of the theorem are deferred to an appendix. We also discuss # Received by the editors August 4, 2000; accepted for publication (in revised form) December 17, 2000; published electronically April 26, 2001. This work was supported by U. S. National Science Foundation grants DMS-9807172 and ECS-9527169, by the U. S. Department of Energy under subcontract B347882 from the Lawrence Livermore Laboratory, by a GAANN fellowship from the Department of Education, and by NASA under contract NAS1-19480 while the authors were in residence at ICASE. http://www.siam.org/journals/sisc/22-6/37619.html Old Dominion University, Norfolk, VA 23529-0162 and ICASE, NASA Langley Research Center, Hampton VA 23681-2199 (hysom@cs.odu.edu, pothen@cs.odu.edu). the role that a subdomain graph constraint plays in the design of the algorithm, show that the preconditioners exist for special classes of matrices, and relate our work to earlier work on this problem. Section 3 contains an analysis that shows that the parallel algorithm is scalable for two-dimensional (2D-) and three-dimensional (3D-)model problems, when they are suitably ordered and partitioned. Section 4 contains computational results on Poisson and convection-di#usion problems. The first subsection shows that the parallel ILU algorithm is scalable on three parallel platforms; the second subsection reports convergence studies. We tabulate how the number of Krylov solver iterations and the number of entries in the preconditioner vary as a function of the preconditioner level for three variations of the algorithm. The results show that fill levels higher than one are e#ective in reducing the number of iterations; the number of iterations is insensitive to the number of subdomains; and the subdomain graph constraint does not a#ect the number of iterations while it makes possible the design of a simpler parallel algorithm. The background needed for ILU preconditioning may be found in several books; see, e.g., [1, 15, 17, 33]. A preliminary version of this paper was presented at Super-computing '99 and was published in the conference proceedings [18]. The algorithm has been revised, additional details have been included, and the proof of the theorem on which it is based has been added. The experimental results in section 4 are new, and most of them have been included in the technical reports [19, 20]. 2. Algorithms. In this section we discuss the Parallel ILU (PILU) algorithm and its underlying theoretical foundations. 2.1. The PILU algorithm. Figure 2.1 describes the steps of the PILU algorithm at a high level; the algorithm is suited for implementation on both message-passing and shared-address space programming models. The PILU algorithm consists of four major steps. In the first step, we create parallelism by dividing the problem into subproblems by means of graph partitioning. In the second step, we preserve the parallelism in the interior of the subproblems by locally scheduling the computations in each subgraph. In the third step, we preserve parallelism in the boundaries of the subproblems by globally ordering the subproblems through coloring a suitably defined graph. In the final step, we compute the preconditioner in parallel. Now we will describe the four steps in greater detail. Step 1: Graph partitioning. In the first step of PILU, we partition the adjacency graph G(A) of the coe#cient matrix A into p subgraphs by removing a small set of edges that connects the subgraphs to each other. Each subgraph will be mapped to a distinct processor that will be responsible for the computations associated with the subgraph. An example of a model five-point grid partitioned into four subgraphs is shown in Figure 2.2. For clarity, the edges corresponding to the coe#cient matrix elements (within each subgraph or between subgraphs) are not shown. The edges drawn correspond to fill elements (elements that are zero in the coe#cient matrix but are nonzero in the incomplete factors) that join the di#erent subgraphs. To state the objective function of the graph partitioning problem, we need to introduce some terminology. An edge is a separator edge if its endpoints belong to di#erent subgraphs. A vertex in a subgraph is an interior vertex if all of its neighbors belong to that subgraph; it is a boundary vertex if it is adjacent to one or more vertices belonging to another subgraph. By definition, an interior vertex in a subgraph is not adjacent to a vertex (boundary or interior) in another subgraph. In Figure 2.2, the first 25 vertices are interior vertices of the subgraph S 0 , and vertices numbered 26 through 36 are its Input: A coe#cient matrix, its adjacency graph, and the number of processors p. Output: The incomplete factors of the coe#cient matrix. 1. Partition the adjacency graph of the matrix into p subgraphs (sub- domains), and map each subgraph to a processor. The objectives of the partitioning are that the subgraphs should have roughly equal work, and there should be few edges that join the di#erent subgraphs. 2. On each subgraph, locally order interior nodes first, and then order boundary nodes. 3. Form the subdomain intersection graph corresponding to the par- tition, and compute an approximate minimum vertex coloring for it. Order subdomains according to color classes. 4. Compute the incomplete factors in parallel. a. Factor interior rows of each subdomain. b. Receive sparsity patterns and numerical values of the nonzeros of the boundary rows of lower-numbered subdomains adjacent to a subdomain (if any). c. Factor boundary rows in each subdomain and send the sparsity patterns and numerical values to higher-numbered neighboring subdomains (if any). Fig. 2.1. High level description of the PILU algorithm. boundary vertices. The goal of the partitioning is to keep the amount of work associated with the incomplete factorization of each subgraph roughly equal, while keeping the communication costs needed to factor the boundary rows as small as possible. There is a di#culty with modeling the communication costs associated with the boundary rows. In order to describe this di#culty, we need to relate this cost more precisely to the separators in the graph. Define the higher degree of a vertex v as the number of vertices numbered higher than v in a given ordering. We assume that upward-looking, row-oriented factorization is used. At each boundary between two subgraphs, elements need to be communicated from the lower numbered subgraph to the higher numbered subgraph. The number of these elements is proportional to the sum of the higher degrees (in the filled graph G(F )) of the boundary vertices in the lower numbered subgraph. But unfortunately, we do not know the fill edges at this point since we have neither computed an ordering of G(A) nor computed a symbolic factorization. We could approximate by considering higher degrees of the boundary vertices in the graph G(A) instead of the filled graph G(F ), but even this requires us to order the subgraphs in the partition. The union of the boundary vertices on all the subgraphs forms a wide vertex separator . This means that the shortest path from an interior vertex in any subgraph to an interior vertex in another subgraph consists of at least three edges; such a path has length at least three. The communication cost in the (forward and backward) triangular solution steps is proportional to the sum of the sizes of the wide vertex separators. None of the publicly available graph partitioning software has the minimization of wide separators as its objective function, but it is possible to modify existing software to optimize this objective. 42 43 44 45 46 93 94 95 96 9799101103 104 105 106 107 108 Fig. 2.2. An example that shows the partitioning, mapping, and vertex ordering used in the PILU algorithm. The graph on the top is a regular 12 - 12 grid with a five-point stencil partitioned into four subdomains and then mapped on four processors. The subdomains are ordered by a coloring algorithm to reduce dependency path lengths. Only the level one and two fill edges that join the di#erent subdomains are shown; all other edges are omitted for clarity. The figure on the bottom right shows the subdomain intersection graph when the subdomain graph constraint is enforced. (This prohibits fill between the boundary nodes of the subdomains S 1 and S 2 , indicated by the broken edges in the top graph.) The graph on the bottom left shows the subdomain intersection graph when the subdomain graph constraint is not enforced. The goal of the partitioning step is to keep the amount of work associated with each subgraph roughly equal (for load balance) while making the communication costs due to the boundaries as small as possible. As the previous two paragraphs show, modeling the communication costs accurately in terms of edge and vertex separators in the initial graph G(A) is di#cult, but we could adopt the minimization of the wide separator sizes as a reasonable goal. This problem is NP-complete, but there exist e#cient heuristic algorithms for partitioning the classes of graphs that occur in practical situations. (Among these graph classes are 2D-finite element meshes and 3D-meshes with good aspect ratios.) Step 2: Local reordering. In the second step, in each subgraph we order the interior vertices before the boundary vertices. This ordering ensures that during the incomplete factorization, an interior vertex in one subgraph cannot be joined by a fill edge to a vertex in another subgraph, as will be shown later. Fill edges between two subgraphs can join only their boundary vertices together. Thus interior vertices corresponding to the initial graph G(A) remain interior vertices in the graph of the factor G(F ). The consequences of this are that the rows corresponding to the interior vertices in each subdomain of the initial problem G(A) can be factored concurrently, and that communication is required only for factoring rows corresponding to the boundary rows. The reader can verify that in each subgraph in Figure 2.2 the interior nodes have been ordered before the boundary nodes. The observation concerning fill edges in the preceding paragraph results from an application of the following incomplete fill path theorem. Given the adjacency graph G(A) of a coe#cient matrix A, the theorem provides a static characterization of where fill entries arise during an incomplete factorization L is the lower triangular incomplete factor, - U is the upper triangular incomplete factor, and E is the remainder matrix. The characterization is static in that fill is completely described by the structure of the graph G(A); no information from the factor is required. We need a definition before we can state the theorem. A fill path is a path joining two vertices i and j, all of whose interior vertices are numbered lower than the end vertices i and j. 1 Recall also the definition of the levels assigned to nonzeros in an incomplete factorization. To discuss the sparsity pattern of the incomplete factors, we consider the filled matrix I. The sparsity pattern of F is initialized to that of A. All nonzero entries in F corresponding to nonzeros in A have level zero, and zero entries have level infinity. New entries that arise during factorization are assigned a level based on the levels of the causative entries, according to the rule The incomplete fill path theorem describes an intimate relationship between fill entries in ILU(k) factors and path lengths in graphs. Theorem 2.1. Let I be the filled matrix corresponding to an incomplete factorization of A, and let f ij be a nonzero entry in F . Then f ij is a level entry if and only if there exists a shortest fill path of length joins i and j in G(A). A proof and a discussion of this theorem are included in the appendix. Now consider the adjacency graph G(A) and a partition of it into subgraphs (subdomains). Any path joining two interior nodes in distinct subdomains must include at least two boundary nodes, one from each of the subgraphs; since each boundary node is numbered higher than (at least one of) the path's end vertices (since these are interior nodes in the subgraph), this path cannot be a fill path. If two interior nodes belonging to separate subgraphs were connected by a fill path and the corresponding fill entry were permitted in F , the interior nodes would be transformed into boundary nodes in G(F ). This is undesirable for parallelism, since then there would be fewer interior nodes to be eliminated concurrently. The local ordering step preserves interior and boundary nodes during the factorization and ensures that a subdomain's interior rows can be factored independently of row updates from any other subdomain. Therefore, when subdomains have relatively large interior/boundary node ratios, and contain approximately equal amounts of computational work, we expect PILU to exhibit a high degree of parallelism. 1 The reader has doubtless noted that interior is used in a di#erent sense here than previously. We trust it will be obvious from the context where interior is used to refer to nodes in paths and where it is used to refer to nodes in subgraphs. Step 3: Global ordering. The global ordering phase is intended to preserve parallelism while factoring the rows corresponding to the boundary vertices. In order to explain the loss of concurrency that could occur during this phase of the algo- rithm, we need the concept of a subdomain intersection graph, which we shall call a subdomain graph for brevity. The subdomain graph S(G, #) is computed from a graph G and its partition subgraphs. The vertex set V s contains a vertex corresponding to every subgraph in the partition; the edge set E s contains edge if there is an edge in G with one endpoint in S i and the other in S j . We can compute a subdomain graph S(A) corresponding to the initial graph G(A) and its partition. (This graph should be denoted S(G(A), #), but we shall write S(A) for simplicity.) We could also compute a subdomain graph S(F ) corresponding to the graph of the factor G(F ). The subdomain graph S(A) corresponding to the partition of the initial graph G(A) (the top graph) in Figure 2.2 is shown in the graph at the bottom right in that figure. We impose a constraint on the fill, the subdomain graph constraint. The sub-domain graph corresponding to G(F ) is restricted to be identical to the subdomain graph corresponding to G(A). This prohibits some fill in the filled graph G(F if two subdomains are not joined by an edge in the original graph G(A), any fill edge that joins those subdomains is not permitted in the graph of the incomplete factor G(F ). The description of the PILU algorithm in Figure 2.1 assumes that the subdomain graph constraint is satisfied. This constraint makes it possible to obtain scalability in the parallel ILU algorithm. Later, we discuss how the algorithm should be modified if this constraint is relaxed. Each subdomain's nodes (in G(A)) are ordered contiguously. Consequently, saying "subdomain r is ordered before subdomain s" is equivalent to saying "all nodes in subdomain r are ordered, and then all nodes in subdomain s are ordered." This permits S(A) to be considered as a directed graph, with edges oriented from lower to higher numbered vertices. Edges in S(F ) indicate data dependencies in factoring the boundary rows of the subdomains. If an edge in S(F ) joins r and s and subdomain r is ordered before subdomain s, then updates from the boundary rows of r have to be applied to the boundary rows of s before the factorization of the latter rows can be completed. It follows that ordering S(F ) so as to reduce directed path lengths reduces serial bottlenecks in factoring the boundary rows. If we impose the subdomain graph constraint, these observations apply to the subdomain graph S(A) as well since then S(A) is identical with S(F ). We reduce directed path lengths in S(A) by coloring the vertices of the subdomain graph with few colors using a heuristic algorithm for graph coloring, and then by numbering the subdomains by color classes. The boundary rows of all subdomains corresponding to the first color can be factored concurrently without updates from any other subdomains. These subdomains update the boundary rows of higher numbered subdomains adjacent to them. After the updates, the subdomains that correspond to the second color can factor their boundary rows. This process continues by color classes until all subdomains have factored their boundary rows. The number of steps it takes to factor the boundary rows is equal to the number of colors it takes to color the subdomain graph. In Figure 2.2, let p i denote the processor that computes the subgraph S i . Then p 0 computes the boundary rows of S 0 and sends them to processors p 1 and p 2 . Similarly, 3 computes the boundary rows of subgraph S 3 and sends them to p 1 and p 2 . The latter processors first apply these updates and then compute their boundary rows. How much parallelism can be gained through subdomain graph reordering? We can gain some intuition through analysis of simplified model problems, although we cannot answer this question a priori for general problems and all possible partitions. Consider a matrix arising from a second order PDE that has been discretized on a regularly structured 2D grid using a standard five-point stencil. Assume that the grid is naturally ordered and that it has been partitioned into square subgrids and mapped into a square grid of p processors. In the worst case, the associated subdomain graph, which itself has the appearance of a regular 2D grid, can have a dependency path of length 2( # p - 1). Similarly, a regularly structured 3D grid discretized with a seven-point stencil that is naturally ordered and then mapped on a cube containing p processors can have a dependency path length of 3( 3 1). However, regular 2D grids with the five-point stencil and regular 3D grids with the seven-point stencil are bipartite graphs and can be colored with two colors. If all subdomains of the first color class are numbered first, and then all subdomains of the second color class are numbered, the longest dependency path in S will be reduced to one. This discussion shows that coloring the subdomain graph is an important step in obtaining a scalable parallel algorithm. Step 4: Preconditioner computation. Now that the subdomains and the nodes in each subdomain have been ordered, the preconditioner can be computed. We employ an upward-looking, row oriented factorization algorithm. The interior of each subdomain can be computed concurrently by the processors, and the boundary nodes can be computed in increasing order of the color classes. Either a level-based ILU(k) or a numerical threshold based ILUT(# , p) algorithm may be employed on each subdomain. Di#erent incomplete factorization algorithms could be employed in di#erent subdomains when appropriate, as in multiphysics problems. Di#erent fill levels could be employed for the interior nodes in a subdomain and for the boundary nodes to reduce communication and synchronization costs. 2.2. Relaxing the subdomain graph constraint. Now we consider how the subdomain graph constraint might be relaxed. Given a graph G(A) and a partition of it into subgraphs, we color the subdomain graph S(A) and order its subdomains as before. Then we compute the graph G(F ) of an incomplete factor and its subdomain graph S(F ). To do this, we need to discover the dependencies in S(F ), but initially we have only the dependencies in S(A) available. This has to be done in several rounds, because fill edges could create additional dependencies between the boundary rows of subdomains, which in turn might lead to further dependences. The number of rounds needed is the length of a longest dependency path in the subdomain graph G(F ), and this could be # p). This discussion applies when an ILU(k) algorithm is employed, with symbolic factorization preceding numerical factorization. If ILUT were to be employed, then symbolic factorization and numerical factorization must be interleaved, as would be done in a sequential algorithm. We can then color the vertices of S(F ) to compute a schedule for factoring the boundary rows of the subdomains. For achieving concurrency in this step the subdomain graph S(F ) should have a small chromatic number (independent of the number of vertices in G(A)). Note that the description of the PILU algorithm in Figure 2.1 needs to be modified to reflect this discussion when the subdomain graph constraint is relaxed. The graph G(F ) in Figure 2.2 indicates the fill edges that join S 1 to S 2 as broken lines. The corresponding subdomain intersection graph S(F ) is shown on the lower left. The edge between S 1 and S 2 necessitates three colors to color S(F the subdomains S 0 and S 3 form one color class; S 1 by itself constitutes the second color class; and S 2 by itself makes up the third color class. Thus three steps are needed for the computation of the boundary rows of the preconditioner when the subdomain graph constraint is relaxed. Note that the processor responsible for the subdomain S 2 can begin computing its boundary rows when it receives an update from either S 0 or S 3 , but that it cannot complete its computation until it has received the update from the Theorem 2.1 has an intuitively simple geometric interpretation. Given an initial node i in G(A), construct a topological "sphere" containing all nodes that are at a distance less than or equal to k edges. Then a fill entry f ij is admissible in an is within the sphere. Note that all such nodes j do not cause fill edges since there needs to be a fill path joining i and j. By applying Theorem 2.1, we can gain an intuitive understanding of the fill entries that may be discarded on account of the subdomain graph constraint. Referring again to Figure 2.2, we see that prohibited edges arise when two nonadjacent subdomains in G(A) have nodes that are joined by a fill path of length less than k zero edge is discarded by the constraint. 2.3. Existence of PILU preconditioners. The existence of preconditioners computed from the PILU algorithm can be proven for some classes of problems. Meijerink and van der Vorst [28] proved that if A is an M-matrix, then ILU factors exist for any predetermined sparsity pattern, and Manteu#el [27] extended this result to H-matrices with positive diagonal elements. These results immediately show that PILU preconditioners with sparsity patterns based on level values exist for these classes of matrices. This is true even when di#erent level values are used for the various subdomains and boundaries. Incomplete Cholesky (IC) preconditioners for symmetric problems could be computed with our parallel algorithmic framework using preconditioners proposed by Jones and Plassmann [21] and by Lin and Mor-e [23] on each subdomain and on the boundaries. The sparsity patterns of these preconditioners are determined by the numerical values in the matrix and by memory constraints. Lin and Mor-e have proved that these preconditioners exist for M- and H-matrices. Parallel IC preconditioners also can be shown to exist for M- and H-matrices. If the subdomain graph constraint is not enforced, then the preconditioner computed in parallel corresponds to a preconditioner computed by the serial algorithm from a reordered matrix. If the constraint is enforced, some specified fill elements are dropped from the Schur complement; it can be shown that the resulting Schur complement matrix is componentwise larger than the former and hence still an M-matrix. 2.4. Relation to earlier work. We now briefly discuss earlier parallel ILU algorithms that are related to the PILU algorithm proposed here. Earlier attempts at parallel algorithms for preconditioning (including approaches other than incomplete are surveyed in [6, 12, 34]; orderings suitable for parallel incomplete factorizations have been studied inter alios in [4, 11, 13]. The surveys also describe the alternate approximate inverse approach to preconditioning. Saad [33, section 12.6.1] discusses a distributed ILU(0) algorithm that has the features of graph partitioning, elimination of interior nodes in a subdomain before boundary nodes, and coloring the subdomains to process the boundary nodes in parallel. Only level 0 preconditioners are discussed there, so that fill between subdomains, or within each subdomain, do not need to be considered. No implementations or results were reported, although Saad has informed us recently of a technical report [24] that includes an implementation and results. Our work, done independently, shows how fill levels higher than zero can be accommodated within this algorithmic framework. We also analyze our algorithm for scalability and provide computational results on the performance of PILU preconditioners. Our results show that fill levels higher than zero are indeed necessary to obtain parallel codes with scalability and good performance. Karypis and Kumar [22] have described a parallel ILUT implementation based on graph partitioning. Their algorithm does not include a symbolic factorization, and they discover the sparsity patterns and the values of the boundary rows after the numerical computation of the interior rows in each subdomain. The factorization of the boundary rows is done iteratively, as in the discussion given above, where we show how the subdomain graph constraint might be relaxed. The partially filled graph of the boundary rows after the interior rows are eliminated is formed, and this graph is colored to compute a schedule for computing the boundary rows. Since fill edges in the boundary rows are discovered as these rows are being factored, this approach could lead to long dependency paths that are #(p). The number of boundary rows is meshes with good aspect ratios. If the cost of factoring and communicating a boundary row is proportional to the number of rows, then this phase of their algorithm could severely limiting the scalability of the algorithm (cf. the discussion in section 3). Recently Magolu monga Made and van der Vorst [25, 26] have reported variations of a parallel algorithm for computing ILU preconditioners. They partition the mesh, linearly order the subdomains, and then permit fill in the interior and the boundaries of the subdomains. The boundary nodes are classified with respect to the number of subdomains they are adjacent to, and are eliminated in increasing order of this number. Since the subdomains are linearly ordered, a "burn from both ends" ordering is employed to eliminate the subdomains. Our approaches are similar, except that we additionally order the subdomains by means of a coloring to reduce dependency path lengths to obtain a scalable algorithm. They have provided an analysis of the condition number of the preconditioned matrices for a class of 2D second order elliptic boundary value problems. They permit high levels of fill (four or greater) as we do, and show that the increased fill permitted across the boundaries enables the condition number of the preconditioned matrix to be insensitive to the number of subdomains (except when the latter gets too great). We have worked independently of each other. A di#erent approach, based on partitioning the mesh into rectangular strips and then computing the preconditioner in parallel steps in which a "wavefront" of the mesh is computed at each step by the processors, was proposed by Bastian and Horton [3] and was implemented for shared memory multiprocessors recently by Vuik, van Nooyen, and Wesseling [36]. This approach has less parallelism than the one considered here. 3. Performance analysis. In this section we present simplified theoretical analyses of algorithmic behavior for matrices arising from PDEs discretized on 2D grids with five-point stencils and 3D grids with seven-point stencils. Since our arguments are structural in nature, we assume ILU(k) is the factorization method used. After a word about nomenclature, we begin with the 2D case. The word grid refers to the grid (mesh) of unknowns for regular 2D and 3D grids with five- and seven-point stencils, respectively; this is identical to the adjacency graph G(A) of the coe#cient matrix of these problems. We use the terms eliminating Fig. 3.1. Counting lower triangular fill edges in a naturally ordered grid. We count the number of edges incident on vertex 9. Considering the graphs from top to bottom, we find that there are two level 0 edges; there is one level 1 edge, due to fill path 9, 3, 4; there is one level 2 edge due to fill path 9, 3, 4, 5; there are two level 3 edges, due to fill paths 9, 3, 4, 5, 6 and 9, 3, 2, 1, 7. We can generalize that two additional fill edges are created for every level greater than three, except near the boundaries. We conclude that asymptotically there are 2k lower triangular edges incident on a vertex in a level k factorization. Since the mesh corresponds to a structurally symmetric problem, there are 2k upper triangular edges incident on a vertex as well. a node and factoring a row synonymously. We assume the grid has been block-partitioned, with each subdomain consisting of a square subgrid of dimension c - c. We also assume the subdomain grid has dimensions # p- # p, so there are p processors in total. There are thus in the grid, and subdomains have at most boundary nodes. If subdomain interior nodes are locally numbered in natural order and k # c, each row in the factor F asymptotically has 2k (strict) upper triangular and 2k (strict) lower triangular nonzero entries. The justification for this statement arises from a consideration of the incomplete fill path theorem; the intuition is illustrated in Figure 3.1. Assuming that the classical ILU(k) algorithm is used for symbolic factorization, both symbolic and numeric factorization of row j entails 4k 2 arithmetic operations. This is because for each lower triangular entry f ji in matrix row j, factorization requires an arithmetic operation with each upper triangular entry in row i. A red-black ordering of the subdomain graph gives an optimal bipartite division. If red subdomains are numbered before black subdomains, our algorithm simplifies to the following three stages. 1. Red processors eliminate all nodes; black processors eliminate interior nodes. 2. Red processors send boundary-row structure and values to black processors. 3. Black processors eliminate boundary nodes. If these stages are nonoverlapping, the cost of the first stage is bounded by the cost of eliminating all nodes in a subdomain. This cost is 4k 2 c The cost for the second stage is the cost of sending structural and numerical values from the upper-triangular portions of the boundary rows to neighboring processors. c, the incomplete fill path theorem can be used to show that, asymptotically, a processor only needs to forward values from c rows to each neighbor. We assume a standard, noncontentious communication model wherein # and # represent message startup and per-word-transfer times, respectively. We measure these times in non-dimensional units of flops by dividing them by the time it takes to execute one flop. The time for an arithmetic operation is thus normalized to unity. Then the cost for the second step is Since the cost of factoring a boundary row can be shown to be asymptotically identical to that for factoring an interior row, the cost for eliminating the 4c boundary nodes is (4k 2 . Speedup can then be expressed as The numerator represents the cost for sequential execution, and the three terms in the denominator represent the costs for the three stages (arithmetic for interior nodes, communication costs, and arithmetic for the boundary nodes) of the parallel algorithm Three implications from this equation are in order. First, for a fixed problem size and number of processors, the parallel computational cost (the first and third terms in the denominator) is proportional to k 2 , while the communication cost (the second term in the denominator) is proportional to k. This explains the increase in e#ciency with level that we have observed. Second, if the ratio N/p is large enough, the first term in the denominator will become preeminent, and e#ciency will approach 100%. Third, if we wish to increase the number of processors p by some factor while maintaining a constant e#ciency, we need only increase the size of the problem N by the same factor. This shows that our algorithm is scalable. This observation is not true for a direct factorization of the coe#cient matrix, where the dependencies created by the additional fill cause loss in concurrency. For the 3D case we assume partitioning into cubic subgrids of dimension c - c - c and a subdomain grid of dimension p 1/3 which gives Subdomains have at most 6c 2 boundary nodes. A development similar to that above shows that, asymptotically, matrix rows in the factor F have 2k 2 (strict) upper and lower triangular entries, so the cost for factoring a row is 4k 4 . Speedup for this case can then be expressed as 4k 4 N 4k 4 N 4. Results. Results in this section are based on the following model problems. Problem 1. Poisson's equation in two or three dimensions: g. Problem 2. Convection-di#usion equation with convection in the xy plane: #x e xy #y e -xy g. Homogeneous boundary conditions were used for both problems. Derivative terms were discretized on the unit square or cube, using 3-point central di#erencing on regularly spaced n x - n y - n z grids (n 2D). The values for # in Problem 2 were set to 1/500 and 1/1000. The problem becomes increasingly unsymmetric, and more di#cult to solve accurately as # decreases. The right-hand sides of the resulting systems, artificially generated as is the all-ones vector. preconditioning is amenable to performance analysis since the nonzero structures of ILU(k) preconditioners are identical for any PDE that has been discretized on a 2D or 3D grid with a given stencil. The structure depends on the grid and the stencil only and is not a#ected by numerical values if pivoting is not needed for numerical stability. Identical structures imply identical symbolic factorization costs, as well as identical flop counts during the numerical factorization and solve phases. In parallel contexts, communication patterns and costs are also identical. While preconditioner e#ectiveness-the number of iterations until the stopping criteria is reached-di#ers with the numerics of the particular problem being modeled, the parallelism available in the preconditioner does not. The structure of ILUT preconditioners, on the other hand, is a function of the grid, the stencil, and the numerics. Changing the problem, particularly for non- diagonally dominant cases, can alter the preconditioner structure, even when the grid and stencil remain the same. We report our performance evaluation for ILU(k) preconditioners, although the parallel algorithmic framework proposed here could just as easily work with ILUT(# , p). We have compared the performance of ILU(k) with ILUT in an earlier report [18]. We report there that for Problem 2 with incurred more fill than ILU(5) on a 2D domain for grid sizes up to 400 - 400; for 3D domains and grid sizes up to 64 - 64 - 64, the same ILUT preconditioner incurred fill between ILU(2) and ILU(3). In addition to demonstrating that our algorithm can provide high degrees of parallelism, we address several other issues. We study the influence of the subdomain graph constraint on the fill permitted in the preconditioner and on the convergence of preconditioned Krylov space solvers. We also report convergence results as a function of the number of nonzeros in the preconditioner. 4.1. Parallel performance. We now report timing and scalability results for preconditioner factorization and application on three parallel platforms: . an SGI Origin2000 at NASA Ames Research Center (AMES); . the Coral PC Beowulf cluster at ICASE, NASA Langley Research Center; . a Sun HPC 10000 Starfire server at Old Dominion University (ODU). 2206 DAVID HYSOM AND ALEX POTHEN Table Time (sec.) required for incomplete (symbolic and numeric) factorization for a 3D scaled problem; 91, 125 unknowns per processor, seven-point stencil, ILU(2) factorization on interior nodes, and ILU(1) factorization on boundary nodes. Dashes (-) for Beowulf and HPC 10000 indicate that the machines have insu#cient cpus to perform the runs. Procs Origin2000 Beowulf HPC 10000 AMES (ICASE) (ODU) 8 2.44 3.11 2.43 Both problems were solved using Krylov subspace methods as implemented in the PETSc [2] software library. Problem 1 was solved using the conjugate gradient method, and Problem 2 was solved using Bi-CGSTAB [35]. PETSc's default convergence criterion was used, which is five orders of magnitude (10 5 ) reduction in the residual of the preconditioned system. We used our own codes for problem generation, partitioning, ordering, and symbolic factorization. Table 4.1 shows incomplete factorization timings for a 3D memory-scaled problem with approximately 91, 125 unknowns per processor. As the number of processors increases, so does the size of the problem. The coe#cient matrix of the problem factored on 216 processors has about 19.7 million rows. ILU(2) was employed for the interior nodes, and ILU(1) was employed for the boundary nodes. Reading down any of the columns shows that performance is highly scalable, e.g., for the SGI Origin2000, factorization for 216 processors and 19.7 million unknowns required only 62% longer than the serial case. Scanning horizontally indicates that performance was similar across all platforms, e.g., execution time di#ered by less than a factor of two between the fastest (Origin2000) and slowest (Beowulf) platforms. Table 4.2 shows similar data and trends for the triangular solves for the scaled problem. Scalability for the solves was not quite as good as for factorization; e.g., the solve with 216 processors took about 2.5 times longer than the serial case. This is expected due to the lower computation cost relative to communication and synchronization costs in triangular solution. We observed that the timings for identical repeated runs on the HPC 10000 and SGI typically varied by 50% or more, while repeated runs on the Beowulf were remarkably consistent. Table 4.3 shows speedup for a constant-sized problem of 1.7 million unknowns. There is a clear correlation between performance and subdomain interior/boundary node ratios; this ratio needs to be reasonably large for good performance. The performances reported in these tables are applicable to any PDE that has been discretized with a seven-point central di#erence stencil since the sparsity pattern of the symbolic factor depends on the grid and the stencil only. 4.2. Convergence studies. Our approach for designing parallel ILU algorithms reorders the coe#cient matrices whose incomplete factorization is being computed. This reordering could have a significant influence on the e#ectiveness of the ILU preconditioners. Accordingly, in this section we report the number of iterations of a preconditioned Krylov space solver needed to reduce the residual by a factor of 10 5 . We compare three di#erent algorithms. Table Time (sec.) to compute triangular solves for 3D scaled problem; 91, 125 unknowns per processor, seven-point stencil, ILU(2) factorization on interior nodes, ILU(1) factorization on boundary nodes. Dashes (-) for Beowulf and HPC 10000 indicate that the machines have insu#cient cpus to perform the runs. Procs Origin2000 Beowulf HPC 10000 Table Speedup for 3D constant-size problem; the grid was 120-120-120 for a total of approximately 1.7 million unknowns; data is for ILU(0) factorization performed on the SGI Origin2000; "I/B ratio" is the ratio of interior to boundary nodes in each subdomain. Procs Unknowns/ I/B Time E#ciency Processor ratio (sec.) (%) 8 216,000 9.3 2.000 100 4.3 .408 62 . Constrained PILU(k) is the parallel ILU(k) algorithm with the subdomain graph constraint enforced. . In unconstrained PILU(k), the subdomain graph constraint is dropped, and all fill edges up to level k between the boundary nodes of di#erent subdomains are permitted, even when such edges join two nonadjacent subdomains of the initial subdomain graph S(A). . In block Jacobi ILU(k) (BJILU(k)), all fill edges joining two di#erent subdomains are excluded. Intuitively, one expects, especially for diagonally dominant matrices, that larger amounts of fill in preconditioners will reduce the number of iterations required for convergence. 4.2.1. Fill count comparisons. For a given problem, the number of permitted fill edges is a function of three components: the factorization level, k; the subdomain and the discretization stencil. While the numerical values of the coe#cients of a particular PDE influence convergence, they do not a#ect fill counts. Therefore, our first set of results consists of fill count comparisons for problems discretized on a using a standard, seven-point stencil. Table 4.4 shows fill count comparisons between unconstrained PILU(k), constrained PILU(k), and block Jacobi ILU(k) for various partitionings and factorization levels. The data shows that more fill is discarded as the factorization level increases, and as subdomain size (the number of nodes in each subdomain) decreases. These two e#ects hold for both constrained PILU(k) and block Jacobi ILU(k) but are much more pronounced for the latter. For example, less than 5% of fill is discarded from unconstrained factors when subdomains contain at least 512 nodes (so that the Table Fill comparisons for the 64 - 64 - 64 grid. U denotes unconstrained, C denotes constrained, and B denotes block Jacobi ILU(k) preconditioners. The columns headed "nzF/nzA" show the ratio of the number of nonzeros in the preconditioner to the number of nonzeros in the original problem and are indicative of storage requirements. The columns headed "constraint e#ects" present another view of the same data: here, the percentage of nonzeros in the constrained PILU(k) and block Jacobi ILU(k) factors are shown relative to that for the unconstrained PILU(k). These columns show the amount of fill dropped due to the subdomain graph constraint. Nodes per Subdom. nzF/nzA Constraint e#ects (%) subdom. count Level U 4 9.73 9.73 9.73 100.00 100.00 3 6.32 6.32 5.70 99.92 90.13 3 6.60 5.64 2.71 85.37 41.04 subgraphs on each processor are not too small), but up to 42% is discarded from block Jacobi factors. Thus, one might tentatively speculate that, for a given subdomain size and level, PILU(k) will provide more e#ective preconditioning than BJILU(k). We have observed similar behavior for 2D problems also. For both 2D and 3D problems, when there is a single subdomain the factors returned by the three algorithms are identical. For the single subdomain case, the ordering we have used corresponds to the natural ordering for these model problems. An important observation to make in Table 4.4 is how the sizes (number of nonze- ros) of the preconditioners depend on levels of fill. For the 3D problems considered here (cube with 64 points on each side, seven-point stencil), a level one preconditioner typically requires twice as much storage as the coe#cient matrix A; when the level is two, this ratio is about three; when the level is three, it is about six; and when the level is four, it is about ten. For 2D problems (square grid with 256 points on a side, Table Iteration comparisons for the 64-64-64 grid. U denotes unconstrained, C denotes constrained, and B denotes block Jacobi ILU(k) preconditioners. The starred entries (*) indicate that, since there is a single subdomain, the factor is structurally and numerically identical to the unconstrained PILU(k). Dashed entries (-) indicate the solutions either diverged or failed to converge after 200 iterations. For Problem 2, when the level zero preconditioners did not reduce the relative error in the solution by a factor of 10 5 at termination; when the level one preconditioners did not do so either. Problem 1 Problem 2 Nodes per Subdom. subdom. count Level U C B U C B U C B 28 78 28 28 28 67 - 4 43 43 64 28 28 - 63 63 - 26 five-point stencil), the growth of fill with level is slower; the ratios are about 1.4 for level one, 1.8 for level two, 2.6 for level three, 3.5 for level four, 4.3 for level five, and 5.4 for level six. In parallel computation fill levels higher than those employed in sequential computing are feasible since modern multiprocessors are either clusters or have virtual shared memory, and these have memory sizes that increase with the number of pro- cessors. Another point to note is that the added memory requirement for these level values is not as prohibitive as it is for a complete factorization. Hence it is practical to trade-o# increased storage in preconditioners for reducing the number of iterations in the solver. 4.2.2. Convergence of preconditioned iterative solvers. The fill results in the previous subsection are not influenced by the actual numerical values of the nonzero coe#cients; however, the convergence of preconditioned Krylov space solvers is influenced by the numerical values. Accordingly, Table 4.5 shows iterations required for convergence for various partitionings and fill levels for the three variant algorithms that we consider. The data in these tables can be interpreted in various ways; we begin by discussing two ways that we think are primarily significant First, by scanning vertically one can see how changing the number of subdomains, and hence, matrix ordering, a#ects convergence. The basis for comparison is the iteration count when there is a single subdomain. The partitioning and ordering for these cases is identical to, and our data in close agreement with, that reported by Benzi, Joubert, and Mateescu [4] for natural ordering. (They report results for Problem 2 with but not for A pleasing property of both the constrained and unconstrained PILU algorithms is that the number of iterations increases only mildly when we increase the number of subdomains from one to 512 for these problems. This insensitivity to the number of subdomains when the number of nodes per subdomain is not too small confirms that the PILU algorithms enjoy the property of parallel algorithmic scalability. For example, Poisson's equation (Problem 1) preconditioned with a level two factorization and a single subdomain required 24 iterations. Preconditioning with the same level, constrained PILU(k) on 512 subdomains needed only two more iterations. Similar results are observed for the convection-di#usion problems also. This property is a consequence of the fill between the subdomains that is included in the PILU algorithm. Similar results have been reported in [26, 36], and the first paper includes a condition number analysis supporting this observation. Increasing the level of fill generally has the beneficial e#ect of reducing the number of iterations needed; this influence is largest for the worse-conditioned convection- di#usion problem with 1/1000. For this problem, level zero preconditioners do not converge for reasonable subdomain sizes. Also, even though level one preconditioners require fewer iteration numbers than level two preconditioners in some cases, when the PETSc solvers terminate because the residual norms are reduced by 10 5 , the relative errors are larger than 10 -5 for the former preconditioners. The relative errors are also large for the convection-di#usion problem with when the level is set to zero. Second, scanning the data in Table 4.5 horizontally permits evaluation of the subdomain graph constraint's e#ects. Again, unless subdomains are small and the factorization level is high; constrained and unconstrained PILU(k) show very similar behavior. Consider, for example, Poisson's equation (Problem 1) preconditioned with a level two factorization and 512 subdomains. The solution with unconstrained required 25 iterations while constrained PILU(k) required 26. We also see that PILU(k) preconditioning is more e#ective than BJILU(k) for all 3D trials. (Recall that the single apparent exception, Problem 2, with 32, 768 nodes per subdomain, has large relative errors at termination.) Again, the extremes of convergence behavior are seen for Problem 2 with with level one preconditioners, BJILU(k) su#ers large relative errors at termination while the other two algorithms do not, when the number of subdomains is 64 or fewer. On 2D domains, while PILU(k) is more e#ective than BJILU(k) for Poisson's equation, BJILU(k) is sometimes more e#ective in the convection-di#usion problems. We also examine iteration counts as a function of preconditioner size graphically. A plot of this data appears in Figure 4.1. In these figures the performance of the 512 nodes per subdomain, 512 subdomains Block Jacobi ILU(k) Const. Unconst. PILU(k)1020304050 4096 nodes per subdomain, 64 subdomains Block Jacobi ILU(k) Const. Unconst. nodes per subdomain, 8 subdomains Block Jacobi ILU(k) Const. Unconst. Fig. 4.1. Convergence comparison as a function of preconditioner size for the convection- di#usion problem, on the 64 - 64 - 64 grid. Data points are for levels through 4. Data points for constrained and unconstrained PILU(k) are indistinguishable in the third graph. constrained and unconstrained PILU algorithms is often indistinguishable. We find again that PILU(k) preconditioning is more e#ective than BJILU(k) for 3D problems for a given preconditioner size; however, this conclusion does not always hold for 2D problems, especially for lower fill levels. As the number of vertices in the subdomains increases, higher fill levels become more e#ective in reducing the number of iterations needed for convergence. We find that fill levels as high as four to six can be the most e#ective when the subdomains are su#ciently large. Fill levels higher than these do not seem to be merited by these problems, even for the di#cult convection-di#usion problems with level four preconditioner reduces the number of iterations below ten. 5. Conclusions. We have designed and implemented a PILU algorithm, a scalable parallel algorithm for computing ILU preconditioners that creates concurrency by means of graph partitioning. The theoretical basis of the algorithm is the incomplete fill path theorem that statically characterizes fill elements in an incomplete factorization in terms of paths in the adjacency graph of the initial coe#cient matrix. To obtain a scalable parallel algorithm, we employ a subdomain graph constraint that excludes fill between subgraphs that are not adjacent in the adjacency graph of the initial matrix. We show that the PILU algorithm is scalable by an analysis for 2D- and 3D-model problems and by computational results from parallel implementations on three parallel computing platforms. We also study the convergence behavior of preconditioned Krylov solvers with preconditioners computed by the PILU algorithm. The results show that fill levels higher than one are e#ective in reducing the number of iterations, that the number of iterations is insensitive to the number of subdomains, and that the subdomain graph constraint does not a#ect the number of iterations needed for convergence while it makes possible the design of a scalable parallel algorithm. Appendix . Proof of the incomplete fill path theorem. Theorem A.1. Let I be the filled matrix corresponding to an incomplete factorization of A, and let f ij be a nonzero entry in F . Then f ij is a level entry if and only if there exists a shortest fill path of length joins i and j in G(A). Proof. If there is a shortest fill path of length we prove that the edge exists by induction on the length of the fill path. Define a chord of a path to be an edge that joins two nonconsecutive vertices on the path. The fill path joining i and j is chordless, since a chord would lead to a shorter fill path. The base case immediate, since a fill path of length one in the graph G(A) is an edge {i, j} in G(A) that corresponds to an original nonzero in A. Now assume that the result is true for all lengths less than k + 1. Let h denote the highest numbered interior vertex on the fill path joining i and j. We claim that the (i, h) section of this path is a shortest fill path in G(A) joining i and h. This section is a fill path by the choice of h since all intermediate vertices on this section are numbered lower than h. If there were a fill path joining i and h that is shorter than the (i, h) section, then we would be able to concatenate it with the (h, section to form a shorter (i, path. Hence the (i, h) section is a shortest fill path joining i and h. Similarly, the (h, j) section of this path is the shortest fill path joining h and j. Each of these sections has fewer than k hence the inductive hypothesis applies. Denote the number of edges in the (i, h) ((h, j)) section of this path by 1. By the inductive hypothesis, the edge {i, h} is a fill edge of level k 1 - 1, and the edge {h, j} is a fill edge of level k 2 - 1. Now by the sum rule for updating fill levels, when the vertex h is eliminated, we have a fill edge {i, j} of level Now we prove the converse. Suppose that {i, j} is a fill edge of level k; we show that there is a fill path in G(A) of length by induction on the level k. The base case immediate, since the edge {i, j} constitutes a trivial fill path of length one. Assume that the result is true for all fill levels less than k. Let h be a vertex whose elimination creates the fill edge {i, j} of level k. Let the edge {i, h} have level k 1 , and let the edge {h, j} have level k 2 ; by the sum rule for computing levels, we have that k. By the inductive hypothesis, there is a shortest fill path of length h, and such a path of length and j. Concatenating these paths, we find a fill path joining i and j of length We need to prove that the (i, fill path in the previous paragraph is a shortest fill path between i and j. Consider the elimination of any another vertex g that causes the fill edge {i, j}. By the choice of the vertex h, if the level of the edge {i, g} is k # 1 and that of {g, j} is k. The inductive hypothesis applies to the (i, g) and (g, j) sections, and hence the sum of their lengths is at least k + 1. This completes the proof. This result is a generalization of the following theorem that characterizes fill in complete factorizations for direct methods, due to Rose and Tarjan [30]. Theorem A.2. Let I be the filled matrix corresponding to the complete factorization of A. only if there exists a fill path joining in the graph G(A). Here we associate level values with each fill edge and relate it to the length of shortest fill paths. The incomplete fill path theorem enables new algorithms for incomplete symbolic factorization that are more e#cient than the conventional algorithm that simulates numerical factorization. We have described these algorithms in an earlier work [29] and the report is in preparation. D'Azevedo, Forsyth, and Tang [9] have defined the (sum) level of a fill edge {i, j} using the length criterion employed here, and hence they were aware of this result. However, the theorem is neither stated nor proved in their paper. Definitions of level that compute levels of fill nonzeros by rules other than by summing the levels of the causative pairs of nonzeros have been used in the literature. The "maximum" rule defines the level of a fill nonzero to be the minimum over all causative pairs of the maximum value of the levels of the causative entries: A variant of the incomplete fill path theorem can be proved for this case, but it is not as simple or elegant as the one for the "sum" rule. Further discussion of these issues will be deferred to a future report. Acknowledgments . We thank Dr. Edmond Chow of CASC, Lawrence Livermore National Laboratory, and Professor Michele Benzi of Emory University for helpful discussions. --R Cambridge University Press http://www. Parallelization of robust multigrid methods: ILU factorization and frequency decomposition method Numerical experiments with parallel orderings for ILU preconditioners Approximate and incomplete factorizations An object-oriented framework for block preconditioning Experimental study of ILU preconditioners of indefinite matrices Ordering methods for preconditioned conjugate gradient methods applied to unstructured grid problems A Graph-Theory Approach for Analyzing the E#ects of Ordering on ILU Preconditioning Ordering strategies and related techniques to overcome the trade-o# between parallelism and convergence in incomplete factorizations Numerical Linear Algebra for High Performance Computers Analysis of parallel incomplete point factorizations Iterative Methods for Solving Linear Systems Parallel incomplete Cholesky preconditioners based on the nonoverlapping data distribution Incomplete Decomposition (ILU): Algorithms Parallel ILU Ordering and Convergence Relationships: Numerical Experiments An improved incomplete Cholesky factorization Parallel threshold-based ILU factorization An incomplete factorization technique for positive definite linear systems An iterative solution method for linear equation systems of which the coe Fast algorithms for incomplete factorization Algorithmic aspects of vertex elimination on directed graphs ILUT: A dual-threshold incomplete LU factorization Iterative Methods for Sparse Linear Systems Parallelism in ILU-preconditioned GM- RES --TR --CTR 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 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, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

preconditioning;parallel preconditioning;incomplete factorization;ILU

587409

Asymptotic Analysis of the Laminar Viscous Flow Over a Porous Bed.

We consider the laminar viscous channel flow over a porous surface. The size of the pores is much smaller than the size of the channel, and it is important to determine the effective boundary conditions at the porous surface. We study the corresponding boundary layers, and, by a rigorous asymptotic expansion, we obtain Saffman's modification of the interface condition observed by Beavers and Joseph. The effective coefficient in the law is determined through an auxiliary boundary-layer type problem, whose computational and modeling aspects are discussed in detail. Furthermore, the approximation errors for the velocity and for the effective mass flow are given as powers of the characteristic pore size $\ep$. Finally, we give the interface condition linking the effective pressure fields in the porous medium and in the channel, and we determine the jump of the effective pressures explicitly.

Introduction Finding eective boundary conditions at the surface which separates a channel ow and a porous medium is a classical problem. Supposing a laminar incompressible and viscous ow, we nd out immediately that the eective ow in a porous solid is described by Darcy's law. In the free uid we obviously keep the Navier-Stokes system. Hence we have two completely dierent systems of partial dierential equations. First, Darcy's law combined with the incompressibility gives a second order equation for the pressure and a rst order system for the velocity. In the Navier-Stokes system, the orders of the corresponding dierential operators are dierent, and it is not clear what kind of conditions one should impose at the interface between the free uid and the porous part. Clearly, due to the incompressibility, the normal mass ux should be continuous. Other classically used conditions are the continuity of the pressure and, for a free uid, the vanishing of the tangential velocity at the interface. Let us discuss the mathematical background of the interface conditions. It is well-known that Darcy's law is a statistical result giving the average of the momentum equation (the Navier-Stokes equations) over the pore structure. Its rigorous derivation involves the weak convergence in div) (respectively the two-scale convergence) of velocities, and only the continuity of the normal velocities is preserved. Other continuity conditions at the interface are generally lost, such that further analysis is required. Concerning other interface conditions used in engineering literature, the vanishing of the tangential velocity is found to be an unsatisfactory approximation, and in [2] a new condition is proposed. The condition reads where ~u e is the eective velocity in the channel, ~v f is the mean ltration velocity given by Darcy's law, ~ is a tangent vector to the interface, ~ is the normal into the Asymptotic analysis of the laminar viscous ow over a porous bed 3 uid, K is the permeability of the porous medium, and the scalar is a function of the geometry of the porous medium. In [2], this law is derived by heuristic arguments and justied experimentally. A theoretical attempt to derive (1.1) is undertaken in [15] and, using a statistical approach, a Brinkman type approximation in the transition layer is derived. A matching argument then allows to obtain the formula The interested reader can also consult the lecture note [4]. Dierent considerations can be found in [5] and [11]. They distinguish two cases: (a) The pressure gradient on the side of the porous solid of the interface is normal to the interface. Consequently, we have a balanced ow on both sides of the interface. Then, using an asymptotic point of view, the following laws are obtained in [11]: ( ~u e on the interface. This case describes the ows in cavities. The mathematical justication is in [9]. We shall not consider it in this paper. (b) The pressure gradient on the side of the porous solid at the interface is not normal. This case is considered in the fundamental paper [5]. After discussing the orders of magnitude of the unknowns it is found out that on the interface the velocity of the free uid is zero, and the pressure is continuous. All results cited above are not mathematically rigorous. Furthermore, dierent approaches give dierent results and two natural questions arise immediately: (Q1) What are the correct matching conditions (i.e. conditions at the interface) between those two ow equations? (Q2) What are the eective constants entering the matching conditions? We are going to answer those questions in the following. In Section 2, we dene our problem and discuss some simple approximations. In Section 3, we introduce an important auxiliary problem of boundary layer type which we need to construct a better approximation, and Section 4 gives additional results for an analogous problem on a nite strip. Then, in Section 7, the eective equations with the Beavers-Joseph type boundary condition are presented together with improved error estimates. Finally, in Section 6, we show how the computation of the constants involved in the interface conditions works in practice. Especially, it turns out that the dierence between Darcy's pressure in the porous part and the eective channel pressure is equal to a constant multiple of the eective channel shear stress, thus contradicting [11]. Asymptotic analysis of the laminar viscous ow over a porous bed 5 e e Y Y * Z * Figure 1: The ow region 2 Setting of the problem This section deals with the equations describing uid motion over a porous bed, under a uniform pressure gradient. We assume the condition of the experiment by Beavers and Joseph [2], i.e. a stationary laminar incompressible viscous ow. For simplicity reasons, we consider a ow over a periodic porous medium with a characteristic pore size ". The ow region of two parts, see Figure 1. The upper part and the lower part 2 is the porous medium which is obtained by putting solid obstacles of size " b into the domain the (permeable) interface betweenand 2 . More precisely, let made of two complementary parts, the solid part Z and the uid part Y . It is assumed that Z is a smooth closed subset of Y , strictly included in Y , Y \Z and Y [Z Now assume that both b and L are integer multiples of ". Then the domain can be covered by a regular mesh of N(") cells of size ". Each cell Y " divided into a uid part "(Y + k) and a solid 6 Willi Jager, Andro Mikelic and Nicolas Neu k). The uid part 2 of the porous medium is therefore "= and the whole ow region is After specifying the geometry, we consider the equations determining the velocity eld and the pressure eld p " in the Beavers{Joseph experiment: in in (@ For small ", this problem is extremely dicult to solve, so that one has to look for approximations. Classically, the system (2.3)-(2.8) used to be approximated by a Poiseuille ow in for and, indeed, it could be shown in [10] that (2.9) is an approximation in the following sense: Z Z Z Asymptotic analysis of the laminar viscous ow over a porous bed 7 The above estimates indicate that in the L 2 -sense in in in in in on and the eective mass ow behaves as H 3 As shown in [2] and [15], this O(") approximation often is not good enough. Therefore, we would like to continue with the asymptotic expansion for ~u " and p " . Energy estimates from [10] imply that in the interior of 1 , but globally in Further, they show that there is an oscillatory boundary layer conned to the neighborhood of which can be represented in the ~ bl;" bl ( x where ~ are solutions to a Stokes problem on an innite strip Z bl , which we discuss in the following Section 3. Z -2 Z bl Figure 2: The boundary layer cell Z bl 3 The auxiliary boundary layer problem Figure 2. For later use, we also introduce the following notation Z <k := Z bl \ (0; 1) (1; l := Z bl \ (0; 1) (l; Now f ~ are the solutions to the problem ~ div y ~ [(r y ~ ~ Asymptotic analysis of the laminar viscous ow over a porous bed 9 Further, in order to dene ! bl uniquely, we require A variational formulation of this problem is: nd a locally square integrable vector eld ~ bl 2 W satisfying Z Z bl r ~ bl Z where W is the function space which contains all y 1 -periodic, divergence-free, locally square integrable vector elds ~z dened on Z bl , having nite viscous energy (i.e. R satisfying the no-slip boundary condition at the solid boundaries [ 1 In [9] (Proposition 3.22, pages 462-463) the following result is proved: Theorem 1 The problem (3.11) has a unique solution ~ bl 2 W . It is locally innitely dierentiable outside of the interface S. Furthermore, a pressure eld ! bl exists, which is unique up to an additive constant and locally innitely dierentiable outside of the interface S, such that (3.5) holds. In a neighborhood of S we have ~ bl The following lemma states some simple properties of f ~ bl l Z Z bl jr ~ Proof. (3.12) follows immediately from (3.6) by Z 1@ bl@y 2 Z 1@ bl@y 1 Then, by integrating the second component of the momentum equation (3.5) over the rectangle (0; 1) (a; b) we obtain and (3.13) follows by applying (3.12). Next, integrate the rst component of the momentum equation (3.5) over (0; 1) with respect to y 1 . Using (3.13) we nd out that linear function of y 2 . Since it is bounded in the limit y it has to be a constant for y 2 0, which proves (3.14). Finally, after taking bl as the test function in (3.11), we obtain (3.15). We expect that the problem (3.5){(3.10) represents a boundary layer. This means that changes of the velocity and the pressure elds are concentrated around the interface S and vanish very rapidly with increasing distance from S. In linear elasticity, results of this type are called Saint-Venant's principle. Saint-Venant's principle is also valid in our case, and we want to discuss this in more detail. We start our considerations with Z (the part of Z bl lled by the uid). Here, one can obtain sharp decay estimates by using results for the decay of solutions of general elliptic equations (see Theorem 10.1 from [12], which is an application of Tartar's lemma). However, in our particular situation we can give a direct proof: Theorem 3 Let Then, for every a > 0, we have Asymptotic analysis of the laminar viscous ow over a porous bed 11 Proof. Applying the curl -operator to equation (3.5), we see that ag : (3.19) Furthermore, by (3.14) and periodicity, we conclude that Consequently, the solution bl of (3.19) may be written as We see, that the leading decay exponent is proportional to 2, such that (3.17) and (3.18) follow. With the help of this estimate for bl , we can prove exponentially fast stabilization of ~ Corollary 4 Let Then for every a > 0 and every < 2 we have ~ and D ~ Proof. We follow [9]: by (3.16) and (3.6) the functions bl bl bl @ Using the explicit form of the right hand side given by (3.20), a variation of constants yields also an explicit representation for bl bl (D 1 (D 1 bl (D 2 1;n (D 2 e 2ny2 (3.27) with the additional relations 4n 4n The representation (3.26),(3.27) allows us to conclude (3.22) and (3.23). For the pressure eld ! bl , we have: Corollary 5 For we have ! bl (y Proof. Taking the divergence of the momentum equation (3.5), we nd that bl is square integrable and the averages over sections fy ag of ! bl C bl are zero, ! bl can be written as from which the assertion follows. Now we turn our attention to the porous part Z . Due to the presence of the solid obstacles, our estimates will be much less precise in this case. Lemma 6 Let the distance between the solid obstacles and the boundary of the unit cell Y be bigger than or equal to 2d . Let ~ bl be the solution of (3.5){(3.10), and let l 2 Z; l < 1. We introduce a function ~ lower by ~ lower bl for y 2 l (3.32) Asymptotic analysis of the laminar viscous ow over a porous bed 13 ~ lower lower bl d Z y1 bl Z l bl Z 2l y 2 l lower bl Z 2l y 2 l bl for Then ~ lower 2 W \ H 1 (Z bl r ~ lower C lower r ~ bl where the constant C lower satises C lower d 3 with C P denoting the Poincare constant appearing in bl r bl If we further assume that the ball B (1=2; 1=2) with radius and center (1=2; 1=2) is contained in Z , an easy calculation in polar coordinates yields the estimate which (together with d 1) results in the estimate r ~ lower r ~ bl Proof. By a straightforward calculation, one veries that ~ lower This calculation also yields the estimates Z dy d 2 bl 2 dy Z bl 2 dy 14 Willi Jager, Andro Mikelic and Nicolas Neu Z @ lower@y 2 dy d 4 Z bl 2 dy Z bl 2 dy +d 4 Z @ lower@y 2 dy Z l l d bl d Z dy 4 Z where Z l;l we use the simple trace estimates Z bl 2 dy Z dy (3.44) Z l l 1 bl Z bl 2 dy Z We insert (3.44),(3.45) and (3.38) into the sum of (3.41){(3.43) and use d 1 to obtain (3.36) and (3.37). With the help of this lemma, we are able to deduce exponential decay of r ~ Proposition 7 Assume that l 2 Z with l < 0. Dene Z <l as in (3.2). Then the solution ~ bl of problem (3.11) fullls r ~ bl r ~ bl e lower where lower =2 ln C lower Proof. If we test (3.11) with some function ~ bl ~ lower , where ~ lower is the function constructed in the previous lemma, this results in the estimate r ~ bl C lower r ~ bl Using the hole-lling technique as in[9], Lemma 2.4 (we add C lower r ~ bl to both sides of (3.48) and evaluate the resulting recursion), we obtain exponential decay with rate lower given by (3.47). Applying local regularity results, one immediately obtains: Asymptotic analysis of the laminar viscous ow over a porous bed 15 Corollary 8 For any a < 0, 2 IN , the solution f ~ exponentially for C(a; )e lower and D ! bl (y C(a; )e lower for all y 2 < a < 0. Proof. From Proposition 3.7 of [9] we know that Z Z l C r ~ bl Z Z l Z Z l+1 C r ~ bl which implies exponentially fast stabilization to a constant (which has to be zero because of (3.10)): Ce lower jlj for l < Pointwise estimates for ~ bl , ! bl and their derivatives can then be obtained as usual by dierentiating the equations obtaining estimates for higher derivatives, which can then be used with the Sobolev embedding theorem. The above results imply that ~ bl is a boundary layer type velocity eld and bl is a boundary layer type pressure. Only the constants C bl 1 and C bl ! from (3.21) and (3.29) will enter the eective ow equations in the channel. They contain the information about the geometry of the porous bed. Remark 9 As we shall see from the numerical examples, in general C bl However, if the geometry of Z is axisymmetric with respect to re ections around the axis y This result is obtained by the following simple argument: Let Z be axisymmetric around the axis y bl be a solution for (3.5){(3.10). Then ( bl a solution. By uniqueness, it must be equal to ( ~ we conclude that Asymptotic analysis of the laminar viscous ow over a porous bed 17 4 Approximation of the boundary layer problem on a nite domain In this section we propose a scheme for computing the actual values of C bl 1 and cases where the geometry of the porous medium is known. Since problem resp. (3.11) is dened on an innite domain Z bl , the rst step is to approximate its solution with solutions of problems dened on nite domains of the form Z k l := Z bl \ (0; 1) (l; bl k;l g be the solution of ~ bl l [ Z k div y ~ bl l [ Z k bl k;l [fr y ~ bl ~ bl bl k;l g is y with the following additional boundary conditions motivated by Corollary 4 and Proposition 7: ~ bl Further, to dene the pressure eld ! bl k;l uniquely, we require Z Analogous to the estimates on ( ~ of the previous section, one can prove the following properties for the solution ( ~ bl k;l ) of (4.1){(4.8): 0 , the solution f ~ bl k;l g of problem (4.1){(4.8) can be represented as (D 1 e 2nk where 1;k;l := Z 1( bl and k;l (y) =X with !;k;l := Z 1! bl Z 1! bl This representation immediately yields, that for every 0 < a < k, 0 < < 2, and are constants C(a; ); C(a; ; ) such that for all y 2 Z k a we have ~ bl D ~ bl Asymptotic analysis of the laminar viscous ow over a porous bed 19 and ! bl !;k;l For ~ bl k;l l Ce lower jmj (4.18) with lower from (3.47). Furthermore, for y 2 Z 0 l k;l (y) Ce lower ~ bl k;l (y) (C bl Ce lower and ! bl Ce lower jy From these estimates we obtain: Proposition 11 Let f ~ bl k;l g be the solution of problem (4.1){(4.8). Then, for every < 2, a constant C exists such that r ~ bl k;l r ~ bl l lower Proof. Let := ~ bl ~ bl k;l . Then (; !) is y 1 -periodic, vanishes on l k=1 (@Z (0; k)) and solves in Z k l . Set lower Jager, Andro Mikelic and Nicolas Neu where lower is derived from in the same way as ~ lower was derived from ~ bl in Lemma 6. Testing (4.23) with ^ , we obtain Z Z l+1 Z 1rr lower = Now note that ! may be replaced by ! c for an arbitrary c 2 IR due to R =const Then we can apply the exponential stabilization results for f ~ k;l g from (3.22), (3.23), (3.30), (4.15), (4.15), and (4.17) to obtain (4.22). The approximation error between C bl 1 and C bl 1;k;l can then be estimated as follows Corollary 12 For every < 2 there is a constant C such that 1;k;l lower Proof. Note that 1;k;l Z bl bl which can be estimated as desired by using Poincare's inequality on Z 0 [ Z 1 together with (4.22). In order to obtain estimates for the pressure dierence ! bl we need the following result: Lemma 13 For each F 2 L 2 (Z k l R l there is a function ~ l ) vanishing on the boundaries y l satisfying div ~ together with the stability estimate l l More generally if F 2 H r (Z k l ) for some integer r 0, then ~ ' can be chosen such that l Asymptotic analysis of the laminar viscous ow over a porous bed 21 Proof. The proof is similar to the proof of Lemma 3.4 in [9]. We search for ~ in the form with div correcting the non-zero boundary values of r. More precisely, let be the solution to @ @ n=l (@Z (0; n)) [ fy Since R l the testing of (4.32) with yields Z l Z l Z l l Z l l l and therefore the estimate l Note that we have used the Poincare inequality l Z l l l with a constant C being independent of k and l. This estimate can easily be proved by extending from Z k l to the rectangle (0; 1) (l; k) and by using the Poincare estimate there. Furthermore, by localizing we also get estimates for higher order derivatives of in the form l l The function # from (4.31) has to correct the non-zero boundary values of and should therefore fulll 22 Willi Jager, Andro Mikelic and Nicolas Neu @# @ @ on n=l (@Z (0; n)) [ fy lg. Since @Z is smooth, such a function can be constructed by a local H r -lift such that l l with C being independent of k and l. The combination of (4.37) and (4.41) then yields the desired regularity estimate for ~ ', and the lemma is proved. With the help of this lemma we obtain: Proposition 14 Let ! bl and ! bl k;l be the pressure elds determined by (3.5){(3.10), resp. (4.1){(4.8). Then, for every < 2, there is a constant C such that k;l l lower For the dierence C bl !;k;l we have the better estimate !;k;l C lower Proof. Set l Z l and let F := k;l !. Since R l yields a function ~ which we can use as test function in the dierence of momentum equations bl ~ bl Inserting ! in the pressure term, testing with ~ and doing a partial integration we obtain Z l r( ~ bl ~ bl k;l )r~' Z l Asymptotic analysis of the laminar viscous ow over a porous bed 23 If we now use the stability estimate (4.29) together with (4.22), this yields l lower Finally, to estimate !, let Z Then, obviously, we have By Theorem 3.7 of [9], we also have r( ~ bl ~ bl which can be used together with (4.21) to estimate lower r( ~ bl ~ bl l Using the triangle inequality in (4.47) together with (4.51) and (4.22), we obtain (4.42). In order to get (4.43), we note that !;k;l such that the estimate follows directly from (4.51) and (4.22). We complete this section with a regularity result, which we will need in the following. Proposition l f; g be the y 1 -periodic solution of in Z k l with boundary conditions l and pressure normalization Z Then we have the estimate l l l l l C l with a constant C independent of k and l. Proof. We rst note that we can bound k 2 k L 2 (Z k l ) and l in terms of krk L 2 (Z k l ) . In the lower region, this follows from Poincare's inequality applied on every cell. In the upper region, we rst have l @ l because setting we also obtain l @ Additionally, the Hardy inequality @ together with an estimate of S 1 (0) by trace inequality and Poincare inequality on l Asymptotic analysis of the laminar viscous ow over a porous bed 25 Next, for deriving a bound for krk L 2 (Z k l ) , we test the momentum equation (4.53) with to obtain l Z l On Z 0 l , we may again use Poincare's inequality to obtain Z l f l l and because of also the estimate Z can be obtained by applying Poincare's inequality. In order to get an estimate for R and write Z Z Z Here, the rst term can be estimated as Z while for the second term we have Z using again (4.62). Combining these estimates, we obtain the desired bound for 26 Willi Jager, Andro Mikelic and Nicolas Neu Next, we localize the problem by multiplying with smooth cut-o functions are identically 1 on Z i and vanish for y 2 Denote with S the region Z i+1 l . By shifting the pressure by a suitable constant we may assume that R (otherwise, the boundary conditions on @Z would be violated), we may also shift by constant multiples of e 1 such that we may also assume that R The resulting function f ~ is a solution of ~ div ~ in S with Dirichlet and periodic boundary conditions in the case i < k 1 and a combination of Dirichlet, periodic and slip boundary conditions when We examine the right hand side of (4.70). Since Z Z Z r ~ we have kr~k H 1 (S) C r ~ . Since R 0, we can apply Proposition 1.2, Ch. I of [16] to obtain C r ~ Then, however, the right hand side of (4.70) is in L 2 while the right hand side of (4.71) is in H 1 , and we can apply Proposition 2.2 of Ch. I of [16] to get that and kr~k L 2 (S) can be bounded in terms of krk L 2 (S) (with a constant depending on the geometry of the porous inclusion). We note that the slip boundary fy easily eliminated by making an even extension for 1 ; and f 1 and an odd extension for 2 and f 2 (the zeroth order re ection). Summing up these local estimates we get that kD 2 k L 2 (Z k l ) and krk L 2 (Z k l ) are bounded by a multiple of krk L 2 (Z k l ) with a constant that does not depend on k and l. Thus, Proposition 15 is proved. Asymptotic analysis of the laminar viscous ow over a porous bed 27 Corollary bl k;l g be the solution to (4.1){(4.8), and let ~ := ~ bl k;l ~ y 22 Then l r! bl k;l l where C can be chosen independent from k and l. 28 Willi Jager, Andro Mikelic and Nicolas Neu 5 Discretization Now, we turn our attention to the discretization of problem (4.1){(4.8). Essen- tially, we use a stabilized nite element discretization in the sense of [8],[3]. Unfor- tunately, in both of these papers only polygonal domains were considered, so that the direct application of these results to our domain Z k l is unsatisfying because of the curved boundaries l We resolve this diculty by using generalized domain partitions given by nonlinear mappings of the elements while essentially keeping the approximation results known for usual domain partitions with linear mappings, see [19], [20], [1], [14]. While this is very convenient theoret- ically, the practical implementation will usually be too complicated. Therefore, it is important that the use of a simpler polygonal approximation of the domain can be interpreted as a perturbation of this approach, see the discussion at the end of this section. Let be a Lipschitz domain, and let T h be a partition of in subsets e, called elements, where each element e is the image of a reference element ^ e under a mapping e is either the reference triangle ^ or the reference quadrangle ^ We require the following properties of the partition 1. 2. For two elements 3. A side of an element e (which is dened as the image of a side of ^ e) is either a subset of @ or the side of exactly one other element e 0 6= e. In the second case, we require that the mapping 1 restricted to the corresponding side of ^ e is linear. 4. The are bi-Lipschitzian mappings with Asymptotic analysis of the laminar viscous ow over a porous bed 29 e e (y)j for some constant C 1 > 0. 5. e 2 W 2;1 , with kD 2 e k 1 C 2 diam(e). 6. For simplicity reasons, we consider only quasiuniform partitions, i.e. a constant exists such that for h := max e2T h diam(e) we have diam(e) Arbitrary ne triangulations of this kind exist, which can be shown by modifying triangulations of polygonal approximations of the domain see [19]. The quality of the domain partition T h is determined by the constants C renement of such a partition results in a new partition which still has the above properties (especially property 5). Next, let where P 1 is the space of linear polynomials, and Q 1 is the span of P 1 and the polynomial x 1 x 2 . With these nite elements, we can dene the space The following approximation result is then the substitute for approximation results on triangulations with linear element mappings. Theorem 17 Let h be as above. Then, for , an u h 2 S h exists such that Second, for , an u h 2 S h exists such that Additionally, if u has zero boundary values on some component of @ , then also u h can be chosen such that it has zero boundary values on that boundary component. In all cases, the constants only depend on the smoothness of the domain and the quality of the domain partition T h . Proof. See [19], [20] for the case of (deformed) triangular elements, and [14] where also the case of (deformed) quadrilaterals is handled by using a generalized interpolation operator of Clement type. Remark In contrast to standard interpolation results for triangulations with linear element mappings, the term kruk L appears on the right hand side of (5.6). This is due to the fact that linear functions are no more interpolated exactly (in contrast to constant functions). Note that this estimate is only possible because the e approximate linear mappings in the limit diam(e) ! 0 (see property 5 required for a partition T h ). let l , and let T h be a domain partition of Z k l (tting across the lateral boundaries, where periodic boundary conditions are prescribed for problem (3.5){(3.10)). Then dene S h analogously to (5.4) as l the ansatz space for the velocity eld as ~ l and the ansatz space for the pressure eld as Z We now search for ( ~ bl being the solution to Z l r ~ bl Z l r! bl k;l;h ~ Z ~ Z l div ~ bl Z r! bl Asymptotic analysis of the laminar viscous ow over a porous bed 31 for all (~' k;l;h ; k;l;h . The second term in (5.11) must be included for the above pair of ansatz spaces to stabilize the discretization, see [8],[3]. The constant can be any positive number. The following error estimates then hold: Proposition 19 Let T h ; ~ be dened as above. Assume that the interior boundary aligned with the sides of the elements of T h , let ( ~ bl be the solution of problem (4.1){(4.8), and let ( ~ bl H h L h be the solution of (5.10){(5.11). Furthermore, let bl r ~ bl k;l l bl k;l bl k;l l r! bl k;l l Then we have an error estimate in the viscous energy norm of the form r( ~ bl k;l;h ~ bl l ChR( ~ bl together with a stability estimate for the pressure gradient of the form@ X r(! bl ChR( ~ bl We further have the following L 2 -error estimate for the pressure k;l;h (! bl k;l Z k l Z k;l dy) l bl and the velocity ~ bl k;l;h ~ bl k;l l bl with a constant C which is independent of k and l. Proof. The proof can be done as in [8], [3], or [7] with the following modi- cations: rst, on our more general domain partitions, the approximation results from theorem 17 replace the standard ones. Second, [8], [3], [7] handle only the Dirichlet case. However, one can easily check, that their proofs can be transfered with almost no changes because the slip boundary condition still allows for partial integration with vanishing boundary terms. For (5.14), the proof uses the assertion of Lemma 13 such that the factor k needs to be included in the estimate. For (5.15), one needs the H 2 -regularity estimate from proposition 15 which introduces the factor k + 1. We now describe the denition of the discrete approximations to C bl 1;k;l and !;k;l . First, we set Z Second, in order to obtain good error estimates for the numerical approximation of !;k;l , we dene C bl !;k;l;h as a smoothly weighted average of the pressure eld in the following be a Lipschitz-continuous function satisfying ~ elsewhere It is obvious that Z ~ Z Z l such that we may approximate C bl !;k;l by Z l ~ Then we can prove: Proposition 20 For C bl !;k;l;h given by (5.16), (5.19) we have the estimates: 1;k;l;h C bl 1;k;l !;k;l;h C bl !;k;l lower Asymptotic analysis of the laminar viscous ow over a porous bed 33 Proof. In order to show (5.20), we use the denitions (4.11) and (5.16) in the momentum equations (4.1) and (5.10) to obtain 1;k;l;h C bl Z k;l;h ~ bl ds Z l r ~ bl k;l r( ~ bl k;l ~ bl Z l r! bl bl k;l ~ bl Z l r( ~ bl k;l ~ bl Z l r ~ bl k;l;h r( ~ bl k;l ~ bl Z l r! bl bl k;l ~ bl Z l r( ~ bl k;l ~ bl dx Z l r(! bl bl Z l r! bl bl k;l ~ bl Here, the rst term is of order O(h 2 ) by (5.12), the second term can be estimated as Z l r(! bl bl Z l r(! bl k;l;h ~ bl k;l by (5.15), and, if one applies (5.11), the third term can be estimated as Z l r! bl bl k;l ~ bl Z l r(! bl bl k;l ~ bl k;l;h )+ Z r! bl k;l;h Next, if we use (5.15) and (5.13), we can see that the resulting terms are of order respectively. Thus, we have shown (5.20). In order to show the estimate (5.21), we rst note that !;k;l C bl Z l (! bl k;l;h (y)) ~ (y) dy +O(e lower jlj ) (5.24) by (4.21), (4.14), and (5.19). Next, let ~ l ) be the vector eld given by Lemma 13 which satises div ~ l , and l 34 Willi Jager, Andro Mikelic and Nicolas Neu Furthermore, theorem 17 yields the existence of an interpolating vector eld ~ ~ H h which fullls l l l l By using this, we can estimate the rst term from the right hand side of (5.24) as follows: Z l (! bl Z l (! bl dy Z l r(! bl dy Z l r(! bl Z l r(! bl Z l r(! bl Z l r( ~ bl k;l ~ bl Here, the rst term is of order C(k by (5.13), (5.26), and (5.25). The second term can be estimated by replacing rst ~ ' h by ~ ' which can be done up to an error of order C(k due to (5.12), (5.27), and (5.25). For the rest, a partial integration yields Z l r( ~ bl k;l ~ bl Z l k;l ~ bl k;l;h )~' dy C(k if one uses (5.15) and (5.25). Thus, also (5.21) is proved. As mentioned at the beginning of this section, the practical implementation of domain partitions with nonlinear element mappings is rather complicated. It is usually easier to approximate a domain with curved boundaries by polygonal domains h which then can be partitioned in triangles and/or quadrangles (see for example [17]). However, it can be shown that, on a discrete level, this simpler approach is equivalent to our theoretically more convenient formulation up to a quadrature error of optimal order, see [18]. Asymptotic analysis of the laminar viscous ow over a porous bed 35 Figure 3: Symmetric cell and its coarsest grid. 6 Numerical Results We are now ready to demonstrate our method for computing the constants C bland C bl on specic examples. First, we consider the symmetric geometry shown on the left part of Figure 3. The boundary @Z is a circle with radius 0:25 and center at (0:5; 0:5). On the right-hand side of Figure 3, the initial polygonal approximation and the initial grid T h0 with quadrangle elements (see section 5) is depicted. T h0 is then uniformly rened, yielding further grids T . By using the discretization described in the previous section, we obtain for every a system of linear equations which must be solved to obtain the discrete solution f ~ bl k;l;h g. Since the arising linear systems are very large, we have applied the multigrid method which is known to be of optimal complexity for a large range of problems, see [6]. However, due to the pressure stabilization and the polygonal approximation of the smooth boundary l k=1 (@Z (0; k)), we are not in a Galerkin setting. In this case, it is known that the simple multigrid V-cycle (one coarse-grid correction between pre- and post-smoothing) does not have to converge independently of the number of levels. We therefore used the W-cycle (two coarse-grid corrections between smoothing). Both pre- and post-smoothing are done by two steps of a block incomplete decomposition where each block contains the unknowns of one grid node (the corners of the elements). And indeed, our numerical observa- 36 Willi Jager, Andro Mikelic and Nicolas Neu (j 1) Table 1: Results for the symmetric cell tions conrm that this method is robust with respect to the number of levels and variations of the parameters k; l, see also Table 5 below. In Table 1, the results for a computation with are shown. Starting from the coarsest level which contains 20 elements, we rene 5 times, which yields a grid with 20480 elements (61440 unknowns). On each level the discretized equation and compute the approximations C bl 1;k;l;h and C bl !;k;l;h given by (5.16) and (5.19) (with the choice (5.17)). The value given for computed by polynomial extrapolation. We see that both C bl 1;k;l;h and !;k;l;h converge to limit values with rate O(h 2 ) which we expected from Proposition 19. As shown in Remark 9, C bl must be zero. Since our grid is symmetric, the solution of the discrete problem also has the same symmetry property, such that the approximations C bl !;k;l;h are zero up to machine accuracy. Let us now assume that the porous part is generated by the unsymmetric cell from Figure 4, where the boundary curve is given by the ellipse 0:5 0:25 For this domain we obtain the results shown in Table 2. Both C bl 1;k;l;h and C bl !;k;l;h converge again with order O(h 2 ), which is in accordance with Proposition 19. The Asymptotic analysis of the laminar viscous ow over a porous bed 37 Figure 4: Unsymmetric cell and coarsest grid. (j 1) (j 1) Table 2: Results for the unsymmetric cell l Table 3: C bl 1;k;l (extrapolated) for varying k; l. l Table 4: C bl !;k;l (extrapolated) for varying k; l. error arising from the cutting of the domain is not noticeable any more already 2. This is shown in Table 3, where the results for varying k and l are given. Note that even the values for are accurate up to the extrapolation error. Figure 5 shows the three solution components bl From here and from the values of C bl in Table 2 it is obvious that a pressure jump occurs inside the boundary layer. Finally, Table 5 shows the convergence rates of our multigrid iteration. As we expected from the discussion above, it is perfectly robust with respect to the number of levels. So far, we did not observe a signicant dependency on k or l, even if one might expect some deterioration, since the error estimates from Proposition 19 are needed in the usual W-cycle convergence theory. The reason might be that sharper error estimates are possible in suitably weighted norms. Asymptotic analysis of the laminar viscous ow over a porous bed 39 Table 5: Multigrid convergence rates. Figure 5: Detail from a picture showing bl Asymptotic analysis of the laminar viscous ow over a porous bed 41 7 The eective equations It turned out that ~ bl stabilizes exponentially fast to a constant vector (C bl (and to 0 for y 2 !1), which translates into ~ bl;" tending to "(C bl for This forces us to consider the corresponding counter- ow in the channel, which is described by the following 2D Oseen-Couette system in d in div ~ in ~ If we assume that Re := is not too big, the problem (7.1){(7.4) has a unique solution in the form of 2D Couette ow ~ Following the ideas from [9], we write down the correction of order " for the velocity. Essentially, it corresponds to the elimination of the tangential component of the normal stress at , caused by the approximation by Poiseuille's ow and its contribution to the energy estimate. The correction reads bl ( x The correction of the velocity eld by the oscillatory boundary layer velocity ~ bl;" involves the introduction of the boundary layer pressure eld ! bl;" . Hence, as usual for ow problems, it is necessary to correct the pressure eld simultaneously. The corresponding pressure correction reads Here, p 1;" is an appropriate regularization of the eective pressure p in the porous 42 Willi Jager, Andro Mikelic and Nicolas Neu bed, dened by in where the permeability tensor K is dened as Z Y Here, the w j are solutions of the auxiliary problems div y ~ ~ Z Y The pressure eld p is a C 1 -function outside the corners. However, due to the discontinuities of the traces at (0; 0) and at (b; 0), its gradient is not square integrable in 2 . We must regularize the values at the upper corners and p 1;" is such a regularization, satisfying "). Let us now introduce the dierence U " 0 between the velocity eld ~u " and its expansion up to order O("), i.e. we set ~ bl;" @v 0@x 2 Then, after constructing the corresponding outer boundary layer, it was proved in [10] that jr ~ Z bj ~ Z bZ Hj ~ Asymptotic analysis of the laminar viscous ow over a porous bed 43 It should be noted that the presence of the logarithmic term is a consequence of the corner singularities in the eective pressure. It was proved in [9] that in the absence of the boundary singularities, the above estimates hold without the log " term. Therefore, in the interior of the domain the expansion (7.5) is of order O(" 3=2 ). Globally, it is of order O(" 3=2 j log "j). The estimates (7.16)-(7.18) are sucient for calculating the next order correction at the interface . The estimate (7.18) gives us a possibility of approximating the velocity values at by an oscillatory velocity eld. Computationally, it is not very useful. In view of the problem setting in [9], the Beavers{Joseph law corresponds to taking into the account the next order corrections for the velocity. In fact, we formally get on the interface : "j log "C bl x x Integrals of the absolute value of the right hand side are not small, since our estimates do not give any pointwise control of ru on . Nevertheless, the right hand side of (7.21) is small in the appropriate Sobolev norm, of a negative order. The precise estimates are in [10]. Hence, we get the eective law which is exactly Saman's modication of the Beavers and Joseph law from (1.2) with 1 K 44 Willi Jager, Andro Mikelic and Nicolas Neu Beavers-Joseph profile bl bl4 Poiseuille profile1 bl bl Figure Beavers-Joseph versus Poiseuille prole. Let us introduce the eective ow equations in 1 through the following boundary value problem: nd a velocity eld ~u e and a pressure eld p e such that in div in Under the same assumption of laminarity as for problem (7.1){(7.4), the problem (7.23)-(7.28) has a unique solution "C bl Asymptotic analysis of the laminar viscous ow over a porous bed 45 (see Figure and the eective mass ow rate through the channel is Z Hu e where C bl By using the theory of the very weak solutions for the Stokes system, the following approximation properties of f~u e ; are proved in [10]: The estimates (7.32)-(7.34) justify Saman's modication of the law by Beavers and Joseph. Furthermore, we are able to calculate their proportionality it is equal toC blK 1=2 . We note that (7.28) is not the only possible interface law. If one replaces Beavers and Joseph's law by u e (0) the estimates remain valid. However, such a condition involves the knowledge of the zeroth order approximation v 0 1 , and is not really an interface condition. At this stage, we can consider the approximation to the channel ow as satis- but we still must determine the ltration velocity and the pressure in the porous medium. We have already mentioned the in uence of corner singularities on the solution of the problem (7.8)-(7.10). In order to avoid the discussion of such eects, we limit ourselves to the behavior in the interior of the porous medium. The inertia eects are negligible and we can use the theory from [9], Theorem 3. In the interior, all boundary layer eects are exponentially small and we have ~ where ~ are dened by (7.12). Hence, the ltration velocity is given through the Darcy law in Far away from the corners, the pressure eld p approximates p " at order O( "). It can only be determined after we have found the eective pressure eld in the channel and the stabilization constant C bl giving the pressure dierence at innities for the auxiliary problem. As shown in Section 6, this stabilization constant is generally dierent from zero and we must use the interface law This shows that, contrary to the intuition, the eective pressure in the system channel ow/porous medium is not always continuous. Thus, the continuity assumption for the eective pressure from [11] is not correct in general. Finally, let us note that we have some freedom in choosing the position of the interface. It could be set at any distance of order O(") from the solid obstacles. Intuitively, the law of Beavers and Joseph should be independent of such a choice. This invariance result can be established rigorously, and we refer for details to [13] where it was proved that a perturbation of order O(") in the position of the interface implies a perturbation in the solution of order O(" 2 ). Consequently, the eective law doesn't change and the physical intuition is conrmed. Asymptotic analysis of the laminar viscous ow over a porous bed 47 --R Finite element methods for problems with rough coe Boundary conditions at a naturally permeable wall Lectures on the Mathematical Theory of Multiphase Flow Equations et ph of Computational Mathematics Eine robuste und e On boundary conditions for uid ow in porous media Some Methods in the Mathematical Analysis of Systems and Their Control PhD thesis On the boundary condition at the interface of a porous medium Theory and Numerical Analysis Curved elements in the --TR

finite elements;unbounded domain;homogenization;multigrid;boundary layer;periodic structures;interface law;beavers-joseph;navier-stokes

587417

Fast Finite Volume Simulation of 3D Electromagnetic Problems with Highly Discontinuous Coefficients.

We consider solving three-dimensional electromagnetic problems in parameter regimes where the quasi-static approximation applies and the permeability, permittivity, and conductivity may vary significantly. The difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing Maxwell's equations in the frequency domain.The present article extends methods we proposed earlier for constant permeability [E. Haber, U. Ascher, D. Aruliah, and D. Oldenburg, J. Comput. Phys., 163 (2000), pp. 150--171; D. Aruliah, U. Ascher, E. Haber, and D. Oldenburg, Math. Models Methods Appl. Sci., to appear.] to handle also problems in which the permeability is variable and may contain significant jump discontinuities. In order to address the problem of slow convergence we reformulate Maxwell's equations in terms of potentials, applying a Helmholtz decomposition to either the electric field or the magnetic field. The null space of the curl operators can then be annihilated by adding a stabilizing term, using a gauge condition, and thus obtaining a strongly elliptic differential operator. A staggered grid finite volume discretization is subsequently applied to the reformulated PDE system. This scheme works well for sources of various types, even in the presence of strong material discontinuities in both conductivity and permeability. The resulting discrete system is amenable to fast convergence of ILU-preconditioned Krylov methods.We test our method using several numerical examples and demonstrate its robust efficiency. We also compare it to the classical Yee method using similar iterative techniques for the resulting algebraic system, and we show that our method is significantly faster, especially for electric sources.

Introduction The need for calculating fast, accurate solutions of three-dimensional electromagnetic equations arises in many important application areas including, among others, geophysical surveys and medical imaging [29, 32, 2]. Conse- quently, a lot of effort has recently been invested in finding appropriate numerical algorithms. However, while it is widely agreed that electromagnetic phenomena are generally governed by Maxwell's equations, the choice of numerical techniques to solve these equations depends on parameter ranges and various other restrictive assumptions, and as such is to a significant degree application-dependent [20, 32, 2]. The present article is motivated by remote sensing inverse problems, e.g. in geophysics, where one seeks to recover material properties - especially conductivity - in an isotropic but heterogeneous body, based on measurements of electric and magnetic fields on or near the earth's surface. The forward model, on which we concentrate here, consists of Maxwell's equations in the frequency domain over a frequency range which excludes high frequencies. Assuming a time-dependence e \Gamma-!t , these equations are written as r r \Theta H \Gamma b r r where - is the magnetic permeability, oe is the conductivity, ffl is the electrical permittivity, J s is a known source current density, and ae is the (unknown) volume density of free charges. In our work we assume that the physical properties - ? 0, can vary with position, and -ffl! 2 L 2 - 1, where L is a typical length scale. The electric field E and the magnetic field H are the unknowns, with the charge density defined by (1c). Note that as long as redundant and can be viewed as an invariant of the system, obtained by taking the r\Delta of (1a). The system (1) is defined over a three-dimensional spatial domain\Omega\Gamma In principle, the domain\Omega is unbounded (i.e. practice, a bounded subdomain of IR 3 is used for numerical approximations. In this paper we have used the boundary conditions H \Theta n although other boundary conditions are possible. A number of difficulties arise when attempting to find numerical solutions for this three-dimensional PDE system. These difficulties include handling regions of (almost) vanishing conductivity, handling different resolutions in different parts of the spatial domain, handling the multiple scale lengths over which the physical properties can vary, and handling regions of highly varying conductivity, magnetic permeability, or electrical permittivity where jumps in solution properties across interfaces may occur. On the other hand, the nature of the data (e.g. measurements of the electric and/or magnetic fields at the surface of the earth) is such that one cannot hope to recover to a very fine detail the structure of the conductivity oe or the permeability -. We therefore envision, in accordance with the inverse problem of interest, a possibly nonuniform tensor product grid covering the domain\Omega\Gamma where b oe and - are assumed to be smooth or even constant inside each grid cell, but they may have significant jump discontinuities that can occur anywhere in\Omega across cell interfaces. The source J s is also assumed not to have jumps across interfaces. The relative geometric simplicity resulting from this modeling assumption is key in obtaining highly efficient solvers for the forward problem. Denoting quantities on different sides of an interface by subscripts 1 and 2, it can be easily shown [33] that across an interface n \Theta These conditions imply that neither E nor H are continuous in the normal direction when b oe, resp. -, have a jump discontinuity across a cell face, and likewise, b oeE and -H are not necessarily continuous in tangential di- rections. Care must therefore be exercised when numerical methods are employed which utilize these variables if they are to be defined where they are double-valued. By far the most popular discretization for Maxwell's equations is Yee's method [37] (see discussions and extensions in [32, 26, 19]). This method employs a staggered grid, necessitating only short, centered first differences to discretize (1a) and (1b). In the more usual application of this method, the electric field components are envisioned on the cell's edges and the magnetic field components are on the cell's faces - see Fig. 1. It is further possible to H z x y z Figure 1: A staggered discretization of E and H in three dimensions: E- components on edges, H-components on faces. eliminate the components of the magnetic field from the discrete equations, obtaining a staggered discretization for the second order PDE in E, r \Theta (- \Gamma1 r \Theta E) \Gamma Related methods include the finite integration technique and certain mixed finite element methods [35, 5, 25]. Although these methods are often presented in the context of time-domain Maxwell's equations the issues arising when applying an implicit time-discretization (a suitable technique under our model assumptions) are often similar to the ones we are faced with here. The popularity of Yee's method is due in part to its conservation properties and other ways in which the discrete system mimics the continuous system [19, 15, 5, 4]. However, iterative methods to solve the discrete system may converge slowly in low frequencies, due to the presence of the rich, nontrivial null space of the curl operator, and additional difficulties arise when highly discontinuous coefficients are present [29, 24, 16]. There are two major reasons for these difficulties. First, the conductivity can essentially vanish (for example, in the air, which forms part of \Omega\Gamma/ from an analytic perspective, the specific subset of Maxwell's equations used typically forms an almost-singular system in regions of almost-vanishing b oe. Even in regions where the conductivity is not close to vanishing, the resulting differential operator is strongly coupled and not strongly elliptic [6, 1]. Second, in cases of large jump discontinuities, care must be taken to handle H and b carefully, since these are located as in Fig. 1 where they are potentially discontinuous In [1], we addressed the often slow convergence of iterative methods when used for the equations resulting from the discretization of (5) by applying a Helmholtz decomposition first, obtaining a potential formulation with a Coulomb gauge condition. This change of variables (used also in [3, 12, 22, 28], among many others) splits the electric field into components in the active and the null spaces of the curl operator. A further reformulation, reminiscent of the pressure-Poisson equation for the incompressible Navier-Stokes equations [14, 31], yields a system of strongly elliptic, weakly coupled PDEs, for which more standard preconditioned Krylov space methods are directly applicable. In [15], we further addressed possible significant jumps in the conductivity while - is assumed constant, by employing a finite volume discretization on a staggered grid, akin to Yee's method with the locations of E- and H- components exchanged, as in Fig. 2. The normal components of E are now double-valued, but this is taken care of in an elegant way by the Helmholtz decomposition of E and by introducing the (generalized) current x y z Figure 2: A staggered discretization of E and H in three dimensions: E- components on faces, H-components on edges. into the equations. The curl operators in (5) are replaced by the vector Laplacian according to the vector identity r \Theta r\Theta for sufficiently smooth vector functions (not E). In this paper we generalize our approach from [1, 15] to the case where the magnetic permeability - may be highly discontinuous as well. This is a realistic case of interest in geophysical applications, although usually the jump in conductivity dominates the jump in permeability. Now the roles of E and H are essentially dual, and it is possible to apply a Helmholtz decomposition to either E or H, keeping the other unknown vector function intact. We choose to decompose the electric field E, referring to Fig. 2 for the locations of the H-unknowns in the ensuing discretization. The major departure from our previous work is in the fact that the identity (7) does not directly extend for the operator r \Theta (- \Gamma1 r\Theta ) appearing in (5). We can, however, stabilize this operator by subtracting r(- this forms the basis for our proposed method. In cases of constant magnetic permeability or electric conductivity the formulation can be reduced to our previous formulation in [15] or a variant thereof. Our approach to dealing with possible discontinuities can be viewed as using fluxes (which are continuous on cell faces) and vorticities (which are continuous on cell edges). The introduction of such unknowns is strongly connected to mixed finite elements which are used for highly discontinuous problems [7, 8, 16]. The paper is laid out as follows. In Section 2, we reformulate Maxwell's equations in a way which enables us to extend our methods. The resulting system is amenable to discretization using a finite-volume technique described in Section 3. The extension and generalization of our approach from [1] through [15] to the present article is not without a price. This price is an added complication in the sparsity structure of the resulting discrete system and a corresponding added cost in the iterative solution of such systems. We briefly describe the application of Krylov space methods to solve the system of algebraic equations in Section 4. We use BICGSTAB (see, e.g., [30]) together with one of two preconditioners: an incomplete block LU-decomposition (which is a powerful preconditioner in the case of diagonally dominant linear systems) and SSOR. The system's diagonal dominance is a direct consequence of our analytic formulation. We present the results of numerical experiments in Section 5 and compare results obtained using our method with those obtained using a more traditional Yee discretization. If the source is not divergence-free, as is the case for electric (but not magnetic) sources, then our method is better by more than two orders of magnitude. The method works well also for a case where the problem coefficients vary rapidly. We conclude with a short summary and further remarks. 2 Reformulation of Maxwell's Equations Maxwell's equations (1a) and (1b) can be viewed as flux balance equations, i.e. each term describes the flux which arises from a different physical con- sideration, and the equations are driven by the conservation of fluxes. (In fact, this was how they were originally developed [23].) Therefore, in both (1a) and (1b) we have a flux term which should be balanced. In (1b) the generalized current density b J defined in (6) is balanced with the source and the flux which arise from magnetic fields, and in (1a) the magnetic flux is balanced with the flux which arises from electric fields. In our context these fluxes are well-defined on cell faces but they may be multi-valued at cell edges. 1 Furthermore, the leading differential operator in (5), say, has a nontrivial null space. Rather than devising iterative methods which directly take this into account (as, e.g., in [2, 16]), we transform the equations before discretization. We decompose E into its components in the active space and in the null space of the curl operator: r We could decompose H instead in a similar way, but we would not decompose both. Here we have chosen to concentrate on the decomposition of E. Substituting equations we obtain r \Theta A \Gamma r \Theta H \Gamma b r Furthermore, (5) becomes r \Theta (- \Gamma1 r \Theta r Note that across an interface between distinct conducting media we have, in addition to (4), n \Theta ae s in (12c) is an electric surface charge density. These conditions and the differential equations (1) imply that while b H(curl;\Omega\Gamma4 see, e.g., [13]. not. Moreover, rOE inherits the discontinuity of E \Delta n, while A is continuous, and both r \Delta A and r \Theta A are bounded (cf. [13]). In [15] we had the relation r \Theta r \Theta A = \Gammar 2 A holding. However, when - varies, the identity (7) does not extend directly, and we must deal with the null space of r \Theta A in a different way. Let us define the Sobolev spaces equipped with the usual norm (see, e.g., [13]) kvk (the (\Omega\Gamma4 and [L are used on the right hand side of (13b)), and Green's formula yields, for any u 2 (r \Theta (- where the usual notation for inner product in L 2(\Omega\Gamma and [L 2(\Omega\Gamma2 3 is used. Thus, for any u 2 W (r \Theta (- \Gamma1 r \Theta u); u) We may therefore stabilize (11a) by subtracting a vanishing term: r \Theta (- \Gamma1 r \Theta obtaining a strongly elliptic operator for A, provided A 2 W 0(\Omega\Gamma4 The latter condition is guaranteed by the choice (17b) below. A similar stabilization (or penalty) approach was studied with mixed success in [10, 20]. However, our experience and thus our recommendation are more positive because of the discretization we utilize. We elaborate further upon this point in the next section. Using (10c), we can write (10b) as r \Theta H \Gamma This may be advantageous in the case of discontinuous -, similarly to the mixed formulation used for the simple div-grad system r in [7, 27, 8, 11] and elsewhere. Our final step in the reformulation is to replace, as in [15], the gauge condition (10c) on A by an indirect one, obtained upon taking r\Delta of (15a) and simplifying using (15b) and (10c). This achieves only a weak coupling in the resulting PDE system. We note that replacing of the gauge condition (10c) is similar to the pressure Poison equation in CFD [31, 14]. The complete system, to be discretized in the next section, can now be written as r \Theta H \Gamma In order to complete the specification of this system we must add appropriate Boundary Conditions (BCs). First, we note that the original BC (3) can be written as (r \Theta A) \Theta n An additional BC on the normal components of A is required for the Helmholtz decomposition (9) to be unique. Here we choose (corresponding to (13c)) A This, together with (9), determines A for a given E. Moreover, since (16d) was obtained by taking the r\Delta of (16a) additional are required on either @OE=@n or OE. For this we note that the original together with the PDE (1b) imply also The latter relation (17c), together with (17b), implies @OE @n 0 at the bound- ary. 23 The above conditions determine OE up to a constant. We pin this constant arbitrarily, e.g. by requiring Z Finally, we note that (17c) together with (16a) and (3), imply in turn that @/ @n 0 at the boundary. Since (16a) and (16d) imply that we obtain / j 0, and thus retrieve (10c), by pinning / down to 0 at one additional point. The system (16) subject to the boundary conditions (17) and / pinned at one point is now well-posed. 3 Deriving a discretization As in [15], we employ a finite volume technique on a staggered grid, where b and A are located at the cell's faces, H is at the cell's edges, and OE and / are located at the cell's center. Correspondingly, the discretization of (16d) and (16c) are centered at cell centers, those of (16e) and (16a) are centered at cell faces, and that of (16b) is centered at cell edges. The variables distribution over the grid cell is summarized in Table 1. To approximate r \Delta u for integrate at first over the cell using the Gauss divergence theoremje i;j;k j Z e i;j;k r \Delta u Z @e i;j;k In cases where the original BC is different from (3) we still use the boundary condition as an asymptotic value for an infinite domain. Alternatively, note the possibility of applying the Helmholtz decomposition to H , although generally there are good practical reasons to prefer the decomposition (9) of E. 3 In our geophysical applications J s \Delta n vanishes at the boundary. A x A y A z J z J z H z Table 1: Summary of the discrete grid functions. Each scalar field is approximated by the grid functions at points slightly staggered in each cell e i;j;k of the grid. and then use midpoint quadrature on each face to evaluate each component of the surface integrals appearing on the right-hand side above. Thus, define and express the discretization of (16d) and (16c) on each grid cell as Note that we are not assuming a uniform grid: each cell may have different widths in each direction. The boundary conditions are used at the end points of (19). Next, consider the discretization at cell faces. Following [15], we define the harmonic average of boe between neighboring cells in the x-direction by where h x )=2. If b oe is assumed constant over each cell, this integral evaluates to 2boe i;j;k 2boe i+1;j;k Then, the resulting approximation for the x-component of (16e) is [15] A x Next, we discretize (16a) as in [37]. Writing the x-component of these equations, where we denote J discretization centered at the center of the cell's x-face results: H z Similar expressions to those in (20c) and (21) can be derived in the yand z-directions. The boundary conditions (3) are used to close (21). Using (20c) we can eliminate b J from (19a) and obtain a discrete equation in which the dominant terms all involve OE. The resulting stencil for OE has 7 points. We also apply the obvious quadrature for the single condition (17d). Finally we discretize the edge-centered (16b). Consider, say, the x-component of (16b), written as Integrating this equation over the surface of a rectangle with corners the expression on the left hand side is integrated using the Gauss curl theo- rem, and - on the right hand side is averaged in both directions to obtain a value on the edge. We do not divide through by - before this integration because we wish to integrate the magnetic flux, which is potentially less smooth around an edge than the magnetic field. This yields Z z k+1 z k If - is assumed to be constant over each cell, this integral evaluates to [h y h z h z h z h z Then, the resulting approximation for the x-component of (16b) is iA y h z A z (22c) Using (19b) as well as (22c) and similar expressions derived in the y- and z-directions, we substitute for H and / in (21) and obtain a discrete system of equations for A. The resulting stencil for A has 19 points and the same structure as for the discretization of the operator r \Theta (- \Gamma1 r\Theta The difference between this discretization and a direct discretization of the latter is that - at the interface is naturally defined as an arithmetic average and not a harmonic average. The discretization described above can be viewed as a careful extension of Yee's method, suitable for discontinuous coefficients and amenable to fast iterative solution methods. It is centered, conservative and second order accu- rate. Note that throughout we have used a consistent, compact discretization of the operators r\Delta ; r\Theta and r. We can denote the corresponding discrete operators by r h and r h and immediately obtain the following identities (cf. [18, 17]), These are of course analogues of vector calculus identities which hold for sufficiently differentiable vector functions. The boundary conditions (17) are discretized using these discrete operators as well. Next, note that upon applying r h \Delta to (21) and using (23b) and (19a) we obtain Moreover, from (21) and (17c), the discrete normal derivative of / vanishes at the boundaries as well. Setting / to 0 arbitrarily at one point, then determines that / j 0 throughout the domain (as a grid function). We obtain another conservation property of our scheme, namely, a discrete divergence-free A: Recall the stabilizing term added in (14). For the exact solution this term obviously vanishes. Now, (24) assures us that the corresponding discretized term vanishes as well (justifying use of the term 'stabilization', rather than 'penalty'). This is not the case for the nodal finite element method which was considered in [20, 10]. For an approximate solution that does not satisfy (24) the stabilization term may grow in size when - varies over a few orders of magnitude, or else r h \Delta A grows undesirably in an attempt to keep - approximately constant across an interface [20]. Our particular way of averaging across discontinuities, namely, arithmetic averaging of - at cell edges and harmonic averaging of b oe at cell faces, can be important. The averaging can be viewed as a careful approximation of the constitutive relationship for discontinuous coefficients. To show that, we look first at the relation across a face whose normal direction is x. This flux flows in series, and therefore an approximate b oe that represents the bulk property of the flow through the volume is given by the harmonic average (corresponding to an arithmetic average of the resistivities). Next, we look at the relation where - is an edge variable and B x is the flux through four cells which share that edge. Here the flow is in parallel, which implies that we need to approximate - on the edge by an arithmetic average. Note also that if we use the more common implementation of Yee's method (i.e. H on the cell's faces and E on the cell's edges) then the roles of b oe and - interchange and we need to average - harmonically and b oe arithmetically. 4 Solution of the discrete system After the elimination of H , / and b J from the discrete equations we obtain a large, sparse system for the grid functions corresponding to A and OE: A OE b A Here is the result of the discretization of the operator r \Theta - corresponds to the discretization of the operator r\Theta ; D likewise corresponds to the discretization of the operator the diagonal matrix S results from the discretization of the operator b oe(\Delta); M and M c similarly arise from the at cell edges and at cell centers respectively; and H represents the discretization of r \Delta (boer(\Delta)). In regions of constant - the simplifies into a discretization of the vector Laplacian. The blocks are weakly coupled through interfaces in -. A typical sparsity structure of H 1 for variable -, as contrasted with constant -, is displayed in Fig. 3. The structure of the obtained system is similar to that in [15], although the main block diagonal is somewhat less pleasant. Note that as long as the frequency is low enough that the diffusion number satisfies (where h is the maximum grid spacing) the matrix is dominated by the diagonal blocks. This allows us to develop a block preconditioner good for low frequencies ! based on the truncated ILU (ILU(t)) decomposition of the major blocks [30]. Thus, we approximate K by the block diagonal matrix and then use ILU(t) to obtain a sparse approximate decomposition of the K. This decomposition is used as a preconditioner for the Krylov solver. Note that although K is complex the approximation - K is real and therefore we need to apply the ILU decomposition only to two real matrices, which saves memory. Note that the block approximation (26) makes sense for our discretization but not for the direct staggered-grid discretization of (1). Thus, the Sparsity structure - constant - Sparsity structure - variable - Figure 3: Sparsity structure of the matrix H 1 corresponding to variable - (right) and to constant - (left). reformulation and subsequent discretization of Maxwell's equations allow us to easily obtain a good block preconditioner in a modular manner, while the discretization of (1) does not. 5 Numerical examples Our goal in this section is to demonstrate the efficacy of our proposed method and to compare it to the more common discretization of the system (1), utilized e.g. in [24, 29], using standard Krylov-type methods and preconditioners for solving the resulting discrete systems. We vary the type of source used, the size of jumps in the coefficients oe and -, the preconditioner and the grid size. In the tables below, 'iterations' denotes the number of BICGSTAB iterations needed for achieving a relative accuracy of the number of giga-flops required; the SSOR parameter (when used) equals and the ILU threshold (when used) equals 10 \Gamma2 . The latter threshold is such that in our current Matlab implementation iterations involving these two preconditioners cost roughly the same. 5.1 Example Set 1 We derive the following set of experiments. Let the air's permeability be its permittivity ffl We assume a cube of constant conductivity oe c and permeability - c embedded in an otherwise homogeneous earth with conductivity oe permeability Fig. 4. In a typical geophysical application, the conductivity may range over four orders of magnitude and more, whereas the permeability rarely changes over more than one order of magnitude. Therefore, we experiment with conductivity oe c ranging from 10 \Gamma2 S/m to 10 3 S/m and permeability - c ranging from - 0 to 1000- 0 . -0.4 Figure 4: The setting of our first numerical experiment. A cube of conductivity oe c and permeability - c is embedded inside a homogeneous earth with conductivity oe e and permeability - e . Also, in the air We experiment with two different sources: (i) a magnetic source (a plane wave); and (ii) an electric dipole source in the x direction centered at (0; 0; 0). The fact that the first source is magnetic implies that it is divergence-free. This source lies entirely in the active space of the curl operator. In contrast, the electric source is not divergence-free. Both sources are assumed to oscillate with different frequencies ranging from 1 to 10 6 Hz. The solution is obtained, unless otherwise noted, on a nonuniform tensor grid (see Fig. 4) consisting of cells. There are 95232 unknowns corresponding to this grid and 128000 (complex) unknowns. We then solve the system using the method described in the previous section. 5.1.1 Example 1a. In order to be able to compare the resulting linear algebra solver with that corresponding to Yee's method we discretize the system r \Theta [- \Gamma1 r \Theta ( b using the staggered grid depicted in Fig. 2, i.e., where b J is on cells' faces, which is similar to the discretization in [24]. This yields the discrete system for the unknowns vector e corresponding to grid values of b J=boe, where the matrices C; M and S are defined in Section 4 and - b depends on the source. In order to solve this system as well as ours we use BICGSTAB and an preconditioner. The comparison between the methods for the case different frequencies is summarized in Table 2. Electric source Magnetic source iterations operations iterations operations 6.3 728 99 113 6.4 7.3 Table 2: Iteration counts and computational effort for our method (A; OE) and the traditional implementation of Yee's method (applied to E, or b J Example Set 1 using both an electric source and a magnetic source. Table 2 shows that our method converges in a moderate number of iterations for both sources, despite the presence of significant jumps in - and oe. On the other hand, the more traditional discretization performs poorly for the electric source and reasonably well for the magnetic source. Slow convergence of the direct staggered discretization of Maxwell's equations in the case of an electric source was also reported in [29], where E was defined on the grid's edges. These results clearly show an advantage of our formulation over the original Yee formulation, even for a simple preconditioning, especially for electric sources and in low frequencies. In such circumstances, the discretized first term on the left hand side of (27) strongly dominates the other term, and the residual in the course of the BICGSTAB iteration has a nontrivial component in the null space of that operator; hence its reduction is very slow. The magnetic source, on the other hand, yields a special situation where, so long as the discrete divergence and the roundoff error are relatively small, the residual component in the null space of the leading term operator is also small, hence the number of iterations using the traditional method is not much larger than when using our method. 5.1.2 Example 1b. Next, we test the effect of discontinuities on our method. We use the electric source and record the number of BICGSTAB iterations needed for our method to converge for various values of ~ using block-ILU(t) preconditioning as described in the previous section. The results are summarized in Table 3. Note that large jump discontinuities in oe do not significantly affect the rate of convergence of the iterative linear system solver for our method, but large jump discontinuities in - have a decisive effect. Results in a similar spirit were reported in [16] regarding the effect of discontinuities in - on a specialized multigrid method for an edge-element discretization. However, even for large discontinuities in - the number of iterations reported in Table 3 remains relatively small compared with similar experiments reported in [29, 9]. We attribute the increase in the number of iterations as the jump in - increases in size to a corresponding degradation in the condition number of K in (25). This degradation, however, does not depend strongly on grid size, as we verify next. Table 3: Iteration counts for different frequencies, conductivities and permeabilities in Example Set 1. The conductivity/permeability structure is a cube in a half-space. 5.1.3 Example 1c. In the next experiment, we use the cube model with the electric source to evaluate the influence of the grid on the number of iterations. We fix and test our method on a case with a modest coefficient jump ~ and on a case with a large jump ~ set of uniform grids in the interval [\Gamma1; 1] 3 is considered. For each grid we record the resulting number of iterations using both the SSOR and the block ILU preconditioners. The results of this experiment are gathered in Table 4. We observe that the number of iterations increases as the number of un- Grid size ILU SSOR ILU SSOR Table 4: Iteration counts for different grids, for two sets of problem coefficients and using two preconditioners. knowns increases. The increase appears to be roughly proportional to the number of unknowns to the power 1=3. The growth in number of iterations as a function of grid size is also roughly similar for both preconditioners, although the block ILU requires fewer iterations (about 1=4 for ~ for each grid. However, ILU requires more memory than SSOR, which may prohibit its use for very large problems. The increase rate is also similar, as expected, for both values of ~ -. Thus, the increase in number of iterations as a function of ~ essentially does not depend on the grid size. Practically, how- ever, this increase is substantial and may be hard to cope with for (perhaps unrealistically) large values of ~ - using the present techniques. 5.2 Example 2 In our next experiment we consider a more complicated earth structure. We employ a random model, which is used by practitioners in order to simulate stochastic earth models [21]. Two distinct value sets oe probability P are assumed for the conductivities and permeabilities: for each cell, the probability of having values oe and the probability of having values . This can be a particularly difficult model to work with, as the conductivity and permeability may jump anywhere from one cell to the next, not necessarily just across a well-defined manifold. A cross-section of such a model is plotted in Fig. 5. We then carry out experiments as before for frequencies ranging from 0 to 10 6 Hz. We use the random model with different conductivities, permeabilities and frequencies and the electric source, with on the domain [\Gamma1; 1] 3 (in km). We employ a uniform grid of size 44 3 and use both the block ILU and the SSOR preconditioners. The results of this experiment in terms Conductivity slice log(S/m) Figure 5: The setting of Example Set 2. of iteration counts are summarized in Table 5. The results show that our solution method is effective even for highly varying conductivity. As before, the method deteriorates when very large variations in - are present. We can also see from Table 5 that the block ILU preconditioner works very well for low frequencies, but it is not very effective for high frequencies. It is easy to check that in all cases where the block ILU preconditioner fails to achieve convergence (denoted 'nc', meaning failure to achieve a residual of \Gamma7 in 800 iterations) the maximum grid spacing h satisfies !-oe h AE 1. In such a case the discretization of the leading terms of the differential operator no longer yields the dominant blocks in the matrix equations (25), and therefore our block ILU preconditioner fails. Thus, for high frequency and high conductivity we require more grid points in order for this preconditioner to be effective. This is also consistent with the physics, as the skin depth [36] decreases and the attenuated wave can be simulated with fidelity only on a finer grid. 6 Summary and further remarks In this paper we have developed a fast finite volume algorithm for the solution of Maxwell's equations with discontinuous conductivity and permeability. The major components of our approach are as follows. ffl Reformulation of Maxwell's equations: The Helmholtz decomposition is applied to E; then a stabilizing term is added, resulting in a strongly elliptic system; the system is written in first order form to allow flexibility ILU SSOR ILU SSOR ILU SSOR Table 5: Iteration counts for different frequencies, conductivities and permeabilities for Example Set 2. in the ensuing discretization; and finally, the divergence-free Coulomb gauge condition is eliminated using differentiation and substitution, which yields a weakly coupled PDE system enabling an efficient preconditioner for the large, sparse algebraic system which results from the discretization. ffl Discretization using staggered grids, respecting continuity conditions and carefully averaging material properties across discontinuities. For this discretization, the stabilizing term vanishes at the exact solution of the discrete equations, which is important for cases with large contrasts in -. ffl Solution of the resulting linear system using fast preconditioned Krylov methods. The resulting algorithm was tested on a variety of problems. We have shown dramatic improvement over the more standard algorithm when the source is electric. Good performance was obtained even when the coefficients - and oe were allowed to vary rapidly on the grid scale - a case which should prove challenging for multigrid methods. The project that has motivated us is electromagnetic inverse problems in geophysical prospecting [34]. Solving the forward problem, i.e. Maxwell's equations as in the present article, is a major bottleneck for the data inversion methods - indeed it may have to be carried out dozens, if not hundreds, of times for each data inversion. Thus, extremely fast solvers of the problem discussed in our paper are needed. Based on the algorithm described here an implementation has been carried out which solves realistic instances of this forward problem in less than two minutes on a single-processor PC, enabling derivation of realistic algorithms at low cost for the inverse problem. Acknowledgment We wish to thank Drs. Doug Oldenburg and Dave Moulton for fruitful discussions and an anonymous referee for valuable comments on the first two sections of our exposition. --R A method for the forward modelling of 3D electromagnetic quasi-static problems Adaptive multilevel methods for edge element discretizations of maxwell's equations. Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism Multigrid solution to elliptic flow problems. Mixed and hybrid finite element methods. Finite Elements Numerical algorithms for the FDITD and FDFD simulation of slowly varying electromagnetic feilds. Theoretical and numerical difficulties in 3D vector potential methods in finite element magnetostatic compu- tations Black box multigrid. Geomagnetic induction in a heterogenous sphere: Azimuthally symmetric test computations and the response of an undulating 660-km discontinuity Finite Element Methods for Navier-Stokes Equations On pressure boundary conditions for the incompressible Navier-Stokes equations Fast modelling of 3D electromagnetic using potentials. Multigrid method for maxwell's equations. Natural discretizations for the divergence gradient and curl on logically rectangular grids. The orthogonal decomposition theorems for mimetic finite difference methods. Mimetic discretizations for Maxwell's equations and equations of magnetic diffusion. The Finite Element Method in Electromagnetics. Electrical conduction in sandstones: Examples of two dimensional and three dimensional percolation. A scalar-vector potential solution for 3D EM finite-difference modeling Theoretical concepts in physics. Three dimensional magnetotelluric modeling using difference equations - theory and comparison to integral equation solutions An analysis of finite volume A convergence analysis of Yee's scheme on nonuniform grids. Numerical implementation of nodal methods. Solutions of 3D eddy current problems by finite elements. Iterartive Methods for Sparse Linear Systems. A multigrid solver for the steady state navier-stokes equations using the pressure-poisson formulation Computational Electrodynamics: the Finite-Difference Time-Domain Method Diadic Green Functions in Electromagnetics. Electromagnetic theory for geophysical applications. Time domain electromagnetic field computation with finite difference methods. Inversion of Magnetotelluric Data for a One Dimensional Conductivity Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media. --TR --CTR Eldad Haber , Stefan Heldmann, An octree multigrid method for quasi-static Maxwell's equations with highly discontinuous coefficients, Journal of Computational Physics, v.223 n.2, p.783-796, May, 2007

krylov methods;mixed methods;coulomb gauge;solution discontinuities;finite volume;helmholtz decomposition;maxwell's equations;preconditioning

587484

Uniform Convergence and Mesh Independence of Newton''s Method for Discretized Variational Problems.

In an abstract framework, we study local convergence properties of Newton's method for a sequence of generalized equations which models a discretized variational inequality. We identify conditions under which the method is locally quadratically convergent, uniformly in the discretization. Moreover, we show that the distance between the Newton sequence for the continuous problem and the Newton sequence for the discretized problem is bounded by the norm of a residual. As an application, we present mesh-independence results for an optimal control problem with control constraints.

Introduction In this paper we study local convergence properties of Newton-type methods applied to discretized variational problems. Our target problem is the variational inequality representing the rst-order optimality conditions in constrained optimal control. In an abstract frame- work, the optimality conditions are modeled by a \generalized equation", a term coined by S. Robinson [12], where the normal cone mapping is replaced by an arbitrary map with closed graph. In this setting, Newton's method solves at each step a linearized generalized equation. When the generalized equation describes rst-order optimality conditions, Newton's method becomes the well-known sequential quadratic programming (SQP) method. We identify conditions under which Newton's method is not only locally quadratically convergent, but the convergence is uniform with respect to the discretization. Moreover, we derive an estimate for the number of steps required to achieve a given accuracy. Under some additional assumptions, which are natural in the context of the target problem, we prove that the distance between the Newton sequences for the continuous problem and the Newton sequence for the discretized problem, measured in the discrete metric, can be estimated by the norm of a residual. Normally, the residual tends to zero when the approximation becomes ner, and the two Newton sequences approach each other. In the context of the target optimal control problem, the residual is proportional to the mesh spacing h, uniformly along the Newton sequence. In particular, this implies that the least number of steps needed to reach a point at distance " from the solution of the discrete problem does not depend on the mesh spacing; that is, the method is mesh-independent. The convergence of the SQP method applied to nonlinear optimal control problems has been studied in several papers recently. In [5, 6] we proved local convergence of the method for a class of constrained optimal control problems. In parallel, Alt and Malanowski obtained related results for state constrained problems [3]. In the same line Troltzsch [13] studied the SQP method for a problem involving a parabolic partial dierential equation. Kelley and Sachs [10] were the rst to obtain a mesh independence result in constrained optimal control; they studied the gradient projection method. More recently, uniform convergence and mesh-independence results for an augmented Lagrangian version of the SQP method, applied to a discretization of an abstract optimization problem with equality constraints, were presented by Kunisch and Volkwein [11]. Alt [2] studied the mesh-independence of Newton's method for generalized equations, in the framework of the analysis of operator equations in Allgower et al. [1]. An abstract theory of mesh independence for innite-dimensional optimization problems with equality constraints, together with applications to optimal control of partial dierential equations and an extended survey of the eld can be found in the thesis of Volkwein [14]. The local convergence analysis of numerical procedures is closely tied to problem's sta- bility. The analysis is complicated for optimization problems with inequality constraints, or for related variational inequalities. In this context, the problem solution typically depends on perturbation parameters in a nonsmooth way. In Section 2 we present an implicit function theorem which provides a basis for our further analysis. In Section 3 we obtain a result on uniform convergence of Newton's method applied to a sequence of generalized equations, while Section 4 presents our mesh-independence results. Although in part parallel, our approach is dierent from the one used by Alt in [2] who adopted the framework of [1]. First, we study the uniform local convergence of Newton's method which is not considered in [2]. In the mesh- independence analysis, we avoid consistency conditions for the solutions of the continuous and the discretized problems; instead, we consider the residual obtained when the Newton sequence of the continuous problem is substituted into the discrete necessary conditions. This allows us to obtain mesh independence under conditions weaker than those in [2] and, in the same time, to signicantly simplify the analysis. In Section 5 we apply the abstract results to the constrained optimal control problem studied in our previous paper [5]. We show that under the smoothness and coercivity conditions given in [5], and assuming that the optimal control of the continuous problem is a Lipschitz continuous function of time, the SQP method applied to the discretized problem is Q-quadratically convergent, and the region of attraction and the constant of the convergence are independent of discretization, for a suciently small mesh size. Moreover, the l 1 distance between the Newton sequence for the continuous problem at the mesh points and the Newton sequence for the discretized problem is of order O(h). In particular, this estimate implies the mesh-independence result in Alt [2]. 2. Lipschitzian localization Let X and Y be metric spaces. We denote both metrics by (; ); it will be clear from the context which metric we are using. B r (x) denotes the closed ball with center x and radius r. In writing \f maps X into Y " we adopt the convention that the domain of f is a (possibly proper) subset of X. Accordingly, a set-valued map F from X to the subsets of Y may have empty values. Denition 2.1. Let map Y to the subsets of X and let x 2 (y ). We say that has a Lipschitzian localization with constants a, b and M around (y ; x ), if the map y 7! (y)\B a single-valued (a function) and Lipschitz continuous in B b (y ) with a Lipschitz constant M . Theorem 2.1. Let G map X into the subsets of Y and let y 2 G(x ). Let G 1 have a Lipschitzian localization with constants a; b, and M around (y ; x ). In addition, suppose that the intersection of the graph of G with B a closed and B a (x ) is complete. Let the real numbers , M , a, m and satisfy the relations Suppose that the function continuous with a constant in the ball B a (x ), that sup and that the set is nonempty. Then the set fx 2 B a consists of exactly one point, x, and for each Proof. Let us choose positive ; a and such that the relations in (1) hold. We rst show that the set T := nonempty. Let x 0 2 and put Take an arbitrary " > 0 such that Choose an y and from the Lipschitzian localization property, there exists x 1 such that We dene inductively a sequence x k in the following way. Let x be already dened for some k 1 in such a way that and Clearly, x 0 and x 1 satisfy these relations. Using the second inequality in (5), we estimate a Thus both x k 1 and x k are in B a (x ) from which we obtain by (2), k. Due to the assumed Lipschitzian localization property of G, there exists x k+1 such that (7), with k replaced by k + 1, is satised and By (6) we obtain and hence (6) with k replaced by k + 1, is satised. The denition of the sequence x k is complete. >From (6) and the condition M < 1, fx k g is a Cauchy sequence. Since all x sequence fx k g has a limit x Passing to the limit in (7), we obtain g(x 00 Hence x 00 2 T and the set T is nonempty. Note that x 00 may depend on the choice of ". If we prove that the set T is a singleton, say ^ x, the point x would not depend on ". Suppose that there exist x 00 2 T and It follows that (g(x); y ) x 00 . >From the denition of the Lipschitzian localization, we obtain which is a contradiction. Thus T consists of exactly one point, ^ x, which does not depend on ". To prove (4) observe that for any choice of k > 1, Passing to the limit in the latter inequality and using (5), we obtain But since x x does not depend on the choice of ", one can take in (8) and the proof is complete. 3. Newton's Method Theorem 2.1 provides a basis for the analysis of the error of approximation and the convergence of numerical procedures for solving variational problems. In this and the following section we consider a sequence of so-called \generalized equations". Specically, for each be a closed and convex subset of a Banach space, let Y N be a linear normed space, let fN : X N 7! Y N be a function, and let FN : X N 7! 2 Y N be a set-valued map with closed graph. We denote by k kN the norms of both X N and Y N . We study the following sequence of problems: Find We assume that there exist constants , , , and L, as well as points x that satisfy the following conditions for each N : (A2) The function fN is Frechet dierentiable in B (x N ) and the derivative rfN is Lipschitz continuous in B (x N ) with a Lipschitz constant L. (A3) The map y 7! has a Lipschitzian localization with constants ; and around the point (z We study the Newton method for solving (9) for a xed N which has the following form: If x k is the current iterate, the next iterate x k+1 satises where x 0 is a given starting point. If the range of the map F is just the origin, then (9) is an equation and (10) becomes the standard Newton method. If F is the normal cone mapping in a variational inequality describing rst-order optimality conditions, then (10) represents the rst-order optimality condition for the auxiliary quadratic program associated with the SQP method. In the following theorem, by applying Theorem 2.1, we obtain the existence of a locally unique solution of the problem (9) which is at distance from the reference point proportional to the norm of the residual z N . We also show that the method (10) converges Q-quadratically and this convergence is uniform in N and in the choice of the initial point from a ball around the reference point x N with radius independent of N . Note that, for obtaining this result we do not pass to a limit and consequently, we do not need to consider sequences of generalized equations. Theorem 3.1. For every there exist positive constants and such that if kz then the generalized equation (9) has a unique solution xN in B (x xN satises Furthermore, for every initial point x N ) there is a unique Newton sequence fx k g, with Newton sequence is Q-quadratically convergent to xN , that is, where is independent of k; N and x Proof. Dene min s We will prove the existence and uniqueness of xN by using Theorem 2.1 with and Observe that a ; b , and b a. By (A3) the map G has a Lipschitzian localization around . One can check that the relations (1) are satised. Further, for N ); we have Obviously, x dened in (3). The assumptions of Theorem 2.1 are satised, hence there exists a unique xN in B (x Hence xN is a unique solution of holds. This completes the rst part of the proof. Given x N ), a point x is a Newton step from x k if and only if x satises the inclusion where G is the same as above, but now The proof will be completed if we show that (14) has a unique solution x this solution satises (12). To this end we apply again Theorem 2.1 with a; b; M; M , and the same as in the rst part of the proof, and with With these identications, it can be checked that the assumptions (1) and (2) hold, and that g is Lipschitz continuous in B (x N ) with a Lipschitz constant . Further, by using the solution xN obtained in the rst part of the proof, we have L The last expression has the estimate Thus xN 2 6= ; and the assumptions of Theorem 2.1 are satised. Hence, there exists a unique Newton step x k+1 in B (x N ) and by Theorem 2.1 and (15) it satises 4. Mesh independence Consider the generalized equation (9) under the assumption (A1)-(A3). We present rst a lemma in which, for simplicity, we suppress the dependence of N . Lemma 4.1. For every , every > 0 and for every suciently small > 0, there exists a positive such that the map is a Lipschitz continuous function from B (z for y and for w. Proof. Let and > 0. We choose the positive constants and as a solution of the following system of inequalities: This system of inequalities is satised by rst taking suciently small, and then taking suciently small. In particular, we have and . We apply Theorem 2.1 with , and We have for all x Hence the function g is Lipschitz continuous with a Lipschitz constant . For +2 Note that a point x is in the set P (y only if g(x) 2 G(x). Since the set dened in (3) is not empty. Hence, from Theorem 2.1 the set P (y consists of exactly one point. Let us call it x 00 . Applying the same argument to an arbitrary point (y that there is exactly one point x Hence x 0 2 and we obtain It remains to prove that P maps B (z From the last inequality with Thus x In the remaining part of this section, we x and 0 < < 1, and we choose the constants and according to Theorem 3.1. For a positive with , let be the constant whose existence is claimed in Lemma 4.1. Note that can be chosen arbitrarily small; we take Also, we assume that kz and consider Newton sequences with initial points In such a way, the assumptions of Theorem 3.1 hold and we have a unique Newton sequence which is convergent quadratically to a solution. Suppose that Newton's method (10) is supplied with the following stopping test: Given " > 0, at the k-th step the point x k is accepted as an approximate solution if Denote by k " the rst step at which the stopping test (18) is satised. Theorem 4.1. For any positive " < , if x k" is the approximate solution obtained using the stopping test (18) at the step and Proof. Choose an " such that 0 < " < . If the stopping test (18) is satised at x k" , then there exists v k " with k v k " such that Let P N be dened as in (16) on the basis of fN and FN . Since Lemma 4.1 implies that The latter inequality yields (19). For all k < k " , we obtain Since x k is a Newton iterate, we have Hence By the denition of the map P N , the Newton step x 1 from x 0 satises while the Newton step x 2 from x 1 is Since P N is Lipschitz continuous with a constant , we have By induction, the 1)-st Newton step x k+1 satises Combining (21) and (22) and we obtain the estimate which yields (20). Our next result provides a basis for establishing the mesh-independence of Newton's method (10). Namely, we compare the Newton sequence x k N for the \discrete" problem and the Newton sequence for a \continuous" problem which is again described by (9) but with us assume that the conditions (A1){(A3) hold for the generalized equation According to Theorem 3.1, for each starting point x 0 0 close to a solution x there is a unique Newton sequence x k which converges to x 0 Q-quadratically. To relate the continuous problem to the discrete one, we introduce a mapping N from X 0 to X N . Having in mind the application to optimal control presented in the following section, X 0 can be thought as a space of continuous functions x() in [0; 1] and, for a given natural number N , t will be the space of sequences fx Ng. In this case the operator N is the interpolation map N Theorem 4.2. Suppose that for every k and N there exists r k and In addition, let for all k and N . Then for all Proof. By denition, we have Using Lemma 4.1 we have By induction we obtain (24). The above result means that, under our assumptions, the distance between the Newton sequence for the continuous problem and the Newton sequence for the discretized problem, measured in the discrete metric, can be estimated by the sup-norm !N of the residual obtained when the Newton sequence for the continuous problem is inserted into the discretized generalized equations. If the sup-norm of the residual tends to zero when the approximation becomes ner, that is, when N !1, then the two Newton sequences approach each other. In the next section, we will present an application of the abstract analysis to an optimal control problem for which the residual is proportional to the mesh spacing h, uniformly along the Newton sequence. For this particular problem Theorem 4.2 implies that the distance between the Newton sequences for the continuous problem and the Newton sequence for the discretized problem is O(h). For simplicity, let us assume that if the continuous Newton process starts from the point N , then the discrete Newton process starts from N (x 0 Also, suppose that for any xed kN (w) N (v)k N ! kw vk 0 as N !1: (25) In addition, let where !N is dened in (23). Letting k tend to innity and assuming that N is a continuous mapping for each N , Theorem 4.2 gives us the following estimate for the distance between the solution xN of the discrete problem and the discrete representation N of the solution x of the continuous problem: Choose a real number " satisfying where is as in Theorem 3.1. Theorem 4.2 yields the following result: Theorem 4.3. Let (25) and (26) hold and let " satisfy (28). Then for all N suciently large, Proof. Let m be such that Choose N so large that1 !N < "=2 and Using Theorem 3.1, Theorem 4.2, (27), and (31), we obtain This means that if the continuous Newton sequence achieves accuracy " (measured by the distance to the exact solution) at the m-the step, then the discrete Newton sequences should achieve the same accuracy " at the (m 1)-st step or earlier. Now we show that the latter cannot happen earlier than at the (m 1)-st step. Choose N so large that and suppose that >From Theorem 3.1, (24), (27), (30) and (31), we get which contradicts the choice of " in (28). 5. Application to optimal control We consider the following optimal control problem: subject to _ U is a nonempty, closed and convex set in IR m , and y 0 is a xed vector in IR n . L the space of essentially bounded and measurable functions with values in IR m and W 1;1 (IR n ) is the space of Lipschitz continuous functions with values in IR n . We are concerned with local analysis of the problem (32) around a xed local minimizer which we assume certain regularity. Our rst standing assumption is the following: Smoothness. The optimal control u is Lipschitz continuous in [0; 1]. There exists a positive number such that the rst three derivatives of ' and g exist and are continuous in the set Dening the Hamiltonian H by it is well known that the rst-order necessary optimality conditions at the solution (y can be expressed in the following way: There exists 2 W 1;1 solution of the variational inequality _ _ where NU (u) is the normal cone to the set U at the point u; that is, NU (u) is empty if u 62 U , while for u 2 U , Although the problem (32) is posed in L 1 and the optimality system (33){(35) is satised almost everywhere in [0; 1], the regularity we assume for the particular optimal solution implies that at (y ; u ; ) the relations (33){(35) hold everywhere in [0; 1]. Dening the matrices yy H(x (t)); where z we employ the following coercivity condition: Coercivity. There exists > 0 such that Let N be a natural number, let 1=N be the mesh spacing, let t the forward dierence operator dened by We consider the following Euler discretization of the optimality system (33){(35): The system (37){(39) is a discrete-time variational inequality depending on the step size h. It represents the rst-order necessary optimality condition for the following discretization of the original problem (32): subject to y 0 In this section we examine the following version of the Newton method for solving the variational system (37)-(39), which corresponds to the SQP method for solving the optimization problem (40). Let x denote the k-th iterate. Let the superscript k and the attached to the derivatives of H and G denote their values at x k . Then the new iterate x solution of the following linear variational inequality for the variable In [5], Appendix 2, it was proved that the coercivity condition (36) is stable under the Euler discretization, then the variational system (41){(43) is equivalent, for x k near x to the following linear-quadratic discrete-time optimal control problem which is expressed in terms of the variables y, u, and z = (y; u): minimize (y N y k r z ' k subject to y 0 A natural stopping criterion for the problem at hand is the following: Given " > 0, a control ~ obtained at the k-th iteration is considered an "-optimal solution if dist(r u H(~y k where ~ i and ~ i are the solutions of the state and the adjoint equations (37) and (38) corresponding to We now apply the general approach developed in the previous section to the discrete-time variational inequality (37){(38). The discrete L 1 N norm is dened by The variable x is the triple (y; u; ) while X N is the space of all nite sequences x with y 0 given, equipped with the L 1 norm. The space Y N is the Cartesian product L 1 N corresponding to the four components of the function fN dened by r With the choice (x the general condition (A1) is satised by taking (z The rst component of z N is estimated in the following way: sup sup ih Since g is smooth and both y and u are Lipschitz continuous, the above expression is bounded by O(h). The same bound applies for the second component of z N , while the third and fourth components are zero. Thus the norm of z N can be made arbitrarily small for all suciently large N . Condition (A2) follows from the smoothness assumption. A proof of condition (A3) is contained in the proof of Theorem 6 in [5]. Applying Theorems 3.1 and 4.1 and using the result from [5], Appendix 2, that the discretized coercivity condition is a second-order sucient condition for the discrete problem, we obtain the following theorem: Theorem 5.1. If Smoothness and Coercivity hold, then there exist positive constants K, " and N with the property that for every N > N there is a unique solution (y of the variational system (37){(39) and (y h ; u h ) is a local minimizer for the discrete problem (40). For every starting point (y there is a unique SQP sequence (y k ; u Q-quadratically convergent, with a constant K, to the solution (y In particular, for the sequence of controls we have Moreover, if the stopping test (44) is applied with an " 2 [0; "], then the resulting "-optimal control u k" satises Note that the termination step k " corresponding to the assumed accuracy of the stopping test can be estimated by Theorem 4.1. Combining the error in the discrete control with the discrete state equation (37) and the discrete adjoint equation (38), yield corresponding estimates for discrete state and adjoint variables. Remark. Following the approach developed in [5] one can obtain an analog of Theorem 5.1 by assuming that the optimal control u is merely bounded and Riemann integrable in [0; 1] and employing the so-called averaged modulus of smoothness to obtain error estimates. The stronger Lipschitz continuity condition for the optimal control is however needed in our analysis of the mesh independence. The SQP method applied to the continuous-time optimal control problem (32) has the following starting point, the iterate x _ _ for a.e. t 2 [0; 1], where the superscript k attached to the derivatives of H and G denotes their values at x k . In particular, (45){(48) is a variational inequality to which we can apply the general theory from the previous sections. We attach the index to the continuous problem and for fy is clearly satised with Condition (A2) follows from the Smoothness assumption. The condition (A3) follows from the Coercivity assumption as proved in [9], Lemma 3 (see also [4], Section 2.3.4 for an earlier version of this result in the convex case). Hence, we can apply Theorem 3.1 obtaining that for any suciently small ball around x (in the norm of X 0 ), if the starting point x 0 is chosen from B, then the SQP method produces a unique sequence x k 2 B which is Q-quadratically convergent to x (in the norm of X 0 ). Moreover, from Theorem 4.1 we obtain an estimate for the number of steps needed to achieve a given accuracy. In order to derive a mesh-independence result from the general theory, we rst study the regularity of the SQP sequence for the continuous problem. Lemma 5.1. There exist positive constants p and q such that for every x continuous in [0; 1], for every Proof. In [5], Section 6, extending a previous result in [7], see also [6], Lemma 2, we showed that the coercivity condition implies pointwise coercivity almost everywhere. In the present circumstances, the latter condition is satised everywhere in [0; 1]; that is, there exists a constant > 0 such that for every v 2 U U and for all t 2 [0; 1] For a positive parameter p consider the SQP sequence x k starting from x the initial control u 0 is a Lipschitz continuous function in [0; 1]. Throughout the proof we will choose suciently small and check the dependence of the constants of p. By (48) the iterate (r uy H(x k (t))(y k+1 (t) y k (t)) for every t 2 [0; 1] and for every u 2 U . Let t are contained in B p (x ) for all k and therefore both y 0k and 0k are bounded by a constant independent of k; hence, y k and k are Lipschitz continuous function in time with Lipschitz constants independent of k. We have from (50) (r uy and the analogous inequality with t 1 and t 2 interchanged. Adding these two inequalities and adding and subtracting the expressions r 2 uy where the function k is dened as By (49), for a suciently small p the right-hand side of the inequality (51) satises Combining (51) and (52) we obtain uy uy uy Let u k be Lipschitz continuous in time with a constant L k . Then the function k is almost everywhere dierentiable and its derivative is given by _ >From this expression we obtain that there exists a constants c 1 , independent of k and t and bounded from above when p ! 0, such that Estimating the expressions in the right-hand side of (54) we obtain that there exists a constant independent of k and t and bounded from above when p ! 0, such that Hence, u k+1 is Lipschitz continuous and, for some constants c of the same kind as c 1 ; c 2 , its Lipschitz constant L k+1 satises Since p can be chosen arbitrarily small, the sequence L i.e. by a constant q. The proof is complete. To apply the general mesh-independence result presented in Theorem 4.2 we need to estimate the residual r k obtained when the SQP sequence of the continuous problem is substituted into the relations determining the SQP sequence of the discretized problem. Specically, the residual is the remainder term associated with the Euler scheme applied to (45){(48); that is, where the subscript i denotes the value at t i . From the regularity of the Newton sequence established in Lemma 5.1, the uniform norm of the residual is bounded by ch, where c is independent of k. Note that the map N (x) dened in Section 4, acting on a function x gives the sequence x(t i Condition (25) is satised because the space X 0 is a subset of the space of continuous functions. Summarizing, we obtain the following result: Theorem 5.2. Suppose that Smoothness and Coercivity conditions hold. Then there exists a neighborhood W, in the norm of X 0 , of the solution x such that for all suciently small step-sizes h, the following mesh-independence property holds: sup where u k () is the control in the SQP sequence (y k (); u k (); k ()) for the continuous problem starting from a point x continuous in [0; 1], and u k h is the control in the SQP sequence (y k h ) for the discretized problem starting from the point Applying Theorem 4.3 to the optimal control problem considered we obtain the mesh- independence property (29) which relates the number of steps for the continuous and the discretized problem needed to achieve certain accuracy. The latter property can be also easily deduced from the estimate (55) in Theorem 5.2, in a way analogous to the proof of Theorem 4.3. Therefore the estimate (55) is a stronger mesh-independence property than (29). Table 1: L 1 error in the control for various choices of the mesh. Iteration Table 2: Error in current iterate divided by error in prior iterate squared. Iteration 5. Numerical examples The convergence estimate of Theorem 5.2 is illustrated using the following example: minimize dt subject to _ This problem is a variation of Problem I in [8] that has been converted from a linear-quadratic problem to a fully nonlinear problem by making the substitution and by adding additional terms to the cost function that degrade the speed of the SQP iteration so that the convergence is readily visible (without these additional terms, the SQP iteration converges to computing precision within 2 iterations). Figures 1{3 show the control iterates for successively ner meshes. The control corresponding to barely visible beneath the Observe that the SQP iterations are relatively insensitive to the choice of the mesh. Specically, suciently large to obtain mesh independence. In Table 1 we give the L 1 error in the successive iterates. In Table 2 we observe that the ratio of the error in the current iterate to the error in the prior iterate squared is slightly larger than 1. Figure 1. SQP iterates for the control with Figure 3. SQP iterates for the control with Figure 2. SQP iterates for the control with --R Discretization and mesh-independence of Newton's method for generalized equa- tions The Lagrange-Newton method for state constrained optimal control problems Lipschitzian stability in nonlinear control and optimiza- tion Variants of the Kuhn-Tucker sucient conditions in cones of non-negative functions Dual approximations in optimal control Multiplier methods for nonlinear optimal control Mesh independence of the gradient projection method for optimal control problems Augmented Lagrangian-SQP techniques and their approxima- tions Mathematical programming: the state of the art (Bonn --TR --CTR Steven J. Benson , Lois Curfman McInnes , Jorge J. Mor, A case study in the performance and scalability of optimization algorithms, ACM Transactions on Mathematical Software (TOMS), v.27 n.3, p.361-376, September 2001

optimal control;newton's method;discrete approximation;sequential quadratic programming;mesh independence;variational inequality

587495

Output Tracking Through Singularities.

Output tracking for nonlinear systems is complicated by the existence of "singular submanifolds." These are surfaces on which the decoupling matrix loses rank. To provide additional control action we identify a class of smooth vector fields whose integral curves can be incrementally tracked using rapidly switched piecewise constant controls. At discrete times the resulting piecewise smooth state trajectories approach the integral curve being tracked. These discontinuous controllers are applied to sliding mode control---we use incremental tracking to move the state toward the sliding surface. The resulting controller achieves approximate output tracking in situations where the usual approach to sliding mode control fails due to the loss of control action on the singular submanifold.

Introduction Tracking in the case where the decoupling matrix loses rank on a \singular submanifold" have been considered by a number of authors (c.f. [2, 5, 6, 7, 9, 15]). In [2] the problem of exact tracking is studied using results on singular ordinary dierential equations and on the multiplicity of solutions. Conditions under which the singular tracking control is smooth or analytic are given in [9], assuming that that the inputs and some of their derivatives are related to the outputs and their derivatives via a singular ordinary dierential equation. In output trajectories which the system can track using continuous open loop controls are identied for systems which satisfy a suitable observability condition and a discontinuous feedback controller is introduced which achieves robust tracking in the face of perturbations. In [5] the relative order is locally This work was supported in part by the Natural Sciences and Engineering Research Council of Canada y Department of Mathematics and Statistics, Queen's University, Kingston, Ontario K7L 3N6, Canada. E-mail: ron@mast.QueensU.CA increased by keeping the state trajectory near a codimension one submanifold. In some sense our approach takes the opposite point of view in that we seek to reduce the relative order by using vibratory controls. These switched controls allow motion in directions other than those of the drift vector eld or vector elds in the Lie Algebra generated by the control vector elds. Recently there has been increased interest in the use of patterns in control. The pioneering work of Brockett [1], Pomet [12], Lui and Sussmann [10] and others looks at curves that can be approached by state trajectories of smooth a-ne systems. For single-input systems these results highlight the very limited class of smooth paths which can be closely approximated by the state trajectory. We introduce the notion of incremental tracking of smooth integral curves by state trajectories. The state trajectories are permitted to move far from the integral curve being tracked but are required to approach them arbitrarily closely arbitrarily often. This weaker notion of approximation by the state trajectory lends itself well to sliding mode control where we wish to steer the state to a sliding surface. This is a surface on which the state evolves so that the tracking errors go to zero. We are not concerned about the path along which the trajectory approaches the sliding surface as long as any large deviations take place in directions which are not seen directly by the output. Sliding mode control utilizing discontinuous feedback controllers can achieve robust asymptotic output tracking (c.f. [16, 13, 14] and the references therein) under the implicit assumption that the state trajectory can always be steered towards the \sliding surface". That is the decoupling matrix is of full rank everywhere (c.f. [8]). In [6] sliding mode control is studied in the case where the decoupling matrix loses rank and there exists a \singular submanifold" near which the state trajectory cannot be steered towards the sliding sur- face. For systems whose singular submanifold satises suitable transversality conditions a class of smooth output functions y d is identied which can be approximately tracked using a truncated sliding mode controller. For these outputs the state trajectory passes through the \singular submanifold" a nite number of times. There are, however, many simple systems where truncated controllers cause the state trajectory to \stick" to the \singular submanifold" so that the state moves ever farther from the sliding surface. For such systems the standard approaches to output tracking are also not very successful. The following example illustrates the di-culties which can arise. Example 1.1 Consider the a-ne nonlinear system in IR 3 _ _ _ (1) Suppose that we wish to regulate the output close to y d (t) while keeping the state vector bounded. If then we can regulate y by keeping the state trajectory on or near to the \sliding y d g. We note that without the term x 2 3 this system is linear with relative order 3 but here and the relative order of y is 2 (c.f. [6, 8]). In particular _ y d y d and . The natural sliding mode controller usm x(t) reaches S p t and stays in S p t after a nite time has elapsed (c.f. [16], [13]). Inherent in this control scheme is the assumption that b does not vanish along the state trajectory. Of course in our case b vanishes on the \sin- gular manifold" hence usm can become unbounded as x(t) approaches N . One natural solution is to use the truncated controller or the simpler controller For linear systems such truncated controllers work on a neighbourhood of the origin which expands as L grows. This is not the case here. In fact, suppose that that we wish to track y negative, and x positive). If we perturb x 3 so that x sm < 0 hence _ returns to N . For x sm > 0 and once again x returns to N . In essence the state trajectory will \stick" to the submanifold 0g. Of course on N we have _ that _ and the state trajectory evolves on N in such a way that Of course we can track y using this approach if the initial state x 2 (0) > 0. The larger x 2 (0) is the more we can insulate the system from the above phe- nomena. On the other hand, if we track y d sin t, even with x 2 (0) > 0, we will inevitably nd that x 2 becomes negative and the above problem dom- inates. This phenomena is illustrated by Figure 1, which shows the results of a simulation performed using Simnon/PCW for Windows Version 2.01 (SSPA Maritime Consulting AB, Sweden). If x 2 (0) < 0 then the divergence of s and e is immediate. With controller (2) with the onset of this divergence is only delayed. y, yd Figure 1: Tracking of sin t using a truncated sliding mode controller. It is of interest to note that if we could enforce s 0 exactly in the case y d 0 then and the resulting \zero dynamics" are unstable. The approximate input-output linearization scheme of [5] applied to this example has similar problems. Tracking schemes which are based on dier- entiating y until u appears come up against this same obstruction. Tomlin and Sastry have observed a similar phenomena in the Ball and Beam example [15], where their switched control scheme is not eective. The above example presents similar obstructions. Instead of taking more derivatives of s to deal with the singular submanifold N we use fewer derivatives. As a result we lose direct control over s (as _ s is independent of u) but avoid the problems associated with the \singular manifold". We introduce a switched periodic controller which causes the state to \incrementally track" the integral curve of a vector eld obtained from Lie Brackets of the drift and control vector elds. The resulting continuous but nonsmooth state trajectory approaches the sliding surface. We will return to this example in Section 4. The rest of the paper is organized as follows: in Section 2 we formulate the sliding mode control problem for single-input single-output a-ne nonlinear systems. In Section 3 we introduce our switched controllers and present our results on approximate trajectory tracking for systems with drift. In Section 4 we state and prove our main results - applications of incremental tracking to sliding mode control - and continue the above example. Finally, some concluding remarks are oered in Section 5. Output tracking and sliding surfaces Suppose that M is a smooth manifold. Given a smooth function and a vector eld X(x) on M , denotes the Lie derivative of h(x) along X(x) and X t curve of X passing through x 0 at so that d Y is a smooth vector eld on M then [X; Y denotes the Lie Bracket of X and Y , and adX LA denote the Lie Algebra generated by i.e. the smallest vector space containing X and Y and closed under Lie Brackets. Suppose that N is a codimension 1 submanifold of M . A vector eld X is transversal to N if X(x) 62 T x N 8x 2 N , where T x N is the tangent space to N at x. If P Q is a submanifold and smooth map of manifolds then f is transversal to P if Image(df x Consider the nonlinear control system model _ where M IR ' is a smooth m dimensional embedded submanifold of IR ' , are smooth vector elds on M , and h is a smooth output function on M . If x 2 M we denote by jjxjj the norm on M which is induced by the standard norm on IR ' . Suppose that y is a smooth function which we wish the output y of (3) to track. The standard approach in sliding mode control (c.f. [13, 16]) is to force the evolution of the output tracking error be governed by a stable dierential equation of the form s(e p linear so that Denition 2.1 The output of (3) can approximately track y d to degree p if, given any - > 0, there exists an admissible input u - and time t - > t 0 such that js(e p (t))j - and the resulting state x(t) is bounded on [t - ; 1). We say that y asymptotically tracks y d to degree p if s(e p and x(t) is bounded on The relative degree r of the output y is the least positive integer for which the derivative y (r) (t) is an explicit function of the input u. More precisely r is the least positive integer for which gf (r 1) h 6 0 (c.f. [7, 8]). For single-input systems the \decoupling matrix" is the 1 1 matrix whose entry is gf (r 1) h. Thus the rank of the decoupling matrix changes where gf (r 1) h vanishes. We choose p r to avoid a possibly singular dierential equation for u. Thus If we set h then s(e p is equivalent to the requirement that s p (x(t); In particular if we let S p t denote the sliding surface then x(t) 2 S p tracking. Similarly if d (t)g (7) t and x(t) d and perfect tracking. Our rst assumption is that S p t is submanifold. t is an embedded codimension 1 submanifold of M for all Remark 2.2 It is straightforward to show that A1 holds if the map h p is transversal to the hyperplane s 1 d (t) (c.f. [6]). The standard sliding mode controller approach (c.f. [6], [13],[16]) is to pick the relative order of the output y. Then u appears explicitly in d d (t)) and h(x). The standard sliding mode controller takes the form usm (x; where K > 0. Using this control d hence, after some nite time t f t 0 , we will have s r . If, in addition, the system has bounded \zero dynamics" on E p t then asymptotic tracking of an output y d will be achieved (c.f. [8]). We note that systems which fail to be strongly observable in the sense of [7] can have unstable zero dynamics (c.f. [6, 15]). Of course the assumption that b does not vanish along the state trajectory is strong. It holds in the linear case but it is rarer in the nonlinear case. Typically b vanishes on the singular submanifold unbounded when the state trajectory reaches N . A natural solution is to use a truncated controller, but the resulting state trajectory can \stick" to N and evolve in such a way that one travels away from S p t on N (such is the case in Example 1.1). We now introduce switched controllers which permit us to move towards the sliding surface even if b(x) vanishes. 3 Incremental Tracking The set of curves which can be approximately tracked by the state trajectories of a-ne systems has been characterized in [12]. For single-input systems the state trajectory can only be made to stay close to integral curves of vector elds of the form f +g where is a smooth function on M . Thus to make the state approach the sliding surface S r (where r is the relative degree of y) we are limited to the standard sliding mode controller and the problems associated with singular submanifolds. We seek instead to identify vector elds whose integral curves can be approached arbitrarily closely at discrete times by the state trajectory. If the deviations from the integral curve are \parallel" to S p for some p r we can use these state trajectories to implement sliding mode controllers for which singular manifolds do not pose a problem. Denition 3.1 The integral curves of a smooth vector eld X are said to be incrementally tracked by the state of (3) if there exist controllers fu n g with the following property: (a) each u n (x; t) is smooth with respect to x and is piecewise constant and periodic with respect to t with period (b) if (t) is an integral curve of X on [0; 1] , x n (t) the state trajectory when su-ciently large, a a a a Figure 2: Incremental Tracking of (t). While not essential, we will assume that vector elds are complete. Let I denote the set of vector elds on M whose integral curves can be incrementally tracked by the state of system (3) and I 0 the subset of I consisting of vector elds X with I for all smooth functions Theorem 3.2 The set of vector elds I and I 0 whose integral curves can be incrementally tracked by the state of (3) have the following properties: I 0 is a Lie Algebra over IR. If X 2 I; Y 2 I 0 then (iii) Suppose that Y 2 I and X; ad k+1 (a) If [ad i ad k (b) If ad 2k (iv) If ad k+1 is odd) and ad k f can be incrementally tracked by the state of (3) using the periodic switched controllers dened by: and Remark 3.3 For the linear system model _ of Theorem 3.2 implies that ad Repeating these steps with ad in place of etc. we nd that b; Ab; ; A n 1 b 2 I 0 and hence these constant vector elds can be incrementally tracked by the state. From this one can deduce the standard linear result on controllability. We also note that (ii) above implies that incremental tracking of the drift vector eld is preserved under smooth static state feedback. We also point out the fact that condition (iv) is nongeneric and will hold only for certain special systems. We are interested in incremental tracking where large deviations of the state trajectory from the integral curve have only a small eect on the output of the system. We now make this notion more precise: Denition 3.4 Suppose that > 0 and X is a vector eld on M whose integral curves can be incrementally tracked by the state fx n (t)g of (3) using controllers fu n g. If, for n su-ciently large, we say that the integral curves of X can be incrementally tracked preserving Let I p denote the set of vector elds on M whose integral curves can be incrementally tracked preserving h p and I p 0 the subset of I p consisting of vector elds X with We assume that p r. Theorem 3.5 The set of vector elds I p and I p 0 have the following properties: 0 is a Lie Algebra over IR. If X 2 I (iii) Suppose that Y 2 I p , X; ad k+1 Then (a) If [ad i k is odd). (b) If ad 2k (iv) If ad k+1 and the output of system (3) has relative order r > p then ad k Example 3.6 (Example 1.1 continued) Here we have 2. Thus condition (iv) of Theorem 3.5 holds and ad 2 Proof (Theorem 3.2) An integral curve of f can be tracked exactly using u 1. In this case the corresponding state trajectory x n hence f 2 I. Now let smooth and set t (x)n, and which approximates (t k ). In particular we can guarantee that jj (t k ) su-ciently large. This means that I hence g 2 I 0 . Note that in both of the above cases x n (t) stays close to (t) 8t 2 [0; 1]. Suppose that X;Y 2 I 0 , (t) is an integral curve for X+Y on [0; 1], and > 0. Then 2X; 2Y 2 I 0 and if ' > 0 we dene the \switched integral curve" It follows that Continuing to switch between integral curves of X and Y we get Here In particular, for ' su-ciently large, jj that given 0 > 0 there exist piecewise constant periodic wrt t controllers with period such that the integral curves of 2Y are incrementally tracked by the corresponding state trajectory x n (t). Thus we have jj 2Y k=n su-ciently large. In particular if we can arrange that su-ciently large. Similarly exists controllers fu 0 with period 0 n =n such that . Thus this concatenation of n g and fu n g results in a piecewise smooth state trajectory ~ x n which achieves n and n su-ciently large. Now we repeat the pattern (u n followed by u 0 n ) to generate a piecewise smooth state trajectory ~ x n for which jj (t k ) ~ applications of the triangle inequality). Thus we can choose to achieve incremental tracking of X +Y , hence X +Y 2 I. Now we can repeat the above argument using X;Y to conclude that (X To show that we argue as above. If ' > 0 then the \switched integral curve" (t) produced by following the integral curve for 'Y for 1=4' units of time, then the integral curve for 'X for 1=4' units of time, then the integral curve for 'Y followed by that of 'X. Then 'X 1=4' (4 'Y 1=4' (4 'X 1=4' (4 so that (Y 1= (Y 1= assuming xed (c.f. [18]). Continuing to switch between these integral curves we generate su-ciently large, where (t) is an integral curve for [X; Y ] on [0; 1], t they can be incrementally tracked using periodic switched controllers fu n g and fu 0 g. We then argue as above to show that Repeating these steps with p aY shows that a[X; Y hence Finally, suppose that X 2 I; Y 2 I 0 . Let (t) be an integral curve for positive integer, and > 0. Then I and we dene the \switched integral curve" t < 1=m' and Continuing to switch between integral curves of X and Y we get so, for ' and m su-ciently large jj Now repeat the argument used to show that I 0 is closed under sums to conclude that (iii) (a): Suppose that Y 2 I, X; ad k+1 by (t) the switched integral curve which results from following the integral curve for X for 1=n 2 units of time where then following the integral curve for Y for 1=n 2 units of time, and nally following the integral curve for X units of time. By construction Noting that ad i an absolutely convergent series for all t (c.f. [17, 18]), we see that where Since X and ad k+1 Algebra from (ii) above, it follows that In particular the integral curve for B (writing B; G for B(n); G(n)) can be incrementally tracked by the state. This means that the switched integral curve which results from following the integral curve for nB for time 1=n 2 followed by the switched integral curve (t) results in Using the Baker-Campbell-Hausdor Formula [17], which converges for n sufciently large, we have [G; From the denitions for G(n) and B(n) and in light of hypotheses (iii)(a) we hence 1 Tedious applications of the Jacobi identity show that 1 a consequence of hypotheses (iii)(a), and the same conclusion applies to the higher order terms in the Baker-Campbell-Hausdor series. In particular we see that Repeating ' times the switched integral curves used to generate we arrive at the state observe that ad k (t) is a switched integral curve of vector elds which can be incrementally tracked by the state of system (3). Furthermore if (x) as n ! 1. If (t) is the integral curve for ad k for n su-ciently large and switched between integral curves of vector elds which can be incrementally tracked. Thus we can repeat the argument used in (ii) above to show that there exist piecewise constant periodic controllers fu n g with periods that su-ciently large and This implies that ad k is odd we re[lace X with X and proceed as above to conclude that ad k I from which we deduce that ad k (iii)(b): This is a particular case of (iii)(a). This result is a consequence of (i) and (iii)(a). If we set as a consequence of (i). Since ad k+1 that ad 2k holds (and also (iii)(b)). In particular we can conclude that ad k can check that the controller u n dened in (iv) is precisely the one used in the proof of (iii)(a). A more direct approach to the proof of (iv) is illuminating and is outlined below. Using the control u n (t) dened in (iv) (and t save accounting) we have Applying the Baker-Campbell-Hausdor formula (c.f. [17]) two times we can In the case (with help of MAPLE V) the expression 72n 3=2 96n 7=2 Because ad 3 it is not hard to show that all terms in X(n) other than ad 2 are multiplied by negative powers of n. In particular lim n!1 X(n) =6 similar situation holds for other values of k , that is lim n!1 Repeating the above we nd that x n (3'=n 2 f is incrementally tracked by the state of system (3). Proof (Theorem 3.5) (i): We can track an integral curve (t) of f exactly using u This means that f 2 I p . As noted in the proof of Theorem 3.2 we can nd controllers u n such that the corresponding state trajectory x n closely follows the integral curves for g for all just for discrete times). Since the state trajectory x n makes no large deviation from the integral curve of g we have incremental tracking preserving h p . (ii): In the proof of Theorem 3.2 (ii) we saw that an integral curve (t) of can be tracked by switched integral curves of X and Y which stay close to (t) for all t 2 [0; 1]. Since X;Y 2 I p we can nd switched controllers n such that the corresponding state trajectories x incrementally track the integral curves of X and Y while preserving h p . Thus the image under h p of the concatenation of x used to incrementally track (t) will stay close to h p ((t)) and we will have incremental tracking of X preserving h p . The same situation holds in the case of [X; Y ]. (iii)(a): In the proof of Theorem 3.2 (iii)(a) we constructed switched integral curves of X and Y which incrementally track integral curves of ad k need not closely approximate these curves except at a discrete set of times. Thus the controllers u n produce state trajectories x n which incrementally track the integral curve making frequent and large deviations from (t). By construction these large motions are along integral curves for the vector elds X and we have (Xad k+1 LA we have Zh In particular B(n)h that the large motions of the state trajectory x n are in directions in which h p does not vary. Thus we achieve incremental tracking of (t) preserving h p . (iii)(b): This is a particular case of (iii)(a). gh (denition of relative order) we have gh and the result follows from (iii)(b) above. 4 Incremental Sliding Mode Controllers In the nonsingular case the simple sliding mode controller (2) gives rise to vector elds f + Lg and f Lg with several noteworthy properties. Given any compact subset C there exists L > 0; (i) On the set C Remark 4.1 Suppose that y d is a smooth function satisfying d on Condition (i) implies that if the state stays in C then the output will asymptotically track y d using the simplied controller (2). In particular if s p (x(t); t) > 0 (so that we are \above" the sliding surface d dt d dt d dt s(y r From the denition of s (s is linear) we have d d This, combined with our assumption that 1 s( _ y r d (t)) or d d (t)) 1 , yields d In particular the state trajectory returns to the sliding surface fs p (x; 0g. A similar situation results when s Remark 4.2 Condition (ii) follows from the denition of the relative order r, since gf i 1. That this is important in sliding mode control can be seen as follows: when the state \slides" on the sliding surface t the trajectory is the integral curve of the \equivalent vector eld" on S r which has the form [3]). Note that Xh r As a consequence along this integral curve the tracking error satises the stable dierential equation s(e r dened by (4). We seek to weaken the above in several ways. First we use the sliding surface S p t where p is allowed be smaller than the relative order r of y. As a consequence of Theorem 3.5 f Lg 2 I p . We relax (ii) by allowing vector elds of the form d I p such that d h only require (i) above to hold on an open subset of Z of M which is invariant under the integral curves of d We summarize these observations as follows: Denition 4.3 Let X be a vector eld on M . An open subset Z of M is said to be invariant with respect to a vector eld X if, for all x 2 Z, the integral curve t 7! X t (x) stays in Z. A2. There exists an open subset Z M invariant with respect to vector elds (i) On Z If A2 holds for constants the following restricted class of desired output functions: fy d j 1 s(y p We will show that these outputs can be approximately tracked. We note that in the nonsingular case A2 holds with for L su-ciently large. If A2 holds with d 2 I p we dene the set-valued map F d (x; t) by F d (x; d where co fd is the closed convex hull generated by the fd Theorem 4.4 Suppose A1, A2 hold for system (3). Then there exist d I p and an open subset Z M such that for all smooth functions y (i) the dierential inclusion _ has a unique solution (ii) for any solution x f to _ Z \ S p (iii) for t t f the curve t 7! y F is a smooth function of t which In particular lim t!1 (y p Proof By construction, F d (x; t) is nonempty, compact and convex and it is straightforward to show that F d is upper semicontinuous with respect to x; t. Thus the basic conditions of [3, p.76] are satised the proof that locally solutions to the dierential inclusion _ exist can be found in [3, pp. 67-68 and pp. 77-78], and is omitted here. That solutions stay in Z follows from A2 i.e. the assumption that Z is strongly invariant with respect to d To establish uniqueness we note that both d + and d are transversal to S P t \ Z as a consequence of A2 (i). Furthermore the limiting vector elds on S P t \ Z which result from d the opposite orientations on S P t \Z. Thus [3, Corollary 2, p.108] implies that there is exactly one solution to this dierential inclusion starting at x(t 0 (ii) Suppose that y d 2 Y d and s p Then, from the deni- tion of Y d , we have 1 s( _ A2(i). Thus d d and s p (x; t) is strictly increasing along integral curves of d strictly decreasing along integral curves of d in fs p > 0g. by (i) we have established (ii). (iii): For t t f x F is a smooth integral curve for the equivalent vector eld X dened in Remark 4.2. Here Xh a consequence of A2 (ii), hence y 1. From Section 2 we know that if y equivalent to s p (x(t); t . In particular, since x F (t) 2 S p t from (ii), we have s(e p A necessary condition for approximate tracking of y d is that both y p d and the state trajectory remain bounded. In the nonsingular case the state trajectory and the solution to the dierential inclusion _ are identical and it su-ces to ensure that solutions to _ remain bounded. In our case the same assumption su-ces. A3. Suppose that A2 holds for system (3) and y d 2 Y d . Then solutions to the dierential inclusion _ with initial state x(t 0 remain bounded for Remark 4.5 Note that in light of Theorem 4.4 (ii) it su-ces to study the trajectory on S p t . Since there is a unique vector eld G(x; t) in co fd that makes @ tangent to S p it su-ces to check that this one integral curve is bounded. A su-cient (but far from necessary) condition for A3 to hold is that Z \ S p t be bounded. Suppose that A1, A2, A3 hold for system (3) with initial state x(t 0 Z, where Z is an open subset of M invariant with respect to vector elds . If we could make the state of (3) exactly track the solution x F (t) to _ Theorem 4.4 would imply asymptotic tracking of y d . We now describe a \digital controller" which allows us to incrementally track x F and approximately track y d . We are motivated by the typical \sample and hold" digital controller with xed sample rate T . That is, if u(x; t) is a smooth function of x and t the digital controller u k (t) takes the form is the state at time t k which results from using the control u k on the time interval We have controllers u are piecewise constant periodic functions of t with periods n =n and respectively, and which cause the state of (3) to incrementally track integral curves of d respectively. Thus we require a digital controller with variable sampling rate. We dene our digital controller for the system (3) as follows: We observe that while u k (x; t) is not constant with respect to t over it is piecewise constant due to the piecewise constant time dependence of u and u n . Theorem 4.6 Under assumptions A1, A2, A3 the switched controller (11) achieves the following property for the closed loop system: if x(t 0 and y d 2 Y d then, for n su-ciently large, the output y of (3) approximately tracks y d to degree p. Proof Let x F (t) denote the solution to the dierential inclusion _ From Theorem 4.4 there exists t f t 0 such that x F (t) 2 Z\S p This implies that x F light of A3, x F (t) is a bounded function of t. We rst consider the case where =n. The vector eld d + is incrementally tracked by the state trajectory produced by u n . We now calculate the rate of change of s p (x(t); t) when x(t) is the integral curve x F (t) of d + but time t is rescaled to match the time rescaling which occurs in incremental tracking. For t < t f we have, from A2 and the linearity of s, d as 0 < n 1. Thus there is some least time t 1 > 0 such that s p positive integer k 1 (depending on n) such that x F We can make jjx F arbitrarily small by increasing n and hence for n su-ciently large. Since x F (t) is incrementally tracked by x n (t) we have 1. Therefore by picking large enough we ensure that jjs p In particular, using the \digital" controller (11) results in a state trajectory x n (t) for (3) with the property that , for n su-ciently large, s )jj < =2. Thus u We can now repeat the above starting from the initial state x(t ' 1 incrementally track the integral curve of d )jj < =2 (increasing n if neces- sary). Because the integral curve x F (t) is bounded we can choose n su-ciently large to continue the above switching and ensure that the state trajectory x n resulting from the controller (11) satises Incremental tracking ensures that x n is close to x F at discrete times ft k g but for t k < t < t k+1 we may have x n (t) far from x F (t). We now use the fact that d 2 I p , and thus are incrementally tracked preserving h p , to show that s p is unaected by these deviations. In particular on [t by denition of incrementally tracking. This allows us to ensure that jjs p su-ciently large. Because 1 this implies that, for n su-ciently large, and x n (t) is bounded, hence the output y of (3) approximately tracks y d to degree p. Let R(x 0 ) denote the set of states which can be reached from the initial state x(t 0 Theorem 4.6 ensures approximate tracking if x 0 2 Z and so it is natural to look for a controller which steers x 0 to the open set Z in nite time. It will often be the case that R(x 0 ) \ Z 6= ;. In particular we need to use the above theorem when the state trajectory \sticks" to the singular submanifold under the naive truncated sliding mode controller. Thus if Z intersects the singular submanifold it is likely reachable from the initial state. Suppose that C is compact, Z an open subset of M , and u 0 (x; t) is a controller for system (3) which transfers the state from x(t 0 to the hybrid controller where u k is the digital controller (11) and k 1. Theorem 4.7 Suppose that A1, A2, A3 hold, C M is compact, and there exists a controller u 0 (x; t) which transfers the state of system (3) to Z \C at su-ciently large, the hybrid switched controller (12) achieves the following property for the closed loop system: if y d 2 Y d then the output y of (3) approximately tracks y d to degree p. Proof For an initial state x(t 1 Theorem 4.6 implies that, for su-ciently large, the controller (12) achieves approximate tracking of y d . From the continuity of solutions to _ with respect to the initial conditions (c.f. [3]) we have approximate tracking of y d for any initial state in some open neighbourhood U 1 of x 1 . Because Z \ C is compact we can obtain a nite open covering [ m U i of Z \ C by such open sets. Thus the hybrid switched controller (12) with n maxfn i results in approximate tracking of y d . Remark 4.8 We note that the hypotheses of Theorem 4.7 are satised for for a-ne systems whose singular set fgf r 1 empty. In this case we use and u 0 is not needed. To verify them for a given a system model one could start by using Theorems 3.2, 3.5 to nd vector elds which the state trajectory can incrementally track. If the natural sliding mode controller has a singular submanifold N , check to see if the vector elds which can be incrementally tracked preserving the output map are su-cient for A2 to hold. Then, If A3 if holds as well (see Remark 4.5), Theorems 4.6, 4.7 yield a controller. Example 1.1 is a case in point. Example 4.9 (Example 1.1 continued) We have seen that f; ad 2 so it is natural to choose a sliding surface with 1. We set (t)g. Clearly the set S 1 t is an embedded submanifold dimensional) for each xed time t so that A1 holds. Here gs(h p 2. To satisfy A2(i) we want (ad 2 some open set Z so it is natural to look for a set invariant with respect to f; ad 2 f and on which x 2 q 2 > 0, x 3 0. For many systems a systematic approach to nding a suitable subset Z may not be possible but for the example under consideration x linear dierential equation. Thus we can nd such a set by constructing a Lyapunov function. In particular let z 1 a 0 a 1 We can nd a Lyapunov function solving Lyapunov's Equations A T I for the positive denite matrix a b where for where q 0 > 0. By construction Z(q 0 ) is invariant with respect to f (a 0 z a where 0 and f 2 I p we have I p as a consequence of Theorem 3.2. Since by construction V is decreasing along the integral curves of d we have Z(q 0 ) invariant with respect to d . Because Z(q 0 ) puts no restrictions on x 1 it is also invariant with respect to d To verify that A2 holds we rst note that s(h p Note that by shrinking q 0 we can ensure that in the set Z(q 0 ) we have x 3 arbitrarily close to 0 and x 2 arbitrarily close to q 2 . In particular, given any constants choose such that on the set Thus A2(i) holds and A2(ii) holds automatically as In light of Remark 4.5 assumption A3 will hold if Z\S p t is bounded. Here is a bounded set by construction and hence A3 holds. Thus A1, A2, A3 hold and Theorem 4.7 implies that we can approximately track to degree 1 the set of output paths fy d j 1 s(y p The construction of the controller u 0 which moves the state into Z is simplied here because Z is the level set fV of a Lyapunov function for _ g. We set u 0 (x; (x). For any x(t 0 ) there will be incrementally track d using u n (x; incrementally track d + using the controller from Theorem 3.2(iv) with namely . If we want y d we can dene Z by choosing q To ensure close tracking we pick 0:1. Figure 3 shows a SIMNON simulation using the controller (12) with Z. The tracking performance is not particularly sensitive to variations in these parameters. Increasing n gives tighter tracking but requires more control eort. y, yd Figure 3: Approximate Tracking of a y d In Figure 4 we show the eect of an initial state which is initially well outside of Z Figure 4: Approximate Tracking of a y d We note that in this situation state trajectories resulting from controllers based on relative degree will stick to the singular manifold send s(e r (t)) !1. Our approach has the state passing back and forth across N . The initial delay is due to the requirement that the state must enter Z before our switched controller can act to reduce s. Conclusions There are situations where it useful to be able to control the state of a system so that it closely approaches a given curve at discrete times. We have introduced the concept of incremental tracking of integral curves where the state trajectory (with re-parametrized time) closely approaches an integral curve at discrete times. These controllers were then applied to sliding mode control where the state trajectory used to reach the sliding surface is not very criti- cal. Our discontinuous \digital sliding mode controller" achieved approximate tracking in situations where the natural truncated sliding mode controller (and the natural truncated smooth controller based on inversion) fails. --R Characteristic phenomena and model problems in non-linear control On the Singular Tracking Problem Nonlinear control via approximate input-output linearization: The ball and beam example Global Sliding Mode Control Global approximate output tracking for nonlinear systems Nonlinear Control Systems On Tracking through singular- ities: regularity of the control Limits of highly oscillatory controls and the approximation of general pathes by admissible trajectories Nonlinear Dynamical Control Systems On the curves that may be approached by trajectories of a smooth a-ne system Applied Nonlinear Control Sliding Controller Design for Nonlinear Systems Sliding Modes in Control Optimization Foundations of di --TR

output tracking;discontinuous state feedback;sliding mode control;lie brackets;singularities

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On a Boundary Control Approach to Domain Embedding Methods.

In this paper, we propose a domain embedding method associated with an optimal boundary control problem with boundary observations to solve elliptic problems. We prove that the optimal boundary control problem has a unique solution if the controls are taken in a finite dimensional subspace of the space of the boundary conditions on the auxiliary domain.Using a controllability theorem due to J. L. Lions, we prove that the solutions of Dirichlet (or Neumann) problems can be approximated within any prescribed error, however small, by solutions of Dirichlet (or Neumann) problems in the auxiliary domain taking an appropriate subspace for such an optimal control problem. We also prove that the results obtained for the interior problems hold for the exterior problems. Some numerical examples are given for both the interior and the exterior Dirichlet problems.

Introduction The embedding or fictitious domain methods which were developed specially in the seventies ([5], [2], [34], [35], [28] or [13]), have been a very active area of research in recent years because of their appeal and potential for applications in solving problems in complicated domains very e#ciently. In these methods, complicated domains # where solutions of problems may be sought, are embedded into larger domains# with simple enough boundaries so that the solutions in this embedded domains can be constructed more e#ciently. The use of these embedding methods are a commonplace these days for solving complicated problems arising in science and engineering. To this end, it is worth mentioning the domain embedding methods for Stokes equations (Borgers [4]), for fluid dynamics and electromagnetics (Dinh et. al. [11]) and for the transonic flow calculation (Young et. al. [36]). In [3], an embedding method is associated with a distributed optimal control problem. There the problem is solved in an auxiliary domain# using a finite element method on a fairly structured mesh which allows the use of fast solvers. The auxiliary domain# contains the domain # and the solution in# is found as a solution of a distributed optimal control problem such that it satisfies the prescribed boundary conditions of the problem in the domain #. The same idea is also used in [9] where a least squares method is used. In [12], an embedding method is proposed in which a combination of Fourier approximations and boundary integral equations is used. Essentially, a Fourier approximation for a solution of the inhomogeneous equation in# is found, and then, the solution in # for the homogeneous equation is sought using the boundary integral methods. # Institute of Mathematics, Romanian Academy of Sciences, P.O. Box 1-764, RO-70700 Bucharest, Romania (e-mail: Department of Mathematics, Texas A&M University, College Station, TX-77843, USA (e-mail: prabir.daripa@math.tamu.edu) In recent years, progress in this field has been substantial, especially in the use of the Lagrange multiplier techniques. In this connection, the works of Girault, Glowinski, Hesla, Joseph, Kuznetsov, Lopez, Pan, Periaux ([14], [15], [16], [17] and [18]) should be cited. There are many problems for which an exact solution on some particular domains may be known or computed numerically very e#ciently. In these cases, an embedding domain method associated with a boundary optimal control problem would allow one to find the solution of the problem very e#ciently in an otherwise complicated domain. Specifically, the particular solution of the inhomogeneous equation can be used to reduce the problem to solving an homogeneous equation in # subject to appropriate conditions on the boundary of the domain #. This solution in the complicated domain # can be obtained via an optimal boundary control problem where one seeks for the solution of the same homogeneous problem in the auxiliary domain# that would satisfy the boundary conditions on the domain #. We mention that the boundary control approach has been already used by Makinen, Neittaanmaki and Tiba for optimal shape design and two-phase Stefan-type problems ([29], [32]). Also, recently there has been an enormous progress in shape optimization using the fictitious domain approaches. We can cite here, for instance, the works of Haslinger, Klarbring, Makinen, Neittaanmaki and Tiba (see [8], [21], [22], [23] and [33]) among many others. In section 2, an optimal boundary control problem involving an elliptic equation is formulated. In this formulation, the solution on the auxiliary domain# is sought such that it satisfies the boundary conditions on the domain #. In general, such an optimal control problem leads to an illposed problem, and consequently it may not have a solution. Using a controllability theorem of J. L. Lions, it is proved here that the solutions of the problems in # can be approximated within any specified error, however small, by the solutions of the problems in# for appropriate values of the boundary conditions. In section 3, it is shown that our optimal control problem has an unique solution in a finite dimensional space. Consequently, considering a family of finite dimensional subspaces having their union dense in the whole space of controls, we can approximate the solution of the problem in # with the solutions of the problems in# using finite dimensional optimal boundary control problems. Since the values of the solutions in are approximately calculated on the boundary of the domain #, we study the optimal control problem with boundary observations in a finite dimensional subspace in section 4. In section 5, we extend the results obtained for the interior problems to the exterior problems. In section 6, we give some numerical examples for both bounded and unbounded domains. The numerical results are presented to show the validity and high accuracy of the method. Finally, in section 7 we provide some concluding remarks. There is still a large room for further improvement and numerical tests. In a future work, we will apply this method in conjunction with a fast algorithm ([6], [7]) to solve other elliptic problems in complicated domains. Controllability Let (i.e. the maps defining the boundaries of the domains and their derivatives are Lipschitz continuous) be two bounded domains in R N such that Their boundaries are denoted by # and #, respectively. In this paper, we use domain embedding and optimal boundary control approach to solve the following elliptic equation: subject to either Dirichlet boundary conditions or Neumann boundary conditions #y #nA (#) h # on #, (2.3) #nA (#) is the outward conormal derivative associated with A. We assume that the operator A is of the form with a ij # C (1),1 ( # and there exists a constant c > 0 such that in# for any (# 1 , , # N ) # R N . Also, we assume that f # L A function y # H 1/2 (#) will be called a solution of the Dirichlet problem (2.1)-(2.2) if it verifies equation (2.1) in the sense of distributions and the boundary conditions (2.2) in the sense of traces in L 2 (#). A function y # H 1/2 (#) will be called a solution of the Neumann problem (2.1), (2.3) if it verifies the equation (2.1) in the sense of distributions and the boundary conditions (2.3) in the sense of traces in H -1 (#) (see [27], Chap. 2, 7). The Dirichlet problem (2.1)-(2.2) has an unique solution and it depends continuously on the data If there exists a constant c 0 > 0 such that a 0 # c 0 in #, then the Neumann problem (2.1), (2.3) has an unique solution and it depends continuously on the data If a #, then the Neumann problem (2.1), (2.3) has a solution if In this case the problem has an unique solution in H 1/2 (#)/R and We also we remark that the solution of problem (2.1)-(2.2) can be viewed (see [27], Chap. 2, as the solution of the problem #n A #) for any # H 2 and that a solution of problem (2.1), (2.3) is also solution of the problem for any # H 2 (# #n A #) where A # is the adjoint operator of A given by (a ji Evidently, the above results also hold for problems in the domain# We consider in the following only the cases in which our problems have unique solutions, i.e. the Dirichlet problems, and we assume in case of the Neumann problems that there exists a constant such that a 0 # c 0 in # Below we use the notations and the notions of optimal control from Lions [26]. First, we will study the controllability of the solutions of the above two problems (defined by (2.1) through (2.3)) in # with the solutions of a Dirichlet problem in # Let be the space of controls. The state of the system for a control v # L 2 (#) will be given by the solution 1/2(# of the following Dirichlet problem in# In the case of the Dirichlet problem (2.1)-(2.2), the space of observations will be and the cost function is given by and y(v) is the solution of problem (2.11). For the Neumann problem given by and (2.3), the space of observations will be and the cost function will be Remark 2.1 Since y(v) # H 2(#5 we have y(v) # H 2 (D) for any domain D which satisfies # (see [30], Chap. 4, 1.2, Theorem 1.3, for instance). Therefore, having the same values on both the sides of #. Also, #y(v) #nA (#) Consequently, above two cost functions make sense. Proposition 2.1 A control u # L 2 (#) satisfies where the control function is given by (2.13), if and only if the solution of (2.11) for #nA (#) and where y is the solution of the Dirichlet problem defined by (2.1) and (2.2) in the domain #. The same result holds if the control function is given by (2.15) and y is the solution of the Neumann problem (2.1) and (2.3). Proof. Let 1/2(# be the solution of problem (2.11) corresponding to an u # L 2 (#) such that with the control function given by (2.13). Consequently, y(u) verifies equation (2.1) in the sense of distributions and the boundary condition (2.2) in the sense of traces. It gives in #. Since y(u) satisfies equation (2.11) # in the sense of distributions, then, evidently, y(u) is a solution the equation in (2.16). From (2.17) and Remark 2.1 we obtain that y(u) also satisfies the two boundary conditions of (2.17). The reverse implication is evident. The same arguments also hold for the Neumann problem defined by (2.1) and (2.3) and the control function given by (2.15). # Since (2.16) is not a properly posed problem, it follows from the above proposition that the optimal control might not exist. However, J. L. Lions proves in [26] (Chap. 2, 5.3, Theorem 5.1) a controllability theorem which can be directly applied to our problem. We mention this theorem below. Lions Controllability Theorem The set { #z0 (v) is dense in H -1 (#), #) is the solution of the problem z z Now, we can easily prove Lemma 2.1 For any g # L 2 (#), the set { #z(v) is dense in H -1 (#) #) is the solution of the problem on #. Proof. Let z # H #) be the solution of the problem on #. Using z z in the Lions controllability Theorem, we get that the set { #(z(v)-z) is dense in H -1 (#). Hence, the lemma follows. # The following theorem proves controllability of the solutions of problems in # by the solutions of Dirichlet problems in # In the proof of this theorem below, we use the spaces # s introduced in Lions and Magenes [27], Chap. 2, 6.3. For the sake of completeness, we give here definitions of these spaces # s . Let #(x) be a function in D( # ) which is positive in# and vanishes on #. We also assume that for any x 0 #, the following limit lim exists and is positive, where d(x, #) is the distance from x # to the boundary #. Then, for , the space # s is defined by With the norm |#s the space # s(# is a Hilbert space, and D(# is dense in # Now, for a positive non-integer real the integer part of s and 0 < # < 1, the space # s is, as in case of the spaces H s , the intermediate space Finally, for negative real values -s, s > 0, the space # -s(# is the dual space of # Theorem 2.1 The set {y(v) |# : v # L 2 (#)} is dense, using the norm of H 1/2 (#), in {y # H 1/2 (#) f in #}, where y(v) # H 1/2(# is the solution of the Dirichlet problem (2.11) for a given v # L 2 (#). Proof. Let us consider y # H 1/2 (#) such that Ay = f in #, and a real number # > 0. We denote the traces of y on # by #nA (#) (#). From the previous lemma, it follows that there exists v # L 2 (#) such that the solution z(v # H - #) of problem (2.18) satisfies | #) be the solution of the Dirichlet problem (2.11) corresponding to v # and let us define y # y on # #. 1/2(# and satisfies in the sense of distributions the equation in# and the boundary conditions Consider, as in Remark 2.1, a fixed domain D such that D(#2 we have 3/2(#, where C(D) depends only on the domain D. Therefore, Taking into account the continuity of the solution on the data (see Lions and Magenes [27], Chap. 2, 7.3, Theorem 7.4), we get Below, the controllability of the solutions of the Dirichlet and the Neumann problems (given by (2.1),(2.2) and (2.1), (2.3) respectively) in # by Neumann problems in# is discussed. Now, as a set of controls we can take the space and for a v # H -1 (#), the state of the system will be the solution y(v) # H 1/2(# of the problem in# = v on #. (2.20) We remark that the following solution of problem (2.11)} # solution of problem (2.20)} establish a bijective correspondence. Consequently, Proposition 2.1 also holds if the space of controls there is changed to H -1 (#) and the states y(v) of the system are solutions of problem (2.20). Theorem 2.1 in this case becomes Theorem 2.2 The set {y(v) |# : v # H -1 (#)} is dense, using the norm of H 1/2 (#), in {y # 1/2(# is a solution of the Neumann problem (2.20) for a 3 Controllability with finite dimensional spaces Let {U # be a family of finite dimensional subspaces of the space L 2 (#) such that given (2.10) as a space of controls with the Dirichlet problems, we have U # is dense in For a v # L 2 (#) we consider the solution y 1/2(# of the problem in# We fix an U # . The cost functions J defined by (2.13) and (2.15) are di#erentiable and convex. Consequently, an optimal control v#U# exists if and only if it is a solution of the equation when the control function is (2.13), and #nA (#) , #y # (v) #y # (v) for any v # U # , (3.5) when the control function is (2.15). Above, y(u # ) is the solution of problem (2.11) corresponding to (v) is the solution of problem (3.2) corresponding to v. If y f # H 2(# is the solution of the problem in# then, for a v # L 2 (#), we have where y(v) and y # (v) are the solutions of problems (2.11) and (3.2), respectively. Therefore, we can rewrite problems (3.4) and (3.5) as and #nA (#) for any v # U # , respectively. Next, we prove the following Lemma 3.1 For a fixed #, let # 1 , , #n# , n # N be a basis of U # and y # i ) be the solution of problem (3.2) for { #y #1 ) #nA (#y #n # ) #nA (#} are linearly independent sets. Proof. From Remark 2.1, we have y # (v) # H 2 (D) for any domain D which satisfies D # and consequently, y # (v) # H 3/2 (#) for any v # L 2 (#). Assume that for # 1 , , # n# R we have and therefore y on #. This implies that #y #1 #1++#n #n # ) From (3.10) and (3.11), we get y #, and therefore, # 1 on #, or # The second part of the statement can be proved using similar arguments. # The following proposition proves the existence and uniqueness of the optimal control when the states of the system are the solutions of the Dirichlet problems. Proposition 3.1 Let us consider a fixed U # . Then, problems (3.8) and (3.9) have unique solutions. Consequently, if the boundary conditions of Dirichlet problems (2.11) lie in the finite dimensional space U # , then there exists an unique optimal control of problem (3.3) corresponding to either the Dirichlet problem (2.1), (2.2) or the Neumann problem (2.1), (2.3). Proof. For a given #, let V # denote the subspace of L 2 (#) generated by {y is a basis of U # , and y # i ) is the solution of problem (3.2) with Since the norms are equivalent to the norm the above lemma then implies that there exists two positive constants c and C such that Consequently, from the Lax-Milgram lemma we get that equation (3.8) has an unique solution. A similar reasoning proves that equation (3.9) also has an unique solution. This time we use the norm equivalence #y # (v) #) | H in the Lax-Milgram lemma. # The following theorem proves the controllability of the solutions of the Dirichlet and Neumann problems in # by the solutions of the Dirichlet problems in # Theorem 3.1 Let {U # be a family of finite dimensional spaces satisfying (3.1). We associate the solution y of the Dirichlet problem (2.1), (2.2) in # with problem (3.3) in which the cost function is given by (2.13). Also, the solution y of the Neumann problem (2.1), (2.3) will be associated with problem (3.3) in which the cost function is given by (2.15). In both the cases, there exists a positive constant C such that for any # > 0, there exists U # such that where u # U # is the optimal control of the corresponding problem (3.3) with # , and y(u # ) is the solution of problem (2.11) with Proof. Let us consider an # > 0 and y # H 1/2 (#) be the solution of problem (2.1), (2.2). From Theorem 2.1, there exists v # L 2 (#) such that y(v # H 1/2(#5 the solution of problem (2.11) with (#. Consequently, there exists a constant C 1 such that U # is dense in L 2 (#), there exists # and v # U # such that |v # - v # | L 2 (# and then, there exists a positive constant C 2 such that From (3.12) and (3.13) we get and consequently, where u # L 2 (#) is the unique optimal control of problem (3.3) on U # with the cost function given by (2.13). Therefore, A similar reasoning can be made for the solution y # H 1/2 (#) of problem (2.1), (2.3). # Using the basis # 1 , , #n# of the space U # we define the matrix and the vector Then problem (3.8) can be written as #,1 , , #,n# R Consequently, using Theorem 3.1, the solution y of problem (2.1), (2.2) can be obtained within any prescribed error by setting the restriction to # of where #,1 , , #,n# ) is the solution of algebraic system (3.16). Above, y f is the solution of problem (3.6) and y # i ) are the solutions of problems (3.2) with An algebraic system (3.16) is also obtained in the case of problem (3.9). This time the matrix of the system is given by #nA (#) , #nA (#) and the free term is #y f #nA (#) , #nA (#) Therefore, using Theorem 3.1, the solution y of problem (2.1), (2.3) can be estimated by (3.17). Also, y f is the solution of problem (3.6) and y # i ) are the solutions of problems (3.2) with The case of the controllability with finite dimensional optimal controls for states of the system given by the solution of a Neumann problem will be treated in a similar way. As in the previous section the space of the controls will be U given in (2.19), and the state of the system y(v) # H be given by the solution of Neumann problem (2.20) for a v # H Let {U # be a family of finite dimensional subspaces of the space H -1 (#) such that U # is dense in This time, the function y # (v) # H appearing in (3.4), (3.5), (3.8) and (3.9) will be the solution of the problem in# #y # (v) = v on #, (3.21) for a v # H appearing in (3.7), (3.8) and (3.9) will be the solution of the problem in# #y f With these changes, Lemma 3.1 also holds in this case, and the proof of the following proposition is similar to that of Proposition 3.1. Proposition 3.2 For a given U # the problems (3.8) and (3.9) have unique solutions. Consequently, if the boundary conditions of Neumann problems (2.20) lie in the finite dimensional space U # , then there exists an unique optimal control of problem (3.3), corresponding to either Dirichlet problem (2.1), (2.2), or Neumann problem (2.1), (2.3). A proof similar to that given for Theorem 3.1 can also be given for the following theorem. Theorem 3.2 Let {U # be a family of finite dimensional spaces satisfying (3.20). We associate the solution y # H 1/2 (#) of problem (2.1), (2.2) with problem (3.3) in which the cost function is given by (2.13). Also, the solution y of problem (2.1), (2.3) will be associated with problem (3.3) in which the cost function is given by (2.15). In both the cases, there exists a constant C such that for any # > 0, there exists # such that where u # U # is the optimal control of the corresponding problem (3.3) with # , and y(u # ) is the solution of problem (2.20) with Evidently, in the case of the controllability with solutions of Neumann problem (2.20) we can also write algebraic systems (3.16) using a basis # 1 , , #n# of a given subspace U # of the space (#). As in the case of the controllability with solutions of the Dirichlet problem (2.11), these algebraic systems have unique solutions. Remark 3.1 We have defined y f as a solution of problems (3.6) or (3.22) in order to have , respectively, on the boundary #. In fact, we can replace y(v) by y # (v) in the cost functions (2.13) and (2.15), with y f # H satisfying only in# , (3.23) and the results obtained in this section still hold. Indeed, the two sets corresponding to y f given by (3.23) and (3.6), y # (v) being the solution of (3.2), are identical to the set {y(v) # H being the solution of (2.11). Also, the two sets corresponding to y f given by (3.23) and (3.22), y # (v) being the solution of (3.21), are identical with the set {y(v) # H being the solution of (2.20). 4 Approximate observations in finite dimensional spaces In practical computing, we calculate the values of y # (v) at some points on # and use in (3.8), some interpolations of these functions. We will see below that using these interpolations, i.e. observations in finite dimensional subspaces, we can still obtain the approximate solutions of problems (2.1), (2.2) and (2.1), (2.3). As in the previous sections we deal at first with the case when the states of the system will be given by the Dirichlet problem (2.11). Let U # be a fixed finite dimensional subspace of the basis # 1 , , #n# . Let us assume that for problem (2.1), (2.2), we choose a family of finite dimensional spaces {H } such that H is dense in Similarly, we choose the finite dimensional spaces {H } such that H is dense in for problem (2.1), (2.3). We notice that H given in (4.1) or (4.2) is a subspace of H given in (2.12) or (2.14), respectively. Let us consider a fixed H , given in (4.1) or (4.2) depending on the problem we have to solve. For a given # i , we will consider the solution y problem (3.2) corresponding to and we will approximate its trace on # by y # ,i . Also, the approximation of #y #nA (#) on # will be denoted by #y # #nA (#) Since the systems (3.16) have an unique solution, the determinants of the matrices # given in and (3.18) are non-zero. Consequently, if |y # i ,i | L 2 (#) or | #y are small enough, then the matrices and #nA (#) , #y # have non-zero determinants. In this case, the algebraic systems have unique solutions. In this system the free term is if the matrix # is given by (4.3), and #y f #nA (#) , #y # ,i if the matrix # is given by (4.4). Above, we have denoted by g # and h # some approximations in H of g # and h # , respectively. Also, y f and #y f #nA (#) are some approximations of y f and #y f #nA (#) in the corresponding H of L 2 (#) and H -1 (#), respectively, with y f # H satisfying (3.23). If we write for a vector # 1 , , # n# R n# , and #y # (#) #nA (#) problems analogous to (3.8) and (3.9) can be written as and #nA (#) (#) #nA (#) for any # R n# , whose solutions # are the optimal control for the following cost functions J (#) =2 |y # and J (#) =2 #y # (#) #nA (#) respectively. The solution y of problems (2.1), (2.2) and (2.1), (2.3) can be approximated with the restriction to # of #,1 , , #,n# ) being the solution of appropriate algebraic system (4.5). For a vector, # 1 , , # n# ), we will use the norm | and the corresponding matrix norm will be denoted by || ||. From (3.17) and (4.14) we have depends only on the basis in U # . Since and from algebraic systems (3.16) and (4.5) we have # l # and l # , we get that there exists C # > 0, depending on the basis in U # , such that In the case of matrices (3.14) and (4.3) and the free terms (3.15) and (4.6), we have Instead, if we take matrices (3.18) and (4.4) and the free terms (3.19) and (4.7), then we get 1#i#n# | #y # ,i #y f #y f 1#i#n# | #y # ,i where C is a constant and C # depends on the basis in U # . In the case when the states of the system will be given by the Neumann problem (2.20), U # will be a subspace of what we said above in the case of the Dirichlet problems in# can be applied in the case of the Neumann problems in # the only di#erence being that this time are the solutions of problems (3.21) with In both cases, when the control is e#ected via Dirichlet and Neumann problems, using Theorems 3.1 and 3.2, and equations (4.15)-(4.18), we obtain Theorem 4.1 Let {U # be a family of finite dimensional spaces satisfying (3.1) if we consider problem (2.11), or satisfying (3.20) if we consider problem (2.20). Also, we associate problem (2.1), (2.2) or (2.1), (2.3) with a family of spaces {H } satisfying (4.1) or (4.2), respectively. Then, for any # > 0, there exists # such that the following holds. (i) if the space H is taken such that |y # i ,i | L 2 (#) , are small enough, y is the solution of problem (2.1), (2.2) and y(u # ) is given by (4.14) in which # is the solution of algebraic system (4.5) with the matrix given in (4.3) and free term in (4.6) then the algebraic system (4.5) has an unique solution and ,i | L 2 (# , (ii) if the space H is taken such that | #y are small enough, y is the solution of problem (2.1), (2.3) and y(u # ) is given by (4.14) in which # is the solution of algebraic system (4.5) with the matrix given in (4.4) and free term in (4.7) then the algebraic system (4.5) has an unique solution and #nA (#y f where C is a constant and C # depends on the basis of U # . Remark 4.1 Since the matrices # given in (4.3) and (4.4) are assumed to be non-singular, it follows that {y # ,i } i=1,,n # and { #y # ,i are some linearly independent sets in L 2 (#) and respectively. Consequently, if m is the dimension of the corresponding subspace H , then 5 Exterior problems In this section, we consider the domain # R N of problems (2.1), (2.2) and (2.1), (2.3) as the complement of the closure of a bounded domain and it lies on only one side of its boundary. The same assumptions will be made on the domain# of problems (2.11) and (2.20), and evidently, # In order to follow the way in the previous sections and to prove that the solutions of the problems in # can be approximated by the solutions of problems in# we have to specify the spaces in which our problems have solutions and also, their correspondence with the trace spaces. First, we notice that the domain# - being bounded, then the Lions controllability Theorem does not need to be extended to unbounded domains. Moreover, we see that the boundaries # and # of the domains # and# are bounded, and consequently, we can use finite open covers of them (as for the bounded domains), to define the traces. In order to avoid the use of the fractional spaces of the spaces in # and# we simply remark that since H 1/2 (#) is dense in L 2 (#), then using the continuity of the solution on the data (of the and the continuity of the conormal derivative operator # on the boundary #, we get from the Lions controllability Theorem that The set #z0 (v) #) is the solution of the problem z z Now, we associate to the operator A the symmetric bilinear form a(y, #y #z # a 0 yz for y, z # H which is continuous on H 1(#1 Evidently, a is also continuous on H 1 (#) H 1 (#). Now, and taking the boundary data g # H 1/2 (#) and h # H -1/2 (#), then problems (2.1), (2.2) and (2.1), (2.3) can be written in the following variational form fz for any z # H 1(#) and z for any z # H 1 (#), (5.2) respectively. Similar equations can also be written for problems (2.11) and (2.20). Therefore, if there exists a constant c 0 > 0 such that a 0 # c 0 in # then the bilinear form a is 1(#6842056# i.e. there exists a constant # > 0 such that #|y| 2 a(y, y) for any y # H 1(#5 It follows from the Lax-Milgram lemma that problems (2.11) and (2.20) have unique weak solutions in 1(#6 Naturally, problems (2.1), (2.2) and (2.1), (2.3) in # also have unique weak solutions given by the solutions of problems (5.1) and (5.2), respectively. We know that there exits an isomorphism and homeomorphism of H (see Theorem 7.53, p. 216, in [1], or Theorem 5.5, p. 99, and Theorem 5.7, p. 103, in [30]), i.e. there are two constants such that we have . For any y # H 1(# , there exists v # H 1/2 (#) such that y | H . . For any v # H 1/2 (#), there exists y # H 1(# such that y = v on # and | y | H Using this correspondence we can easily prove the continuous dependence of the solutions on data. For instance, for problems (2.1), (2.2) and (2.1), (2.3) we have and respectively. Therefore, if there exists a constant c 0 > 0 such that a 0 # c 0 in # then we can proceed in the same manner and obtain similar results for the exterior problems to those obtained in the previous sections for the interior problems. Evidently, in this case we take as a space of the controls for problem (2.11), in place of that given in (2.10), and the space of controls for problem (2.20) will be taken as in place of the space given in (2.19). If a in # the domain being unbounded, then our problems might not have solutions in the classical Sobolev spaces (see [10]), and we have to introduce the weighted spaces which take into account the particular behavior of the solutions to infinity. For domains in R 2 , we use the weighted spaces introduced in [24, 25], specifically where D # is the space of the distributions on # and r denotes the distance from the origin. The norm on W 1(# is given by | v | W (L For domains in R N , N # 3, appropriate spaces, introduced in [20] and used in [19, 31], are with the norm | v | W (L We remark that the space H 1(# is continuously embedded in W 1(#8 and the two spaces coincide for the bounded domains. We use W 1 to denote the closure of D(# in W Concerning the space of the traces of the functions in W 1(#6 we notice that the boundary # being bounded, these traces lie in H 1/2 (#). This fact immediately follows considering a bounded domain D # such that #D and taking into account that W 1 (D) and H 1 (D) are identical. Assuming that and using the spaces W 1 in place of the spaces H 1 , we can rewrite the problems (5.1) and (5.2), and also, similar equations for problems (2.11) and (2.20). For 2, the bilinear form a(y, z) generates on W 1 an equivalent norm with that induced by (see [24]). Also, the bilinear form a(y, z) generates on W 1(# /R a norm which is equivalent to the standard norm. For N # 3, the above introduced norm on W 1 (R N ) is equivalent to that generated by a (see [20]). Now, if we extend the functions in W 1 0(# with zero in R N - # we get that the bilinear form a(y, z) also generates on W 1 a norm equivalent to that induced by W 1(#1 Moreover, using the fact that the domain# is the complement of a bounded set, it can be proved that the bilinear form a(y, z) generates in W 1(# a norm equivalent to the above introduced norm. Therefore, we can conclude that, in the case of a our exterior problems have unique solutions in the spaces W 1 if N # 3. If 2, the Dirichlet problems have unique solutions in W 1 , and the Neumann problems have unique solutions in W 1 /R. Using the fact that on the bounded domains D the spaces W 1 (D) and H 1 (D) coincide, the continuous embedding of H 1(# in W 1(#7 and the homeomorphism and isomorphism between we can easily prove that there exits an homeomorphism and isomorphism between W 1(# and W Consequently, we get the following continuous dependence on the data of the solution y of problem (2.1), (2.2). and Concerning the problem (2.1), (2.3), we have and Therefore, we can prove in a manner similar to the previous sections that when a and N # 3, the solutions of the Dirichlet and Neumann problems in # can be approximated with solutions of both the Dirichlet and the Neumann problems in # Naturally, the controls will be taken in the appropriate space (5.3) or (5.4). If a on# and 2, the solutions of the Dirichlet problems in # can be approximated with solutions of the Dirichlet in # the Neumann problems not having unique solutions. 6 Numerical Results In this section, we consider some fixed U # and H , and we drop the subscripts # and . First, we summarize the results obtained in the previous sections concerning the algebraic system we have to solve to obtain the solutions, within a prescribed error, of problems (2.1), (2.2) or (2.1), (2.3) using the solutions of problems (2.11) or (2.20). We saw that, if, for both the bounded and unbounded domains, there exists a constant c 0 > 0 such that the coe#cient a 0 of the operator A satisfies a 0 # c 0 in # then the solutions of problems (2.1), (2.2) or (2.1), (2.3) can be estimated by the solutions of both problems (2.11) and (2.20). If a in# for both the bounded and the unbounded domains, then the solutions of problems (2.1), (2.2) can be estimated by the solutions of problems (2.11). If a the domains are unbounded and N # 3, then the solutions of problems (2.1), (2.3) can be obtained from the solutions of problems (2.20). Actually, we have to solve an algebraic system (4.5) which we rewrite as Some remarks on the computing of the elements of the matrix # and the free term l are made below. a) Depending on the problem in # we choose the finite dimensional subspace of controls U # U . If we use problem (2.11), U is L 2 (#) if# is bounded and is H 1/2 (#) if# is unbounded. Also, U is if# is bounded and is H -1/2 (#) if# is unbounded, if we use problem (2.20). Let # 1 , , #n , N, be the basis of U . b) Depending on the problem in# and the coe#cient a 0 of the operator A, we calculate the values of y # i n, at the nodes of a mesh on #, the boundary of #. For problem (2.11), we calculate y # i solutions of problems (3.2). For problem (2.20), if there exists a constant c 0 > 0 such that a 0 # c 0 in # or if a and# is unbounded, we calculate y # i solutions of problems (3.21). c) Using the values of y # i calculated at the nodes of the mesh in #, we will compute the elements of the matrix # which are some inner products in we have to solve problem (2.1), (2.2), or in we have to solve problem (2.1), (2.3). We notice that the inner product in H -1 (#) is given by where -# is the Laplace-Beltrami operator on # is the tangential gradient on #, I # is the identity operator, and Evidently, use of this inner product implies the solving of n problems of the above type to find the corresponding of #y #nA (#) , and one problem to find the corresponding of h #y f #nA (#) . The finite dimensional subspace H # H depends on the numerical integration method that we use. We remark that the matrix # is symmetric and full. d) The elements of the free term l will also be some inner products in the corresponding space of observations H. In these inner products we use a solution y f of equation (3.23), and the data # or h # in the boundary conditions of the problem we have to solve, (2.1), (2.2) or (2.1), (2.3), respectively. e) The elements of the matrix # and the free term l depend on the problems in# and #, and also, on the coe#cient a 0 of the operator A. For problem (2.1), (2.2), the matrix # and the free term l are given in (4.3) and (4.6), respectively. For problem (2.1), (2.3), if there exists a constant c 0 > 0 such that a 0 # c 0 in # or if a and# is unbounded, the matrix # and the free term l are given in (4.4) and (4.7), respectively. Evidently, in these equations, y # , the approximations in H of y # i on the problem in # Finally, if # 1 , , # n ) is the solution of algebraic system (6.1), and y is the solution of the problem we have to solve, then its approximation is the restriction to # of If we use the finite element method to calculate the functions y f and y # i is not necessary to adapt the meshes in# to the geometry of #. The values of these functions at the points of # mentioned in the items b) and d) above can be found by interpolation using their values at the mesh nodes. In our numerical examples we use an explicit formula for these functions. Actually, we can find explicit formulae for the solutions of most problems in simple shaped domains We saw that the matrices # given in (3.14) and (3.18) are non-singular and therefore, problems have unique solutions. Also, algebraic systems (6.1) have unique solutions if their matrices are good approximations in H of the matrix and the free term of the algebraic systems (3.16), respectively. In fact, this approximation depends on the numerical integration on #. Also, from Remark 4.1 we must take n # m, n being the dimension of U and m the dimension of H . However, as we saw in Section 2, the problem in infinite space may not have a solution. Consequently, for very large n, we might obtain algebraic systems (3.16) almost singular. These algebraic systems can be solved by an iterative method, the conjugate gradient method, for instance, but we wanted to see whether the algebraic system is non-singular and we applied the Gauss method, checking the diagonal elements during the elimination phase. Below, we show that our numerical results are encouraging. Our numerical tests refer to both the interior and exterior Dirichlet problems on #, where # R 2 is either the interior or exterior domain of the square # centered at the origin, with the sides parallel to the axes and of length 2 unit. The approximate solution of this problem is given by the solutions of the Dirichlet problems in# in which the domain# is either the disc # centered at the origin with radius equal to 2, or the exterior domain of the disc # centered at the origin with radius equal to 0.99. The solutions of these interior and exterior Dirichlet problems in# are found by the Poisson formulae 2#r #|=r dS # . The square # is discretized with m equidistant points and H is taken as the space of the continuous piecewise linear functions. The circle # is similarly discretized with n equidistant points and U is taken as the space of the piecewise constant functions. The values of the integrals in the Poisson equation at the points on # are calculated using the numerical integration with 3 nodes. The integrals in the inner products in L 2 (#) are calculated by an exact formula, i.e., if we have on # two continuous piecewise linear functions y 1 and y 2 , such that for x # [x k , x k+1 then [y k being the mesh size on #. It is worth mentioning right at the outset that all computations below used fifteen significant digits (double precision). Numerical experiments were carried out with three sets of data for the problems in #: g # only for the interior problem, g # In the case of the interior problems, exact known solutions are compared with the computed ones at 19 equidistant points on a diagonal of the square: (-1.4,-1.4), ,(0,0), ,(1.4,1.4). The maximum of the relative error between exact and computed solutions is denoted in the tables below by err d . In the case of the unbounded domains, we do not know the exact solution but we can directly compute the error between the values of the boundary conditions g # and the values of the computed solution given in (6.2), at the considered points on #. The maximum of the relative error at these points is denoted in these tables by err b . The errors err d and err b in the three examples for the interior problem are almost the same. In Table 6.1, we give an example for g(x 1 , x -4. In this example, corresponding to the mesh size 0.1 on #. In all these cases, err d < err b as shown in the Table 6.1. For these simple examples, it is not necessary to compute y f numerically as we used y f 36 .67121E-12 .11648E-06 Table 6.1. Tests for the interior Dirichlet problem. The smaller diagonal element during the Gauss elimination method is of the order 10 -17 for and of the order 10 -14 for It is greater than 10 -10 for We should mention that in both cases with 72, the last pivot is of the order However, we notice an increase in error for n > 60 (see Table 6.1), and these cases should be cautiously considered. In all these cases the error err b , which can be calculated for any example, is a good indicator of the computational accuracy. In Table 6.2, we give an example for the exterior problem with In this example, corresponding to the mesh size of 1/15 on #. 90 0.18828E+00 Table 6.2. Tests for the exterior Dirichlet problem. The smaller diagonal element during the Gauss elimination method is of order 10 -15 for of the order 10 -14 for and it is greater than 10 -12 for Conclusions In this paper we studied, for both interior and exterior problems, the approximation of the solutions of the Dirichlet (Neumann) problems in # with the solutions of the Dirichlet (Neumann) in# by means of an optimal boundary control problem in# with observations on the boundary of the domain #. As we saw in Section 2, such an optimal boundary control problem might lead to an illposed problem if the space of the controls is infinite dimensional. We proved that if the controls are taken in a finite dimensional subspace, then our problem has an unique optimal control. Using the J. L. Lions controllability theorem we also proved that the set of the restrictions to # of the solutions of the Dirichlet (Neumann) problems in# is dense in the set of the solutions of the Dirichlet (Neumann) problems in #. It is natural to take a family of finite dimensional spaces whose union is dense in the space of the boundary conditions of the problem in # Then the set of restrictions to # of the solutions of the Dirichlet (Neumann) problems in# which assume values of the boundary conditions in that union is dense in the set of solutions of these problems in #. Consequently, the optimal boundary control problem in# in which the controls are taken in a finite dimensional space of such a family, will provide a solution of a Dirichlet (Neumann) problem in# whose restriction to # will approximate the solution of the Dirichlet (Neumann) problem in #. Actually, such an optimal control problem whose controls are taken in a finite dimensional space leads to the solution of a linear algebraic system. Since in the practical applications, the values of the solutions in# are approximately calculated on the boundary of the domain #, we also studied the optimal control problem with boundary observations in a finite dimensional subspace. Our primary goal in this paper has been to present this method and provide some theory and calculations in order to bring forth some of the merits of this method as well as to provide some theoretical support. However, we feel at this point that it is perhaps worthwhile to make some remarks on this method against the backdrop of a somewhat di#erent technique within the same framework: namely, Lagrange multiplier technique. For reasons mentioned below, we think that the boundary control approach is simpler, more flexible, and can be more accurate than the Lagrange multiplier approach to domain embedding methods. As we mentioned earlier in the section on Introduction, very good results have been obtained in recent years by using the Lagrange multiplier approach to the domain embedding methods. In the Lagrange multiplier approach to domain embedding methods, values of Lagrange multipliers are sought such that the solution of the problem in the domain# satisfies the specified boundary conditions of the problem in #. In fact, values of the Lagrange multipliers are essentially the jump at the boundary # of #, in the normal derivative of the solution in # In the boundary control approach to domain embedding methods proposed in this paper, boundary values of the solution in# are sought such that this solution satisfies the specified boundary conditions of the problem in #. Consequently, the numbers of supplementary unknowns introduced in the two methods are equivalent. In the Lagrange multipliers approach, one solves a saddle-point problem for the nodal values of the solution in# and those of the multipliers. In the method proposed in this paper, the problem is reduced to solving a linear algebraic system. The construction of this linear system needs the solution of many problems in # but these problems, being completely independent, can be simultaneously solved on parallel machines. We point out that in the conjugate gradient method associated with the Lagrange multipliers method, one has to solve at each iteration a problem in# which has an additional term arising from the Lagrange multipliers. In contrast, our method requires solutions of simpler problems, o#ers good parallelization opportunities, and consequently a low computational complexity. In our approach, we solve several problems in the simple shaped domain# and a linear combination of these solutions provides the final desired solution. Since fast solution techniques are usually available for many problems in simpler domains, we can expect to get very accurate solutions in our approach. In fact, our numerical results for both, the interior and the exterior Dirichlet, problems confirm a high accuracy of the proposed method. In the Lagrange multipliers method, problem formulation introduces an additional term (which is an integral on #) which almost always forces one to use the finite element method to solve for desired solutions even in regular domains. Very good approximate solutions in the presence of such additional term usually requires additional computational complexity such as the use of finer meshes. This increases the dimension of problems in each iteration. In a future work, we will apply the proposed method in conjunction with fast algorithms to solve general elliptic partial di#erential equation in complex geometries. There we will also make numerical comparisons between the results obtained with the method proposed in this paper and that using the Lagrange multipliers method. Acknowledgment : The second author (Prabir Daripa) acknowledges the financial support of the Texas Advanced Research Program (Grant No. TARP-97010366-030). --R Methods of fictitious domains for a second order elliptic equation with natural boundary conditions Domain embedding methods for the Stokes equations The direct solution of the discrete Poisson equation on irregular regions A fast algorithm to solve nonhomogeneous Cauchy-Riemann equations in the complex plane Singular Integral Transforms and Fast Numerical Algorithms: I Les espaces du type Beppo-Levi A spectral embedding method applied to the advection-di#usion equation analysis of a finite element realization of a fictitious domain/domain decomposition method for elliptic problems On the solution of the Dirichlet problem for linear elliptic operators by a distributed Lagrange multiplier method Joseph D. On the coupling boundary integral and finite element methods for the exterior Stokes problem in 3-D Espaces de Sobolev avec poids. Shape optimization of materially non-linear bodies in con- tact Equations int Optimal control of systems governed by partial di A boundary controllability approach in optimal shape design problems On the approximation of the boundary control in two-phase Stefan-type problems An embedding of domain approach in free boundary problems and optimal design Capacitance matrix methods for the Helmholtz equation on general three-dimensional regions On the numerical solution of Helmholtz equation by the capacitance matrix method A locally refined finite rectangular grid finite element method. --TR

optimal control;domain embedding methods

587569

Properties of a Multivalued Mapping Associated with Some Nonmonotone Complementarity Problems.

Using the homotopy invariance property of the degree and a newly introduced concept of the interior-point-$\varepsilon$-exceptional family for continuous functions, we prove an alternative theorem concerning the existence of a certain interior-point of a continuous complementarity problem. Based on this result, we develop several sufficient conditions to assure some desirable properties (nonemptyness, boundedness, and upper-semicontinuity) of a multivalued mapping associated with continuous (nonmonotone) complementarity problems corresponding to semimonotone, P$(\tau, \alpha, \beta)$-, quasi-P*-, and exceptionally regular maps. The results proved in this paper generalize well-known results on the existence of central paths in continuous P0 complementarity problems.

Introduction . Consider the nonlinear complementarity problem (NCP) where f is a continuous function from R n into itself. This problem has now gained much importance because of its many applications in optimization, economics, engi- neering, etc. (see [8, 12, 16, 18]). There are several equivalent formulations of the NCP in the form of a nonlinear equation F is a continuous function from R n into R n . Given such an equation F the most used technique is to perturb F to a certain F # , where # is a positive parameter, and then consider the equation F # has a unique solution denoted by x(#) and x(#) is continuous in #, then the solutions {x(#)} describe, depending on the nature of F # (x), a short path denoted by {x(# (0, - or a long path {x(# (0, #)}. If a short path {x(# (0, - #]} is bounded, then for any subsequence {# k } with # k # 0, the sequence {x(# k )} has at least one accumulation point, and by the continuity each of the accumulation points is a solution to the NCP. Thus, a path can be viewed as a certain continuous curve associated with the solution set of the NCP. Based on the path, we may construct various computational methods for solving the NCP, such as interior-point path-following methods (see, e.g., [15, 25, 26, 27, 28, 32, 39]), regularization methods (see [8, 10, 11, 41]), and noninterior path-following methods (see [1, 2, 3, 5, 7, 17, 21]). The most common interior-point path-following method is based on the central path. The curve (0, #)} is said to be the central path if for each # > 0 the vector x(#) is the unique solution to the system (1) # Received by the editors September 28, 1998; accepted for publication (in revised form) March 14, 2000; published electronically September 20, 2000. http://www.siam.org/journals/sicon/39-2/34519.html Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, P.O. Box 2719, Beijing 100080, China (zyb@lsec.cc.ac.cn, ybzhao@se.cuhk. edu.hk). # Department of Mathematics and Computer Sciences, Royal Military College of Canada, Kingston, Ontario, K7K 7B4, Canada (isac-g@rmc.ca). 572 YUN-BIN ZHAO AND GEORGE ISAC is continuous on (0, #). In the case when f is a monotone function and the NCP is strictly feasible (i.e., there is a vector u # R n such that u > 0 and f(u) > 0), the existence of the central path is well known (see, for example, [14, 25, 30, 31]). This existence result has been extended to some nonmonotone complementarity problems. Kojima, Mizuno, and proved that the central path exists if f is a uniform-P function. If f is a -function satisfying a properness condition and the NCP is strictly feasible, Kojima, Megiddo, and Noma [25] showed that there exists a class of interior-point trajectories which includes the central path as a special case. If f is a P 0 -function and NCP has a nonempty and bounded solution set, Chen, Chen, and Kanzow [4] and Gowda and Tawhid [13] proved that the NCP has a short central path {x(# (0, - #)}. Under a certain properness condition, Gowda and Tawhid [13] showed that the NCP with a P 0 -function has a long central path [13, Theorem 9]. It should be pointed out that noninterior-point trajectories have also been extensively studied in the recent literature (see [1, 2, 3, 5, 10, 11, 13, 17, 35, 37]). However, for a general complementarity problem, the system (1) may have multiple solutions for a given # > 0, and even if the solution is unique, it is not necessarily continuous in #. As a result, the existence of the central path is not always guaranteed. We define the (multivalued) mapping U : (0, # S(R n (2) ++ ) is the set of all subsets of R n ++ , the positive orthant of R n . The main contribution of this paper is to describe several su#cient conditions which ensure that the multivalued mapping U(#) has the following desirable properties. (a) U(# for each # (0, #). (b) For any fixed - # > 0, the set #(0,-#] U(#) is bounded. (c) If U(#, then U(#) is upper-semicontinuous at #. (That is, for any su#- ciently small # > 0, we have that #= U(# U(#B for all # su#ciently close to #, where 1} is the Euclidean unit ball.) (d) If U(-) is single-valued, then U(#) is continuous at # provided that U(#. If the mapping U(-) satisfies properties (a), (b), and (c), then the set can be viewed as an "interior band" associated with the solution set of the NCP. The "interior band" can be viewed as a generalization of the concept of the central path. Indeed, if U(-) satisfies properties (a), (b), and (d), then the set #(0,#) U(#) coincides with the central path of the NCP. There exist several ways of generating the central path of the NCP, including maximal monotone methods [14, 30], minimization methods [31], homeomorphism techniques [6, 14, 15, 25, 33], the parameterized Sard theorem [42], and weakly univalent properties of continuous functions [13, 35, 37]. In this paper, we develop a di#erent method for the analysis of the existence of the central path. By means of the homotopy invariance property of the degree and a newly introduced concept of interior-point-exceptional family for continuous functions, we establish an alternative theorem for the nonemptyness of the mapping U(#). For a given # > 0, the result states that there exists either an interior-point-exceptional family for f or U(#. Consequently, to show the nonemptyness of the mapping U(-), it is su#cient to verify conditions under which the function f possesses no interior-point-exceptional family for any # > 0. Along with this idea, we provide several su#cient conditions that guarantee the aforementioned desirable properties of the multivalued mapping U(-). PROPERTIES OF A MULTIVALUED MAPPING 573 These su#cient conditions are related to several classes of (nonmonotone) functions such as semimonotone, quasi-P # -, P(#)-, and exceptionally regular maps. The results proved in the paper include several known results on the central path as special instances. This paper is organized as follows. In section 2, we introduce some definitions and some basic results that will be utilized in the paper. In section 3, we show an essential alternative theorem that is useful in later derivations. In section 4, we establish some su#cient conditions to guarantee the nonemptyness, boundedness, and upper-semicontinuity of the map U(#), and the existence of the central path. Some concluding remarks are given in section 5. Notations: R n (respectively, R n denotes the space of n-dimensional real vectors with nonnegative components (respectively, positive components), and R n-n stands for the space of n - n matrices. For any x # R n , we denote by #x# the Euclidean norm of x, by x i the ith component of x for the vector whose ith component is max{0, x i }. When x # R n (R n ++ ), we also write it as x # 0 for simplicity. 2. Preliminaries. We first introduce the concept of an E 0 -function, which is a generalization of an E 0 -matrix, i.e., semimonotone matrix, (see [8]). Recall that an is said to be an E 0 -matrix if for any 0 #= x # 0, there exists a component x i > 0 such that (Mx) i # 0. M is a strictly semimonotone matrix if for any 0 #= x # 0, there exists a component x i > 0 such that (Mx) i > 0. Definition 2.1. A function f : R n R n is said to be an E 0 -function (i.e., semimonotone function) if for any x #= y and x # y in R n , there exists some i such that x i > y i and f i (x) # f i (y). f is a strictly semimonotone function if for any x #= y and x # y in R n , there exists some i such that x i > y i and f i (x) > f i (y). It is evident that is an E 0 -function if and only if M is an E 0 -matrix. We recall that a function f is said to be a P 0 (P)- function if for any x #= y in R n Clearly, a P 0 -function is an E 0 -function. However, the converse is not true (see [8, Example 3.9.2]). Thus the class of E 0 -functions is larger than that of P 0 -functions. Definition 2.2. (D1) [23, 24]. A map f : R n R n is said to be quasi monotone if for x #= y in R n , f(y) T implies that f(x) T R n is said to be a P # -map if there exists a scalar # 0 such that for any x #= y in R n we have where I I - (x, [26]. M is said to be a P # -matrix if there exists a scalar # 0 such that Clearly, for a linear map is a P # -map if and only if M is a showed that the class of P # -matrices coincides with the class of su#cient matrices [8, 9]. A new equivalent definition of the P # -matrix is given in [46]. The next concept is a generalization of the quasi monotone function and the Definition 2.3. [46] A function f : R n R n is said to be a quasi-P # -map if there exists a constant # 0 such that the following implication holds for all x #= y in R n . defined by (3). From the above definition, it is evident that the class of quasi-P # -maps includes quasi monotone functions and P # -maps. (see [46] for details). The following concept of a P (#)-map is also a generalization of the P # -map. In [46], it is pointed out that monotone functions and P # -maps are special cases of P(#)-maps. Definition 2.4. [46] A mapping f : R n R n is said to be a P(#)-map if there exist constants # 0, # 0, and 0 # < 1 such that the following inequality holds for all x #= y in R 1#i#n 1#i#n The concept of exceptional regularity that we are going to define next has a close relation to such concepts as copositive, R 0 -, P 0 -, and E 0 -functions. It is shown that the exceptional regularity is a weak su#cient condition for the nonemptyness and the boundedness of the mapping U(#) (see section 4.4 for details). Definition 2.5. Let f be a function from R n into R n . f is said to be exceptionally regular if, for each # 0, the following complementarity problem has no solution of norm 1: The following two results are employed to prove the main result of the next section. Let S be an open bounded set of R n . We denote by S and #(S) the closure and boundary of S, respectively. Let F be a continuous function from S into R n . For any y # R n such that y # F (#(S)), the symbol deg(F, S, y) denotes the topological degree associated with F, S, and y (see [34]). Lemma 2.1. [34] Let S # R n be an open bounded set and F, G be two continuous functions from S into R n . (i) Let the homotopy H(x, t) be defined as and let y be an arbitrary point in R n . If y / then deg(G, S, (ii) If deg(F, S, y) #= 0, then the equation F y has a solution in S. The following upper-semicontinuity theorem of weakly univalent maps is due to Ravindran and Gowda [35]. PROPERTIES OF A MULTIVALUED MAPPING 575 Lemma 2.2. [35] Let g : R n R n be weakly univalent; that is, g is continuous and there exist one-to-one continuous functions uniformly on every bounded subset of R n . Suppose that q # R n such that g -1 (q # ) is nonempty and compact. Then for any given scalar # > 0 there exists a scalar # > 0 such that for any weakly univalent function h : R n R n and for any q # R n with sup -# we have where B denotes the open unit ball in R n 3. Interior-point-exceptional family and an alternative theorem. We now introduce the concept of the interior-point-exceptional family for a continuous function, which brings us to a new idea, to investigate the properties of the mapping U(#) defined by (2), especially the existence of the central path for NCPs. This concept can be viewed as a variant of the exceptional family of elements which was originally introduced to study the solvability of complementarity problems and variational inequalities [19, 20, 36, 43, 44, 45, 46]. Definition 3.1. Let f : R n R n be a continuous function. Given a scalar # > 0, we say that a sequence {x r ++ is an interior-point-exceptional family for f if #x r # as r # and for each x r there exists a positive number x r for all Based on the above concept, we can prove the following result which plays a key role in the analysis of the paper. Theorem 3.1. Let f be a continuous function from R n into R n . Then for each there exists either a point x(#) such that or an interior-point-exceptional family for f . Proof. Let F be the Fischer-Burmeister function of f defined by It is well known that x solves the NCP if and only if x solves the equation F Given # > 0, we perturb F (x) to F # (x) given by It is easy to see that x(#) solves the equation F # only if x(#) satisfies the system (5). We now consider the convex homotopy between the mapping F # (x) and the identity mapping, that is, 576 YUN-BIN ZHAO AND GEORGE ISAC Let r > 0 be an arbitrary positive scalar. Consider the open bounded set S r}. The boundary of S r is given by #S There are only two cases. Case 1. There exists a number r > 0 such that 0 / [0, 1]}. In this case, by (i) of Lemma 2.1, we have that deg(I, S r , where I is the identity mapping. Since deg(I, S r , 1, from the above equation and (ii) of Lemma 2.1, we deduce that the equation F # has a solution, denoted by x(#), which satisfies the system (5). Case 2. For each r > 0, there exists some point x r #S r and t r # [0, 1] such that then the above equation reduces to F # implies that x(#) := x r satisfies the system (5). We now verify that t r #= 1. In fact, if t 0, then from (7) we have that x which is impossible since x r #S r . Therefore, it is su#cient to consider the case of 0 < t r < 1 for all r > 0. In this case, it is easy to show that f actually has an interior-point-exceptional family. Indeed, in this case, (7) can be written as x r Squaring both sides of the above and simplifying, we have x r 1), the above equation implies that x r We see from the above equation that x r We further show that x r ++ . In fact, it follows from (8) that x r On the other hand, by using (9) we obtain x r x r Combining (10) and the above equation yields x r ++ . Since #x r it is clear that #x r # as r #. Consequently, the sequence {x r } is an interior-point- exceptional family for f . The above result shows that if f has no interior-point-exceptional family for each # > 0, then property (a) of the mapping U(-) holds. From the result, it is interesting to study various practical conditions under which a continuous function does not possess an interior-point-exceptional family for every # (0, #). In the next section, we provide several such conditions ensuring the aforementioned desirable properties of the mapping U(-). PROPERTIES OF A MULTIVALUED MAPPING 577 4. Su#cient conditions for properties of U(-). 4.1. -function. In this section, we prove that the multivalued mapping U(-) has properties (a) and (b) if f is a continuous E 0 -function satisfying a certain properness condition. Moreover, if F # (x) given by (6) is weakly univalent, then property (c) also holds. Applied to P 0 complementarity problems, this existence result extends a recent result due to Gowda and Tawhid [13]. The following lemma is quite useful. Lemma 4.1. Let f : R n R n be an E 0 -function. Then for any sequence ++ with #u k #, there exist an index i and a subsequence of {u k denoted by {u k j }, such that u k j Proof. This proof has appeared in several works, see [11, 13, 35, 38]. Let {u k ++ be a sequence satisfying #u k #. Choosing a subsequence if necessary, we may suppose that there exists an index set I # {1, . , n} such that u k for each i } is bounded for each i / R n be a vector constructed as follows: Thus, } is a bounded sequence. Clearly, u k is an E 0 -function, there exist an index i # I and a subsequence of {u k }, denoted by {u k j }, such that Note that the right-hand side of the above inequality is bounded. The desired result follows. To show the main result of this subsection, we will make use of the following assumption which is weaker than several previously known conditions. Condition 4.1. For any sequence {x k satisfying # and [-f(x k # 0, and (ii) for each index i with x k i #, the corresponding sequence {f i above, and (iii) there exists at least one index i 0 such that x k it holds that 1#i#n for some subsequence {x k l As we see in the following result the above condition encompasses several particular cases; we omit the details. Proposition 4.1. Condition 4.1 is satisfied if one of the following conditions holds. (C1) For any positive sequence {x k ++ with #x k # and [-f(x k 0, it holds that max 1#i#n x k l subsequence {x k l (C2) For any sequence {x k ++ with #x k # and min 1#i#n f i 0, it holds that max 1#i#n x k l subsequence {x k l [22, 29] For any sequence {x k } with #x k #, # 0, and # 0, it holds that lim inf 578 YUN-BIN ZHAO AND GEORGE ISAC [13] For any sequence {x k } with #x k #, lim inf min 1#i#n x k # 0, and lim inf there exist an index j and a subsequence {x k l } such that x k l is a R 0 -function. monotone and the NCP is strictly feasible. is a uniform P-function. Remark 4.1. The condition (C1) of the above proposition is weaker than each of the conditions (C2) through (C7). (C2) is weaker than each of the conditions (C4) through (C7). The concept of the R 0 -function, a generalization of the R 0 -matrix [8], was introduced in [39] and later modified in [6]. In what follows, we show under a properness condition that the short "interior band" #(0,-#] U(#) is bounded for each given - # > 0. The boundedness is important because it implies that the sequence {x(# k )}, where is bounded and each accumulation point of the sequence is a solution to the NCP provided that f is continuous. We impose the following condition on f . Condition 4.2. For any positive sequence {x k ++ such that #x k #, and the sequence {f i is bounded for each index i with #, it holds that 1#i#n for some subsequence {x k l Clearly, Condition 4.2 is weaker than Condition 4.1 and thereby weaker than all conditions listed in Proposition 4.1. We now prove the boundedness of the short "interior band" under the above condition. Lemma 4.2. Suppose that Condition 4.2 is satisfied. If U(# for each # > 0, then for any - # > 0 the set #(0,-#] U(#) is bounded, i.e., property (b) holds. Particu- larly, U(#) is bounded for each # > 0. Proof. Suppose that there exists some - #(0,-#] U(#) is unbounded. Then there exists a sequence {x(# k )}, where # k # (0, - #], such that #x(# k )# as and that for all Thus, for each i such that x i #, the sequence {f i (x(# k ))} is bounded. By Condition 4.2, we deduce that there exists a subsequence {x(# k l )} such that 1#i#n This is a contradiction since x i # for all The main result on E 0 -functions is given as follows. Even for P 0 -functions, this result is new. Theorem 4.1. Suppose that f is a continuous E 0 -function and Condition 4.1 is satisfied. Then the properties (a) and (b) of the mapping U(#) hold. Moreover, if F # (x) defined by (6) is weakly univalent in x, then the mapping U(-) is upper-semicontinuous, i.e., property (c) also holds. PROPERTIES OF A MULTIVALUED MAPPING 579 Proof. To prove property (a), by Theorem 3.1, it su#ces to show that there exists no interior-point-exceptional family of f for any # > 0. Assume to the contrary that for certain # > 0 the function f has an interior-point-exceptional family {x r #x r #, {x r ++ , and f is an E 0 -function, by Lemma 4.1 there exist some index m and a subsequence {x r j }, such that x r j From (4), we have bounded below, the right-hand side of the above equation is bounded below. It follows that lim j# - r On the other hand, we note that for any 0 < - < 1 the function # -- -# is monotonically decreasing with respect to the variable t # (0, #). Passing through a subsequence, we may suppose that there exists an index set I # {1, . , n} such that i } is bounded for each i / # I. # I, then there exists some scalar C > 0 such that x r j #(t) is decreasing and - r j # 1, we have Thus, for all su#ciently large j, we have # I. using (4) and the facts - r #, we have which implies that Therefore, [-f(x r it follows from (4) that which implies that {f i (x r j )} is bounded above for all i # I. Since m # I and {f m is bounded below, the sequence {f m (x r j )} is indeed bounded. From Condition 4.1, there is a subsequence of {x r j denoted also by {x r j }, such that 1#i#n However, from (4) we have for all i # {1, . , n}. This is a contradiction. Property (a) of U(#) follows. Since Condition 4.1 implies Condition 4.2, the boundedness of the set follows immediately from Lemma 4.2. It is known that x(# U(#) if and only if x(#) is a solution to the equation F # # (0). Since U(#) is bounded, the set F -1 # (0) is bounded (in fact, compact, since f is continuous). If F # (x) is weakly univalent in x, by Lemma 2.2, for each scalar # > 0 there is a # > 0 such that for any weakly univalent function h : R n sup x# -# we have It is easy to see that for the given # > 0 there exists a scalar # > 0 such that sup x# -# Setting h(x) := F # (x) in (13) and (14), we obtain that #= F -1 for all |#, i.e., U(# U(#B for all # su#ciently close to #. Thus, U(#) is upper-semicontinuous. Ravindran and Gowda [35] showed that if f is a P 0 -function, then F # (x) given by (6) is a P-function in x, and hence the equation F # has at most one solution x(#). In this case, the upper-semicontinuity of U(-) reduces to the continuity of x(#). By the fact that every P 0 -function is an E 0 -function and is weakly univalent, we have the following result from Theorem 4.1. Corollary 4.1. Suppose that f : R n R n is a continuous P 0 -function and Condition 4.1 is satisfied. Then the central path exists and any slice of it is bounded, i.e., for each # > 0 there exists a unique x(#) satisfying the system (1), x(#) is continuous on (0, #), and the set {x(# (0, - #]} is bounded for each - When f is a P 0 -function, Gowda and Tawhid [13, Theorem 9] showed that the (long) central path exists if condition (C4) of Proposition 4.1 is satisfied. Corollary 4.1 can serve as a generalization of the Gowda and Tawhid result. It is worth noting that the consequences of Corollary 4.1 remain valid if condition (C1) or (C2) of Proposition 4.1 holds. 4.2. Quasi-P # -maps. The concept of the quasi-P # -map that is a generalization of the quasi monotone function and the P # -map was first introduced in [46] to study the solvability of the NCP. Under the strictly feasible assumption as well as the following condition, we can show the nonemptyness and the boundedness of U(-) if f is a continuous quasi-P # -map . Condition 4.3. For any sequence {x k ++ such that #, lim and {f(x k )} is bounded, it holds that 1#i#n for some subsequence {x k l PROPERTIES OF A MULTIVALUED MAPPING 581 Clearly, the above condition is weaker than Conditions 4.1 and 4.2. It is also weaker than Condition 3.8 in [4] and Condition 1.5(iii) in [25]. The following is the main result of this subsection. Theorem 4.2. Let f be a continuous quasi-P # -map with the constant # 0 (see Definition 2.3). Suppose that Condition 4.3 is satisfied. If the NCP is strictly feasible, then property (a) of U(#) holds. Moreover, if Condition 4.2 is satisfied, then property (b) holds, and if F # (x) is weakly univalent in x, then property (c) also holds. While the nonemptyness of U(#) is ensured under Condition 4.3, it is not clear if the boundedness of U(#) can follow from this condition. However, from the implications Condition 4.1# Condition 4.2 # Condition 4.3, we have the next consequence. Corollary 4.2. Suppose that f is a continuous quasi-P # -map and F # (x) is weakly univalent in x. If the NCP is strictly feasible and Condition 4.1 or 4.2 is satisfied, then the mapping U(-) has properties (a), (b), and (c). The proof of Theorem 4.2 is postponed until we have proved two technical lemmas. Lemma 4.3. Let f satisfy Condition 4.3. Assume that {x r } r>0 is an interior- point-exceptional family for f . If there exists a subsequence of {x r }, denoted by {x rk }, such that for some 0 < # < 1, lim #x rk then we have lim 1#i#n x rk Proof. Suppose that {x rk } is an arbitrary subsequence of {x r } such that (15) holds. Since #(t) defined by (11) is decreasing on (0, #), for each i # {1, . , n} we have 1#i#n x rk min 1#i#n x rk and 1#i#n x rk Suppose to the contrary that there exists a subsequence of {x rk denoted also by {x rk }, such that min 1#i#n x rk is a constant. We derive a contradiction. Indeed, since - rk - 1 < 0, from (16) we have min 1#i#n x rk # for all From (17) and the above relation, we obtain 1#i#n x rk Since #x rk #, we deduce from (15) that lim 1#i#n x rk Therefore, it follows from (18) that there exists a scalar c such that c # f i for all Condition 4.3, there exists a subsequence of {x rk denoted still by {x rk }, such that max 1#i#n x rk However, from (12) we have that x rk This is a contradiction. Lemma 4.4. Let f satisfy Condition 4.3. Assume that {x r } is an interior-point- #-exceptional family for f . Let u > 0 be an arbitrary vector in R n . Then for any subsequence {x rk (where r k # as k #) there exists a subsequence of {x rk denoted still by {x rk }, such that f(x rk su#ciently large k. Proof. Let {x rk } be an arbitrary subsequence of {x r (where r k # as k #). By using (4) we have #x rk x rk #x rk x rk # . We suppose that f(x rk su#ciently large k. We derive a contra- diction. From (19), we have #x rk Since #x rk #, for all su#ciently large k we have #x rk which implies that lim #x rk #x rk for any scalar 0 < # < 1. Thus, we see from Lemma 4.3 that min 1#i#n x rk Notice #x rk for all su#ciently large k. From (19), (20), and the above inequality, we have x rk min 1#i#n x rk PROPERTIES OF A MULTIVALUED MAPPING 583 for all su#ciently large k. This is a contradiction. We are now ready to prove the results of Theorem 4.2. Proof of Theorem 4.2. To show property (a) of the mapping U(#), by Theorem 3.1, it su#ces to show that f has no interior-point-exceptional family for any # > 0. Assume to the contrary that there exists an interior-point-exceptional family for f , denoted by {x r }. By the strict feasibility of the NCP, there is a vector u > 0 such that f(u) > 0. Consider two possible cases. Case (A). There exists a number r 0 > 0 such that 1#i#n In this case, the index set I #, it is easy to see that for all su#ciently large r. Since f is a quasi-P # -map and I is empty, the above inequality implies that f(x r su#ciently large r. How- ever, by Lemma 4.4 there exists a subsequence of {x r }, denoted by {x rk }, such that su#ciently large k. This is a contradiction. Case (B). There exists a subsequence of {x r } denoted by {x r j as j #, such that 1#i#n By using (4), for each i we have # . There exist a subsequence of {x r j denoted also by {x r j }, and a fixed index m such that 1#i#n For each i such that x r j #, (21) implies that A (r j ) j, we deduce that {x r j m} is bounded, i.e., there is a constant - # such that 0 < x r j for all j. um , setting in (21), we have um # 1# 1 If um < x r j #, setting in (21), we obtain 584 YUN-BIN ZHAO AND GEORGE ISAC We consider two subcases, choosing a subsequence whenever it is necessary. Subcase 1. - r j # 1. From (22) and (23), for all su#ciently large j we have um Thus, for all su#ciently large j, we obtain 1#i#n um /2)} The last inequality above follows from the fact that f(u) > 0, {x r j ++ , and #. Since f is a quasi-P # -map, the above inequality implies that f(x r su#ciently large j, which is impossible according to Lemma 4.4. Subcase 2. There exists a subsequence of {- r j }, denoted also by {- r j }, such that 1. In this case, from (22) and (23), we have # . It follows from (4) that 1#i#n . We now show that T (r su#ciently large j. noting that - r j # and #x r j #, we obtain ), by the same argument as the above, we can show that for all su#ciently large j. Thus, by the quasi-P # -property of f , we deduce from su#ciently large j. It is a contradiction since {x r j #, and f(u) > 0. The above contradictions show that f has no interior-point-exceptional family for each # > 0. By Theorem 3.1, the set U(# for any # > 0. The boundedness of PROPERTIES OF A MULTIVALUED MAPPING 585 the short "interior band " follows from Lemma 4.2, and the upper-semicontinuity of U(#) follows easily from Lemma 2.2. The class of quasi-P # -maps includes the quasi monotone functions as particular cases. The following result is an immediate consequence of Theorem 4.2. Corollary 4.3. Suppose that f is a continuous quasi monotone (in particular, pseudomonotone) function, and the NCP is strictly feasible. (i) If Condition 4.3 is satisfied, then property (a) of U(#) holds. (ii) If Condition 4.2 is satisfied, then properties (a) and (b) of U(#) hold. In the case when F # (x) is univalent (continuous and one-to-one) in x, the equation has at most one solution. Combining this fact and Theorem 4.2, we have the following result concerning the existence of the central path of the NCP. To our knowledge, this result can be viewed as the first existence result on the central path for the NCP with a (generalized) quasi monotone function. Up to now, there is no interior-point type algorithms designed for solving (generalized) quasi monotone complementarity problems. Corollary 4.4. Let f be a quasi-P # -map, and F # (x) is univalent in x. If the NCP is strictly feasible and Condition 4.2 is satisfied, then the central path exists and the set {x(# (0, - #]} is bounded for any given - # > 0. Particularly, if f is a P 0 -function, then F # (x) is univalent in x (see [35]). We have the following result. Corollary 4.5. Let f be a continuous P 0 and quasi-P # -map. If the NCP is strictly feasible and Condition 4.2 is satisfied, then the conclusions of Corollary 4.4 are valid. 4.3. P (#)-maps. It is well known (see [14, 25, 30, 31]) that the monotonicity combined with strict feasibility implies the existence of the central path. In this section, we extend the result to a class of nonmonotone complementarity problems. Our result states that if f is a P(#) and P 0 -map (see Definition 2.4), the central path exists provided that the NCP is strictly feasible. This result gives an answer to the question "What class of nonlinear functions beyond P # -maps can ensure the existence of the central path if the NCP is strictly feasible?" We first show properties of the mapping U(-) when f is a P (#)-map. Theorem 4.3. Let f be a continuous P (#)-map. If the NCP is strictly fea- sible, then properties (a) and (b) of U(#) hold. Moreover, if F # (x) is weakly univalent in x, property (c) also holds. Proof. Suppose that there exists a scalar # > 0 such that f has an interior-point- #-exceptional family denoted by {x r ++ and #x r # as r #, there exist some p and a subsequence denoted by {x r j such that #x r j # and 1#i#n Clearly, x r j On the other hand, there exists a subsequence of {x r j denoted also by {x r j }, such that for some fixed index m and for all j we have 1#i#n By the definition of the P(#)-map, we have 586 YUN-BIN ZHAO AND GEORGE ISAC 1#i#n 1#i#n - u# . From (4), we have that f p p , and hence It is easy to see that Combining (25) and (26) leads to From min := min 1#i#n we deduce that min # as j #. We now show that {x r j m} is bounded. Assume that there exists a subsequence of {x r j denoted still by {x r j m}, such that x r j #. Then, from (21), we have and hence for all su#ciently large j we have 1#i#n By (27) and the above relation, we obtain min # as j #. However, since f is a P(#)-map, we have min #, which contradicts (28). This contradiction shows that the sequence {x r j m} is bounded. By using (4) and (24), we have - u# . PROPERTIES OF A MULTIVALUED MAPPING 587 Multiplying both sides of the above inequality by 1/(x r j rearranging terms, and using (26), we have For all su#ciently large j, the left-hand side of the above inequality is negative, but the right-hand side tends to f p (u) > 0 as j #. This is a contradiction. The contradiction shows that f has no interior-point-exceptional family for every # > 0. By Theorem 3.1, property (a) of U(#) follows. The proof of the boundedness of the set #(0,-#] U(#) is not straightforward. It can be proved by the same argument as the above. Indeed, we suppose that {x(# k )} 0<#k<-#(0,-#] U(#) is an unbounded sequence. Replacing {x r j } by {x(# k )}, using instead of (4), and repeating the aforementioned proof, we can derive a contradic- tion. The upper-semicontinuity of U(-) can be obtained by Lemma 2.2. The proof is complete. The class of P(#)-maps includes several particular cases such as P (#, 0)-, P(#, 0, 0)-, and P(0, #)-maps. It is shown in [46] that the class of P(#, 0, 0)-maps coincides with the class of P # -maps. Therefore, f is said to be a P # -map if and only if there exists a nonnegative scalar # 0 such that 1#i#n 1#i#n Particularly, a matrix M # R n-n is a P # -matrix if and only if there is a constant # 0 such that 1#i#n 1#i#n This is an equivalent definition of the concept of a P # -matrix (su#cient matrix) introduced by Kojima et al. [26] and Cottle, Pang, and Venkateswaran [9]. The following result follows immediately from Theorem 4.3. Corollary 4.6. Let f be a continuous P 0 and P(#)-map. If the NCP is strictly feasible, then the central path exists and any slice of it is bounded. It is worth noting that each P # -map is a P 0 and a P(#)-function. The following result is a straightforward consequence of the above corollary. Corollary 4.7. Let f be a continuous P # -map. If the NCP is strictly feasible, then the central path exists and any slice of it is bounded. It should be pointed out that P # -maps are also special instances of quasi-P # - maps. A result similar to Corollary 4.3 can be stated for P # -maps. However, as we have shown in Corollary 4.7, the additional conditions such as Conditions 4.1, 4.2, and 4.3 are not necessary for a P # -map to guarantee the existence of the central path. While P # -maps and quasi monotone functions are contained in the class of quasi-P # - maps, Zhao and Isac [46] gave examples to show that a P # -map, in general, is not a quasi monotone function, and vice versa. 588 YUN-BIN ZHAO AND GEORGE ISAC 4.4. Exceptionally regular functions. In section 4.1, we study the properties of the mapping U(#) for -functions satisfying a properness condition, i.e., Condition 4.1. In sections 4.2, we show properties of U(#) for quasi-P # -maps under the strictly feasible condition as well as some properness conditions. In the above section, properness assumptions are removed, and properties of U(#) for P (#)-maps are proved under the strictly feasible condition only. In this section, removing both the strictly feasible condition and properness conditions, we prove that properties of U(#) hold if f is an exceptionally regular function. The exceptional regularity of a function (see Definition 2.5) was originally introduced in [46] to investigate the existence of a solution to the NCP. Definition 4.1. [16] A map v : R n R n is said to be positively homogeneous of degree # > 0 if the above concept reduces to the standard concept of positive homogeneity. Under the assumption of positively homogeneous of degree # > 0, we can show that properties (a) and (b) of U(#) hold if f is exceptionally regular. See the following result. Theorem 4.4. Let f be a continuous and exceptionally regular function from R n into R n . If positively homogeneous of degree # > 0, then properties (a) and (b) of U(#) hold. Moreover, if F # (x) is weakly univalent, property (c) also holds. Proof. Suppose that there is a scalar # > 0 such that f has an interior-point- exceptional family {x r We derive a contradiction. Indeed, since G(x) is positively homogeneous of degree # > 0, we have #) - f(0)). Without loss of generality, assume that x r /#x r x. From the above relation, we have lim r# From (4), we have2 x r for all # and x r we deduce that x r for each i # I (-x). We now show that lim #x r for some - It is su#cient to show the existence of the above limit. Indeed, for each using (30) and (29) we have lim #x r r# #x r x r #x r x r Thus, (31) holds, with PROPERTIES OF A MULTIVALUED MAPPING 589 Now, we consider the case of i / (-x). In this case, - using (4), (31), and (29), we see from x r # 0 that r# x r r# #x r r# #x r #x r i.e., Combining (32) and the above relation implies that f is not exceptionally regular. This is a contradiction. The contradiction shows that f has no interior-point- exceptional family for each # > 0, and hence property (a) of U(#) follows from Theorem 3.1. Property (b) of U(#) can be easily proved. Actually, suppose that there exists a sequence {x(# k )} 0<#k<-# with #x(# k )#, where loss of generality, let x(# k )/#x(# k )# - 1. As in the proof of (29) we have Therefore, which contradicts the exceptional regularity of f(x). It is not di#cult to see that a strictly copositive map and a strictly semimonotone function are special cases of exceptionally regular maps. Hence, we have the following result. Corollary 4.8. Suppose that positively homogeneous of degree # > 0. Then conclusions of Theorem 4.4 are valid if one of the following conditions holds. (i) f is an E 0 -function, and for each 0 #= x # 0 there exists an index i such that (ii) f is strictly copositive, that is, x (iii) f is a strictly semimonotone function. Proof. Since each of the above conditions implies that f(x) is exceptionally reg- ular, the result follows immediately from Theorem 4.4. Motivated by Definition 2.5, we introduce the following concept. 4.2. M # R n-n is said to be an exceptionally regular matrix if for all #I is an R 0 -matrix. It is evident that an exceptionally regular matrix is an R 0 -matrix, but the converse is not true. The following result is an immediate consequence of Theorem 4.4 and its corollary. Corollary 4.9. Let is an arbitrary vector in R n . If one of the following conditions is satisfied, then properties (a) and (b) of the mapping U(#) hold: 590 YUN-BIN ZHAO AND GEORGE ISAC (i) M # R n-n is an exceptionally regular matrix. (ii) M is a strictly copositive matrix. (iii) M is a strictly semimonotone matrix. (iv) M is an E 0 -matrix, and for each 0 #= x # 0 there exists an index i such that (possibly, (Mx) i < 0). Furthermore, if M is also a P 0 -matrix, then the central path of a linear complementarity problem exists and any slice of it is bounded. The R 0 -property of f has played an important role in the complementarity theory. We close this section by considering this situation. The concept of a nonlinear R 0 - function was first introduced by Tseng [38] and later modified by Chen and Harker [6]. We now give a definition of the R 0 -function that is di#erent from those in [38] and [6]. R n is said to be an R 0 -function if is the unique solution to the following complementarity problem: This concept is a natural generalization of the R 0 -matrix [8]. In fact, for the linear function it is easy to see that f is an R 0 -function if and only if M is an R 0 -matrix. In the case when f is an E 0 -function, we have shown in Theorem 4.1 that there exists a subsequence {- rk } such that - rk # 1. Moreover, if G is positively homogeneous, then from (31) we deduce that - using these facts and the above R 0 -property and repeating the proof of Theorem 4.4, we have the following result. Theorem 4.5. Suppose that for each scalar t # 0 and x # R n , and that f is an E 0 and R 0 -function. Then the conclusions of Theorem 4.4 remain valid. Moreover, if f is a P 0 and R 0 -function, the central path exists and any slice of it is bounded. 5. Conclusions. We introduced the concept of the interior-point-exceptional family for continuous functions, which is important since it strongly pertains to the existence of an interior-point x(# U(#) and the central path, even to the solvability of NCPs. By means of this concept, we proved that for every continuous NCP the set U(#) is nonempty for each scalar # > 0 if there exists no interior-point-exceptional family for f . Based on the result, we established some su#cient conditions for the assurance of some desirable properties of the multivalued mapping U(#) associated with certain nonmonotone complementarity problems. Since properties (a) and (b) of U(#) imply that the NCP has a solution, the argument of this paper based on the interior-point-exceptional family can serve as a new analysis method for the existence of a solution to the NCP. It is worth noting that any point in U(#) is strictly feasible, i.e., x(#) > 0 and Therefore, the analysis method in this paper can also be viewed as a tool for investigating the strict feasibility of a complementarity problem. In fact, from Theorems 3.1, 4.1, 4.4, and 4.5, we have the following result. Theorem 5.1. Let f be a continuous function. Then the complementarity problem is strictly feasible whenever one of the following conditions holds. (i) There exists a scalar # > 0 such that f has no interior-point-exceptional family. (ii) f is an E 0 -function and Condition 4.1 is satisfied. positively homogeneous of degree # > 0 and f is exceptionally regular. PROPERTIES OF A MULTIVALUED MAPPING 591 is an E 0 and R 0 -matrix. It should be pointed out that the results and the argument of this paper can be easily extended to other interior-point paths. For instance, we can consider the existence of the path (where b and a > 0 are fixed vectors in R n ) first studied by Kojima, Megiddo, and the above path reduces to the central path). This path can be studied by the concept of interior-point-#(a, b)-exceptional family. For a continuous we say that a sequence {x r ++ is an interior- b)-exceptional family for f if #x r # as r #, and for each x r there exists a positive number - r # (0, 1) such that for each i x r Using and arguing as in the same proof of Theorem 3.1, we can show that for any # > 0 there exists either a point x(#) satisfying (33) or an interior-point-#(a, b)-exceptional family for f . This result enables us to develop some su#cient conditions for the existence of the path (33). Acknowledgments . The authors would like to thank the referees and Professor Jim Burke for their helpful suggestions and comments on an earlier version of this paper, which helped the authors to correct some mistakes and improve the presentation of the manuscript. They also thank Dr. Mustapha Ait Rami for his valuable comments. --R The global linear convergence of a non-interior-point path following algorithm for linear A global and local superlinear continuation-smoothing method for P 0 and R 0 NCP or monotone NCP A Penalized Fischer-Burmeister NCP-Function: Theoretical Investigation and Numerical Results Smooth approximations to A. class of smoothing functions for nonlinear and mixed The Linear Complementarity Problem Beyond monotonicity in regularization methods for Engineering and economic applications of Existence and limiting behavior of trajectories associated with A survey of theory Global convergence of a class of non-interior-point algorithms using Chen-Harker-Kanzow functions for Lecture Notes in Math. Functions without exceptional families of elements and Some nonlinear continuation methods for linear New NCP-functions and their properties Complementarity problems over cones with monotone and pseudomonotone maps Seven kinds of monotone maps Homotopy continuation methods for A. Unified Approach to Interior Point Algorithms for Linear A. new continuation method for A. polynomial-time algorithm for linear A new class of merit functions for the nonlinear complementarity problem The complementarity problem for maximal monotone multifunctions Pathways to the optimal set in linear programming Interior path following primal dual algorithms Properties of an interior-point mapping for mixed Iterative Solution of Nonlinear Equations in Several Variables Regularization of P 0 A. solution condition for Growth behavior of a class of merit functions for the An infeasible path-following method for monotone An algorithm for the linear complementarity problem with a P 0 On Constructing Interior-Point Path-Following Methods for Certain Semimonotone Linear Existence of a solution to nonlinear variational inequality under generalized positive homogeneity Exceptional family of elements for a variational inequality problem and its applications --TR --CTR Y. B. Zhao , D. Li, A New Path-Following Algorithm for Nonlinear P*Complementarity Problems, Computational Optimization and Applications, v.34 n.2, p.183-214, June 2006

generalized monotonicity;weakly univalent maps;nonlinear complementarity problems;interior-point-va-exceptional family;central path

587580

Second Order Methods for Optimal Control of Time-Dependent Fluid Flow.

Second order methods for open loop optimal control problems governed by the two-dimensional instationary Navier--Stokes equations are investigated. Optimality systems based on a Lagrangian formulation and adjoint equations are derived. The Newton and quasi-Newton methods as well as various variants of SQP methods are developed for applications to optimal flow control, and their complexity in terms of system solves is discussed. Local convergence and rate of convergence are proved. A numerical example illustrates the feasibility of solving optimal control problems for two-dimensional instationary Navier--Stokes equations by second order numerical methods in a standard workstation environment.

Introduction This research is devoted to the analysis of second methods for solving optimal control problems involving the time dependent Navier Stokes equations. Thus we consider min J(y; u) over (y; u) subject to> > < @y @ in Here is a bounded domain in R 2 , with su-ciently smooth boundary @ The nal time T > 0 and the initial condition y 0 are xed. The vector valued variable y and the scalar valued variable p represent the velocity and the pressure of the uid. Further u denotes the control variable and B the control operator. The precise functional analytic setting of problem (1.1), (1.2) will be given in Section 2. For the moment it su-ces to say that typical cost functionals include tracking type and functionals involving the vorticity of the uid jcurl y(t; )j 2 where > 0 and z are given. For the following discussion it will be convenient to formally represent all the equality constraints involved in (1.2) by ^ e(y; that (1.1), (1.2) can be expressed in the form min J(y; u) over (y; u) subject to In this form solving (1.1), (1.2) appears at rst to be a standard task, see [AM, SECOND ORDER METHODS IN FLOW CONTROL 3 and the references given there. However, the formidable size of (1.1), (1.2) and the goal of analyzing second order methods necessitate an independent analysis. For second order methods applied to optimal control problems two classes can be distinguished depending on whether (y; p) in (1.1), (1.2) are considered as independent variables or as functions of the control variable u. In the former case represents an explicit constraint for the optimization problem whereas in the latter case serves the purpose of describing the evaluation of (y; p) as a function of u. In fact (P ) can be expressed as the reduced problem where y(u) is implicitly dened via To obtain a second order method in the case when (y; p) are considered as independent variables one can derive the optimality system for (P ) and apply the Newton algorithm to the optimality system. This is referred to as the sequential quadratic programming (SQP){method. Alternatively, if (y; p) are considered as functions of u, then Newton's method can be applied to directly. The relative merits of these two approaches will be discussed in Section 4. To anticipate some of this discussion let us point out that the dierence in numerical eort between these two methods is rather small. In fact, after proper rearrangements the dierence in computational cost per iteration of the SQP{method for (P ) and the Newton method for solving either the linearized equation (1.2) or the full nonlinear equation itself. In view of the time{dependence of either of these two equations an iterative procedure is used for their solution so that the dierence between solving the linearized and nonlinear equation per sweep is not so signi- cant. A second consideration that may in uence the choice between SQP{method or Newton{method applied to and u 0 for (y; p; u) can clearly be used independently of each other in the SQP{ method, where the states are decoupled from the controls. It is sometimes hinted at that this decoupling is not only important for the initialization but also during the iteration and that as a consequence the SQP{method may require fewer iterations than Newton's method for As we shall see below, the variables y and p can be initialized independently from u 0 also in the Newton method. Specically, are available it is not necessary to abandon (y As for the choice of the initial guess (y is to rely on one of the suboptimal strategies that were developed in the recent past to obtain approximate solutions to (1.1), (1.2). We mention reduced order techniques [IR], POD{based methods [HK, KV, LT] and the instantaneous control method [CTMK, BMT, CHK]. As another possibility one can carry out some gradient steps before one switches to the Newton iteration. Let us brie y comment on some related contributions. In [AT] optimality systems are derived for problems of the type (1.1), (1.2). A gradient technique is proposed in [GM] for the solution of (1.1), (1.2). Similarly in [B] gradient techniques are analyzed for a boundary control problem related to (1.1), (1.2). In [FGH] the authors analyze optimality systems for exterior boundary control problems. One of the few contributions focusing on second order methods for optimal control of uids are given in [GB, H]. These works are restricted to stationary problems, however. This paper, on the other hand focuses on second order methods for time dependent problems. We show that despite the di-culties due to the size of (1.1), (1.2) and the fact that the optimality systems contains a two point boundary value problem, forward in time for the primal- and backwards in time for the adjoint vari- ables, second order methods are computationally feasible. We establish that the initial approximation to the reduced Hessian is only a compact perturbation of the Hessian at the minimizer. In addition we give conditions for second order su-cient optimality conditions of tracking type problems. These results imply superlinear convergence of quasi{Newton as well as SQP{methods. While the present paper focuses on distributed control problems in a future paper we plan to address the case of velocity control along the boundary. The paper is organized as follows. Section 2 contains the necessary analytic prerequisites. First and second order derivatives of the cost functional with respect to the control are computed in Section 3. The fourth section contains a comparison of second order methods to solve (1.1), (1.2). In Section 5 convergence of quasi{ Newton method and SQP{methods applied to P ) is analyzed. Numerical results for the Newton{method and comparisons to a gradient method are contained in Section 6. 2. The optimal control problem In this section we consider the optimal control problem (1.1), (1.2) in the abstract subject to e(y; To dene the spaces and operators arising in (2.1) we assume to be a bounded domain in R 2 with Lipschitz boundary and introduce the solenoidal spaces with the superscripts denoting closures in the respective norms. Further we dene W endowed with the norm H equipped with the norm H denoting the dual space of V . Here is an abbreviation for L 2 (0; that up to a set of measure zero in (0; T ) elements can be identied with elements in C([0; T can be identied with elements in C([0; T ]; V ). In (2.1) further U denotes the Hilbert space of controls R is the cost functional which is assumed to be bounded from below, weakly lower semi-continuous, twice Frechet dierentiable with locally Lipschitzean second derivative, and radially unbounded in u, i.e. . Furthermore, the control space U is identied with SECOND ORDER METHODS IN FLOW CONTROL 5 its dual U . To simplify the notation for the second derivative we also assume that the functional J can be decomposed as The nonlinear mapping is dened by Comparing (1.1), (1.2) to (2.1) we note that the conservation of mass, as well as the boundary condition are realized in the choice of the space W while the dynamics are described by the condition e(y; In variational form the constraints in (2.1) can be equivalently expressed as: given u 2 U nd y 2 W such that The following existence result for the Navier{Stokes equations in dimension two is well known ([CF, L, T], Chapter III). Proposition 2.1. There exists a constant C such that for every u 2 U there exists a unique element and U From Proposition (2.1) we conclude that with respect to existence (2.1) is equivalent to subject to u 2 U; Theorem 2.1. Problem (2.1) admits a solution (y Proof. With the above formalism the proof is quite standard and we only give a brief outline. Since J is bounded from below there exists a minimizing sequence un g in W U . Due to the radial unboundedness property of J in u and Proposition 2.1 the sequence f(y n ; un )g is bounded in W U and hence there exists a subsequence with a weak limit (y ; u ) 2 W U . Weak lower semi-continuity of (y; u) ! J(y; u) implies that and it remains to show that y This can be achieved by passing to the limit in (2.3) with (y; u) replaced by (y(u n ); un ). We shall also require the following result concerning strong solutions to the Navier-Stokes equation, ([T], Theorem III. 3.10). Proposition 2.2. If y then for every u 2 U the solution Moreover, for every bounded set U in U is bounded in H 2;1 (Q): 6 MICHAEL HINZE AND KARL KUNISCH We shall frequently refer to the linearized Navier-Stokes system and the adjoint equations given next: in a.e. on (0; T ]; and in a.e. on [0; T Proposition 2.3. Let y 3 ]. Then (2.5) admits a unique variational solution v 2 W and (2.6) has a unique variational solution w and the rst equation in (2.6) holding in L (V ) \ W . Moreover, the following estimates hold. iii. jwj L 2 If in addition y iv. jwj L 2 For @ solutions v of (2.5) and w of (2.6) are elements of H 2;1 (Q) and satisfy the a-priori estimates v. and vi. Proof. The proof is sketched in the Appendix. 3. Derivatives In this section representations for the rst and second derivatives of ^ J appropriate for the treatment of (2.4) by the Newton and quasi{Newton method are derived. We shall utilize the notation Proposition 3.1. The operator continuously differentiable with Lipschitz continuous second derivative. The action of the rst two derivatives of e 1 are given by he 1 where (v; r) 2 X . SECOND ORDER METHODS IN FLOW CONTROL 7 Proof. Since e 2 is linear we restrict our attention to e 1 . Let dened by and recall that, due to the assumption that for all (u; v; To argue local Lipschitz continuity of e, let pZ Tjy ~ Here and below C denotes a constant independent of x; ~ x and . Due to the continuous embedding of W into L 1 (H) we have jx ~ R T Using Holder's inequality this further implies the estimate jx ~ and consequently, after redening C one last time This estimate establishes the local Lipschitz continuity of e. To verify that the formula for e x given above represents the Frechet - derivative of e we estimate R T sup R T Cjy ~ R T and Frechet - dierentiability of e follows. To show Lipschitz continuity of the rst derivative let x; ~ x and (v; r) be in X and estimate R T R T Cjy ~ The expression for the second derivative can be veried by an estimate analogous to the one for the rst derivative. The second derivative is independent of the point at which it is taken and thus it is necessarily Lipschitz continuous. From (3.2) it follows that for 2 L 2 (V ) and w 2 W the mapping is an element of W . In Section 4 we shall use the fact that can also be identied with an element of L 4 Lemma 3.1. For 2 L 2 (V ) and w 2 W the functional can be identied with an element in W \ L 4=3 (V ). Proof. To argue that 2 L 4=3 using (3.2) where k is the embedding constant of V into H . This gives the claim . Proposition 3.2. Let is a homeo- morphism. Moreover, if the inverse of its adjoint e is applied to an element setting (w; w we have w and w is the variational solution to (2.6). Proof. Due to Proposition 3.1, e y (x) is a bounded linear operator. By the closed range theorem the claim follows once it is argued that (2.5) has a unique solution . This is a direct consequence of Proposition 2.3, i. and ii. The assertion concerning the adjoint follows from the same proposition, iii. and its proof. As a consequence of Propositions 3.1 and 3.2 and the implicit function theorem the rst derivative of the mapping u ! y(u) at u in direction -u is given by y (x)e u (x)-u; u). By the chain rule we thus obtain Introducing the variable we obtain utilizing Proposition 2.3 iii. with representation for the rst derivative of Here is the variational solution of where the rst equation holds in L 4=3 (V ) \ W . The computation of the second derivative of ^ J is more involved. Let (-u; -v) 2 U U and note that the second derivative of u ! y(u) from U to W can be expressed as SECOND ORDER METHODS IN FLOW CONTROL 9 By the chain rule, and since W We introduce the Lagrangian and the matrix operator y (x)e u (x) We observe that the second derivative of L with respect to x can be expressed as The above computation for ^ J 00 (u) together with (3.4) imply that 4. Second order methods This section contains a description and a comparison of second order methods to solve (2.1). Throughout u denotes a (local) solution to (2.1). 4.1. Newton{and quasi{Newton algorithm. For the sake of reference let us specify the Newton algorithm. Algorithm 4.1. (Newton Algorithm). 1. Choose u 2. Do until convergence ii) update u Let us consider the linear system in 2. i). Its dimension is that of the control space U . From the characterization of the Hessian ^ we conclude that its evaluation requires as many solutions to the linearized Navier{Stokes equation (3.4) with appropriate right hand sides as is the dimension of U . If U is innite dimensional then an appropriate discretization must be carried out. Let us assume now that the dimension of U is large so that direct evaluation of ^ not feasible. In this case 2. i) must be solved iteratively, e. g. by a conjugate gradient technique. We shall then refer to 2. i) as the "inner " loop as opposed to the do{loop in 2. which is the "outer" loop of the Newton algorithm. The inner loop at iteration level k of the outer loop requires to with (ii) iteratively evaluate the action of ^ j , the j{th iterate of the inner loop on the k{th level of the outer loop. The iterate j can be evaluated by successively applying the steps a) solve in L 2 b) evaluate J yy (x)v c) solve in W for w d) and nally set q := J uu -u +B w. We recall that 1 2 L 2 (V ) and that for s 2 W he 1 Z TZ Moreover, by Lemma 3.1 the functional appearing in b) is an element of W \ Hence by Proposition 2.3 the adjoint equation in c) can equivalently be rewritten as where the rst equation holds in W \ L 4=3 (V ). Summarizing, for the outer iteration of the Newton method one Navier{Stokes solve for y(u k ) and one linearized Navier{Stokes solve for are required. For the inner loop one forward ({in time) as well as one backwards linearized Navier{Stokes solve per iteration is necessary. Concerning initialization we observe that if initial guesses (y are available (with y 0 not necessarily y(u 0 )) then alternatively to the initialization in Algorithm 4.1 this information can be used advantageously to compute the adjoint variable 1 required for the initial guess for the right hand side of the linear system as well as to carry out steps a) - c) for the evaluation of the Hessian. There is no necessity to recompute y(u 0 ) from u 0 . To avoid the di-culties of evaluating the action of the exact Hessian in Algorithm 4.1 one can resort to quasi{Newton algorithms. Here we specify one of the most prominent candidates, the BFGS{method. For w and z in U we dene the rank{one operator z 2 L(U ), the action of which is given by (w In the BFGS{method the Hessian ^ J 00 at u is approximated by a sequence of operators Algorithm 4.2. (BFGS{Algorithm) 1. Choose u SECOND ORDER METHODS IN FLOW CONTROL 11 2. Do until convergence ii) update u Note that the BFGS{algorithm requires no more system solves than the gradient algorithm applied to (2.1), which is one forward solution of the nonlinear equation to obtain y(u k ) and one backward solve of the linearized equation (3.7) obtain the adjoint variable (u k ). In order to compare Newton's method to the SQP method derived in the next section we rewrite the update step 2. i) in Algorithm 4.1. To begin with we observe that the right hand side in the update step can be written with the help of the adjoint variable from and the operator T (x) dened in (3.10) as J J where we dropped the iteration indices. As a consequence, with (3.3) the update can be written as -y J so that -y J holds. Since e x closed and we have the sequence of identities x Thus there exists - 2 Z such that e -y J Using this equation together with the denition of -y, Newton's update may be rewritten as 2-y 4.2. Basic SQP{method. Here we regard (2.1) as a minimization problem of the functional J over the space X subject to the explicit constraint SQP-algorithm consists in applying Newton's method to the rst order optimality system where the Lagrangian L is dened in (3.9). With x denoting a solution to problem (P), e x surjective by Proposition 3.2, and hence there exists a Lagrange multiplier 2 Z which is even unique such that (4.4) holds. The SQP{method will be well dened and locally second order convergent, if in addition to the surjectivity of e x the following second order optimality condition holds. There exists > 0 such that If (H1) holds then, due to the regularity properties of e there exists a neighborhood uniformly positive denite on ker(e x (x)) for every Algorithm 4.3. (SQP{algorithm) 1. Choose 2. Do until convergence solve ii) update Just as for Newton's method step 2. i.) is the di-cult one. While in contrast to Newton's method neither the Navier{Stokes equation nor its linearization needs to be solved, the dimension of the system matrix which is twice the dimension of the state plus the dimension of the control space is formidable for applications in uid mechanics. In addition from experience with Algorithm 4.3 for other optimal control problems, see [KA, V] for example, it is well known that preconditioning techniques must be applied to solve (4.5) e-ciently. As a preconditioner one might consider the (action of the) operator H is the inverse to the (discretized) instationary Stokes operator or the (discretized) linearization of the Navier{Stokes equation at the state y k , either one with homogenous boundary conditions. One iteration of the preconditioned version of Algorithm 4.3 therefore requires two linear parabolic solves, one forward and one backwards in time. As a con- sequence, even with the application of preconditioning techniques, the numerical expense counted in number of parabolic system solves is less for the SQP{method than for Newton's method. However, the number of iterations of iterative methods applied to solve the system equations in Algorithms 4.1 and 4.3 strongly depends on the system dimension, which is much larger for Algorithm 4.3 than for Algorithm 4.1. To further compare the structure of the Newton and the SQP{methods let us assume for an instance that x k is feasible for the primal equation, i. e. e(x k and feasible for the adjoint equation (3.5), i. e. e SECOND ORDER METHODS IN FLOW CONTROL 13 Then the right hand side of (4.5) has the form@J u A and comparing to the computation at the end of section 4.1 we observe that the linear systems describing the Newton and the SQP{methods coincide. In general the nonlinear primal and the linearized adjoint equation will not be satised by the iterates of the SQP{method and we therefore refer to the SQP{method as an outer or unfeasible method, while the Newton method is a feasible one. 4.3. Reduced SQP{method. The idea of the reduced SQP{method is to replace (4.5) with an equation in ker e x (x), so that the reduced system is of smaller dimension than the original one. To develop the reduced system we follow the lines of [KS]. Recall the denition of T Note that A is a right{inverse to e x (x). In fact, we have y (x)e u (x)v By Proposition 3.2 and due to B 2 L(U; L 2 (V )) the operator T (x) is an isomorphism from U to ker e x (x) and hence the second equality in (4.5) given by can be expressed as Using this in the rst equality of (4.5) we nd x Applying T (x) to this last equation and ii) from above implies that if -u is a solution coordinate of (4.5) then it also satises Once -u is computed from (4.8) then -y and - can be obtained from (4.7) (which requires one forward linear parabolic solve) and the rst equation in (4.5) (another backwards linear parabolic solve). Let us note that if x is feasible then the rst term on the right hand side of (4.8) is zero and (4.8) is identical to step 2. i) in Newton's Algorithm 4.1. This again re ects the fact that Newton's method can be viewed as an SQP{ method that obeys the feasibility constraint It also points at the fact that the amount of work (measured in equation solves) for the inner loop coincides for both the Newton and the reduced SQP{methods. The signicant dierence between the two methods lies in the outer iteration. To make this evident we next specify the reduced SQP{algorithm. 14 MICHAEL HINZE AND KARL KUNISCH Algorithm 4.4. (Reduced SQP{algorithm). 1. Choose x 2. Do until convergence i) Lagrange multiplier update: solve e ii) Solve iii) update Note that in the algorithm that we specied we did not follow the procedure outlined above for the update of the Lagrange variable. In fact for reduced SQP{methods there is no "optimal" update strategy for . The two choices described above are natural and frequently used. To implement Algorithm 4.4 two linear parabolic systems have to be solved in steps 2. i) and 2. ii) ) and, in addition two linear parabolic systems are necessary to evaluate the term involving the operator A on the right hand side of 2. ii) ). In applications this term is often neglected since it vanishes at x . The reduced SQP{method and Newton's method turn out to be very similar. Let us discuss the points in which they dier: Most signicantly the velocity eld is updated by means of the nonlinear equation in Newton's method and via the linearized equation in the reduced SQP{method. ii) The right hand sides of the linear systems dier due to the appearance of the term involving the operator A. As mentioned above this term is frequently not implemented. iii) Formally there is a dierence in the initialization procedure in that y 0 is chosen independently from u 0 in the reduced SQP{method and y Newton's method. However, as explained in section 4.1 above, if a good initial guess y 0 independent from y(u 0 ) is available, it can be utilized in Newton's method as well. 5. Convergence analysis We present local convergence results for the algorithms introduced in Section 4 for cost functionals of separable type (2.2). For this purpose it will be essential to derive conditions that ensure positive deniteness of ^ (H1). The key to these conditions are the a-priori estimates of Proposition 2.3. We shall also prove that the dierence ^ compact. This property is required for the rate of convergence analysis of quasi-Newton methods. In our rst result we assert positive deniteness of the Hessian provided that J y (x) is su-ciently small, a condition which is applicable to tracking-type problems. SECOND ORDER METHODS IN FLOW CONTROL 15 Lemma 5.1. (Positive deniteness of Hessian) Let u 2 U and assume that J yy positive semi-denite and J uu (x) 2 L(U) be positive denite, where Then, the Hessian (u) is positive denite provided that jJ y (x)j L 2 (V ) is su-ciently small. Proof: We recall from (3.11) that is the solution to (3.7). It follows that e Here we note that for -u 2 U the functional is an element of W . Since J yy (x) is assumed to be positive denite and J uu (x) is positive denite the result will follow provided the operator norm of R := e can be bounded by jJ y (x)j L 2 (V ) . Straightforward estimation gives From Proposition 2.3 we conclude that To estimate yy (x)(; ); 1 (x)ik L(W;W ) we recall that for he 1 ZZ Using (3.2) and the continuity of the embedding W ,! L 1 (H) we may estimate with a constant C independent of g and h. Therefore, where we applied iii. in Proposition 2.3 to (3.7). Lemma 5.2. Let x 2 X and denote by the function dened in (3.5). Then, under the assumptions of Lemma 5.1 on J condition (H1) is satised with replaced by (x; ). Proof. Let (v; u) 2 N (e x (x)). Then v solves (2.5) with to Proposition 2.3, v 2 W and satises be chosen such that J uu (x)(u; u) -juj 2 U for all u 2 U . We nd pT Here and below C denotes a generic constant independent of (v; u) and Due to (3.5) and Proposition 2.3 These estimates imply and combined with (5.4) the claim follows. Lemma 5.3. If B 2 L(U; L 2 (H)), then the dierence is compact for every u 2 U . Proof. Utilizing (5.2) we may rewrite It will be shown that both summands dene compact operators on U . For this purpose let U be a bounded subset of U . Utilizing Proposition 2.3 we conclude that y (x)e is a bounded subset of W and hence of L 2 (V ). Since by assumption J is twice continuously Frechet dierentiable with respect to y from L 2 (V ) to R it follows that J yy (S) is a bounded subset of L 2 (V ). Proposition 2.3, iii. implies that consequently e y (J yy (S)) is bounded in W 2 )g. Since W 2 4=3 is compactly embedded in L 2 (H) [CF] and B 2 L(U; L 2 (H)) it follows from the fact that e that is pre-compact in U . Let us turn to the second addend in (5.5). Due to Lemma 3.1 and its proof the set is a bounded subset of W \ L 4=3 (V ). It follows, utilizing Proposition 2.3 that is a bounded subset of W 2 4=3 H . As above the assumption that B 2 L(U; L 2 (H)) implies that SECOND ORDER METHODS IN FLOW CONTROL 17 is precompact in U and the lemma is veried. The following lemma concerning the operators T (x) and A(x) dened in (3.10) and (4.6) will be required for the analysis of the reduced SQP-method. Lemma 5.4. The mappings Let x 7! A(x) from X to L(Z ; X) and x 7! T (x) from X to L(U; X) are Frechet dierentiable with Lipschitz continuous derivatives. Proof. An immediate consequence of i., ii. in Proposition 2.3 and the identities ii) and iii) in Section (4.3) together with the dierentiability properties of the mapping x 7! e x (x). We are now in the position to prove local convergence for the algorithms discussed in Section 4. Throughout we assume that (y ; u ) is a local solution to (2.1) and set y In addition to the general conditions on J , B and e we require positive semi-denite, J uu positive denite, and jJ y su-ciently small. With (H2) holding (H1) is satised due to Lemma 5.1. In particular a second order su-cient optimality condition holds and (y ; u ) is a strict local solution to (2.1). The following theorem follows from well known results on Newton's algorithm Theorem 5.1. If (H2) holds then there exist a neighbourhood U(u ) such that for every u 0 2 U(u ) the iterates fu n gn2N of Newton's Algorithm 4.1 converge quadratically to u . Theorem 5.2. If (H2) holds then there exist a neighbourhood U(u ) and > 0 such that for all u positive denite operators H 0 2 L(U) with the BFGS method of Algorithm 4.2 converges linearly to u . If in addition B 2 then the convergence is super-linear. Proof: For the rst part of the theorem we refer to [GR, Section4], for example. For the second claim we observe that the dierence ^ by Lemma 5.3, so that the claim follows from [GR, Theorem 5.1], see also [KS1]. Theorem 5.3. Assume that (H2) holds and let be the Lagrange multiplier associated to x . Then there exist a neighbourhood U(x ; ) X Z such that for all 4.3 is well dened and the iterates converge quadratically to Proof: Since J and e are twice continuously dierentiable with Lipschitz continuous second derivative, e x surjective by Proposition 3.2 and (H1) is satised, second order convergence of the SQP-method follows from standard results, see for instance [IK]. We now turn to the reduced SQP-method. Theorem 5.4. Assume that (H1) holds and let denote the Lagrange multiplier associated to x . Then there exist a neighbourhood U(x ) X such that for all reduced SQP-algorithm 4.4 is well dened and its iterates fx n gn2N converge two-step quadratically to x , i.e. for some positive constant C independent of k 2 N. Proof: First note that (H1) implies positive deniteness of T in a neighbourhood ~ U(x ) of x . By Lemma 5.4 the mappings x 7! T (x) and x 7! A(x) are Frechet dierentiable with Lipschitz continuous derivatives. Fur- thermore, utilizing Proposition 2.3, iii. and Lemma A.2 it can be shown that the mapping x 7! (x) is locally Lipschitz continuous, where is dened through (3.5). This, in particular, implies for the Lagrange multiplier updates k the estimate where the constant C is positive and depends on x and on supfjJ yy (x)j L(L 2 (V );L 2 (V U(x )g. Altogether, the assumptions for Corollary 3.6 in [K] are met and there exists a neighbourhood ^ claim follows. 6. Numerical results Here we present numerical examples that should rst of all demonstrate the feasibility of utilizing Newton's method for optimal control of the two-dimensional instationary Navier-Stokes equations in a workstation environment despite the formidable size of the optimization problem. The total number of unknowns (primal-, adjoint-, and control variables) in Example 1 below, for instance, is of order 2.210 6 . The time horizon could still be increased or the mesh size decreased by utilizing reduced storage techniques at the expense of additional cpu-time, but we shall not pursue this aspect here. The control problem is given by (1.1), (1.2) with cost function J dened by Qo Qc with c and subsets of denoting the control and observation volumes, respectively. In our examples Re =400 and B is the indicator function of Q c . The results for Newton's method will be compared to those of the gradient algorithm, which we recall here for the sake of convenience. Algorithm 6.1. Gradient Algorithm 1. choose u 0 , 2. Set d d) 3. Set 4. 2. SECOND ORDER METHODS IN FLOW CONTROL 19 Given a control u the evaluation of the gradient of J at a point u amounts to solving (1.2) for the state y and (3.7) for the adjoint variable . Implementing a stepsize rule to determine an approximation of is numerically expensive as every evaluation of the functional J at a control u requires solving the instationary Navier-Stokes equations with right hand side Bu. We compare two possibilities for computing approximations to the optimal step size . For this purpose let us consider for search direction d 2 U the solutions of the systems and is the associated adjoint variable. 1. For a given search direction d 2 U interpolate the function I() by a quadratic polynomial using the values I(0); I 0 (0) and I 00 (0), i.e. and use the unique zero of the equation I 0 as approximation of , with w given by (6.3). 2. Use the linearization of the mapping 7! y(u + d) at in the cost functional J . This results in the quadratic approximation I 2 () := J(y(u) d) of the functional I(). Now use the unique root of the equation I 0 as approximation of , with v given in (6.2). The denominator of (u)d; di U . From (5.1) with u replaced by u it follows as in the proof of Theorem 5.1 that it is positive, provided that the state y(u) is su-ciently close to z in L 2 (H). Let us note that the computation of 1 requires the solution of linearized Navier-Stokes equations forward and backward in time, whereas that of only requires one solve of the linearized Navier-Stokes equations. In addition, a numerical comparison shows that the step-size guess performs better than both with respect to the number of iterations in the gradient method and with respect to computational time. For the numerical results presented below we therefore use the step size proposal . Thus, every iteration of the gradient algorithm amounts to solving the nonlinear Navier-Stokes equations forward in time and the associated adjoint equations backward in time for the computation of the gradient, and to solving linearized Navier-Stokes equations forward in time for the step size proposal. The inner iteration of Newton's method is performed by the conjugate gradient method, the choice of which is justied in a neighbourhood of a local solution u of the optimal control problem by the positive deniteness of ^ desired state z is su-ciently close to the optimal state y(u ). For the numerical tests the target ow z is given by the Stokes ow with boundary condition z tangential direction, see Fig. 1. The termination criterion for the j-th iterate u k j in the conjugate gradient method is chosen as min< The initialization for Newton's method was u 0 := 0. Figure 1. Control target, Stokes ow in the cavity The discretization of the Navier-Stokes equations, its linearization and adjoint was carried out by using parts of the code developed by Bansch in [BA], which is based on Taylor-Hood nite elements for spatial discretization. As time step size we took which resulted in 160 grid points for the time grid, and 545 pressure and 2113 velocity nodes for the spatial discretization. All computations were performed on a DEC-ALPHA TM station 500. Iteration CG-steps 6 19 4.686819e-6 0.032 1.480534e-3 Table 1. Performance of Newton's method for Example 1 Example 1 We rst present the results for c Table conrms super-linear convergence of the in-exact Newton method. To SECOND ORDER METHODS IN FLOW CONTROL 21 achieve the the same accuracy as Newton's method the gradient algorithm requires iterations. The computing time with Newton's method is approximately minutes whereas the gradient method requires 110 minutes. This demonstrates the superiority of Newton's method over the gradient algorithm for this example. For larger values of and coarser time and space grids the dierence in computing time is less drastic. In fact this dierence increases with decreasing and increasing mesh renement. As expected a signicant amount of computing time is spent for read-write actions of the variables to the hard-disc in the sub-problems. In Figures 2, 3, 4 the evolution of the cost functional, the dierence to the Stokes ow and the control as a function of time are documented. It can be observed that Newton's method tends to over-estimate the control in the rst iteration step, whereas the gradient algorithm approximates the optimal control from below, see Figure 4. Graphically there is no signicant change after the second iteration for Newton's method. These comments hold for quite a wide range of values for . In Fig. 5 the uncontrolled ow together with the controlled ow and the control action at the end of the time interval are presented.2.8e-02 Figure 2. Newton's method (6 Iterations) (top) versus Gradient algorithm (96 Iterations), Re=400, Evolution of cost functional for relative In the previous example the observation volume and the control volume c each cover the whole spatial domain. From the practical point of view this is not feasible. However, from the numerical standpoint this is a complicated situation, since the inhomogeneities in the primal and adjoint equations are large. 22 MICHAEL HINZE AND KARL KUNISCH1.4e-02 Figure 3. Newton's method (6 Iterations) (top) versus Gradient algorithm (96 Iterations), Re=400, Evolution of dier- ence to Stokes ow for relative We next present two numerical examples with dierent observation and control volumes. This results in smaller control and observation volumes than in Example 1, and thus the primal and adjoint equations are numerically simpler to solve. Example 2 Here and 1:). The spatial and temporal discretizations as well as the parameter are the same as in Example 1. Newton's method takes 15 minutes cpu-time and its convergence statistics are presented in Tab. 2. The gradient algorithm needs 25 iterations and 26 minutes cpu to reduce the value of the cost functional from ^ to Iteration CG-steps Table 2. Performance of Newton's method for Example 2 Example 3 Here and 0:7). Again, the discretization of the spatial and the time domain as well as the parameter are the SECOND ORDER METHODS IN FLOW CONTROL 231.8e+00 Figure 4. Newton's method (6 Iterations) (top) versus Gradient algorithm (96 Iterations), Re=400, Evolution of control for relative same as in Example 2. The gradient algorithm needs 38 iterations to reduce the value of the cost functional from ^ It takes about 80 minutes cpu-time. We also implemented the Polak-Ribiere variant of the conjugate gradient algorithm. It converges after 37 iterations and yields a slightly better reduction of the residual. The amount of of cpu-time needed is nearly equal to that taken by the gradient algorithm. Newton's method is faster. It converges within 7 iterations to the approximate solution and needs cpu-time. The average cpu-time for the inner iteration loop is 7.5 minutes. As in the previous examples the average cost of a conjugate gradient iteration in the inner loop decreases with decreasing residual of the outer-iteration loop. The results are depicted in Tab. 3. Appendix A. Proof of Proposition 2.3 In the proof of Proposition 2.3 we make frequent use of the following Lemma. Lemma A.1. For Z Figure 5. Results for from top to bottom: uncontrolled controlled ow at control force at Iteration CG-steps Table 3. Performance of Newton's method for Example 3 there exists a positive constant C such that with a positive constant C. Proof. In [T1]. Lemma A.2. There exists a positive constant C such that Proof. Since the proof is identical to that of Lemma 3.1. Note that the power 4=3 in the previous estimate cannot be improved by requiring Su-cient conditions for (ru) t v +(u r)v 2 L 2 (V ) are given by requiring in addition that u or v 2 L 1 (V ). Proof of Proposition 2.3. Existence and uniqueness of a solution to (2.5) can be shown following the lines of the existence and uniqueness proof for the instationary two-dimensional Navier-Stokes equations in [T, Chap.III]. In the following we sketch the derivation of the necessary a-priori estimates. i. Test with respect to time, use b(u; v; estimate using Young's inequality and the rst estimate in (A.1). This results in After integration from 0 to t Gronwall's inequality gives Using (A.3) in (A.2), the Cauchy-Schwarz inequality yields 26 MICHAEL HINZE AND KARL KUNISCH Combining (A.3) and (A.4) yields the rst claim. ii. Test (2.5) with 2 V pointwise in time and estimate using the Cauchy-Schwarz inequality and the rst estimate in (A.1). This gives Z which implies This, together with y 2 W L 1 (H), and the estimates (A.3), (A.4) gives ii. Combining i. and ii. implies iii. For y 2 W we introduce the bounded linear operator A(y) 2 L(W;Z ) by Note that A(y) coincides with e y (x) of Section 3. Due to i. this operator admits a continuous inverse A(y) 1 2 L(Z ; W ). For the adjoint A(y) 2 there exists for every solution (w; to From i. and ii. together with the fact that we have By Lemma A.2 and the assumption that g 2 L 4=3 (V ) the mapping is an element of L (V ), with 2 [1; 4=3]. From (A.8) we therefore conclude that w t 2 L (V ). Together with w [DL5]. From (A.8) we deduce that the rst equation in (2.6) is well dened in L (V ). Referring to (A.8) a third time and utilizing the fact that w(T ) is well dened in H it follows that w(T there exists a constant C such that Combining this estimate with (A.9) implies the estimate in iii. iv. If y utilizing (A.8) we nd that w t 2 L 2 (V ). Moreover, by (A.1) we have Together with (A.9) this gives the desired estimate in iv. v. Test (2.5) with v pointwise in time and utilize Young's inequality and the last estimate in (A.1) to obtain Integration from 0 to t together with (A.4) results in so that Gronwall's inequality gives Using this in (A.11) yields To estimate jv t j L 2 (H) test (2.5) with 2 V and use the last estimate in (A.1). This gives Z so that y 2 together with (A.13) and (A.14) implies Therefore, which is v. The estimation for jwj L 1 2 ) is similar to that for In order to cope with b(; in the estimation of utilizes the third estimate in (A.1) to obtain the estimate vi. 28 MICHAEL HINZE AND KARL KUNISCH --R The Lagrange-Newton method for state constrained optimal control problems On some control problems in uid mechanics Numerical solution of a ow-control problem: vorticity reduction by dynamic boundary action Instantaneous control of backward-facing-step ows Preprint No Feedback control for unsteady ow and its application to the stochastic Burgers equation Mathematical Analysis and Numerical Methods for Science and Technology Boundary value problems and optimal boundary control for the Navier-Stokes system: the two-dimensional case Numerical Methods for Nonlinear Variational Problems Optimal control of two-and three-dimensional incompressible Navier-Stokes Flows The local convergence of Broyden-like methods in Lipschitzean problems in Hilbert spaces The velocity tracking problem for Navier-Stokes ows with bounded distributed controls Formulation and analysis of a sequential quadratic programming method for the optimal Dirichlet boundary control of Navier-Stokes ow Control strategies for uid ows - optimal versus suboptimal control Augmented Lagrangian-SQP-methods for nonlinear optimal control problems of tracking type Optimal control of thermally convected uid ow Reduced SQP methods for parameter identi Mesh independence of the gradient projection method for optimal control problems Control of Burgers equation by a reduced order approach using Proper Orthogonal Decomposition Bericht Nr. Mathematical Topics in Fluid Mechanics I Modelling and control of physical processes using proper orthogonal decomposition --TR --CTR Kerstin Brandes , Roland Griesse, Quantitative stability analysis of optimal solutions in PDE-constrained optimization, Journal of Computational and Applied Mathematics, v.206 n.2, p.908-926, September, 2007

optimal control Navier-Stokes equations;newton method;second order sufficient optimality;SQP method

587583

Stability of Perturbed Delay Differential Equations and Stabilization of Nonlinear Cascade Systems.

In this paper the effect of bounded input perturbations on the stability of nonlinear globally asymptotically stable delay differential equations is analyzed. We investigate under which conditions global stability is preserved and if not, whether semiglobal stabilization is possible by controlling the size or shape of the perturbation. These results are used to study the stabilization of partially linear cascade systems with partial state feedback.

Introduction The stability analysis of the series (cascade) interconnection of two stable nonlinear systems described by ordinary differential equations is a classical subject in system theory ([13], [14], [17]). stable stable Contrary to the linear case, the zero input global asymptotic stability of each subsystem does not imply the zero input global asymptotic stability of the interconnection. The output of the first subsystem acts as a transient input disturbance which can be sufficient to destabilize the second subsys- tem. In the ODE case, such destabilizing mechanisms are well understood. they can be subtle but are almost invariably associated to a finite escape time in the second subsystem (Some states blow up to infinity in a finite time). The present paper explores similar instability mechanisms generated by the series interconnection of nonlinear DDEs. In particular we consider the situation where the destabilizing effect of the interconnection is delayed and examine the difference with the ODE situation. In the first part of the paper we study the effect of external (affine) perturbations w on the stability of nonlinear time delay systems whereby we assume that globally asymptotically stable. We consider perturbations which belong to both L 1 and L1 and investigate the region in the space of initial conditions which give rise to bounded solutions under various assumptions on the system and the perturbation First, we consider global results: in the ODE-case, an obstruction is formed by the fact that arbitrary small input perturbations can cause the state to escape to infinity in a finite time, for instance when the interconnection term \Psi(z) is nonlinear in z. This is studied extensively in the literature in the context of stability of cascades, i.e. when the perturbation in (1) is generated by another ODE, see e.g. [15] [13] and the references therein. Even though delayed perturbations do not cause a finite escape time, we explain a similar mechanism giving rise to unbounded solutions, caused by nonlinear delayed interconnection terms. In a second part, we allow situations whereby unbounded solutions are inevitable and we investigate under which conditions trajectories can be bounded semi-globally in the space of initial conditions, in case the perturbation is parametrized, i.e. a). Hereby we let the parameter a regulate the L 1 or L1 norm of the perturbation. We also consider the effect of concentrating the perturbation in an arbitrary small time-interval. As an application, we consider the special case whereby the perturbation is generated by a globally asymptotically stable ODE. This allows us to strengthen the previous results by the application of a generalization of LaSalle's theorem [7] to the DDE-case: the convergence to zero of a solution is implied by its boundedness. We also show that the origin of the cascade is stable. We will concentrate on the following synthesis problem: the stabilization of a partial linear cascade,! with the SISO-system (A; B; C) controllable, using only partial state feed-back laws of the form which allows to influence the shape and size of the input 'perturbation' y to the nonlinear delay equation. In the ODE-case this stabilization problem is extensively studied in the literature, for instance in [16][1][15][8]. Without any structural assumption on the interconnection term, achieving global stabilization is generally not possible because the output of the linear subsystem, which acts as a destabilizing disturbance to the nonlinear subsystem, can cause trajectories to escape to infinity in a finite time. Therefore one concentrates on semi-global stabilization, i.e. the problem of finding feedback laws making the origin asymptotically stable with a domain of attraction containing any pre-set compact region in the state space. An instructive way to do so is to drive the 'perturbation' y fast to zero. However, a high-gain control, placing all observable eigenvalues far into the LHP, will not necessarily result in large stability regions, because of the fast peaking obstacle [15] [13]. Peaking is a structural property of the -subsystem whereby achieving faster convergence implies larger overshoots which can in turn destabilize the cascade. Semi-global stability results are obtained when imposing structural assumptions on the -subsystem (a nonpeaking system) or by imposing conditions on the z-subsystem and the interconnection term \Psi: for example in [15] one imposes a linear growth restriction on the interconnection term and requires global exponential stability of the z-subsystem. In this paper the classical cascade results are obtained and analysed in the more general framework of bounded input perturbations and generalized to the time-delay case. Preliminaries The state of the delay equation (1) at time t can be described as a vector z(t) 2 R n or as a function segment z t defined by z Therefore delay equations form a special class of functional differential equations [3][5][6]. We assume that the right-hand side of (1) is continuous in all of its arguments and Lipschitz in z and z(t \Gamma ). Then a solution is uniquely defined by specifying as initial condition a function segment z 0 whereby z 0 2 C([\Gamma; 0]; R n ), the Banach space of continuous bounded functions mapping the delay-interval [\Gamma; 0] into R n and equipped with the supremum-norm k:k s . Sufficient conditions for stability of a functional differential equation are provided by the theory of Lyapunov functionals [3] [6], a generalization of the classical Lyapunov theory for ODEs: for functional differential equations of the form a mapping R is called a Lyapunov functional on a set G if V is continuous on G and 0 on G. Here V is the upper-right-hand derivative of V along the solutions of (2), i.e. The following theorem, taken from [3], provides sufficient conditions for stability: Theorem 1.1 Suppose z = 0 is a solution of (2) and and there exist nonnegative functions a(r) and b(r) such that a(r) !1 as r !1 and Then the zero solution is stable and every solution is bounded. If in addition, b(r) is positive definite, then every solution approaches zero as t !1. Instead of working with functionals, it is also possible to use classical Lyapunov functions when relaxing the condition This approach, leading to the so-called Razumikhin-type theorems [6], is not considered in this paper In most of the theorems of the paper, the condition of global asymptotic stability for the unperturbed system (equation (1) with cient. When the dimension of the system is higher than one, we sometimes need precise information about the interaction of different components of the state z(t). This information is captured in the Lyapunov functional, associated with the unperturbed system. Therefore, when necessary, we will restrict ourself to a specific class of functionals, with the following assumption Assumption 1.2 The unperturbed system delay-independent globally asymptotically stable (i.e. GAS for all values of the delay) with a Lyapunov-functional of the form radially unbounded and such that the conditions of theorem 1.1 (with b(r) positive definite) are satisfied. This particular choice is motivated by the fact that such functionals are used for a class of linear systems [3][6] and, since we assume delay-independent stability, the time-derivative of the functional should not depend explicitly on . Choosing a delay-independent stable unperturbed system also allows us to investigate whether the results obtained in the presence of perturbations are still global in the delay. Note that in the ODE-case (3) reduces to hardly forms any restriction because under mild conditions its existence is guaranteed by converse theorems. The perturbation j(t) 2 L p ([0; 1)) when 9M such that kjk We assume j in (1) to be continuous and to belong to both L 1 and L1 . When the perturbation is generated by an autonomous ODE, b() with a and b continuous and locally Lipschitz, with is globally asymptotically and locally exponentially stable (GAS and LES), these assumptions are satisfied. In the paper we show that when the unperturbed system is delay-independent stable with a functional of the form (3) and the initial condition is bounded (i.e. kz 0 k s R ! 1), arbitrary small perturbations can cause unbounded trajectories provided the delay is large enough. Therefore it is instructive to treat the delay as the (n + 1)-th state variable when considering semi-global results: with a parametrized perturbation j(t; a), we say for instance that the trajectories of (1) can be bounded semi-globally in z and semi-globally in the delay if for each compact region\Omega ae R n , and 8 a positive number a such that all initial conditions z with z 0 (') , give rise to bounded trajectories when a a. belongs to class , if it is strictly increasing and The symbol k:k is used for the Euclidean norm in R n and by kx; yk we mean . 2 The mechanism of destabilizing perturbations In contrast to linear systems, small perturbations (in the L 1 or L1 sense) are sufficient to destabilize nonlinear differential equations. In the ODE-case, the nonlinear mechanism for instability is well known: small perturbations suffice to make solutions escape to infinity in a finite time, for instance when the interconnection term \Psi is nonlinear in z. This is illustrated with the following example: which can be solved analytically for z to give ds escapes to infinity in a finite time t e which is given by log This last expression shows that the escape time becomes smaller as the initial conditions are chosen larger, and, as a consequence, however fast j(t) would be driven to zero in the first equation of (4), z(0) could always be chosen large enough for the solution to escape to infinity in finite time. In the simple example (4), the perturbation is the output of a stable linear system. Its initial condition j(0) dictates the L1-norm of the per- turbation, while the parameter a controls its L 1 -norm. Making these norms arbitrary small does not result in global stability. This is due to the nonlinear growth of the interconnection term. One may wonder whether the instability mechanism encountered in the ODE situation (4) will persist in the DDE situation ae In contrast to (4), system (7) exhibits no finite escape time. This can be proven by application of the method of steps, i.e. from the boundedness of of and thus of z('). Nevertheless the exponentially decaying input j still causes unbounded solutions in (7): this particular system is easily seen to have an exponential solution z e at . The instability mechanism can be explained by the superlinear divergence of the solutions of Theorem 2.1 has solutions which diverge faster than any exponential function. Proof 2.1 Take as initial condition a strictly positive solution segment z 0 over [\Gamma; 0] with z(0) ? 1. For t 0, the trajectory is monotonically increasing. This means that in the interval [k; z The solution at point k; k 1 is bounded below by the sequence satisfying z which has limit +1. The ratio R z satisfies R and consequently (R tends to infinity. However for an exponential function e at , R = e a and (R \Gamma 1)R is constant. As a consequence, for the system (7), arbitrarily fast exponential decay of cannot counter the blow-up caused by the nonlinearity in z(t \Gamma ), and hence the system is not globally asymptotically stable. The instability mechanism illustrated by (4) and (7) can be avoided by imposing suitable growth restrictions on the interconnection term \Psi. When the unperturbed system is scalar, it is sufficient to restrict the interconnection term to have linear growth in both of its arguments, i.e. This linearity condition is by itself not sufficient however, if the unperturbed system has dimension greater than one. In that case, the interaction of the different components of the state z(t) can still cause "nonlinear" effects leading to unbounded solutions. An illustration of this phenomenon is given by the following system! which was shown in [13] to have unbounded solutions, despite the linearity of the interconnection. The instability is caused by the mutual interaction between z 1 and z 2 when j 6= 0. The following theorem, inspired by theorem 4.7 in [13], provides sufficient conditions for bounded solutions. To prevent the instability mechanism due to interacting states, conditions are put on the Lyapunov functional of the unperturbed system. Theorem 2.2 Assume that the system satisfies Assumption 1.2 and that the interconnection term \Psi(z; grows linearly in its arguments, i.e. satisfies (8). Furthermore if the perturbation (ii) jj dk dz jj jjzjj ck(z), then all trajectories of the perturbed system are bounded, for all values of the time delay. Condition (ii) is sometimes called a polynomial growth condition because it is satisfied if k(z) is polynomial in z. Proof 2.2 Along a trajectory z(t) we have: dz kzk cff 1=fl r ff 2=fl' cff 1=fli cff 1=fli For cannot escape to infinity because k(z(t\Gamma )) is bounded (calculated from the initial condition) and the above estimate can be integrated over the interval since the right hand side is linear in V and j 2 L 1 . For t we can use the estimate k(z(t \Gamma Because this estimate for V is increasing in both of its argument, an upper bound for V (t) along the trajectory is described by with as initial condition W (z the method of steps, it is clear that W cannot escape to infinity in a finite time. From monotonically increasing. As a consequence, for t 2 , W (t) W and and this estimate can be integrated leading to boundedness of lim t!1 sup V (t) because Hence the trajectory z(t) is bounded. Remark: When the interconnection term \Psi is undelayed, i.e. \Psi(z), condition (i) in theorem 2.2 can be dropped [13]. 3 Semi-global results for parametrized perturbation Although no global results can be guaranteed in the absence of growth con- ditions, the examples in the previous section suggest that one should be able to bound the solutions semi-globally in the space of initial conditions by decreasing the size of the perturbation. Therefore we assume that the perturbation is parametrized, We will consider two cases: a) parameter a controls the L 1 - or the L1-norm of j and b) a regulates the shape of a perturbation with fixed L 1 -norm. 3.1 Modifying the L 1 and the L1 norm of the perturbation We first assume that the L 1 -norm of j is parametrized. We have the following result: Theorem 3.1 Consider the system and suppose that the unperturbed system is GAS with the Lyapunov functional Assumption 1.2. If furthermore kj(t; a !1, then the trajectories can be bounded semi-globally both in z and the delay , by increasing a. Proof 3.1 Let 0 be fixed and denote by\Omega the desired stability domain in R n , i.e. such that all trajectories starting in z 0 with z 0 (') 2\Omega for ' 2 [\Gamma; 0] are bounded. Let V c , sup z0 As long as V (t) 2V c , z(t) and belong to a compact set. Hence When a ! 1, the increase of V tends to zero. As a consequence the assumption is valid for 8t 0. Hence the trajectories with initial condition in\Omega are bounded. Note that for a fixed region\Omega increases with and this influences both the value M in the estimation of jk 0 (z)\Psi(z; z(t \Gamma )j and the critical value a of a in order to bound the trajectories. However when belongs to a compact interval [0; ], we can take a sup 2[0; hence bound the trajectories semi-globally in both the state and the delay. The result given above is natural because for a given initial condition, a certain amount of energy is needed for destabilization, expressed mathematically by kjk 1 . However global stability in the state is not possible because the required energy can become arbitrary small provided the initial condition is large enough, see for instance example (4). Later we will discuss why the trajectories cannot be bounded globally in the delay. Now we consider the case whereby the L1-norm of the perturbation is parametrized. Theorem 3.2 Consider the system Suppose that the unperturbed system is GAS with the Lyapunov functional Assumption 1.2. If kj(t; a)k trajectories of the perturbed system can be bounded semi-globally in both z and the delay . Proof 3.2 As in the proof of Theorem 3.1, it is sufficient to prove semi-global stability in the state for a fixed 0. Let\Omega and V c be defined as in Theorem 3.1. \Omega , with ffl ? 0 small. The time derivative of V satisfies When z(t) n\Omega ffl we have, since b is positive definite, Mkjk1 \GammaN for some number N ? 0 provided kjk 1 is small enough. Only when z(t) the value of V can increase with the estimate Mkjk1 . Now we prove by contradiction that all trajectories with initial condition in\Omega are bounded for small suppose that a solution starting in\Omega (with it has to cross the level set 2V c . Assume that this happens for the first time at t . Note that for small kjk 1 , t is large. During the interval [t increase and decrease, but increases, z(t) 2\Omega ffl and the increase \DeltaV is limited: \DeltaV Mkjk1 . When z(t) would be outside\Omega ffl for a time-interval Hence by reducing kj(t; a)k 1 we can make the time-interval \Deltat arbitrary small. On the other hand (for large a), dz when z t is inside\Omega 2 , because f and \Psi map bounded sets into bounded sets. Hence with jt \Deltat we have L\Deltat. Because of (12) we can increase a (reduce kj(t; a)k 1 ) such that L\Deltat ffl and consequently we have: If ffl was chosen such lies inside\Omega , we have a contradiction because this implies W (t ) W c . Hence a trajectory can never cross the level set 2W c and is bounded. The results of Theorems 3.1 and 3.2 are not global in the delay, although the unperturbed system is delay-independent stable. Global results in the delay are generally not possible: we give an example whereby it is impossible to bound the trajectories semi-globally in the state and globally in the delay, even if we make the size of the perturbation arbitrary small w.r.t. the L 1 and L1 -norm. Example 3.1 Consider the following system: z The unperturbed system, i.e. (13) with delay-independent stable. This is proven with the Lyapunov functional Its time derivative \Gammaz 2(z 1 z 2+1 z 2+1 is negative definite: when z 1 62 [1; 3], both terms are negative and in the other case the second term is dominated, because it saturates in z 2 . From this it follows that the conditions of Assumption 1.2 are satisfied. With the perturbation whereby increasing a leads to a reduction of both kjk 1 and kjk 1 , we can not bound the trajectories semi-globally in the state and globally in : for each value of a we can find a bounded initial condition (upper bound independent of a), leading to a diverging solution, provided is large enough: the first equation of (13) has a solution z \Gammaff is the real solution of equation boundedness in of this solution over the interval [\Gamma; 0] (initial condition) is guaranteed. Choose z 2 The above solution for z 1 satisfies: when ff log 5 3 ] and thus A rather lengthy calculation shows that with z 2 and the perturbation (14), the solution of (15) always escapes to infinity in a finite time t f (a). Hence this also holds for the solution of the original system when the delay is large enough such log 5 This result is not in contradiction with the intuition that a perturbation with small L 1 -norm can only cause escape in a finite time when the initial condition is far away from the origin, as illustrated with example (4): in the system (13) with driven away from the origin as long as By increasing the delay in the first equation, we can keep z 1 in this interval as long as desired. Thus the diverging transient of the unperturbed system is used to drive the state away from the origin, far enough to make the perturbation cause escape. 3.2 Modifying the shape of the perturbation We assume that the shape of a perturbation with a fixed L 1 -norm can be controlled and consider the influence of an energy concentration near the origin. In the ODE case this does not allow to improve stability proper- ties. This is illustrated with the first equation of example (4): instability occurs when z(0) 1 R te \Gammas j(s)ds and by concentrating the perburbation the stability domain may even shrink, because the beneficial influence of damping is reduced. In the DDE-case however, when the interconnection term is linear in the undelayed argument, it behaves as linear during one delay interval preventing escape. Moreover, starting from a compact region of initial conditions, the reachable set after one delay interval can be bounded independently of the shape of the perturbation (because of the fixed L 1 -norm). After one delay interval we are in the situation treated in Theorem 3.1. This is expressed in the following theorem. As in Theorem 2.2 the polynomial growth condition prevents hidden nonlinearities due to interacting states. Theorem 3.3 Consider and suppose that the unperturbed system is GAS with the Lyapunov functional Assumption 1.2. Let k(z) satisfy the polynomial growth condition k dk dz kkzk ck(z). Assume that \Psi has linear growth in z(t), kj(t; a)k independent of a and lim a!1 the trajectories of (16) can be bounded semi-globally in z and for all Proof 3.3 Consider a fixed let\Omega be the desired stability domain in R n and let R be such that z 0 (') The interconnection term has linear growth in z, i.e. there exist two class- functions fl 1 and fl 2 such that The time-derivative of the Lyapunov function V satisfies dz jj(t; a)j cV (z t )jj(t; a)j During the interval [0; to\Omega . Therefore, when kzk R, one can bound by a factor c 2 independent of a. Thus and when a trajectory leaves the set fz : kzk Rg at t , because t jj(s;a)jds for some constant M , independently of a. In the above expression, V As a consequence, also k(z) and kz(t)k can be bounded, uniformly in a. Hence at time the state z , i.e. z(t); t 2 [0; ] belongs to a compact region\Omega 2 independently of a. Now we can translate the original problem over one delay interval: at time the initial conditions belong to the bounded region\Omega 2 and with t jj(s; a)jds Because of Theorem 3.1, we can increase a such that all solutions starting in\Omega 2 are bounded. Until now we assumed a fixed . But because is compact, we can take the largest threshold of a for bounded solutions over this interval. Applications: stability of cascades and partial state feedback In this section we use the results given above to study the stability of cascade systems: whereby the subsystem with a functional of the form (3), globally asymptotically and locally exponentially stabilizable, h() is continuous and locally Lipschitz, partial state feedback laws investigate in which situations the equilibrum (z; can be made semiglobally asymptotically stable. 4.1 LaSalle's theorem Because the 'perturbation' y in (17) is generated by a GAS ODE, we can strengthen the boundedness results in the previous section to asymptotic stability results, by applying a generalization to the time-delay case of the classical LaSalle's theorem [7]: Theorem 4.1 (LaSalle Invariance Principle) Let\Omega be a positively invariant set of an autonomous ODE, and suppose that every solution starting in\Omega converges to a set E 2\Omega . Let M be the largest invariant set contained in E, then every bounded solution starting in\Omega converges to M as t !1. This theorem is generalized to functional differential equations by Hale [3]: with the following definition of an invariant set, Definition 4.1 Let fT (t); t 0g be the solution semigroup associated to the functional differential equation , then a set Q ae C is called invariant if LaSalle's theorem can be generalized to: Theorem 4.2 If V is a Lyapunov functional on G and x t is a bounded solution that remains in G, then x t converges to the largest invariant set in We will now outline its use in the context of stability of cascades. Suppose that in the cascade (17)-(18), with a particular feedback law the -subsystem is GAS. Hence there exists a Lyapunov function V () such that we use this Lyapunov function as a Lyapunov functional for the whole cascade, we can conclude from Theorem 4.2 that every solution that remains bounded converges to the largest invariant set where and by the GAS of this is the equilibrium point (z; solutions are either unbounded or converge to the origin. By means of a local version of Theorem 3.1 one can show that the origin of (17)-(18) is stable. Hence the theorems of the previous section are strengthened to asymptotic stability theorems. For example under the conditions of Theorem 3.1, one can achieve semi-global asymptotic stability in both the state and the delay. 4.2 Stabilization of partially linear cascades In the rest of this paper we assume a SISO linear driving system (the - with controllable and consider linear partial state feedback control laws . From the previous sections it is clear that the input y of the z-subsystem can act as a destabilizing disturbance. However, the control can drive the output of the linear system fast to zero. We will investigate under which conditions this is sufficient to stabilize the whole cascade. An important issue in this context is the so-called fast peaking phenomenon [15]. This is a structural property of the -system whereby imposing faster convergence of the output to zero implies larger overshoots which can in turn destabilize the cascade and may form an obstacle to both global and semi-global stabilizability. We start with a short description of the peaking phenomenon, based on [15], and then apply the results of the previous section to the stabilization of the cascade system. 4.2.1 The peaking phenomenon When in the system, the pair (A; B) is controllable, one can always find state feedback laws F resulting in an exponential decay rate with exponent \Gammaa. Then the output of the closed loop system satisfies where fl depends on the choice of the feedback gain. We are interested in the lowest achievable value of fl among different feedback laws and its dependence upon a. Denote by F(a) the collection of all stabilizing feedback laws with the additional property that all observable 1 eigenvalues of (C; A F ), with A \Gammaa. For a given a and F 2 F(a), define the smallest value of fl in (22) as ky(t)ke at where the supremum is taken over all t 0 and all initial conditions satisfying k(0)k 1. Now denote by (a) = inf F2F(a) F . The output of system (21) is said to have peaking exponent s when there exists constants ff such that In [15] one places all eigenvalues to the left of the line for large a. When the output is said to be nonpeaking. The peaking exponent s is a structural property related to the zero- dynamics: when the system has relative degree r, it can be transformed (in- cluding a preliminary feedback transformation) into the normal form [4][1]: ae which can be interpreted as an integrator chain linearly coupled with the zero-dynamics subsystem . Using state feedback the output of an integrator chain can be forced to zero rapidly without peaking [13]. Because of the linear interconnection term, stability of the zero-dynamics subsystem implies stability of the whole cascade. On the contrary, when the zero dynamics are unstable, some amount of energy, expressed by is needed for its stabilization and therefore the output must peak. More precisely we have the following theorem, proven in the appendix. Theorem 4.3 The peaking exponent s equals the number of eigenvalues in the closed RHP of the zero-dynamics subsystem. The definition of the peaking exponent (23) is based on an upper bound of the exponentially weighted output, while its L 1 -norm is important in most of the theorems of section 3. But because the overshoots related to peaking occur in a fast time-scale ( at), there is a connection. For instance we have the following theorem, based on a result of Braslavsky and Middleton [10]: Theorem 4.4 When the output y of system (21) is peaking (s 1), ky(t)k 1 can not be reduced arbitrarily. Proof 4.1 Denote by z 0 an unstable eigenvalue of the zero-dynamics of (21). When a feedback is stabilizing the relation between y and in the Laplace-domain is given by with . The first term vanishes at z 0 because the eigenvalues of the zero dynamics appear as zeros in the corresponding transfer function H(s) and since the feedback F is stabilizing, no unstable pole-zero cancellation occurs at z 0 . Hence 4.2.2 Nonpeaking cascades When the -subsystem is minimum-phase and thus nonpeaking, one can find state feedback laws resulting in and the L 1 -norm of the output can be made arbitrary small. So by Theorem 3.1, the cascade (20) can be stabilized semi-globally in the state and in the delay. 4.2.3 Peaking cascades When the -subsystem is nonminimum phase, the peaking phenomenon forms an obstacle to semi-global stabilizability, because the L 1 -norm of the output cannot be reduced (Theorem 4.4). For ODE-cascades, we illustrate the peaking obstruction with the following example: Example 4.1 In the cascade, the peaking exponent of the -subsystem is 1 (zero dynamics cascade cannot be stabilized semi-globally since the explicit solution of the first equation is given by whereby Hence the solution reaches infinity in a For DDE-cascades, we consider two cases: Case 1: Peaking exponent=1 We can apply theorem 3.3 and obtain semi-global stabilizability in the state and in the delay, when the interconnection term is linear in the undelayed argument: besides (25) the L 1 -norm of y can also be bounded from above since there exists feedback laws such that and because of the fast time-scale property, the energy can be concentrated since 8t ? 0: jy(s)jds ds ! 0 as a !1: Case 2: Peaking exponent ? 1 In this case, we expect the L 1 -norm of y to grow unbounded with a, as suggested by the following example: Example 4.2 When k is considered as the output of the integrator chain, the peaking exponent is be reduced arbitrarily by achieving a faster exponential decay rate. In Proposition 4.32 of [13], it is shown that with the feedback-law k=1 a n\Gammak+1 q solutions of there exists a constant c independent of a such that hence the particular feedback gain is able to achieve an upper bound which corresponds to definition (23), for each choice of the output . It is also shown in [13] that with the same feedback and with as initial condition 1 d such that s such that As a consequence, while the peaking exponent of output With the two following examples, we show that when the energy of an exponentially decaying input perturbation ( e \Gammaat ) grows unbounded with a, an interconnection term which is linear in the undelayed argument, is not sufficient to bound the solutions semi-globally in the state. Because it is hard to deal in general with outputs generated by a linear system with peaking exponent s ? 1, we use an artificial perturbation a s e \Gammaat , which has both the fast time-scale property and the suitable growth-rate of the energy (a Example 4.3 The solutions of equation can not be bounded semi-globally in z by increasing a, for any ? 0, if the 'peaking exponent' s is larger than one. \Sigma Proof: Equation (26) has an exponential solution z e (t), z e ( a a s ff e a Consider the solution z(t) with initial condition z 0 j L ? 0 on [\Gamma; 0]. For and consequently coincides on [o; ] with For large a, expression (27) describes a decreasing lower bound on [; 2 ], since y(t) reaches its maximum in t (a) with t ! 0 as a ! 1. Thus imposing and from this one can argue 2 that z(t) z e (t); t . Thus the trajectory starting with initial condition L on [\Gamma; 0] is unbounded when Le L ff a a a s ff e 3a When s ? 2, for each value of L, the solution is unstable for large a, thus the attraction domain of the stable zero solution shrinks to zero. When a solution starting from L ? \Theta 3 ffff is unstable for large a. Even when the interconnection term contains no terms in z(t), but only delayed terms of z, semi-global results are still not possible in general, as shown with the following example. 2 Intersection at t would imply Example 4.4 The solutions of the system with otherwise, can not be bounded semi-globally in z by increasing a, for any ? 0, when the 'peaking exponent' s is greater than one. \Sigma Proof: When z 1, equation (28) reduces to: which has the following explicit solution, z l (t) = at When the initial condition of (28) is L on [\Gamma; 0], during one delay-interval, one can find an lower bound of the solution by integrating with solution z When a is chosen such that b 1, the expression for z l (t) is valid for t 0. When z u (2) ? z l (2 ), one can argue that for large a, z u (t) ? z l (t); t 2 [; 2 ] and z u (t) describes a lower bound for the solution starting in L for reaches is maximum in t (a) with t ! 0 as 1). Consequently, the trajectory with initial condition L on [\Gamma; 0], is unbounded when 4.2.4 Zero dynamics with eigenvalues on the imaginary axis The situation where the zero dynamics possess eigenvalues on the imaginary axis but no eigenvalues in the open RHP deserves special attention. According to Theorem 4.3, the system is peaking, that is, the L 1 norm of the output cannot be reduced arbitrarily. However this energy can be 'spread out' over a long time interval: it is indeed well known that a system with all its ei- genalues in the closed LHP can be stabilized with a low-gain feedback, as expressed by the following theorem, taken from [13]: Theorem 4.5 If a system stabilizable and the eigenvalues of A 0 are in the closed left half plane, then it can be stabilized with a low-gain control law which for large a satisfies: a The infinity-norm of such a low-gain control signal can be arbitrary reduced, which results, by theorem 3.2, in satisfactory stabilizability results when it also acts as an input disturbance of a nonlinear system. This suggests not to force the output of (21) exponentially fast ( e \Gammaat ) to zero, which results in peaking, but to drive it rapidly without peaking to the manifold on which the dynamics are controlled by the low-gain control action. Mathematically, with and a feedback transformation the normal form of the -subsystem is transformed into Using a high-gain feedback driving e(t) to zero without peaking, as proven in [13], proposition 4.37, one can always force the output to satisfy the constraint a with fl independent of a. A systematic treatment of such high-low gain control laws can be found in [8]. For instance the system, ae is weakly minimum-phase (zero-dynamics 0). With the high-low gain feedback the explicit solution of (30) for large a can be approximated by: a a a Perturbations satisfying constraint (29) can be decomposed in signals with vanishing L 1 and L1 -norm. This suggests the combination of theorems 3.1 and 3.2 to: Theorem 4.6 Consider the interconnected system Suppose that the z-subsystem is GAS with the Lyapunov functional V (z t ) satisfying Assumption 1.2 and the zeros of the -subsystem are in the closed LHP. Then the interconnected system can be made semi-globally asymptotically stable in both [z; ] and the delay, using only partial-state feedback. Proof 4.2 As argumented in the beginning of section 4, the origin (z; (0; Let\Omega be the desired region of attraction in the (z; )- space and choose R such that for all (z R. Because of the assumption on the -subsystem, there exist partial-state feedback laws such that with fl independent of a. Consider the time-interval [0; 1]. Because lim a one can show, as in the proof of theorem 3.1, that by taking a large, the increase of V can be limited arbitrary. Hence for t 1, the trajectories can be bounded inside a compact region\Omega 2 . We can now translate the original problem over one time-unit and since sup flR(e \Gammaat +a we can, by theorem 3.2, increase a until the stability domain Hence all trajectories starting in\Omega are bounded and converge to the origin, because of LaSalle's theorem. Conclusions We studied the effect of bounded input-perturbations on the stability of nonlinear delay equations of the form (1). Global stability results are generally not possible without structural assumptions on the interconnection term because arbitrary small perturbations can lead to unbounded trajectories, even when they are exponentially decaying. In the ODE-case this is caused by the fact that superlinear destabilizing terms can drive the state to infinity in a finite time. Superlinear delayed terms cannot cause finite-escape but can still make trajectories diverge faster than any exponential function. In a second part we dropped most of the structural assumptions on the unperturbed system and the interconnection term and considered semi-global results when the size of the perturbation can be reduced arbitrary. Here we assumed that the unperturbed system is delay-independent stable. When the L 1 or the L1 norm of the perturbations is brought to zero, trajectories can be bounded semi-globally in both the state and the delay. By means of examples we explained mechanisms prohibiting global results in the delay. We also considered the effect of concentrating a perturbation with a fixed L 1 -norm near the origin. This leads to semi-global stabilizability in the state and compact delay-intervals not containing the origin, when the interconnection term is linear in its undelayed arguments. As an application, we studied the stabilizability of partial linear cascades using partial state feedback. When the interconnection term is nonlinear, output peaking of the linear system can form an obstruction to semi-global stabilizability because the L 1 -norm of the output cannot be reduced by achieving a faster exponential decay rate. If we assume that the interconnection term is linear in the undelayed argument and the peaking exponent is one, we have semi-global stabilizability results, because the of the output can be bounded from above while concentrating its energy. Even with the above assumption on the interconnection term, higher peaking exponents may form an obstruction. When the zeros of the linear driving system are in the closed left half plane, we have satisfactory stability results when using a high-low gain feedback, where the output of the linear subsystem can be decomposed in two signals with vanishing L 1 and L1 norm respectively. The main contribution of this paper lies in placing the classical cascade results in the more general framework of bounded input perturbations and its generalization to a class of functional differential equations. Acknowledgements The authors thank W.Aernouts for fruitful discussions on the results presented in the paper. This paper presents research results of the Belgian programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture (IUAP P4/02). The scientific responsibility rests with its authors. --R Asymptotic stability of minimum phase non-linear systems Sufficient conditions for stability and instability of autonomous functional-differential equations Nonlinear control systems. Introduction to the theory and application of functional differential equations Stability of Functional Differential Equations Stability theory of ordinary differential equations. Robustness of nonlinear delay equations w. How violent are fast controls? Slow peaking and low-gain design for global stabilization of nonlinear systems Constructive Nonlinear Control. On the input-to-state stability properties The peaking phenomenon and the global stabilization of nonlinear systems. Tools for semiglobal stabilization by partial state and output feedback. nonlinear small gain theorem for the analysis of control systems with saturation. --TR

nonlinear control;delay equations;cascade systems

587590

Affine Invariant Convergence Analysis for Inexact Augmented Lagrangian-SQP Methods.

An affine invariant convergence analysis for inexact augmented Lagrangian-SQP methods is presented. The theory is used for the construction of an accuracy matching between iteration errors and truncation errors, which arise from the inexact linear system solvers. The theoretical investigations are illustrated numerically by an optimal control problem for the Burgers equation.

Introduction . This paper is concerned with an optimization problem of the following type: minimize J(x) subject to are su#ciently smooth functions and X , Y are real Hilbert spaces. These types of problems occur, for example, in the optimal control of systems described by partial di#erential equations. To solve (P) we use the augmented (sequential quadratic programming) technique as developed in [11]. In this method the di#erential equation is treated as an equality constraint, which is enforced by a Lagrangian term together with a penalty functional. We present an algorithm, which has second-order convergence rate and depends upon a second-order su#cient optimality condition. In comparison with SQP methods the augmented Lagrangian-SQP method has the advantage of a more global behavior. For certain examples we found it to be less sensitive with respect to the starting values, and the region for second-order convergence rate was reached earlier, see e.g. [11, 15, 17]. We shall point out that the penalty term of the augmented Lagrangian functional need not to be implemented but rather that it can be realized by a first-order Lagrangian update. Augmented Lagrangian-SQP methods applied to problem (P) are essentially Newton type methods applied to the Kuhn-Tucker equations for an augmented optimization problem. Newton methods and their behavior under di#erent linear transformations were studied by several authors, see [5, 6, 7, 8, 10], for instance. In this paper, we combine both lines of work and present an a#ne invariant setting for analysis and implementation of augmented Lagrangian-SQP methods in Hilbert spaces. An a#ne invariant convergence theory for inexact augmented Lagrangian-SQP methods is presented. Then the theoretical results are used for the construction of an accuracy matching between iteration errors and truncation errors, which arise from the inexact linear system solvers. The paper is organized as follows. In -2 the augmented Lagrangian-SQP method is introduced and necessary prerequisites are given. The a#ne invariance is introduced Karl-Franzens-Universit?t Graz, Institut f?r Mathematik, Heinrichstra-e 36, A-8010 Graz, Austria Konrad-Zuse-Zentrum f?r Informationstechnik Berlin (ZIB), Takustra-e 7, D- 14195 Berlin, Germany (weiser@zib.de). The work of this author was supported by Deutsche Forschungsgemeinschaft (DFG), Sonderforschungsbereich 273. S. VOLKWEIN AND M. WEISER in -3. In -4 an a#ne invariant convergence result for the augmented Lagrangian-SQP method is presented. Two invariant norms for optimal control problems are analyzed in -5, and the inexact Lagrangian-SQP method is studied in -6. In the last section we report on some numerical experiments done for an optimal control problem for the Burgers equation, which is a one-dimensional model for nonlinear convection-di#usion phenomena. 2. The augmented Lagrangian-SQP method. Let us consider the following constrained optimal control problem minimize J(x) subject to are real Hilbert spaces. Throughout we do not distinguish between a functional in the dual space and its Riesz representation in the Hilbert space. The Hilbert space X - Y is endowed with the Hilbert space product topology and, for brevity, we set Let us present an example for (P) that illustrates our theoretical investigations and that is used for the numerical experiments carried out in -7. For more details we refer the reader to [18]. Example 2.1. Let# denote the interval (0, 1) and set -# for given We define the space W (0, T ) by which is a Hilbert space endowed with the common inner product. For controls the state y # W (0, T ) is given by the weak solution of the unsteady Burgers equation with Robin type boundary conditions, i.e., y satisfies and #y x (t, -)# y(t, -)y x (t, - f(t, -) # denotes the duality pairing between H 1 and its dual. We suppose that f # L Recall that W (0, T ) is continuously embedded into the space of all continuous functions from [0, T ] into L 2(#2 denoted by C([0, T ]; L see e.g. [3, p. 473]. Therefore, (2.1a) makes sense. With every controls u, v we associate the cost of tracking type J(y, u, where z # L 2 (Q) and # > 0 are fixed. Let We introduce the bounded operator whose action is defined by #-e(y, u, v), # L 2 (0,T ;H e(y, u, where for given g # H 1(# the mapping (-# 1(# is the Neumann solution operator associated with . Now the optimal control problem can be written in the form (P). # For c # 0 the augmented Lagrange functional associated with (P) is defined by Y . The following assumption is rather standard for SQP methods in Hilbert spaces, and is supposed to hold throughout the paper. Assumption 1. Let x # X be a reference point such that a) J and e are twice continuously Fr-echet-di#erentiable, and the mappings J # and e # are Lipschitz-continuous in a neighborhood of x # , b) the linearization e # of the operator e at x # is surjective, c) there exists a Lagrange multiplier # Y satisfying the first-order necessary optimality conditions where the Fr-echet-derivative with respect to the variable x is denoted by a prime, and d) there exists a constant # > 0 such that for all # ker e # where ker e # denotes the kernel or null space of e # Remark 2.2. In the context of Example 2.1 we write x proved in [18] that Assumption 1 holds provided #y # - z# L 2 (Q) is su#ciently small. # The next proposition follows directly from Assumption 1. For a proof we refer to [12] and [13], for instance. Proposition 2.3. With Assumption 1 holding x # is a local solution to (P). Furthermore, there exists a neighborhood of is the unique solution of (2.2) in this neighborhood. The mapping x # L c (x, # ) can be bounded from below by a quadratic func- tion. This fact is referred to as augmentability of L c and is formulated in the next proposition. For a proof we refer the reader to [11]. 4 S. VOLKWEIN AND M. WEISER Proposition 2.4. There exist a neighborhood - U of x # and a constant - c # 0 such that the mapping x # L # c (x, # ) is coercive on the whole space X for all x # - U and Remark 2.5. Due to Assumption 1 and Proposition 2.4 there are convex neighborhoods of # such that for all (x, # a) J(x) and e(x) are twice Fr-echet-di#erentiable and their second Fr-echet-deri- vatives are Lipschitz-continuous in U(x # ), c) L # 0 (x, #) is coercive on the kernel of e # (x), d) the point z is the unique solution to (2.2) in U , and there exist - # > 0 and - c # 0 such that for all # X and c # - c. # (2.3) To shorten notation let us introduce the operator # for all (x, # U. Then the first-order necessary optimality conditions (2.2) can be expressed as To find x # numerically we solve (OS) by the Newton method. The Fr-echet-derivative of the operator F c in U is given by denotes the adjoint of the operator e # (x). Remark 2.6. With Assumptions 1 holding there exists a constant C > 0 satisfying (see e.g. in [9, p. 114]), where B(Z) denotes the Banach space of all bounded linear operators on Z. # Now we formulate the augmented Lagrangian-SQP method. Algorithm 1. a) Choose c) Solve for (#x, #) the linear system d) Set back to b). Remark 2.7. Since X and Y are Hilbert spaces, equivalently be obtained from solving the linear system and setting #). Equation (2.7) corresponds to a Newton step applied to (OS). This form of the iteration requires the implementation of e # c) of Algorithm 1 do not - see [11]. In case of Example 2.1 this requires at least one additional solve of the Poisson equation. # 3. A#ne invariance. Let - be an arbitrary isomorphism. We transform the x variable by By. Thus, instead of (P) we study the whole class of equivalent transformed minimization problems By) subject to e( - with the transformed solutions - By Setting I # and G c (y, By, #), the first-order necessary optimality conditions have the form Applying Algorithm 1 to ( OS) we get an equivalent sequence of transformed iterates. Theorem 3.1. Suppose that Assumption 1 holds. Let be the starting iterates for Algorithm 1 applied to the optimality conditions (OS) and ( OS), respectively. Then both sequences of iterates are well-defined and equivalent in the sense of By Proof. First note that the Fr-echet-derivative of the operator G c is given by By, #). (3.3) To prove (3.2) we use an induction argument. By assumption the identity (3.2) holds Now suppose that (3.2) is satisfied for k # 0. This implies - By Using step b) of Algorithm 1 it follows that - By From (3.3), we conclude that (#y, #) Utilizing step d) of Algorithm 1 we get the desired result. Due to the previous theorem the augmented Lagrangian-SQP method is invariant under arbitrary transformations - B of the state space X . This nice property should, of course, be inherited by any convergence theory and termination criteria. In -4 we develop such an invariant theory. Example 3.2. The usual local Newton-Mysovskii convergence theory (cf. [14, p. 412]) is not a#ne invariant, which leads to an unsatisfactory description of the domain of local convergence. Consider the optimization problem subject to # with unique solution x associated Lagrange multiplier Note that the Jacobian #F 0 does not depend on # here, but only on (#). In the context of Remark 2.5 we choose the neighborhood 6 S. VOLKWEIN AND M. WEISER x d) c) a) x x Fig. 3.1. Illustration for Example 3.2. a) Contour lines of the cost functional, the constraint, and the areas occupied by the other subplots. b) Neighborhood U(x # ) (gray) and Kantorovich ball of theoretically assured convergence (white) for the original problem formulation. c) U(x # ) and Kantorovich ball for the "better" formulation. d) U(x # ) and Kantorovich ball for the "better" formulation plotted in coordinates of the original formulation. Defining the Newton-Mysovskii theory essentially guarantees convergence for all starting points in the Kantorovich region Here, # denotes the spectral norm for symmetric matrices and # 2 is the Euclidean norm. For our choice of U , resulting in # 1.945 and a section of the Kantorovich region at # is plotted in Figure 3.1-b). A di#erent choice of coordinates, however, yields a significantly di#erent result. With the transformation problem (3.4) can be written as subject to For the same neighborhood U , the better constants # 1.859 and result. Again, a section of the Kantorovich region at # is shown in Figure 3.1-c). Transformed back to (#) space, Figure 3.1-d) reveals a much larger domain of theoretically assured convergence. This "better" formulation of the problem is, however, not at all evident. In contrast, a convergence theory that is invariant under linear transformations, automatically includes the "best" formulation. # Remark 3.3. The invariance of Newton's method is not limited to transformations of type (3.1). In fact, Newton's method is invariant under arbitrary transformations of domain and image space, i.e., it behaves exactly the same for AF c Because F c has a special gradient structure in the optimization context, meaningful transformations are coupled due to the chain rule. Meaningful transformations result from transformations of the underlying optimization problem, i.e., transformations of the domain space and the image space of the constraints. Those are of the type x # . For such general transformations there is no possibility to define a norm in an invariant way, since both the domain and the image space of the constraints are transformed x). For this reason, di#erent types of transformations have been studied for di#erent problems, see e.g. [6, 7, 10]. # 4. A#ne invariant convergence theory. To formulate the convergence theory and termination criteria in terms of an appropriate norm, we use a norm that is invariant under the transformation (3.1). Definition 4.1. Let z # U . Then the norms # z : Z # R, z # U , are called a#ne invariant for (OS), if #F c (-z)#z# z for all - z # U and #z # Z. (4.1) We call {# z } z#U a #-continuous family of invariant norms for (OS), if for every r, #z # Z and z # U such that z +#z # U . Using a#ne invariant norms we are able to present an a#ne invariant convergence theorem for Algorithm 1. Theorem 4.2. Assume that Assumption 1 holds and that there are constants #-continuous family of a#ne invariant norms {# z } z#U , such that the operator #F c satisfies z for s, # [0, 1], z # U , and #z # Z such that co{z, z +#z} # U , where co A denotes the convex hull of A. For k # N let h Suppose that h 0 < 2 and that the level set L(z 0 ) is closed. Then, the iterates stay in U and the residuals converge to zero at a rate of k . 8 S. VOLKWEIN AND M. WEISER Additionally, we have #F c (z k+1 )# z k #F c (z k )# z k . (4.5) Proof. By induction, assume that L(z k ) is closed and that h k < 2 for k # 0. Due to Remark 2.5 the neighborhood U is assumed to be convex, so that z all # [0, 1]. From #F c (z k )#z we conclude that ds ds for all # [0, 1]. Applying (4.2), (4.3), h z k ds holds. If z k +#z k there exists an - # [0, 1] such that z k i.e., which is a contradiction. Hence, z k+1 z k /2. Thus, we have h k+1 # h 2 closed, every Cauchy sequence in L(z k+1 ) converges to a limit point in L(z k ), which is, by (4.4) and the continuity of the norm, also contained in L(z k+1 ). Hence, L(z k+1 ) is closed. Finally, using in (4.6), the result (4.5) is obtained. Remark 4.3. We choose simplicity over sharpness here. The definition of the level set L(z) can be sharpened somewhat by a more careful estimate of the term Theorem 4.2 guarantees that lim k# h To ensure that z k z # in Z as k # we have to require that the canonical norm # Z on Z can be bounded appropriately by the a#ne invariant norms # z Corollary 4.4. If, in addition to the assumptions of Theorem 4.2, there exists a constant - C > 0 such that for all # Z and z # U, then the iterates converge to the solution z Proof. By assumption and Theorem 4.2 we have Thus, {z k } k#N is a Cauchy sequence in L(z 0 ) # U . Since L(z 0 ) is closed, the claim follows by Remark 2.5-d). For actual implementation of Algorithm 1 we need a convergence monitor indicating whether or not the assumptions of Theorem 4.2 may be violated, and a termination criterion deciding whether or not the desired accuracy has been achieved. From (4.5), a new iterate z k+1 is accepted, whenever #F c (z k+1 )# z k < #F c (z k )# z k . (4.7) Otherwise, the assumptions of Theorem 4.2 are violated and the iteration is considered as to be non-convergent. The use of the norm # z k for both the old and the new iterate permits an e#cient implementation. Since in many cases the norm #F c (z k+1 )# z k is defined in terms of #z derivative need not be evaluated at the new iterate. If a factorization of #F c (z k ) is available via a direct solver, it can be reused at negligible cost even if the convergence test fails. If an iterative solver is used, #z k+1 in general provides a good starting point for computing #z k+1 , such that the additional cost introduced by the convergence monitor is minor. The SQP iteration will be terminated with a solution z k+1 as soon as with a user specified tolerance TOL. Again, the use of the norm # z k allows an e#cient implementation. 5. Invariant norms for optimization problems. What remains to be done is the construction of a #-continuous family of invariant norms. In this section we introduce two di#erent norms. 5.1. First invariant norm. The first norm takes advantage of the parameter c in the augmented Lagrangian. As we mentioned in Remark 2.5, there exists a - c # 0 such that L # c (z) is coercive on X for all z # U and c # - c. Hence, the operator belongs to B(Z) for all c # - c. Let us introduce the operator I # for all z # U and c # 0. (5.1) Since L # c (z) is self-adjoint for all z # U , S c (z) is self-adjoint as well. Due to (2.3) the operator S c (z) is coercive for all z # U and c # - c. Thus, for all z # U is a norm on Z for c # - c. Proposition 5.1. Let c # - c. Then, for every z # U the mapping S. VOLKWEIN AND M. WEISER defines an a#ne invariant norm for (2.2). Proof. Let z # U be arbitrary. Since #S 1/2 defines a norm on Z for c # - c and #F c (z) is continuously invertible by Remark 2.6, it follows that # z is a norm on Z. Now we prove the invariance property (4.1). Let - L c denote the augmented Lagrangian associated with the transformed problem (3.1). Then we have - setting - #r# From (3.3) we conclude that with # U . Using (5.3) and (5.4) we obtain which gives the claim. In order to show the #-continuity (4.2) required for Theorem 4.2, we need the following lemma. Lemma 5.2. Suppose that c # - c and that there exists a constant # 0 such that for all # Z, z # U and #z # Z such that z + #z # U . Then we have where Y Proof. Let # Z and z # U . From (5.1) and (5.2) we infer By assumption S c (z) is continuously invertible. Utilizing the Lipschitz assump- tion (5.5) the second additive term on the right-hand side can be estimated as #. Note that Y This implies Inserting (5.7) into (5.6) the claim follows. Proposition 5.3. Let all hypotheses of Lemma 5.2 be satisfied. Then {# z } z#U is a #(3 )/2-continuous family of invariant norms with #F c (z)# z (5.8) for all # Z and z # U , where - introduced in (2.3). Proof. From (5.3) it follows that We estimate the additive terms on the right-hand side separately. Using Lemma 5.2 we find Applying (5.3) and (5.5) we obtain Hence, using 12 S. VOLKWEIN AND M. WEISER and it follows that {# z } z#U is a #(3 +C e )/2-continuous family of invariant norms. Finally, from z Z we infer (5.8). 5.2. Second invariant norm. In -5.1 we introduced an invariant norm provided the augmentation parameter in Algorithm 1 satisfies c # - c. But in many applications the constant - c is not explicitly known. Thus, L # c (x, #) -1 need not to be bounded for c # [0, - c), so that S c (x, #) given by (5.1) might be singular. To overcome these di#culties we define a second invariant norm that is based on a splitting X , such that at least the coercivity of L # 0 (x, #) on ker e # (x) can be utilized. Even though the thus defined norm can be used with larger value of c may improve the global convergence properties - see [16, Section 2.3]. To begin with, let us introduce the bounded linear operator T c Lemma 5.4. For every (x, # U and c # 0 the operator T c (x, #) is an isomorphism Proof. Let r # X be arbitrary. Then the equation T c (x, equivalent with Due to Remark 2.6 the operator #F c (x, #) is continuously invertible for all (x, # U and c # 0. Thus, # is uniquely determined by (5.9), and the claim follows. We define the bounded linear operator R c I # for (x, # U and c # 0. (5.10) Note that R c (x, #) is coercive and self-adjoint. Next we introduce the invariant norm z Y for z # U and (r 1 , r 2 ) T # Z. To shorten notation, we write #R c (z) 1/2 T c (z) the first additive term. Proposition 5.5. For every z # U the mapping given by (5.11) is an a#ne invariant norm for (OS), which is equivalent to the usual norm on Z. Proof. Let z # U be arbitrary. Since R c (z) is coercive and T c (z) is continuously invertible, it follows that # z defines a norm which is indeed equivalent to the usual norm on Z. Now we prove the invariance property (4.1). For (x, By, # U we have I # . (5. Utilizing (3.3), (5.11) and (5.12) the invariance property follows. The following proposition guarantees that {# z } z#U is a #-continuous family of invariant norms for (OS). Proposition 5.6. Suppose that there exists a constant # 0 such that for all # Z, z # U and #z # Z such that z + #z # U . Then we have For the proof of the previous proposition, we will use the following lemmas. Lemma 5.7. With the assumption of Proposition 5.6 holding and z = (x, #) it follows that for all # ker e # Proof. Let Using (5.10) and (5.11) we obtain For all c # 0 the operator R c (z) is continuously invertible. Furthermore, R c (z) is self-adjoint. Thus, applying (5.13) and the second additive term on the right-hand side of (5.14) can be estimated as Inserting this bound in (5.14) the claim follows. Lemma 5.8. Let the assumptions of Theorem 5.6 be satisfied. Then # z for all r # X. Proof. For arbitrary r # X we set # 1 , Using (5.9) and (5.13) we estimate # z # z 14 S. VOLKWEIN AND M. WEISER so that the claim follows. Proof of Proposition 5.6. Let z, z Utilizing (5.11), Lemmas 5.7 and 5.8 we find # z # z and therefore z . Hence, {# z } z#U is a 3#/2-continuous family of invariant norms. Remark 5.9. Note that the Lipschitz constant of the second norm does not involve C e and hence is independent of the choice of c. In contrast, choosing c too small may lead to a large Lipschitz constant of the first norm and thus can a#ect the algorithm. # Example 5.10. Let us return to Example 3.2. Using the second norm with the theoretically assured, a#ne invariant domain of convergence is shown in Figure 5.1, to be compared with Figures 3.1 b) and d). Its shape and size is clearly more similar to the non-invariant domain of convergence for the "better" formulation, and, by definition, does not change when the coordinates change. # x Fig. 5.1. Illustration for Examples 3.2 and 5.10. Neighborhood U(x # ) (gray) and a#ne invariant domain of theoretically assured convergence (white). 5.3. Computational e#ciency. The a#ne invariance of the two norms developed in the previous sections does not come for free: the evaluation of the norms is more involved than the evaluation of some standard norm. Nevertheless, the computational overhead of the first norm defined in -5.1 is almost negligible, since it can in general be implemented by one additional matrix vector multiplication. It requires, however, a su#ciently large parameter c. On the other hand, the second norm defined in -5.2 works for arbitrary c # 0, but requires one additional system solve with the same Jacobian but di#erent right hand side. In case a factorization of the matrix is available, the computational overhead is negligible - compare the CPU times of the exact Newton method in -7. If, however, the system is solved iteratively, the additional system solve may incur a substantial cost, in which case the first norm should be preferred. 5.4. Connection to the optimization problem. When solving optimization problems of type (P), feasibility optimality are the relevant quan- tities. This is well reflected by the proposed norms # z . Let z = (x, #) and Using Taylor's theorem (see [19, p. 148]) and the continuity of L # 0 , we obtain for the first norm z Y Y Y Y The second norm is based on the partitioning F c (x, correspondingly on a splitting of the Newton correction into a optimizing direction tangential to the constraints manifold and a feasibility direction #F c (x, # 1 , we have for z Y Y Y Y Y Recall that Thus, in the proximity of the solution, both a#ne invariant norms measure the quantities we are interested in when solving optimization problems, in addition to the error in the Lagrange multiplier and the optimizing direction's Lagrange multiplier component, respectively. 6. Inexact augmented Lagrangian-SQP methods. Taking discretization errors or truncation errors resulting from iterative solution of linear systems into account, we have to consider inexact Newton methods, where an inner residual remains z Such inexact Newton methods have been studied in a non a#ne invariant setting by Dembo, Eisenstat, and Steihaug [4], and Bank and Rose [1]. S. VOLKWEIN AND M. WEISER With slightly stronger assumptions than before and a suitable control of the inner residual, a similar convergence theory can be established as in -4. Note that exact a#ne invariance is preserved only in case the inner iteration is a#ne invariant, too. Theorem 6.1. Assume that Assumption 1 holds and that there are constants #-continuous family of a#ne invariant norms {# z } z#U , such that the operator #F c satisfies z (6.2) for s, # [0, 1], z # U , and #z # Z such that z and define the level sets Suppose that z 0 # U and that L(z 0 ) is closed. If the inner residual r k resulting from the inexact solution of the Newton correction (6.1) is bounded by where then the iterates stay in U and the residuals converge to zero as k # at a rate of #F c (z k+1 )# z k+1 #F c (z k )# z k . (6.5) Proof. Analogously to the proof of Theorem 4.2, one obtains ds (6.6) for all # [0, 1]. Using (6.6), (6.2), (4.2), and (6.3), we find for # [0, 1] z k ds z k z k . From (6.1) and (6.3) we have and thus, setting # in (6.7) and #F c (z k )#z k z k and using (6.4) it follows that From (6.4) we have #F c (z k )#z k If z k+1 # U , then there is some # [0, 1] such that co{z k , z k #/(2#F c (z k )# z k , which contradicts (6.10). Thus, z k+1 # U . Furthermore, inserting and therefore L(z k+1 ) # L(z k ) is closed. The next corollary follows analogously as Corollary 4.4. Corollary 6.2. If, in addition to the assumptions of Theorem 6.1, there exists a constant - C > 0 such that C#F c (z)# z for all # Z and z # U , then the iterates converge to the solution z of (OS). For actual implementation of an inexact Newton method following Theorem 6.1 we need to satisfy the accuracy requirement (6.4). Thus, we do not only need an error estimator for the inner iteration computing # k , but also easily computable estimates [#] and [#] for the Lipschitz constants # and # in case no suitable theoretical values can be derived. Setting in (6.6), we readily obtain z k and hence a lower bound z k #. Unfortunately, the norms involve solutions of Newton type systems and therefore cannot be computed exactly. Assuming the relative accuracy of evaluating the norms are - respectively, we define the actually computable estimate S. VOLKWEIN AND M. WEISER We would like to select a # k such that the accuracy matching condition (6.4) is Unfortunately, due to the local sampling of the global Lipschitz constant # and the inexact computation of the norms, the estimate [#] k is possibly too small, translating into a possibly too large tolerance for the inexact Newton correction. In order to compensate for that, we introduce a safety factor # < 1 and require the approximate accuracy matching condition to hold. An obvious choice for # would be (1 From Propositions 5.3 and 5.6 we infer that # is of the same order of magnitude as #. Thus we take the estimate currently ignoring C e when using the first norm. Again, the convergence monitor (4.7) can be used to detect non-convergence. In the inexact setting, however, the convergence monitor may also fail due to # k chosen too large. Therefore, whenever (4.7) is violated and a reduction of # k is promising (e.g. , the Newton correction should be recomputed with reduced # k . Remark 6.3. If an inner iteration is used for approximately solving the Newton equation (6.1) which provides the orthogonality relation (#z k , #z k in a scalar product (-) z k that induces the a#ne invariant norm, the estimate (6.11) can be tightened by substituting k . Furthermore, the norm #z k of the exact Newton correction is computationally available, which permits the construction of algorithms that are robust even for large inaccuracies # k . The application of a conjugate gradient method that is confined to the null space of the linearized constraints [2] to augmented Lagrangian-SQP methods can be the focus of future research. # 7. Numerical experiments. This section is devoted to present numerical tests for Example 2.1 that illustrate the theoretical investigations of the previous sections. To solve (P) we apply the so-called "optimize-then-discretize" approach: we compute an approximate solution by discretizing Algorithm 1, i.e., by discretizing the associated system (2.6). In the context of Example 2.1 we have x (#y, To reduce the size of the system we take advantage of a relationship between the SQP steps #u, #v for the controls and the SQP step # for the Lagrange multiplier. In fact, from we infer that of Algorithm 1. Inserting (7.1) into (2.6) we obtain a system only in the unknowns (#y, #). Note that the second Fr-echet-derivative of the Lagrangian is given by . The solution (#y, #u, #v, #) of (2.6) is computed as follows: First we solve #y x (-, #y x (-, in# , - z in Q, in# , y and #. Then we obtain #u and #v from (7.1). For more details we refer the reader to [18]. For the time integration we use the backward Euler scheme while the spatial variable is approximated by piecewise linear finite elements. The programs are written in MATLAB, version 5.3, executed on a Pentium III 550 MHz personal computer. Run 7.1 (Neumann control). In the first example we choose The grid is given by 50 for 50 for To solve (2.1) for we apply the Newton method at each time step. The algorithm needs one second CPU time. The value of the cost functional is 0.081. Now we turn to the optimal control problem. We choose and the desired state is z(t, In view of the choice of z and the nonlinear convection term yy x in (2.1b) we can interprete this problem as determining u in such a way that it counteracts the uncontrolled dynamics which smoothes the discontinuity at transports it to the left as t increases. The discretization of (7.2) leads to an indefinite system # . (7.3) As starting values for Algorithm 1 we take y S. VOLKWEIN AND M. WEISER t-axis x-axis optimal Optimal controls u (t) and v (t) t-axis Fig. 7.1. Run 7.1: residuum t #y(t, - z(t, -)# L 2(# and optimal controls. Table Run 7.1-(i): decay of #Fc (z k+1 )# z k for the first norm. (i) First we solve (7.3) by an LU-factorization (MATLAB routine lu) so that the theory of -4 applies. According to -4 we stop the SQP iteration if In case #F c (z 0 )# z 0 is very large, the factor 10 -3 on the right-hand side of (7.4) might be too big. To avoid this situation Algorithm 1 is terminated if (7.4) and, in addition, holds. The augmented Lagrangian-SQP method stops after four iterations. The CPU times for di#erent values of c can be found in Tables 7.6 and 7.7. Let us mention that for the algorithm needs 102.7 seconds and for divergence of Algorithm 1. As it was proved in [15] the set of admissible starting values reduces whenever c enlarges. The value of the cost functional is 0.041. In Figure 7.1 the residuum t #y(t, - z(t, -)# L 2(# for the solution of (2.1) for as for the optimal state is plotted. Furthermore, the optimal controls are presented. The decay of #F c (z k+1 )# z k , for the first invariant norm given by (5.3) and for di#erent values of c is shown in Table 7.1. Recall that the invariant norm is only defined for c # - c. Unfortunately, the constant - c # 0 is unknown. We proceed as follows: Choose a fixed value for c and compute Table Run 7.1-(i): values of [#] k for di#erent c. Table Run 7.1-(i): decay of #Fc (z k+1 )# z k for the second norm. in each level of the SQP iteration. Whenever [#] k is greater than zero, we have coercivity in the direction of the SQP step. Otherwise, c needs to be increased. In Table 7.2 we present the values for [#] k . We observed numerically that [#] k is positive increased if c increased. Next we tested the second norm introduced in (5.11) for Again, the augmented method stops after four iterations and needs 97.4 seconds CPU time. Thus, both invariant norms lead to a similar performance of Algorithm 1. The decay of #F c (z k+1 )# z k can be found in Table 7.3. (ii) Now we solve (7.3) by an inexact generalized minimum residual (GMRES) method (MATLAB routine gmres). As a preconditioner for the GMRES method we took an incomplete LU-factorization of the matrix by utilizing the MATLAB function luinc(D,1e-03). Here, the matrix P is the discretization of the heat operator y t - #y xx with the homogeneous Robin boundary conditions #y x (-, The same preconditioner is used for all Newton steps. We chose # In -6 we introduced estimators for the constants # and #, denoted by [#] k and [#] k , respectively. Thus, for k # 0 we calculate [#] k and [#] k , and then we determine # k+1 as follows: while do For the first norm #F c (z k )#z k z k is already determined by the computation of the previous Newton correction. Thus we have but in case of the second norm, #F c (z k )#z k z k has to be calculated with a given tolerance - # k . In our tests we take - # k for all k # 0. As starting values we choose # We test four strategies for the choice of - for It turns out that for - we obtain the best performance with respect to CPU times. Hence, in the following 22 S. VOLKWEIN AND M. WEISER Table Run 7.1-(ii): decay of #Fc (z k )# z k for the first norm with # Table Run 7.1-(ii): values of [#] k for # test examples we take # The decay of #F (z k )# z k is presented in Table 7.4. Algorithm 1 stops after at most seven iterations. Let us mention that for c the estimates [#] k for the coercivity constant are positive. In particular, for the augmented Lagrangian-SQP method has the best performance. In Table 7.5 the values of the estimators [#] k are presented. In Table 7.6 the CPU times for the first norm are presented. It turns out that the performance of the inexact method does not change significantly for di#erent values of # k . Since we have to solve an additional linear system at each level of the SQP iteration in order to compute the second norm, the first norm leads to a better performance of the inexact method with respect to the CPU time. Compared to part (i) the CPU time is reduced by about 50% if one takes the first norm. In case of the second norm the reduction is about 45% for 7.7. Finally we test the inexact method using decreasing # k . We choose # # 1. It turns out that this strategy speeds up the inexact method for both norms, as can be expected from the theoretical complexity model developed in [7]. Run 7.2 (Robin control). We choose -10 in # 0, T and y The desired state was taken to be z(t, (i) First we again solve (7.3) by an LU-factorization. We take the same starting values and stopping criteria as in Run 7.1. The augmented Lagrangian-SQP method stops after four iteration and needs 105 seconds CPU time. The discrete optimal solution is plotted in Figure 7.2. From Table 7.8 it follows that (4.7) is satisfied exact 97.5 96.8 96.9 Table Run 7.1-(ii): CPU times in seconds for the first norm. first norm second norm exact 97.5 97.4 Table Run 7.1-(ii): CPU times in seconds for both norms and numerically. Let us mention that [#] 0 , . , [#] 3 are positive for c For the needed CPU times we refer to Tables 7.10 and 7.11. (ii) Now we solve (7.3) by an inexact GMRES method. As a preconditioner we take the same as in Run 7.1. We choose # k. The decay of #F (z k )# z k is presented in Table 7.9. As in part (i) we find that [#] k > 0 for all test runs. The needed CPU times are shown in Tables 7.10 and 7.11. As we can see, the inexact augmented Lagrangian-SQP method with GMRES is much faster than the exact one using the LU-factorization. For the first norm the CPU time is reduced by about 55%, and for the second norm by about 50% for # k # {0.3, 0.4, 0.5, 0.6, 0.7}. Moreover, for our example the best choice for c is . For smaller values of # k the method does not speed up significantly. As in Run 7.1 we test the inexact method using decreasing # k . Again we choose # 1. As in Run 7.1, this strategy speeds up the inexact method significantly for both norms. The reduction is about 9% compared to the CPU times for fixed # k , compare Table 7.11. Table Run 7.2-(i): decay of #Fc (z k+1 )# z k for di#erent c. S. VOLKWEIN AND M. WEISER0.501 -0.50.5t-axis Optimal state y * (t,x) -113Optimal controls u (t) and v * (t) t-axis Fig. 7.2. Run 7.2: optimal state and controls. Table Run 7.2-(ii): decay of #Fc (z k )# z k for # exact 105.1 105.7 105.7 Table Run 7.2-(ii): CPU times in seconds for the first norm. first norm second norm exact 105.1 105.5 Table Run 7.2-(ii): CPU times in seconds for both norms and --R Global approximate newton methods A subspace cascadic multigrid method for Mortar elements. Mathematical Analysis and Numerical Methods for Science and Technology Newton Methods for Nonlinear Problems. Local inexact Newton multilevel FEM for nonlinear elliptic problems Finite Element Methods for Navier-Stokes Equations Inexact Gauss Newton Methods for Parameter Dependent Nonlinear Problems Augmented Lagrangian-SQP-methods in Hilbert spaces and application to control in the coe#cient problems Optimization by Vector Space Methods First and second-order necessary and su#cient optimality conditions for infinite-dimensional programming problems Iterative solution of nonlinear equations in several variables Nonlinear Functional Analysis and its Applications --TR --CTR Anton Schiela , Martin Weiser, Superlinear convergence of the control reduced interior point method for PDE constrained optimization, Computational Optimization and Applications, v.39 n.3, p.369-393, April 2008 S. Volkwein, Lagrange-SQP Techniques for the Control Constrained Optimal Boundary Control for the Burgers Equation, Computational Optimization and Applications, v.26 n.3, p.253-284, December

affine invariant norms;burgers' equation;nonlinear programming;multiplier methods

587626

On Reachability Under Uncertainty.

The paper studies the problem of reachability for linear systems in the presence of uncertain (unknown but bounded) input disturbances that may also be interpreted as the action of an adversary in a game-theoretic setting. It defines possible notions of reachability under uncertainty emphasizing the differences between reachability under open-loop and closed-loop control. Solution schemes for calculating reachability sets are then indicated. The situation when observations arrive at given isolated instances of time leads to problems of anticipative (maxmin) or nonanticipative (minmax) piecewise open-loop control with corrections and to the respective notions of reachability. As the number of corrections tends to infinity, one comes in both cases to reachability under nonanticipative feedback control. It is shown that the closed-loop reach sets under uncertainty may be found through a solution of the forward Hamilton--Jacobi--Bellman--Isaacs (HJBI) equation. The basic relations are derived through the investigation of superpositions of value functions for appropriate sequential maxmin or minmax problems of control.

Introduction Recent developments in real-time automation have promoted new interest in the reachability problem-the computation of the set of states reachable by a controlled process through available controls. Being one of the basic problems of control theory, it was studied from the very begining of investigations in this field (see [18]). The problem was usually studied in the absence of disturbances, under complete information on the system equations and the constraints on the control variables. It was shown, in particular, that the set of states Research supported by National Science Foundation Grant ECS 9725148. We thank Oleg Botchkarev for the figures. reachable at given time t under bounded controls is one and the same, whether one uses open-loop or closed-loop (feedback) controls. It was also indicated that these "reachability sets" could be calculated as level sets for the (perhaps generalized) solutions to a "forward" Hamilton- Jacobi-Bellman equation [18], [19], [3], [15], [17]. However, in reality the situation may be more complicated. Namely, if the system is subject to unknown but bounded disturbances, it may become necessary to compute the set of states reachable despite the disturbances or, if exact reachability is impossible, to find guaranteed errors for reachability. These questions have implicitly been present in traditional studies on feedback control under uncertainty for continuous-time systems, [10], [28], [4], [9], [12]. They have also appeared in studies on hybrid and other types of transition systems [1], [29], [21], [5]. This leads us to the topic of the present paper which is the investigation of reachability under uncertainty for continuous-time linear control systems subjected to unknown input disturbances, with prespecified geometric (hard) bounds on the controls and the unknowns. The paper indicates various notions of reachability, studies the properties of respective reach sets and indicates routes for calculating them. The first question here is to distinguish, whether reachability under open-loop and closed-loop controls yield the same reach sets. Indeed, since closed-loop control is based on better information, namely, on the possibility of continuous on-line observations of the state space variable (with no knowledge of the disturbance), it must produce, generally speaking, a result which is at least "not worse," for example, than the one by an open-loop control which allows no such observations, but only the knowledge of the initial state, with no knowledge of the disturbance. An open-loop control of the latter type is further referred to as "nonanticipative." However, there are many other possibilities of introducing open-loop or piecewise open-loop controls, with or without the availability of some type of isolated on-line measurements of the state space variable, as well as with or without an "a priori" knowledge of the disturbance. Thus, in order to study the reachability problem in detail, we introduce a hierarchy of reachability problems formulated under an array of different "intermediate" information conditions. These are formulated in terms of some auxiliary extremal problems of the maxmin or minmax type. Starting with open-loop controls, we first distinguish the case of anticipative control from nonanticipative control. The former, for example, is when a reachable set, from a given initial state x 0 , at given time , is defined as the set of such states x, that for any admissible disturbance given in advance, for the whole interval under con- sideration, there exists an admissible control that steers the system to a -neighborhood g. Here the respective auxiliary extremal problem is of the maxmin type.(Maximum in the disturbance and minimum in the control). On the other hand, for the latter the disturbance is not known in advance. Then the reachability set from a given initial state is defined as the set X of such states x whose -neighborhoods B (x) may be reached with some admissible control, one and the same for all admissible disturbances, whatever they be. Now the respective auxiliary problem is of the minmax type. It is shown that always X and that the closed-loop reach set X attained under nonanticipative, but feedback control lies in between, namely, There also are some intermediate situations when the observations of the state space variable arrive at given N isolated instants of time. In that case one has to deal with reachability under possible corrections of the control at these N time instants. Here again we distinguish between corrections implemented through anticipative control (when the future disturbance is known for each time interval in between the corrections) and nonanticipative control, when it is unknown. The respective extremal problems are of sequential maxmin and types accordingly and the controls are piecewise open-loop: at isolated time instants of correction comes information on the state space variable, while in between these the control is open-loop (either anticipative or not). Both cases produce respective sequences reach sets". The relative positions of the reach sets in the hierarchical scheme are as follows Finally, in the limit, as the number of corrections N tends to infinity, both sequences of reachability sets converge to the closed-loop reach set 1 . The adopted scheme is based on constructing superpositions of value functions for open-loop control problems. In the limit these relations reflect the Priciple of Optimality under set-membership uncertainty. This principle then allows one to describe the closed loop reach set as a level set for the solution to the forward HJBI (Hamilton-Jacobi-Bellman-Isaacs) equation. The final results are then presented either in terms of value functions for this equation or in terms of set-valued relations. Schemes of such type have been used in synthesizing solution strategies for differential games and related problems, and were constructed in backward time, [23], [11], [27], [28]. The topics of this paper were motivated by applied problems and also by the need for a theoretical basis for further algorithmic schemes. As indicated in the sequel, this is true when all the sets involved are nonempty and when the problems satisfy some regularity conditions. dynamics. Reachability under open loop controls In this section we introduce the system under consideration and define two types of open-loop reachability sets. Namely, we discuss reachability under unknown but bounded disturbances for the system with continuous matrix coefficients A(t); B(t); C(t). Here x 2 IR n is the state and u 2 IR p is the control that may be selected either as an open loop control OLC-a Lebesgue-measurable function of time t, restricted by the inclusion or as a closed-loop control CLC-a set-valued strategy Here v 2 IR q is the unknown input disturbance with values P(t); Q(t) are set-valued continuous functions with convex compact values. The class of OLC's u(\Delta) bounded by inclusion (2) is denoted by UO and the class of input disturbances v(\Delta) bounded by (4) as VO . The strategies U are taken to be in U C -the class U C of CLC's that are multivalued maps U(t; x) bounded by the inclusion (3), which guarantee the solutions to equation (1), (which now turns into a differential inclusion), for any Lebesgue-measurable function v(\Delta). 2 We distinguish two types of open loop reach sets-the maxmin type and the minmax type. As we will see in the next Section, the names maxmin and minmax assigned to these sets are due to the underlying optimization problems used for their calculation. Definition 1.1 An open loop reach set (OLRS) of the maxmin type (from set X is the set vectors x such that for every disturbance there exist an initial state x 0 2 X 0 and an OLC u(t) 2 P(t) which steer the trajectory , from state x The set X 0 is assumed convex and compact (X 2 For example, the class of set-valued functions with values in compIR n , upper semicontinuous in x and continuous in t. turns to be empty, one may introduce the open loop -reachable set as in Definition 1.1 except that (5) is replaced by Here is the ball of radius with center x. Thus the OLRS of the maxmin type is the set of points x 2 IR n that can be reached, for any disturbance v(t) 2 Q(t) given in advance, for the whole interval t 0 t , from some point x(t The open loop -reach set is the set of points x 2 IR n whose -neighborhood may be reached, for any disturbance v(t) given in advance, through some x(t 0 By taking 0 large enough, we may assume to be the unique trajectory corresponding to x(t 0 u(\Delta) and disturbance v(\Delta). Then is the reach set in the variable u(\Delta) 2 UO (at time t from set X 0 ) with fixed disturbance input v(\Delta). Lemma 1.1 This formula follows from Definition 1.1. Recall the definition of the geometrical difference P \GammaQ of sets Then directly from (1) one gets Z S(s; )P(s)ds Z S(s; )(\GammaQ(s))ds: (7) Here S(s; t) stands for the matrix solution of the adjoint equation In other words the set is the geometric difference of two "ordinary" reach sets, namely, the set X(t; t taken from X(t 0 calculated in the variable u, with v(t) j 0, and the set taken from x(t 0 calculated in the variable v, with u(\Delta) j 0. This simple geometrical interpretation is of course due to the linearity of (1). For the -reachable set, we have the following lemma. Lemma 1.2 The set may be expressed as also Remark 1.1. Definition (8) of may also be rewritten as We now define another class of open-loop reach sets under uncertainty-the OLRS of the Definition 1.2 An open loop -reach set (OLRS) of the minmax type (from set is the set X for each of which there exists a control u(t) 2 P(t) that assigns to each v(t) 2 Q(t) a vector x such that the respective trajectory ends in x[ Thus the -OLRS of minmax type consists of all x whose -neighborhood B (x) contains the states x[ ] generated by system (1) under some control u(t) 2 P(t) and all fv(t) 2 selected depending on u; v. 3 A reasoning similar to the above leads to the following lemma. Lemma 1.3 The set X may be expressed as and usually turns out that X Remark 1.2. Definition may be rewritten as Direct calculation, based on the properties of set-valued operations, allows to conclude the following. Lemma are both nonempty for some ? 0, we have We shall now calculate the open-loop reach sets defined above, using the techniques of convex analysis ([25], [12], [15]). 2 The calculation of open-loop reach sets Here we shall calculate the two basic types of open-loop reach sets. The relations of this section will also serve as the basic elements further constructions which will be produced as some superposititions of the relations of this section. The calculations of this section and especially of later sections related to reachability under feedback control require a number of rather cumbersome calculations of geometrical (Minkowski) differences and their support functions. In order to simplify these calculations we transform system (1) to a simpler form. Taking the transformation gets Keeping the previous notations loss of generality, to the system with the same constraints on u; v as before. For equation (10) consider the following two problems: Problem (I) Given a set X 0 and x 2 IR n , find min u min under conditions Problem (II) Given a set X 0 and x 2 IR n , find min under conditions Here and G is a closed set in IR n . Thus where h+ (Q; G) is the the Hausdorff semidistance between compact sets Q; G; defined as min z The Hausdorff distance is h(Q; Q)g. In order to calculate the function explicitly, we use the relations and (see [10], [15] for the next formula) where is the support function of G [15]. (For compact G, sup may be substituted by max.) We thus need to calculate min u min which gives, after an application of (11), and an interchange of min u ; min x() and max l (see Z Due to (11), the last formula says simply that V \Gamma is given by where Z B(t)P(s)ds Z (\GammaC(s))Q(s)ds; It then follows that and so (12) implies that x 2 Z This gives, from the definitions of support function and geometrical difference, ae l Z B(s)P(s)ds Z (\GammaC(s)Q(s))ds which, interpreted as integrals of multivalued functions, again results in (14). Theorem 2.1 The set its support function It is clear that if the difference Z B(t)P(s)ds Z C(s)(\GammaQ(s))ds 6= ;; Note that function may be also defined as the solution to Problem min u min Direct calculations then produce the formula which gives the same result as Problem (I). Similarly, we may calculate min Taking into account the minimax theorem of [7] and the fact that l l we come to l Z ae(ljB(s)P(s))ds Z ae(\GammaljC(s)Q(s))ds: Here (conc h)(l) is the closed concave hull of h(l). Note that where (conv (l) is the closed convex hull and also the Fenchel second conjugate h (l) of h(l) (see [25], [12] for the definitions). Therefore where Z (\GammaC(s))Q(s)ds Z B(s)P(s)ds: It then follows that Similarly, (18) implies that Z so that the support function l Z B(s)P(s)ds l Z (\GammaC(s))Q(s)ds Theorem 2.2 The set X by (20) and its support function It can be seen from (22) that X may be empty. At the same time, in order that it is sufficient that Z (\GammaC(s))Q(s)ds 6= ;; which holds for ? 0 sufficiently large. It is worth mentioning that a minmax OLRS may be also be specified through an alternative definition. Definition 2.1 An open loop -reach set (OLRS) of the minmax type (from set X 0 , at time t 0 ) is the union where for some u(\Delta) 2 UP with 0 given and each set X This leads to Problem (II ). Given set X 0 , and vector x 2 IR n , find min under conditions x(t 0 Direct calculations here lead to the formula the same result as Problem II . The equivalence of Problems II ; II means that definitions 1.2 and 2.1 both lead to the same set X As we shall see, this is not so for the problem of reachability with corrections. A similar observation holds for problems I ; I . Remark 2.1. For the case that X is a singleton, one should recognize the following. The OLRS of the maxmin type is the set of points reachable at time from a given point x 0 for any disturbance v(\Delta) 2 VO , provided function v(t); t 0 t is communicated to the controller in advance, before the selection of control u(t). As mentioned above, the control u(\Delta) is then selected through an anticipative control procedure. On the other hand, for the construction of the the \Gammareach set of the minmax type there is no information provided in advance on the realization of v(\Delta), which becomes known only after the selection of u. Indeed, given point one has to select the control u(t) for the whole time interval t 0 t , whatever be the unknown v(t) over the same interval. The control u(\Delta) is then selected through a nonanticipative control procedure. Such a definition allows to specify an OLRS as consisting of points x each of which is complemented by a neighborhood B (x) so that for a certain control u(\Delta) 2 UO . This requires ? 0 to be sufficiently large. As a first step towards reachability under feedback, we consider piecewise open-loop controls with possibility of corrections at fixed instants of time. 3 Piecewise open-loop controls: reachability with corrections Here we define and calculate reachability sets under a finite number of corrections. This is done either through the solution of problems sequential maxmin and minmax or through operations on set-valued integrals. Taking a given instant of time t that divides the interval T in two, namely, consider the following sequential maxmin problem. Problem min u min and then find min u min The latter is a problem on finding a sequential maxmin with one "point of correction" . Using the notation [1; Let us find using the technique of convex analysis. According to section 2, (see (11)), we have (ae(ljB(s)P(s))\Gammaae(\GammaljC(s)Q(s))dsj(l; l) 1g where B(s)P(s)ds (\GammaC(s))Q(s))ds: Substituting this in (24), we have min u min l Z Continuing the calculation, we come to l Z (ae(ljB(s)P(s))ds Z where (ae(ljB(s)P(s))ds \Gamma ae(\GammaljC(s)Q(s)))ds: is the support function of the set B(s)P(s)ds (\GammaC(s)Q(s))ds: Together with (25) this allows us as in Section 2, to express V \Gamma(; x; [1; 2]) as where "' B(s)P(s)ds (\GammaC(s))Q(s)ds Z B(s)P(s)ds Z (\GammaC(s))Q(s))ds: Formula (26) shows that (defined as the level set of 1 ) is also the reach set with one correction. In particular, X \Gamma(; t consists of all states x that may be reached for any function v(\Delta) 2 VP , whose values are communicated in two stages, through two consecutive selections of some open-loop control u(t) according to the following scheme. Stage (1): given at time t 0 are the initial state x 0 and function v(t) for ,-select at time t 0 the control u(t) for Then at instant of correction t additional information for stage (2). Stage (2): given at time t are the state x(t ) and function v(t) for ,-select at time the control u(t) for This proves Theorem 3.1. Theorem 3.1 The set is the maxmin OLRS with one correction at instant t oe and is given by formula (26). We refer to as the maxmin OLRS with one correction at instant The two-stage scheme may be further propagated to the class of piecewise open-loop controls with k corrections. Taking the interval, introduce a partition so that the interval T is now divided into k where are the points of correction. Consider also a nondecreasing continuous function (t) 0; denoting and also Problem Solve the following consecutive optimization problems. Find min u min then find min u min then consecutively, for min u min and finally min u min Direct calculation gives with B(s)P(s)ds (\GammaC(s))Q(s)ds; then with "' B(s)P(s)ds (\GammaC(s)Q(s)ds B(s)P(s)ds (\GammaC(s)Q(s)ds; then consecutively with B(s)P(s)ds (\GammaC(s))Q(s))ds B(s)P(s)ds (\GammaC(s))Q(s)ds and finally where (0)+ B(s)P(s)ds (\GammaC(s))Q(s))ds B(s)P(s)ds (\GammaC(s))Q(s)ds Z B(s)P(s)ds Z (\GammaC(s))Q(s)ds We refer to as the maxmin OLRS with k corrections at points Theorem 3.2 The set is given by formula (29). We denote and also introduce additional notations for the functions emphasizing the dependence of on the initial condition We further assume Note that the number of nodes j in any partition \Sigma k is k 1. The partition applied to a function V k is precisely \Sigma k . Consequently, the increment is presented as a sum of k once it is applied to a function V k with index k. A sequence of partitions \Sigma k is monotone in k if for every contains all the nodes j of partition \Sigma k 1 Theorem 3.3 Given are a monotone sequence of partitions \Sigma continuous nondecreasing function (t) 0; that generates for any partition \Sigma k a sequence of numbers Given also are a sequence of value functions each of which is formed by the partition \Sigma k and a sequence is the index of Then the following relations are true. (i) For any fixed ; x, one has (ii) For any fixed ; x and index i 2 [1; k] one has (iii) The following inclusions are true for where the sets are defined by (30). The proofs are based on the following properties of the geometrical (Minkowski) sums and differences of sets and the fact that in general a maxmin does not exceed a min max. Direct calculations indicate that the following superpositions will also be true. Lemma 3.1 The functions k satisfy the following property This follows from Theorem 3.2 and the definitions of the respective functions Remark 3.1 Formula (34) reflects a semigroup property, but only for the selected points of correction The reasoning above indicates, for example, that is the set of states that may be reached for any function v(\Delta) 2 VO , whose values are communicated in advance in stages, through k consecutive selections of some open-loop control u(t) according to the following scheme. Stage (1): given at time t 0 are initial state x 0 and function v(t) for select at time Then at instant of correction j comes additional information for stage (j 1). Stage (j), (j = 2; :::; k): given at time j are state x( j ) and function v(t) for select at time control u(t) for t 2 T j+1 . Remark 3.2. There is a case when all the functions taken for all the integers coincide. This is when system (10) satisfies the so-called matching conditions: We now pass to the problem of sequential minmax, with one correction at instant t 0 using the notations for Problem (I 1 ). This is Problem min then find min The latter is a problem of finding a sequential minmax with one point of correction Denoting let us find X +(; t using the techniques of convex analysis (as above, with obvious changes). This gives where (\GammaC(s))Q(s)ds B(s)P(s)ds: Continuing the calculations, we have min where Z (\GammaC(s))Q(s)ds Z B(s)P(s)ds: This proves Theorem 3.4 Theorem 3.4 The set is the minmax OLRS with one correction at instant Here the problem is again solved in two stages, according to the following scheme. Stage (1): given at time t 0 are set X 0 and x 2 IR n . Select control u(t) (one and the same for all v) and for each v(t); t 2 T 1 , assign a vector x(t 0 that jointly with u; v produces Then at instant of correction t additional information for stage (2). Stage (2): given at time t are x(t ) and vector x 2 IR n . Select control u(t); t 2 T 2 (one and the same for all v) and for each v(t); t 2 T 2 assign a vector x(t that jointly with u; v steers the system to state x() 2 B 2 (x). We now propagate this minmax procedure to a sequential minmax problem in the class of piecewise open-loop controls with k corrections, using the notations of Problem (I k ). Problem (II k ). Solve the following consecutive optimization problems. Find min then consecutively, for min u min and finally min u This time direct calculation gives where (\GammaC(s))Q(s)ds \Gamma::: (\GammaC(s))Q(s)ds \Gamma::: Z (\GammaC(s))Q(s)ds Z B(s)P(s)ds We refer to X as the maxmin OLRS with k corrections at points k . Theorem 3.5 The set is then the mimax OLRS with one corection and is given by formula (38). Denote assuming Under the assumptions and notations of Theorem 3.3, the last results may be summarized in the following proposition. Theorem 3.6 (i) For any fixed values ; x one has (ii) For any fixed ; x and index i 2 [1; k] one has (iii) The following inclusions are true for (iv) The following superpositions will also be true In this section we have considered problems with finite number of possible corrections and additional information coming at fixed instants of time, having presented a hierarchy of piecewise open-loop reach sets of the anticipative (maxmin) or of the nonanticipative type. These were presented as level sets for value functions which are superpositions of "one- stage" value functions calculated in Section 2. A semigroup-type property (34) for these value functions was indicated which is true only for the points of correction (Remark 3.1). In the continuous case, however, we shall need this property to be true for any points. Then it would be possible to formulate the Principle of Optimality under uncertainty for our class of problems. We shall therefore investigate some limit transitions with number of corrections tending to infinity. This will allow a further possibility of continuous corrections of the control under unknown disturbances. 4 The alternated integrals and the value functions We observed above that the open-loop reach sets of both types (maxmin and minmax) are described as the level sets of some value functions, namely 4 We now propagate this approach, based on using value functions, to systems with continuous measurements of the state to allow continuous corrections of the control. First note that inequality is always true with equality attained, for example, under the following assumption. Assumption 4.1 There exists a scalar function ffl(t) ? 0 such that for all In order to simplify the further explanations, we shall further deal in this section with the case omitting the last symbol 0 in the notations for Now note that Lemmas 3.1, 3.2 indicate that each of the functions may be determined through a sequential procedure, and a similar one for . How could one express this procedure in terms of set-valued For a given partition \Sigma k we have (j i) in view of the previous relations (see (27) -(29)), we may formulate a set-valued analogy of Lemma 3.1. 4 Here, without abuse of notation for , we shall use symbol (\Delta) rather than the earlier emphasizing the function (t); used in the respective constructions. 5 The case (\Delta) 6= 0 would add to the length of the expressions, but not to the essence of the scheme. This case could be treated similarly, with obvious complements. Lemma 4.1 The following relations are true In terms of set-valued integrals (43) is precisely the equivalent of (29). Moreover, min u min u oe l Similarly, for the sequential minmax, we have Using notations identical to (42),(43), but with minus changed to plus in the symbols for k , we have Lemma 4.2. Lemma 4.2 The following relations are true In terms of set-valued integrals, formula (46) is precisely the equivalent of (38), provided Moreover, min u oe l 0g. It is important to emphasize that until now all the relations were derived for a fixed partition 6 Also note that under Assumption 4.1, with X 0 single-valued, one may treat the sets X as the Hausdorff limits What would happen, however, if k increases to infinity with and would the result depend on the type of partition? Our further discussion will require an important nondegeneracy assumption. Assumption 4.2 There exist continuous vector functions and a number ffl ? 0 such that (a) for all the sets and for all the sets whatever be the partition \Sigma k . This last assumption is further taken to be true without further notice. 7 Observing that (29), (38) have the form of certain set-valued integral sums, ("the alternated sums"), we introduce the additional notation Let us now proceed with the limit operation. Take a monotone sequence of partitions 1. Due to inclusions (33) and the boundedness of the sequence from below by any of the sets X the sequence I \Gamma (; t a set-valued limit. Similarly, the inclusions (40) and the boundedness of the sequence above ensure that it also has a set-valued limit. A more detailed investigation of this scheme along the lines of [23] would indicate that under assumption 4.2 (a); (b) these set-valued limits do not depend on the type of partition \Sigma k . This leads to Theorem 4.1. 7 If at some stage this assumption is not fulfilled, it may be applied to sets of type sufficiently large. Theorem 4.1 There exist Hausdorff limits I \Gamma (; t with These limits do not depend on the type of partition \Sigma k . Morerover, so that We refer to I(; t as the alternated reach set. 8 The proofs of the convergence of the alternated integral sums to their Hausdorff limits and of the equalities (52) are not given here. They follow the lines of those given in detail in [14] for problems on sequential maxmimin and minmax considered in backward time (see also [23], [22], [13]). Let us now study the behavior of the function According to (38), (31) the sequence increasing in i with i !1. This sequence is pointwise bounded in x by any solution of Problem (II k ) and therefore has a pointwise limit. Due to (29), Theorem 4.1, and the continuity of the distance function d(x; M) in lim and therefore we may conclude that under condition (48). This yields Theorem 4.2. Theorem 4.2 Under condition (48) there exists a pointwise limit lim limit does not depend on the type of partititon The alternated integral is the level set of the function 8 A maxmin construction of the indicated type had been introduced in detail in [23], where it was constructed in backward time, becoming known as the alternated integral of Pontryagin. does not depend on the partition \Sigma k and due to the properties of minmax we also come to the following conclusion. Theorem 4.3 The function satisfies the semigroup property: for The following inequality is true min u Similarly, for the decreasing sequence of functions we have Theorem 4.4. Theorem 4.4 (i) Under condition (48) there exists a pointwise limit lim limit does not depend on the type of partititon (ii) The alternated integral is the level set of the function V I (iii) The function satisfies the semigroup property: for (iv) The following inequality is true A consequence of (52) is the basic assertion, Theorem 4.5. Theorem 4.5 With the initial condition the following equality is true The function V(; x) satisfies the semigroup property The last relation follows from (59), (54), (57). Thus, under the nondegeneracy Assumption 4.2 the two forward alternated integrals I coincide and so do the value functions Relations (55), (58), (59) allow us to construct a partial differential equation for the function V(t; x)-the so-called HJBI (Hamilton-Jacobi-Bellman-Isaacs) equation. We now investigate the existence of the total derivative dV(t; x)=dt along the trajectories of system (10). Due to (59), (13), we have Observing that for d(x; X(t; t of (61) is unique and taking l 0 (t; may apply the rules for differentiating a "maximum"-type function [6], to get Direct calculations indicate that the respective partials exist and are continuous in the domain D[intD 0 , where and intD 0 stands for the interior of the respective set. To find the value of the total derivative take inequalities (58), (55), which may be rewritten as and min u Dividing both relations by oe ? 0 and passing to the limit with oe ! 0, we get Since in Theorem 4.5 we had for the linear system (10) we have which results in the next proposition. Theorem 4.6 In the domain D [ intD 0 the value function V(t; x) satisfies the "forward" equation over Equation (63) may be rewritten as The last theorem indicates that the HJBI equation (63) is satisfied everywhere in the open domain D [ intD 0 . However, the continuity of the partials @V=@x; @V=@t on the boundary of the domains D; D 0 was not investigated and in fact may not hold. But it is not difficult to check that with boundary condition (65) the function V(t; x) will be a minmax solution to equation (66) in the sense of [26], which is equivalent to the statement that V(t; x) is a viscosity solution ([3], [20]) to (66), (67). This particularly follows from the fact that function V(t; x) is convex, being a pointwise limit of convex functions Let us note here that the problem under discussion may be treated not only as above but also within the notion of classical solutions to equation (66), (65). Indeed, although all the results above were proved for the criterion d(x(t in the respective problems, the following assertion is also true. Assertion 4.1 Theorems 3.1-3.6, 4.1-4.6, are all true with the criterion d(x(t in the respective problems substituted by d 2 (x(t This assertion follows from direct calculations, as in paper [13], with formula (11) substituted by The respective value function similar to V(t; x), denoted further as V 1 (t; x), will now be a solution to (66) with boundary condition together with its first partials, turns out to be continuous in t; x 2 D[D 0 . Thus we come to Theorem 4.7 The function V 1 (t; x)-a classical solution to (66), (67)-satisfies the relation We have constructed the set X(t; t as the limit of OLRS and the level set of function V(t; x), (or function V 1 (t; x))-the sequential maxmin or minmax of function d(t; function d 2 (t; X 0 restriction It remains to show that X(t; t precisely the set of points that may be reached from X 0 with a certain feedback control strategy U(t; x), whatever be the function v(t). Prior to the next section, we wish to note the following. Function V(t; may be interpreted as the value function for the following Problem (IV): find the value function U x(\Delta) is a CLC (see Section 1) and X U (\Delta) is the set of all solutions to the differential inclusion generated by taken within the interval Its level set is precisely the closed-loop reach set. It is the set of such points x 2 IR n that there exists a strategy U 2 U C which for any solution x(t) of (69), ensures the inequality . Due to the structure of (69), (A(t) j 0), this is equivalent to the following definition of closed-loop reachability sets. Definition 4.1 A closed-loop reachability set X () is the set of such points x 2 IR n for each of which there exists a strategy U 2 U C that for every v(\Delta) 2 VO assigns a point x such that every solution x[t] of the differential inclusion satifies the inequality d(x[ ]; x) . Once the Principle of Optimality (60) is true, it may also be used directly to derive equation - the HJBI equation for the function V(t; x). Therefore, set X () (if nonempty), will be nothing else than the set X(; t defined earlier as the limit of open-loop reach sets. 5 Closed loop reachability under uncertainty We shall now show that each point of X(t; t may be reached from X 0 with a certain feedback control strategy U(t; x), whatever be the function v(t). In order to do this, we shall need the notion of solvability set, (or, in other terms, "the backward reachability set", see [11], [27], [15]) - a set similar to X(t; t in backward time. We first recall from [13] some properties of these sets. Consider Problem find the value function U x(\Delta) where M is a given convex compact set (M 2 convIR n ) and X U is the variety of all trajectories x(\Delta) of the differential inclusion (69), generated by a given strategy U 2 U C . The formal HJBI equation for the value V (t; x) is @t @x with boundary condition Equation (71) may be rewritten as @t @x @x 0: (73) An important feature is that function V (t; x) may be interpreted as a sequential maxmin similar to the one in section 3. Namely, taking the interval t t 1 , introduce a partition similar to that of Section 3. For the given partition, consider the recurrence relations min u min u min u almost everywhere in the respective intervals. Lemma 5.1 ([13]) With there exists a pointwise limit that does not depend upon the type of partition \Sigma k . The function We shall refer to as the sequential maxmin. This function enjoys properties similar to those of its "forward time" counterpart, the function of section 3. A similar construction is possible for a "backward" version of the sequential minmax. The level set is referred to as the closed loop solvability set CLSS at time t, from set M. It may be presented as an alternated integral of Pontryagin, - the Hausdorff limit of the sequence I (t; Z Z Z under conditions (74). Also presumed is a nondegeneracy assumption similar to Assumption 4.2. Assumption 5.1 For a given set M 2 convIR n there exists a continuous function fi 3 (t) 2 and a number ffl ? 0, such that for any whatever be the partition \Sigma k . This assumption is presumed in the next lemma. Lemma 5.2 Under condition (76) there exists a Hausdorff limit I (t; I (t; 0: This limit does not depend on the type of partition \Sigma k and coincides with the CLSS, I (t; From the theory of control under uncertainty and differential games it is known that if there exists a feedback strategy U(t; x) 2 U C that steers system (10) from state x(t) = x to set M whatever be the unknown disturbance v(\Delta) ([11], [29], [15]). Therefore, assuming our assumptions, we just have to prove the inclusion or, in view the properties of V(t; x); V (t; x), that is the solution to equation (71) with boundary condition (Recall that V(t 1 Due to the definition of the geometrical difference and of the integral I check that We thus have to prove the inclusion Under assumptions 4.2 (a) taken for 0, or under assumption 4.2, it is possible to observe, through direct calculation, using the properties of integrals I (see formulas (29),(75)), that the following holds: I (t; where and we arrive at Lemma 5.3. Lemma 5.3 The following inclusion is true moreover, Inclusion (79) implies the existence of a feedback strategy U (t; x) that brings system (10) from x Theorem 5.1 Under assumptions 4.1(a), there exists a closed-loop strategy U (t; x) ' U C that steers system (10) from x The strategy U (t; x) may be found through the solution V(t; x) of equation (71), with boundary condition (78), as U (t; (if the gradient @V (t; x)=@x does exist at ft; xg), or, more generally, as U (t; 0g. This is verified by differentiating V(t; x) with respect to t and checking that a.e. dV dt (see [10], [15]). The previous theorem ensures merely that some point of may be reached from x . In order to demonstrate that any point x may be reached from position ft; x g, we have to prove the inclusion for any x ? is a solution to (66) with boundary condition But inclusions (82), (83) again follow from the properties of I assuming both of these set-valued integrals are nonempty. The latter, in its turn, is again ensured by either assumptions 4.2(a), Assumption 4.1. This leads to Theorem 5.2. Theorem 5.2 Under either Assumptions 4.1(a), or 4.1 there exists a closed-loop strategy U ? (t; x) ' U C that steers system (10) from x to point The strategy U ? (t; x) may be found through the solution V ? (t; x) of equation (71), with boundary condition (84), as (if the gradient @V ? (t; x)=@x does exist at ft; xg), or, more generally, as 0g. Remark 5.1. Assumptions 4.1(a), are ensured by Assumption 4.1. If this does not hold, it is possible to go through all the procedures taking \Gamma neighborhoods of sets X(\Delta); W (\Delta) rather than the sets themselves. Then one has to look for the (\Delta)-reach sets -solvability sets W (t; sufficiently large, so that X(t; t would surely be nonempty. Remark 5.2. The emphasis of this paper is to discuss the issue of reachability under uncertainty governed by unknown but bounded disturbances. This topic was studied here through a reduction to the calculation of value functions for a successive problems on sequential and maxmin of certain distance functions or their squares. The latter problems were dealt with via techniques of convex analysis and set-valued calculus. However the solution schemes of this paper naturally allow a more general situation which is to substitute the distance function d(x; M) by any proper convex function OE(x), for example, with similar results passing through. The more general problems then reduce to those of this paper. Thus, given terminal cost function OE(x), it may readily generate a terminal set M as a level set some ff, with support function ([25]) The given formalisms for decribing reachability are not the only ones available. We further indicate yet another formal scheme. 6 Reachability and the funnel equations In this section we briefly indicate some connections between the previous results and those that can be obtained through evolution equations of the "funnel type" [2], [15]. Consider the evolution equations lim oe with initial condition and lim oe with Under some regularity assumptions (similar to Assumption 4.1) which ensure that all the sets that appear in (87), (88) are nonempty, these equations have solutions which turn to be set-valued. The solutions almost everywhere. But they need not be unique. However, the property of uniqueness may be restored if we presume that are the (inclusion) maximal solutions, (see [15], Sections 1.3, 1.7). solution X 0 (t) to a funnel equation of type (87), (88) is maximal if it satisfies the inclusion other solution X (t) to the respective equation with the same initial condition). Equations (87), (88) may be interpreted as some limit form of the recurrence equations \Gammaoe(\GammaC and \Gammaoe(\GammaC (t)Q(t) 6= ;: Indeed, taking, for example solving the recurrence equation (89) for values of time oe (0j) to oe (kj ), we observe that oe (kj) is similar to I \Gamma (; t selected with constant oe \Gamma(\GammaC is similar to (29)(when and to (43). Under Assumtion 4.1 a direct calculation leads to the next conclusions. Lemma 6.1 The following relations are true with oe oe is a maximal solution to equation (87) with Therefore, the closed-loop reach set X (; t may be also calculated through the funnel equation (87) which therefore also describes the dynamics of the level sets of the value function V(; x) -the solution to the forward HJBI equation (66). Remark 6.1. As we have seen, equation (87) describes the evolution of the alternated integral Similarly to that, equation (88) describes the evolution of the alternated recurrence equations (89), (90) may then serve to be the basis of numerical schemes for calculating the reach sets. 7 Example Consider the system defined on the interval [0; ], with hard bounds on the control u and the uncertain disturbance v. As is known (see, for example, [16]), a parametric representation of the boundary of the reach set X(; t of system (91) without uncertainty (v(t) j 0) is given by two curves (see external set in fig.1, generated for x and where oe 0 is the parameter, (the values oe ? 0 correspond to the vertices of Similarly, the reach set X(; t in the variable v is given by the curves According to (7) , the set which leads to a parametrization of the boundary of this set in the form (see internal set in fig.1, generated for so that the OLRS under uncertainty is smaller than X(; 0; x 0 jP(\Delta); f0g)-the reach set without uncertainty. Let us now look for the OLRS one correction at time Taking may figure out that set has to be bounded by two curves (fig.2) which gives we come to intX Continuing with r We also observe that with r 2 ! 1=2 we have sufficiently large. As indicated above, sets X turn to be empty unless is sufficiently large. Continuing our example further, for \GammaX (2; 0; 0jf0g; Q(\Delta))g; and The last set is nonvoid if is such that B (0) \GammaX (2; 0; 0; jf0g; ;. The smallest value 0 of all such ensures For all 0 it is then possible to compare sets that the latter is smaller than the former (see fig.3, where X shown by the internal continuous curve, by the external continuous curve and by the dashed curve). Fig.1 xFig.2 -226 xFig.3 8 Conclusion In this paper we deal with one of the recent problems in reachability analysis which is to specify the sets of points that can be reached by a controlled system despite the unknown but bounded disturbances in the system inputs. The paper gives a description of several notions of such reachability and indicates schemes to calculate various types of reach sets. We consider systems with linear structure and closed-loop controls that are generally nonlinear. In particular, we emphasize the difference between reachability under open-loop and closed-loop controls. We distinguish open-loop controls of the anticipative type, which presume the disturbances to be known in advance, and of the nonanticipative type, which presume no such knowledge. The nonanticipative open-loop reach set is smaller than the one for anticipative open-loop controls and the closed loop reach set (which is always nonanticipative) lies in between. Intermediate reach sets are those generated by piecewise closed-loop controls that allow on-line measurements of the state space variable at isolated instants of time - the points of correction. Increasing the number of corrections to infinity and keeping them dense within the interval under consideration, we came to the case of continuous corrections - the solution to the problem of reachability under closed-loop (feedback control). The various types of reach sets introduced here were calculated through two alternative presentations, namely, either through operations on set-valued integrals or as level sets for value functions in sequential problems on maxmin or minmax for certain distance functions. For the closed-loop reachability problem the corresponding value function defines a mapping that satifies the semigroup property. This property allowed us to formulate the Principle of Optimality under Uncertainty for the class of problems considered here. The last Principle allowed to demonstrate that the closed-loop reach set under uncertainty is the level set for the solution to a forward HJBI equation. On the other hand, the feedback control strategy that steers a point to its closed-loop reach set (whatever be the disturbance) may be found from the solution to a backward HJBI equation whose boundary condition is taken from the solution of the earlier mentioned forward HJBI equation. This paper leaves many issues for further investigation. For example, there is a strong demand from many applied areas to calculate reach sets under uncertainty. However, the given solutions to the problem are not simple to calculate. Among the nearest issues may be the calculation of the reach sets of this paper through ellipsoidal approximations along the schemes of [15], [16]. Then, of course, comes the propagation of the results to nonlinear systems. Here the application of the HJBI technique seems to allow some progress. Needless to say, similar problems could also be posed for systems with uncertainty in its parameters or in the model itself, as well as for other types of controlled transition systems. --R KUPFERMAN O. Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations Optimal Control and Related Minimax Design Prob- lems Stochastic Transition Systems. RUBINOV A. Topics in Control Theory. SUBBOTIN A. Control and Observation Under Uncertainty. Pontryagin's alternated integral in the theory of control synthesis. N. MARKUS L. Optimality and reachability with feedback controls. SOUGANIDIS P. Controllers for reachability specifications for hybrid systems. IVANOV G. WETS R. Generalized Solutions of First Order PDE's: the Dynamic Optimization Perspective. On the Existence of Solutions to a Differential Game. Existence of Saddle Points in Differential Games. Reach set Computation Using Optimal Control. --TR

uncertainty;differential games;reach sets;alternated integral;funnel equations;differential inclusions;closed-loop control;HJBI equation;open-loop control;reachability;dynamic programming

587740

On Lagrangian Relaxation of Quadratic Matrix Constraints.

Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. These problems are in general NP hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Such relaxations are equivalent to semidefinite programming relaxations.For several special cases of QQP, e.g., convex programs and trust region subproblems, the Lagrangian relaxation provides the exact optimal value, i.e., there is a zero duality gap. However, this is not true for the general QQP, or even the QQP with two convex constraints, but a nonconvex objective. In this paper we consider a certain QQP where the quadratic constraints correspond to the matrix orthogonality condition XXT=I. For this problem we show that the Lagrangian dual based on relaxing the constraints XXT=I and the seemingly redundant constraints XT X=I has a zero duality gap. This result has natural applications to quadratic assignment and graph partitioning problems, as well as the problem of minimizing the weighted sum of the largest eigenvalues of a matrix. We also show that the technique of relaxing quadratic matrix constraints can be used to obtain a strengthened semidefinite relaxation for the max-cut problem.

Introduction . Quadratically constrained quadratic programs (QQPs) play an important modeling role for many diverse problems. They often provide a much improved model compared to the simpler linear relaxation of a problem. However, very large linear models can be solved e#ciently, whereas QQPs are in general NP-hard and numerically intractable. Lagrangian relaxations often provide good approximate solutions to these hard problems. Moreover these relaxations can be shown to be equivalent to semidefinite programming (SDP) relaxations, and SDP problems can be solved e#ciently, i.e., they are polynomial time problems; see, e.g., [31]. relaxations provide a tractable approach for finding good bounds for many hard combinatorial problems. The best example is the application of SDP to the max-cut problem, where a 87% performance guarantee exists [11, 12]. Other examples include matrix completion problems [23, 22], as well as graph partitioning problems and the quadratic assignment problem (references given below). In this paper we consider several quadratically constrained quadratic (nonconvex) programs arising from hard combinatorial problems. In particular, we look at the orthogonal relaxations of the quadratic assignment and graph partitioning problems. We show that the resulting well-known eigenvalue bounds for these problems can be obtained from the Lagrangian dual of the orthogonally constrained relaxations, # Received by the editors June 9, 1998; accepted for publication (in revised form) by P. Van Dooren July 30, 1999; published electronically May 31, 2000. http://www.siam.org/journals/simax/22-1/34029.html Department of Management Sciences, University of Iowa, Iowa City, IA 52242-1000 (kurt-anstreicher@uiowa.edu). # University of Waterloo, Department of Combinatorics and Optimization, Waterloo, Ontario N2L 3G1, Canada (henry@orion.uwaterloo.ca). This author's research was supported by Natural Sciences and Engineering Research Council of Canada. 42 KURT ANSTREICHER AND HENRY WOLKOWICZ but only if the seemingly redundant constraint X T I is explicitly added to the orthogonality constraint XX Our main analytical tool is a strong duality result for a certain nonconvex QQP, where the quadratic constraints correspond to the orthogonality conditions XX We also show that the technique of applying Lagrangian relaxation to quadratic matrix constraints can be used to obtain a strengthened SDP relaxation for the max-cut problem. Our results show that current tractable (nonconvex) relaxations for the quadratic assignment and graph partitioning problems can, in fact, be found using Lagrangian relaxations. converse statement is well known, i.e., the Lagrangian dual is equivalent to an (tractable) SDP relaxation.) Our results here provide further evidence to the following conjecture: the Lagrangian relaxation of an appropriate QQP provides the strongest tractable relaxation for QQPs. 1.1. Outline. We complete this section with the notation used in this paper. In section 2, we present several known results on QQPs. We start with convex QQPs where a zero duality gap always holds. Then we look at the minimum eigenvalue problem and the trust region subproblem, where strong duality continues to hold. We conclude with the two trust region subproblem, the max-cut problem, and general nonconvex QQPs where nonzero duality gaps can occur. The main results are in section 3. We show that strong duality holds for a class of orthogonally constrained quadratic programs if we add seemingly redundant constraints before constructing the Lagrangian dual. In section 4 we apply this result to several problems, i.e., relaxations of quadratic assignment and graph partitioning problems, and a weighted sum of eigenvalue prob- lem. In section 5 we present strengthened semidefinite relaxations for the max-cut problem. In section 6 we summarize our results and describe some promising directions for future research. 1.2. Notation. We now describe the notation used in the paper. Let S n denote the space of n-n symmetric matrices equipped with the trace inner product, positive semidefiniteness positive definiteness) and A # B denote A - B # 0, i.e., S n is equipped with the L-owner partial order. We let P denote the cone of symmetric positive semidefinite matrices; M m,n denotes the space of general m- n matrices also equipped with the trace inner product, #A, denotes the space of general m-m matrices; O denotes the set of orthonormal (orthogonal) matrices; # denotes the set of permutation matrices. We let Diag(v) be the diagonal matrix formed from the vector v; its adjoint operator is diag(M ), which is the vector formed from the diagonal of the matrix M. For , the vector formed (columnwise) from M . The Kronecker product of two matrices is denoted A# B, and the Hadamard product is denoted A # B. We use e to denote the vector of all ones, and ee T to denote the matrix of all ones. 2. Some known results. The general QQP is x. We now present several QQP problems where the Lagrangian relaxation is important and well known. In all these cases, the Lagrangian QUADRATIC MATRIX CONSTRAINTS 43 dual provides an important theoretical tool for algorithmic development, even where the duality gap may be nonzero. 2.1. Convex quadratic programs. Consider the convex quadratic program where all q i (x) are convex quadratic functions. The dual is min x If # is attained at # , x # , then a su#cient condition for x # to be optimal for CQP is primal feasibility and complementary slackness, i.e., In addition, it is well known that the Karush-Kuhn-Tucker (KKT) conditions are su#cient for global optimality, and under an appropriate constraint qualification the KKT conditions are also necessary. Therefore strong duality holds if a constraint qualification is satisfied, i.e., there is no duality gap and the dual is attained. However, surprisingly, if the primal value of CQP is bounded, then it is attained and there is no duality gap; see, e.g., [44, 36, 34, 35] and, more recently, [26]. However, the dual may not be attained, e.g., consider the convex program and its dual min x Algorithmic approaches based on Lagrangian duality appear in, e.g., [19, 25, 31]. 2.2. Rayleigh quotient. Suppose that A = A T . It is well known that the smallest eigenvalue # 1 of A is obtained from the Rayleigh quotient, i.e., Since A is not necessarily positive semidefinite, this is the minimization of a nonconvex function on a nonconvex set. However, the Rayleigh quotient forms the basis for many algorithms for finding the smallest eigenvalue, and these algorithms are very e#cient. In fact, it is easy to see that there is no duality gap for this nonconvex problem, i.e., min x To see this, note that the inner minimization problem in (2.2) is unconstrained. This implies that the outer maximization problem has the hidden semidefinite constraint (an ongoing theme in the paper) i.e., # is at most the smallest eigenvalue of A. With # set to the smallest eigenvalue, the inner minimization yields the eigenvector corresponding to # 1 . Thus, we have an example of a nonconvex problem for which strong duality holds. Note that the problem (2.1) has the special norm constraint and a homogeneous quadratic objective. 44 KURT ANSTREICHER AND HENRY WOLKOWICZ 2.3. Trust region subproblem. We will next see that strong duality holds for a larger class of seemingly nonconvex problems. The trust region subproblem (TRS) is the minimization of a quadratic function subject to a norm constraint. No convexity or homogeneity of the objective function is assumed. Assuming that the constraint in TRS is written "#," the Lagrangian dual is min x This is equivalent to (see [43]) the (concave) nonlinear semidefinite program s.t. Q+ #I # 0, Moore-Penrose inverse. It is shown in [43] that strong duality holds for TRS, i.e., there is a zero duality gap - # , and both the primal and dual are attained. Thus, as in the eigenvalue case, we see that this is an example of a nonconvex program where strong duality holds. Extensions of this result to a two-sided general, possibly nonconvex constraint are discussed in [43, 28]. An algorithm based on Lagrangian duality appears in [40] and (implicitly) in [29, 41]. These algorithms are extremely e#cient for the TRS problem, i.e., they solve this problem almost as quickly as they can solve an eigenvalue problem. 2.4. Two trust region subproblem. The two trust region subproblem (TTRS) consists of minimizing a (possibly nonconvex) quadratic function subject to a norm and a least squares constraint, i.e., two convex quadratic constraints. This problem arises in solving general nonlinear programs using a sequential quadratic programming approach and is often called the Celis-Dennis-Tapia (CDT) problem; see [4]. In contrast to the above single TRS, the TTRS can have a nonzero duality gap; see, e.g., [33, 47, 48, 49]. This is closely related to quadratic theorems of the alterna- tive, e.g., [5]. In addition, if the constraints are not convex, then the primal may not be attained; see, e.g., [26]. In [27], Martinez shows that the TRS can have at most one local and nonglobal optimum, and the Lagrangian at this point has one negative eigenvalue. Therefore, if we have such a case and add another ball constraint that contains the local, nonglobal optimum in its interior and also makes this point the global optimum, we obtain a TTRS where we cannot close the duality gap due to the negative eigenvalue. It is uncertain what constraints could be added to close this duality gap. In fact, it is still an open problem whether TTRS is an NP-hard or a polynomial-time problem. 2.5. Max-cut problem. Suppose that E) is an undirected graph with vertex set and weights w ij on the edges (v i , The max-cut problem consists of finding the index set I # {1, 2, . , n}, in order to maximize the weight of the edges with one end point with index in I and the other in the complement. This is equivalent to the following discrete optimization problem with a quadratic objective: We equate x I and x otherwise. Define the homogeneous quadratic objective where Q is an n - n symmetric matrix. Then the MC problem is equivalent to the QQP This problem is NP-hard, i.e., intractable. Since the above QQP has many nonconvex quadratic constraints, a duality gap for the Lagrangian relaxation is expected and does indeed occur most of the time. However, the Lagrangian dual is equivalent to the SDP relaxation (upper bound) e, which has proven to have very strong theoretical and practical properties, i.e., the bound has an 87% performance guarantee for the problem MC and a 97% performance in practice; see, e.g., [12, 18, 15]. Other theoretical results for general objectives and further relaxed constraints appear in [30, 46]. In [38], several unrelated, though tractable, bounds for MC are shown to be equivalent. These bounds include the box relaxation -e # x # e, the trust region relaxation an eigenvalue relaxation. Furthermore, these bounds are all shown to be equivalent to the Lagrangian relaxation; see [37]. Thus we see that the Lagrangian relaxation is equivalent to the best of these tractable bounds. 2.6. General QQP. The general, possibly nonconvex QQP has many applications in modeling and approximation theory; see, e.g., the applications to SQP methods in [21]. Examples of approximations to QQPs also appear in [9]. The Lagrangian relaxation of a QQP is equivalent to the SDP relaxation and is sometimes referred to as the Shor relaxation; see [42]. The Lagrangian relaxation can be written as an SDP if one takes into the account the hidden semidefinite constraint, i.e., a quadratic function is bounded below only if the Hessian is positive semidefinite. The SDP relaxation is then the Lagrangian dual of this semidefinite program. It can also be obtained directly by lifting the problem into matrix space using the fact that relaxing xx T to a semidefinite matrix X. One can relate the geometry of the original feasible set of QQP with the feasible set of the SDP relaxation. The connection is through valid quadratic inequalities, i.e., nonnegative (convex) combinations of the quadratic functions; see [10, 20]. 3. Orthogonally constrained programs with zero duality gaps. Consider the orthonormal constraint (The set of such X is sometimes known as the Stiefel manifold; see, e.g., [7]. Applications and algorithms for optimization on orthonormal sets of matrices are discussed in [7].) In this section we will show that for holds for a certain (nonconvex) quadratic program defined over orthonormal matrices. Because of the similarity of the orthonormality constraint to the norm constraint x T the result of this section can be viewed as a matrix generalization of the strong duality result for the Rayleigh quotient problem (2.1). Let A and B be n - n symmetric matrices, and consider the orthonormally constrained homogeneous QQP This problem can be solved exactly using Lagrange multipliers (see, e.g., [14]) or using the classical Ho#man-Wielandt inequality (see, e.g., [3]). We include a simple proof for completeness. Proposition 3.1. Suppose that the orthogonal diagonalizations of A, B are respectively, where the eigenvalues in # are ordered nonincreasing and the eigenvalues in # are ordered nondecreasing. Then the optimal value of QQPO is - O = tr # and the optimal solution is obtained using the orthogonal matrices that yield the diagonalizations, i.e., Proof. The constraint G(X) := XX T - I maps M n to S n . The Jacobian of the constraint at X acting on the direction h is . The adjoint of the Jacobian acting on S # S n is J # tr But J # is one-one for all X orthogonal. Therefore, J is onto, i.e., the standard constraint qualification holds at the optimum. It follows that the necessary conditions for optimality are that the gradient of the Lagrangian is 0, i.e., Therefore, i.e., AXBX T is symmetric, which means that A and XBX T commute and so are mutually diagonalizable by the orthogonal matrix U . Therefore, we can assume that both A and B are diagonal and we choose X to be a product of permutations that gives the correct ordering of the eigenvalues. The Lagrangian dual of QQPO is min tr AXBX T However, there can be a nonzero duality gap for the Lagrangian dual; see [50] for an example. The inner minimization in the dual problem (3.3) is an unconstrained quadratic minimization in the variables vec (X), with Hessian I# S. this minimization is unbounded if the Hessian is not positive semidefinite. In order to close the duality gap, we need a larger class of quadratic functions. Note that in QQPO the constraints XX are equivalent. Adding the redundant constraints X T we arrive at Using symmetric matrices S and T to relax the constraints XX respectively, we obtain a dual problem s.t. (B# A), Theorem 3.2. Strong duality holds for QQPOO and DQQPOO , i.e., - and both primal and dual are attained. Proof. Let and U are orthonormal matrices whose columns are the eigenvectors of A and B, respectively, # and # are the corresponding vectors of eigenvalues, and Diag(#). Then for any S and U# V is nonsingular, tr S, and tr T , the dual problem DQQPOO is equivalent to s.t. However, since # and # are diagonal matrices, (3.4) is equivalent to the ordinary linear program: But LD is the dual of the linear assignment problem: s.t. Assume without loss of generality that # 1 Then LP can be interpreted as the problem of finding a permutation #(-) of {1, . , n} so that But the minimizing permutation is then Proposition 3.1 the solution value - D is exactly - O . 48 KURT ANSTREICHER AND HENRY WOLKOWICZ 4. Applications. We now present three applications of the above strong duality result. 4.1. Quadratic assignment problem. Let A and B be n - n symmetric ma- trices, and consider the homogeneous quadratic assignment problem (QAP) (see, e.g., [32]), QAP min tr AXBX T s.t. X #, where # is the set of n - n permutation matrices. The set of orthonormal matrices contains the permutation matrices, and the orthonormally constrained problem (3.1) is an important relaxation of QAP. The bounds obtained are usually called the eigenvalue bounds for QAP; see [8, 13]. Theorem 3.2 shows that the eigenvalue bounds are in fact obtained from a Lagrangian relaxation of (3.1) after adding the seemingly redundant constraint XX 4.2. Weighted sums of eigenvalues. Consider the problem of minimizing the weighted sum of the k largest eigenvalues of an n - n symmetric matrix Y , subject to linear equality constraints. An SDP formulation for this problem involving 2k semidefiniteness constraints on n - n matrices is given in [1, section 4.3]. We will show that the result of section 3 can be applied to obtain a new SDP formulation of the problem having only k semidefiniteness constraints on n - n matrices. For convenience we consider the equivalent problem of maximizing the weighted sum of the k minimum eigenvalues of Y . Let w are interested in the problem s.t. A vec (Y are the eigenvalues of Y , and A is a matrix. From Proposition 3.1 it is clear that, for any Y , tr Y XWX T , and therefore from Theorem 3.2 the problem WEIG is equivalent to the problem s.t. A vec (Y Note that, for any Y , the matrix W# Y is block diagonal, with the final n- k blocks identically zero. Since I# S is also block diagonal, and tr T is a function of the diagonal of T only, it is obvious that T can be assumed to be a diagonal matrix Writing the problem (4.1) in terms of t, and separating the block diagonal constraints, results in the SDP We have thus obtained an SDP representation for the problem WEIG with semidefiniteness constraints on n - n matrices, as claimed. 4.3. Graph partitioning problem. Let E) be an edge-weighted undirected graph with node set The graph partitioning (GP) problem consists of partitioning the node set N into k disjoint subsets S 1 , . , S k of specified sizes as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. This problem is well known to be NP-hard. GP can be modeled as a quadratic problem z := min tr X T LX where L is the Laplacian of the graph and P is the set of n - k partition matrices (i.e., each column of X is the indicator function of the corresponding set; node i is in set j and 0 otherwise). The well-known Donath-Ho#man bound [6] z DH # z for GP is z DH := max are the eigenvalues of We will now show that the Donath-Ho#man bound can be obtained by applying Lagrangian relaxation to an appropriate QQP relaxation of GP. (An SDP formulation for this bound is given in [1].) Clearly, if P is a partition matrix, then i is the ith row of X. Moreover, the columns of X are orthogonal with one another, and the norm of the jth column of X is # m j . It follows that if X is a partition matrix, there is an n - n orthogonal matrix - X such that where M is the k - k matrix # . In addition, note that x T is the ith diagonal element of XX T , so the constraint equivalent to - i is the ith row of - X. Since tr X T 50 KURT ANSTREICHER AND HENRY WOLKOWICZ tr LXX T , a lower bound z 1 # z can be defined by We will now obtain a second bound z 2 # z 1 by applying a Lagrangian procedure to all of the constraints in (4.2). Using symmetric matrices S and T for the constraints respectively, and a vector of multipliers u i for the constraints u,S,T min tr L - Theorem 4.1. z Proof. Rearranging terms and using Kronecker product notation, the definition of z 2 can be rewritten as u,S,T +min X), and we are using the fact that solves the implicit minimization problem in the definition of z 2 , and if this constraint fails to hold, the minimum is -#. Using this hidden semidefinite constraint, we can write Note that if u M for any scalar #, then M# I), M# I). In addition, tr T It follows that we may choose any normalization for e T u without a#ecting the value of z 2 . Choosing e T arrive at QUADRATIC MATRIX CONSTRAINTS 51 However, as in the previous section, Proposition 3.1 and Theorem 3.2 together imply that for any U , the solution value in the problem is exactly Therefore, we immediately have z SDP relaxations for the GP problem are obtained via Lagrangian relaxation in [45]. A useful corollary of Theorem 4.1 is that any Lagrangian relaxation based on a more tightly constrained problem than (4.2) will produce bounds that dominate the Donath-Ho#man bounds. A problem closely related to the orthogonal relaxation of GP is the orthogonal Procrustes problem on the Stiefel manifold; see [7, section 3.5.2]. This problem has a linear term in the objective function, and there is no known analytic solution for the general case. 5. A strengthened relaxation for max-cut. As discussed above, the SDP relaxation for MC performs very well in practice and has strong theoretical proper- ties. There have been attempts at further strengthening this relaxation. For example, a copositive relaxation is presented in [39]. Adding cuts to the SDP relaxation is discussed in [15, 16, 17, 18]. These improvements all involve heuristics, such as deciding which cuts to choose or solving a copositive problem, which is NP-hard in itself. The relaxation in (2.3) is obtained by lifting the vector x into matrix space using . Though the matrix X in the lifting is not an orthogonal matrix, it is a partial isometry up to normalization, i.e., We will now show that we can improve the semidefinite relaxation presented in section 2.5 by considering Lagrangian relaxations using the matrix quadratic constraint (5.1). In particular, consider the relaxation of MC s.t. diag e, where X is a symmetric matrix. Note that if X and diag(X 2 ne. As a result, the above relaxation is equivalent to the relaxation tr QX 2 ith row of X, and x 0 is a scalar. (Note that if replacing X with -X leaves the objective and constraints in (5.2) unchanged.) We will obtain an upper bound - 2 # - 1 by applying a Lagrangian procedure to all of the constraints in (5.2). Using multipliers u i for the 52 KURT ANSTREICHER AND HENRY WOLKOWICZ constraints x T for the constraint x 2 matrix S for the matrix equality X 2 we obtain a Lagrangian problem tr QX 2 Letting - x problem can be written in Kronecker product form as ne T u Q-x, where Applying the hidden semidefinite constraint - we obtain an equivalent problem, ne T u Note that if we take clearly optimal and the problem reduces to s.t. -Q+ U # 0, which is exactly the dual of (2.3), the usual SDP relaxation for MC. It follows that we have obtained an upper bound - 2 which is a strengthening of the usual SDP bound, The strengthened relaxation (5.3) involves a semidefiniteness constraint on a (n 2 as opposed to an n-n matrix in the usual SDP relaxation (2.3). This dimensional increase can be mitigated by taking note of the fact that X in (5.2) must be a symmetric matrix, and therefore (5.2) can actually be written as a problem over a vector x of dimension n(n + 1)/2. In addition, alternative relaxations can be obtained by not making the substitutions based on (5.1) used to obtain the problem (5.2). The e#ect of these alternatives on the performance of strengthened SDP bounds for MC is the topic of ongoing research; for up-to-date developments, see the URL http://orion.uwaterloo.ca/-hwolkowi/henry/reports/strngthMC.ps.gz. 6. Conclusion. In this paper we have shown that a class of nonconvex quadratic problems with orthogonal constraints can satisfy strong duality if certain seemingly redundant constraints are added before the Lagrangian dual is formed. As applications of this result we showed that well-known eigenvalue bounds for QAP and GP problems can actually be obtained from the Lagrangian dual of QQP relaxations of these problems. We also showed that the technique of relaxing quadratic matrix constraints can be used to obtain strengthened SDP relaxations for the max-cut problem. Adding constraints to close the duality gap is akin to adding valid inequalities in cutting plane methods for discrete optimization problems. In [2, 24] this approach, in QUADRATIC MATRIX CONSTRAINTS 53 combination with a lifting procedure, is used to solve discrete optimization problems. In our case we add quadratic constraints. The idea of quadratic valid inequalities has been used in [10]; and closing the duality gap has been discussed in [20]. Our success in closing the duality gap for the QQPO problem considered in section 3, where we have the special Kronecker product in the objective function, raises several interesting questions. For example, can the strong duality result for QQPO be extended to the same problem with an added linear term in the objective, or are there some other special classes of objective functions where this is possible? Another outstanding question is whether it is possible to add quadratic constraints to close the duality gap for the TTRS. --R Interior point methods in semidefinite programming with applications to combinatorial optimization Perturbation Bounds for Matrix Eigenvalues An alternative theorem for quadratic forms and extensions Lower bounds for the partitioning of graphs The geometry of algorithms with orthogonality constraints Approximation algorithms for quadratic programming Semidefinite programming relaxation for nonconvex quadratic pro- grams Semidefinite programming in combinatorial optimization Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming Bounds for the quadratic assignment problems using continuous optimization A new lower bound via projection for the quadratic assignment problem An Interior Point Method for Semidefinite Programming and Max-Cut Bounds Fixing Variables in Semidefinite Relaxations A spectral bundle method for semidefinite programming An interior-point method for semidefinite programming On the convergence of the method of analytic centers when applied to convex quadratic programs Cones of Matrices and Successive Convex Relaxations of Non-convex Sets A tour d'horizon on positive semidefinite and Euclidean distance matrix completion problems An Analytic Center Based Column Generation Algorithm for Convex Quadratic Feasibility Problems On the Extension of Frank-Wolfe Theorem Local minimizers of quadratic functions on Euclidean balls and spheres Interior Point Polynomial Algorithms in Convex Program- ming The quadratic assignment problem: A survey and recent developments Optimality conditions for the minimization of a quadratic with two quadratic constraints Duality in quadratic programming and l p-approximation Duality in quadratic programming and l p-approximation Duality in quadratic programming and l p-approximation A recipe for semidefinite relaxation for (0 Convex relaxations of 0-1 quadratic programming Copositive relaxation for general quadratic programming A semidefinite framework for trust region subproblems with applications to large scale minimization A New Matrix-Free Algorithm for the Large-Scale Trust-Region Subproblem Nauk SSSR Tekhn. Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations On l p programming Semidefinite relaxations for the graph partitioning problem Approximating quadratic programming with bound and quadratic constraints Some Properties of Trust Region Algorithms for Nonsmooth Optimization On a subproblem of trust region algorithms for constrained optimization A dual algorithm for minimizing a quadratic function with two quadratic constraints Semidefinite programming relaxations for the quadratic assignment problem --TR --CTR Henry Wolkowicz , Miguel F. Anjos, Semidefinite programming for discrete optimization and matrix completion problems, Discrete Applied Mathematics, v.123 n.1-3, p.513-577, 15 November 2002

quadratically constrained quadratic programs;lagrangian relaxations;quadratic assignment;max-cut problems;graph partitioning;semidefinite programming

587742

Optimal Kronecker Product Approximation of Block Toeplitz Matrices.

This paper considers the problem of finding n n matrices and Bk that minimize $||T - \sum A_k \otimes B_k||_F$, where $\otimes$ denotes Kronecker product and T is a banded n n block Toeplitz matrix with banded n n Toeplitz blocks. It is shown that the optimal and Bk are banded Toeplitz matrices, and an efficient algorithm for computing the approximation is provided. An image restoration problem from the Hubble Space Telescope (HST) is used to illustrate the effectiveness of an approximate SVD preconditioner constructed from the Kronecker product decomposition.

Introduction . A Toeplitz matrix is characterized by the property that its entries are constant on each diagonal. Toeplitz and block Toeplitz matrices arise naturally in many signal and image processing applications; see, for example, Bunch [4] and Jain [17] and the references therein. In image restoration [21], for instance, one needs to solve large, possibly ill-conditioned linear systems in which the coefficient matrix is a banded block Toeplitz matrix with banded Toeplitz blocks (bttb). Iterative algorithms, such as conjugate gradients (cg), are typically recommended for large bttb systems. Matrix-vector multiplications can be done efficiently using fast Fourier transforms [14]. In addition, convergence can be accelerated by preconditioning with block circulant matrices with circulant blocks (bccb). A circulant matrix is a Toeplitz matrix in which each column (row) can be obtained by a circular shift of the previous column (row), and a bccb matrix is a natural extension of this structure to two dimensions; c.f. Davis [10]. Circulant and bccb approximations are used extensively in signal and image processing applications, both in direct methods which solve problems in the "Fourier domain" [1, 17, 21], and as preconditioners [7]. The optimal circulant preconditioner introduced by Chan [8] finds the closest circulant matrix in the Frobenius norm. Chan and Olkin [9] extend this to the block case; that is, a bccb matrix C is computed to minimize bccb approximations work well for certain kinds of bttb matrices [7], especially if the unknown solution is almost periodic. If this is not the case, however, the performance of bccb preconditioners can degrade [20]. Moreover, Serra-Capizzano and Tyrtyshnikov [6] have shown recently that it may not be possible to construct a bccb preconditioner that results in superlinear convergence of cg. Here we consider an alternative approach: optimal Kronecker product approxi- mations. A Kronecker product A\Omega B is defined as a Raytheon Systems Company, Dallas, y Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322 (nagy@mathcs.emory.edu). J. KAMM AND J. NAGY In particular, we consider the problem of finding matrices A k , B k to minimize s A where T is an n 2 \Theta n 2 banded bttb matrix, and A k , B k are n \Theta n banded Toeplitz matrices. A general approach for constructing such an optimal approximation was proposed by Van Loan and Pitsianis [25] (see also Pitsianis [23]). Their approach, which we describe in more detail in Section 2, requires computing principal singular values and vectors of an n 2 \Theta n 2 matrix related to T . An alternative approach for computing a Kronecker product approximation T - A\Omega B for certain deconvolution problems was proposed by Thirumalai [24]. A similar approach for banded bttb matrices was considered by Nagy [22]. As opposed to the method of Van Loan and Pitsianis, the schemes described in [22, 24] require computing principal singular values and vectors of an array having dimension at most n \Theta n, and thus can be substantially less expensive. Moreover, Kamm and Nagy [20] show how these approximations can be used to efficiently construct approximate svd preconditioners. Numerical examples in [20, 22, 24] indicate that this more efficient approach can lead to preconditioners that perform better than bccb approximations. However, theoretical results establishing optimality of the approximations, such as in equation (1.1), were not given. In this paper, we provide these results. In particular, we show that some modifications to the method proposed in [22, 24] are needed to obtain an approximation of the form (1.1). Our theoretical results lead to an efficient algorithm for computing Kronecker product approximations of banded bttb matrices. This paper is organized as follows. Some notation is defined, and a brief review of the method proposed by Van Loan and Pitsianis is provided in Section 2. In Section 3 we show how to exploit the banded bttb structure to obtain an efficient scheme for computing terms in the Kronecker product decomposition. A numerical example from image restoration is given in Section 4. 2. Preliminaries and Notation. In this section we establish some notation to be used throughout the paper, and describe some previous work on Kronecker product approximations. To simplify notation, we assume T is an n \Theta n block matrix with n \Theta n blocks. 2.1. Banded bttb Matrices. We assume that the matrix T is a block banded Toeplitz matrix with banded Toeplitz blocks (bttb), so it can be uniquely determined by a single column t which contains all of the non-zero values in T ; that is, some central column. It will be useful to define an n \Theta n array P as operator transforms matrices into vectors by stacking columns as follows: \Theta a 1 a 2 \Delta \Delta \Delta an a 1 a 2 an TOPELITZ KRONECKER PRODUCT APPROXIMATION 3 Suppose further that the entry of P corresponding to the diagonal of T is known 1 . For example, suppose that where the diagonal of T is located at (i; is the sixth column of T , and we write In general, if the diagonal of T is then the upper and lower block bandwidths of are respectively. The upper and lower bandwidths of each Toeplitz block are In a similar manner, the notation is used to represent a banded point Toeplitz matrix X constructed from the vector x, where x i corresponds to the diagonal entry. For example, if the second component of the vector corresponds to the diagonal element of a banded Toeplitz matrix X , then 2.2. Kronecker Product Approximations. In this subsection we review the work of Van Loan and Pitsianis. We require the following properties of Kronecker products: (A\Omega B)(C\Omega (AC)\Omega (BD), ffl If U 1 and U 2 are orthogonal matrices, then U A more complete discussion and additional properties of Kronecker products can be found in Horn and Johnson [16] and Graham [13]. Loan and Pitsianis [25] (see also, Pitsianis [23]) propose a general technique for an approximation involving Kronecker products where jjT \Gamma minimized. By defining the transformation to tilde space of a block matrix T , In image restoration, P is often referred to as a "point spread function", and the diagonal entry is the location of the "point source". See Section 4 for more details. 4 J. KAMM AND J. NAGY as ~ it is shown in [23, 25] that s s (~ a k ~ where ~ a Thus, the Kronecker product approximation problem is reduced to a rank-s approximation problem. Given the svd of ~ it is well known [12] that the rank-s approximation ~ which minimizes jj ~ ~ . Choosing ~ a ~ ~ a k ~ jj F over all rank-s approximations, and thus one can construct an approximation - This general technique requires computing the largest s singular triplets of an which may be expensive for large n. Thirumalai [24] and Nagy [22] show that a Kronecker product approximation of a banded bttb matrix T can be found by computing the largest s singular triplets of the n \Theta n array P . However, this method does not find the Kronecker product which minimizes the Frobenius norm approximation problem in equation (1.1). In the next section we show that if T is a banded bttb matrix, then this optimal approximation can be computed from an svd of a weighted version of the n \Theta n array P . 3. bttb Optimal Kronecker Product Approximation. Recall that the Van Loan and Pitsianis approach minimizes for a general (un- structured) matrix T , by minimizing jj ~ (~ a k ~ k )jj F . If it is assumed that A k are banded Toeplitz matrices, then the array P associated with the central column of T can be weighted and used to construct an approximation which minimizes k=1 (~ a k ~ Theorem 3.1. Let T be the n 2 \Theta n 2 banded bttb matrix constructed from P , is the diagonal element of T (therefore, the upper and lower block bandwidths of T are and the upper and lower bandwidths of each Toeplitz block are be an n \Theta n banded Toeplitz matrix with upper lower bandwidth be an n \Theta n banded Toeplitz matrix with upper bandwidth lower bandwidth n \Gamma j. Define a k and b k such that A ~ ~ a ~ TOPELITZ KRONECKER PRODUCT APPROXIMATION 5 s ~ a k ~ s (W a a k )(W b b k ) T Proof. See Section 3.1. 2 Therefore, if A k and B k are constrained to be banded Toeplitz matrices, then can be minimized by finding a k , b k which minimize jjP (W a a k )(W b b k ) T jj F . This is a rank-s approximation problem, involving a matrix of relatively small dimension, which can be constructed using the svd of Pw . Noting that W a and W b are diagonal matrices which do not need to be formed explicitly, the construction of - are banded Toeplitz matrices, can be computed as follows: ffl Define the weight vectors w a and w b based on the (i; location (in P ) of the diagonal entry of T : \Theta p \Theta p ffl Calculate its svd , where ": " denotes point-wise multiplication. ffl Calculate a where "./" denotes point-wise division. The proof of Theorem 3.1 is based on observing that ~ T has at most n unique rows and n unique columns, which consist precisely of the rows and columns of P . This observation will become clear in the following subsection. 3.1. Proof of Theorem 3.1. To prove Theorem 3.1, we first observe that if a matrix has one row which is a scalar multiple of another row, then a rotator can be constructed to zero out one of these rows, i.e., ff If this is extended to the case where more than two rows are repeated, then a simple induction proof can be used to establish the following lemma. 6 J. KAMM AND J. NAGY Lemma 3.2. Suppose an n \Theta n matrix X has k identical rows: x Then a sequence of k \Gamma1 orthogonal plane rotators can be constructed such that thereby zeroing out all the duplicate rows. It is easily seen that this result can be applied to the columns of a matrix as well, using the transpose of the plane rotators defined in Lemma 3.2. Lemma 3.3. Suppose an n \Theta n matrix X contains k identical columns: Then an orthogonal matrix Q can be constructed from a series of plane rotators such that \Theta p The above results illustrate the case where the first occurrence of a row (column) is modified to zero out the remaining occurrences. However, this is for notational convenience only. By appropriately constructing the plane rotators, any one of the duplicate rows (columns) may be selected for modification, and the remaining rows (columns) zeroed out. These rotators can now be applied to the matrix ~ T . Lemma 3.4. Let T be the n 2 \Theta n 2 banded bttb matrix constructed from P , where ij is the diagonal entry of T . In other words, define ~ TOPELITZ KRONECKER PRODUCT APPROXIMATION 7 Then orthogonal matrices Q 1 and Q 2 can be constructed such that ~ Proof. By definition, representing ~ T using the n \Theta n 2 submatrices ~ ~ ~ ~ ~ it is clear that ~ contains only n unique rows, which are ~ t T n , and that the i th submatrix, ~ contains all the unique rows, i.e., ~ Furthermore, it can be seen that there are occurrences of ~ t T of ~ t T occurrences of ~ t T occurrences of ~ t T of ~ t T . Therefore, a sequence of orthogonal plane rotators can be constructed to zero 8 J. KAMM AND J. NAGY out all rows of ~ T except those in the submatrix ~ ~ W a ~ partitioning ~ ~ \Theta ~ T in where each ~ ij is an n \Theta n submatrix, it can be seen that ~ contains only n unique columns, which are the columns of P , and that the j th submatrix ~ contains all the unique columns, i.e., ~ Furthermore, the matrix ~ occurrences of p 1 of occurrences of pn . Therefore, a sequence of orthogonal plane rotators can be constructed such that ~ :The following properties involving the vec and toep2 operators are needed. Lemma 3.5. Let T , ~ T , and P be defined as in Lemma 3.4. Further, let A k be an n \Theta n banded Toeplitz matrix with upper bandwidth lower bandwidth and let B k be an n \Theta n banded Toeplitz matrix with upper bandwidth lower bandwidth j. Define a k and b k such that A Then 1. are any two matrices of the same size, TOPELITZ KRONECKER PRODUCT APPROXIMATION 9 2. toep2(x; are any two vectors of the same length, 3. toep2fvec[( 4. Proof. Properties 1 and 2 are clear from the definitions of the vec and toep2 operators. Property 3 can be seen by considering the banded Toeplitz matrices toep(a; i) and noting that the central column of all the non-zero entries is an b 1 an Therefore, property 3 holds when both sides are banded bttb matrices constructed from the same central column, and can be extended to applying property 2. Property 4 follows from properties 2 and 3. 2 Using these properties, Lemma 3.4 can be extended to the matrix ~ ~ a k ~ k . Lemma 3.6. Let T be the n 2 \Theta n 2 banded bttb matrix constructed from P , where ij is the diagonal entry of T . Further, let A k be an n \Theta n banded Toeplitz matrix with upper bandwidth lower bandwidth be an n \Theta n banded Toeplitz matrix with upper bandwidth j. Define a k and b k such that a Let ~ T , W a , and W b be defined as in Lemma 3.4. Then orthogonal matrices Q 1 and can be constructed such that s ~ a k ~ Proof. Using Lemma 3.5, s A s a By definition of the transformation to tilde space, s A s ~ a k J. KAMM AND J. NAGY Applying Lemma 3.4 to T \Gamma s ~ a k ~ a :The proof of Theorem 3.1 follows directly from Lemma 3.6 by noting that s ~ a k ~ s ~ a k ~ s a s (W a a k )(W b b k ) T 3.2. Further Analysis. It has been shown how to minimize when the structure of - T is constrained to be a sum of Kronecker products of banded Toeplitz matrices. We now show that if T is a banded bttb matrix, then the matrix - must adhere to this structure. Therefore, the approximation minimizes when T is a banded bttb matrix. If T is a banded bttb matrix, then the rows and columns of ~ T have a particular structure. To represent this structure, using an approach similar to Van Loan and Pitsianis [25], we define the constraint matrix S n;! . Given an n \Theta n banded Toeplitz matrix T , with upper and lower bandwidths is an matrix such that S T be a 4 \Theta 4 banded Toeplitz matrix with bandwidths ! TOPELITZ KRONECKER PRODUCT APPROXIMATION 11 and Note that S T n;! clearly has full row rank. Given the matrix T in (2.2), ~ and the rows and columns of ~ ~ ~ . Using the structure of ~ T , the matrix - A minimizing must be structured such that A i and B i are banded Toeplitz matrices, as the following sequence of results illustrate. Lemma 3.7. Let \Theta a 1 a 2 \Delta \Delta \Delta an be the n \Theta n matrix whose structure is constrained by S T n;! a be the svd of A, where n;! Proof. Given the svd of A, n;! J. KAMM AND J. NAGY By definition, S T n;! Applying this result to A T , it is clear that the right singular vectors of A satisfy if the rows of A are structured in the same manner. Lemma 3.8. Let A =6 6 6 4 a be the n \Theta n matrix whose structure is constrained by S T n;! a i be the svd of A, where Theorem 3.9. Let T be an n \Theta n banded block Toeplitz matrix with n \Theta n banded Toeplitz blocks, where the upper and lower block bandwidths of T are and the upper and lower bandwidths of each Toeplitz block are \Theta fl u fl l . Then the matrices A i and B i minimizing banded Toeplitz matrices, where the upper and lower bandwidths of A i are given by !, and the upper and lower bandwidths of B i are given by fl. Proof. Recall that (~ a i where . The structure of T results in rank( ~ ~ ~ Letting ~ i be the svd of ~ (~ a i ~ is minimized by ~ a Therefore, A i is an n \Theta n banded Toeplitz matrix with upper and lower bandwidths given by !, and B i is an n \Theta n banded Toeplitz matrix with upper and lower bandwidths given by fl. 2 3.3. Remarks on Optimality. The approach outlined in this section results in an optimal Frobenius norm Kronecker product approximation to a banded bttb matrix. The approximation is obtained from the principal singular components of an array Pw = W a PW b . It might be interesting to consider whether it is possible to compute approximations which are optimal in another norm. In particular, the method considered in [20, 22, 24] uses a Kronecker product approximation computed from the principal singular components of P . Unfortunately we are unable to show that this leads to an optimal norm approximation. However, there is a very close relationship between the approaches. Since W a and W b are full rank, well-conditioned diagonal matrices, P and Pw have the same rank. Although it is possible to establish bounds on the singular values of products of matrices (see, for example, Horn and Johnson [15]), we have not been able to determine a precise relationship between the Kronecker product approximations obtained from the two methods. However we have found through extensive numerical results that both methods give similarly good approximations. Since numerical comparisons do not provide any additional insight into the quality of the approximation, we omit such results. Instead, in the next TOPELITZ KRONECKER PRODUCT APPROXIMATION 13 section we provide an example from an application that motivated this work, and illustrate how a Kronecker product approximation might be used in practice. We note that further comparisons with bccb approximations can be found in [20, 24]. 4. An Image Restoration Example. In this section we consider an image restoration example, and show how the Kronecker product approximations can be used to construct an approximate svd preconditioner. Image restoration is often modeled as a linear system: where b is an observed blurred, noisy image, T is a large, often ill-conditioned matrix representing the blurring phenomena, n is noise, and x is the desired true image. If the blur is assumed to be spatially invariant, then T is a banded bttb matrix [1, 21]. In this case, the array P corresponding to a central column of T is called a point spread function (psf). The test data we use consists of a partial image of Jupiter taken from the Hubble Space Telescope (hst) in 1992, before the mirrors in the Wide Field Planetary Camera were fixed. The data was obtained via anonymous ftp from ftp.stsci.edu, in the directory pub/stsdas/testdata/restore/data/jupiter. Figure 4.1 shows the observed image. Also shown in Figure 4.1 is a mesh plot of the psf, P , where the peak corresponds to the diagonal entry of T . The observed image is 256 \Theta 256, so T is 65; 536 \Theta 65; 536. 50 100 150 200 250100200020406010305000.040.08a. Observed, blurred image. b. psf, P . Fig. 4.1. Observed hst image and point spread function. We mention that if T is ill-conditioned, which is often the case in image restora- tion, then regularization is needed to suppress noise amplification in the computed solution [21]. Although T is essentially too large to compute its condition number, certain properties of the data indicate that T is fairly well conditioned. For instance, we observe that the psf is not very smooth (smoother psfs typically indicate more 14 J. KAMM AND J. NAGY ill-conditioned T ). Another indication comes from the fact that the optimal circulant approximation of T , as well as our approximate svd of T (to be described below) are well conditioned; specifically these approximations have condition numbers that are approximately 20. We also mention that if the psf can be expressed as (i.e., it has rank 1), then the matrix T is separable. Using Theorem 3.1, A\Omega B, where oev). Efficient numerical methods that exploit the Kronecker product structure of T (e.g., [2, 5, 11]) can then be used. However, as can be seen from the plot of the singular values of P in Figure 4.2, for this data, P is not rank one, and so T is not separable. We therefore suggest con- values of the psf, P . structing an approximate svd to use as a preconditioner, and solve the least squares problem using a conjugate gradient algorithm, such as cgls; see Bj-orck [3]. This preconditioning idea was proposed in [20], and can be described as follows. Given s A an svd approximation of T can be constructed as A . Note that the number of terms s only affects the setup cost of calculating \Sigma. For s - 1, clearly solves the minimization problem min TOPELITZ KRONECKER PRODUCT APPROXIMATION 15 over all diagonal matrices \Sigma and therefore produces an optimal svd approximation, given a fixed UA\Omega UB and VA\Omega VB . This is analogous to the circulant and bccb approximations discussed earlier, which provide an optimal eigendecomposition given a fixed set of eigenvectors (i.e., the Fourier vectors). In our tests, we use cgls to solve the ls problem using no preconditioner, our approximate svd preconditioner (with terms in equation (4.1)) and the optimal circulant preconditioner. Although we observed that T is fairly well conditioned, we should still be cautious about noise corrupting the computed restorations. There- fore, we use the conservative stopping tolerance jjT T Table 4.1 shows the number of iterations needed for convergence in each case, and in Figure 4.3 we plot the corresponding residuals at each iteration. The computed solutions are shown in Figure 4.4, along with the hst observed, blurred image for comparison. Table Number of cgls and pcgls iterations needed for convergence. cgls, no prec. pcgls, circulant prec. pcgls, svd prec. 43 12 4 iteration residual 2-norm no prec. circulant prec. svd prec Fig. 4.3. Plot of the residuals at each iteration. 5. Concluding Remarks. Because the image and psf used in the previous section come from actual hst data, we cannot get an analytical measure on the accuracy of the computed solutions. However, we observe from Figure 4.4 that all solutions appear to be equally good restorations of the image, and from Figure 4.3 we see that the approximate svd preconditioner is effective at reducing the number of iterations needed to obtain the solutions. Additional numerical examples comparing the accuracy of computed solutions, as well as computational cost of bccb and the approximation svd preconditioner, can be found in [19, 20]. A comparison of J. KAMM AND J. NAGY 50 100 150 200 25010020050 100 150 200 250100200a. hst blurred image. b. cgls solution, 43 iterations. 50 100 150 200 25010020050 100 150 200 250100200c. pcgls solution, circ. prec., 12 its. d. pcgls solution, svd prec., 4 its. Fig. 4.4. The observed image, along with computed solutions from cgls and pcgls. computational complexity between bccb preconditioners and the approximate svd preconditioner depends on many factors. For example: ffl What is the dimension of P (i.e., the bandwidths of T )? ffl Is a Lanczos scheme used to compute svds of P , A 1 and ffl Do we take advantage of band and Toeplitz structure when forming matrix-matrix products involving UA , UB , VA , VB and A k , B k , TOPELITZ KRONECKER PRODUCT APPROXIMATION 17 ffl How many terms, s, do we take in the Kronecker product approximation? ffl For bccb preconditioners: is n a power of 2? If we assume T is set up and application of the approximate svd preconditioner is at most O(n 3 ). If we further assume that n is a power of 2, then the corresponding cost for bccb preconditioners is at least O(n 2 log 2 n). It should be noted that the approximate svd preconditioner does not require complex arithmetic, does not require n to be a power of 2, or any zero padding. Moreover, decomposing T into a sum of Kronecker products, whose terms are banded Toeplitz matrices, might lead to other fast algorithms (as has occurred over many years of studying displacement structure [18]). In this case, the work presented in this paper provides an algorithm for efficiently computing an optimal Kronecker product approximation. --R Restoration of images degraded by spatially varying pointspread functions by a conjugate gradient method Stability of methods for solving Toeplitz systems of equations Application of ADI iterative methods to the image restoration of noisy images Any circulant-like preconditioner for multilevel Toeplitz matrices is not superlinear Conjugate gradient methods for Toeplitz systems An optimal circulant preconditioner for Toeplitz systems Preconditioners for Toeplitz-block matrices Algorithms for the regularization of ill-conditioned least squares problems with tensor product structure Matrix Computations Kronecker Products and Matrix Calculus: with Applications Restoration of atmospherically blurred images by symmetric indefinite conjugate gradient techniques Matrix Analysis Fundamentals of Digital Image Processing Theory and applications Singular value decomposition-based methods for signal and image restoration Kronecker product and SVD approximations in image restoration Iterative Identification and Restoration of Images Decomposition of block Toeplitz matrices into a sum of Kronecker products with applications in image restoration The Kronecker Product in Approximation and Fast Transform Generation High performance algorithms to solve Toeplitz and block Toeplitz matrices Approximation with Kronecker products --TR --CTR S. Serra Capizzano , E. Tyrtyshnikov, How to prove that a preconditioner cannot be superlinear, Mathematics of Computation, v.72 n.243, p.1305-1316, July

conjugate gradient method;block Toeplitz matrix;singular value decomposition;kronecker product;image restoration;preconditioning

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Computing Symmetric Rank-Revealing Decompositions via Triangular Factorization.

We present a family of algorithms for computing symmetric rank-revealing VSV decompositions based on triangular factorization of the matrix. The VSV decomposition consists of a middle symmetric matrix that reveals the numerical rank in having three blocks with small norm, plus an orthogonal matrix whose columns span approximations to the numerical range and null space. We show that for semidefinite matrices the VSV decomposition should be computed via the ULV decomposition, while for indefinite matrices it must be computed via a URV-like decomposition that involves hypernormal rotations.

Introduction . Rank-revealing decompositions of general dense matrices are widely used in signal processing and other applications where accurate and reliable computation of the numerical rank, as well as the numerical range and null space, are required. The singular value decomposition (SVD) is certainly a decomposition that reveals the numerical rank, but what we have in mind here are the RRQR and (i.e., URV and ULV) decompositions which can be computed and, in particular, updated more eciently than the SVD. See, e.g., [7, xx2.7.5{2.7.7], [20, x2.2] and [33, Chapter 5] for details and references to theory, algorithms, and applications. The key to the eciency of RRQR and UTV algorithms is that they consist of an initial triangular factorization which can be tailored to the particular matrix, followed by a rank-revealing post-processing step. If the matrix is mn with m n and with numerical rank k, then the initial triangular factorization requires O(mn 2 ) ops, while the rank-revealing step only requires O((n k)n 2 ) ops if k n, and O(kn 2 ) ops if n. The updating can always be done in O(n 2 ) ops, when implemented properly. We refer to the original papers [9], [10], [16], [18], [19], [23], [31], [32] for details about the algorithms. For structured matrices (e.g., Hankel and Toeplitz matrices), the initial triangular factorization in the RRQR and UTV algorithms has the same complexity as the rank-revealing step, namely, O(mn) ops; see [7, x8.4.2] for signal processing aspects. However, accurate principal singular values and vectors can also be computed by means of Lanczos methods in the same complexity, O(mn) ops [13]. Hence the advantage of a rank-revealing decomposition depends on the matrix structure and the numerical rank of the matrix. Rank-revealing decompositions of general sparse matrices are also in use, e.g., in optimization and geometric design [27]. For sparse matrices, the initial pivoted triangular factorization can exploit the sparsity of A. However, the UTV post-processors may produce a severe amount of ll, while the ll in the RRQR post-processor is P. Y. Yalamov was supported by a fellowship from the Danish Rectors' Conference and by Grants MM-707/97 and I-702/97 from the National Scientic Research Fund of the Bulgarian Ministry of Education and Science. y Department of Mathematical Modelling, Technical University of Denmark, Building 321, DK- 2800 Lyngby, Denmark (pch@imm.dtu.dk). z Center of Applied Mathematics and Informatics, University of Rousse, 7017 Rousse, Bulgaria (yalamov@ami.ru.acad.bg). P. C. HANSEN AND P. Y. YALAMOV restricted to lie in the columns that are permuted to the right of the triangular factor [7, Thm. 6.7.1]. An alternative sparse URL decomposition A = U RL, where U is orthogonal and R and L are upper and lower triangular, respectively, was proposed in [26]. This decomposition can be computed with less ll, at the expense of working with only one orthogonal matrix. Numerically rank-decient symmetric matrices also arise in many applications, notably in signal processing and in optimization algorithms (such as those based on interior point and continuation methods). In both areas, fast computation and efcient updating are key issues, and sparsity is also an issue in some optimization problems. Symmetric rank-revealing decompositions enable us to compute symmetric rank-decient matrix approximations (obtained by neglecting blocks in the rank- revealing decomposition with small norm). This is important, e.g., in rank-reduction algorithms in signal processing where one wants to compute rank-decient symmetric semidenite matrices. In addition, utilization of symmetry leads to faster algorithms, compared to algorithms for nonsymmetric matrices. In spite of this, very little work has been done on symmetric rank-revealing decom- positions. Luk and Qiao [24] introduced the term VSV decomposition and proposed an algorithm for symmetric indenite Toeplitz matrices, while Baker and DeGroat [2] presented an algorithm for symmetric semi-denite matrices. The purpose of this paper is to put the work in [2] and [24] into a broader perspective by surveying possible rank-revealing VSV decompositions and algorithms, including the underlying theory. Our emphasis is on algorithms which, in addition to revealing the numerical rank, provide accurate estimates of the numerical range and null space. We build our algorithms on existing methods for computing rank- revealing decompositions of triangular matrices, based on orthogonal transformations. Our symmetric decompositions and algorithms inherit the properties of these underlying algorithms which are well understood today. We emphasize that the goal of this paper is not to present detailed implementations of our VSV algorithms, but rather to set the stage for such implementations. The papers [4] and [28] clearly demonstrate that careful implementations of ecient and robust mathematical software for numerically rank-decient problems requires a major amount of research which is outside the scope of the present paper. Our paper is organized as follows. After brie y surveying general rank-revealing decompositions in x2, we dene and analyze the rank-revealing VSV decomposition of a symmetric matrix in x3. Numerical algorithms for computing VSV decompositions of symmetric semi-denite and indenite matrices are presented in x4, and we conclude with some numerical examples in x5. 2. General Rank-Revealing Decompositions. In this paper we restrict our attention to real square n n matrices. The singular value decomposition (SVD) of a square matrix is given by where u i and v i are the columns of the orthogonal matrices U and V , and with . The numerical rank k of A, with respect to the threshold , is the number of singular values greater than or equal to , i.e., k > k+1 [20, x3.1]. The RRQR, URV, and ULV decompositions are given by Here, Q, UR , UL , VR , and VL are orthogonal matrices, is a permutation matrix, T and R are upper triangular matrices, and L is a lower triangular matrix. Moreover, if we partition the triangular matrices as then the numerical rank k of A is revealed in the triangular matrices in the sense that are k k and F The rst k columns of the left matrices Q, UR , and UL span approximations to the numerical range of A, dened as spanfu g, and the last n k columns of the right matrices VR and VL span approximations to the numerical null-space of A, dened as spanfv g. See, e.g., [20, x3.1] for details. Precise denitions of RRQR decompositions and algorithms are given by Chandrasekaran and Ipsen [11], Gu and Eisenstat [19] and Hong and Pan [23], and associated large-scale implementations are available in Fortran [4]. Denitions of UTV decompositions and algorithms are given by Stewart [31], [32]. Matlab software for both RRQR and UTV decompositions is available in the UTV Tools package [17]. 3. Symmetric Rank-Revealing Decompositions. For a symmetric n n matrix A, we need rank-revealing decompositions that inherit the symmetry of the original matrix. In particular this is true for the eigenvalue decomposition (EVD) are the right singular vectors, while Corresponding to the UTV decompositions, Luk and Qiao [24] dened the following VSV decomposition where VS is an orthogonal matrix, and S is a symmetric matrix with partitioning in which S 11 is k k. We say that the VSV decomposition is rank-revealing if 4 P. C. HANSEN AND P. Y. YALAMOV This denition is very similar to the denition used by Luk and Qiao, except that they use ktriu(S 22 )k 2 F instead of kS 22 k 2 F , where \triu" denotes the upper triangular part. Our choice is motivated by the fact that kS 22 k 2 n as kS 12 Given the VSV decomposition in (3.2), the rst k columns of VS and the last columns of VS provide approximate basis vectors for the numerical range and null space, respectively. Moreover, given the ill-conditioned problem b, we can compute a stabilized \truncated VSV solution" x k by neglecting the three blocks in consists of the rst k columns of VS . We return to the computation of x k in x4.4. Instead of working directly with the matrix S, it is more convenient to work with a symmetric decomposition of S and, in particular, of S 11 . The form of this decomposition depends on both the matrix A (semi-denite or indenite) and the rank-revealing algorithm. Hence, we postpone a discussion of the particular form of S to the presentation of the algorithms. Instead, we summarize the approximation properties of the VSV decomposition. Theorem 3.1. Let the VSV decompositions of A be given by (3.2), and partition the matrix S as in (3.3) where k is the numerical rank. Then the singular values of diag(S 11 ; S 22 ) are related to those of A as Moreover, the angle between the subspaces spanned by the rst k columns of V and VS , dened by sin bounded as sin Proof. The bound (3.4) follows from the standard perturbation bound for singular values: where we use that the singular values of the symmetric \perturbation matrix" appear in pairs. To prove the upper bound in (3.5), we partition columns. Moreover, we write k. If we insert these partitionings as well as (3.1) and (3.2) into the product AV S;0 then we obtain Multiplying from the left with V T k we get from which we obtain Taking norms in this expression and inserting sin we get sin 1 which immediately leads to the upper bound in (3.5). To prove the lower bound, we use that Taking norms and using sin k+1 , we obtain the left bound in (3.5). We conclude that if there is a well-dened gap between k and kS 22 k 2 , and if the norm kS 12 k 2 of the o-diagonal block is suciently small, then the numerical rank k is indeed revealed in S, and the rst k columns of VS span an approximation to the singular subspace spanfv g. The following theorem shows that a well-dened gap is also important for the perturbation bounds. Theorem 3.2. Let e S , and let denote the angle between the subspaces spanned by the rst k columns of VS and e sin 4 g. Proof. The bound follows from Corollary 3.2 in [14]. We see that a small upper bound is guaranteed when kAk 2 as well as and k+1 are somewhat smaller than k . 4. Algorithms for Symmetric Rank-Revealing Decompositions. Similar to general rank-revealing algorithms, the symmetric algorithms consist of an initial triangular factorization and a rank-revealing post-processing step. The purpose of the latter step is to ensure that the largest k singular values are revealed in the leading submatrix S 11 and that the corresponding singular subspace is approximated by the span of the rst k columns of VS . For a semi-denite matrix A, our initial factorization is the symmetrically pivoted where P is the permutation matrix, and C is the upper triangular (or trapezoidal) factor. The numerical properties of this algorithm are discussed by Higham in [22]. If A is a symmetric semi-denite Toeplitz matrix, then there is good evidence (although no strict proof) that the Cholesky factor can be computed eciently and reliably without the need for pivoting by means of the standard Schur algorithm [30]. When A is indenite, then it would be convenient to work with an initial factorization of the form P T C where C is again triangular and diag(1). Unfortunately such factorizations are not guaranteed to exist. Therefore our initial factorization is the symmetrically pivoted LDL T factorization where P is the permutation matrix, L is a unit lower triangular matrix, and D is a block diagonal matrix with 11 and 22 blocks on the diagonal. The state-of-the-art in LDL T algorithms is described in [1], where it is pointed out that special care must be taken in the implementation to avoid large entries in L when A is ill conditioned. Alternatively, one could use the factorization G 6 P. C. HANSEN AND P. Y. YALAMOV Table The four post-processing rank-revealing steps for a symmetric semi-denite matrix. Post-proc. Decomposition Symmetric matrix R T 22 R 22 RRQR 22 T 22 22 L 21 L T 22 L 22 described in [29], where G is block triangular. If A is a symmetric indenite Toeplitz matrix, then the currently most reliable approach to computing the LDL T factorization seems to be via orthogonal transformation to a Cauchy matrix [21]. The reason why we need the post-processing step is that the initial factorization may not reveal the numerical rank of A| there is no guarantee that small eigenvalues of A manifest themselves in small diagonal elements of C or in small eigenvalues of D. In particular, since 2 , we obtain 2showing that a small n may not be revealed in D when L is ill conditioned. 4.1. Algorithms for Semi-Denite Matrices. For symmetric semi-denite matrices there is a simple relationship between the SVDs of A and C. Theorem 4.1. The right singular vectors of P T AP are also the right singular vectors of C, and Proof. The result follows from inserting the SVD of C into P T Hence, once we have computed the initial pivoted Cholesky factorization (4.1), we can proceed by computing a rank-revealing decomposition of C, and this can be done in several ways. Let E denotes the exchange matrix consisting of the columns of the identity matrix in reverse order, and write P T AP as Then we can compute a URV or RRQR decomposition of C, a ULV decomposition of ECE, or an RRQR decomposition of (ECE) T , as shown in the left part of Table 4.1. The approach using the URV decomposition of C was suggested in [2]. Table 4.1 also shows the particular forms of the resulting symmetric matrix S, as derived from the following relations: R The rst, third and fourth approaches lead to a symmetric matrix S that reveals the numerical rank of A by having both an o-diagonal block S 12 and a bottom right block S 22 with small norm. The second approach does not produce blocks S 12 and S 22 with small norm; instead (since T 11 is well conditioned) this algorithm provides a permutation P that is guaranteed to produce a well-conditioned leading The remaining three algorithms yield approximate bases for the range and null spaces of A, due to Theorem 3.1. It is well known that among the rank-revealing decompositions, the ULV decomposition can be expected to provide the most accurate bases for the right singular subspaces, in the form of the columns of VL ; see, e.g., [32] and [15]). Therefore, the algorithm that computes the ULV decomposition of ECE is to be preferred. We remark that the matrix UL in the ULV decomposition need not be computed. In terms of the blocks S 12 and S 22 , the ULV-based algorithm is the only algorithm that guarantees small norms of both the o-diagonal block S 22 and the bottom right block S 22 L 22 , because the norms of both L 12 and L 22 are guaranteed to be small. From Theorem 4.1 and the denition of the ULV decomposition we have and therefore kS 12 k 2 ' kS 22 k 2 ' k+1 . For a sparse matrix the situation is dierent, because the UTV post-processors may produce severe ll, while the RRQR post-processor produces only ll in the n k rightmost columns of T . For example, if A is the upper bidiagonal matrix in which B p is an upper bidiagonal p p matrix of all ones, and e p is the pth column of the identity matrix, then URV with threshold produces a full k k upper triangular R 11 , while RRQR with the same threshold produces a k k upper bidiagonal T 11 . Hence, for sparsity reasons, the UTV approaches may not be suited for computing the VSV decomposition, depending on the sparsity pattern of A. An alternative is to use the algorithm based on RRQR decomposition of the transposed and permuted Cholesky factor (ECE) and we note that the permutation matrix is not needed. In terms of the matrix S, only the bottom right submatrix of S is guaranteed to have a norm of the order k+1 , because of the relations kS and kS 22 k In practice the situation can be better, because the RRQR-algorithm|when applied to the matrix E C T E |may produce an o-diagonal block T 12 whose norm is smaller than what is guaranteed (namely, of the order 1=2 The reason is that the initial Cholesky factor C often has a trailing (n whose norm is close to 1=2 k+1 , which may produce a norm kS 12 k 2 close to k+1 . From the partitionings 22 En k En k C T and the fact that the RRQR post-processor leaves column norms unchanged and may permute the leading n k columns of E C T E to the back, we see that the norm of the resulting o-diagonal block T 12 in the RRQR decomposition can be bounded by Our numerical examples in x5 illustrate this. However, we stress that in the RRQR approach we can only guarantee that kS 12 k 2 is of the order 1=2 , and this point is illustrated by the matrix 8 P. C. HANSEN AND P. Y. YALAMOV 1. Compute the eigenvalue decomposition 2. Write as jj 1=2 . 3. Compute an orthogonal W such that C is lower triangular. Fig. 4.1. Interim processor for symmetric indenite matrices. where K is the \infamous" Kahan matrix [7, p. 105] that is left unchanged by QR factorization with ordinary column pivoting, yet its numerical rank is factorization with symmetric pivoting computes the Cholesky factor and when we apply RRQR to E C T E we obtain an upper triangular matrix T in which only the (n; n)-element is small, while kT 12 n . 4.2. Algorithms for Indenite Matrices. No matter which factorization is used for an indenite matrix, such as (4.2) or (4.3), there is no simple relationship between the singular values of A and the matrix factors. Hence the four \intrinsic" decompositions from Table 4.1 do not apply here, and the diculty is to develop a new factorization from which the numerical rank can be determined. All rank-revealing algorithms currently in use maintain the triangular form of the matrix in consideration, but when we apply the algorithms to the matrix L in the LDL T factorization (4.2) we destroy the block diagonal form of D. We can avoid this diculty by inserting an additional interim stage between the initial LDL T factorization and the rank-revealing post-processor, in which the middle block-diagonal matrix D is replaced by the signature matrix diag(1). At the same time, L is replaced by the product of an orthogonal matrix and a triangular matrix. The interim processor, which is summarized in Fig. 4.1, thus computes the factorization where W is orthogonal and C is upper triangular The interim processor is simple to implement and requires at most O(n 2 ) opera- tions, because W and W are block diagonal matrices with the same block structure as D. For each 1 1 block d ii in D the corresponding 1 1 blocks in W , jj 1=2 , and W are equal to 1, jd ii j 1=2 , and 1, respectively. For each 2 2 block in D we compute the eigenvalue decomposition d ii d i;i+1 d i;i+1 d i+1;i+1 then the corresponding 22 block in W is W ii , and the associated 22 block in W is a Givens rotation chosen such that C stays triangular. If A is sparse, then some ll may be introduced in C by the interim processor, but since the Givens transformations are applied to nonoverlapping 22 blocks, ll introduced in the treatment of a particular block does not spread during the processing of the other blocks. The same type of processor can also be applied to the G G T factorization (4.3) in order to turn the block triangular matrix G into triangular form. Future developments of rank-revealing algorithms for more general matrices than the triangular ones may render the interim processor super uous. It may also be possible to compute the factorization (4.5) directly. We shall now explore the possibilities for using triangular rank-revealing post-processors similar to the ones for semi-denite matrices, but modied such that they yield a decomposition of C in which the leftmost matrix U is hypernormal with respect to the signature matrices and b i.e., we require U Hypernormal matrices and the corresponding transformations are introduced in [8] in connection with up- and downdating of symmetric indenite matrices. Here we use them to maintain the triangular form of the matrix C. The following theorem shows that a small singular value of A is guaranteed to be revealed in the triangular matrix C. Theorem 4.2. If n (C) denotes the smallest singular value of C in the interim factorization (4.5), then Proof. We have 1 , from which the result follows. Unfortunately, there is no guarantee that n (C) does not underestimate 1=2 dramatically, neither does it ensure that the size of n is revealed in S. We illustrate this with a small 5 5 numerical example from [1] where A is given by with and . The singular values of A are such that A has full rank with respect to the threshold . The corresponding matrix C has singular values Thus, 5 (C) is not a good approximation of 1=2 5 , and if we base the rank decision on 5 (C) and the threshold wrongly conclude that A is numerically rank decient. The conclusion is that for indenite matrices, a well conditioned C ensures that A is well conditioned, but we cannot rely solely on C for determination of the numerical rank of A. This rules out the use of RRQR factorization of C and ECE. The following theorem (which expands on results in [24]) shows how to proceed instead. Theorem 4.3. Let wn be an eigenvector of C C corresponding to the eigenvalue n that is smallest in absolute value, and let ~ wn be an approximation to wn . Moreover, choose the orthogonal matrix b V such that b the last column of the identity matrix, and partition the matrix P. C. HANSEN AND P. Y. YALAMOV such that S 11 is (n 1) (n 1). Then ks wn wn k 2 and wn wn Proof. Let C and consider rst the quantity A ~ wn s 22 Next, write ~ A ~ Au wn A u: Combining these two results we obtain s 22 n A n I) u and taking norms we get ks Both ks 12 k 2 2 and js 22 n j are lower bounds for the left-hand side, while kuk 2 is bounded above by tan . Combining this with the bound k ^ obtain the two bounds in the theorem. The above theorem shows that in order for n to reveal itself in S, we must compute an approximate null vector of C apply Givens rotations to this vector to transform it into e n , and accumulate these rotations from the right into C. At the same time, we should apply hypernormal rotations from the left in order to keep C upper triangular. Theorem 4.3 ensures that if ~ wn is close enough to wn then ks 12 k 2 is small and s 22 approximates n . We note that hypernormal transformations can be numerically unstable, and in our implementations we use the same stabilizations as in the stabilized hyperbolic rotations [7, x3.3.4]. Once this step has been performed, we de ate the problem and apply the same technique to the (n 1)(n 1) submatrix S are the leading submatrices of the updated factors. This is precisely the algorithm from [24]. When the process stops (because all the small singular values of A are revealed) we have computed the URV-like decomposition R such that U T R and the middle rank-revealing matrix is given by R Tb where 1 is k k. The condition estimator used in the URV-like post-processor must be modied, compared to the standard URV algorithm, because we must now estimate the smallest Table Summary of approaches for symmetric indenite matrices. Note that the RRQR approaches do not reveal the numerical rank, and that the ULV-like approach is impractical. Post-proc. Decomposition Comments to decomposition R R R 11 RRQR singular value of the matrix C C. In our implementation we use one step of inverse iteration applied to C C, with starting vector from the condition estimator of the ordinary URV algorithm applied to C. Finally consider a ULV-like approach applied to ECE. Again we must compute an approximate null vector of C C and transform it into the form e n by means of an orthogonal transformation. This transformation is applied from the right to ECE, and a hypernormal transformation from the left is then required to resotre the lower triangular form of To de ate this factorization, note that the leading (n 1) (n 1) block of L is given by L 11 and This shows that we cannot merely work on the block L 11 ; also the 1 (n 1) block ' T 21 is needed. Hence, after the de ation step we must work with trapezoidal matices instead of triangular matrices. This fact renders the ULV-like approach impractical. To summarize, for symmetric indenite matrices only the approach using the URV-like post-processor leads to a practical algorithm for revealing the numerical rank of A. Moreover, a well conditioned C signals a well conditioed A, but C cannot reveal A's numerical rank. Our analysis is summarized in Table 4.2, and the URV- based algorithm is summarized in Fig. 4.2 (following the presentations from [17]), where is the rank-decision tolerance for A. 4.3. Updating the VSV Decomposition. One of the advantages of the rank- revealing VSV decomposition over the EVD and SVD is that it can be updated eciently when A is modied by a rank-one change v v T . From the relation we see that the updating of A amounts to updating the rank-revealing matrix S by the rank-one matrix ww T with v, i.e., S . This can be done in while the EVD/SVD updating requires O(n 3 ) operations. Consider rst the semi-denite case, and let M denote one of the triangular 12 P. C. HANSEN AND P. Y. YALAMOV 1. Let k n and compute an initial factorization P T 2. Apply the interim processor to compute P T 3. Condition estimation: let e k estimate k C(1: k; 1: and let w k estimate the corresponding right singular vector. 4. If e k > 1=2 then exit. 5. Revealment: determine an orthogonal Q k such that Q T 6. update C(1: k; 1: 7. update C(1: k; 1: the hypernormal is chosen such that the updated C is 8. De ation: let k k 1. 9. Go to step 3. Fig. 4.2. The URV-based VSV algorithm for symmetric indenite matrices. matrices R, L, or T T from the algorithms in Table 4.1. Then and we see that the VSV updating is identical to standard updating of a triangular RRQR or UTV factor, which can be done stably and eciently by means of Givens transformation as described in [5], [31] and [32]. Next we consider the indenite case (4.9), where the updating takes the form showing that the VSV updating now involves hypernormal rotations. Hence, the up-dating is computationally similar to UTV downdating, whose stable implementation is discussed in [3] and [25]. Downdating the VSV decomposition will, in both cases, also involve hypernormal rotations. 4.4. Computation of Truncated VSV Solutions. Here we brie y consider the computation of the truncated VSV solution which we dene as consists of the rst k columns of VS . Both URV-based decompositions are simple to use. For the ULV-based decomposition we have S and we can safely neglect the term L T whose norm is at most of the order k+1 . Finally, for the RRQR-based decomposition we can use the following theorem. Theorem 4.4. If T is the triangular QR factor of (T then Alternatively, if the columns of the matrix form an orthonormal basis for the null space of (T I W 1 (4. Table Numerical results for the rank-revealing VSV algorithms. Post-processor (semi-def.) ULV mean 2:7 (semi-def.) RRQR mean 1:5 (semi-def.) URV-like mean (indef.) Proof. If T is a QR factorization then S T and S 1 T whici is (4.11). The same relation leads to S 1 denotes the pseudoinverse. In [6] used that which, combined with the relation (I W W T immediately leads to (4.12). The rst relation (4.11) in Theorem 4.4 can be used when k n, while the second relation (4.12) is more useful when k n. Note that W can be computed by orthonormalization of the columns of the matrix I This approach is particularly useful for sparse matrices because we only introduce ll when working with the \skinny" n (n 5. Numerical Examples. The purpose of this section is to illustrate the theory derived in the previous sections by means of some test problems. Although robust- ness, eciency and op counts are important practical issues, they are also tightly connected to the particular implementation of the rank-revealing post-processor, and not the subject of this paper. All our experiments were done in Matlab, and we use the implementations of the ULV, URV, and RRQR algorithms from the UTV Tools package [17]. The condition estimation in all three implementations is the Cline-Conn-Van Loan (CCVL) estimator [12]. The modied URV algorithm used for symmetric indenite matrices is based on the URV algorithm from [17], augmented with stabilized hypernormal rotations when needed, and with a condition estimator consisting of the CCVL algorithm followed by one step of inverse iteration applied to the matrix C C. Numerical results for all the rank-revealing algorithms are shown in Table 5.1, where we present mean and maximum values of the norms of various submatrices associated with the VSV decompositions. In particular, X or T 12 , and X 22 denotes either R 22 , L 22 , or T 22 . The results are computed on the basis of randomly generated test matrices of size 64, 128, and 256 (100 matrices of each size), each with n 4 eigenvalues geometrically distributed between 1 and 10 4 , 14 P. C. HANSEN AND P. Y. YALAMOV Table Numerical results with improved singular vector estimates. URV mean 1:8 (semi-def.) URV-like mean 8:6 (indef.) and the remaining four eigenvalues given by the numerical rank with respect to the threshold The test matrices were produced by generating random orthogonal matrices and multiplying them to diagonal matrices with the desired eigenvalues. For the indenite matrices the signs of the eigenvalues were chosen to alternate. Table 5.1 illustrates the superiority of the ULV-based algorithm for semi-denite matrices, for which the norm kS 12 k 2 of the o-diagonal block in S is always much smaller than the norm kS 22 k 2 of the bottom right submatrix. This is due to the fact that the ULV algorithm produces a lower triangular matrix L whose o-diagonal block L 21 has a very small norm (and we emphasize that the size of this norm depends on the condition estimator). The second best algorithm for semi-denite matrices is the one based on the RRQR algorithm, for which kS 12 k 2 and kS 22 k 2 are of the same size. Note that it is the latter algorithm which we recommend for sparse matrices. The URV-based algorithm for semi-denite matrices produces results that are consistently less satisfactory than the other two algorithms. All these results are consistent with our theory. For the indenite matrices, only the URV-like algorithm can be used, and the results in Table 5.1 show that this algorithm also behaves as expected from the theory. In order to judge the backward stability of this algorithm, which uses hypernormal rotations, we also computed the backward error kA k 2 for all three hundred test problems. The largest residual norm was 1:9 10 11 , and the average is We conclude that we loose a few digits of accuracy due to the use of the hypernormal rotations. It is well known that the norm of the o-diagonal block in the triangular URV factor depends on the quality of the condition estimator | the better the singular vector estimate, the smaller the norm. Hence, it is interesting to see how much the norms of the o-diagonal blocks in R and S decrease if we improve the singular vector estimates by means of one step of inverse iteration (at the expense of additional ops). In the semi-denite case we now apply an inverse iteration step to the CCVL estimate, and in the indenite case we use two steps of inverse iteration applied to C C instead of one. The results are shown in Table 5.2 for the same matrices as in Table 5.1. As expected, the norms of the o-diagonal blocks are now smaller, at the expense of more work. The average backward errors kA VS S V T did not change in this experiment. 6. Conclusion. We have dened and analyzed a class of rank-revealing VSV decompositions for symmetric matrices, and proposed algorithms for computing these decomposition. For semi-denite matrices, the ULV-based algorithm is the method of choice for dense matrices, while the RRQR-based algorithm is better suited for sparse matrices because it preserves sparsity better. For indenite matrices, only the URV-based algorithm is guaranteed to work. --R Accurate Symmetric Inde A correlation-based subspace tracking algorithm An algorithm and a stability theory for downdating the ULV decomposition On rank-revealing QR factorizations Generalizing the LINPACK condition estima- tor Perturbation analysis for two-sided (or complete) orthogonal decompositions Bounding the subspaces from rank revealing two-sided orthogonal decompositions Matlab templates for rank- revealing UTV decompositions Rank and null space calculations using matrix decomposition without column interchanges Transformation techniques for Toeplitz and Toeplitz-plus-Hankel matrices II Analysis of the Cholesky decomposition of a semi-de nite matrix The rank revealing QR decomposition and SVD A symmetric rank-revealing Toeplitz matrix decomposition A Sparse URL Rather Than a URV Factorization Sparse multifrontal rank revealing QR factorization Cholesky factorization of semi-de nite Toeplitz matrices An updating algorithm for subspace tracking Updating a rank-revealing ULV decomposition Matrix Algorithms Vol. --TR

rank-revealing decompositions;hypernormal rotations;matrix approximation;symmetric matrices

587766

Twice Differentiable Spectral Functions.

A function F on the space of n n real symmetric matrices is called spectral if it depends only on the eigenvalues of its argument. Spectral functions are just symmetric functions of the eigenvalues. We show that a spectral function is twice (continuously) differentiable at a matrix if and only if the corresponding symmetric function is twice (continuously) differentiable at the vector of eigenvalues. We give a concise and usable formula for the Hessian.

Introduction In this paper we are interested in functions F of a symmetric matrix argument that are invariant under orthogonal similarity transformations: orthogonal U and symmetric A : Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Email: aslewis@math.uwaterloo.ca. Research supported by NSERC. y Department of Combinatorics & Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada. Email: hssendov@barrow.uwaterloo.ca. Research supported by NSERC. Every such function can be decomposed as the map that gives the eigenvalues of the matrix A and f is a symmetric function. (See the next section for more details). We call such functions F spectral functions (or just functions of eigenvalues) because they depend only on the spectrum of the operator A. Classical interest in such functions arose from their important role in quantum mechanics [7], [20]. Nowadays they are an inseparable part of optimization [11], and matrix analysis [4, 5]. In modern optimization the key example is \semidenite programming", where one encounters problems involving spectral functions like log det(A), the largest eigenvalue of A, or the constraint that A must be positive denite. There are many examples where a property of the spectral function F is actually equivalent to the corresponding property of the underlying symmetric function f . Among them are rst-order dierentiability [9], convexity [8], generalized rst-order dierentiability [9, 10], analyticity [26], and various second-order properties [25], [24], [23]. It is also worth mentioning the \Chevalley Restriction Theorem", which in this context identies spectral functions that are polynomials with symmetric polynomials of the eigen- values. Second-order properties of matrix functions are of great interest for optimization because the application of Newton's method, interior point methods [13], or second-order nonsmooth optimality conditions [19] requires that we know the second-order behaviour of the functions involved in the mathematical model. The standard reference for the behaviour of the eigenvalues of a matrix subject to perturbations in a particular direction is [6]. Second-order properties of eigenvalue functions in a particular direction are derived in [25]. The problem that interests us in this paper is that of when a spectral function is twice dierentiable (as a function of the matrix itself, rather than in a particular direction) and when its Hessian is continuous. Analyticity is discussed in [26]: thus our result lies in some sense between the results in [9] and [26]. Smoothness properties of some special spectral functions (such as the largest eigenvalue) on certain manifolds are helpful in perturbation theory and Newton-type methods: see for example [15, 16, 18, 17, 22, 21, 14]. We show that a spectral function is twice (continuously) dierentiable at a matrix if and only if the corresponding symmetric function is twice (con- tinuously) dierentiable at the vector of eigenvalues. Thus in particular, a spectral function is C 2 if and only if its restriction to the subspace of diagonal matrices is C 2 . For example, the Schatten p-norm of a symmetric matrix is the pth root of the function (where the i s are the eigenvalues of the matrix). We see that this latter function is C 2 for p 2, although not analytic unless p is an even integer. As part of our general result, we also give a concise and easy-to-use formula for the Hessian: the results in [26], for analytic functions, are rather implicit. The paper is self-contained and the results are derived essentially from scratch, making no use of complex-variable techniques as in [2], for ex- ample. In a parallel paper [12] we give an analogous characterization of those spectral functions that have a quadratic expansion at a point (but that may not be twice dierentiable). Notation and preliminary results In what follows S n will denote the Euclidean space of all n n symmetric matrices with inner product hA; will be the vector of its eigenvalues ordered in nonincreasing order. By O n we will denote the set of all nn orthogonal matrices. For any vector x in R n , Diag x will denote the diagonal matrix with the vector x on the main diagonal, and x will denote the vector with the same entries as x ordered in nonincreasing order, that is x 1 x n . Let R n # denote the set of all vectors x in R n such that x 1 x 2 x n . Let also the dened by diag m=1 will denote a sequence of symmetric matrices converging to 0, and m=1 will denote a sequence of orthogonal matrices. We describe sets in R n and functions on R n as symmetric if they are invariant under coordinate permutations. denotes a function, dened on an open symmetric set, with the property permutation matrix P and any x 2 domainf: We denote the gradient of f by rf or f 0 , and the Hessian by r 2 f or f 00 . Vectors are understood to be column vectors, unless stated otherwise. Whenever we denote by a vector in R n # we make the convention that Thus r is the number of distinct eigenvalues. We dene a corresponding partition and we call these sets blocks. We denote the standard basis in R n by e is the vector with all entries equal to 1. We also dene corresponding matrices For an arbitrary matrix A, A i will denote its i-th row (a row vector), and A i;j will denote its (i; j)-th entry. Finally, we say that a vector a is block rened by a vector b if implies a We need the following result. R be a symmetric function, twice dierentiable at the point 2 R n # , and let P be a permutation matrix such that Then In particular we have the representation a a a r1 E r1 a r2 E r2 a rr R rr where the E uv are matrices of dimensions jI u j jI v j with all entries equal to i;j=1 is a real symmetric matrix, b := (b 1 ; :::; b n ) is a vector which is block rened by , and J u is an identity matrix of the same dimensions as Proof. Just apply the chain rule twice to the equality order to get parts (i) and (ii). To deduce the block structure of the Hessian, consider the block structure of permutation matrices P such that when we permute the rows and the columns of the Hessian in the way dened by P , it must stay unchanged. Using the notation of this lemma, we dene the matrix Note 2.2 We make the convention that if the i-th diagonal block in the above representation has dimensions 1 1 then we set a (). Otherwise the value of b k i is uniquely determined as the dierence between a diagonal and an o-diagonal element of this block. Note also that the matrix B and the vector b depend on the point and the function f . Lemma 2.3 For 2 R n # and a sequence of symmetric matrices Mm ! 0 we have that (Diag +Mm Proof. Combine Lemma 5.10 in [10] and Theorem 3.12 in [3]. The following is our main technical tool. Lemma 2.4 Let fMmg be a sequence of symmetric matrices converging to 0, such that Mm =kMm k converges to M . Let be in R n # and Um ! U 2 O n be a sequence of orthogonal matrices such that Diag (Diag +Mm ) (2) Then the following properties hold. (i) The orthogonal matrix U has the form l is an orthogonal matrix with dimensions jI l j jI l j for all l. (ii) If i 2 I l then lim p2I l (U i;p 0: (iii) If i and j do not belong to the same block then lim (U i;j (iv) If i 2 I l then l Diag (X T l MX l ) l (v) If l , and p 62 I l then lim U i;p 0: (vi) For any indices i 6= j such that lim p2I l U i;p 0: (vii) For any indices i 6= j such that l Diag (X T l MX l ) l (viii) For any three indices i, j, p in distinct blocks, lim U i;p 0: (ix) For any two indices i, j such that i 2 I l , lim p2I l U i;p ks p2Is U i;p Proof. (i) After taking the limit in equation (2) we are left with (Diag The described representation of the matrix U follows. (ii) Let us denote We use Lemma 2.3 in equation (2) to obtain Diag (Diag hm )U T and the equivalent form (Diag )Um We now divide both sides of these equations by kMm k and rearrange: Diag Um (Diag )U T (Diag hm )U T and Diag U T (Diag )Um Diag hm Notice that the right hand sides of these equations converge to a nite limit as m increases to innity. If we call the matrix limit of the right hand side of the rst equation L, then clearly the limit of the second equation is U T LU . We are now going to prove parts (ii) and (iii) together inductively, by dividing the orthogonal matrix Um into the same block structure as U . We begin by considering the rst row of blocks of Um . Let i be an index in the rst block, I 1 . Then the limit of the (i; i)-th entry in the matrix at the left hand side of equation (4) is lim (U i;p ks p2Is (U i;p Now recall that and because V 1 is an orthogonal matrix, notice that i(Diag (X T We now sum equation (6) over all i in I 1 to get lim (U i;p ks (U i;p 0: Notice here, that the coe-cients in front of the k l in the numerator sum up to zero. That is, U i;p r U i;p us choose a number such that and add to every coordinate of the vector thus \shifting" it. The coordinates of the shifted vector that are in the rst block are strictly bigger than zero, and the rest are strictly less than zero. By our comment above, the last limit remains true if we \shift" in this way. If we rewrite the last limit for the \shifted" vector, because all summands are positive, we immediately see that we must have lim (U i;p and lim (U i;p The rst of these limits can be written as lim (U i;p and because all the summands are positive, we conclude that lim (U i;p The second of the limits implies immediately that lim (U i;p Thus we proved part (ii) for i 2 I 1 and part (iii) for the cases specied above. Here is a good place to say a few more words about the idea of the proof. As we said, we divide the matrix Um into blocks complying with the block structure of the vector (exactly as in part (i) for the matrix U ). We proved part (ii) and (iii) for the elements in the rst row of blocks of this division. What we are going to do now is prove the same thing for the rst column of blocks. In order to do this we x an index i in I 1 and consider the (i; i)-th entry in the matrix at the left hand side of equation (5), and take the limit: lim (U p;i ks p2Is (U p;i Using also the block-diagonal structure of the matrix U , we again have So we proceed just as before in order to conclude that lim (U p;i and lim (U p;i We are now ready for the second step of our induction. Let i be an index in I 2 . Then the limit of the (i; i)-th entry in the matrix at the left hand side of equation (4) is lim U i;p U i;p r ks p2Is U i;p Analogously as above we have so summing the above limit over all i in I 2 we get lim U i;p U i;p r ks U i;p 0: We know from (8) that lim (U i;p 0: So now we choose a number such that and as before exchange with its shifted version. Just as before we conclude that lim (U i;p and lim (U i;p We repeat the same steps for the second column of blocks in the matrix Um and so on inductively until we exhaust all the blocks. This completes the proof of parts (ii) and (iii). (iv) For the proof of this part, one needs to consider the (i; i)-th entry of the right hand side of equation (4). Because the diagonal of the left hand side converges to zero (by (ii) and (iii)), taking the limit proves the statement in this part. (v) This part follows immediately from part (iii). (vi) Taking the limit in equation (4) gives lim s6=l ks p2Is U i;p p2I l U i;p where L i;j is the (i; j)-th entry of the limit of the right hand side of equation (4). Note that the coe-cients of ks again sum up to zero: s6=l p2Is U i;p p2I l U i;p because Um is an orthogonal matrix. Now by part (v) we have s6=l p2Is U i;p p2I l U i;p as required, and moreover L (vii) The statement of this part is the detailed way of writing the fact, proved in the previous part, that L (viii) This part follows immediately from part (iii). (In fact the expression in part (viii) is identical to the one in part (v), re-iterated with dierent index conditions for later convenience.) (ix) We again take the limit of the (i; j)-th entry of the matrices on both sides of equation (4). lim t6=l;s U i;p p2I l U i;p ks p2Is U i;p By part (viii) we have that all but the l-th and the s-th summand above converge to zero. On the other hand Mm (Diag hm )U T i;j lim Diag hm because U i and U j are rows in dierent blocks and (Diag hm )=kMm k converges to a diagonal matrix. Now we have all the tools to prove the main result of the paper. 3 Twice dierentiable spectral functions In this section we prove that a symmetric function f is twice dierentiable at the point (A) if and only if the corresponding spectral function f - is twice dierentiable at the matrix A. Recall that the Hadamard product of two matrices of the same size is the matrix of their elementwise product A - Let the symmetric function f : R n ! R be twice dierentiable at the point 2 R n # , where, as before, We dene the vector as in Lemma 2.1. Specically, for any index i, (say i 2 I l for some l 2 f1; 2; :::; rg) we dene ii (); if jI l pq (); for any p 6= q 2 I l : Lemma 2.1 guarantees that the second case of this denition doesn't depend on the choice of p and q. We also dene the matrix A(): A i;j Notice the similarity between this denition and classical divided dierence constructions in Lowner theory (see [1, Chap. V], for example). For simplic- ity, when the argument is understood by the context, we will write just b i and A i;j . The following lemma is Theorem 1.1 in [9]. Lemma 3.1 Let A 2 S n and suppose (A) belongs to the domain of the is dierentiable at the point (A) if and only if f - is dierentiable at the point A. In that case we have the for any orthogonal matrix U satisfying We recall some standard notions about twice dierentiability. Consider a function F from S n to R. Its gradient at any point A (when it exists) is a linear functional on the Euclidean space S n , and thus can be identied with an element of S n , which we denote rF (A). Thus rF is a map from S n to S n . When this map is itself dierentiable at A we say F is twice dierentiable at A. In this case we can interpret the Hessian r 2 F (A) as a symmetric, bilinear function from S n S n into R. Its value at a particular point will be denoted r 2 F (A)[H; Y ]. In particular, for xed H, the function r 2 F (A)[H; ] is again a linear functional on S n , which we consider an element of S n , for brevity denoted by r 2 F (A)[H]. When the Hessian is continuous at A we say F is twice continuously dierentiable at A. In that case the following identity holds: t=0 The next theorem is a preliminary version of our main result. Theorem 3.2 The symmetric function f : R n ! R is twice dierentiable at the point 2 R n # if and only if f - is twice dierentiable at the point Diag . In that case the Hessian is given by Hence Proof. It is easy to see that f must be twice dierentiable at the point whenever f - is twice dierentiable at Diag because by restricting f - to the subspace of diagonal matrices we get the function f . So the interesting case is the other direction. Let f be twice dierentiable at the point 2 R n and suppose on the contrary that either f - is not twice dierentiable at the point Diag , or equation (10) fails. Dene a linear operator by (Lemma 3.1 tells us that f - is at least dierentiable around Diag .) So, for this linear operator there is an > 0 and a sequence of symmetric matrices fMm g 1 m=1 converging to 0 such that loss of generality we may assume that the sequence fMmg 1 m=1 is such that Mm =kMm k converges to a matrix M , because some subsequence of fMm g 1 m=1 surely has this property. Let fUm g 1 m=1 be a sequence of orthogonal matrices such that Diag (Diag +Mm ) Without loss of generality we may assume that Um ! U 2 O n , or otherwise we will just take subsequences of fMmg 1 m=1 and fUmg 1 m=1 . The above inequality shows that for every m there corresponds a pair (or more precisely at least one pair) of indices (i; j) such that i;j So at least for one pair of indices, call it again (i; j), we have innitely many numbers m for which (i; j) is the corresponding pair, and because if necessary we can again take a subsequence of fMmg 1 m=1 and fUmg 1 m=1 we may assume without loss of generality that there is a pair of indices (i; j) for which the last inequality holds for all :::. Dene the symbol hm again by equation (3). Notice that using Lemma 3.1, Lemma 2.3, and the fact that rf is dierentiable at , we get We consider three cases. In every case we are going to show that the left hand side of inequality (11) actually converges to zero, which contradicts the assumption. Case I. If using equation (12) the left hand side of inequality (11) is less that or equal to Diag rf() Diag r 2 f()hm We are going to show that each summand approaches zero as m goes to innity. Assume that i 2 I l for some l 2 f1; :::; rg. Using the fact that the vector block renes the vector rf() (Lemma 2.1, part (i)) the rst term can be written askMmk f 0 l p2I l U i;p s:s6=l ks p2Is U i;p We apply now Lemma 2.4 parts (ii) and (iii) to the last expression. We now concentrate on the second term above. Using the notation of equation (1) (that is, r B+Diag b) this term is less than or equal to Diag ((Diag b)h m ) (Diag b)(diag Mm ) As m approaches innity, we have that U i . We dene the vector h to be: hm taking limits, expression (13) turn into: (Diag b)(diag M) We are going to investigate each term in this sum separately and show that they are both actually equal to zero. For the rst, we use the block structure of the matrix B (see Lemma 2.1) and the block structure of the vector h to obtain r a qs tr (X T Using the fact that i 2 I l and that V l is orthogonal we get l Diag (Bh) l l (Diag (Bh))X l l r a ls tr (X T l r a ls tr (X T (Bdiag M) which shows that the rst term is zero. For the second term, we use the block structure of the vector b, to write (Diag In the next to the last equality below we use part (iv) of Lemma 2.4: l Diag ((Diag b)h) l l (Diag ((Diag b)h))X l l l Diag b k l l MX l ) l (Diag b)(diag M) We can see now that the second term is also zero. Case II. If i 6= j but I l for some l 2 f1; 2; :::rg, then using equation (12) the left hand side of inequality (11) becomes Diag rf() Using the fact that block renes vector rf(), we can write the rst summand above askMm k s6=l ks p2Is U i;p l p2I l U i;p We use parts (v) and (vi) of Lemma 2.4 to conclude that this expression converges to zero. We are left with Substituting r 2 Diag b we get Diag ((Diag b)h m ) Recall the notation from Lemma 2.1 used to denote the entries of the matrix B. Then the limit of the rst summand above can be written as lim r a sl tr (X T l MX l ) p2Is U i;p U j;p because clearly p2Is U i;p U :::rg. We are left with the following limit lim Diag ((Diag b)h m ) Using Lemma 2.4 part (vii) we observe that the right hand side is zero. Case III. If i 2 I l and j 2 I s , where l 6= s, then using equation (12), the left hand side of inequality (11) becomes (up to o(1)) Diag rf() Diag r 2 f()hm l ks ks We start with the second term above. Its limit is lim because in our case, U i has nonzero coordinates where the entries of U j are zero. We are left with lim Diag rf() l ks ks We expand the rst term in this limit. Diag rf() l p2I l U i;p ks p2Is U i;p t6=l;s U i;p Using Lemma 2.4 part (viii) we see that the third summand above converges to zero as m goes to innity. Part (ix) of the same lemma tells us that lim p2I l U i;p ks p2Is U i;p In order to abbreviate the formulae we introduce the following notation l p2I l U i;p Substituting everything in (14) we get the following equivalent limit: lim l ks l ks ks l ks s Simplifying we get lim ks l ks ks Notice now that r l because Um is an orthogonal matrix and the numerator of the above sum is the product of its i-th and the j-th row. Next, Lemma 2.4, part (viii) says that lim t6=l;s so lim which completes the proof. We are nally ready to give and prove the full version of our main result. Theorem 3.3 Let A be an nn symmetric matrix. The symmetric function twice dierentiable at the point (A) if and only if the spectral function f - is twice dierentiable at the matrix A. Moreover in this case the Hessian of the spectral function at the matrix A is where W is any orthogonal matrix such that A = W Diag (A) dened by equation (9). Hence diag ~ Hi: Proof. Let W be an orthogonal matrix which diagonalizes A in an ordered fashion, that is Let Mm be a sequence of symmetric matrices converging to zero, and let Um be a sequence of orthogonal matrices such that Diag (A) +W T Then using Lemma 3.1 we get We also have that goes to innity. Because W is an orthogonal matrix we have kWXW T matrix X. It is now easy to check the result by Theorem 3.2. 4 Continuity of the Hessian Suppose now that the symmetric function f : R n ! R is twice dierentiable in a neighbourhood of the point (A) and that its Hessian is continuous at the point (A). Then Theorem 3.3 shows that f - must be twice dierentiable in a neighbourhood of the point A, and in this section we are going to show that r 2 (f - ) is also continuous at the point A. We dene a basis, fH ij g, on the space of symmetric matrices. If i all the entries of the matrix H ij are zeros, except the (i; j)-th and (j; i)-th, which are one. If we have one only on the (i; i)-th position. It su-ces to prove that the Hessian is continuous when applied to any matrix of the basis. We begin with a lemma. Lemma 4.1 Let 2 R n # be such that and let the symmetric function f : R n ! R be twice continuously dieren- tiable at the point . Let f m g 1 m=1 be a sequence of vectors in R n converging to . Then lim Proof. For every m there is a permutation matrix Pm such that P T m . (See the beginning of Section 2 for the meaning of the bar above a vector.) But there are nitely many permutation matrices (namely n!) so we can form n! subsequences of f m g such that any two vectors in a particular subsequence can be ordered in descending order by the same permutation matrix. If we prove the lemma for every such subsequence we will be done. So without loss of generality we may assume that P T for every m, and some xed permutation matrix P . Clearly, for all large enough m, we have Consequently the matrix P is block-diagonal with permutation matrices on the main diagonal, and dimensions matching the block structure of , so Consider now the block structure of the vectors f m g. Because there are nitely many dierent block structures, we can divide this sequence into subsequences such that the vectors in a particular subsequence have the same block structure. If we prove the lemma for each subsequence we will be done. So without loss of generality we may assume that the vectors f m g have the same block structure for every m. Next, using the formula for the Hessian in Theorem 3.3 we have and Lemma 2.1 together with Theorem 3.2 give us These equations show that without loss of generality it su-ces to prove the lemma only in the case when all vectors f m g are ordered in descending order, that is, the vectors m all block rene the vector . In that case we have and We consider four cases. Case I. If lim Diag r just because r 2 f() is continuous at . Case II. If i 6= j, but belong to the same block for m , then i, j will be in the same block of as well and we have lim again because r 2 f() is continuous at . Case III. If i and j belong to dierent blocks of m but to the same block of , then lim and So we have to prove that lim ii (See the denition of b i () in the beginning of Section 3.) For every m we dene the vectors _ m and _ Because we conclude that both sequences f _ converge to , because f m g 1 does so. Below we are applying the mean value theorem twice: is a vector between m and _ is a vector between _ m and . Notice that vector m is obtained from m by swapping the i-th and the j-th coordinate. Then using the rst part of Lemma 2.1 we see that f 0 Finally we just have to take the limit above and use again the continuity of the Hessian of f at the point . Case IV. If i and j belong to dierent blocks of m and to dierent blocks of , then lim because rf() is continuous at and the denominator is never zero. Now we are ready to prove the main result of this section. Theorem 4.2 Let A be an nn symmetric matrix. The symmetric function continuously dierentiable at the point (A) if and only if the spectral function f - is twice continuously dierentiable at the matrix A. Proof. We know that f - is twice dierentiable at A if and only if f is twice dierentiable at (A), so what is left to prove is the continuity of the Hessian. Suppose that f is twice continuously dierentiable at (A) and that f - is not twice continuously dierentiable at A, that is, the Hessian not continuous at A. Take a sequence, fAmg 1 m=1 , of symmetric matrices converging to A such that for some > 0 we have for all m. Let fUm g 1 m=1 be a sequence of orthogonal matrices such that Without loss of generality we may assume that Um ! U , where U is orthogonal and then (Otherwise we take subsequences of fAmg and fUmg.) Using the formula for the Hessian given in Theorem 3.3 and Lemma 4.1 we can easily see that lim for every symmetric H. This is a contradiction. The other direction follows from the chain rule after observing This completes the proof. 5 Example and Conjecture As an example, suppose we require the second directional derivative of the function f - at the point A in the direction B. That is, we want to nd the second derivative of the function at W be an orthogonal matrix such that A = W(Diag (A))W T . Let ~ We dierentiate twice: Using Lemma 3.1 and Theorem 3.3 at Diag rf((A)) diag ~ In principle, if the function f is analytic, this second directional derivative can also be computed using the implicit formulae from [26]. Some work shows that the answers agree. As a nal illustration, consider the classical example of the power series expansion of a simple eigenvalue. In this case we consider the function f given by the k-th largest entry in x; and the matrix # and Then we have so for the function our results show the following formulae (familiar in perturbation theory and quantum mechanics): This agrees with the result in [6, p. 92]. We conclude with the following natural conjecture. Conjecture 5.1 A spectral function f - is k-times dierentiable at the matrix A if and only if its corresponding symmetric function f is k-times dierentiable at the point (A). Moreover, f - is C k if and only if f is C k . --R Matrix Analysis. Derivations, derivatives and chain rules. Sensitivity analysis of all eigen-values of a symmetric matrix Matrix Analysis. Topics in Matrix Analysis. A Short Introduction to Perturbation Theory for Linear Op- erators The Fundamental Principles of Quantum Mechanics. Convex analysis on the Hermitian matrices. Derivatives of spectral functions. Nonsmooth analysis of eigenvalues. Eigenvalue optimization. Quadratic expansions of spectral func- tions On minimizing the maximum eigenvalue of a symmetric matrix. Second derivatives for optimizing eigenvalues of symmetric matrices. Towards second-order methods for structured nonsmooth optimization WETS. Variational Analysis. Quantum Mechanics. First and second order analyis of nonlinear semid On eigenvalue optimization. Valeurs propres de matrices sym On analyticity of functions involving eigenvalues. --TR --CTR Xin Chen , Houduo Qi , Liqun Qi , Kok-Lay Teo, Smooth Convex Approximation to the Maximum Eigenvalue Function, Journal of Global Optimization, v.30 n.2-3, p.253-270, November 2004 Lin Xiao , Stephen Boyd , Seung-Jean Kim, Distributed average consensus with least-mean-square deviation, Journal of Parallel and Distributed Computing, v.67 n.1, p.33-46, January, 2007

symmetric function;semidefinite program;spectral function;twice differentiable;perturbation theory;eigenvalue optimization

587777

Means and Averaging in the Group of Rotations.

In this paper we give precise definitions of different, properly invariant notions of mean or average rotation. Each mean is associated with a metric in SO(3). The metric induced from the Frobenius inner product gives rise to a mean rotation that is given by the closest special orthogonal matrix to the usual arithmetic mean of the given rotation matrices. The mean rotation associated with the intrinsic metric on SO(3) is the Riemannian center of mass of the given rotation matrices. We show that the Riemannian mean rotation shares many common features with the geometric mean of positive numbers and the geometric mean of positive Hermitian operators. We give some examples with closed-form solutions of both notions of mean.

where is the angle of rotation of R. The kth root exp(1 Log R) is the one for which the eigenvalues have the largest positive real part, and is the only one we denote by R1=k. In the case it is the only square root with positive real part. 2.2. Metrics in SO(3). A straightforward way to dene a distance function in SO(3) is to use the Euclidean distance of the ambient space M(3), i.e., if R1 and are two rotation matrices then where k kF is the Frobenius norm which is induced by the Euclidean inner product, known as the Frobenius inner product, dened by hR1; It is easy to see that this distance is bi-invariant in SO(3), i.e., dF (P R1Q; P dF (R1; R2) for all P ; Q in SO(3). Another way to dene a distance function in SO(3) is to use its Riemannian structure. The Riemannian distance between two rotations is the length of the shortest geodesic curve that connects R1 and given by 4 M. MOAKHER Note that the geodesic curve of minimal length may not be unique. If RT1 R2 is an involution, in other words if (RT1 rotation through an angle , then and can be connected by two curves of equal length. In such a case, the rotations and are said to be antipodal points in SO(3) and is said to be the cut point of R1 and vice versa. The Riemannian distance (2.6) is also bi-invariant in SO(3). Indeed, using the fact [3], we can show that dR(P R1Q; P Remark 2.1. The Euclidean distance (2.5) represents the chordal distance between and R2, i.e., the length of the Euclidean line segment in the space of M(3) (except for the end points R1 and R2, this line segment does not lie in SO(3)), whereas the Riemannian distance (2.6) represents the arc-length of the shortest geodesic curve (great-circle arc), which lies entirely in SO(3), passing through R1 and R2. Remark 2.2. If denotes the angle of rotation of RT1 Therefore, when theprotations R1 and are suciently close, i.e., is small, we have dF (R1; R2) 2 dR(R1; R2). 2.3. Covariant derivative and Hessian. We recall that the tangent space at a point R of SO(3) is the space of all matrices such that RT is skew symmetric and that the normal space (associated with the Frobenius inner product) at R consists of all matrices N such that RT N is symmetric [5]. For a real-valued function f(R) dened on SO(3), the covariant derivative rf is the unique tangent vector at R such that d dt where Q(t) is a geodesic emanating from R in the direction of , i.e., R exp(tA) and The Hessian of f(R) is given by the quadratic form d2 dt2 where Q(t) is a geodesic and is in the tangent space at R as above. 2.4. Geodesic convexity. We recall that a subset A of a Riemannian manifold M is said to be convex if the shortest geodesic curve between any two points x and y in A is unique in M and lies in A. A real-valued function dened on a convex subset A of M is said to be convex if its restriction to any geodesic path is convex, i.e., if its domain for all x 2 M and u 2 Tx(M),where exp is the exponential map at x. x With these denitions, one can readily see that any geodesic ball Br(Q) in SO(3) of radius r less than around Q is convex and that the real-valued function f denedon Br(Q) by when r is less than 2 . Geodesic balls with radius greater or equal than are not convex.3. Mean rotation. For a given set of N points xn; in IRd the mean x is given by the barycenter of the N points. The mean also has a variational property; it minimizes the sum of the squared distances to the given points xn; x2IRd n=1 where here de(; ) represents the usual Euclidean distance in IRd. One can also use the arithmetic mean to average N positive real numbers xn > and the mean is itself a positive number. In many applications, however, it is more appropriate to use the geometric mean to average positive numbers, which is possible because positive numbers form a multiplicative group. The geometric 1=N 1=N mean also has a variational property; it minimizes the sum of the squared hyperbolic distances to the given data where dh(x; log yj is the hyperbolic distance1 between x and y. As we have seen, for the set of positive real numbers dierent notions of mean can be associated with dierent metrics. In what follows, we will extend these notions of mean to the group of proper orthogonal matrices. By analogy with IRd, a plausible denition of the mean of N rotation matrices is that it is the minimizer in SO(3) of the sum of the squared distances from that rotation matrix to the given rotation matrices represents a distance in SO(3). Now the two distance functions (2.5) and (2.6) dene the two dierent means. Definition 3.1. The mean rotation in the Euclidean sense, i.e., associated with the metric (2.5), of N given rotation matrices is dened as Definition 3.2. The mean rotation in the Riemannian sense, i.e., associated with the metric (2.6), of N given rotation matrices is dened as The minimum here is understood to be the global minimum. We remark that in IRd, or in the set of positive numbers, the objective functions to be minimized are convex over their domains, and therefore the means are well dened and unique. However, in SO(3), as we shall see, the objective functions in (3.3) and (3.4) are not (geodesically) convex, and therefore the means may not be unique. Before we proceed to study these two means, we note that both satisfy the following desirable properties that one would expect from a mean in SO(3), and that are counterparts of properties of means of numbers, namely, 1We borrow this terminology from the hyperbolic geometry of the Poincare upper half-plane. In fact, the hyperbolic length of the geodesic segment joining the points P (a; y1) and Q(a; y2), y1; y2 > 0 is j log y1 j, (see [26]). 6 M. MOAKHER 1. Invariance under permutation: For any permutation of the numbers 1 through N, we have 2. Bi-invariance: If R is the mean rotation of fRng; is the mean rotation of fP RnQg; every P and Q in SO(3). This property follows immediately from the bi-invariance of the two metrics dened above. 3. Invariance under transposition: If R is the mean rotation of fRng; then RT is the mean rotation of fRT We remark that the bi-invariance property is in some sense the counterpart of the homogeneity property of means of positive numbers (but here left and right multiplication are both needed because the rotation group is not commutative). 3.1. Characterization of the Euclidean mean. The following proposition gives a relation between the Euclidean mean and the usual arithmetic mean. Proposition 3.3. The mean rotation is the orthogonal projection of R = onto the special orthogonal group SO(3). In other words, the mean rotation in the Euclidean sense is the projection of the arithmetic mean R of in the linear space M(3) onto SO(3). Proof. As are all orthogonal, it follows that On the other hand, the orthogonal projection of R onto SO(3) is given by "XN XN Rn RTm XN RTn # Because of Proposition 3.3, the mean in the Euclidean sense will be termed the projected arithmetic mean to reect the fact that it is the orthogonal projection of the usual arithmetic mean in M(3) onto SO(3). Remark 3.4. The projected arithmetic mean can now be seen to be related to the classical orthogonal Procrustes problem [10], which seeks the orthogonal matrix that most closely transforms a given matrix into a second one. Proposition 3.5. If det R is positive, then the mean rotation in the Euclidean sense given by the unique polar factor in the polar decomposition [10] of R. Proof. Critical points of the objective function dened on SO(3) and corresponding to the minimization problem (3.3) are those elements of SO(3) for which the covariant derivative of (3.5) vanishes. Using (2.8) we get Therefore, critical points of (3.5) are the rotation matrices R such that PNn=1 R RTn R RT equivalently, for which the matrix S dened by is symmetric. Since R is orthogonal, and both S and are symmetric, it follows that N2M. Therefore, there exists an orthogonal matrix U such that of M. The eight possible square roots of M are UT diag( To determine the square root of N2M that corresponds to the minimum of (3.5) we require that the Hessian of the objective function (3.5) at R given by (3.6) be positive for all tangent vectors at R. From (2.9) we obtain Hess therefore at R given by (3.6) we have Hess where a; b; c are such that = UT RBU and b a 0 As we are looking for a proper rotation matrix, i.e., an orthogonal matrix with determinant one, it follows from (3.6) that det that Hess F(; ) is positive for all tangent vectors at R if and only positive and In fact, (3.5) has four cpriticapl poinpts belonging to which consist of apminimpum [(1p; 2; Hence, the projected arithmetic mean is given by which, when det R > 0, coincides with the polar factor of the polar decomposition of R. Of course uniqueness fails when the smallest eigenvalue of M is not simple. Remark 3.6. The case where det is a degenerate case. However, if R has rank 2, i.e., when nd a unique closest proper orthogonal matrix to R (see [6] for details), and hence can dene the mean rotation in the Euclidean sense. 3.2. Characterization of the Riemannian mean. First, we compute the derivative of the real-valued function H(P to t where P is the geodesic emanating from R in the direction of RA. As is in the tangent space at R, we have be the angle of rotation of QT P (t), i.e., such that 8 M. MOAKHER Dierentiate (3.8) to get d H(P is the dt t=0 sin angle of rotation of QT R and we have used the fact that H(P Recall that, since A is skew symmetric, symmetric matrix S. It follows that tr(QT Then, with the help of (2.4) we obtain d H(P dt t=0 fore, the covariant derivative of H is given by The second derivative of (3.8) gives d2 sin cos Let U be an orthogonal matrix and B the skew-symmetric matrix such that Then, as tr(QT it is easy to see that d2 sin The RHS of (3.10) is always positive for arbitrary a, b, c in IR and 2 (; ). It follows that Hess H(; ) is positive for all tangent vectors . denote the objective function of the minimization problem (3.4), i.e., Using the above, the covariant derivative of G is found to be Therefore, a necessary condition for regular extrema of (3.11) is By (3.10) we conclude that the Hessian Hess G(; ) of the objective function (3.11) is positive for all tangent vectors . Therefore, equation (3.12) characterizes local minima of (3.11) only. As a matter of fact, local maxima are not regular points, i.e., they are points where (3.11) is not dierentiable. It is worth noting that, as R = R , the characterization for the Riemannian mean given in (3.12) is similar to the characterization of the geometric mean (3.2) of positive numbers. However, while in the scalar case the characterization (3.13) has the geometric mean as unique solution, the characterization (3.12) has multiple solutions, and hence is a necessary but not a sucient condition to determine the Riemannian mean. The lack of uniqueness of solutions of (3.12) is akin to the fact that, due to the existence of a cut point for each element of SO(3), the objective function (3.11) is not convex over its domain. In general, closed-form solutions to (3.12) cannot be found. However, for some special cases solutions can be given explicitly. In the following subsections, we will present some of these special cases. Remark 3.7. The Riemannian mean of may also be called the Riemannian barycenter of which is a notion introduced by Grove, Karcher and Ruh [11]. In [17] it was proven that for manifolds with negative sectional curva- ture, the Riemannian barycenter is unique. 3.2.1. Riemannian mean of two rotations. Intuitively, in the case the mean rotation in the Riemannian sense should lie midway between R1 and along the shortest geodesic curve connecting them, i.e., it should be the rotation R2)1=2. Indeed, straightforward computation shows that R1(RT1 R2)1=2 does satisfy condition (3.12). Alternatively, equation (3.12) can be solved analytically as follows. First, we rewrite it as then we take the exponential of both sides to obtain multiplying both sides with RT1 R we get (RT1 Such an equation has two solutions in SO(3) that correspond to local minima of (3.11). However, the global minimum is the one that corresponds to taking the square root of the above equation that has eigenvalues with positive real part, i.e., (RT1 R2)1=2. Therefore, for two non- antipodal rotation matrices R1 and R2, the mean in the Riemannian sense is given explicitly by The second equality can be easily veried by pre-multiplying R1(RT1 R2)1=2 by R2RT2 which is equal to I. This makes it clear that G is symmetric with respect to R1 and R2, i.e., G(R1; 3.2.2. Riemannian mean of rotations in a one-parameter subgroup. In the case where all matrices Rn; belong to a one-parameter subgroup of SO(3), i.e., they represent rotations about a common axis, we expect that their mean is also in the same subgroup. Further, one can easily show that equation (3.12) Y reduces to saying that R is an Nth root of Rn. Therefore, the Riemannian mean is the Nth root that yields the minimum value of the objective function (3.11). In this case, all rotations lie on a single geodesic curve. One can show that the geometric mean G(R1; R2; R3) of three rotations R1, and R3 such that 3, is the rotation that is located at 32 of the length of the shortest geodesic segment connecting R1 and G(R2; R3), i.e., the rotation have M. MOAKHER This explicit formula does not hold in the general case due to the inherent curvature of SO(3), see the discussion at the end of Example 2 below. When the rotations belong to a geodesic segment of length less than and centered at the identity, the above formula reduces to 1=N 1=N Once again we see the close similarity between the geometric mean of positive numbers and the Riemannian mean of rotations. This is to be expected since both the set of positive numbers and SO(3) are multiplicative groups, and we have used their intrinsic metrics to dene the mean. For this reason, we will call the mean in the Riemannian sense the geometric mean. 3.3. Equivalence of both notions of mean of two rotations. In the follow- ing, we show that for two rotations the projected arithmetic mean and the geometric mean coincide. First, we prove the following lemma. Lemma 3.8. Let R1 and R2 be two elements of SO(3), then det(R1 Proof. Consider the real-valued function dened on [0; 1] by We see that this function is continuous with Assume that f(1) < 0, i.e., det(R1 there exists in [0; 1] such that Hence, must be in the spectrum of RT2 R1 which is a proper orthogonal matrix. But this cannot happen, which contradicts the assumption that det(R1 In general, the result of the above lemma does not hold for more than two rotations matrices. We will see examples of three rotation matrices for which the determinant of their sum can be negative. Proposition 3.9. The polar factor of the polar decomposition of and are two rotation matrices, is given by R1(RT1 R2)1=2. Proof. Let Q be the proper orthogonal matrix and S be the positive-denite matrix such that QS is the unique polar decomposition of R1. One can easily verify that (RT1 R2)1=2 (RT1 R2)1=2 is the positive-denite square root of 2I and that the inverse of this square root is given by Hence, the polar factor is Since the polar decomposition is unique, the result of this proposition together with the previous lemma shows that both notions of mean agree for the case of two rotation matrices. For more than two rotations, however, both notions of mean coincide only in special cases that present certain symmetries. In Example 2 of x 4 below, we shall consider a two-parameter family of cases illustrating this coincidence. 4. Analytically solvable examples. In this section we present two cases in which we can solve for both the projected arithmetic mean and the geometric mean explicitly. These examples help us gain a deeper and concrete insight to both notions of mean. Furthermore, Example 2 conrms our intuitive idea that for \symmetric" cases, both notions of mean agree. 4.1. Example 1. We begin with a simple example where all rotation matrices for which we want to nd the mean lie in a one-parameter subgroup of SO(3). Using the bi-invariance property we can reduce the problem to that of nding the mean of MEAN ROTATION 11 Projected arithmetic mean: The arithmetic sum of these matrices has a positive determinant n)2. Hence, the projected arithmetic mean of the given matrices is given by the polar factor of the polar decomposition of their sum. After performing such a decomposition we nd thatN 2cos a sin a >< cos > sin a = sin n: Such a mean is well dened as long as This mean agrees with the notionof directional mean used in the statistics literature for circular and spherical data [20, 7, 9, 8]. The quantity 1 r=N, which is called the circular variance, is a measure of dispersion of the circular data . The direction dened by the angle a is called the mean direction of the directions dened by Geometric mean: Solutions of (3.12) are given by cos l sin l 0 N 4sin l cos l 05; where The geometric mean of these rotation matrices is therefore the solution that yields the minimum value of the objective function (3.11). Of course, as we have seen in x 3, the geometric mean is given explicitly by (3.15). Note that, even though elements of a one-parameter subgroup commute, the two rotations (3.15) and (3.16) are dierent. This is due to choice of the kth root of a rotation matrix to be the one with eigenvalues that have the largest positive real parts. To see this, consider the case where P is a rotation of an angle about the z-axis while R If the rotation matrices Rn are such that n < certain number 2 IR, then their geometric mean is a rotation about the z-axis of an angle The geometric mean rotation of the rotations given by (4.1) coincides with the concept of median direction of circular data [20, 7]. Remark 4.1. When neither the projected arithmetic mean nor the geometric mean is well dened. On the one hand so the projected arithmetic mean is not dened, while on the other hand the objective function (3.11) for the geometric mean has two local minima with the same value, namely, R1(RT1 R2)1=2 and its cut value andtherefore the global minimum is not unique. Let F~ and G~ be the functions dened on [; ] such that any rotation R about the z-axis through an angle , i.e., F~ and G~ are the restrictions of the objective functions (3.5) and (3.11) to the subgroup considered in this example. In Fig. 4.1 we give the plots of F~ and G~ for the sets of data takes several dierent values. It is clear that neither (3.5) nor (3.11) is convex. While the function (3.5) M. MOAKHER Projected Arithmetic Mean Geometric Mean10 ~ a=p/4 ~ a=p/4 a=p Fig. 4.1. Plots of the objective functions F~() and G~() for dierent values of . Note that is constant and G~ has four local minima with an equal value. Consequently, neither the projected arithmetic mean nor the geometric mean is well dened. is smooth the function (3.11) has cusp points but only at local maxima. However, if the given rotations are located in a geodesic ball of radius less than =2, i.e., in this example have angles i such that ji jj < then the objective functions restricted to this geodesic ball are convex and hence the means are well dened. Such case is illustrated in Fig. 4.2 which shows plots of F~ and G~ for the following sets of data takes several dierent values. Projected Arithmetic Mean Geometric Mean8 ~ a=p/4 ~ a=p/4 Fig. 4.2. Plots of the objective functions F~() and G~() for dierent values of . Restricted to [=4; 3=4], i.e., between the dashed lines, the objective functions are indeed convex. 4.2. Example 2. In the second example we consider N elements of SO(3) that represent rotations through an angle about the axes dened by the unit vectors sin sin n; cos ]T , where Projected arithmetic mean: Straightforward computations show that the projected arithmetic mean is given by cos a sin a 0 ><cos a cos a 05; By using half-angle tangent formulas in the above we obtain the following simple relation between a and a Geometric mean: Since the rotation axes are symmetric about the z-axis, and the rotations share the same angle, we expect that their geometric mean is a rotation about the z-axis through a certain angle g. Furthermore, because of this symmetry we also expect that the mean in the Euclidean sense agrees with the one in the Riemannian sense. From the Campbell-Baker-Hausdor formula for elements of SO(3) [23] we have where the coecients a; b; c and are given by a Therefore, the characterization (3.12) of the geometric mean reduces to a Log Rn bN Log R This is a matrix equation in so(3), which is equivalent to a system of three nonlinear equations. Because the axes of rotation of Rn are symmetric about the z-axis we have cos It follows that [Log Rn; Log R. Therefore, this system reduces to the following single equation for the angle g which when compared with (4.2) indeed shows that a = g and therefore the projected arithmetic mean and the geometric mean coincide. This example provides a family of mean problems parameterized by and where the projected arithmetic and geometric mean coincide. We now further examine the problem of nding the mean of three rotations about the three coordinate axes 14 M. MOAKHER through the same angle , which, by the bi-invariance property of both means, can be considered as a special case of this two-parameter family with .Therefore the mean of these three rotations is a rotatiopn through an angle about the axis generated by the vector [1; 1; 1]T with tan . The rotations R1, and R3 form a geodesic equilateral triangle in SO(3). By symmetry arguments the geometric mean should be the intersection of the three geodesic medians, i.e., the geodesic segments joining the vertices of the geodesic triangle to the midpoints of the opposite sides. In at geometry, this intersection is located at two-thirds from the vertices of the triangle. However, in the case of SO(3), due to its intrinsic curvature, this is not true. The ratio of the length of the geodesic segment joining one rotation and the geometric mean, to the length of the geodesic median joining this rotation and the midpoint of the geodesic curve joining the two other rotations is plotted as a function of the angles in Fig. 4.3.0.65g0.620 p/4 p/2 3p/4 p Fig. 4.3. Plot of the ratio of the geodesic distance from one vertex to the barycenter over the geodesic distance from this vertex to the midpoint of the opposed edge in the geodesic equilateral triangle in SO(3). The departure of from 2/3, which is due to the curvature of SO(3), increases with the length, , of the sides of the triangle. 5. Weighted means and power means. Our motivation of this work was to construct a lter that smooths the rotation data giving the relative orientations of successive base pairs in a DNA fragment, see [19] for details. Such a lter can be a generalization of moving window lters, which are based on weighted averages, used in linear spaces to smooth noisy data. The construction of such lters and the direct analogy we have found between the arithmetic and geometric means in the group of positive numbers, and the projected arithmetic and geometric means in the group of rotations, have led us to the introduction of weighted means and power means of rotations that we discuss next. Definition 5.1. The weighted projected arithmetic mean of N given rotations dened as This mean satises the bi-invariance property. Using similar arguments as for the projected arithmetic mean one can show that the weighted projected arithmetic mean is given by the polar factor of the polar decomposition of the matrix MEAN ROTATION 15 provided that det A is positive. Definition 5.2. The weighted geometric mean of N rotations weights dened as This mean also satises the bi-invariance property. Using arguments similar to those used for the geometric mean, we can show that the weighted geometric mean is characterized by n=1 wn Log(Rn Definition 5.3. For a real number s such that 0 < jsj 1, we dene the weighted s-th power mean rotation of N rotations h is We note that Mw 1 this is the weighted projected arithmetic mean. Because elements of SO(3) are orthogonal, and the trace operation is invariant under transposition, the weighted s-th power mean is the same as the weighted (s)-th power mean. Therefore, it is immediate that the weighted projected harmonic mean, dened by coincides with the weighted projected arithmetic mean. This is a natural generalization of the s-th power mean of positive numbers and it is in line with the fact that for positive numbers the s-th power mean is given by the s-th root of the arithmetic mean of One has to note, however, that for s such that 0 < jsj < 1 this mean is not invariant under the action of elements of SO(3). This is not a surprise as the power mean of positive numbers also does not satisfy the homogeneity property. For the set of positive numbers [12] and similarly for the set of Hermitian denite positive operators [25], there is a natural ordering of elements and the classical mean inequalities holds. Furthermore, it is well known [12, 25] that the s-th power mean converges to the geometric mean as s goes to 0. However, for the group of rotations such a natural ordering does not exists. Nonethe- less, one can show that if all rotations belong to a geodesic ball of radius less than centered at the identity, then the projected power mean indeed convergesto the geometric mean as s tends to 0. Analysis of numerical algorithms for computing the geometric mean rotation and the use of the dierent notions of mean rotation for smoothing three-dimensional orientation data will be published elsewhere. Acknowledgment . The author is grateful to Professor J. H. Maddocks for suggesting this problem and for his valuable comments on this paper. He also thanks the anonymous referee for his helpful comments. --R Dierential Geometry: Manifolds Matrix Computations Jacobi elds and Finsler metrics on compact Lie groups with an application to dierentiable pinching problem optimization and Dynamical Systems Maximum likelihood estimation for the matrix von Mises-Fisher and Bingham distributions Fitting smooth paths to spherical data Riemannian center of mass and mollier smoothing The von Mises-Fisher matrix distribution in orientation statis- tics A continuum rod model of sequence-dependent DNA structure Statistics of directional data A Mathematical Introduction to Robotic Manipula- tion Fitting smooth paths to rotation data Geometrical Methods in Robotics optimization techniques on Riemannian manifolds Hermitian semidenite matrix means and related matrix inequalities-an intro- duction Convex functions and optimization methods on Riemannian manifolds Equatorial distributions on a sphere --TR --CTR Doug L. James , Christopher D. Twigg, Skinning mesh animations, ACM Transactions on Graphics (TOG), v.24 n.3, July 2005 Kavan , Steven Collins , Ji ra , Carol O'Sullivan, Skinning with dual quaternions, Proceedings of the 2007 symposium on Interactive 3D graphics and games, April 30-May 02, 2007, Seattle, Washington Xavier Pennec, Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements, Journal of Mathematical Imaging and Vision, v.25 n.1, p.127-154, July 2006 Christophe Lenglet , Mikal Rousson , Rachid Deriche , Olivier Faugeras, Statistics on the Manifold of Multivariate Normal Distributions: Theory and Application to Diffusion Tensor MRI Processing, Journal of Mathematical Imaging and Vision, v.25 n.3, p.423-444, October 2006 Peter J. Basser , Sinisa Pajevic, Spectral decomposition of a 4th-order covariance tensor: Applications to diffusion tensor MRI, Signal Processing, v.87 n.2, p.220-236, February, 2007

special orthogonal group;rotation;geodesics;averaging;operator means

587780

On Positive Semidefinite Matrices with Known Null Space.

We show how the zero structure of a basis of the null space of a positive semidefinite matrix can be exploited to determine a positive definite submatrix of maximal rank. We discuss consequences of this result for the solution of (constrained) linear systems and eigenvalue problems. The results are of particular interest if A and the null space basis are sparse. We furthermore execute a backward error analysis of the Cholesky factorization of positive semidefinite matrices and provide new elementwise bounds.

Introduction . The Cholesky factorization A exists for any symmetric positive semidenite matrix A. In fact, R is the upper triangular factor of the QR factorization of A 1=2 [11, x10.3]. R can be computed with the well-known algorithm for positive denite matrices. However, zero pivots may appear. As zero pivots come with a zero row/column in the reduced A, a zero pivot implies a zero row in R. To actually compute a numerically stable Cholesky factorization of a positive semidenite matrix one is advised to apply diagonal pivoting [11]. A semidenite matrix A may be given implicitly, in factored form pn is of full row rank that does not need to be a exposes the singularity of A explicitly as In this case both the linear system and the eigenvalue problem can be solved eciently and elegantly by working directly on the matrix F , never forming the matrix A explicitly. In fact, in some applications, not assembling the matrix A but its factor F is the most important step in the overall process of the numerical computation. One obvious reason is that the (spectral) condition number of F is the square root of the condition number of A. In nite element computation, F is the so called natural factor of the stiness matrix A [2]. In the framework of linear algebra, every symmetric positive semidenite matrix is the Gram matrix of some set of vectors, the columns of F . Another possibility to have the singularity of A explicit is to have available a basis of its null space N (A). This is the situation that we want to investigate in this note. We will see that knowing a basis of N (A) allows to determine a priori when the zero pivots will occur in the Cholesky factorization. It also permits to give a positive denite submatrix of A right away. These results are of particular interest if A and the null space basis are sparse. This is the case in the application from electromagnetics that prompted this study [1]. There, a vector that is orthogonal to the null space corresponds to a discrete electric eld that is divergence-free. Our ndings permit to work with the positive denite part of A and to compute a rank revealing Cholesky factorization A = R T R where the upper trapezoidal R has full row rank. What is straightforward in exact arithmetic amounts to simply replacing by zero potentially inaccurate small numbers. We analyze the error that is introduced by this procedure. Swiss Federal Institute of Technology (ETH), Institute of Scientic Computing, CH-8092 Zurich, Switzerland (arbenz@inf.ethz.ch) y University of Zagreb, Departement of Mathematics, Bijenicka 30, HR-10000 Zagreb, Croatia (drmac@math.hr). The work of this author was supported by the Croatian Ministry of Science and Technology grant 037012. We complement this note with some implications of the above for solving eigenvalue problems and constrained systems of equations. 2. Cholesky factorization of a positive semidenite matrix with known null space. In this section we consider joint structures of a semidenite matrix A and its null space. Theorem 2.1. Let A = R T R be the Cholesky factorization of the positive semidenite matrix A 2 R nn . Let Y 2 R nm with R(Y m. These are the only zero entries on the diagonal of R. Proof. Notice that the assumptions imply that Y := [y has full rank. By Sylvester's law of inertia R has precisely m zeros on its diagonal. Further, (Ry whence r n i n i If only n 1 n 2 nm , Y , ipped upside-down, can be transformed into column-echelon form in order to obtain strong inequalities. The Cholesky factor R appearing in Theorem 2.1 is an n n upper triangular matrix with m zero rows. These rows do not aect the product R T R. Therefore, they can be removed from R to yield an (n m) n matrix b R with b If the numbers n i are known, it is convenient to permute the rows of Y and accordingly the rows and columns n i of A to the end. Then Theorem 2.1 can be applied with m+i. The last m rows of R in Theorem 2.1 vanish. So, b R is upper trapezoidal. After the just mentioned permutation the lowest mm block of Y is non-singular, in fact, upper triangular. This consideration leads to an alternative formulation of Theorem 2.1. Theorem 2.2. Let A = R T R be the Cholesky factorization of the positive semidenite matrix A 2 R nn . Let Y 2 R nm with R(Y (A). If the last m rows of Y are linearly independent, then the leading principal (n m) (n m) submatrix of A is positive denite and R can be taken (n m) n upper triangular. Proof. Let O Y 2 consists of the last m rows of Y . W is therefore invertible. Applying a congruence transformation with W on A gives A 11 A 12 A 21 A 22 O Y 2 A 11 O O O By Sylvester's law of inertia A 11 must be positive denite. Let A R 11 be the Cholesky factorization of A 11 . Then, the Cholesky factor of the matrix in (2.2) is [R 11 ; O] 2 R (n m)n . Therefore, the Cholesky factor of A is [R Theorem 2.2 is applicable as long as the last m rows of Y form an invertible matrix. of Y are linearly independent, we can permute Y such that these rows become the last ones. In particular, if we want A 11 to be as sparse as possible, we may choose to be the m most densely populated rows/columns of A with the following greedy algorithm: If we have determined choose i k+1 to be the index of the densest column of A such that rows are linearly independent. In this way we can hope for an A 11 with sparse Cholesky factors. Remark 2.1. The equation in @ in a simply connected domain is satised by all constant functions u. The discretization of (2.3) with nite elements of Lagrange type [4] leads to a positive semidenite matrix A with a one dimensional null space spanned by the vector e with all entries equal to 1. Theorem 2.1 now implies that, no matter how we permute A, in the factorization the single zero on the diagonal of R will not appear before the very last elimination step. Example 2.1. Let A and Y be given by O. As the last two rows of Y are linearly independent, Theorem 2.2 states that the principal 33 submatrix of A is positive denite and that its Cholesky factor is 3 5 upper triangular. In fact, R =4 Let P be the permutation matrix, that exchanges 2nd with 4th and 3rd with 5th entry of a 5-vector. Then, Now we have according to Theorem 2.1 the Cholesky factor R 1 of A 1 has zero diagonal elements at positions 3 and 5. Indeed, pp 3. Consistent semidenite systems. In this section we discuss how to solve where A, R, and Y are as in Theorem 2.1. Without loss of generality, we can assume that m. We split matrices and vectors in (3.1), A 11 A 12 A T R TR T with is obtained from A by deleting rows and columns m. The factorization (3.2) yields Although A 11 is invertible, its condition number can be arbitrarily high. To reduce ll-in during factorization [8] any symmetric permutations can be applied to A 11 without aecting the sequel. As R T has full rank, O or diagonal elements. Because the right side b of (3.1) has to satisfy It is now easy to show that a particular solution of (3.1) is given by x with components In fact, employing (3.3){(3.5) the second block row in (3.2) is A T The manifold S of the solutions of (3.1){(3.2) is The vector a can be determined such that the solution x satises some constraints In particular, if then x is perpendicular to the null space of A. Let now A be given implicitly as a Gram matrix and Y 2 R nm be as above. (This may require renumbering the columns of F .) As and as Y 2 is nonsingular, the block F 2 depends linearly on F 1 . Therefore, the QR factorization of F has the form R 11 R 12 O O RR T , the factor equals the upper trapezoidal Cholesky factor in (3.2). 4. Error Analysis. In this section we give backward error analyses for the semidenite Cholesky factorization and for the null space basis. 4.1. Semidenite Cholesky factorization. The oating-point computation of the Cholesky factorization of a semidenite matrix is classied as unstable by Higham [11, x10.3.2]. The principal problem is the determination of the rank of the matrix. If we assume, as we do in this note, that a basis of the null space of the matrix under consideration is known a priori then, of course, its rank is known. Let A be partitioned as in (3.2). We assume that A 11 2 R rr is positive denite numerically, i.e. that the Cholesky factorization does not break down in oating point arithmetic with round-o unit u. Due to a result by Demmel [5] (see also [11, Thm.10.14]) this is the case if, min where min () denotes the minimal eigenvalue, kk is the spectral norm, and If (4.1) does not hold, A 11 is not numerically denite. Note that positive denite with unit diagonal. The assumption on min can be relaxed if, for instance, we use double precision accumulation during the factorization. Then f(r) can be replaced by a small integer for all r not larger than 1=u. We assume, however, that 2rf(r)u < 1. The Cholesky decomposition of A is computed as indicated in (3.3). The Cholesky factor of A 11 is computed rst. Then the matrix R 12 is obtained as the solution of the matrix equation R T Let e R 11 denote the computed oating-point Cholesky factor of A 11 . Then the following two important facts are well-known. (1) There exists a symmetric A 11 such that A 11 R R 11 and 1i;jr f(r)u: (4.2) This is the backward error bound by Demmel [5], [11, Theorem 10.5]. (2) Let imply that the Frobenius norm of assumption (4.1) implies show [7] that there exists an upper triangular matrix such that e s s <p: Let e R 12 be the oating-point solution of the matrix equation e R T is the computed approximation of the exact Cholesky factor Let e R T e R be partitioned conforming with (3.2). Since A+A is positive semidenite and of rank r by construction, the equation e A A 1e A 12 holds. If we compute e R 12 column by column, then, using Wilkinson's analysis of triangular linear systems [11, Theorem 8.5], R 6 PETER ARBENZ AND ZLATKO DRMA where the matrix absolute values and the inequality are to be understood entry-wise. Thus, we can write e R 12 as e R T R 12 j: (4.3) Also, if we R T R T 11 j, we have e R T R T Further, from the inequality j e using the M-matrix property of I we obtain Hence, relations (4.2), (4.3), (4.5) imply that the backward error for all (i; j) in the in (3.2) is bounded by A 22 We rst observe that k j j k and that s k. Note that our assumptions imply that r 2: It remains to estimate the backward error in the (2; 2) block of the partition (3.2). Using relation (4.4), we compute A 22 = e R R R T R 1e R T Using the inequalities from relations (4.4), (4.5) we obtain, for all (i; j), We summarize the above analysis in the following Theorem 4.1. Let A be a nn positive semidenite matrix of rank r with block partition (3.2), where the r r matrix A 11 is positive denite with the property (4.1). Then the oating-point Cholesky factorization with roundo u will compute an upper trapezoidal matrix e R of rank r such that e R T e A is a symmetric backward perturbation with the following bounds: A ii A jj A ii A jj ~ A ii A jj ; r < In the last estimate, s k. Further, if e is the exact Cholesky factor of A, then e R 11 R R ~ Here, the matrix is upper triangular and is to the rst order j j . Further, let the Cholesky factorization of A 11 be computed with pivoting so that (R 11 ) ii k=i (R 11 Then, the error R R 11 R 11 is also row-wise small, that is Remark 4.1. Note that Theorem 4.1 also states that in the positive denite case the Cholesky factorization with pivoting computes the triangular factor with small column- and row-wise relative errors. This aects the accuracy of the linear equation solver (forward and backward substitutions following the Cholesky factorization) not only by ensuring favorable condition numbers but also by ensuring that the errors in the coecients of the triangular systems are small. 4.2. Null space error. We now derive a backward error for the null space Y of A. We seek an n (n r) full rank matrix e Y such that Y is small and e A e As the null space and the range of A change simultaneously (being orthogonal complements of each other), the size of Y necessarily depends on a certain condition number of A; and the relevant condition number will depend on the form of the perturbation A. The equation that we investigate is e equivalently, e RY . If e R is suciently close to R (to guarantee invertibility of e RR RR RR Though simple this equation is instructive. First of all, only the components of the columns of R that lie in the null space N (A) aect the value of Y . Also, Y keeps the full column rank of Y . Finally, Y T kY k= min (Y ). It is easy to modify Y such that min (Y ) 1, e.g., if Y Thus, kY k measures the angle between the true null space and the null space of the perturbed matrix e A. In the sequel we try to bound kY k. If we rewrite (4.7) as we get, after some manipulations, Proposition 4.2. Let D be nonsingular matrix and let If kR 0 (R Here, y denotes the orthogonal projection onto the null space of A. We will discuss choices for D later. The Proposition indicates that the crucial quantity for bounding kY k is kR 0 Y k. The following two examples detail this fact. Example 4.1. Let be big, of the order of 1=u, and let pp The null space of A is spanned by which means that deleting any row and column of A leaves a nonsingular 22 matrix. Let's choose it be the last one, and let us follow the algorithm. For the sake of simplicity, let the only error be committed in the computation of the (1; 1) entry of e R 11 which is instead of 3. Then we solve the lower triangular system for e R 12 and obtain e Thus, If we take perform the computation in Matlab where u 2:22 10 O() such that the angle between Y and Y is small. Example 4.2. We alter the (1; 1) entry 3 of R of the previous example to get , Again, we delete the last row and column of A and proceed as in Example 4.1. Let us again assume that the only error occurs in the entry of R 11 which becomes =(1 e and Again, O(1). But now also kY In fact, in computations with Matlab, we observe an angle as large as O(10 2 ) between Y and Y . Remark 4.2. Interestingly, if we set Example 4.1, the Matlab function chol() computes the Cholesky factor e R =4 It is clear that the computed and stored A is a perturbation of the true A. Therefore, numerically, it can be positive denite. It is therefore quite possible to know the rank r < n of A exactly, to have a basis of the null space of A and a numerically stored positive denite oating-point A. Strictly speaking, this is a contradiction. Certainly, from an application or numerical point of view, it is advisable to be very careful when dealing with semideniteness. In Examples 4.1 and 4.2 we excluded the largest diagonal entry of A. In fact, we can give an estimate that relates the error in R 12 to the size of the deleted entries. Suppose we managed the deleted diagonal entries of A to be the smallest ones. Can we then guarantee that the relevant error in R will be small, and can we check the stability by a simple, inexpensive test? According to Theorem 4.1, the matrix R 11 is computed with row-wise small relative error, provided that the Cholesky factorization of A 11 is computed with pivoting. If that is the case, then it remains to estimate the row-wise perturbations of R 12 . If is as in Theorem 4.1, then the inequality holds for all with some i 2 (0; =2]. The angle i has a nice interpretation. Let A = F T F be any factorization of A, with full column rank and F T Then i is the angle between F 1 e i and the span of fF 1 e g. (This is easily seen from the QR factorization of F 1 .) The following Proposition states that well-conditioned dominance of A 11 over A 22 ensure accurate rows of the computed matrix e R. Proposition 4.3. With the notation of Theorem 4.1, let A (and accordingly Y ) be arranged such that If the Cholesky factorization of A 11 is computed with (standard) pivoting, then sin i where sin i is dened in (4.10). Proof. This follows from relations (4.6), (4.9), (4.10) and the assumption (4.11). We only note that in (4.9) and (4.10) we can replace Remark 4.3. If spans N partition of A s satises condition (4.11). If we apply the preceeding analysis to A s and SY , we get an estimate for Y in the elliptic norm generated by S. Note that Proposition 4.2 is true for any diagonal D as long as k(R 0 moderately big and kR 0 k is small. We have just seen that R 0 is nicely bounded if we choose has an inverse nicely bounded independent of A 11 because [11, x10] Here the function h(r) is in the worst case dominated by 2 r and in practice one usually observes an O(r) behaviour. In any case, k(D 1 R 11 is at most r times larger than sophisticated pivoting can make sure that the behaviour of h(r) is not worse than Wilkinson's pivot growth factor. We skip the details for the sake of brevity. To conclude, if the Cholesky factorization of A 11 is computed with pivoting and relation (4.11) holds, then the backward error in Y can be estimated using (4.8) and (4.12), where 4.3. Computation with implicit A. We consider now the backward stability of the computation with A given implicitly as rank r. Thus, the Cholesky factorization of A is accomplished by computing the QR factorization of F . In the numerical analysis of the QR factorization we use the standard, well-known backward error analysis which can be found e.g. in [11, x18]. The simplest form of this analysis states that the backward error in the QR factorization is column-wise small. For instance, if we compute the Householder (or Givens) QR factorization of F in oating point arithmetic with roundo u, then the backward error F satises n) is a polynomial of moderated degree in the matrix dimensions. Our algorithm follows the same ideas as in the direct computation of R from A. The knowledge of a null space basis admits that we can assume that F is in the the p r matrix F 1 is of rank r, see section 3. We then apply r Householder re ections to F which yields, in exact arithmetic, the matrix R 11 R 12 O R 22 where R 11 2 R rr is upper triangular and nonsingular. If conforming with F , then F is the QR factorization of F 1 . In oating point computation, R 22 is unlikely to be zero. Our algorithm simply sets to zero whatever is computed as approximation of R 22 . As we shall see, the backward error (in F ) of this procedure depends on a certain condition number of the Theorem 4.4. Let F 2 R pn have rank r and be partitioned in the form pr has the numerically well determined full rank r. More specically, if is obtained from F 1 by scaling columns to have unit Euclidean norm, then we assume that Let the QR factorization of F be computed as described above, and let e be the computed upper trapezoidal factor. Then there exist a backward perturbation F and an orthogonal matrix b Q such that F R is the QR factorization of F +F . The matrix F +F has rank r. are partitioned as F , and Q 1 := b then c k; e R 11 R bounds the roundo. Proof. Let e F (r) be the matrix obtained after r steps of the Householder QR factorization. Then there exist an orthogonal matrix b Q and a backward perturbation F such that e R 11 e R 12 O e R 22 Our assumption on the numerical rank of F 1 implies that F 1 e R 11 is the QR factorization with nonsingular e R 11 . Now, setting e R 22 to zero is, in the backward error sense, equivalent to the QR factorization of a rank r matrix, R 11 e R 12 O O O O O e R 22 It remains to estimate b First note that F the i-th column of R 12 has the same norm as the corresponding column of F 2 . Then, and we can write To estimate Q 1 , we rst note that F e R 11 imply that (R 11 e and that R R R Thus, e R 11 is the Cholesky factor of I +E, where Now, by [7], kEkF < 1=2 implies that e R I +, where is upper triangular and <p: Hence, R 11 e Finally note that e R 11 R We remark that e which means that we can nicely bound R 12 R 12 R 12 . We have, for instance, If we use entry-wise backward analysis of the QR factorization (jF then we can also write where the matrix absoulute values and inequalities are understood entry-wise, and " 2 is dened similarly as " 1 . From the above analysis we see that the error in the computed matrix e R is bounded in the same way as in Theorem 4.1. Also, the QR factorization can be computed with the standard column pivoting and R 11 can have additional structure just as in the Cholesky factorization of A 11 . Therefore, the analysis of the backward null space perturbation based on e R T holds in this case as well. However, the bounds of Theorem 4.4 are sharper than those of Theorem 4.1. 5. Constrained systems of equations. Let again be Y 2 R nm having full rank. Let C 2 R nm be a matrix with full rank. Systems of equations of the form A C x y c appear at many occasions, e.g. in mixed nite element methods [3], or constrained optimization [12]. They have a solution for every right side if R which is the case if H := Y T C is nonsingular. In computations of Stokes [3] or Maxwell equations [1] the second equation in (5.1) with imposes a divergence-free condition on the ow or electric eld, respectively. To obtain a solution of (5.1) we rst construct a particular solution of the rst block row. Pre-multiplying it by Y T yields As b Cy 2 R(A) we can proceed as in section 3 to obtain a vector ~ x with A~ Cy. The solution x of (5.1) is obtained by setting determining a such that C T Thus, x). This procedure can be described in an elegant way if a congruence transformation as in (6.2) is applied. Multiplying (5.1) by W T I m , cf. (2.2), yields4 A 11 O C 1 O O H a n;r b: Notice that ~ From (5.2) we read that This geometric approach diers from the algebraic one based on the factorization4 A 11 A 12 C 1 A T R T O O R T O I m54 R O O C O C T where the LU factorization of C employed to solve (5.1). In the geometric approach the LU factorization of H is used instead. Of course, there is a close connection between the two approaches: Using (3.4) we get C T . Notice that the columns of C or Y can be scaled such that the condition numbers of H or C 2 R T are not too big. Notice also that Y can be chosen such that Y in which case C T perturbation analysis of (5.1){(5.3) remains to be done in our future work. Golub and Greif [9] use the algebraic approach to solve systems of the form (5.1) if the positive semidenite A has a low-dimensional null space. As they do not have available a basis for the null space they apply a trial-and-error strategy for nding a permutation of A such that the leading rr principal submatrix becomes nonsingular. They report that usually the rst trial is successful. This is intelligible because n the basis of the null space is dense which is often the case. If the null space of A is high-dimensional then Golub and Greif use an augmented Lagrangian approach. They modify (5.1) such that the (1; 1) block becomes positive denite, x y c Here, is some symmetric positive denite matrix, e.g. a multiple of the identity. A+CC T is positive denite if Y T C is nonsingular. The determiniation of a good is dicult. Golub and Grei thoroughly discuss how to choose and how the 'penalty aects the condition of the problem. In contrast to this approach where a term is added to A that is positive denite on the null space of A, N (A) can be avoided right away if a basis of it is known. 6. Eigenvalue problems. Let us consider the eigenvalue problem where A is symmetric positive semidenite with positive denite. We assume that the last m rows of Y are linearly independent such that W in (2.1) is nonsingular. Then, A 11 O O O where Using the decomposition O O H I O I with the Schur complement S := M 11 C 1 , and noting that P T W T is easy to see that the positive eigenvalues of (6.1) are the eigenvalues of A 11 14 PETER ARBENZ AND ZLATKO DRMA Notice that S is dense, in general, whence, in sparse matrix computations, it should not be formed explicitly. If y is an eigenvector of (6.4) then y y is an eigenvector of (6.1). By construction, C T to the null space of A. We now consider the situation when A and M are given in factored form, such that the rank of F 1 equals the rank of A. Let us nd an implicit formulation of the reduced problem (6.4). With W from (2.1) we have As before, A R 11 is computed by the QR factorization of F 1 . It remains to compute a Cholesky factor of the Schur complement S, but directly from the matrix B. To that end we employ the QL factorization ('backward' QR factorization) of BW , whence, with (6.3), Straightforward calculation now reveals that Thus, the eigenvalues of the matrix pencil are the squares of the generalized singular values [10] of the matrix pair (R equivalently, the squares of the singular values of R 11 L 1 11 . An eigenvector y corresponds to a right singular vector y. The blocks L 21 and L 22 come into play when the eigenvectors of (6.1) are to be computed: using (6.7) equation (6.5) becomes y 22 L 21 y It is known that the GSVD of (R 11 ; L 11 ) can be computed with high relative accuracy if the matrices (R 11 ) c and (L 11 ) c are well conditioned [6]. Here, (R 11 ) c and are obtained by R 11 and L 11 , respectively, by scaling their columns to make them of unit length. Obviously, 2 ((R 11 is the spectral condition number. It remains to determine 2 ((L 11 ) c ). From (6.6) we get whence . Let the diagonal matrix D 1 be such that (B 1 1has columns of unit length. Further, let (B 1 be the QR factorization of As Q 1 is orthogonal we have k(L 11 where is the largest principal angle [10] between r minD=diagonal 2 (L 11 D) [13] [11, Thm.7.5], we have So, we have identied condition numbers that do not depend on column scalings and that have a nice geometric interpretation. If the perturbations are column-wise small, then these condition number are the relevant ones. 7. Concluding remarks. In this paper we have investigated ways to exploit the knowledge of an explicit basis of the null space of a symmetric positive semidenite matrix. We have considered consistent systems of equations, constrained systems of equations and generalized eigenvalue problems. First of all, the knowledge of a basis of the null space of a matrix A permits to extract a priori a maximal positive semidenite submatrix. The rest of the matrix is redundant information and is needed neither for the solution of systems of equations nor for the eigenvalue computation. The order of the problem is reduced by the dimension of the null space. In iterative solvers it is not necessary to complement preconditioners with projections onto the complement of the null space. Our error analysis shows that a backward stable positive semidenite Cholesky factorization exists if the principal r r submatrix, This does however not mean that the computed Cholesky factor ~ R has a null space that is close to the known null space of R, A = R T R. We observed that the backward error in the null space is small if the error in the Cholesky factor is (almost) orthogonal to the null space of A. We show that this is the case if the positive denite principal r r submatrix after scaling is well conditioned and if its diagonal elements dominate those of the remaining diagonal block. For systems of equations and eigenvalue problems, we considered the case when is rectangular. This leads to interesting variants of the original algorithms and most of all leads to more accurate results. What remains to be investigated is the relation between extraction of a positive denite matrix and ll-in during the Cholesky factorization. In future work we will use the new techniques in applications and, if possible, extend the theory to matrix classes more general than positive semidenite ones. --R A comparison of solvers for large eigenvalue problems originating from Maxwell's equations Mixed and Hybrid Finite Element Methods The Finite Element Method for Elliptic Problems On oating point errors in Cholesky Computer Solution of Large Sparse Positive De Techniques for solving general KKT systems Matrix Computations Accuracy and Stability of Numerical Algorithms Condition numbers and equilibration of matrices --TR

positive semidefinite matrices;null space basis;cholesky factorization

587787

Joint Approximate Diagonalization of Positive Definite Hermitian Matrices.

This paper provides an iterative algorithm to jointly approximately diagonalize K Hermitian positive definite matrices ${\bf\Gamma}_1$, \dots, ${\bf\Gamma}_K$. Specifically, it calculates the matrix B which minimizes the criterion $\sum_{k=1}^K n_k [\log \det\diag(\B\C_k\B^*) - \log\det(\B\C_k\B^*)]$, nk being positive numbers, which is a measure of the deviation from diagonality of the matrices BCkB*$. The convergence of the algorithm is discussed and some numerical experiments are performed showing the good performance of the algorithm.

Introduction The problem of diagonalizing jointly approximately several positive definite matrices has arisen in (at least) two different contexts. The first one is the statistical problem of common principal components in k group introduced by Flury (1984). He considers k populations of multivariate observations of size n 1 obeying the Gaussian distribution with zero means and covariance matrices and assume that \Gamma k can be written as B k B for some orthogonal matrix B and some diagonal matrices k (the symbol denotes the transpose). The problem is to estimate the matrix B (the colum of which are the common principal components) from the sample covariance matrices C k of the populations. As it is well known that n k C k , are distributed independently according to the Wishart distribution of n k degrees of freedom and covariance matrices \Gamma k (see for example Seber, 1984), the log likelihood function of based on them is where C is a constant term and tr denotes the trace. Therefore the log likelihood method for estimating B and the k amounts to minimizing For fixed B, it is not hard to see that the above expression is minimized with respect to the k when the notation diag(M) denoting the diagonal matrix with the same diagonal as M. Thus, one is led to the minimization (with respect to B) of which is the same as that of since B has unit determinant. But (1.1) is precisely a measure of the global deviation of the matrices from diagonality, since, from the Hadamard inequality (Noble and Daniel, 1977, exercise 11.51), det M - det diag(M) with equality if and only if M is diagonal. Thus minimizing (1.1) can be viewed as trying to find a matrix B which diagonalizes jointly the matrices C much as it can. More recently, several authors (Cardoso and Souloumiac, 1993, Belouchrami et al., 1977, Pham and Garat, 1997) have introduced the joint approximate diagonalization as a method for the separation of source problem. In this problem, there are K sensors which record each a linear mixture of K sources, so that denoting by X(t) and S(t) the vectors of measurements and of sources at time t, one has A. The goal is to extract the sources from the observations and in the so called blind separation one does not have any specific knowledge about the sources other than that they are statistically independent. Thus a sensible method is to try to find a matrix B such that the components of BX(t) (which represent the reconstructed sources) are as independent as possible. As it is easier to work with non correlation rather than independence, a simple method would be to try to make the cross-correlation, eventually lagged, between the sources, vanish. This would lead to the joint approximate diagonalization of a certain set of covariance matrices, as proposed in Belouchrami et al. (1977). Note that Pham and Garat (1997) also consider joint diagonalization but they use only two matrices and then the diagonalization can be exact (see for ex. Golub and Van Loan, 1989). Cardoso and Souloumiac (1993), on the other hand, do not consider lagged covariance but use higher order cumulants between the sources instead. They construct certain set of matrices in which such cumulants appear as off diagonal elements. The separation of source is then solved through a joint approximate diagonalization of these matrices. It should be pointed out that the above authors use a different measure of deviation to diagonality than that of Flury. Their measure is simply the sum of squares of the off-diagonal elements of the considered matrices. But there is a common feature in all above works in that the diagonalizing matrix B is taken to be orthogonal. In this work we shall drop this restriction. The orthogonality condition is part of the assumption of Flury (1984) but there is no clear reason that it should be satisfied. This condition is justified in the works of Cardoso and Souloumiac (1993) and Belouchrami et al. (1997), since these authors have pre-normalized their observations to be uncorrelated and have unit variance. We want to avoid this pre- normalizing stage in the separation of source procedure, which can adversely affect its performance since the statistical error committed in this stage cannot be corrected in the following "effective separation" stage. By dropping the orthogonality restriction, we obtain a single-stage separation procedure which is simpler and can perform better. Note that without the orthogonality restriction, exact joint diagonalization is possible for two matrices (see for ex. Golub and Van Loan, 1989). But for more than two one can only achieve approximate joint diagonalization, relative to some measure of deviation to diagonality. We take this measure to be (1.1) for two following reasons. Firstly, it can be traced back to the likelihood criterion, widely used in statistics. Secondly, this criterion is invariant with respect to scale change: it remains the same if the matrices to be diagonalized are pre- and post-multiplied by a diagonal matrix. The other measure which consists in taking the sum of squares of the off-diagonal elements of the matrices, does not have this nice invariant property. Of course, one can introduce this property by first normalizing the matrices so that they have unit diagonal element, but then the resulting criterion would be very hard to manipulated. The main result of this paper is the derivation of an algorithm to perform the joint approximate diagonalization in the sense of the criterion (1.1) and without the restriction that the diagonalizing matrix be orthogonal. Our algorithm has some similarity with that of Cardoso and Souloumiac (1993) and even more with that of Flury and Gautschi (1996): it also operates through successive transformations on pairs of rows and columns of the matrices to be diagonalized. However the convergence proof is completely different since we can no longer rely on the orthogonality property. Incidentally our method of proof can be easily adapted to prove the convergence result in Flury and Gautschi (1996), in a much simpler way. 2 The algorithm As one frequently encounters complex data in signal processing applications, we shall consider complex Hermitian (instead of real symmetric) positive definite matrices . (Note that Cardoso and Soulomiac (1993) and Bellouchrani et al. (1997) also work in a complex setting.) The goal is to find a complex matrix B such that the matrices are as close to diagonal as possible, the notation now denoting the transpose complex conjugated. The measure of deviation to diagonality is taken to be (1.1) where the n k are positive weighs (they need not be integers). Note that since the C k do not depend of B, the minimization criterion (1.1) can be reduced to The algorithm consists in performing successive transformations, each time on a pair of rows of B, the i-th row B i\Delta and the j-th row say, according to a b c d and in such a way that the criterion is sufficiently decreased. Whether the decrease is sufficient is a question which we shall returned in next section. Once this is done, the procedure is repeated with another pair of indices until convergence is achieved. Denote by u ij the general element of the matrix the decrease of the criterion associated with the transformation (2.1) is du ii )=(u (k) ii u (k) A natural idea is to chose a, b, c, d to maximize the above decrease. However, this maximization cannot be done analytically. Our idea is to maximize a lower bound of it instead. Since the logarithm function is convex, for any two sequences of positive 1, one has Applying this inequality, the above decrease is bounded below by ii ii )=u (k) . Introducing the matrices ii =n this lower bound can be rewritten as [a b]P - a log [c d]Q d The maximization of (2.3) can be done analytically as it will be shown below. Since (2.3) vanishes at a = clear that its maximum is non negative and can be zero only if this maximum is also attained a Thus the decrease of the criterion, associated with the transformation using the values of a, b, c, d realizing the maximum of (2.3), is positive unless attains it maximum at a = Let us return to the maximization of (2.3). For any given a can always parameterize a; b; c; d as a b c d a provided that the last matrix is non singular. Put a a a a one can express (2.3) as log ja 0 d The first term of the above right hand side is no other than (2.3) evaluated at the point (a Therefore, a necessary and sufficient condition that this point realizes the maximum of (2.3) is that the last term in the above right side is non negative, for all -, ffl, j. But for this term can be seen to be equivalent to n[(-fflp 0 Hence a necessary condition for this condition to hold is that This is the same that the matrices P and Q be jointly diagonalized by a . Further, for the right hand side of (2.6) reduces to Again, for 0, the last term in the above expression can be seen to be equivalent to jfflj 2 p 0 j. Therefore it is also necessary that this quadratic form in the variables ffl; - j be non negative. This requirement is satisfied if and only if p 0 2 . On the other hand, using the inequality can be seen to be bounded above by log ja 0 d But for p 0 2 , the quadratic form j-fflj (in the variables - -ffl; -j) is non negative, entailing that the above expression is bounded above by n log[ja choices of -; ffl; j. Thus we have proved that a necessary and sufficient condition for a to realize the maximum of (2.3) is that a jointly diagonalizes P and Q and that the diagonal terms 2 of the diagonalized matrix satisfy p 0 . (Note that if the last condition is not satisfied, one simply needs to permute a In the appendix, the problem of jointly diagonalizing two Hermitian matrices P and Q of size two (not necessarily positive definite) is completely solved. It is shown there that if P and Q are positive definite and not proportional, the solution exists and is unique up to a permutation and a scaling, that is all solutions can be obtained from a representative one by pre-multiplying it by a diagonal matrix and a permutation matrix. Denoting by the diagonal and the upper off diagonal elements of P and Q and putting a representative solution is given by 2ff D and is positive. We shall prove below that (i) p 2 q 1 - 1 with equality if and only if the matrices P and Q are proportional and (ii) the p 0 defined by (2.5) with a are such that p 0 1 has the same sign as As obviously the above results show that the solution to the maximization of (2.3) is given by [a b] / [D 2fl], [c d] / [2ff D], the meaning proportional to. Of course, this result doesn't apply in the case where the two matrices P and Q are proportional. But in this case, diagonalize one would diagonalize the other. As it has been proved before, this would be enough to ensure that (2.3) be maximized. Thus in this case, there exists an infinite number of solutions (even after eliminating the ambiguity associated with scaling). Note The Flury and Gautschi (1996) algorithm operates on a similar principle. However, these authors iterate the transformation (2.1) with a fixed pair (i; convergence and only then he changes to another pair. We feel that this is less efficient, because by using the same pair, the decrease of the criterion tends to be smaller each time while by changing it one can get big decrease in the first few iterations. Our algorithm is also simpler to program. We now proved the results (i) and (ii) announced above. By formula (2.2), one has l ii ii l ii ii ii ii l hi u (k) ii ii ii ii It follows that p 2 q 1 ? 1 with equality if and only if u (1) ii =u (1) ii =u (K) jj . In the last case P and Q are proportional. This proves the result (i). On the other hand, for a one gets from (2.5) Hence the product p 0 denoting the real part. The product p 0 2 can be obtained from the above formula by interchanging Therefore, the difference p 0 The last three terms of the above expression may be regrouped as D)]: putting \Delta, one has and hence, noting that fi)=2: Further Therefore, combining the above result, one gets fi)]g: As fi is purely imaginary, the last term in the above expression equals One thus obtains finally p 0 which has the same sign as Note The computation of ff, fi and fl could be subjected to large relative error if the matrices P and Q are nearly proportional. But this doesn't matter as long as the matrices P and Q are diagonalized with sufficient accuracy so that the criterion be adequately decreased. To this end, note that the solution to the problem remains the same if one replaces Q by arbitrary ae. One can chose ae such that when P and Q are nearly proportional, R is almost zero. The numerical calculation of R will then be subjected to large relative error, but this is the only main source of error since there will be no near cancellation is subsequent calculation. More precisely, let ~ R be the calculated R, the algorithm would diagonalize P and ~ R. Hence it would diagonalize ~ the absolute error ~ small (it is the relative error which is large), P and Q are still accurately diagonalized. The above argument can be repeated with the role of P and Q interchanged, that is one takes jointly diagonalizes Q and R. This alternative would be preferable, from the numerical accuracy point of view, if it leads to a smaller (in absolute value) of ae. A simple rule to chose ae is to require that it results in a zero diagonal element of R and is as small as possible. Thus is defined as is defined as we want ae to be small, we chose the first possibility if q 1 and the second otherwise. We shall assume that as it is the case here. (But the case can be handled in a similar way). Thus the diagonal and upper off diagonal elements and r of R are and one jointly diagonalizes P and R in the first case and Q and R in the second case. Hence in the first case, we compute ff, fi, fl as and in the second case as The second right hand sides in the above formula provide more efficient computations without loss of accuracy. Further, they remain applicable even in if the matrices P and Q are proportional, by replacing by an arbitrary non zero number (ff in the first case or fl in the second case can be taken arbitrary too). Indeed the above computation ensures that in the first case this is sufficient to entail that the matrix P is diagonalized (see the Proof of Proposition A1), and hence so is Q as it is proportional to P. Similarly, in the second case, the above computation ensures that Q is diagonalized and hence so is P. 3 Convergence of the algorithm We have shown in previous section that our algorithm decreases the criterion at each step, unless (2.5) is maximized at a = But from the results of this section, this implies that the matrix P and Q are diagonal (we already proved that 1). If this occurs for a pair of indexes (i; j) then one would skip this pair and continue the algorithm with another pair. But if this occurs for all pairs, then the algorithm stops. Explicitly, it stops when One may recognize that the above condition is no other than the condition that B be a stationary point of the criterion (1.1). Indeed, consider a small change in B of the form ffiB (hence the matrix ffi represents a relative change), then the corresponding change of (1.1) is log[(u ri r rs - denoting the general elements of ffi. Expanding this expression with respect to ffi, up to the first order, one gets Thus, the vector with components 2n- ij , i 6= j can be viewed as the relative gradient vector of the criterion (1.1). We now prove that the decrease in the criterion at each step of the algorithm is sufficient to ensure the convergence to zero of the above gradient vector. As it has been proved in previous section, by parameterizing a; b; c; d as in (2.4) with a being the point realizing the maximum of (2.3), one can express (2.3) as are defined by (2.5). Take -; ffl; j such that the left hand side or (2.4) is the identity matrix. Then (2.3) vanishes. Thus (2:6 0 ) must also vanish, hence its upper bound derived in section 2 must be positive. Therefore, noting that p 0 , one has log ja 0 d where -; ffl; j; -, are the elements of the inverse of the matrix a . The left hand side of (3.4) is no other than the value of (2.3) at a . Further, the transformation (2.1) for this step of the algorithm uses precisely these a Thus the decrease of the criterion associated with this transformation, which is bounded below by (2.3), must be at least as large as the left hand side of (3.4). But from the definition of -; ffl; -; j and (2.5), one has hence, noting that the right hand side of (3.4) can be seen to equal The last expression can be rewritten as Therefore, noting that the middle matrix in this quadratic form has eigenvalues 1 \Sigma ae and 0 - ae - 1, it is bounded below by n[(q 0 This is also the lower bound of the decrease of our criterion at this step. Since the criterion is always decreased during our algorithm, it must converge to a limit. Therefore the decrease of the criterion at each step of the algorithm must converge to zero, implying that (q 0 tends to zero. Note that p, q are no other than - ij , - ji defined in (3.1) and 2n- ij are the components of the relative gradient vector at this step of the algorithm. Still, the above result proved the convergence to zero of this vector. The difficulty is due to the lack of normalization. Indeed, our algorithm constructs the transformation only up to a scaling of its rows, hence a row of B can be arbitrary large or arbitrary small and this has an effect on the gradient, even when relative gradient is considered. To avoid this, we shall renormalize the transformation matrices B after each step of the algorithm. Any reasonable normalization procedure will do but for simplicity and definiteness, we will consider the normalization which makes the rows of B having unit norm. Then u ii will be bounded between the smallest and the largest eigenvalue of C k . Therefore, letting m and M be the minimum of the smallest eigenvalues and the maximum of the largest eigenvalues, of C respectively, one has m - u for all for all i all k. Note that m ? 0 since the matrices are all positive definite. Therefore, from (2.2) and (2.5), [a mn a 0 Mn a 0 and the same inequalities hold for q 0 1 , and similar inequalities, with a 0 , b 0 replaced by 2 . Thus both p 0 2 can be bounded above by M=m and below by m=M . It follows that the relative gradient vector of the criterion evaluated at each step of the algorithm, which has components 2n- ij , converges to zero. The above result shows that if the algorithm converges, then the limit must be a stationary point of the criterion. Further, since the algorithm always decreases the criterion, this point is actually a local minimum, unless the algorithm is started at a stationary point, in which case it stops immediately. Note that, the sequence of transformation matrices constructed by the algorithm, being normalized and hence lying on a compact set, will admit a convergent subsequence and this in fact also holds for any of its subsequence. Therefore, if the criterion admit an unique local minimum, the algorithm will converge to it. However, Flury and Gautschi (1996) has shown that in some extreme cases, the minimization of the criterion (1.1) under the orthogonality constraint, admits more than one local minimum. Therefore, it seems likely that in our problem, where the maximization is without constraint, the uniqueness of the local minimum is also not satisfied in all cases. Nevertheless, if there are only several local minima, one can still expect that the algorithm converge to one of them. Indeed, if this is not so, then since we have proved that the gradient vector converge to 0, the algorithm must jump continually from one local minimum to another, a quite implausible thing. The existence of a finite number of local clearly does not hold if the matrices C are all proportional to a single matrix, but this is a very extreme case. We conclude this section by showing that our algorithm behaves near the solution very much like to the Newton-Rhapson iteration, provided that the matrices can be nearly jointly diagonalized. To derive the Newton-Rhapson iter- ation, one makes a second order Taylor expansion of the criterion around the current point, then minimizes this expansion (instead of the true criterion) to obtain the new point. As we have already computed the change of the criterion corresponding to a change ffiB of B, resulting in the formula (3.2), we need only to expand it up to second order in ffi. Note that the first order expansion has already been given by (3.3), we need only to pursue the expansion up to second order, yielding r s rs =u (k) ii r s ir =u (k) ii )!(ffi is - is =u (k) Assume that the matrices C can be nearly jointly diagonalized, then near the solution, the off diagonal term u (k) rs , r 6= s and u (k) ir , r 6= i, of the matrices would be small relative to the diagonal term u (k) ii . Hence we may neglect, in the above expression the term of second order in ffi containg the factor u (k) rs =u (k) ii , r 6= s or u (k) ir =u (k) ii , r 6= i. With this approximations, the above expansion for (3.2) (up to second order in reduces to The (approximate) Newton-Rhapson algorithm consists in minimizing (3.2') with respect to ffi, then change B into B being the solution to the above minimization. Note that (3:2 0 ) can be written as the sum over all (unordered) pairs The minimization of this expression with respect to [ffi ij can be easily done, yielding - It is worthwhile to note that the diagonal elements of ffi do not appear in (3:2 0 ) so they can be anything as long as they are small. For convenience, we take them to be 0. This is further justified by the fact that by dividing the i-th row of B+ ffiB by 1 one is led to the matrix 0 has zero diagonal element and (i; diagonal element which is about the same as Note also that because the ffi ij are small, the matrix B + ffiB can be obtained through successive transformations of the form (2.1) with a associated with all distinct pairs of indexes (i; j), i 6= j. Reverting to the notation defined by (2.2), are no other than p 2 , q 1 , p and - q. Thus On the other hand, for small p and q, fi and D, as defined in (2.7) and (2.8), can both be approximated by p 1 Hence, since one gets that b and c equal approximately 2fi=(fi +D) and +D). The above results show that the new matrix B resulting from one step Newton-Rhapson algorithm is about the same as that resulting from a "sweep" of the algorithm of section 2. constituted by successive steps associated with all distinct pairs of indexes.) Threfore, our algorithm has about the same quadratic convergence speed of the Newton-Rhapson iteration, near the solution. But the Newton-Rhapson method may converge badly, even not at all, if it is started at a point far from the solution. Our algorithm would have better convergence behavior in this case since it always decreases the criterion. 4 Some numerical examples We consider the same example as in Flury and Gautschi (1996). The following 6 \Theta 6 matrices are to be diagonalized \Gamma12:5 27:5 \Gamma4:5 \Gamma:5 2:04 \Gamma3:72 \Gamma:5 \Gamma4:5 24:5 \Gamma9:5 \Gamma3:72 \Gamma2:04 \Gamma4:5 \Gamma:5 \Gamma9:5 24:5 3:72 2:04 \Gamma2:04 2:04 \Gamma3:72 3:72 54:76 \Gamma4:68 3:72 \Gamma3:72 \Gamma2:04 2:04 \Gamma4:68 51:247 7 7 7 7 5 We take our algorithm with B being the identity matrix. The following table reports the values of criterion after each sweep, that is after steps associated with each of the 15 possible pairs of indexes. Criterion 0:809676 0:189367 0:00562301 The last sweep produces a zero value of the criterion up to machine precision, the slightly negative value we have got comes from the rounding errors. Note that since there are only 2 matrices, exact joint diagonalization can be achieved. Actually, after 3 sweeps (sweep 0 corresponds to the initial matrices) the diagonalization is already quite good. We have 50:0000 \Gamma0:0198 \Gamma0:0013 \Gamma0:0001 0:0000 0:0000 \Gamma0:0013 0:0001 29:8099 0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0001 0:0000 0:0000 39:0333 \Gamma0:0000 \Gamma0:0000 0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 20:1550 \Gamma0:0000 0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 10:04497 7 7 7 7 5 20:0000 0:0177 0:0052 0:0000 \Gamma0:0000 \Gamma0:0000 0:0177 10:0000 \Gamma0:0003 \Gamma0:0000 0:0000 0:0000 0:0052 \Gamma0:0003 40:0120 \Gamma0:0000 0:0000 0:0000 0:0000 \Gamma0:0000 \Gamma0:0000 30:7912 0:0000 0:0000 \Gamma0:0000 0:0000 0:0000 0:0000 59:1714 0:0000 \Gamma0:0000 0:0000 0:0000 0:0000 0:0000 48:47167 7 7 7 7 5 which corresponds to the transformation matrix (For definiteness, the rows of B have been normalized to have unit norm.) The fourth sweep zeros all off diagonal elements of C 1 and C 2 (at least up to 4 digits after the decimal point) without changing their diagonal elements. The transformation matrix B is also almost unchanged: 0:5000 0:5000 \Gamma0:5000 \Gamma0:5000 0:0000 \Gamma0:0000 0:5000 0:5000 0:5000 0:5000 \Gamma0:0000 0:0000 0:5527 \Gamma0:5527 0:4206 \Gamma0:4206 0:1688 \Gamma0:0817 0:3977 \Gamma0:3977 \Gamma0:5752 0:5752 \Gamma0:0664 \Gamma0:1324 0:0073 \Gamma0:0073 \Gamma0:0272 0:0272 0:6082 0:79287 7 7 7 7 5 One can see that our algorithm converges quite fast. The Flury and Gautschi needs 4 to 5 sweeps to converge. Moreover it makes several iterations for each pair of indexes while we make only one. However, our algorithm does not solves the same problem, since we do not require the transformation matrix to be orthogonal. A simple way to implement the orthogonality constraint, at least approximately, is to add another matrix C 3 which is the identity matrix and give it a large weigh n 3 . For 1), the values of the criterion after each sweep are given below Criterion 0:809676 0:226183 0:0291083 0:0290463 0:0290454 0:0290454 The criterion does not decrease further after 4 sweeps. The change in the transformation produced by the fifth sweep is also very slight, affecting only the last digit and never more than 2 units. This matrix, after sweep 5, is and the corresponding matrices C 1 , C 2 are 50:0000 0:0000 0:0000 0:0000 \Gamma0:0000 \Gamma0:0000 0:0000 29:9224 0:0000 \Gamma1:8497 2:2318 0:1111 0:0000 0:0000 60:0000 0:0000 \Gamma0:0000 \Gamma0:0000 0:0000 \Gamma1:8497 0:0000 39:7221 \Gamma0:7727 1:0432 \Gamma0:0000 2:2318 \Gamma0:0000 \Gamma0:7727 20:2390 \Gamma0:0385 \Gamma0:0000 0:1111 \Gamma0:0000 1:0432 \Gamma0:0385 10:02407 7 7 7 7 5 20:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 40:2097 \Gamma0:0000 \Gamma2:3088 4:4428 1:7605 \Gamma0:0000 \Gamma0:0000 10:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma0:0000 \Gamma2:3088 \Gamma0:0000 31:7746 \Gamma1:2795 7:2126 \Gamma0:0000 4:4428 \Gamma0:0000 \Gamma1:2795 59:3949 0:5032 \Gamma0:0000 1:7605 \Gamma0:0000 7:2126 0:5032 48:34577 7 7 7 7 5 These results are very similar to that of Flurry and Gautsch (1986). (Note that our matrix B is the transposed of their.) Of course, the orthogonality constraint is not exactly satisfied here. We have 1:0000 \Gamma0:0000 0:0000 \Gamma0:0000 0:0000 0:0000 \Gamma0:0000 1:0000 \Gamma0:0000 0:0119 \Gamma0:0185 \Gamma0:0047 0:0000 \Gamma0:0000 1:0000 \Gamma0:0000 0:0000 0:0000 \Gamma0:0000 0:0119 \Gamma0:0000 1:0000 0:0060 \Gamma0:0253 0:0000 \Gamma0:0185 0:0000 0:0060 1:0000 \Gamma0:0007 0:0000 \Gamma0:0047 0:0000 \Gamma0:0253 \Gamma0:0007 1:00007 7 7 7 7 5 but the difference of this matrix from the identity matrix is slight. We should mention here that our algorithm is not designed to enforce orthogonality, the above numerical results are given only as examples showing its good convergence property. Appendix Joint diagonalization of two Hermitian matrices of size two The following results provide the explicit and complete solutions to the problem of joint diagonalization of two non proportional Hermitian matrices of order two. (Proportionality should here be understood in the large sense so that a null matrix is proportional to any other one.) Note that if the matrices are proportional, then the problem degenerates to the diagonalization of a single matrix. Proposition Let P and Q be two non proportional Hermitian matrices of order two, with diagonal elements diagonal elements respectively. Then are not all zero and real and the matrix a b c d diagonalizes simultaneously P and Q if and only if (i) [a b] or [c d] vanishes or (ii) [a and proportional to [2ff fi + ffi] or to [c d] 6= [0 0] and proportional to [2ff ffi] or to [fi where ffi is any one of the two square roots of \Delta. Remark: proportional to [2ff fi implies that it is proportional to conversely, since then [a b] proportional to [2ff fi reduces to it being proportional to [0 1], which is the same as it being proportional to [fi \Gamma ffi 2fl] only if ffi has been chosen to be fi and not \Gammafi . Similar conclusion holds if Also in the condition (ii) of the Proposition, one has to exclude the case [a since then [c d] needs not be proportional to anything. Similarly, the case [c has to be excluded. hence [c d] / [a b]. Thus P and Q are not only be diagonalized but are actually transformed by a singular matrix into the null matrix. Proof Since P and Q are non proportional, the vectors [p 1 are linearly independent. This entails that ff, fi, fl are not all zero. To prove that real, we expand it as Consider now the solutions to the joint diagonalization problem. The condition that the transformed matrices be diagonal can be written as [a b]P [a b]Q d or [c d]P [c d]Q - a If we exclude the trivial solutions [a then the above equations imply that the matrices in their left hand side have zero determinants. Thus and the same equation holds with a, b replaced by c, d. After expansion, one gets the equations ffb The solution [a b] to the equation ffb determined only up to a multiplicative factor. Thus, if ff 6= 0 then either [a a 6= 0 and b=a is the root of the quadratic polynomial ffz is a square root of \Delta. Therefore [a b] is proportional to Similarly, if fl 6= 0, then [a b] must be proportional to We have chosen here the minus sign before ffi so that this solution is the same as the previous one (with the same ffi) in the case where ff 6= 0. Note that these solutions still apply in the case since then they reduce to that [a b] being proportional to [0 fi] or [fi 0] which is the same as ab = 0. Similar calculations apply to the equation ffd solutions [c d] must be proportional to [2ff is a square root of (not necessarily the same as ffi). We have shown that the rows [a b] and [c d] of a matrix jointly diagonalizing P and Q must have the above form. But we haven't proved the converse. Further, our results yield two choices for [a b] (modulo a constant multiple), depending on the choice of the square root ffi of \Delta, and similarly there are two other choices for [c d]. Hence one must determine which choice of the later must be associated with one of the former. For this purpose, we shall consider, for each choice [a [a the possible corresponding choices for [c d] of the form [ff fi \Sigma - ffi] or [fi \Upsilon - ffi 2fl] and see if there is one for which [a b]P[c d] will be more convenient here to write the two square roots of \Delta in the form \Sigma - ffi.) Suppose that ff 6= 0, then we need to consider the choice [a since the other is proportional. For [c ffi], we has [a b]P[c d] d The last expression can be expanded as Note that p 1 - hence the first term in the above expression using the equality - fi)=2 or 2-ff-p+ - reduces to This expression vanishes if the minus sign is used in \Sigma. Hence [a b]Q[c d] for the choice [c ffi]. A similar calculation, with q 1 , q 2 , q in place of shows that the same choice leads to [a b]Q[c d] One needs not consider further choices. Indeed if there exists another choice [~c ~ d] not proportional to [2ff fi such that [a b]P[~c ~ since one already has [a b]P[2ff and the vectors [~c ~ d] and [2ff fi are linearly independent, one must have [a By a similar argument, one also must have [a these two equalities [a can be easily seen to imply that P and Q are proportional, which contradicts our assumption. The calculations are similar in the case fl 6= 0, interchanging a with b, c with d, ff with fl, p 1 with reversing the sign of ffi. Finally, if the possible choice for [a b] is either proportional to [0 1] or to [1 0] according to ffi equal plus or minus fi. But it can be easily seen that this case can happen if and only if either p-q is real, as P and Q are non proportional. Then, it can be checked that the only possible corresponding choice for [c d] is, in the first case, proportional to [1 0] or to [0 1] and in the second case, to [0 1] or to [1 0]. This completes the proof of the Proposition. Corollary With the same notation and assumption of the Proposition and assume further that at least one of the matrices P and Q has positive determinant. Then and the matrix a b c d jointly diagonalizes P and Q if only if it equals denotes the sign function) pre-multiplied by a permutation and a diagonal matrix. Proof We first show that \Delta ? 0 if det P ? 0 or det Q ? 0. We shall prove the result only for the case det Q ? 0, the proof for the other case is similar. As the last two terms in the above right hand side can be written as Thus the first condition implies that -. Then the second condition implies that Thus P is proportional to Q, contradicting our assumption. As its roots are real and we denote as usual \Delta the positive one. \Delta so that fi since one obtains from Proposition A1 that a b c d equals pre-multiplied by a diagonal matrix. On the other hand, for \Delta so that by a similar calculation, the Proposition shows that a b c d equals a diagonal matrix times . This yields the result of the corollary. --R A blind source separation technique using second-order statistics Matrix computations. An algorithm for the simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly orthogonal form. Common principal components in k groups. Applied linear Algebra. Multivariate Observations. --TR --CTR Ale Holobar , Milan Ojsterek , Damjan Zazula, Distributed Jacobi joint diagonalization on clusters of personal computers, International Journal of Parallel Programming, v.34 n.6, p.509-530, December 2006 Y. Moudden , J.-F. Cardoso , J.-L. Starck , J. Delabrouille, Blind component separation in wavelet space: application to CMB analysis, EURASIP Journal on Applied Signal Processing, v.2005 n.1, p.2437-2454, 1 January 2005 Ch. Servire , D. T. Pham, Permutation correction in the frequency domain in blind separation of speech mixtures, EURASIP Journal on Applied Signal Processing, v.2006 n.1, p.177-177, 01 January Andreas Ziehe , Motoaki Kawanabe , Stefan Harmeling , Klaus-Robert Mller, Blind separation of post-nonlinear mixtures using linearizing transformations and temporal decorrelation, The Journal of Machine Learning Research, v.4 n.7-8, p.1319-1338, October 1 - November 15, 2004 Andreas Ziehe , Motoaki Kawanabe , Stefan Harmeling , Klaus-Robert Mller, Blind separation of post-nonlinear mixtures using linearizing transformations and temporal decorrelation, The Journal of Machine Learning Research, 4, 12/1/2003 Andreas Ziehe , Pavel Laskov , Guido Nolte , Klaus-Robert Mller, A Fast Algorithm for Joint Diagonalization with Non-orthogonal Transformations and its Application to Blind Source Separation, The Journal of Machine Learning Research, 5, p.777-800, 12/1/2004

principal components;separation of sources;diagonalization

587790

Accurate Solution of Weighted Least Squares by Iterative Methods.

We consider the weighted least-squares (WLS) problem with a very ill-conditioned weight matrix. WLS problems arise in many applications including linear programming, electrical networks, boundary value problems, and structures. Because of roundoff errors, standard iterative methods for solving a WLS problem with ill-conditioned weights may not give the correct answer. Indeed, the difference between the true and computed solution (forward error) may be large. We propose an iterative algorithm, called MINRES-L, for solving WLS problems. The MINRES-L method is the application of MINRES, a Krylov-space method due to Paige and Saunders [SIAM J. Numer. Anal., 12 (1975), pp. 617--629], to a certain layered linear system. Using a simplified model of the effects of roundoff error, we prove that MINRES-L ultimately yields answers with small forward error. We present computational experiments for some applications.

Introduction Consider the weighted least-squares (WLS) problem In this formula and for the remainder of this article, k \Delta k indicates the 2-norm. We make the following assumptions: D is a diagonal positive definite matrix and rank A = n. These assumptions imply that (1) is a nonsingular linear system with a unique solution. The normal equations for (1) have the form A T Weighted least-squares problems arise in several application domains including linear programming, electrical power networks, elliptic boundary value problems and structural analysis, as observed by Strang [21]. This article focuses on the case when matrix D is severely ill-conditioned. This happens in certain classes of electrical power networks. In this case, A is a node-arc adjacency matrix, D is matrix of load conductivities, b is the vector of voltage sources, and x is the vector of voltages of the nodes. Ill-conditioning occurs when resistors are out of scale, for instance, when modeling leakage of current through insulators. Ill-conditioning also occurs in linear programming when an interior-point method is used. To compute the Newton step for an interior-point method, we need to solve a weighted least-squares equation of the form (2). Since some of the slack variables become zero at the solution, matrix D always becomes ill-conditioned as the iterations approach the boundary of the feasible region. In Section 9, we cover this application in more detail. Ill-conditioning also occurs in finite element methods for certain classes of boundary value problems, for example, in the heat equilibrium equation r \Delta thermal conductivity field c varies widely in scale. An important property of problem (1) or (2) is the norm bound on the solution, which was obtained independently by Stewart [20], Todd [22] and several other authors. See [6] for a more complete bibliography. Here we state this result as in the paper by Stewart. Theorem 1 Let D denote the set of all positive definite m \Theta m real diagonal matrices. Let A be an m \Theta n real matrix of rank n. Then there exist constants -A and - -A such that for any D 2 D Note that the matrix appearing in (3) is the solution operator for the normal equations (2), in other words, (2) can be rewritten as Since the bounds (3), (4) exist, we can hope that there exist algorithms for (2) that possess the same property, namely, the forward error bound does not depend on D. We will call these algorithms stable, where stability, as defined by Vavasis [23], means that forward error in the computed solution x satisfies where ffl is machine precision and f(A) is some function of A not depending on D. Note that the underlying rationale for this kind of bound is that the conditioning problems in (1) stem from an ill-conditioned D rather than an ill-conditioned A. This stability property is not possessed by standard direct methods such as QR factorization, Cholesky factorization, symmetric indefinite factoriza- tion, range-space and null-space methods, nor by standard iterative methods such as conjugate gradient applied to (2). The only two algorithms in literature that are proved to have this property are the NSH algorithm by Vavasis [23] and the complete orthogonal decomposition (COD) algorithm by Hough and Vavasis [12], both of them direct. See Bj-orck [1] for more information about algorithms for least-squares problems. We would like to have stable iterative methods for this problem because iterative methods can be much more efficient than direct methods for large sparse problems, which is the common setting in applications. This article presents an iterative algorithm for WLS problems called MINRES-L. MINRES-L consists of applying the MINRES algorithm of Paige and Saunders [14] to a certain layered linear system. We prove that MINRES- satisfies (5). This proof of the forward error bound for MINRES-L is based on a simplified model of how roundoff error affects Krylov space methods. This analysis is then confirmed with computational experiments in Section 8. (The simplified model itself is described in Section 5.) An analysis of round-off in MINRES-L starting from first principles is not presented here because the effect of roundoff on the MINRES iteration is still not fully understood. MINRES-L imposes the additional assumption on the WLS problem instance that D is "layered." This assumption is made without loss of generality (i.e., every weighted least-squares problem can be rewritten in layered form), but the MINRES-L algorithm is inefficient for problems with many layers. This article is organized as follows. In Section 2 we state the layering assumption, and also the layered least-squares (LLS) problem. In Section 3 we consider previous work. In Section 4 we describe the MINRES-L method for two-layered WLS problems. In Section 5 we analyze the convergence in the two-layered case using the simplifying assumptions about roundoff error. In Section 6 and Section 7 we extend the algorithm and analysis to the case of p layers. In Section 8 we present some computational experiments in support of our claims. In Section 9 we consider application of MINRES-L to interior-point methods for linear programming. 2 The Layering Assumption Recall that we have already assumed that the weight matrix D appearing in (1) is diagonal, positive definite and ill-conditioned. For the rest of this article we impose an additional "layering" assumption: we assume, after a suitable permutation of the rows of (A; b) and corresponding symmetric permutation of D, that D has the structure where each D k is well-conditioned and scaled so that its smallest diagonal entry is 1, and where denote the maximum diagonal entry among D . The layering assumption is that - is not much larger than 1. Note that this assumption is made without any loss of generality (and we could assume since we could place each diagonal entry of D in its own layer. Unfortunately, the complexity of our algorithm grows quadratically with p. Furthermore, our upper bound on the forward error degrades as p increases (see (39) below). Thus, a tacit assumption is that the number of layers p is not too large. From now on, we write A in partitioned form as A pC C A to correspond with the partitioning of D. We partition similarly. Under this assumption, we say that (1) is a "layered WLS" problem. In the context of electrical networks, this assumption means that there are several distinct classes of wires in the circuit, where the resistance of wires in class l is of order 1=ffi l . For instance, one class of wires might be transmission lines, whereas the other class might consist of broken wires (open lines) where the resistance is much higher. In the context of the heat equilibrium equation, the layering assumption means that the object under consideration is composed of a small number of different materials. Within each material the conductivity ffi l is constant, but the different materials have very different conductivities. In linear programming, taking means that the some of the slack variables at the current interior-point iterate are "small" while others are "large." A limiting case of layered WLS occurs when the gaps between the ffi l 's tend to infinity, that is, ffi 1 is infinitely larger than ffi 2 and so on. As the weight gaps tend to infinity, the solution to (1) tends to the solution of the following problem, which we refer to as layered least squares (LLS). Construct a sequence of nested affine subspaces L 0 oe L 1 oe \Delta \Delta \Delta oe L p of R n . These spaces are defined recursively: L fminimizers of kD 1=2 l Finally, x, the solution to the LLS problem, is the unique element in L p . The layered least-squares problem was first introduced by Vavasis and Ye [25] as a technique for accelerating the convergence of interior-point methods. They also established the result mentioned above in this paragraph: the solution to the WLS problem in the limit as ffi l+1 =ffi l ! 0 for all l converges to the solution of the LLS problem. Combining this result with Theorem 1 yields the following corollary, also proved by Vavasis and Ye. Corollary 1 Let x be the solution to the LLS problem posed with matrix A and right-hand side vector b. Then kxk -Akbk and kAxk - -Akbk for any choice of diagonal positive definite weight matrices D 3 Previous Work The standard iterative method for least-squares problems, including WLS problems, is conjugate gradient (see Golub and Van Loan [7] or Saad [18]) applied to the normal equations (2). This algorithm is commonly referred to as CGNR, which is how we will denote it here. There are several variants of CGNR in the literature; see, e.g., Bj-orck, Elfving, and Strako-s [2]. Note that in most variants one does not form the triple product A T DA when applying CG to (2); instead, one forms matrix-vector products involving matrices A T , D and A. This trick can result in a substantial savings in the running time since A T DA could be much denser than A alone. The same trick is applicable to our MINRES-L method and was used in our computational experiments. The difficulty with CGNR is that an inaccurate solution can be returned because A T DA can be ill-conditioned when D is ill-conditioned. To understand the difficulty, consider the two-layered WLS problem, which is obtained by subtituting (6) in the case Observe that if sequence A T Db; constructed by CGNR is very close to In other words, information about A 2 , D 2 and b 2 is lost when forming the Krylov sequence. A different framework for interpreting this difficulty is described in Section 5. Another iterative method for least-squares problems is LSQR due to Paige and Saunders [15]. This method shares the same difficulty with CGNR because it works in the same Krylov space. A standard technique for handling ill-conditioning in conjugate gradient is reorthogonalization; see, for example, Paige [16] and Parlett and Scott [17]. Reorthogonalization, however, cannot solve the difficulty with ill-conditioning in (2) because even the act of forming the first Krylov vector A T Db causes a loss of information. Another technique for addressing ill-conditioned linear systems with iterative methods is called "regularization"; a typical regularization technique modifies the ill-conditioned system with additional terms. See Hanke [10]. Regularization does not appear to be a good approach for solving (1) because (1) already has a well-defined solution (in particular, Theorem 1 implies that solutions are not highly sensitive to perturbation of the data vector b). A regularization technique would compute a completely different solution. In our own previous work [3], we proposed an iterative method for (2) based on "correcting" the standard CGNR search directions. We have since dropped that approach because we found a case that seemingly could not be handled or detected by that algorithm. 4 MINRES-L for Two Layers In this section and the next we consider the two-layered case, that is, in (6). We consider the two-layered case separately from the p-layered case because the two-layered case contains all the main ideas of the general case but is easier to write down and analyze. (In the our algorithm reduces to MINRES applied to (2) and hence is not novel.) Furthermore, the case is expected to occur commonly in practice. We mention also that the two-layered WLS and LLS problems were considered in x22 of Lawson and Hanson [13]. As noted in the preceding section, the two-layered WLS problem is written in the form (7), in which the diagonal entries of D 1 ; D 2 on the order of 1 and us introduce a new variable v such that A T Note that this equation always has a solution v because the right-hand side is in the range of A T 1 . Multiplying (8) by ffi 2 and adding to (7) yields A T Putting (8) and (9) together, we get A T !/ x A T Our algorithm, which we call MINRES-L (for MINRES "layered"), is the application of the MINRES iteration due to Paige and Saunders [14] to (10). Note that (10) is a symmetric linear system. In general, this linear system is rank deficient because if (x; v) is a solution solution. Thus, (10) is rank deficient whenever the rank of A 1 is less than n. This means we must address existence and uniqueness of a solution. Existence follows because the original WLS problem (7) is guaranteed to have a solution. Uniqueness of x is established as follows: if we add times the first row of (10) to ffi 1 times the second row, we recover the original WLS problem (7). Since (7) has a unique solution, (10) must uniquely determine x. Since x is uniquely determined, so is A 1 v. The question arises whether MINRES (in exact arithmetic) will find a solution of (10). MINRES can find a solution only if it lies in the Krylov space, which (because of rank deficiency) is not necessarily full dimensional. This question was answered affirmatively by Theorem 2.4 of Brown and Walker [4]. (Their analysis concerns GMRES, but the same result applies to MINRES in exact arithmetic.) Furthermore, their result states that, assuming the initial guess is 0, the computed solution (x; v) will have minimum norm over all possible solutions. Since x is uniquely determined, their result implies that will have minimum norm. Recall from Section 3 that the problem with applying conjugate gradient directly to (7) is that the linear system may be ill-conditioned when and hence conjugate gradient may return an inaccurate answer. Thus, it may seem paradoxical that we remedy a problem caused by ill-conditioning with an iterative method based on a truly rank-deficient system. One explanation of this paradox concerns the limiting behavior as 1. In this case, (7) tends to the linear system A T This system will, in general, not have a unique solution (because A 1 is not assumed to have rank n), so CGNR will compute some solution that may have nothing to do with . Thus, the CGNR solution is not expected to have the forward accuracy that we demand. On the other hand, as we see that (10) tends to A T !/ x A T This system is easily seen to be the Lagrange multiplier conditions for the two-layered LLS problem: recall from Section 2 that the two-layered LLS problem is minimize kD 1=2 subject to A T This is the correct limiting behavior: the WLS solution tends to the LLS solution as explanation of MINRES-L's convergence behavior follows. Convergence Analysis for Two Layers In this section we consider convergence of MINRES-L in the presence of roundoff error for the case 2. As mentioned in the introduction, we make a simplifying assumption concerning the effect of roundoff error in Krylov space methods. The assumption concerns either CG or MINRES applied to the symmetric linear system In our use of these algorithms, there is no preconditioner, and the initial guess is x Further, in our use of MINRES, c lies in the range-space of M (i.e., the system is consistent). In our use of CG, M is positive definite. With these restrictions in mind, our assumption about the effect of roundoff is that after a sufficient number of iterations, either method will compute an iterate - x satisfying where C is a modest constant, ffl is machine epsilon, and x is the true solution. (If multiple solutions exist, we take x to be the minimum-norm solution.) As far as we know, this bound has not been rigorously proved, but it is related to a bound proved by Greenbaum [9] in the case of conjugate gradient. In particular, Greenbaum's result implies that (11) would hold for CG if we were guaranteed that the recursively updated residual drops to well below machine precision, which always happens in our test cases. As for MINRES, less is known, but a bound like (11) is known to hold for GMRES implemented with Householder transformations [5]. GMRES is equivalent to MINRES augmented with a full reorthogonalization process. We are content to assert (11) for MINRES, with evidence coming from our computational experiments. This bound sheds light on why MINRES-L can attain much better accuracy than CGNR. For CGNR, the error bound (11) implies that kA T A T DA- xk gets very small, where - x is the computed solution. This latter quantity is the same as x)k. But recall that we are seeking a bound on the forward error, that is, on xk. In this case, the factor greatly skew the norm when is close to zero, so there is no bound on kx \Gamma - xk independent of ffi 1 =ffi 2 , that is, (5) is not expected to be satisfied by CGNR. This is confirmed by our computational experiments. In contrast, an analysis of MINRES-L starting from (11) does yield the accuracy bound (5). We need the following preliminary lemma. A be an m \Theta n matrix of rank n and - A an r \Theta n submatrix of A. Suppose the linear system - consistent. Here, c is a given vector, and - D is a given diagonal positive definite matrix. Then for any solution x, -A \Delta kck (12) and Furthermore, there exists a solution x satisfying -A Proof. First, note the following preliminary result. Let H;K be two symmetric n \Theta n matrices such that H is positive semidefinite and K is positive definite. Let b be an n-vector in the range space of H. Then (H converges to a solution of . This is proved by reducing to the diagonal case using simultaneous diagonalization of H;K. Let D be the extension of - D to an m \Theta m diagonal matrix obtained by filling in zeros, so that A T A. Since A T the limit of solution x of - as noted in the preceding paragraph. Let M be an m \Theta m diagonal matrix with 1's in diagonal positions corresponding to - D and zeros elsewhere. We have ck ck (15) ffl?0 -A \Delta kck: The last line was obtained by the transpose of (4). This proves (12). Note that this holds for all x satisfying - c, since this latter equation uniquely determines - Ax. Similarly, to demonstrate (13), we start from (15): ck ffl?0 ck For the second part of the proof, observe by the first part that A T and thus ffl?0 Axk: Combining this with (12) proves (14). To resume the analysis of MINRES-L, we define where (- x; - v) is the solution computed by MINRES-L. Then (11) applied to yields the bounds In this formula, H 2 is shorthand for the coefficient matrix of (10). We can extract another equation from (16) and (17); in particular, if we multiply (16) by multiply (17) by ffi 1 and then add, we eliminate the terms involving - v: Let x be the exact solution to the WLS problem. The last two terms of this equation can be replaced with terms involving x by using (7). Interchanging the left- and right-hand sides yields The goal is to derive an accuracy bound like (5) from (18) and (19). We start by bounding the quantity on the right-hand side of (18). Note that can be bounded by because the largest entries in D 1 ; D 2 are bounded by -. We can bound kxk by -Akbk using Theorem 1. Next we turn to bounding kvk in (18). Recall that, as mentioned in the preceding section, v is not uniquely determined, but MINRES will find the minimum-norm v satisfying (10). Recall that v is determined by the constraint One way to pick such a v is to make it minimize kA 2 vk subject to the above constraint. In this case, v is a layered least-squares solution with right-hand side data (b yields the bound for this choice of v. (The factor - can be improved to - -A by using the analysis of Gonzaga and Lara [8].) Combining the x and v contributions means that we have bounded the right-hand side of (18); let us rewrite (18) with the new bound: Next, we write new equations for r . Observe that r 1 lies in the range of A Tand A T, so we can find h 1 satisfying Similarly, by (17) there exists h 2 satisfying By applying (13) to r 1 and r 2 separately, with "A T c" in the lemma taken to be first r 1 and then r 2 , we conclude from (21) and (22) that Substituting (21) and (22) into (19) yields Notice (by analogy with (7)) that the preceding equation is exactly a weighted least-squares computation where the "unknown" is - and the right-hand side data is Thus, by Theorem 1, We now build a chain of inequalities: the right-hand side of the preceding inequality is bounded by (23) and (24), and the right-hand side of (23) and (24) is bounded by (20). Combining all of this yields To obtain the preceding inequality, we used the facts that assumption) and that kdiag(D \Gamma1 by assumption, since the smallest entry in each D i is taken to be 1). Thus, we have an error bound of the form (5) as desired; in particular, there is no dependence of the error bound on ffi 2 =ffi 1 . Note that this bound depends on -. Recall that - is defined to be the maximum entry in D and is assumed to be small. Indeed, as noted in Section 2, we can always assume that if we are willing to divide the problem into many layers. 6 MINRES-L for p Layers In this section we present the MINRES-L algorithm for the p-layered WLS problem. The algorithm is the application of MINRES to the symmetric linear system H p is a square matrix of size (1 1)=2)n \Theta (1 +p(p \Gamma 1)=2)n, c p is a vector of that order, and w is the vector of unknowns. Matrix H p is partitioned into (1 blocks each of size n \Theta n. Vectors c p and w are similarly partitioned. The WLS solution vector is the first subvector of w. In more detail, the vector w is composed of x concatenated with p(p\Gamma1)=2 n-vectors that we denote v i;j , where i lies in lies in Recall that the p-layered WLS problem may be written Let x be the solution to this equation. Then we see from this equation that A T lies in the span of [A T Therefore, there exists a solution [v to the equation A T This equation is the first block-row of H In other words, the first block row of H p contains one copy of each of the matrices A T and the first block of c p is A T In general, the (p 1)th block-row of H the equation A T A T A T This completes the description of block-rows . We now establish some properties of these block-rows, and we postpone the description of block-rows Lemma Suppose w is a solution to the linear equation (28) for each denotes the concatenation of x and all of the v i;j 's. Then x is the solution to the WLS problem (26). Proof. For each i, multiply (28) by ffi i and then sum all p equations obtained in this manner. Observe that all the v i;j terms cancel out and we end up exactly with (26). We also need the converse to be true. Lemma 3 Suppose x is the solution to (26). Then there exist vectors v i;j such that (28) is satisfied for each Proof. The proof is by induction on (decreasing) We assume that we have already determined v i;j for all that (28) is satisfied for now we must determine v k;j for for the particular value k. The base case of the induction is that we can select v to satisfy (28) in the case lies in the range of [A T because of (26). Now for the induction case of k ! p. Rewrite (28) for the case multiply through by A T Recall that our goal is to choose v k;j for to make this equation valid. Multiply (28) for each and add this to (29). After rearranging and summations and cancelling common terms on the left-hand side, we end up with Dividing through by ffi k and separating out the v k;j terms from the second summation yields: A T A T A T But from (26) we know that lies in the range of the rightmost summation of (31) also lies in the same range. Therefore, there exist v k;j for But then these same choices will make (29) valid because the algebraic steps used to derive (31) from (29) can be reversed. This proves the lemma. Note that the preceding proof actually demonstrates a strengthened version of the lemma. The strengthened version states that if we are given x satisfying (26) and, for some k, vectors v i;j for k - that satisfy (28) for all then we can extend the given data to a solution of (28) for all strengthened version is needed below. We now explain the remaining p(p \Gamma 1)=2 block-rows of H p . These rows exist solely for the purpose of making H p symmetric. First, we have to order the variables and equations correctly. The variables will be listed in the order (x; v The first equations will be listed in the order (28) for 1. This means that the first p rows of H p have the format [S and T p is a p \Theta (p \Gamma 1)(p \Gamma 2)=2 matrix. Furthermore, it is easily checked that S p is symmetric: its first block-row and first block-column both consist of A T listed in the order 1)st entry of its main diagonal is \Gamma(ffi p =ffi i )A T all its other blocks are zeros. Then we define H p to be We define c p as A T A T where there are p(p \Gamma 1)=2 blocks of zeros. For example, the following linear system is H 3 A T A T A T A T x A T A T We now must consider whether has any solutions; in particular, we must demonstrate that the new group of equations T T with the first p rows. Here w 0 denotes the first p blocks of w, that is, Studying the structure of T p , we see that there are indexed by (i; correspondence with the columns of T p , which correspond to variables v i;j for in that range). The row indexed by (i; j) has exactly two nonzero block entries that yield the equation A T A T Our task is therefore to show that we can simultaneously satisfy (28) for Our approach is to select the v p;j 's in the order v In particular, assuming v are already selected, we define v p;j to be any solution to A T The following lemma shows that this linear system is consistent. Lemma 4 If the v p;j 's are chosen in reverse order to satisfy (33), then at each step the linear system is consistent, and (32) is satisfied. Proof. The proof is by reverse induction on j. The base case is which case (33) has a solution because, as noted above, A T lies in the span of [A T In the case vacuously true: there is no i in the specified range. Now consider the case any i in the range Start with the version of (33) satisfied by v p;i , which holds by the induction hypothesis: A T Move the terms of the first summation to the right-hand side: A T A T A T A T A T The second line was obtained from the first by applying (32) inductively (with "j" in (32) taken to be k). The third line was obtained by merging the two summations on the right. But notice that the preceding equation means that v p;i satisfies the same linear system as v p;j , that is (33), except with the right-hand side scaled by This proves that (33) is consistent for the j case since we have constructed a solution to it. Although this linear system does not necessarily have a unique solution, a linear system of the form A T uniquely determines Ax. Thus, we have also proved that for all This result is actually a strengthening of (32) for j; for that equation we need only the specific case of The reader may have noticed that the preceding proof is apparently too complicated and that we could establish the result more simply by solving for v p;p\Gamma1 in (33) with setting v 2. This simpler approach does not yield the bounds on kv p;j k needed in the next section. This proof shows that the above method for selecting v consistent and satisfies (32). We also see that (27) is satisfied; this follows immediately from taking (33). To complete the proof that there is a solution to H p need only verify (28) in the case But recall from the proof of Lemma 3 that the remaining v i;j 's for can be determined sequentially by using the construction in the proof. Thus, the arguments of this section have established the following theorem. Theorem 2 There exists at least one solution w to H p more, any such solution has as its first n entries the vector x that solves (26). 7 Convergence Analysis for p Layers The convergence analysis for p layers follows the same basic outline as the convergence analysis for two layers. In particular, we use (11) as the starting point for the error analysis. Observe that (11) has the norm of the true solution on the right-hand side. Thus, to apply that bound, we must get a norm bound on v i;j for all We start with bounds on v p;j for Apply Lemma 1 to (33) in the case In the lemma, take - As noted above, A T lies in the range of [A T so (33) is consistent. The right-hand side of (33) in the case has the form A T c with Note that kD p (b 1)kbk. Thus, from (12), -A To derive the third line from the second, we used the facts that kD \Gamma1 for each i and Now we use the same line of reasoning to get a bound on v p;p\Gamma2 based on (33) for the case 2. In this case, the right-hand side of (33) has the Thus, kck is bounded by ffi p\Gamma2 (- A which is at most We continue this argument inductively. Each time the bound grows by a factor 2- A to take into account the fact that v p;i appears on the right-hand side for the equation determining v p;i\Gamma1 . In the end we conclude that Next we must bound v i;j for 1 These vectors are determined by (28). We can find a solution to (28) by first solving for z i , where - is already known to be consistent. Furthermore, in the preceding equation. We set v Using (12), we conclude that for each We now claim that p. This is proved by induction on decreasing i using recurrence (36). The on the right-hand side of (36) is bounded by (35), and the remaining terms are bounded by the induction hypothesis. We omit the details. For the right-hand side of (11) we need a bound on kv i;j k. Note that up to now we have not uniquely determined v i;j itself. Recall that in each case Lemma 1 was used to bound kA k v i;j k. We can force unique determination by choosing the v i;j as in the proof of Lemma 1, yielding by (14). Note that MINRES does not necessarily select this v i;j , but because of its minimization property (that is, Theorem 2.4 of Brown and Walker [4] described in Section 4), it will select v i;j whose norm is no larger than in the preceding bound. We now can apply (11). The other factor on the right-hand side, namely, easily seen to be bounded by p w be the solution computed by MINRES-L, and let substituting (37) on the right-hand side of (11) yields Let r be the first p block-entries of r. Note that r j must lie in the span of [A in order for the equation H to have a solution, because it can be seen from (28) that the (p 1)st block-row of us find h i that solves r for each i. By (13) we know that k[A Let - x be the first n entries of - that is, the computed WLS solution. If we multiply the (p \Gamma i+1)st block row of H and add these p rows, we obtain A T The third line was obtained from the second by interchanging the order of summation. Thus, we see from the third line above that - solves a WLS problem in which the ith entry of the data vector is A i in this range, we conclude that the data vector is bounded in norm by k. Then Theorem 1 implies that A Substituting (38) yields A \Delta (4- A ) This is a bound of the form (5) as desired. Computational Experiments In this section we present computational experiments on MINRES-L and CGNR to compare their accuracy and efficiency. The first few tests involve a small node-arc adjacency matrix. The remaining tests are on matrices arising in linear programming and boundary value problems. All tests were conducted in Matlab 4.2 running on an Intel Pentium under Microsoft Windows NT 4.0. Matlab is a software package and programming language for numerical computation written by The Mathworks, Inc. All computations are in IEEE double precision with machine epsilon approximately 2:2 Matlab sparse matrix operations were used in all tests. Our implementation of CGNR is based on CGLS1 as in (3.2) of Bj-orck, Elfving and Strako-s [2]. These authors conclude that CGLS1 is a good way to organize CGNR. There are two matrix-vector products per CGLS1 it- eration, one with matrix A T D 1=2 and one with D 1=2 A. In our implemen- tation, the CGNR iteration terminates when the scaled computed residual ks k k=kA T Dbk drops below 10 \Gamma13 . Our implementation of MINRES is based on [14], except Givens rotations were used instead of 2 \Theta 2 Householder matrices (so that there are some inconsequential sign differences). The MINRES-L iteration terminates when the scaled computed residual The first matrix A used in the following tests is the reduced node-arc adjacency matrix of the graph depicted in Figure 1. A "node-arc adjacency" matrix contains one column for each node of a graph and one row for each edge. Each row contains exactly two nonzero entries, a +1 and a \Gamma1 in the columns corresponding to the endpoints of the edge. (The choice of which endpoint is assigned +1 and which is assigned \Gamma1 induces an orientation on the edge, but often this orientation is irrelevant for the application.) A reduced node-arc incidence (RNAI) matrix is obtained from a node-arc incidence matrix by deleting one column. RNAI matrices arise in the analysis of an electrical network with batteries and resistors; see [23]. They also arise in network flow problems. In the case of Figure 1, the column corresponding to Figure 1: An based on this graph was used for the first group of tests. The column corresponding to the top node is deleted. Edges marked with heavy lines are weighted 1, and edges marked with light lines are weighted varies from test to test. the top node was deleted. Thus, A is an 9 matrix. It is well known that the RNAI matrix for a connected graph always has full rank. RNAI matrices are known to have small values of -A and - -A [23]. In all these tests, the weight matrix has two layers. We took vary from experiment to experiment. The rows of A in correspondence with D 2 are drawn as thinner lines in Figure 1. Finally, the right-hand side b was chosen to be the first prime numbers. The results are displayed in Table 1, and the cases when are plotted in Figure 2. The scaled error that is tabulated and plotted in all cases is defined to be k- x \Gamma xk=kbk. We choose this particular scaling for the error because our goal is to investigate stability bound (5). The true solution x is computed using the COD method [12]. Note that the accuracy of CGNR decays as ffi 2 gets smaller, whereas MINRES-L's accuracy stays constant. MINRES-L requires many more flops than CGNR because the system matrix is larger. The running time of CGNR is about the same for the first four rows of the table as the ill-conditioning increases. In the last two rows the running time of CGNR drops because the matrix A T DA masquerades as a low-rank matrix for small values of ffi 2 , causing early termination of the Lanczos process. Besides returning an inaccurate solution, CGNR has the additional difficulty that its residual (the quantity normally measured in practical use of this algorithm) does not reflect the forward error, so there is no simple way Table 1: Behavior of the two-layered MINRES-L algorithm compared to CGNR for decreasing values of ffi 2 . The error reported is the scaled error defined in the text. Note that the CG accuracy degrades while the MINRES- accuracy stays about the same. MINRES-L MINRES-L MINRES-L CGNR CGNR CGNR Iterations Error Flops Iterations Error to determine whether CGNR is computing good answers. In contrast, the error and residual in MINRES-L are closely correlated. This correlation is predicted by our theory. The next computational test involved a larger matrix A taken from the Netlib linear programming test set, namely, the matrix in problem AFIRO, which is 51 \Theta 27. We used a matrix D with 1's in its first 27 diagonal positions its remaining 24 positions (i.e., D The right-hand side vector b was chosen to contain the first primes. MINRES-L required 137 iterations and 250 kflops and yielded a solution - x with scaled error 3:0 with respect to the true solution computed by the COD method. For this matrix, -A and - -A are not known. CGNR on this problem required 69 iterations and 61 kflops and returned an answer with scaled error 2:2 . The convergence plots are depicted in Figure 3. The excessive number of iterations required by MINRES is apparently caused by a loss of orthogonality in the Lanczos process. To verify this hypothesis, we ran GMRES on the same layered matrix. GMRES [19] on a symmetric matrix is equivalent to MINRES with full reorthogonalization. (In exact arithmetic the two algorithms are identical.) We call this algorithm GMRES-L. The same termination tests were used. The result is depicted in Figure 4. In this case, GMRES-L ran for 50 iterations (fewer than (1 returned a more accurate answer, one with forward error . However, the number of flops was higher, 350 k, because of the scaled error scaled residual scaled error scaled residual Figure 2: Convergence behavior of CGNR and MINRES-L for the RNAI test case. The plots are for In these plots and all that follow, the x-axis is the iteration number. For both algorithms the computed (i.e., recursively updated) residual is plotted rather than the true residual. Other experiments (not reported here) indicate that these are usually indistinguishable. The \Theta on the y-axis indicates the cutoff below which the CGNR scaled residual must drop in order for (11) to be true . The ffi on the y-axis is the analog for MINRES-L. Figure 3: Convergence behavior of CGNR and MINRES-L for AFIRO. The curves are labeled as in Figure 2. Gram-Schmidt process in the GMRES main loop. The next computational test involves a larger matrix A arising from finite-element analysis. The application is the solution of the boundary value problem r \Delta on the polygonal domain depicted in Figure 5 with Dirichlet boundary conditions. The conductivity field c is 1 on the outer part of the domain and is 10 12 on the darker triangles. As discussed in [24], this type of problem gives rise to a weighted least-squares problem in which A encodes information about the geometry and D encodes the ill-conditioned conductivity field. The values of -A and - -A for this matrix are not known, although bounds are known for variants of these parameters. The particular matrix A is 652 \Theta 136. The right-hand side vector b was chosen according to the Dirichlet boundary conditions described in [24]. The MINRES-L method for this problem gave scaled error of 1:3 iterations and 6.5 mflops. To compute the true solution, we used the NSHI method in [24]. In this case, surprisingly, CGNR gave almost as accurate an answer, but the termination test was never activated. (We cut off CGNR after 10n iterations.) The residual of CGNR is quite oscillatory as depicted in Figure 6. In the finite-element literature, CGNR would be referred to as conjugate gradient on the assembled stiffness matrix, which is A T DA. A cause of this odd behavior of CGNR is as follows. Note that the region of high conductivity is not incident on the boundary of the domain so Figure 4: Convergence behavior of GMRES-L (- and \Delta \Delta Figure 5: Domain and finite element mesh used for the finite element exper- iment. Conductivity in the dark triangles is 10 12 and in the light triangles is Figure Convergence of CGNR and MINRES-L for the finite element test problem. The curves are labeled as in Figure 2. Thus, A T starts from a right-hand side that is already almost zero. Furthermore, this right-hand side is nearly orthogonal to the span of A T dominates the stiffness matrix A T DA. Thus, CGNR has trouble making progress. The surprisingly accurate answer from CGNR in this example is not so useful in practice because there is no apparent way to detect that convergence is underway. The final test is a three-layered problem based on the matrix A from ADLITTLE of the Netlib test set, a 138 \Theta 56 matrix. Matrix D has as its first 28 diagonal entries 1, its next 28 diagonal entries 10 \Gamma8 and its last 82 entries . The right-hand side vector is the first 138 prime numbers. The convergence is depicted in Figure 7. As expected, the scaled error of MINRES-L decreased to while the scaled error of CGNR was 0:3. Note the excessive number of iterations required by MINRES-L. Again, this is apparently due to loss of orthogonality because the number of iterations was only 118 for GMRES-L to achieve a scaled error of 9:4 In fact, for this test GMRES-L was more efficient than MINRES-L in terms of flop count. In most cases we see that the MINRES-L algorithm performs essentially as expected, except for the two cases in which a loss of orthogonality causes many more iterations than expected. In every case, MINRES-L's running time is higher than CGNR's, but CGNR can produce bad solutions as measured by forward error. Figure 7: Convergence of CGNR and MINRES-L for ADLITTLE. The curves are labeled as in Figure 2. Note the excessive number of iterations for MINRES-L caused by a loss of orthogonality. 9 An Issue for Interior-Point Methods In this section we describe an issue that arises when using the MINRES-L algorithm in an interior-point method for linear programming. Full consideration of this matter is postponed to future work. It is well known that the system of equations for the Newton step in an interior-point method can be expressed as a weighted least-squares problem. To be precise, consider the linear programming problem subject to A T whose dual is subject to Ay (which is standard form, except we have transposed A to be consistent with least-squares notation). A primal-dual method starting at a feasible interior point problem computes an update \Deltay to y satisfying is an algorithm-dependent parameter usually in [0; 1], - is the duality gap, and e is the vector of all 1's. See Wright [26]. Since (40) has the form of a WLS problem, we can obtain \Deltay using the MINRES-L algorithm. One way to compute \Deltas is via \Deltas := \GammaA\Deltay. This method is not stable because \Deltas has very small entries in positions where s has very small en- these small entries must be computed accurately with respect to the corresponding entry of s. In contrast, the error in all components of \Deltas arising from the product A\Deltay is on the order of ffl \Delta ksk (where ffl is machine- epsilon). A direct method for accurately computing all components of \Deltas was proposed by Hough [11], who obtains a bound of the form \Deltas for each i. We will consider methods for extending MINRES-L to accurate computation of \Deltas in future work. As noted by Hough, \Deltax is easily computed from \Deltas with a similar accuracy bound assuming \Deltas satisfies (41). Conclusions We have presented an iterative algorithm MINRES-L for solving weighted least squares. Theory and computational experiments indicate that the method is more accurate than CGNR when the weight matrix is highly ill- conditioned. This work raises a number of questions. 1. Is there an iterative method that does not require the layering assumption 2. If layering is indeed required, can we get a more parsimonious layered linear system when p - 3? In particular, is there a 3n \Theta 3n system of equations with all the desired properties for the 3-layered case (instead of the 4n \Theta 4n system that we presented)? 3. What is the best way to handle loss of orthogonality in MINRES that was observed in Section 8? 4. Can this work be extended to stable computation of \Deltax and \Deltas in an interior-point method? (This question was raised in Section 9.) 5. What about preconditioning? In most of our computational tests, we ran both MINRES and CG for more than n iterations because our aim was to compute the solution vector as accurately as possible. In prac- tice, one hopes for convergence in much fewer than n iterations. What are techniques for preconditioning WLS problems? Note that the analysis of MINRES-L's accuracy in Section 5 and Section 7 presupposes that no preconditioner is used. Acknowledgments We had helpful discussions of this work with Anne Greenbaum and Mike Overton of NYU; Roland Freund, David Gay, and Margaret Wright of Bell Labs; Patty Hough of Sandia; Rich Lehoucq and Steve Wright of Argonne; Homer Walker of Utah State; and Zden-ek Strako-s of the Czech Academy of Sciences. We thank Patty Hough and Gail Pieper for carefully reading an earlier draft of this paper. In addition, we received the Netlib linear programming test cases in Matlab format from Patty Hough. --R Numerical methods for least squares problems. Stability of conjugate gradient and Lanczos methods for linear least squares problems. Iterative methods for weighted least squares. GMRES on (nearly) singular systems. Numerical stability of GMRES. On linear least-squares problems with diagonally dominant weight matrices Matrix Computations A note on properties of condition numbers. Estimating the attainable accuracy of recursively computed residual methods. Conjugate gradient type methods for ill-posed problems Stable computation of search directions for near-degenerate linear programming problems Complete orthogonal decomposition for weighted least squares. Solving Least Squares Problems. Solution of sparse indefinite systems of linear equations. LSQR: An algorithm for sparse linear equations and sparse least squares. Practical use of the symmetric Lanczos process with re- orthogonalization The Lanczos algorithm with selective reorthog- onalization Iterative methods for sparse linear systems. GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems. On scaled projections and pseudoinverses. A framework for equilibrium equations. A Dantzig-Wolfe-like variant of Karmarkar's interior-point linear programming algorithm Stable numerical algorithms for equilibrium systems. Stable finite elements for problems with wild coefficients. A primal-dual interior point method whose running time depends only on the constraint matrix --TR

krylov-space;iterative method;conjugate gradient;MINRES;weighted least squares;achievable accuracy

587793

Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems.

Numerical solution of ill-posed problems is often accomplished by discretization (projection onto a finite dimensional subspace) followed by regularization. If the discrete problem has high dimension, though, typically we compute an approximate solution by projecting the discrete problem onto an even smaller dimensional space, via iterative methods based on Krylov subspaces. In this work we present a common framework for efficient algorithms that regularize after this second projection rather than before it. We show that determining regularization parameters based on the final projected problem rather than on the original discretization has firmer justification and often involves less computational expense. We prove some results on the approximate equivalence of this approach to other forms of regularization, and we present numerical examples.

Introduction . Linear, discrete ill-posed problems of the form (1) or min x equivalently, A (2) arise, for example, from the discretization of first-kind Fredholm integral equations and occur in a variety of applications. We shall assume that the full-rank matrix A in (2) and in (1). In discrete ill-posed problems, A is ill-conditioned and there is no gap in the singular value spectrum. Typically, the right hand side b contains noise due to measurement and/or approximation error. This noise, in combination with the ill-conditioning of A, means that the exact solution of (1) or (2) has little relationship to the noise-free solution and is worthless. Instead, we use a regularization method to determine a solution that approximates the noise-free solution. Regularization methods replace the original operator by a better-conditioned but related one in order to diminish the effects of noise in the data and produce a regularized solution to the original problem. Sometimes this regularized problem is too large to solve exactly. In that case, we typically compute an approximate solution by projection onto an even smaller dimensional space, perhaps via iterative methods based on Krylov subspaces. The conditioning of the new problem is controlled by one or more regularization parameters specific to the method. A large regularization parameter yields a new well-conditioned problem, but its solution may be far from the noise-free solution since the new operator is a poor approximation to A. A small regularization parameter generally yields a solution very close to the noise-contaminated exact solution of (1) or (2), and hence its distance from the noise-free solution also can be large. Thus, This work was supported by the National Science Foundation under Grants CCR 95-03126 and CCR-97-32022 and by the Army Research Office, MURI Grant DAAG55-97-1-0013. y Dept. of Computer and Electrical Engineering, Northeastern University, Boston, MA 02115 z Dept. of Computer Science and Institute for Advanced Computer Studies, University of Mary- land, College Park, MD 20742 (oleary@cs.umd.edu). a key issue in regularization methods is to choose a regularization parameter that balances the error due to noise with the error due to regularization. A wise choice of regularization parameter is obviously crucial to obtaining useful approximate solutions to ill-posed problems. For problems small enough that a rank- revealing factorization or singular value decomposition of A can be computed, there are well-studied techniques for computing a good regularization parameter. These techniques include the Discrepancy Principle [8], generalized cross-validation (GCV) [9], and the L-curve [15]. For larger problems treated by iterative methods, though, the parameter choice is much less understood. If regularization is applied to the projected problem that is generated by the iterative method, then there are essentially two regularization parameters: one for the standard regularization algorithms, such as Tikhonov or truncated SVD, and one controlling the number of iterations taken. One subtle issue is that the standard regularization parameter that is correct for the discretized problem may not be the optimal one for the lower-dimensional problem actually solved by the iteration, and this observation leads to the research discussed in this paper. At first glance, there can appear to be a lot of work associated with the selection of a good regularization parameter, and many algorithms proposed in the literature are needlessly complicated. But by regularizing after projection by the iterative method, so that we are regularizing the lower dimensional problem that is actually being solved, much of this difficulty vanishes. The purpose of this paper is to present parameter selection techniques designed to reduce the regularization work for iterative methods such as Krylov subspace tech- niques. Our paper is organized as follows. In x2, we will give an overview of the regularization methods we will be considering, and we follow up in x3 by surveying some methods for choosing the corresponding regularization parameters. In x4, we show how parameter selection techniques for the original problem can be applied instead to a projected problem obtained from an iterative method, greatly reducing the cost without much degradation in the solution. We give experimental results in x5 and conclusions and future work in x6. 2. Regularization background. In the following we shall assume that e, where b true denotes the unperturbed data vector and e denotes zero-mean white noise. We will also assume that b true satisfies the discrete Picard condition; that is, the spectral coefficients of b true decay faster, on average, than the singular values. Under these assumptions, it is easy to see why the exact solution to (1) or (2) is hopelessly contaminated by noise. Let " denote the singular value decomposition of A, where the columns of " U and " V are the singular vectors, and the singular values are ordered as oe 1 oe . Then the solution to (1) or (2) is given by " " As a consequence of the white noise assumption, j"u i ej is roughly constant for all i, while the discrete Picard condition guarantees that j"u than oe i does . The matrix A is ill-conditioned, so small singular values magnify the corresponding coefficients " e in the second sum, and it is this large contribution of noise from the approximate null space of A that renders the exact solution x defined in (3) worthless. The following regularization methods try in different ways to lessen the contribution of noise to the solution. For further information on these methods, see, for example, [17]. 2.1. Tikhonov regularization. One of the most common methods of regularization is Tikhonov regularization [34]. In this method, the problem (1) or (2) is replaced with the problem of solving where L denotes a matrix, often chosen to be the identity matrix I or a discrete derivative operator, and is a positive scalar regularization parameter. For ease in notation, we will assume that Solving (4) is equivalent to solving In analogy with (3) we have In this solution, the contributions from noise components " u e for values of oe are much smaller than they are in (3), and thus x can be closer to the noise-free solution than x is. If is too large, however, A I is very far from the original operator A A, and x is very far from x true , the solution to (2) when Conversely, if is too small, the singular values of the new operator A A are close to those of A A; thus x x, so small singular values again greatly magnify noise components. 2.2. Truncated SVD. In the truncated SVD method of regularization, the regularized solution is chosen simply by truncating the expansion in (3) as " Here the regularization parameter is ', the number of terms to be dropped from the sum. Observe that if ' is small, very few terms are dropped from the sum, so x ' resembles x in that the effects of noise are large. If ' is too large, however, important information could be lost; such is the case if " An alternative, yet related, approach to TSVD is an approach introduced by Rust [31] where the truncation strategy is based on the value of each spectral coefficient itself. The strategy is to include in the sum (3) only those terms corresponding to a spectral coefficient " b whose magnitude is greater than or equal to some tolerance ae, which can be regarded as the regularization parameter. 2.3. Projection and iterative methods. Solving (5) or (7) can be impractical if n is large, but fortunately, regularization can be achieved through projection onto a subspace; see, for example, [7]. The truncated SVD is an example of one such projection: the solution is constrained to lie in the subspace spanned by the singular vectors corresponding to the largest values. Other projections can be more economical. In general, we constrain our regularized solution to lie in some k-dimensional subspace of C n , spanned by the columns of an n \Theta k matrix Q (k) . For example, we choose x (k) min kAQ or equivalently The idea is that with an appropriately chosen subspace, the operator (Q (k) ) A AQ (k) will be better conditioned than the original operator and hence that x (k) reg will approximate x true well on that subspace. This projection is often achieved through the use of iterative methods such as conjugate gradients, GMRES, QMR, and other Krylov subspace methods. The matrix contains orthonormal columns generated via a Lanczos tridiagonalization or bidiagonalization process [27, 1]. In this case, Q (k) is a basis for some k-dimensional Krylov subspace (i.e., the subspace K k (c; K) spanned by the vectors c; for some matrix K and vector c). The regularized solutions x (k) reg are generated iteratively as the subspaces are built. Krylov subspace algorithms such as CG, CGLS, GMRES, and LSQR tend to produce, at early iterations, solutions that resemble x true in the subspace spanned by (right) singular vectors of A corresponding to the largest singular values. At later iterations, however, these methods start to reconstruct increasing amounts of noise into the solution. This is due to the fact that for large k, the operator (Q (k) ) A AQ (k) approaches the ill-conditioned operator A A. There- fore, the choice of the regularization parameter k, the stopping point for the iteration and the dimension of the subspace, is very important. 1 2.4. Hybrid methods: projection plus regularization. Another important family of regularization methods, often referred to as hybrid methods [17], was introduced by O'Leary and Simmons [27]. These methods combine a projection method with a direct regularization method such as TSVD or Tikhonov regularization. The problem is projected onto a particular subspace of dimension k, but typically the restricted operator in (9) is still ill-conditioned. Therefore, a regularization method is applied to the projected problem. Since the dimension k is usually small relative to n, regularization of the restricted problem is much less expensive. Yet, with an appropriately chosen subspace, the end results can be very similar to those achieved by applying the same direct regularization technique to the original problem. We will become more precise about how "similar" the solutions are in x4.5. Because the projected problems are usually generated iteratively by a Lanczos method, this approach is useful when A is sparse or structured in such a way that matrix-vector products can be handled efficiently with minimal storage. 3. Existing parameter selection methods. In this section, we discuss a sampling of the parameter selection techniques that have been proposed in the literature. They differ in the amount of a priori information required as well as in the decision criteria. 3.1. The Discrepancy Principle. If some extra information is available - for example, an estimate of the variance of the noise vector e - then the regularization parameter can be chosen rather easily. Morozov's Discrepancy Principle [25] says that if ffi is the expected value of kek 2 , then the regularization parameter should be chosen so that the norm of the residual corresponding to the regularized solution x reg is that is, Usually, small values of the regularization parameter correspond to a closer solution to the noisy equation, but despite this, we will call k, rather than 1=k, the regularization parameter. x l ||Fig. 1. Example of a typical L-curve. This particular L-curve corresponds to applying Tikhonov regularization to the problem in Example 2 predetermined real number. Note that as Other methods based on knowledge of the variance are given, for example, in [12, 5]. 3.2. Generalized Cross-Validation. The Generalized Cross-Validation (GCV) parameter selection method does not depend on a priori knowledge about the noise variance. This idea of Golub, Heath, and Wahba [9] is to find the parameter that minimizes the GCV functional denotes the matrix that maps the right hand side b onto the regularized solution x . In Tikhonov regularization, for example, A ] is GCV chooses a regularization parameter that is not too dependent on any one data measurement [11, 12.1.3]. 3.3. The L-Curve. One way to visualize the tradeoff between regularization error and error due to noise is to plot the norm of the regularized solution versus the corresponding residual norm for each of a set of regularization parameter values. The result is the L-curve, introduced by Lawson and popularized by Hansen [15]. Figure 1 for a typical example. As the regularization parameter increases, noise is damped, so that the norm of the solution decreases while the residual increases. Intuitively, the best regularization parameter should lie on the corner of the L-curve, since for values higher than this, the residual increases without reducing the norm of the solution much, while for values smaller than this, the norm of the solution increases rapidly without much decrease in residual. In practice, only a few points on the L-curve are computed and the corner is located by approximate methods, estimating the point of maximum curvature [19]. Like GCV, this method of determining a regularization parameter does not depend on specific knowledge about the noise vector. 3.4. Disadvantages of these parameter choice algorithms. The appropriate choice of regularization parameter - especially for projection algorithms - is a difficult problem, and each method has severe flaws. Basic cost Added Cost Disc. GCV L-curve Rust's TSVD O(mn 2 ) O(m log m) O(m log m) O(m log m) Projection Table Summary of additional flops needed to compute the regularization parameter for each four regularization methods with various parameter selection techniques. Notation: q is the cost of multiplication of a vector by A. p is the number of discrete parameters that must be tried; k is the dimension of the projection. m and n are problem dimensions. The Discrepancy Principle is convergent as the noise goes to zero, but it relies on knowing information that is often unavailable or incorrectly estimated. Even with a correct estimate of the variance, the solutions tend to be oversmoothed [20, pg. 96] (see also the discussion in x6.1 of [15]). One noted difficulty with GCV is that G can have a very flat minimum, making it difficult to determine the optimal numerically [35]. The L-curve is usually more tractable numerically, but its limiting properties are nonideal. The solution estimates fail to converge to the true solution as n !1 [36] or as the error norm goes to zero [6]. All methods that assume no knowledge of the error norm - including GCV - have this latter property [6]. For further discussion and references about parameter choice methods, see [5, 17]. The cost of these methods is tabulated in Table 1. 3.5. Previous work on parameter choice for hybrid methods. At first glance, it appears that for Tikhonov regularization, multiple systems of the form (5) must be solved in order to evaluate candidate values of for the Discrepancy Principle or the L-curve. Techniques have been suggested in the literature for solving these systems using projection methods. Chan and Ng [4], for example, note that the systems involve the closely related matrices matrices suggest solving the systems simultaneously using a Galerkin projection method on a sequence of "seed" systems. Although this is economical in storage, it can be unnecessarily expensive in time because they do not exploit the fact that for each fixed k, the Krylov subspace K k the same for all values of . Frommer and Maass [8] propose two algorithms for approximating the that satisfies the Discrepancy Principle (10). The first is a "truncated cg" approach in which they use conjugate gradients to solve k systems of the form (5), truncating the iterative process early for large and using previous solutions as starting guesses for later problems. Like Chan and Ng, this algorithm does not exploit any of the redundancy in generating the Krylov-subspaces for each i . The second method they propose, however, does exploit the redundancy so that the CG iterates for all k systems can be updated simultaneously with no extra matrix-vector products. They stop their "shifted cg" algorithm when kAx for one of their values. Thus the number of matrix-vector products required is twice the number of iterations for this particular system to converge. We note that while the algorithms we propose in x4 for finding a good value of are based on the same key observation regarding the Krylov subspace, our methods will usually require less work than the shifted cg algorithm. Calvetti, Golub, and Reichel [3] compute upper and lower bounds on the L-curve generated by the matrices C() using a Lanczos bidiagonalization process. From this, they approximate the best parameter for Tikhonov regularization without projection. In x4, we choose instead to approximate the best parameter for Tikhonov regularization on the projected problem, since this is the approximation to the continuous problem that is actually being used. Kaufman and Neumaier [21] suggest an envelope guided conjugate gradient approach for the Tikhonov L-curve problem. Their method is more complicated than the methods we propose because they maintain nonnegativity constraints on the variables. Substantial work has also been done on TSVD regularization of the projected problems. Bjorck, Grimme, and van Dooren [2] use GCV to determine the truncation point for the projected SVD. Their emphasis is on stable ways to maintain an accurate factorization when many iterations are needed, and they use full reorthogonalization and implicit restart strategies. O'Leary and Simmons [27] take a somewhat different viewpoint that the problem should be preconditioned appropriately so that a massive number of iterations is unnecessary. That viewpoint is echoed in this current work, so we implicitly assume that the problem has been left-preconditioned or "filtered" [27]. For example, in place of (4), we solve min x 2for a square preconditioner M . See [14, 26, 24, 23] for preconditioners appropriate for certain types of ill-posed problems. Note that we could alternately have considered right preconditioning, which amounts to solving, in the Tikhonov case, min y A I for y then setting . Note that either left or right preconditioning effectively changes the balance between the two terms in the minimization. 4. Regularizing the projected problem. In this section we develop nine approaches to regularization using Krylov methods. Many Krylov methods have been proposed; for ease of exposition we focus on just two of these: the LSQR algorithm of Paige and Saunders [29] and the GMRES algorithm of Saad and Schultz [33]. The LSQR algorithm of Paige and Saunders [29] iteratively computes the bidiag- onalization introduced by Golub and Kahan [10]. Given a vector b, the algorithm is as follows [29, Alg. Bidiag 1]: Compute a scalar fi 1 and a vector u 1 of length one so that fi 1 Similarly, determine ff 1 and v 1 so that ff 1 For where the non-negative scalars ff i+1 and fi i+1 are chosen so that u i+1 and v i+1 have length one. End for The vectors u are called the left and right Lanczos vectors respectively. The algorithm can be rewritten in matrix form by first defining the matrices . ff k denoting the ith unit vector, the following relations can be established: A T U where the subscript on I denotes the dimension of the identity. Now suppose we want to solve min where S denotes the k-dimensional subspace spanned by the first k Lanczos vectors . The solution we seek is of the form x vector y (k) of length k. to be the corresponding residual. From the relations above, observe that in exact arithmetic Since U k+1 has, in exact arithmetic, orthonormal columns, we have Therefore, the projected problem we wish to solve is min y Solving this minimization problem is equivalent to solving the normal equations involving the bidiagonal matrix: Typically k is small, so reorthogonalization to combat the effects of inexact arithmetic might or might not be necessary. The matrix B k may be ill-conditioned because some of its singular values approximate some of the small singular values of A. Therefore solving the projected problem might not yield a good solution y (k) . However, we can use any of the methods of Section 3 to regularize this projected problem; we discuss options in detail below. As alluded to in x4, the idea is to generate y (k) reg , the regularized solution to (18), and then to compute a regularized solution to (16) as reg . If we used the algorithm GMRES instead of LSQR, we would derive similar relations. Here, though, the U and V matrices are identical and the B matrix is upper Hessenberg rather than bidiagonal. Conjugate gradients would yield similar relationships. For cost comparisons for these methods, see Tables 1 and 2. Storage comparisons are given in Tables 3 and 4. 4.1. Regularization by projection. As mentioned earlier, if we terminate the iteration after k steps, we have projected the solution onto a k dimensional subspace and this has a regularizing effect that is sometimes sufficient. Determining the best value of k can be accomplished, for instance, by one of our three methods of parameter choice: 1. Discrepancy Principle. In this case, we stop the iteration for the smallest value of k for which kr k k ffi . Both LSQR and GMRES have recurrence relations for determining kr k k using scalar computations, without computing either r k or x k [29, 32]. 2. GCV. For the projected problems (see x4.1) defined by either LSQR or GMRES, the operator AA ] is given by U is the pseudo-inverse of the matrix B k . Thus from (11), the GCV functional is [17] We note that there are in fact two distinct definitions for B y and hence two definitions for the denominator in G(k); for small enough k, the two are comparable, and the definition we use here is less expensive to calculate [18, x7.4]. 3. L-Curve. To determine the L-curve associated with LSQR or GMRES, estimates of are needed for several values of k. Using either algorithm, we can compute kr k k 2 with only a few scalar calculations. Paige and Saunders give a similar method for computing kx k k 2 [29], but, with GMRES, the cost for computing In using this method or GCV, one must go a few iterations beyond the optimal k in order to verify the optimum [19]. 4.2. Regularization by projection plus TSVD. If projection alone does not regularize, then we can compute the TSVD regularized solution to the projected problem (19). We need the SVD of the . This requires O(k 3 ) operations, but can also be computed from the SVD of B k\Gamma1 in O(k 2 ) operations [13]. Clearly, we still need to use some type of parameter selection technique to find a good value of '(k). First, notice that it is easy to compute the norms of the residual and the solution resulting from neglecting the ' smallest singular values. If jk is the component of e 1 in the direction of the j-th left singular vector of B k , and if fl j is the j-th singular value (ordered largest to smallest), then the residual and solution 2-norms are and fi 1@ k\Gamma'(k) X Using this fact, we can use any of our three sample methods: 1. Discrepancy Principle. Let r (k) denote the quantity b \Gamma Ax (k) and note that by (13) and orthonor- mality, kr (k) k 2 is equal to the first quantity in (20). Therefore, we choose '(k) to be the largest value for which kr (k) if such a value exists. 2. GCV. Another alternative for choosing '(k) is to use GCV to compute '(k) for the projected problem. The GCV functional for the kth projected problem is obtained by substituting B k for A and B for A ] , and substituting the expression of the residual in (20) for the numerator in (11): 3. L-Curve. We now have many L-curves, one for each value of k. The coordinate values in (20) form the discrete L-curve for a given k, from which the desired value of '(k) can be chosen without forming the approximate solutions or residuals. As k increases, the value '(k) chosen by the Discrepancy Principle will be monotonically nondecreasing. 4.3. Regularization by projection plus Rust's TSVD. As in standard TSVD, to use Rust's version of TSVD for regularization of the projected problem requires that we compute the SVD of the . Using the previous notation, Rust's strategy is to set y ae ae ik where q (k) are the right singular vectors of B k and I (k) aeg. We focus on three ways to determine ae: 1. Discrepancy Principle. Using the notation from the previous section, the norm of the regularized solution is given by fi 1 ( ae ik According to the discrepancy principle, we must choose ae so that the residual is less than ffi . In practice, this would require that the residual be evaluated by sorting the values j ik j and adding terms in that order until the residual norm is less than ffi . 2. GCV. Let us denote by card(I (k) ae ) the cardinality of the set I (k) ae . From (11), it is easy to show that the GCV functional corresponding to the projected problem for this regularization technique is given by ae ik ae In practice, for each k we first sort the values j ik smallest to largest. Then we define k discrete values ae j to be equal to these values with ae 1 being the smallest. We set ae that because the values of are the sorted magnitudes of the SVD expansion coefficients, we have (j Finally, we take the regularization parameter to be the ae j for which G k (ae j ) is a minimum. 3. L-Curve. As with standard TSVD, we now have one L-curve for each value of k. For fixed k, if we define the ae as we did for GCV above and we reorder the fl i in the same way that the j ik j were reordered when sorted, then we have When these solution and residual norms are plotted against each other as functions of ae, the value of ae j corresponding to the corner is selected as the regularization parameter. 4.4. Regularization by projection plus Tikhonov. Finally, let us consider using Tikhonov regularization to regularize the projected problem (18) for some integer k. Thus, for a given regularization parameter , we would like to solve min y or, equivalently, min y I The solution y to either formulation satisfies Using (13) and (15), we see that y (k) also satisfies A AV k A b: Therefore, y A I Using x , we have Thus as k ! n, the backprojected regularized solution x approaches the solution to (4). We need to address how to choose a suitable value of . 1. Discrepancy Principle. Note that in exact arithmetic, we have r Hence kB k y (k) . Therefore, to use the Discrepancy Principle requires we choose so that kr (k) discrete trial values j . For a given k, we take to be the largest value j for which kr (k) it exists; if not, we increase k and test again. 2. GCV. Let us define (B k ) y to be the operator mapping the right hand side of the projected problem onto the regularized solution of the projected problem: Given the SVD of B k as above, the denominator in the GCV functional defined for the projected problem (refer to (11)) is@ k The numerator is simply kr (k) 2 . For values of k n, it is feasible to compute the singular values of B k . 3. L-Curve. The L-curve is comprised of the points (kB k y (k) using (25) and the orthonormality of the columns of V k , we see these points are precisely (kr (k) discrete values of , the quantities kr (k) k 2 can be obtained by updating their respective estimates at the (k \Gamma 1)st iteration. 2 4.5. Correspondence between Direct Regularization and Projection Plus Regularization. In this section, we argue why the projection plus regularization approaches can be expected to yield regularized solutions nearly equivalent to the direct regularization counterpart. The following theorem establishes the desired result for the case of Tikhonov vs. projection plus Tikhonov. Theorem 4.1. Fix ? 0 and define x to be the kth iterate of conjugate gradients applied to the Tikhonov problem Let y (k) be the exact solution to the regularized projected problem are derived from the original problem A A = A b, and set z (k) Then z Proof: By the discussion at the beginning of x4.4 and equations (23) and (24), it follows that y (k) solves A b: Now the columns of V k are the Lanczos vectors with respect to the matrix A A and right-hand side A b. But these are the same as the Lanczos vectors generated with respect to the matrix A I and right-hand side A b. Therefore V k y (k) is precisely the kth iterate of conjugate gradients applied to pg. 495]. Hence z (k) . 2 2 The technical details of the approach are found in [28, pp. 197-198], from which we obtain . The implementation details for estimating kx (k) k and kr (k) were taken from the Paige and Saunders algorithm at http://www.netlib.org/linalg/lsqr. Projection plus - Disc. GCV L-curve Table Summary of flops for projection plus inner regularization with various parameter selection techniques, in addition to the O(qk) flops required for projection itself. Here k is the number of iterations (ie. the size of the projection) taken and p is the number of discrete parameters that must be tried. Let us turn to the case of TSVD regularization applied to the original problem vs. the projection plus TSVD approach. Direct computation convinces us that the two methods compute the same regularized solution if and arithmetic is exact. An approximate result holds in exact arithmetic when we take k iterations, with n. Let the singular value decomposition of B k be denoted by and define the s \Theta j matrix W s;j as I Then the regularized solution obtained from the TSVD regularization of the projected problem is reg denotes the leading j \Theta j principle submatrix of \Gamma k . If k is taken to be enough larger than j so that V k Q k W k;j " U T and the leading principle submatrix of \Sigma, then we expect x (k) reg to be a good approximation to x ' . This is made more precise in the following theorem. Theorem 4.2. Let k ? j such that contain the first j columns of " U respectively. Let Then reg kbk: Proof: Using the representations x 2 )b, we obtain reg and the conclusion follows from bounding each term. 2 Note that typically oe j AE oe n so that 1=oe j is not too large. For some results relating to the value of k necessary for the hypothesis of the theorem to hold, the interested reader is referred to theory of the Kaniel-Paige and Saad [30, x12.4]. Basic cost Added Cost Disc. GCV L-curve TSVD O("q) O(1) O(m) O(m) Rust's TSVD O("q) O(m) O(m) O(m) Projection O(kn) O(1) O(k) O(k) Table Summary of additional storage for each of four regularization methods under each of three parameter selection techniques. The original matrix is m \Theta n with q nonzeros, p is the number of discrete parameters that must be tried, k iterations are used in projection, and the factorizations are assumed to take " q storage. Projection plus - Disc. GCV L-curve Rust's TSVD O(k) O(k Table Summary of storage, not including storage for the matrix, for projection plus inner regularization approach, various parameter selection techniques. Here p denotes the number of discrete parameters tried. Each of these regularization methods also requires us to save the basis V or else regenerate it in order to reconstruct x. 5. Numerical results. In this section, we present two numerical examples. All experiments were carried out using Matlab and Hansen's Regularization Tools [16], with IEEE double precision floating point arithmetic. Since the exact, noise-free solutions were known in both examples, we evaluated the methods using the two- norm difference between the regularized solutions and the exact solutions. In both examples when we applied Rust's method to the original problem, the ae i were taken to be the magnitudes of the spectral coefficients of b sorted in increasing order. 5.1. Example 1. The 200\Theta200 matrix A and true solution x true for this example were generated using the function baart in Hansen's Regularization Toolbox. We generated true and then computed the noisy vector b as b + e, where e was generated using the Matlab randn function and was scaled so that the noise level, kb truek . The condition number of A was on the order of 10 19 . Many values of were tested: log displays the values of the regularization parameters chosen when the three parameter selection techniques were applied together with one of the four regularization methods on the original problem. Since 5:3761E\Gamma4, we set ffi that defines the discrepancy principle as the very close approximation 5:5E\Gamma4. The last column in the table gives the value of the parameter that yielded a regularized solution with the minimum relative error when compared against the true solution. The relative error values for regularized solutions corresponding to the parameters in Table 5 are given in Table 6. Note that using GCV to determine a regularization parameter for Rust's TSVD resulted in an extremely noisy solution with huge error. The corners of the L-curves for the Tikhonov, projection, and TSVD methods were determined using Hansen's lcorner function, with the modification that points corresponding to solution norms greater than 10 6 for the TSVD methods were not Rust's TSVD ae 1:223E\Gamma4 9:645E\Gamma7 1:223E\Gamma4 1:259E\Gamma4 or 1:223E\Gamma4 Projection Table Example 1: parameter values selected for each method. Disc. GCV L-curve optimal Rust's TSVD .1213 7E+14 .1213 .1213 Projection .1134 .1207 .1134 .1134 Table Example 1: comparison of kx true for each of 4 regularization methods on the original problem, where the regularization method was chosen using methods indicated. considered (otherwise, a false corner resulted). Next, we projected using LSQR and then regularized the projected problem with one of the three regularization methods considered. For each of the three methods, we computed regularization parameters for the projected problem using Discrepancy, GCV, and L-curve, then computed the corresponding regularized solutions; the parameters that were selected in each case at iterations 10 and 40 are given in Tables 7 and 9 respectively. As before, the lcorner routine was used to determine the corners of the respective L-curves. Comparing Table 6 and 8, we observe that computing the regularized solution via projection plus Tikhonov for projection size of 10 using either the Discrepancy Principle or the L-curve to find the regularization parameter gives results as good as if those techniques had been used with Tikhonov on the original problem to determine a regularized solution. Similar statements can be made for projection plus TSVD and projection plus Rust's TSVD. We should also note that for Tikhonov, with and without projection, none of the errors in the tables is optimal; that is, no parameter selection techniques ever gave the parameter for which the error was minimal. 5.2. Example 2. The 255 \Theta 255 matrix A for this example was a symmetric Toeplitz matrix with bandwidth 16 and exponential decay across the band. 3 The true solution vector x true is displayed as the top picture in Figure 2. We generated true and then computed the noisy vector b as b + e, where e was generated using the Matlab randn function and was scaled so that the noise level, kek kb truek , was . The vector b is shown in the bottom of Figure 2. The condition number of A was We generated our discrete i using log \Gamma1. The norm of the noise vector was 7:16E\Gamma2, so we took the value of ffi that defines the discrepancy principle to be 8:00E\Gamma2. In this example, it took 61 iterations for LSQR to reach a minimum relative error of 9:48E\Gamma2, and several more iterations were needed for the L-curve method to 3 It was generated using the Matlab command Rust's TSVD ae(k) 1:679E\Gamma4 1:773E\Gamma4 1:679E\Gamma5 1:679E\Gamma5 Table Example 1, iteration 10: regularization parameters selected for projection plus Tikhonov, TSVD, and Rust's TSVD. Disc. GCV L-curve optimal Rust's TSVD .1213 .1663 .1213 .1213 Table Example 1, iteration 10: comparison of kx true projection plus Tikhonov, TSVD, and Rust's TSVD. Disc. GCV L-curve optimal Rust's TSVD ae(k) 9:201E\Gamma5 1:225E\Gamma4 9:201E\Gamma5 9:201E\Gamma5 Table Example 1, iteration 40: regularization parameters selected for projection plus Tikhonov, TSVD, and Rust's TSVD. Disc. GCV L-curve optimal Rust's TSVD .1162 .1162 .1162 .1162 Table Example 1, iteration 40: comparison of kx true projection plus Tikhonov, TSVD, and Rust's TSVD. 50 100 150 200 250 -22610exact solution 50 100 150 200 250 -226 Fig. 2. Example 2: Top: exact solution. Bottom: noisy right hand side b. Rust's TSVD ae 2:183E\Gamma2 2:586E\Gamma6 1:477E\Gamma2 1:527E\Gamma2 Projection Table Example 2: parameter values selected for each method. The projection was performed on a left preconditioned system. Disc. GCV L-curve optimal Rust's TSVD Projection Table Example 2: comparison of kx true for each of 4 regularization methods on the original problem. estimate a stopping parameter. Likewise, the dimension k of the projected problem had to be around 60 to obtain good results with the projection-plus-regularization ap- proaches, and much larger than 60 for the L-curve applied to the projected, Tikhonov regularized problem to give a good estimate of the corner with respect to the Tikhonov regularized original problem. Therefore, for the projection based techniques, we chose to work with a left preconditioned system (refer to the discussion at the end of x 3.5). Our preconditioner was chosen as in [22] where the parameter defining the preconditioner was taken to be The values of the regularization parameters chosen when the three parameter selection techniques were applied together with one of the four regularization methods on the original problem are given in Table 11. The last column in the table gives the value of the parameter that gave a regularized solution with the minimum relative error over the range of discrete values tested, with respect to the true solution. The relative errors that resulted from computing solutions according to the parameters in Table 11 are in Table 12. We note that GCV with TSVD and Rust's TSVD were ineffective. The corners of the L-curves for the Tikhonov, projection, and TSVD methods were determined using Hansen's lcorner function, with the modification that points corresponding to the largest solution norms for the TSVD methods were not considered (otherwise, a false corner was detected by the lcorner routine). Next, we projected using LSQR (note that since the matrix and preconditioner were symmetric, we could have used MINRES as in [22]) and then regularized the projected problem with one of the three methods considered. For each of the three methods, we computed regularization parameters for the projected problem using Dis- crepancy, GCV, and L-curve, then computed the corresponding regularized solutions; the parameters that were selected in each case at iterations 15 and 25 are given in Tables 13 and 15, respectively. The relative errors of the regularized solutions generated accordingly are given in Tables 14 and 16. Again, we used the lcorner routine to determine the corners of the respective L-curves, except in the case of Rust's TSVD method. In the latter case, there was Rust's TSVD ae(k) 3:558E\Gamma2 3:558E\Gamma2 3:558E\Gamma2 3:558E\Gamma2 Table Example 2, iteration 15: regularization parameters selected for projection plus Tikhonov, TSVD, and Rust's TSVD. Disc. GCV L-curve optimal Rust's TSVD Table Example 2, iteration 15: comparison of kx true projection plus Tikhonov, TSVD, and Rust's TSVD. always a very sharp corner that could be picked out visually. Comparing Table 11 with Tables 13 and 15, we see that the parameter chosen by applying the L-curve method to projected-plus-Tikhonov problem was the same parameter chosen by applying the L-curve to the original problem. Moreover, a comparison of Table 12 with Tables 14 and 16 shows that relative errors of the regularized solutions computed accordingly are comparable to applying Tikhonov to the original problem with that same parameter. Similar results are shown for the other cases, with the exception that the discrepancy principle did not work well for the projection- plus-TSVD problems, and GCV was not effective for the projected problems when 6. Conclusions. In this work we have given methods for determining the regularization parameter and regularized solution to the original problem based on regularizing a projected problem. The proposed approach of applying regularization and parameter selection techniques to a projected problem is economical in time and storage. We presented results that in fact the regularized solution obtained by backprojecting the TSVD or Tikhonov solution to the projected problem is almost equivalent to applying TSVD or Tikhonov to the original problem, where "almost" depends on the size of k. The examples indicate the practicality of the method, and illustrate that our regularized solutions are usually as good as those computed using the original system and can be computed in a fraction of the time, using a fraction of the storage. We note that similar approaches are valid using other Krylov subspace methods for computing the projected problem. In this work, we did not address potential problems from loss of orthogonality as the iterations progress. In this discussion, we did, however, assume that either k was naturally very small compared to n or that preconditioning had been applied to enforce this condition. Possibly for this reason, we found that for modest k, round-off did not appear to degrade either the LSQR estimates of the residual and solution norms or the computed regularized solution in the following sense: the regularization parameters chosen via the projection-regularization and the corresponding regularized solutions were comparable to those chosen and generated for the original discretized problem. For the Tikhonov approach in this paper, we have assumed that the regularization Disc. GCV L-curve optimal Rust's TSVD ae(k) 4:828E\Gamma2 7:806E\Gamma3 4:828E\Gamma2 4:828E\Gamma2 Table Example 2, iteration 25: regularization parameters selected for projection plus Tikhonov, TSVD, and Rust's TSVD. Disc. GCV L-curve optimal Rust's TSVD Table Example 2, iteration 25: comparison of kx true projection plus Tikhonov, TSVD, and Rust's TSVD. operator L was the identity or was related to the preconditioning operator; this allowed us to efficiently compute kr (k) k and kx (k) k for multiple values of efficiently for each k. If L is not the identity but is invertible, we can first implicitly transform the problem to "standard form" [17]. With Lx, we can solve the equivalent system min Then the projection plus regularization schemes may be applied to this transformed problem. Clearly the projection based schemes will be useful as long as solving systems involving L can be done efficiently. --R Estimation of the L-curve via Lanczos bidiagonal- ization Galerkin projection method for solving multiple linear systems The 'minimum reconstruction error' choice of regularization pa- rameters: Some more efficient methods and their application to deconvolution problems Using the L-curve for determining optimal regularization pa- rameters Equivalence of regularization and truncated iteration in the solution of ill-posed image reconstruction problems Fast CG-based methods for Tikhonov-Phillips regularization Generalized cross-validation as a method for choosing a good ridge parameter Calculating the singular values and pseudo-inverse of a matrix Matrix Computations Theory of Tikhonov Regularization for Fredholm equations of the First Kind A stable and fast algorithm for updating the singular value decom- position Preconditioned iterative regularization for ill-posed problems Analysis of discrete ill-posed problems by means of the L-curve a Matlab package for analysis and solution of discrete ill-posed problems The use of the L-curve in the regularization of discrete ill-posed problems Regularization for Applied Inverse and Ill-Posed Problems Regularization of ill-posed problems by envelope guided conjugate gradients Symmetric Cauchy-like preconditioners for the regularized solution of 1-d ill-posed problems Pivoted Cauchy-like preconditioners for regularized solution of ill-posed problems On the solution of functional equations by the method of regularization Iterative image restoration using approximate inverse preconditioning A bidiagonalization-regularization procedure for large scale discretization of ill-posed problems Algorithm 583 The Symmetric Eigenvalue Problem Truncating the singular value decomposition for ill-posed problems Iterative Methods for Sparse Linear Systems GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems Solutions of Ill-Posed Problems Pitfalls in the numerical solution of linear ill-posed problems --TR --CTR Angelika Bunse-Gerstner , Valia Guerra-Ones , Humberto Madrid de La Vega, An improved preconditioned LSQR for discrete ill-posed problems, Mathematics and Computers in Simulation, v.73 n.1, p.65-75, 6 November 2006 E. T. F. Santos , A. Bassrei, L- and -curve approaches for the selection of regularization parameter in geophysical diffraction tomography, Computers & Geosciences, v.33 n.5, p.618-629, May, 2007 G. Landi, The Lagrange method for the regularization of discrete ill-posed problems, Computational Optimization and Applications, v.39 n.3, p.347-368, April 2008 Alexander B. Konovalov , Vitaly V. Vlasov , Olga V. Kravtsenyuk , Vladimir V. Lyubimov, Space-varying iterative restoration of diffuse optical tomograms reconstructed by the photon average trajectories method, EURASIP Journal on Applied Signal Processing, v.2007 n.1, p.18-18, 1 January 2007

projection;l-curve;truncated singular value decomposition;regularization;discrepancy principle;iterative methods;ill-posed problems;tikhonov;krylov subspace

587794

Data Fitting Problems with Bounded Uncertainties in the Data.

An analysis of a class of data fitting problems, where the data uncertainties are subject to known bounds, is given in a very general setting. It is shown how such problems can be posed in a computationally convenient form, and the connection with other more conventional data fitting problems is examined. The problems have attracted interest so far in the special case when the underlying norm is the least squares norm. Here the special structure can be exploited to computational advantage, and we include some observations which contribute to algorithmic development for this particular case. We also consider some variants of the main problems and show how these too can be posed in a form which facilitates their numerical solution.

Introduction . Let A 2 R mn arise from observed data, and for Then a conventional tting problem is to minimize krk over x 2 R n , where the norm is some norm on R m . This involves an assumption that A is exact, and all the errors are in b, which may not be the case in many practical situations; the eect of errors in A as well as b has been recognized and studied for many years, mainly in the statistics literature. One way to take the more general case into account is to solve the problem subject to where the matrix norm is one on (m(n+1)) matrices. This problem, when the matrix norm is the Frobenius norm, was rst analyzed by Golub and Van Loan [10], who used the term total least squares and developed an algorithm based on the singular value decomposition of [A : b]. Since then, the problem has attracted considerable attention: see, for example, [18], [19]. While the formulation (1.1) is often satisfactory, it can lead to a solution in which the perturbations E or d are quite large. However, it may be the case that, for exam- ple, A is known to be nearly exact, and the resulting correction to A may therefore be excessive. In particular, if bounds are known for the size of the perturbations, then it makes sense to incorporate these into the problem formulation, and this means that the equality constraints in (1.1) should be relaxed and satised only approxi- mately. These observations have motivated new parameter estimation formulations where both A and b are subject to errors, but in addition, the quantities E and d are bounded, having known bounds. This idea gives rise to a number of dierent, but Received by the editors May 25, 1999; accepted for publication (in revised form) by L. El Ghaoui December 5, 2000; published electronically April 6, 2001. http://www.siam.org/journals/simax/22-4/35659.html y Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland (gawatson@ maths.dundee.ac.uk). closely related, problems and algorithms and analysis for problems of this type based on least squares norms are given, for example, in [1], [2], [3], [4], [5], [8], [9], [15], [17]. The general problem (1.1) is amenable to analysis and algorithmic development for a wide class of matrix norms, known as separable norms, a concept introduced by Osborne and Watson [13]. The main purpose of this paper is to show how problems with bounded uncertainties also can be considered in this more general setting. In particular, it is shown how such problems can be posed in a more computationally convenient form. As well as facilitating their numerical solution, this enables connections with conventional data tting problems to be readily established. Motivation for extending these ideas beyond the familiar least squares setting is provided by the important role which other norms can play in more conventional data tting contexts. We continue this introductory section by dening separable norms and by introducing some other necessary notation and tools. We rst introduce the concept of the dual norm. Let k:k be a norm on R m . Then for any v 2 R m , the dual norm is the norm on R m dened by r T v: The relationship between a norm on R m and its dual is symmetric, so that for any r r T v: Definition 1.1. A matrix norm on m n matrices is said to be separable if given vectors u there are vector norms k:k A on R m and k:k B on R n such that Most commonly occurring matrix norms (operator norms, orthogonally invariant norms, norms based on an l p vector norm on the elements of the matrix treated as an extended vector in R mn ) are separable. A result which holds for separable norms and will be subsequently useful is that see, for example, [13] or [20]. Another valuable tool is the subdierential of a vector norm, which extends the idea of the derivative to the nondierentiable case. A useful characterization of the subdierential (for k:k A ) is as follows. Definition 1.2. The subdierential or set of subgradients of krkA is the set A 1g: If the norm is dierentiable at r, then the subdierential is just the unique vector of partial derivatives of the norm with respect to the components of r. The main emphasis of this paper is on problems which address the eects of worst case perturbations. This gives rise to problems of min-max type. In section 2, we consider problems where separate bounds on kEk and kdkA are assumed known, and in section 3, we consider a similar problem except that a single bound on kE : dk is given. In both cases, the matrix norm is assumed to be separable. In section 4, some variants of the original problems are considered, and nally, in section 5 we consider a related class of problems which are of min-min rather than min-max type. 1276 G. A. WATSON 2. Known bounds on kEk and kdkA . Suppose that the underlying problem is such that we know bounds on the uncertainties in A and b so that where the matrix norm is a separable norm, as in Denition 1.1. Then instead of forcing the equality constraints of (1.1) to be satised, we wish to satisfy them approximately by minimizing the A-norm of the dierence between the left- and right-hand sides, over all perturbations satisfying the bounds. This leads to the problem Therefore x minimizes the worst case residual, and this can be interpreted as permitting a more robust solution to be obtained to the underlying data tting problem: for an explanation of the signicance of the term robustness in this context, in the least squares case, see, for example, [9], where a minimizing x is referred to as a robust least squares solution. Another interpretation of the problem being solved is that it guarantees that the eect of the uncertainties in the data will never be overestimated, beyond the assumptions made by knowledge of the bounds. We now show that (2.1) can be restated in a much simpler form as an unconstrained problem in x alone. Theorem 2.1. For any x, the maximum in (2:1) is attained when where otherwise u is arbitrary but 1. The maximum value is Proof. We have for any E; d such that kEk , kdkA d , Now let E and d be as in the statement of the theorem. Then and further The result follows. An immediate consequence of this result is that the problem (2.1) is solved by minimizing with respect to x kAx bk A and it is therefore appropriate to analyze this problem. In particular, we give conditions for x to be a solution and also conditions for that solution to be results are a consequence of standard convex analysis, as is found, for example, in [14]. Theorem 2.2. The function (2:2) is minimized at x if and only if there exists @kxkB such that Theorem 2.3. Let there exist v 2 @kbkA so that Proof. For to give a minimum we must have v 2 @kbkA so that (2.3) is satised with kwk 1. The result follows. 2.1. Connections with least norm problems. We will next establish some connections between solutions to (2.2) and solutions to traditional minimum norm data tting problems. In [9], coincidence of solutions in the least squares case is said to mean that the usual least squares solution may be considered to be robust. Consider the least norm problem minimize kAx bk A : Then x is a solution if and only if there exists v 2 @kAx bk A such that solves (2.4), then clearly it also solves (2.2). (Note that we can take (2.3).) Otherwise, if k:k A is smooth, solutions to (2.2) and to (2.4) can coincide only is unique). In this case, let b be any solution to denote the minimum A-norm solution to A T Then if x minimizes (2.2), (2.3) is satised with w 2 @kxkB and kvk A 1, otherwise v is unrestricted. Because it follows that we must have A In other words, A is also a solution to (2.2) only if b 2 range(A) and A For example, if both norms are least squares norms, then this condition is Note that if b is the minimum B-norm solution to immediately solves (2.2), and so there must exist w such that this inequality is satised independently of . 1278 G. A. WATSON The case when k:k A is nonsmooth is more complicated. Example 2.1. Let corresponds to the separable norm being the sum of moduli of the components.) Then (2:4) is solved by any x; 1 x 2. Further a solution to (2:2) provided that 2. To summarize, we can augment Theorem 2.3 by the following, which can be interpreted as a generalization of a result of [9]. Theorem 2.4. If b 2 range(A) and any solution to provided that A We can also prove the following, which connects Theorems 2.3 and 2.4. Theorem 2.5. Let b 2 range(A), and A min Proof. Let otherwise arbitrary. It follows by denition of A + and A T Thus and Now using (2.5) and (2.6). The result follows. A consequence of the above results is that if b 2 range(A) and A min then any point in the convex hull of f0; A + bg is a solution. 2.2. Methods of solution. From a practical point of view, it is obviously of importance to e-ciently solve (2.1) (or, equivalently, (2.2)) in appropriate cases. Let Most commonly occurring norms are either smooth (typied by l p norms, 1 < p < 1) or polyhedral (typied by the l 1 and l 1 norms). If the norms in the denition of f are smooth, then derivative methods are natural ones to use. A reasonable assumption in most practical situations is that range(A), so that would then give the only derivative discontinuity. If is not a solution, then f is dierentiable in a neighborhood of the solution, and once inside that neighborhood, derivative methods can be implemented in a straightforward manner. Theorem 2.3 tells us when is a solution; the following theorem gives a way of identifying a descent direction at that point in the event that it is not. It applies to arbitrary norms. Theorem 2.6. Assume that vk B be such that g. Then d is a descent direction for f at Proof. Let the stated conditions be satised, and let d be dened as in the statement of the theorem. By theorem 2.3, is not a solution. For d to be a descent direction at the directional derivative of f at in the direction d must be negative, that is, arbitrary. Then vk < 0: The result follows. If k:k A is smooth, then ^ v is unique, and the construction of d using this result is straightforward. If the norm on E is the norm given by 1=p or then f becomes G. A. WATSON In fact, the minimization of f for any p; q satisfying 1 < can be readily achieved by derivative methods, using Theorem 2.6 to get started. Indeed, it is normally the case that second derivatives exist and can easily be calculated so that Newton's method (damped if necessary) can be used following a step based on Theorem 2.6. The Hessian matrix of f is positive denite (because f is convex), so that the Newton direction is a descent direction away from a minimum. Some numerical results are given in [21]. For polyhedral norms (typied by l 1 and l 1 norms), the convex objective function (2.2) is a piecewise linear function. Therefore, it may be posed as a linear programming problem, and solved by appropriate methods. Arguably, the most interesting case from a practical point of view is the special case when both k:k A and k:k B are the least squares norm, so that This case has particular features which greatly facilitate computation, and Chandrasekaran et al. [1], [3] exploit these in a numerical method. In contrast to the problems considered above, which involve a minimization problem in R n , special features of the l 2 case can be exploited so that the problem reduces to one in R. When (2.7) is dierentiable, (2.3) becomes or where Let the singular value decomposition of A be where U 2 R mm and V 2 R nn are orthogonal and ng is the matrix of singular values in descending order of magnitude. Let It will be assumed in what follows that A has rank n, and further that is not a solution (which means, in particular, that b 1 6= (which means that b 2 6= 0). From Theorem 2.3, we require that < Then it is shown in [3] that satises the equation where This can be rearranged as where It is also shown in [3] that (2.8) is both necessary and su-cient for (2.10) to have exactly one positive root . In addition, G 0 ( ) > 0. Dierent methods can be used for nding in this case. One possibility which is suggested by (2.9) is the simple iteration process and it is of interest to investigate whether or not this is likely to be useful. It turns out that this method is always locally convergent, as the following result shows. Theorem 2.7. Let < Then (2:10) has exactly one positive root and (2:11) is locally convergent to . Proof. Let satisfy (2.12). Then (2.10) has a unique positive root . Dieren- tiating G() gives and so using G( Now g() and G() are related by and so using g( G. A. WATSON Table Simple iteration: stack loss data. 9 Substituting from (2.13) gives 0: It follows using (2.15) and G 0 ( ) > 0 that and the result is proved. Indeed, simple iteration seems to be remarkably eective, and in problems tried, it converged in a satisfactory way from obvious starting points. For example, for the stack loss data set of Daniel and Wood [6] 4), performance for dierent values of is shown in Table 1, where the iteration is terminated when the new value of diers from the previous one by less than 10 6 . Another example is given by using the Iowa wheat data from Draper and Smith 9). The performance of simple iteration in this case is shown in Table 2. Although simple iteration is in some ways suggested by the above formulation, of course higher order methods can readily be implemented, such as the secant method or Newton's method. Actual performance will depend largely on factors such as the nature and size of the problem and the relative goodness of starting points. 3. A known bound on kE : dk. Suppose now that the underlying problem is such that we know upper bounds on the uncertainties in A and b, in the form where and the (separable) matrix norm are given. Consider the problem of deter- mining Table Simple iteration: Iowa wheat data. 6 5.912536 12.925885 30.839779 9 12.927008 30.881951 14 30.882685 where the A-norm on R m is dened by the particular choice of separable norm (or vice versa). This problem and variants have been considered, for example, by El Ghaoui and Lebret [8], [9], where the matrix norm is the Frobenius norm, so that both the Aand B-norms are least squares norms. Arguing as in Theorem 2.1 gives the following result. Theorem 3.1. For any x, the maximum in (3:1) is attained when where any vector with 1. The maximum value is The problem (3.1) is therefore equivalent to the problem of minimizing with respect to x kAx bk A Standard convex analysis then gives the following result. Theorem 3.2. The function (3:4) is minimized at x if and only if there exists where u 1 denotes the rst n components of u. 3.1. Connection with least norm problems. As before, it is of interest to establish connections with the corresponding least norm problems. If (2.4), then it will also minimize (3.4) for monotonic norms k:k B . (k:k B is a monotonic norm on R n+1 if kck B kdkB whenever jc does not solve (2.4), then just as before when k:k A is smooth, solutions to this problem and (3.1) cannot coincide unless b 2 range(A). In that case, as in section 2.1 let a solution to denote the minimum A-norm solution to A T For a solution to (3.4), there must exist v; kvk (otherwise unrestricted) so that 1284 G. A. WATSON where consists of the rst n components of u Therefore, and so A In other words the solutions will coincide if b 2 range(A) and A Note that if k:k B is smooth, then u is unique. For example, when both norms are least squares norms, this gives as given in [9]. The situation when k:k A is not smooth is, of course, once again more complicated. Consider again Example 2.1 where > 0 is arbitrary. Recall that (2.4) is solved by any x; 1 x 2: the unique solution to the problem of minimizing (3.4) is 3.2. Connection with total approximation problems. The nature of the bound in (3.1) means that there is a connection to be made with the total approximation problem (1.1). It is known [13], [20] that a minimum value of (1.1) coincides with the minimum of the problem subject to the smallest -generalized singular value of the matrix [A : b]. In particular, if the vector norms are least squares norms, then this is just the smallest singular value of at a minimum of (1.1) is obtained from a z = z T at a minimum of (3.7) by scaling so that z T corresponds to nonexistence of a solution to (1.1).) It is known also that a minimizing pair E; d is given by and consider the problem (3. or equivalently, so if k:k A is smooth, x T is a solution to this problem provided that A as a consequence of the previous analysis. For example, when both norms are least squares norms, this gives (see also [9]) For the least squares case, El Ghaoui and Lebret [8] suggest using robust methods in conjunction with total approximation to identify an appropriate value of . The idea is rst to solve the total approximation problem. Then (3.8) is constructed from the total approximation solution and solved with set to T , the minimum value in (3.7), that is, Of course if T does not exceed the right-hand side of (3.9), there is nothing to solve. 3.3. Methods of solution. For the special case of (3.4) when the norms k:k A and k:k B are (possibly dierent) l p norms, we have derivative methods may again be used. Let us again make the (reasonable) assumption that there is no x which makes kAx bk so that kAx bk p is dierentiable for all x. Then in contrast to the earlier problem, since the second term cannot be identically zero, f is dierentiable for all x. We can easily compute rst and second derivatives of f , and so Newton's method, for example, can be implemented. A line search in the direction of the Newton step will always guarantee descent, because f is convex, so eventually we must be able to take full steps and get a second order convergence rate. Some numerical results are given in [21]. For polyhedral norms occurring in (3.4), linear programming techniques may be used. Now consider the special case when 2. An analysis similar to that given in section 2.2 can be given in this case, leading to a similar numerical method. This particular problem is considered by El Ghaoui and Lebret [8], [9]. The main emphasis of those papers is on structured perturbations, which is a harder problem, and an exact solution to that problem is obtained. For the present case, the method suggested is similar to that given for the problem of section 2 by Chandrasekaran et al. in [1], [3]. Let A have singular value decomposition as before and have full rank. Assume also that range(A). Then optimality conditions are 1286 G. A. WATSON or where It can be shown as before that satises the equation where This can as before be rearranged as where with G() dened as in (2.10). It is easily seen that H() has at least one positive root for any > 0. As in [3], it may be shown that H() in fact has exactly one positive root, ^ , with H(^) > 0: Note that here there is no restriction on except that it should be positive. Consider the simple iteration process Theorem 3.3. The iteration scheme (3:11) is locally convergent to ^ . Proof. We can rst show that 2: We can then show that h() and H() are related by h() where Thus using Substituting from (3.12) gives It follows using (3.13) and H 0 (^) > 0 that and the result is proved. The performance of simple iteration in this case is, of course, similar to the same method applied in the previous situation. Other methods like the secant method, or Newton's method, are more complicated but can give potentially better performance. 4. Some modications. There are dierent ways in which additional information may be incorporated into the problems of the last two sections, resulting in appropriate modications of these problems. For example, some components of A or b may be exact, in which case the corresponding components of E or d will be zero. The bounds may take dierent forms and may be on submatrices of E rather than E itself. Also the perturbation matrices may have known structure, which we want to preserve. Examples of all these possibilities are considered in this section. 4.1. Exact columns and rows. Some problems are such that some of the columns and possibly rows of A are known to be exact (see, for example, [3]). A treatment can be given for both the problems of sections 2 and 3, and we will demonstrate only for those of section 2; the appropriate requirements for the problems of section 3 are obvious. We begin by considering the case when certain columns only of A are known to be exact. In that case (following suitable reordering of columns if necessary) the general problem is to minimize and the (separable) matrix norm is one dened on m t matrices. We can partition x as x arguing as in Theorem 2.1, we have the following. Theorem 4.1. For any x, the maximum in (4:1) is attained when where otherwise u is arbitrary, but 1. The maximum value is 1288 G. A. WATSON Therefore, the problem is solved by minimizing with respect to x kAx bk A Now consider the case when some columns and rows of A are exact. This corresponds to the requirement to perturb only a submatrix of A. Assume this to be the lower right-hand s t submatrix. An appropriate problem is then to minimize (b d) A where A 2 and A 4 have t columns, A 3 and A 4 have s rows, and the matrix norm is a separable norm on s t matrices. Unfortunately, the separable norm is dened in terms of two vector norms k:k A on R s and k:k B on R t , and k:k A as used in (4:3) is on R m . We get around this potential con ict by assuming that k:k A is dened for any length of vector; we will also assume that the introduction of additional zero components does not change the value of the norm. The attainment of the maximum in (4.3) is not quite so straightforward as before. However, we can prove the following result. Theorem 4.2. Let denote the rst m s components of r, and let r 2 denote the last s components. Let x solve the problem subject to r Then x solves (4:3): Proof. Arguing as in previous results, an upper bound for the maximum (subject to the constraints) in (4.3) is Now dene the set For any x 2 X, dene where u rst (m s) components zero, and last s components forming the vector u 2 with arbitrary except that A 3 A 4 +E (b d) A A A The result is proved. Of course the set X may be empty. In that case, while the problem (4.3) is still well dened, it is not clear that a matrix E and a vector d can be dened such that the maximum in the problem is attained. That being the case, there is no obvious equivalent simpler problem. 4.2. Bounded columns of E. Suppose that the columns of E are individually bounded so that where e i is the ith unit vector, and consider the problem of nding As for Theorem 2.1, we can prove the following result. Theorem 4.3. For any x, the maximum in (4:5) is attained when otherwise u is arbitrary but 1. The maximum value is Even in the least squares case, this objective function is not normally dieren- tiable, being a combination of a least squares norm and a weighted l 1 norm. It can be reposed as a smooth constrained optimization problem, and solved by standard techniques. 4.3. Structured problems. In some applications, the perturbation matrices have known structure, as in the following problem considered by El Ghaoui and Lebret [9]. Given A min k-k x A where k:k A is a given norm on R m and k:k is a given norm on R p . Dene for any Consider the maximum in (4.6), which will be attained at the solution to the problem maximize kr 0 +M-kA subject to assuming that - maximizing kr exceeds in norm. Because the functions involved are convex, necessary conditions for a solution can readily be given: these are that there exists R such that 1290 G. A. WATSON Using these conditions, it is easily seen that Therefore, an equivalent (in a sense dual) problem is subject to Consider the special case when both norms are least squares norms. Then and so the necessary conditions can be written Thus provided that I F is nonsingular. A way of solving this problem based on those results is given by El Ghaoui and Lebret [9]. They also consider the problem when k:k is the Chebyshev norm. Extending the ideas to more general norms, however, does not look straightforward. 5. A min-min problem. The problems (2.1) and (3.1) are examples of min-max problems: minimization is carried out with respect to x over all allowed perturbations in the data. This is justied if the emphasis is on robustness. However, from other considerations it may be su-cient to minimize with respect to x while simultaneously minimizing with respect to the perturbations. This gives rise to a min-min problem, as considered (least squares case) in [2], [3], [5]. In this nal section, we will brie y consider this problem. Again there are two versions, consistent with those treated in sections 2 and 3. To illustrate the ideas involved, we will consider nding min (5. In contrast to the min-max case, here we are seeking to nd a solution x which gives the smallest possible error over allowable perturbations. Again the problem can be replaced by an equivalent unconstrained optimization problem. Theorem 5.1. Let be small enough that Then (5:1) is equivalent to the problem of minimizing with respect to x kAx bk A k[x Proof. Let (5.2) be satised and let x be arbitrary. Let otherwise arbitrary. Then Now x where otherwise u is arbitrary with and further using (5.2). The result follows. There are two important dierences between (5.3) and (3.4): rst, the relationship leading to (5.3) requires a condition on , and second, the resulting problem is not a convex problem. The nonconvexity of (5.3) is interpreted in [2] as being equivalent to using an \indenite" metric, in the spirit of recent work on robust estimation and ltering: see, for example, [11], [12], [16]. The condition (5.2) is satised if that is, if does not exceed T (see section 3.2). If attained at Indeed if is set to any local minimum of (3.7), with value T , then the corresponding point x T generated from the local minimizer z T is a stationary point of (5.1), as the following argument shows. Necessary conditions for x to solve (3.7) are that there exist v and a Lagrange multiplier such that G. A. WATSON Multiplying through by z T T shows that Now the relationship z implies that sign()v 2 @kAx T bk A and In other words, there exist v 2 @kAx T bk A , w 2 denotes the rst n components of w. It follows from standard convex analysis that x T is a stationary point of the problem of minimizing A similar treatment can be given if (5.1) is replaced by the related problem of nding min Provided that is small enough that then this is equivalent to the problem of nding fkAx bk A kxkB g: An algorithm is given in [2] for solving the least squares case of this problem. It has similarities to the algorithms given before, involving the solution of a nonlinear equation for and a linear system for x. Indeed it is clear that many of the ideas which apply to min-max problems carry over to problems of the present type. However, we do not consider that further here. 6. Conclusions. We have given an analysis in a very general setting of a range of data tting problems, which have attracted interest so far in the special case when least squares norms are involved. While this case is likely to be most useful in practice, consideration of other possibilities can be motivated by the valuable role that other norms play in a general data tting context. The main thrust of the analysis has been to show how the original problems may be posed in a simpler form. This permits the numerical treatment of a wide range of problems involving other norms, for example, l p norms. We have also included some observations which contribute to algorithmic development for the important least squares case. Acknowledgment . I am grateful to the referees for helpful comments which have improved the presentation. --R Parameter estimation in the presence of bounded modeling errors Parameter estimation in the presence of bounded data uncertainties The degenerate bounded errors-in-variables model Fitting Equations to Data Applied Regression Analysis Robust solutions to least squares problems with uncertain data Robust solutions to least-squares problems with uncertain data An analysis of the total least squares problem Recursive linear estimation in Krein spaces-Part I: Theory Filtering and smoothing in an H 1 setting An analysis of the total approximation problem in separable norms Convex Analysis Estimation in the presence of multiple sources of uncertainties with applications Inertia conditions for the minimization of quadratic forms in inde Estimation and control in the presence of bounded data uncertainties ed., Recent Advances in Total Least Squares Techniques and Errors-in- Variables Modeling The Total Least Squares Problem: Computational Aspects and Analysis Choice of norms for data Solving data --TR

robustness;data fitting;minimum norm problems;bounded uncertainties;separable matrix norms

587796

Inversion of Analytic Matrix Functions That are Singular at the Origin.

In this paper we study the inversion of an analytic matrix valued function A(z). This problem can also be viewed as an analytic perturbation of the matrix A0=A(0). We are mainly interested in the case where A0 is singular but A(z) has an inverse in some punctured disc around z=0. It is known that A-1(z) can be expanded as a Laurent series at the origin. The main purpose of this paper is to provide efficient computational procedures for the coefficients of this series. We demonstrate that the proposed algorithms are computationally superior to symbolic algebra when the order of the pole is small.

Introduction Let fA k g k=0;1;::: ' R n\Thetan be a sequence of matrices that de-nes the analytic matrix valued function The above series is assumed to converge in some non-empty neighbourhood of z = 0. We will also say that A(z) is an analytic perturbation of the matrix A Assume the inverse matrices A \Gamma1 (z) exist in some (possibly punctured) disc centred at In particular, we are primarily interested in the case where A 0 is singular. In this case it is known that A \Gamma1 (z) can be expanded as a Laurent series in the form A and s is a natural number, known as the order of the pole at z = 0. The main purpose of this paper is to provide eOEcient computational procedures for the Laurent series coeOEcients As one can see from the following literature review, few computational methods have been considered in the past. This work was supported in part by Australian Research Council Grant #A49532206. y INRIA Sophia Antipolis, 2004 route des Lucioles, B.P.93, 06902, Sophia Antipolis Cedex, France, e-mail: k.avrachenkov@sophia.inria.fr z Department of Statistics, The Hebrew University, 91905 Jerusalem, Israel and Department of Economet- rics, The University of Sydney, Sydney, NSW 2006, Australia, e-mail: haviv@mscc.huji.ac.il x CIAM, School of Mathematics, The University of South Australia, The Levels, SA 5095, Australia, e-mail: The inversion of nearly singular operator valued functions was probably -rst studied in the paper by Keldysh [22]. In that paper he studied the case of a polynomial perturbation are compact operators on Hilbert space. In particular, he showed that the principal part of the Laurent series expansion for the inverse operator A \Gamma1 (z) can be given in terms of generalized Jordan chains. The generalized Jordan chains were initially developed in the context of matrix and operator polynomials (see [13, 26, 30] and numerous references therein). However, the concept can be easily generalized to the case of an analytic perturbation (1). Following Gohberg and Sigal [15] and Gohberg and Rodman [14], we say that the vectors Jordan chain of the perturbed matrix A(z) at for each 0 - k - r \Gamma 1. Note that ' 0 is an eigenvector of the unperturbed matrix A 0 corresponding to the zero eigenvalue. The number r is called the length of the Jordan chain and ' 0 is the initial vector. Let f' (j) j=1 be a system of linearly independent eigenvectors, which span the null space of A 0 . Then one can construct Jordan chains initializing at each of the eigenvectors ' (j) 0 . This generalized Jordan set plays a crucial role in the analysis of analytic matrix valued functions A(z). Gantmacher [11] analysed the polynomial matrix (3) by using the canonical Smith form. Vishik and Lyusternik [37] studied the case of a linear perturbation showed that one can express A \Gamma1 (z) as a Laurent series as long as A(z) is invertible in some punctured neighbourhood of the origin. In addition, an undetermined coeOEcient method for the calculation of Laurent series terms was given in [37]. Langenhop [25] showed that the coeOEcients of the regular part of the Laurent series for the inverse of a linear perturbation form a geometric sequence. The proof of this fact was re-ned later in Schweitzer [33, 34] and Schweitzer and Stewart [35]. In particular, the paper [35] proposed a method for computing the Laurent series coeOEcients. However the method of [35] cannot be applied (at least imme- diately) to the general case of an analytic perturbation. Many authors have obtained existence results for operator valued analytic and meromorphic functions [3, 15, 23, 27, 29, 36]. In par- ticular, Gohberg and Sigal [15], used a local Smith form to elaborate on the structure of the principal part of the Laurent series in terms of generalized Jordan chains. Recently, Gohberg, Kaashoek and Van Schagen [12] have re-ned the results of [15]. Furthermore, Bart, Kaashoek and Lay [5] used their results on the stability of the null and range spaces [4] to prove the existence of meromorphic relative inverses of -nite meromorphic operator valued functions. The ordinary inverse operator is a particular case of the relative inverse. For the applications of the inversion of analytic matrix functions see for example [8, 9, 20, 23, 24, 28, 31, 32, 36]. Howlett [20] provided a computational procedure for the Laurent series coeOEcients based on a sequence of row and column operations on the coeOEcients of the original power series (1). Howlett used the rank test of Sain and Massey [32] to determine s, the order of the pole. He also showed that the coeOEcients of the Laurent series satisfy a -nite linear recurrence relation in the case of a polynomial perturbation. The method of [20] can be considered as a starting point for our research. The algebraic reduction technique which is used in the present paper was introduced by Haviv and Ritov [17, 18] in the special case of stochastic matrices. Haviv, Ritov and Rothblum [19] also applied this approach to the perturbation analysis of semi-simple eigenvalues. In this paper we provide three related methods for computing the coeOEcients of the Laurent series (2). The -rst method uses generalized inverse matrices to solve a set of linear equations and extends the work in [17] and [20]. The other two methods use results that appear in [2, 17, 18, 19] and are based on a reduction technique [6, 10, 21, 23]. All three methods depend in a fundamental way on equating coeOEcients for various powers of z. By substituting the series (1) and (2) into the identity A(z)A I and collecting coeOE- cients of the same power of z, one obtains the following system which we will refer to as the fundamental equations: A similar system can written when considering the identity A I but of course the set of fundamental equations (4:0); suOEcient. Finally, for matrix operators each in-nite system of linear equations uniquely determines the coeOEcients of the Laurent series (2). This fact has been noted in [3, 20, 23, 37, 36]. Main results Let us de-ne the following augmented matrix A (t) 2 R (t+1)n\Theta(t+1)n A (t) =6 6 6 6 6 4 A t A and prove the following basic lemma. s be the order of the pole at the origin for the inverse function A \Gamma1 (z). Any eigenvector \Phi 2 R (s+1)n of A (s) corresponding to the zero eigenvalue possesses the property that its -rst n elements are zero. Proof: Suppose on the contrary that there exists an eigenvector \Phi 2 R (s+1)n such that A and not all of its -rst n entries are zero. Then, partition the vector \Phi into s blocks and rewrite (5) in the form with ' 0 6= 0. This means that we have found a generalized Jordan chain of length s + 1. However, from the results of Gohberg and Sigal [15], we conclude that the maximal length of a generalized Jordan chain of A(z) at z = 0 is s. Hence, we came to a contradiction and, consequently, direct proof of Lemma 1 is given in Appendix 1. vectors \Phi 2 R (s+j+1)n in the null space of the augmented matrix A (s+j) , j - 0, possess the property that the -rst (j + 1)n elements are zero. The following theorem provides a theoretical basis for the recursive solution of the in-nite system of fundamental equations (4). Theorem 1 Each coeOEcient X k , k - 0 is uniquely determined by the previous coeOEcients and the set of s fundamental equations Proof: It is obvious that the sequence of Laurent series coeOEcients fX i i=0 is a solution to the fundamental equations (4). Suppose the coeOEcients X i , determined. Next we show that the set of fundamental equations (4.k)-(4.k+s) uniquely determines the next coeOEcient X k . Indeed, suppose there exists another solution ~ are both solutions, we can write A (s)6 4 ~ ~ and A (s)6 4 where the matrix J i is de-ned as follows: ae and where ~ X k+s are any particular solutions of the nonhomogenous linear system (4.k)-(4.k+s). Note that (6) and (7) have identical righthand sides. Of course, the dioeerence between these two righthand sides, [ ~ is in the right null space of A Invoking Lemma 1, the -rst n rows of [ ~ are hence zero. In other words, ~ which proves the theorem.Using the above theoretical background, in the next section we provide three recursive computational schemes which are based on the generalized inverses and on a reduction tech- nique. The reduction technique is based on the following result. A weaker version of this result was utilized in [17] and in [19]. Theorem 2 Let fC k g t suppose that the system of t equations is feasible. Then the general solution is given by (R where C y 0 is the Moore-Penrose generalized inverse of C 0 and Q 2 R m\Thetap is any matrix whose columns form a basis for the right null space of C 0 . Furthermore, the sequence of matrices solves a reduced -nite set of t matrix equations where the matrices D k 2 R p\Thetap and S k 2 R p\Thetan , are computed by the following recursion. Set U Then, where M 2 R p\Thetam is any matrix whose rows form a basis for the left null space of C 0 . Proof: The general solution to the matrix equation (7.0) can be written in the form arbitrary matrix. In order for the equation to be feasible, we need that the right hand side R belongs to R(C 0 is where the rows of M form a basis for N(C T Substituting expression (13) for the general solution into the above feasibility condition, one -nds that W 0 satis-es the equation which can be rewritten as Thus we have obtained the -rst reduced fundamental equation (9.0) with Next we observe that the general solution of equation (7.1) is represented by the formula (R with . Moving on and applying the feasibility condition to equation (7.2), we obtain and again the substitution of expressions (13) and (14) into the above condition yields (R which is rearranged to give The last equation is the reduced equation (9.1) with . Note that this equation imposes restrictions on W 1 as well as on By proceeding in the same way, we eventually obtain the complete system of equations with coeOEcients given by formulas (11) and (12) each of which can be proved by induction in a straightforward way.Remark 3 In the above theorem it is important to observe that the reduced system has the same form as the original but the number of matrix equations is decreased by one and the coeOEcients are reduced in size to matrices in R p\Thetap , where p is the dimension of N(C 0 ) or, equivalently, the number of redundant equations de-ned by the coeOEcient C 0 . In the next section we use this reduction process to solve the system of fundamental equations. Note that the reduction process can be employed to solve any appropriate -nite subset of the fundamental equations. 3 Solution methods In this section we discuss three methods for solving the fundamental equations. The -rst method is based on the direct application of Moore-Penrose generalized inverses. The second method involves the replacement of the original system of the fundamental equations by a system of equations with a reduced dimension. In the third method we show that the reduction process can be applied recursively to reduce the problem to a non-singular system. Since all methods depend to some extent on the prior knowledge of s, we begin by discussing a procedure for the determination of s. A special procedure for determining this order for the case where the matrices A(z) are stochastic and the perturbation series is -nite is given in [16]. It is based on combinatorial properties (actually, network representation) of the processes and hence it is a stable procedure. However, as will be seen in Section 3.4, it is possible to use the third method without prior knowledge of s. Actually, the full reduction version of our procedure determines s as well. Of course, as in any computational method which is used to determine indices which have discrete values, using our procedures in order to compute the order of singularity might lack stability. 3.1 The determination of the order of the pole The rank test on the matrix A (t) proposed by Sain and Massey in [32] is likely to be the most eoeective procedure for determining the value of s. The calculation of rank is essentially equivalent to the reduction of A (t) to a row echelon normal form and it can be argued that row operations can be used successively in order to calculate the rank of A (0) ,A (1) ,A (2) and -nd the minimum value of t for which rankA (t\Gamma1) +n. This minimum value of t equals s, the order of the pole. Note that previous row operations for reducing A (t\Gamma1) to row echelon form are replicated in the reduction of A (t) and do not need to be repeated. For example, if a certain combination of row operations reduces A 0 to row echelon form, then the same operations are used again as part of the reduction of to row echelon form. 3.2 Basic generalized inverse method In this section we obtain a recursive formula for the Laurent series coeOEcients X k , k - 0 by using the Moore-Penrose generalized inverse of the augmented matrix A (s) . y be the Moore-Penrose generalized inverse of A (s) and de-ne the matrices G G G 0s G Furthemore, we would like to note that in fact we use only the -rst n rows of the generalized namely, [G Proposition 1 The coeOEcients of the Laurent series (2) can be calculated by the following recursive formula s G 0s and the matrix J i is de-ned by ae Proof: According to Theorem 1, once the coeOEcients X i , are determined, the next coeOEcient X k can be obtained from the (4.k)-(4.k+s) fundamental equations. A (s)6 4 The general solution to the above system is given in the form6 6 6 4 ~ ~ G 0s G 1s G where the -rst block of matrix \Phi is equal to zero according to Lemma 1. Thus, we immediately obtain the recursive expression (15). In particular, applying the same arguments as above to the -rst s we obtain that 0s .Note that the matrices J j+k in the expression (15) disappear when the regular coeOEcients are computed. Remark 4 The formula (15) is a generalization of the recursive formula for the case where A 0 is invertible. In this case, while initializing with Remark 5 Probably from the computational point of view it is better not to compute the generalized inverse G (s) beforehand, but rather to -nd the SVD or LU decomposition of A and then use these decompositions for solving the fundamental equations (3:k)-(3:k + s). This is the standard approach for solving linear systems with various righthand sides. 3.3 The one step reduction process In this section we describe an alternative scheme that can be used in the case where it is relatively easy to compute the bases for the right and for the left null spaces of A 0 . Speci-cally, be the dimension of the null space of A 0 , let Q 2 IR n\Thetap be a matrix whose columns form a basis for the right null space of A 0 and let M 2 IR p\Thetan be a matrix whose p rows form a basis for the left null space of A 0 . Of course, although possible, we are interested in the singular case where p - 1. Again, as before, we suppose that the coeOEcients X i , determined. Then, by Theorem 1, the next coeOEcient X k is the unique solution to the subsystem of fundamental equations The above system is like the one given in (9) with C and with R Therefore, we can apply the reduction process described in Theorem 2. This results in the system where the coeOEcients D i and S i , can be calculated by the recursive formulae (11) and (12). Remark 6 Note that in many practical applications p is much less than n and hence the above system (17) with D i 2 IR p\Thetap is much smaller than the original system (16). Now we have two options. We can either apply the reduction technique again (see the next subsection for more details) or we can solve the reduced system directly by using the generalized inverse approach. In the latter case, we de-ne and 0t Then, by carrying out a similar computation to the one presented in the proof of Proposition 1, we obtain Once W 0 is determined it is possible to obtain X k from the formula Furthermore, substituting for S i , 0 - i - s\Gamma1, from (12) and changing the order of summation gives A y s Note that by convention the sum disappears when the lower limit is greater than the upper limit. Now, substituting R into the expression (18), we obtain the explicit recursive formula for the Laurent series coeOEcients A y s A (J k+j \Gamma for all k - 1. In particular, the coeOEcient of the -rst singular term in (2) can be given by the 3.4 The complete reduction process As was pointed out in the previous section, the reduced system has essentially the same structure as the original one and hence one can apply again the reduction step described in Theorem 2. Note that each time the reduction step is carried out, the number of matrix equations is reduced by one. Therefore one can perform up to s reduction steps. We now outline how these steps can be executed. We start by introducing the sequence of reduced systems. The fundamental matrix equations for the l-th reduction step are A (l) A (l) A (l) s\Gammal X (l) s\Gammal one gets the original system of fundamental equations and with gets the reduced system for the -rst reduction step described in the previous subsection. Initializing with R (0) I and with A (0) s, the matrices A (l) and R (l) for each reduction step 1 - l - s, can be computed successively by a recursion similar to (11) and (12). In general we have U (l) A (l\Gamma1) A (l) R (l) U (l) where Q (l) and M (l) are the basis matrices for the right and left null spaces respectively of the matrix A (l\Gamma1) 0 and where A (l\Gamma1)y 0 is the Moore-Penrose generalized inverse of A (l\Gamma1) . After s reduction steps, one gets the -nal system of reduced equations A is a unique solution to the subsystem of fundamental equations (4.0)-(4.s) and Theorem 2 states the equivalence of the l-th and (l 1)-st systems of reduced equations, the system (22) possesses a unique solution, and hence matrix A 0 is invertible. Thus, The original solution X 0 can be now retrieved by the backwards recursive relationship Now by taking R (0) one gets the algorithm for computing the Laurent series coeOEcients recursive formulae similar to (15) and (19) can be obtained, but they are quite complicated in the general case. The order s of the pole can also be obtained from the reduction process by continuing the process until A (l) becomes non-singular. The number of reduction steps equals the order of the pole. Note also that the sequence of matrices A (l) can be computed irrespectively of the right hand sides. Once s is determined, one can compute R (l) Computational complexity and comparison with symbolic algebr In this section we compare the computational complexity of the one-step-reduction process when applied to compute X 0 with the complexity of symbolic algebra. In particular, we show that the former comes with a reduced complexity in the case where the pole has a relatively small order. The computational complexity of the other two procedures can be determined similarly. To compute the coeOEcients D i , of the reduced fundamental system (17), one needs to perform O(s 2 n 3 ) operations. The total number of reduced equations is sp (recall that p is the dimension of the null space of A 0 ). Hence, the computational complexity for determining X 0 by the one-step-reduction process is O(maxfs 2 g). The Laurent series (2) in general, and the coeOEcient X 0 in particular, can also be computed by using symbolic algebra. This for example can be executed by MATLAB symbolic toolbox and is done as follows. Since X 0 is uniquely determined by the -rst s equations one needs to do in order to compute X 0 is it to invert symbolically the following matrix polynomial Symbolic computations here mean performing operations, such as multiplication and division, over the -eld of rational functions (and not over the -eld of the reals). In particular, if the degrees of numerators and of denominators of rational functions do not exceed q, then each operation (multiplication or division) which is performed in the -eld of rational functions translates into qlog(q) operations in the -eld of real numbers [1]. Note that during the symbolic inversion of the polynomial matrix (25), the degree of rational functions does not exceed sn. The latter fact follows from Cramer's rule. Thus, the complexity of the symbolic inversion of (25) equals O(n 3 ) \Theta log(sn)). As a result, one gets a matrix A \Gamma1 (z) whose elements are rational functions of z. The elements of the matrix X 0 can then be immediately calculated by dividing the leading coe-cients of the numerator and denominator. Finally, one can see that if s !! n and p !! n, which is typically the case, then our method comes with a reduced computational burden. Concluding remarks In this paper we have shown that the Laurent series for the inversion of an analytic matrix valued function can be computed by solving a system of fundamental linear equations. Furthermore, we demonstrated that the system of fundamental equations can be solved recur- sively. In particular, the coeOEcient X k is determined by the previous coeOEcients X and the next s is the order of the pole. We suggest three basic methods, one without any reduction (see (15)), one with a single reduction step (see and (20)), and one using a complete reduction process with s steps (see (23) and (24)). Of course, an intermediate process with the number of reductions between 1 and s could be used too. We note that when the complete reduction process is used the order of the pole can be determined through the execution of the algorithm. When s !! n and p !! n, the proposed algorithms by far outperform the method based on symbolic algebra. Acknowledgement The authors are grateful to Prof. Jerzy A. Filar for his helpful advice. Also the authors would like to thank anonymous referees for their valuable suggestions and for directing us to some existing literature. Apendix 1: Another proof of Lemma 1 A direct proof of Lemma 1 can be carried out using augmented matrices. Speci-cally, de-ne are the coeOEcients of the Laurent series (2). Then it follows from the fundamental systems (4) and (5) that the augmented matrices A (t) and X (t) satisfy the relationship where the augmented matrix E (t) 2 R (t+1)n\Theta(t+1)n is de-ned by setting p;q=0 where n\Thetan and ae I for s: Now, as before, let \Phi 2 R (s+1)n satisfy the equation A If we multiply equation (27) from the left by X reduces to The vector E (s) \Phi has ' 0 as the (s 1)-st block, which gives the required result. Apendix 2: A Numerical example Let us consider the matrix valued function where 2. Construct the augmented matrices and note that which is the dimension of the original coeOEcients A 0 and A 1 . Therefore, according to the test of Sain and Massey [32], the Laurent expansion for A \Gamma1 (z) has a simple pole. Alternatively, we can compute a basis for which in this particular example consists of only one vector \Theta The -rst three zero elements in q (1) con-rm that the Laurent series has a simple pole. Next we compute the generalized inverse of A (1) given by G (1) 1=3 \Gamma5=12 \Gamma1=12 1=8 1=8 \Gamma1=8 1=3 \Gamma5=12 \Gamma1=12 \Gamma3=8 \Gamma3=8 3=8 Consequently, \Gamma3 \Gamma3 Alternatively, we know that X 0 is uniquely determined by the fundamental equations After one reduction step these equations reduce to where \Theta and \Theta Hence, \Theta and \Gamma3 \Gamma3 The latter expression is identical with (28) and coincides with the one computed by expanding A \Gamma1 (z) with the help of the MATLAB symbolic toolbox. Note that even for this three dimensional example the direct symbolic calculation of the Laurent series takes a relatively long time. --R The design and analysis of computer algorithms The fundamental matrix of singularly perturbed Markov chains Meromorphic operator valued functions. Stability properties of Relative inverses of meromorphic operator functions and associated holomorphic projection functions Generalized Inverses of Linear Transformation iSingular systems of dioeerential equationsj iSingular systems of dioeerential equations IIj A reduction process for perturbed Markov chains The theory of matrices On the local theory of regular analytic matrix functions Matrix Polynomials Analytic matrix functions with prescribed local data An operator generalization of the logarithmic residue theorem and the theorem of Rouch Mean passage times and nearly uncoupled Markov chains Series expansions for Matrix Anal. iTaylor expansions of eigenvalues of perturbed matrices with applications to spectal radii of nonnegative matrices Input retrieval in Perturbation theory for linear operators On the characteristic values and characteristic functions of certain classes of non-selfadjoint equations Mathematical foundations of the state lumping of large systems iInversion of lambda-matrices and application to the theory of linear vibrationsj The Laurent expansion for a nearly singular matrix Introduction to the spectral theory of polynomial operator pencils iSpectral properties of a polynomial op- eratorj Theory of Suboptimal Decisions An introduction to operator polynomials The Laurent expansion of a generalized resolvent with some applications Invertibility of linear time invariant dynamical systems The Laurent expansion for a nearly singular pencil Perturbation series expansions for nearly completely-decomposable Markov chains The Laurent expansion of pencils that are singular at the origin Theory of branching of solutions of non-linear equations The solution of some perturbation problems in the case of matrices and self-adjoint and non-self-adjoint dioeerential equations --TR --CTR Jerzy A. Filar, Controlled Markov chains, graphs, and Hamiltonicity, Foundations and Trends in Stochastic Systems, v.1 n.2, p.77-162, January 2006

matrix inversion;matrix valued functions;analytic perturbation;laurent series

587798

Multiple-Rank Modifications of a Sparse Cholesky Factorization.

Given a sparse symmetric positive definite matrix $\mathbf{AA}\tr$ and an associated sparse Cholesky factorization $\mathbf{LDL}\tr$ or $\mathbf{LL}\tr$, we develop sparse techniques for updating the factorization after either adding a collection of columns to A or deleting a collection of columns from A. Our techniques are based on an analysis and manipulation of the underlying graph structure, using the framework developed in an earlier paper on rank-1 modifications [T. A. Davis and W. W. Hager, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 606--627]. Computationally, the multiple-rank update has better memory traffic and executes much faster than an equivalent series of rank-1 updates since the multiple-rank update makes one pass through L computing the new entries, while a series of rank-1 updates requires multiple passes through L.

Introduction . This paper presents a method for evaluating a multiple rank update or downdate of the sparse Cholesky factorization LDL T or LL T of the matrix AA T , where A is m by n. More precisely, given an m r matrix W, we evaluate the Cholesky factorization of AA T either is +1 (corresponding to an update) and W is arbitrary, or is 1 (corresponding to a downdate) and W consists of columns of A. Both AA T and AA T +WW T must be positive denite. It follows that in the case of an update, and n r m in the case of a downdate. One approach to the multiple rank update is to express it as a series of rank-1 updates and use the theory developed in [10] for updating a sparse factorization after a rank-1 change. This approach, however, requires multiple passes through L as it is updated after each rank-1 change. In this paper, we develop a sparse factorization algorithm that makes only one pass through L. For a dense Cholesky factorization, a one-pass algorithm to update a factorization is obtained from Method C1 in [18] by making all the changes associated with one column of L before moving to the next column, as is done in the following algorithm that overwrites L and D with the new factors of AA T performs oating-point operations. Algorithm 1 (Dense rank-r update/downdate). to r do end for do to r do This work was supported by the National Science Foundation. y davis@cise.u .edu/~davis, PO Box 116120, Department of Computer and Information Science and Engineering, University of Florida, Gainesville, FL 32611-6120. Phone (352) 392-1481. Fax (352) 392-1220. TR-99-006 (June 1999, revised Sept. 2000) z hager@math.u .edu/~hager, PO Box 118105, Department of Mathe- matics, University of Florida, Gainesville, FL 32611-8105. Phone (352) 392-0281. Fax (352) 392-8357. A. DAVIS AND WILLIAM W. HAGER end for do to r do l end for end for end for We develop a sparse version of this algorithm that only accesses and modies those entries in L and D which can change. For the theory in our rank-1 paper [10] shows that those columns which can change correspond to the nodes in an elimination tree on a path starting from the node k associated with the rst nonzero element w k1 in W. For r > 1 we show that the columns of L which can change correspond to the nodes in a subtree of the elimination tree, and we express this subtree as a modication of the elimination tree of AA T . Also, we show that with a reordering of the columns of W, it can be arranged so that in the inner loop where elements in row p of W are updated, the elements that change are adjacent to each other. The sparse techniques that we develop lead to sequential access of matrix elements and to e-cient computer memory tra-c. These techniques to modify a sparse factorization have many applications including the Linear Program Dual Active Set Algorithm least-squares problems in statistics, the analysis of electrical circuits and power systems, structural mechanics, sensitivity analysis in linear programming, boundary condition changes in partial dierential equations, domain decomposition methods, and boundary element methods (see [19]). Section 2 describes our notation. In section 3, we present an algorithm for computing the symbolic factorization of AA T using multisets, which determines the location of nonzero entries in L. Sections 4 and 5 describe our multiple rank symbolic update and downdate algorithms for nding the nonzero pattern of the new factors. Section 6 describes our algorithm for computing the new numerical values of L and D, for either an update or downdate. Our experimental results are presented in Section 7. 2. Notation and background. Given the location of the nonzero elements of AA T , we can perform a symbolic factorization (this terminology is introduced by George and Liu in [15]) of the matrix to predict the location of the nonzero elements of the Cholesky factor L. In actuality, some of these predicted nonzeros may be zero due to numerical cancellation during the factorization process. The statement will mean that l ij is symbolically nonzero. The main diagonals of L and D are always nonzero since the matrices that we factor are positive denite (see [26, p. 253]). The nonzero pattern of column j of L is denoted while L denotes the collection of patterns: Similarly, A j denotes the nonzero pattern of column j of A, while A is the collection of patterns: The elimination tree can be dened in terms of a parent map (see [22]). For any node j, (j) is the row index of the rst nonzero element in column j of L beneath the diagonal element: where \min X" denotes the smallest element of i: Our convention is that the min of the empty set is zero. Note that j < (j) except in the case where the diagonal element in column j is the only nonzero element. The children of node j is the set of nodes whose parent is j: The ancestors of a node j, denoted P(j), is the set of successive parents: for each j, the ancestor sequence is nite. The sequence of nodes j, (j), ((j)), , forming P(j), is called the path from j to the associated tree root, the nal node on the path. The collection of paths leading to a root form an elimination tree. The set of all trees is the elimination forest. Typically, there is a single tree whose root is m, however, if column j of L has only one nonzero element, the diagonal element, then j will be the root of a separate tree. The number of elements (or size) of a set X is denoted jX j, while jAj or jLj denote the sum of the sizes of the sets they contain. 3. Symbolic factorization. For a matrix of the form AA T , the pattern L j of column j is the union of the patterns of each column of L whose parent is j and each column of A whose smallest row index of its nonzero entries is j (see [16, 22]): min Ak=j To modify (3.1) during an update or downdate, without recomputing it from scratch, we need to keep track of how each entry i entered into L j [10]. For example, if (c) changes, we may need to remove a term L c n fcg. We cannot simply perform a set subtraction, since we may remove entries that appear in other terms. To keep track of how entries enter and leave the set L j , we maintain a multiset associated with column j. It has the form 4 TIMOTHY A. DAVIS AND WILLIAM W. HAGER where the multiplicity m(i; j) is the number of children of j that contain row index i in their pattern plus the number of columns of A whose smallest entry is j and that contain row index i. Equivalently, for i 6= j, For we increment the above equation by one to ensure that the diagonal entries never disappear during a downdate. The set L j is obtained from L ] by removing the multiplicities. We dene the addition of a multiset X ] and a set Y in the following way: where Similarly, the subtraction of a set Y from a multiset X ] is dened by where The multiset subtraction of Y from X ] undoes a prior addition. That is, for any multiset X ] and any set Y , we have In contrast ((X [ Y) n Y) is equal to X if and only if X and Y are disjoint sets. Using multiset addition instead of set union, (3.1) leads to the following algorithm for computing the symbolic factorization of AA T . Algorithm 2 (Symbolic factorization of AA T , using multisets). do for each c such that do end for for each k where min A do end for end for 4. Multiple rank symbolic update. We consider how the pattern L changes when AA T is replaced by AA T +WW T . Since we can in essence augment A by W in order to evaluate the new pattern of column in L. According to (3.1), the new pattern L j of column j of L after the update is min Ak=j A where W i is the pattern of column i in W. Throughout, we put a bar over a matrix or a set to denote its new value after the update or downdate. In the following theorem, we consider a column j of the matrix L, and how its pattern is modied by the sets W i . Let L ] j denote the multiset for column j after the rank-r update or downdate has been applied. Theorem 4.1. To compute the new multiset L ] j and perform the following modications: Case A: For each i such that to the pattern for column Case B: For each c such that (c is a child of j in both the old and new elimination tree). Case C: For each c such that (c is a child of j in the new tree, but not the old one). Case D: For each c such that (c is a child of j in the old tree, but not the new one). Proof. Cases A{D account for all the adjustments we need to make in L j in order to obtain L j . These adjustments are deduced from a comparison of (3.1) with (4.1). In case A, we simply add in the W i multisets of (4.1) that do not appear in (3.1). In case B, node c is a child of node j both before and after the update. In this case, we must adjust for the deviation between L c and L c . By [10, Prop. 3.2], after a rank-1 update, L c L c . If w i denotes the i-th column of W, then Hence, updating AA T by WW T is equivalent to r successive rank-1 updates of AA T . By repeated application of [10, Prop. 3.2], L c L c after a rank-r update of AA T . It 6 TIMOTHY A. DAVIS AND WILLIAM W. HAGER follows that L c and L c deviate from each other by the set L c n L c . Consequently, in case B we simply add in L c n L c . In case C, node c is a child of j in the new elimination tree, but not in the old tree. In this case we need to add in the entire set L c n fcg since the corresponding term does not appear in (3.1). Similarly, in case D, node c is a child of j in the old elimination tree, but not in the new tree. In this case, the entire set L c nfcg should be deleted. The case where c is not a child of j in either the old or the new elimination tree does not result in any adjustment since the corresponding L c term is absent from both (3.1) and (4.1). An algorithm for updating a Cholesky factorization that is based only on this theorem would have to visit all nodes j from 1 to m, and consider all possible children c < j. On the other hand, not all nodes j from 1 to m need to be considered since not all columns of L change when AA T is modied. In [10, Thm. 4.1] we show that for the nodes whose patterns can change are contained in P(k 1 ) where we dene . For a rank-r update, let P (i) be the ancestor map associated with the elimination tree for the Cholesky factorization of the matrix Again, by [10, Thm. 4.1], the nodes whose patterns can change during the rank-r update are contained in the union of the patterns Although we could evaluate each i, it is di-cult to do this e-ciently since we need to perform a series of rank-1 updates and evaluate the ancestor map after each of these. On the other hand, by [10, Prop. 3.1] and [10, Prop. 3.2], P (i) (j) P (i+1) (j) for each and j, from which it follows that P (i) Consequently, the nodes whose patterns change during a rank-r update are contained in the set 1ir Theorem 4.2, below, shows that any node in T is also contained in one or more of the sets P (i) (k i ). From this it follows that the nodes in T are precisely those nodes for which entries in the associated columns of L can change during a rank-r update. Before presenting the theorem, we illustrate this with a simple example shown in Figure 4.1. The left of Figure 4.1 shows the sparsity pattern of original matrix AA T , its Cholesky factor L, and the corresponding elimination tree. The nonzero pattern of the rst column of W is 2g. If performed as a single rank-1 update, this causes a modication of columns 1, 2, 6, and 8 of L. The corresponding nodes in the original tree are encircled; these nodes form the path P (1) 8g from node 1 to the root (node 8) in the second tree. The middle of Figure 4.1 shows the matrix after this rank-1 update, and its factor and elimination tree. The entries in the second 1 that dier from the original matrix AA T are shown as small pluses. The second column of W has the nonzero pattern W 7g. As a rank-1 update, this aects columns P (2) of L. These columns form a single path in the nal elimination tree shown in the right of the gure. For the rst rank-1 update, the set of columns that actually change are P (1) 8g. This is a subset of the path in the nal tree. If we use P(1) to guide the work associated with column 1 of W, we visit all the columns after second update after first update elimination tree Elimination tree After first update Elimination tree Original factor L Factor after second update Factor after first update Original matrix After second update1 A T A T6743 A T Original A Fig. 4.1. Example rank-2 update that need to be modied, plus column 7. Node 7 is in the set of nodes P(3) aected by the second rank-1 update, however, as shown in the following theorem. Theorem 4.2. Each of the paths contained in T and conversely, if contained in P (i) Proof. Before the theorem, we observe that each of the paths contained in T . Now suppose that some node j lies in the tree T . We need to prove that it is contained in P (i) s be the largest integer such that P(k s ) contains j and let c be any child of j in T . If c lies on the path P(k i ) for some i, then j lies on the path P(k i ) since j is the parent of c. Since j does not lie on the path P(k i ) for any i > s, it follows that c does not lie on the path P(k i ) for any i > s. Applying this same argument recursively, we conclude that none of the nodes on the subtree of T rooted at j lie on the path P(k i ) for any i > s. Let T j denote the subtree of T rooted at j. Since contained in P(k i ) for each i, none of the nodes of T j lie on any of the paths Thm. 4.1], the patterns of all nodes outside the path are unchanged for each i. Let L (i) c be the pattern of column c in the Cholesky factorization of (4.2). Since any node c contained in T j does not lie 8 TIMOTHY A. DAVIS AND WILLIAM W. HAGER (d,c) (b,e) e f c d a Fig. 4.2. Example rank-8 symbolic update and subtree T on any of the paths c for all i, l s. Since k s is a node of T j , the path P must include j. Figure 4.2 depicts a subtree T for an example rank-8 update. The subtree consists of all those nodes and edges in one or more of the paths P(k 1 These paths form a subtree, and not a general graph, since they are all paths from an initial node to the root of the elimination tree of the matrix L. The subtree T might actually be a forest, if L has an elimination forest rather than an elimination tree. The rst nonzero positions in w 1 through w 8 correspond to nodes k 1 through k 8 . For this example node k 4 happens to lie on the path P (1) (k 1 ). Nodes at which paths rst intersect are shown as smaller circles, and are labeled a through f . Other nodes along the paths are not shown. Each curved arrow denotes a single subpath. For example, the arrow from nodes b to e denotes the subpath from b to e in P(b). This subpath is denoted as P(b; e) in Figure 4.2. The following algorithm computes the rank-r symbolic update. It keeps track of an array of m \path-queues," one for each column of L. Each queue contains a set of path-markers in the range 1 to r, which denote which of the paths P(k 1 ) through next. If two paths have merged, only one of the paths needs to be considered (we arbitrarily select the higher-numbered path to represent the merged paths). This set of path-queues requires O(m Removing and inserting a path-marker in a path-queue takes O(1) time. The only outputs of the algorithm are the new pattern of L and its elimination tree, namely, L ] and (j) for all columns are aected by the rank-r update. We dene L and node j not in T . Case C will occur for c and j prior to visiting column (c), since We thus place c in the lost-child-queue of column (c) when encountering case C for nodes c and j. When the algorithm visits node (c), its lost-child-queue will contain all those nodes for which case D holds. This set of lost-child-queues is not the same as the set of path-queues (although there is exactly one lost-child-queue and one path-queue for each column j of L). Algorithm 3 (Symbolic rank-r update, add new matrix W). Find the starting nodes of each path to r do place path-marker i in path-queue of column k i end for Consider all columns corresponding to nodes in the paths P(k 1 to m do if path-queue of column j is non-empty do for each path-marker i on path-queue of column j do Let c be the prior column on this path (if any), where do Case A: j is the rst node on the path P(k i ), no prior c else if Case B: c is an old child of j, possibly changed else Case C: c is a new child of j and a lost child of (c) place c in lost-child-queue of column (c) endif end for Case D: consider each lost child of j for each c in lost-child-queue of column j do end for Move up one step in the path(s) Let i be the largest path-marker in path-queue of column j Place path-marker i in path-queue of column (j) if path-queue of column j non-empty end for The optimal time for a general rank-r update is A. DAVIS AND WILLIAM W. HAGER The actual time taken by Algorithm 3 only slightly higher, namely, because of the O(m) book-keeping required for the path-queues. In most practical cases, the O(m) term will not be the dominant term in the run time. Algorithm 3 can be used to compute an entire symbolic factorization. We start by factorizing the identity matrix I = II T into LDL III. In this case, we have j. The initial elimination tree is a forest of m nodes and no edges. We can now determine the symbolic factorization of I +AA T using the rank-r update algorithm above, with m. This matrix has identical symbolic factors as AA T . Case A will apply for each column in A, corresponding to the min Ak=j term in (3.1). Since (c) = 0 for each c, cases B and D will not apply. At column j, case C will apply for all children in the elimination tree, corresponding to the term in (3.1). Since duplicate paths are discarded when they merge, we modify each column j once, for each child c in the elimination tree. This is the same work performed by the symbolic factorization algorithm, Algorithm 2, which is O(jLj). Hence, Algorithm 3 is equivalent to Algorithm 2 when we apply it to the update I +AA T . Its run time is optimal in this case. 5. Multiple rank symbolic downdate. The downdate algorithm is analogous. The downdated matrix is AA T WW T where W is a subset of the columns of A. In a downdate, P(k) P(k), and thus rather than following the paths P(k i ), we follow the paths P(k i ). Entries are dropped during a downdate, and thus L j L j and (j) (j). We start with L ] j and make the following changes. Case A: If then the pattern W i is removed from column j, Case B: If then c is a child of j in both the old and new tree. We need to remove from L ] entries in the old pattern L c but not in the new pattern L c , Case C: If for some node c, then c is a child of j in the old elimination tree, but not the new tree. We compute MULTIPLE-RANK MODIFICATIONS 11 Case D: If for some node c, then c is a child of j in the new tree, but not the old one. We compute Case C will occur for c and j prior to visiting column (c), since We thus place c in the new-child-queue of (c) when encountering case C for nodes c and j. When the algorithm visits node (c), its new-child-queue will contain all those nodes for which case D holds. Algorithm 4 (Symbolic rank-r downdate, remove matrix W). Find the starting nodes of each path to r do place path-marker i in path-queue of column k i end for Consider all columns corresponding to nodes in the paths P(k 1 to m do if path-queue of column j is non-empty do for each path-marker i on path-queue of column j do Let c be the prior column on this path (if any), where do Case A: j is the rst node on the path P(k i ), no prior c else if Case B: c is an old child of j, possibly changed else Case C: c is a lost child of j and a new child of (c) place c in new-child-queue of column (c) endif end for Case D: consider each new child of j for each c in new-child-queue of j do end for Move up one step in the path(s) Let i be the largest path-marker in path-queue of column j Place path-marker i in path-queue of column (j) if path-queue of column j non-empty end for A. DAVIS AND WILLIAM W. HAGER The time taken by Algorithm 4 is which slightly higher than the optimal time, In most practical cases, the O(m) term in the asymptotic run time for Algorithm 4 will not be the dominant term. 6. Multiple rank numerical update and downdate. The following numerical rank-r update/downdate algorithm, Algorithm 5, overwrites L and D with the updated or downdated factors. The algorithm is based on Algorithm 1, the one-pass version of Method C1 in [18] presented in Section 1. The algorithm is used after the symbolic update algorithm (Algorithm 3) has found the subtree T corresponding to the nodes whose patterns can change, or after the symbolic downdate algorithm (Algorithm 4) has found T . Since the columns of the matrix W can be reordered without aecting the product WW T , we reorder the columns of W using a depth-rst search [6] of T (or T ) so that as we march through the tree, consecutive columns of W are utilized in the computations. This reordering improves the numerical up- date/downdate algorithm by placing all columns of W that aect any given subpath next to each other, eliminating an indexing operation. Reordering the columns of a sparse matrix prior to Cholesky factorization is very common [3, 22, 23, 25]. It improves data locality and simplies the algorithm, just as it does for reordering W in a multiple rank update/downdate. The depth rst ordering of the tree changes as the elimination tree changes, so columns of W must be ordered for each update or downdate. To illustrate this reordering, consider the subtree T in Figure 4.2 for a rank-8 update. If the depth-rst-search algorithm visits child subtrees from left to right, the resulting reordering is as shown in Figure 6.1. Each subpath in Figure 6.1 is labeled with the range of columns of W that aect that subpath, and with the order in which the subpath is processed by Algorithm 5. Consider the path from node c to e. In Figure 4.2, the columns of L corresponding to nodes on this subpath are updated by columns 2, 8, 3, and 5 of W, in that order. In the reordered subtree (Figure 6.1), the columns on this subpath are updated by columns 5 through 8 of the reordered W. Algorithm 5 (Sparse numeric rank-r modication, add WW T ). The columns of W have been reordered. to r do end for for each subpath in depth-rst-search order in T Let c 1 through c 2 be the columns of W that aect this subpath for each column j in the subpath do do e f c d a 13th 2nd 1st 6th 3rd 4th 7th 8th 9th 10th 11th 12th 3 47-885th Fig. 6.1. Example rank-8 update after depth-rst-search reordering end for for all do l end for end for end for end for The time taken by r rank-1 updates [10] is O@ r where L (i) j is the pattern of column j after the i-th rank-1 update. This time is asymptotically optimal. A single rank-r update cannot determine the paths but uses P(k i ) instead. Thus, the time taken by Algorithm 5 for a rank-r update is O@ r This is slightly higher than (6.1), because 14 TIMOTHY A. DAVIS AND WILLIAM W. HAGER Table Dense matrix performance for 64-by-64 matrices and 64-by-1 vectors operation M ops DGEMM (matrix-matrix multiply) 171.6 DGEMV (matrix-vector multiply) 130.0 DTRSV (solve DAXPY (the vector computation DDOT (the dot product the i-th column of W does not necessarily aect all of the columns in the path P(k i ). If w i does not aect column j, then w ji and will both be zero in the inner loop in Algorithm 5. An example of this occurs in Figure 4.1, where column 1 of W does not aect column 7 of L. We could check this condition, and reduce the asymptotic run time to O@ r In practice, however, we found that the paths dier much. Including this test did not improve the overall performance of our algorithm. The time taken by Algorithm 5 for a rank-r downdate is similar, namely, O@ r The numerical algorithm for updating and downdating LL T is essentially the same as that for LDL T [4, 24]; the only dierence is a diagonal scaling. For either LL T or LDL T , the symbolic algorithms are identical. 7. Experimental results. To test our methods, we selected the same experiment as in our earlier paper on the single-rank update and downdate [10], which mimics the behavior of the Linear Programming Dual Active Set Algorithm [20]. The rst consists of 5446 columns from a larger 6071- arising in an airline scheduling problem (DFL001) [13]. The 5446 columns correspond to the optimal solution of the linear programming problem. Starting with an initial LDL T factorization of the matrix T , we added columns from B (corresponding to an update) until we obtained the factors of 10 6 I +BB T . We then removed columns in a rst-in-rst-out order (corresponding to a downdate) until we obtained the original factors. The LP DASA algorithm would not perform this much work (6784 updates and 6784 down- dates) to solve this linear programming problem. Our experiment took place on a Sun Ultra Enterprise running the Solaris 2.6 operating system, with eight 248 Mhz UltraSparc-II processors (only one processor was used) and 2GB of main memory. The dense matrix performance in millions of oating-point operations per second (M ops) of the BLAS [12] is shown in Table 7.1. All results presented below are for our own codes (except for colmmd, spooles, and the BLAS) written in the C programming language and using double precision oating point arithmetic. We rst permuted the rows of B to preserve sparsity in the Cholesky factors of BB T . This can be done e-ciently with colamd [7, 8, 9, 21], which is based on an Table Average update and downdate performance results ops r in seconds update downdate update downdate 9 14 approximate minimum degree ordering algorithm [1]. However, to keep our results consistent with our prior rank-1 update/downdate paper [10], we used the same permutation as in those experiments (from colmmd [17]). Both colamd and Matlab's colmmd compute the ordering without forming BB T explicitly. A symbolic factorization of BB T nds the nonzero counts of each column of the factors. This step takes an amount of space this is proportional to the number of nonzero entries in B. It gives us the size of a static data structure to hold the factors during the updating and downdating process. The numerical factorization of BB T is not required. A second symbolic factorization nds the rst nonzero pattern L. An initial numerical factorization computes the rst factors L and D. We used our own non-supernodal factorization code (similar to SPARSPAK [5, 15]), since the update/downdate algorithms do not use supernodes. A supernodal factorization code such as spooles [3] or a multifrontal method [2, 14] can get better performance. The factorization method used has no impact on the performance of the update and downdate algorithms. We ran dierent experiments, each one using a dierent rank-r update and downdate, where r varied from 1 to 16. After each rank-r update, we solved the sparse linear system LDL T using a dense right-hand side b. To compare the performance of a rank-1 update with a rank-r update (r > 1), we divided the run time of the rank-r update by r. This gives us a normalized time for a single rank-1 update. The average time and M ops rate for a normalized rank-1 update and downdate for the entire experiment is shown in Table 7.2. The time for the update, downdate, or solve increases as the factors become denser, but the performance in terms of M ops is fairly constant for all three operations. The rst rank-16 update when the factor L is sparsest takes 0.47 seconds (0.0294 seconds normalized) and runs at 65.5 M ops compared to 65.1 M ops in Table 7.2 for the average speed of all the rank-16 updates. The performance of each step is summarized in Table 7.3. A rank-5 update takes about the same time as using the updated factors to solve the sparse linear system even though the rank-5 update performs 2.6 times the work. The work, in terms of oating-point operations, varies only slightly as r changes. With rank-1 updates, the total work for all the updates is 17.293 billion A. DAVIS AND WILLIAM W. HAGER Table Dense matrix performance for 64-by-64 matrices and 64-by-1 vectors Operation Time (sec) M ops Notes colamd ordering 0.45 - Symbolic factorization (of BB T Symbolic factorization for rst L 0.46 - 831 thousand nonzeros Numeric factorization for rst L (our code) 20.07 24.0 Numeric factorization for rst L (spooles) 18.10 26.6 Numeric factorization of BB T (our code) 61.04 18.5 not required Numeric factorization of BB T (spooles) 17.80 63.3 not required Average rank-16 update 0.63 65.1 compare with rank-1 Average rank-5 update 0.25 51.0 compare with solve step Average rank-1 update 0.084 30.3 Average solve LDL T point operations, or 2.55 million per rank-1 update. With rank-16 updates (the worst case), the total work increases to 17.318 billion oating-point operations. The downdates take a total of 17.679 billion oating-point operations (2.61 million per rank-1 downdate), while the rank-16 downdates take a total of 17.691 billion operations. This conrms the near-optimal operation count of the multiple rank update/downdate, as compared to the optimal rank-1 update/downdate. Solving when L is sparse and b is dense, and computing the sparse LDL T factorization using a non-supernodal method, both give a rather poor computation- to-memory-reference ratio of only 2/3. We tried the same loop unrolling technique used in our update/downdate code for our sparse solve and sparse LDL T factorization codes, but this resulted in no improvement in performance. A sparse rank-r update or downdate can be implemented in a one-pass algorithm that has much better memory tra-c than that of a series of r rank-1 modications. In our numerical experimentation with the DFL001 linear programming test problem, the rank-r modication was more than twice as fast as r rank-1 modications for r 11. The superior performance of the multiple rank algorithm can be explained using the computation-to-memory-reference ratio. If c in Algorithm 5 (a subpath aected by only one column of W), it can be shown that this ratio is about 4/5 when L j is large. The ratio when c aected by 16 columns of W) is about 64/35 when L j is large. Hence, going from a rank-1 to a rank-16 update improves the computation-to-memory-reference ratio by a factor of about 2.3 when column j of L has many nonzeros. By comparison, the level-1 BLAS routines for dense matrix computations (vector computations such as DAXPY and DDOT) [11] have computation-to-memory-reference ratios between 2/3 and 1. The level-2 BLAS (DGEMV and DTRSV, for example) have a ratio of 2. 8. Summary . Because of improved memory locality, our multiple-rank sparse update/downdate method is over twice as fast as our prior rank-1 update/downdate method. The performance of our new method (65.1 M ops for a sparse rank-16 update) compares favorably with both the dense matrix performance (81.5 M ops to solve the dense system and the sparse matrix performance (18.0 M ops to solve the sparse system and an observed peak numerical factorization of 63.3 ops in spooles) on the computer used in our experiments. Although not strictly optimal, the multiple-rank update/downdate method has nearly the same operation count as the rank-1 update/downdate method, which has an optimal operation count. MULTIPLE-RANK MODIFICATIONS 17 --R An approximate minimum degree ordering algorithm Vectorization of a multiprocessor multifrontal code SPOOLES: an object-oriented sparse matrix library A Cholesky up- and downdating algorithm for systolic and SIMD architectures SPARSPAK: Waterloo sparse matrix package Introduction to Algorithms A column approximate minimum degree ordering algorithm A column approximate minimum degree ordering algorithm Modifying a sparse Cholesky factorization Philadelphia: SIAM Publications A set of level-3 basic linear algebra subprograms Distribution of mathematical software via electronic mail The multifrontal solution of inde Computer Solution of Large Sparse Positive De A data structure for sparse QR and LU factorizations Sparse matrices in MATLAB: design and implementation Methods for modifying matrix factorizations Updating the inverse of a matrix An approximate minimum degree column ordering algorithm The role of elimination trees in sparse factorization A supernodal Cholesky factorization algorithm for shared-memory multiprocessors New York --TR --CTR W. Hager, The Dual Active Set Algorithm and Its Application to Linear Programming, Computational Optimization and Applications, v.21 n.3, p.263-275, March 2002 Ove Edlund, A software package for sparse orthogonal factorization and updating, ACM Transactions on Mathematical Software (TOMS), v.28 n.4, p.448-482, December 2002 Matine Bergounioux , Karl Kunisch, Primal-Dual Strategy for State-Constrained Optimal Control Problems, Computational Optimization and Applications, v.22 n.2, p.193-224, July 2002 Nicholas I. M. Gould , Jennifer A. Scott , Yifan Hu, A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations, ACM Transactions on Mathematical Software (TOMS), v.33 n.2, p.10-es, June 2007

numerical linear algebra;matrix updates;cholesky factorization;sparse matrices;mathematical software;direct methods

587801

On Algorithms For Permuting Large Entries to the Diagonal of a Sparse Matrix.

We consider bipartite matching algorithms for computing permutations of a sparse matrix so that the diagonal of the permuted matrix has entries of large absolute value. We discuss various strategies for this and consider their implementation as computer codes. We also consider scaling techniques to further increase the relative values of the diagonal entries. Numerical experiments show the effect of the reorderings and the scaling on the solution of sparse equations by a direct method and by preconditioned iterative techniques.

Introduction We say that an n \Theta n matrix A has a large diagonal if the absolute value of each diagonal entry is large relative to the absolute values of the off-diagonal entries in its row and column. Permuting large nonzero entries onto the diagonal of a sparse matrix can be useful in several ways. If we wish to solve the system where A is a nonsingular square matrix of order n and x and b are vectors of length n, then a preordering of this kind can be useful whether direct or iterative methods are used for solution (see Olschowka and Neumaier (1996) and Duff and Koster (1997)). The work in this report is a continuation of the work reported by Duff and Koster (1997) who presented an algorithm that maximizes the smallest entry on the diagonal and relies on repeated applications of the depth first search algorithm MC21 (Duff 1981) in the Harwell Subroutine Library (HSL 1996). In this report, we will be concerned with other bipartite matching algorithms for permuting the rows and columns of the matrix so that the diagonal of the permuted matrix is large. The algorithm that is central to this report computes a matching that corresponds to a permutation of a sparse matrix such that the product (or sum) of the diagonal entries is maximized. This algorithm is already mentioned and used in Duff and Koster (1997), but is not fully described. In this report, we describe the algorithm in more detail. We also consider a modified version of this algorithm to compute a permutation of the matrix that maximizes the smallest diagonal entry. We compare the performance of this algorithm with that of Duff and Koster (1997). We also investigate the influence of scaling of the matrix. Scaling can be used before or after computation of the matching to make the diagonal entries even larger relative to the off-diagonals. In particular, we look at a sparse variant of a bipartite matching and scaling algorithm of Olschowka and Neumaier (1996) that first maximizes the product of the diagonal entries and then scales the matrix so that these entries are one and all other entries are no greater than one. The rest of this report is organized as follows. In Section 2, we describe some concepts of bipartite matching that we need for the description of the algorithms. In Section 3, we review the basic properties of algorithm MC21. MC21 is a relatively simple algorithm that computes a matching that corresponds to a permutation of the matrix that puts as many entries as possible onto the diagonal without considering their numerical values. The algorithm that maximizes the product of the diagonal entries is described in Section 4. In Section 5, we consider the modified version of this algorithm that maximizes the smallest diagonal entry of the permuted matrix. In Section 6, we consider the scaling of the reordered matrix. Computational experience for the algorithms applied to some practical problems and the effect of the reorderings and scaling on direct and iterative methods of solution are presented in Sections 7 to 7.2. The effect on preconditioning is also discussed. Finally, we consider some of the implications of this current work in Section 8. matching be a general n \Theta n sparse matrix. With matrix A, we associate a bipartite graph E) that consists of two disjoint node sets V r and V c and an edge set E, where (u; v) 2 E implies that . The sets V r and V c have cardinality n and correspond to the rows and columns of A respectively. Edge (i; only if a ij 6= 0. We define the sets ROW c . These sets correspond to the positions of the entries in row i and column j of the sparse matrix respectively. We use both to denote the absolute value and to signify the number of entries in a set, sequence, or matrix. The meaning should always be clear from the context. A subset M ' E is called a matching (or assignment) if no two edges of M are incident to the same node. A matching containing the largest number of edges possible is called a maximum cardinality matching (or simply maximum matching). A maximum matching is a perfect matching if every node is incident to a matching edge. Obviously, not every bipartite graph allows a perfect matching. However, if the matrix A is nonsingular, then there exists a perfect matching for GA . A perfect matching M has cardinality n and defines an n \Theta n permutation matrix so that both PA and AP are matrices with the matching entries on the (zero-free) diagonal. Bipartite matching problems can be viewed as a special case of network flow problems (see, for example, Ford Jr. and Fulkerson (1962)). The more efficient algorithms for finding maximum matchings in bipartite graphs make use of augmenting paths. Let M be a matching in GA . A node v is matched if it is incident to an edge in M . A path P in GA is defined as an ordered set of edges in which successive edges are incident to the same node. A path P is called an M-alternating path if the edges of P are alternately in M and not in M . An M-alternating path P is called an M - augmenting path if it connects an unmatched row node with an unmatched column node. In the bipartite graph in Figure 2.1, there exists an M-augmenting path from column node 8 to row node 8. The matching M (of cardinality 7) is represented by the thick edges. The black entries in the accompanying matrix correspond to the matching and the connected matrix entries to the M-augmenting path. If it is clear from the context which matching M is associated with the M-alternating and M-augmenting paths, then we will simply refer to them as alternating and augmenting paths. Let M and P be subsets of E. We define If M is a matching and P is an M-augmenting path, then M \Phi P is again a matching, and jM \Phi is an M-alternating cyclic path, i.e., an alternating path whose first and last edge are incident to the same node, then M \Phi P is also a matching and jM \Phi Figure 2.1: Augmenting path r1368425791368In the sequel, a matching M will often be represented by a pointer array Augmenting paths in a bipartite graph G can be found by constructing alternating trees. An alternating tree subgraph of G rooted at a row or column node and each path in T is an M-alternating path. An alternating tree rooted at a column node j 0 can be grown in the following way. We start with the initial alternating tree (;; fj 0 g; ;) and consider all the column nodes j 2 T c in turn. Initially . For each node j, we check the row nodes i 2 COL(j) for which an alternating path from i to j 0 does not yet exist. If node i is already matched, we add row node i, column node to T . If i is not matched, we extend T by row node i and edge (and the path in T from node i to the root forms an augmenting path). A key observation for the construction of a maximum or perfect matching is that a matching M is maximum if and only if there is no augmenting path relative to M . Alternating trees can be implemented using a pointer array c such that, given an edge (i; is either the root node of the tree, or the edges are consecutive edges in an alternating path towards the root. Augmenting paths in an alternating tree (provided they exist) can thus easily be obtained from p and m. Alternating trees are not unique. In general, one can construct several alternating trees starting from the same root node that have equal node sets, but different edge sets. Different alternating trees in general will contain different augmenting paths. The matching algorithms that we describe in the next sections impose different criteria on the order in which the paths in the alternating trees are grown in order to obtain augmenting paths and maximum matchings with special properties. Matching The asymptotically fastest currently known algorithm for finding a maximum matching is by Hopcroft and Karp (1973). It has a worst-case complexity of O( the number of entries in the sparse matrix. An efficient implementation of this algorithm can be found in Duff and Wiberg (1988). The algorithm MC21 implemented by Duff (1981) has a theoretically worst-case behaviour of O(n- ), but in practice it behaves more like O(n Because this latter algorithm is simpler, we concentrate on this in the following although we note that it is relatively straightforward to use the algorithm of Hopcroft and Karp (1973) in a similar way to how we will use MC21 in later sections. MC21 is a depth-first search algorithm with look-ahead. It starts off with an empty matching M , and hence all column nodes are unmatched initially. See Figure 3.1. For each unmatched column node j 0 in turn, an alternating tree is grown until an augmenting path with respect to the current matching M is found (provided one exists). A set B is used to mark all the matched row nodes that have been visited so far. Initially, First, the row nodes in COL(j 0 ) are searched (look-ahead) for an unmatched node i 0 . If one is found, the singleton path is an M-augmenting path. If there is no such unmatched node, then an unmarked matched node i is marked, the nodes i 0 and , and the edges (i are added to the alternating tree (by setting The search then continues with column node . For node j 1 , the row nodes in COL(j 1 ) are first checked for an unmatched node. If one exists, say then the path forms an augmenting path. If there is no such unmatched node, a remaining unmarked node i 1 is picked from is set to j 1 , , and the search moves to node j 2 . This continues in a similar (depth-first search) fashion until either an augmenting path (with nodes j 0 and i k unmatched) or until for some k ? 0, COL(j k ) does not contain an unmarked node. In the latter case, MC21 backtracks by resuming the search at the previously visited column node j k\Gamma1 for some remaining unmarked node i 0 Backtracking for if MC21 resumes the search at column node j 0 and COL(j 0 ) does not contain an unmarked node, then an M-augmenting path starting at node j 0 does not exist. In this case, MC21 continues with the construction of a new alternating tree starting at the next unmatched column node. (The final maximum matching will have cardinality at most n \Gamma 1 and hence will not be perfect.) Figure 3.1: Outline of MC21. do repeat if there exists i 2 COL(j) and i is unmatched then else if there exists else until iap 6= null or if iap 6= null then augment along path from node iap to node j 0 end for Weighted matching In this section, we describe an algorithm that computes a matching for permuting a sparse matrix A such that the product of the diagonal entries of the permuted matrix is maximum in absolute value. That is, the algorithm determines a matching that corresponds to a permutation oe that maximizes n Y This maximization multiplicative problem can be translated into a minimization additive problem by defining matrix log a where a is the maximum absolute value in column j of matrix A. Maximizing (4.1) is equal to minimizing log log a oe i log ja ioe i (log a oe i log ja ioe i Minimizing (4.2) is equivalent to finding a minimum weight perfect matching in an edge weighted bipartite graph. This is known in literature as the bipartite weighted matching problem or (linear sum) assignment problem in linear programming and combinatorial optimization. Numerous algorithms have been proposed for computing minimum weight perfect matchings, see for example Burkard and Derigs (1980), Carpaneto and Toth (1980), Carraresi and Sodini (1986), Derigs and Metz (1986), Jonker and Volgenant (1987), and Kuhn (1955). A practical example of an assignment problem is the allotment of tasks to in the cost matrix C represents the cost or benefit of assigning person i to task j. be a real-valued n \Theta n E) be the corresponding bipartite graph each of whose edges (i; . The weight of a matching M in GC , denoted by c(M ), is defined by the sum of its edge weights, i.e., (i;j)2M A perfect matching M is said to be a minimum weight perfect matching if it has smallest possible weight, i.e., c(M) - c(M 0 ), for all possible maximum matchings M 0 . The key concept for finding a minimum weight perfect matching is the so-called shortest augmenting path. An M-augmenting path P starting at an unmatched column node j is called shortest if c(M \Phi P possible M-augmenting paths P 0 starting at node j. We define as the length of alternating path P . A matching M is called extreme if and only if it does not allow any alternating cyclic path with negative length. The following two relations hold. First, a perfect matching has minimum weight if it is extreme. Second, if matching M is extreme and P is a shortest M-augmenting path, then M \Phi P is extreme also. The proof for this goes roughly as follows. Suppose M \Phi P is not extreme. Then there exists an alternating cyclic path Q such that c((M \Phi P is extreme, there must exist a subset forms an M-augmenting path and is shorter than P . Hence, P is not a shortest M-augmenting path. This contradicts the supposition. These two relations form the basis for many algorithms for solving the bipartite weighted matching problem: start from any (possibly empty) extreme matching M and successively augment M along shortest augmenting paths until M is maximum (or perfect). In the literature, the problem of finding a minimum weight perfect matching is often stated as the following linear programming problem. Find matrix minimizing X subject to X If there is a solution to this linear program, there is one for which x ij 2 f0; 1g and there exists a permutation matrix X such that 1g is a minimum weight perfect matching (Edmonds and Karp 1972, Kuhn 1955). Furthermore, M has minimum weight if and only if there exist dual variables u i and v j with Using the reduced weight matrix the reduced weight c(M) of matching M equals the reduced length l(P ) of any M-alternating path P equals (i;j)2P nM and if M \Phi P is a matching, the reduced weight of M \Phi P equals Thus, finding a shortest augmenting path in graph GC is equivalent to finding an augmenting path in graph G C , with minimum reduced length. edge contains no alternating paths P with negative length, leading subpath P 0 of P . Shortest augmenting paths in a weighted bipartite graph E) can be obtained by means of a shortest alternating path tree. A shortest alternating path tree T is an alternating tree each of whose paths is a shortest path in G. For any node i we define d i as the length of the shortest path in T from node i to the root node (d if no such path exists). T is a shortest alternating path tree if and only if d for every edge (i; nodes i, j, An outline of an algorithm for constructing a shortest alternating path tree rooted at column node j 0 is given in Figure 4.1. Because the reduced weights c ij are non-negative, and graph G C contains no alternating paths with negative length, we can use a sparse variant of Dijkstra's algorithm (Dijkstra 1959). The set of row nodes is partitioned into three sets B, Q, and W . B is the set of (marked) nodes whose shortest alternating paths and distances to node j 0 are known. Q is the set of nodes for which an alternating path to the root is known that is not necessarily the shortest possible. W is the set of nodes for which an alternating path does not exist or is not known yet. (Note that since W is defined implicitly as V r n (B [Q), it is not actually used in Figure 4.1.) The algorithm starts with shortest alternating tree and extends the tree until an augmenting path is found that is guaranteed to be a shortest augmenting path with respect to the current matching M . Initially, the length of the shortest augmenting path lsap in the tree is set to infinity, and the length of the shortest alternating path lsp from the root to any node in Q is set to zero. On each pass through the main loop, another column node j is chosen that is closest to the root j 0 . Initially Each row node i 2 COL(j) whose shortest alternating path to the root is not known yet (i 62 B), is considered. If P j 0 !j!i , the shortest alternating path from the root node 0 to node j (with length lsp) extended by edge (i; j) from node j to node i (with length longer than the tentative shortest augmenting path in the tree (with length lsap), then there is no need to modify the tree. If P j 0 !j!i has length smaller than lsap, and i is unmatched, then a new shorter augmenting path has been found and lsap is updated. If i is matched and P j 0 !j!i is also shorter than the current shortest alternating path to (with length d i ), then a shorter alternating path to node i has been found and the tree is updated, d i is updated, and if node i has not been visited previously, i is moved to Q. Next, if Q is not empty, a node i 2 Q is determined that is closest to the root. Since all weights c ij in the bipartite graph are non-negative, there cannot be any other alternating path to node i that is shorter than the current one. Node i is marked (by adding it to B), and the search continues with column node j This continues until there are no more column nodes to be searched or until no new augmenting path can be found whose length is smaller than the current shortest one (line lsap - lsp). The original Dijkstra algorithm (intended for dense graphs) has O(n 2 ) complexity. For sparse problems, the complexity can be reduced to O(- log n) by implementing the set Q as a k-heap in which the nodes i are sorted by increasing distance d i from the root (see for example Tarjan (1983) and Gallo and Pallottino (1988)). The running time of the algorithm is dominated by the operations on the heap Q of which there are O(n) delete operations, O(n) insert operations, and O(-) modification operations (these are necessary each time a distance d i is updated). Each insert and modification operation runs in O(log k n) time, a delete operation runs in O(k log k n) time. Consequently, the algorithm for finding a shortest augmenting path in a sparse bipartite graph has run time n) and the total run time for the sparse bipartite weighted algorithm is n). If we choose 2, the algorithm uses binary heaps and we obtain a time bound of O(n(- +n) log 2 n). If we choose we obtain a bound of O(n- log -=n n). The implementation of the heap Q is similar to the implementation proposed in Derigs and Metz (1986). Q is a pair (Q is an array that contains all the row nodes for which the distance to the root is shortest (lsp), and By separating the nodes in Q that are closest to the root, we may reduce the number of operations on the heap, especially in those situations where the cost matrix C has only few different numerical values and many alternating paths have the same length. Deleting a node from Q for which d i is smallest (see Figure 4.1), now consists of choosing an (arbitrary) element from Q 1 . If Q 1 is empty, then we first move all the nodes in Q 2 that are closest to the root to Q 1 . After the augmentation, the reduced weights c ij have to be updated to ensure that alternating paths in the new G have non-negative length. This is done by modifying the Figure 4.1: Construction of a shortest augmenting path. while true do dnew if dnew ! lsap then unmatched then lsap := dnew; isap := else choose if lsap - lsp then exit while-loop; if lsap 6= 1 then augment along path from node isap to node j dual vectors u and v. If is the shortest alternating path tree that was constructed until the shortest augmenting path was found, then u i and v j are updated as The updated dual variables u and v satisfy (4.3) and the new reduced weights c ij are non-negative. The running time of the weighted matching algorithm can be decreased considerably by means of a cheap heuristic that determines a large initial extreme matching M . We use the strategy proposed by Carpaneto and Toth (1980). We calculate Inspecting the sets COL(j) for each column node j in turn, we determine a large initial matching M of edges for which for each remaining unmatched column node j, every node i 2 COL(j) is considered for which and that is matched to a column node other than j, say j . If an unmatched row node can be found for which c i in M is replaced by (i; having repeated this for all unmatched columns, the search for shortest augmenting paths starts with respect to the current matching. Finally, we note that the above weighted matching algorithm can also be used for maximizing the sum of the diagonal entries of matrix A (instead of maximizing the product of the diagonal entries). To do this, we again minimize (4.2), but we redefine matrix C as 0; otherwise: Maximizing the sum of the diagonal entries is equal to minimizing (4.2), since a oe i (a oe i 5 Bottleneck matching We describe a modification of the weighted bipartite matching algorithm from the previous section for permuting rows and columns of a sparse matrix A such that the smallest ratio between the absolute value of a diagonal entry and the maximum absolute value in its column is maximized. That is, the modification computes a permutation oe that maximizes min 1-i-n a oe i where a j is the maximum absolute value in column j of the matrix A. Similarly to the previous section, we transform this into a minimization problem. We define the matrix a j Then maximizing (5.1) is equal to minimizing 1-i-n a oe i a oe i 1-i-n Given a matching M in the bipartite graph E), the bottleneck value of M is defined as (i;j)2M The problem is to find a perfect (or maximum) bottleneck matching M for which c(M) is minimal, i.e. c(M) - c(M 0 ), for all possible maximum matchings M 0 . A matching M is called extreme if it does not allow any alternating cyclic path P for which c(M \PhiP The bottleneck algorithm starts off with any extreme matching M . The initial bottleneck value b is set to c(M ). Each pass through the main loop, an alternating tree is constructed until an augmenting path P is found for which either c(M \Phi P or c(M \Phi P small as possible. The initializations and the main loop for constructing such an augmenting path are those of Figure 4.1. Figure 5.1 shows the inner-loop of the weighted matching algorithm of Figure 4.1 modified to the case of the bottleneck objective function. The main differences are that the sum operation on the path lengths in Figure 4.1 is replaced by the "max" operation and, as soon as an augmenting path P is found whose length lsap is less than or equal to the current bottleneck value b, the main loop is exited, P is used to augment M , and b is set to max(b; lsap). The bottleneck algorithm does not modify the edge weights c ij . Similarly to the implementation discussed in Section 4, the set Q is implemented as a now the array Q 1 contains all the nodes whose distance to the root is less than or equal to the tentative bottleneck value b. Q 2 contains the nodes whose distance to the root is larger than the bottleneck value but not infinity. Q 2 is again implemented as a heap. Figure 5.1: Modified inner loop of Figure 4.1 for the construction of a bottleneck augmenting path. dnew := if dnew ! lsap then unmatched then lsap := dnew; isap := if lsap - b then exit while-loop; else A large initial extreme matching can be found in the following way. We define as the smallest entry in row i and column j, respectively. A lower bound b 0 for the bottleneck value is An extreme matching M can be obtained from the edges (i; j) for which c ij - b 0 ; we scan all nodes in turn and for each node i 2 COL(j) that is unmatched and for which is added to M . Then, for each remaining unmatched column node j, every node i 2 COL(j) is considered for which c ij - b and that is matched to a column node other than j, say j . If an unmatched row node i 1 2 COL(j 1 ) can be found for which c i is replaced by (i; having done this for all unmatched columns, the search for shortest augmenting paths starts with respect to the current matching. Other initialization procedures can be found in the literature. For example, a slightly more complicated initialization strategy is used by Finke and Smith (1978) in the context of solving transportation problems. For every use as the number of admissible edges incident to row node i and column node j respectively. The idea behind using g i and h j is that once an admissible edge (i; j) is added to M , all the other admissible edges that are incident to nodes i and j are no longer candidates to be added to M . Therefore, the method tries to pick admissible edges such that the number of admissible edges that become unusable is minimal. First, a row node i with minimal g i is determined. From the set ROW (i) an admissible entry (i; (provided one exists) is chosen for which h j is minimal and (i; j) is added to M . After deleting the edges and the edges (k; j), k 2 COL(j), the method repeats the same for another row node i 0 with minimal g i 0 . This continues until all admissible edges are deleted from the graph. Finally, we note that instead of maximizing (5.1) we also could have maximized the smallest absolute value on the diagonal. That is, we maximize min 1-i-n and define the matrix C as Note that this problem is rather sensitive to the scaling of the matrix A. Suppose for example that the matrix A has a column containing only one nonzero entry whose absolute value v is the smallest absolute value present in A. Then, after applying the bottleneck algorithm, the bottleneck value b will be equal to this small value. The smallest entry on the diagonal of the permuted matrix is maximized, but the algorithm did not have any influence on the values of the other diagonal values. Scaling the matrix prior to applying the bottleneck algorithm avoids this. In Duff and Koster (1997), a different approach is taken to obtain a bottleneck matching. Let A ffl denote the matrix that is obtained by setting to zero in A all entries a ij for which ja ij denote the matching obtained by removing from matching M all the entries (i; j) for which ja ij Throughout the algorithm, fflmax and fflmin are such that a maximum matching of size jM j does not exist for A fflmax but does exist for A fflmin . At each step, ffl is chosen in the interval (fflmin; fflmax), and a maximum matching for the matrix A ffl is computed using a variant of MC21. If this matching has size jM j, then fflmin is set to ffl, otherwise fflmax is set to ffl. Hence, the size of the interval decreases at each step and ffl will converge to the bottleneck value. After termination of the algorithm, M 0 is the computed bottleneck matching and ffl the corresponding bottleneck value. 6 Scaling Olschowka and Neumaier (1996) use the dual solution produced by the weighted matching algorithm to scale the matrix. Let u and v be such that they satisfy relation (4.3). If we define then we have Equality holds when that is (i; In words, D 1 AD 2 is a matrix whose diagonal entries are one in absolute value and whose off-diagonal entries are all less than or equal to one. Olschowka and Neumaier (1996) call such a matrix an I-matrix and use this in the context of dense Gaussian elimination to reduce the amount of pivoting that is needed for numerical stability. The more dominant the diagonal of a matrix, the higher the chance that diagonal entries are stable enough to serve as pivots for elimination. For iterative methods, the transformation of a matrix to an I-matrix is also of interest. For example, from Gershgorin's theorem we know that the union of all discs contains all eigenvalues of the n \Theta n matrix A. Disc K i has center at a ii and radius that is equal to the sum of the absolute off-diagonal values in row i. Since the diagonal entries of an I-matrix are all one, all the n disks have center at 1. The estimate of the eigenvalues will be sharper as A deviates less from a diagonal matrix. That is, the smaller the radii of the discs, the better we know where the eigenvalues are situated. If we are able to reduce the radii of the discs of an I-matrix, i.e. reduce the off-diagonal values, then we tend to cluster the eigenvalues more around one. In the ideal case, all the discs of an I-matrix have a radius smaller than one, in which case the matrix is strictly row-wise diagonally dominant. This guarantees that many types of iterative methods will converge (in exact even simple ones like the Jacobi and Gauss-Seidel method. However, if at least one disc remains with radius larger than or close to one, zero eigenvalues or small eigenvalues are possible. A straightforward (but expensive) attempt to decrease large off-diagonal entries of a matrix is by row and column equalization (Olschowka and Neumaier 1996). Let A be an I-matrix. We define matrix simplicity we assume that A contains no zero entries.) Equalization consists of repeatedly equalizing the largest absolute value in row i and the largest absolute values in column i: for k := do for to n do For and thus, if we define d 1 the algorithm minimizes the largest off-diagonal absolute value in matrix D 1 AD 2 . The diagonal entries do not change. Note that the above scaling strategy does not guarantee that all off-diagonal entries of an I-matrix will be smaller than one in absolute value, for example if the I-matrix A contains two off-diagonal entries a kl and a lk , k 6= l, whose absolute values are both one. 7 Experimental results In this section, we discuss several cases where the reorderings algorithms from the previous section can be useful. These include the solution of sparse equations by a direct method and by an iterative technique. We also consider its use in generating a preconditioner for an iterative method. The set of matrices that we used for our experiments are unsymmetric matrices taken from the Harwell-Boeing Sparse Matrix Test Collection (Duff, Grimes and Lewis 1992) and from the sparse matrix collection at the University of Florida (Davis 1997). All matrices are initially row and column scaled. By this we mean that the matrix is scaled so that the maximum entry in each row and in each column is one. The computer used for the experiments is a SUN UltraSparc with 256 Mbytes of main memory. The algorithms are implemented in Fortran 77. We use the following acronyms. MC21 is the matching algorithm from the Harwell Subroutine Library for computing a matching such that the corresponding permuted matrix has a zero free-diagonal (see Section 3). BT is the bottleneck bipartite matching algorithm from Section 5 for permuting a matrix such that the smallest ratio between the absolute value of a diagonal entry and the maximum absolute value in its column is maximized. BT' is the bottleneck bipartite matching algorithm from Duff and Koster (1997). MPD is the weighted matching algorithm from Section 4 and computes a permutation such that the product of the diagonal entries of the permuted matrix is maximum in absolute value. MPS is equal to the MPD algorithm, but after the permutation, the matrix is scaled to an I-matrix (see Section 6). Table 7.1 shows for some large sparse matrices the order, number of entries, and the time for the algorithms to compute a matching. The times for MPS are not listed, because they are almost identical to those for MPD. In general, MC21 needs the least time to compute a matching, except for the ONETONE and TWOTONE matrices. For these matrices, the search heuristic that is used in MC21 (a depth-first search with look-ahead) does not perform well. This is probably caused by the ordering of the columns (variables) and the entries inside the columns of the matrix. A random permutation of the matrix prior to applying MC21 might lead to other results. There is not a clear winner between the bottleneck algorithms BT and BT', although we note that BT' requires the entries inside the columns to be sorted by value. This sorting can be expensive for relatively dense matrices. MPD is in general the most expensive algorithm. This can be explained by the more selective way in which this algorithm constructs augmenting paths. 7.1 Experiments with a direct solution method For direct methods, putting large entries on the diagonal suggests that pivoting down the diagonal might be more stable. Indeed, stability can still not be guaranteed, but if we have a solution scheme like the multifrontal method of Duff and Reid (1983), where a symbolic phase chooses the initial pivotal sequence and the subsequent factorization phase then modifies this sequence for stability, it can mean that the modification required is less than if the permutation were not applied. In the multifrontal approach of Duff and Reid (1983), later developed by Amestoy and Duff (1989), an analysis is performed on the structure of A+A T to obtain an ordering that reduces fill-in under the assumption that all diagonal entries will be numerically suitable for pivoting. The numerical factorization is guided by an assembly tree. At each node of the tree, some steps of Gaussian elimination are performed on a dense submatrix whose Schur complement is then passed to the parent node in the tree where it is assembled Table 7.1: Times (in seconds) for matching algorithms. Order of matrix is n and number of entries - . Matrix GOODWIN 7320 324784 0.27 2.26 4.17 1.82 (or summed) with Schur complements from the other children and original entries of the matrix. If, however, numerical considerations prevent us from choosing a pivot then the algorithm can proceed, but now the Schur complement that is passed to the parent is larger and usually more work and storage will be needed to effect the factorization. The logic of first permuting the matrix so that there are large entries on the diagonal, before computing the ordering to reduce fill-in, is to try and reduce the number of pivots that are delayed in this way thereby reducing storage and work for the factorization. We show the effect of this in Table 7.2 where we can see that even using MC21 can be very beneficial although the other algorithms can show significant further gains. In Table 7.3, we show the effect of this on the number of entries in the factors. Clearly this mirrors the results in Table 7.2. In addition to being able to select the pivots chosen by the analysis phase, the multifrontal code MA41 will do better on matrices whose structure is symmetric or nearly so. Here, we define the structural symmetry for a matrix A as the number of entries a ij for which a ji is also an entry, divided by the total number of entries. The structural symmetry after the permutations is shown in Table 7.4. The matching orderings in some cases increase the symmetry of the resulting reordered matrix, which is particularly apparent when we have a very sparse system with many zeros on the diagonal. In that case, the reduction in number of off-diagonal entries in the reordered matrix has an influence on the symmetry. Notice that, in this respect, the more sophisticated matching algorithms may actually cause problems since they could reorder a symmetrically structured matrix with a zero-free diagonal, whereas MC21 will leave it unchanged. Table 7.2: Number of delayed pivots in factorization from MA41. An "-" indicates that MA41 needed more than 200 MBytes of memory. Matrix Matching algorithm used None MC21 BT MPD MPS GOODWIN 536 1622 427 53 41 Table 7.3: Number of entries (10 3 ) in the factors from MA41. Matrix Matching algorithm used None MC21 BT MPD MPS ONETONE2 14,083 2,876 2,298 2,170 2,168 GOODWIN 1,263 2,673 2,058 1,282 1,281 Table 7.4: Structural symmetry after permutation. Matrix Matching algorithm used None MC21 BT MPD/MPS GEMAT11 Finally, Table 7.5 shows the effect on the solution times of MA41. We sometimes observe a dramatic reduction in time for the solution when preceded by a permutation. Table 7.5: Solution time required by MA41. Matrix Matching algorithm used None MC21 BT MPD MPS GOODWIN 3.64 14.63 7.98 3.56 3.56 Our implementations of the algorithms described in this paper have been used successfully by Li and Demmel (1998) to stabilize sparse Gaussian elimination in a distributed-memory environment without the need for dynamic pivoting. Their method decomposes the matrix into an N \Theta N block matrix A[1 : by using the notion of unsymmetric supernodes (Demmel, Eisenstat, Gilbert, Li and Liu 1995). The blocks are mapped cyclically (in both row and column dimensions) onto the nodes (processors) of a two-dimensional rectangular processor grid. The mapping is such that at step k of the numerical factorization, a column of processors factorizes the block column A[k : N; k], a row of processes participates in the triangular solves to obtain the block row U and all processors participate in the corresponding multiple-rank update of the remaining The numerical factorization phase in this method does not use (dynamic) partial pivoting on the block columns. This allows for the a priori computation of the nonzero structure of the factors, the distributed data structures, the communication pattern, and a good load balancing scheme, which makes the factorization more scalable on distributed-memory machines than factorizations in which the computational and communication tasks only become apparent during the elimination process. To ensure a solution that is numerically stable, the matrix is permuted and scaled before the factorization to make the diagonal entries large compared to the off-diagonal entries, any tiny pivots encountered during the factorization are perturbed, and a few steps of iterative refinement are performed during the triangular solution phase if the solution is not accurate enough. Numerical experiments demonstrate that the method (using the implementation of the MPS algorithm) is as stable as partial pivoting for a wide range of problems. 7.2 Experiments with iterative solution methods For iterative methods, simple techniques like Jacobi or Gauss-Seidel converge more quickly if the diagonal entry is large relative to the off-diagonals in its row or column, and techniques like block iterative methods can benefit if the entries in the diagonal blocks are large. Additionally, for preconditioning techniques, for example for diagonal preconditioning or incomplete LU preconditioning, it is intuitively evident that large diagonals should be beneficial. 7.2.1 Preconditioning by incomplete factorizations In incomplete factorization preconditioners, pivots are often taken from the diagonal and fill-in is discarded if it falls outside a prescribed sparsity pattern. (See Saad (1996) for an overview.) Incomplete factorizations are used so that the resulting factors are more economical to store, to compute, and to solve with. One of the reasons why incomplete factorizations can behave poorly is that pivots can be arbitrarily small (Benzi, Szyld and van Duin 1997, Chow and Saad 1997). Pivots may even be zero in which case the incomplete factorization fails. Small pivots allow the numerical values of the entries in the incomplete factors to become very large, which leads to unstable and therefore inaccurate factorizations. In such cases, the norm of the residual U will be large. (Here, - L and - U denote the computed incomplete A way to improve the stability of the incomplete factorization, is to preorder the matrix to put large entries onto the diagonal. Obviously, a successful factorization still cannot be guaranteed, because nonzero diagonal entries may become very small (or even zero) during the factorization, but the reordering may mean that zero or small pivots are less likely to occur. Table 7.6 shows some results for the reorderings applied prior to incomplete factorizations of the form ILU(0), ILU(1), and ILUT and the iterative methods GMRES(20), BiCGSTAB, and QMR. In some cases, the method will only converge after the permutation, in others it greatly improves the convergence. However, we emphasize that permuting large entries to the diagonal of matrix will not always improve the accuracy and stability of incomplete factorization. An inaccurate factorization can also occur in the absence of small pivots, when many (especially large) fill-ins are dropped from the incomplete factors. In this respect, it may be beneficial to apply a symmetric permutation after the matching reordering to reduce fill-in. Another kind of instability in incomplete factorizations, which can occur with and without small pivots, is severe ill-conditioning of the triangular factors. (In this situation, jjRjj F need not be very large, but jjI \Gamma A( - will be.) This is also a common situation when the coefficient matrix is far from diagonally dominant. Table 7.6: Number of iterations required by some preconditioned iterative methods after permutation. Matrix and method Matching algorithm QMR 72 21 12 12 MAHINDAS WEST0497 We also performed a set of experiments in which we first permuted the columns of the matrix A by using a reordering computed by one of the matching algorithms, followed by a symmetric permutation of A generated by the reverse Cuthill-McKee ordering (Cuthill and McKee 1969) applied to A . The motivation behind this is that the number of entries that is dropped from the factors can be reduced by applying a reordering of the matrix that reduces fill-in. In the experimental results, we noticed that the additional permutation sometimes has a positive as well as a negative effect on the performance of the iterative solvers. Table 7.7 shows some results for the three iterative methods from Table 7.6 preconditioned by ILUT on the WEST matrices from the Harwell-Boeing collection. Table 7.7: Number of iterations required by some ILUT-preconditioned iterative methods after the matching reordering with and without reverse Cuthill-McKee. Matrix and method Matching algorithm Matching algorithm without RCM with RCM 7.2.2 Experiments with a block iterative solution method The Jacobi method is not a particularly current or powerful method so we focussed our experiments on the block Cimmino implementation of Arioli, Duff, Noailles and Ruiz (1992), which is equivalent to using a block Jacobi algorithm on the normal equations. In this implementation, the subproblems corresponding to blocks of rows from the matrix are solved by the sparse direct method MA27 (HSL 1996). We show the effect of this in Table 7.8 on the problem MAHINDAS from Table 7.6. The matching algorithm was followed by a reverse Cuthill-McKee algorithm to obtain a block tridiagonal form. The matrix was partitioned into 2, 4, 8, and 16 blocks of rows and the accelerations used were block CG algorithms with block sizes 1, 4, and 8. The block rows are chosen of equal (or nearly equal) size. Table 7.8: Number of iterations of block Cimmino algorithm for the matrix MAHINDAS. Acceleration Matching algorithm # block rows None MC21 BT MPD MPS In general, we noticed in our experiments that the block Cimmino method often was more sensitive to the scaling (in MPS) and less to the reorderings. The convergence properties of the block Cimmino method are independent of row scaling. However, the sparse direct solver MA27 (HSL 1996) used for solving the augmented systems, performs numerical pivoting during the factorizations of the augmented matrices. Row scaling might well change the choice of the pivot order and affect the fill-in in the factors and the accuracy of the solution. Column scaling should affect convergence of the method since it can be considered as a diagonal preconditioner. For more details see (Ruiz 1992). 8 Conclusions and future work We have considered, in Sections 3-4, techniques for permuting a sparse matrix so that the diagonal of the permuted matrix has entries of large absolute value. We discussed various criteria for this and considered their implementation as computer codes. We also considered in Section 6 possible scaling strategies to further improve the weight of the diagonal with respect to the off-diagonal values. In Section 7, we then indicated several cases where such a permutation (and scaling) can be useful. These include the solution of sparse equations by a direct method and by an iterative technique. We also considered its use in generating a preconditioner for an iterative method. The numerical experiments show that for a multifrontal solver and preconditioned iterative methods, the effect of these reorderings can be dramatic. The effect on the block Cimmino iterative method seems to be less dramatic. For this method, the discussed scaling tends to have a more important effect. While it is clear that reordering matrices so that the permuted matrix has a large diagonal can have a very significant effect on solving sparse systems by a wide range of techniques, it is somewhat less clear that there is a universal strategy that is best in all cases. One reason for this is that increasing the size of the diagonal only is not always sufficient to improve the performance of the method. For example, for the incomplete preconditioners that we used for the numerical experiments in Section 7, it is not only the size of the diagonal but also the amount and size of the discarded fill-in plays an important role. We have thus started experimenting with combining the strategies mentioned in Sections 3-4 and, particularly for generating a preconditioner and the block Cimmino approach, with combining our unsymmetric ordering with symmetric orderings. Another interesting extension to the discussed reorderings is a block approach to increase the size of diagonal blocks instead of only the diagonal entries and use for example a block Jacobi preconditioner on the permuted matrix. This is of particular interest for the block Cimmino method. One could also build other criteria into the weighting for obtaining a bipartite matching, for example, to incorporate a Markowitz cost so that sparsity would also be preserved by the choice of the resulting diagonal as a pivot. Such combination would make the resulting ordering suitable for a wider class of sparse direct solvers. Finally, we notice in our experiments with MA41 that one effect of the matching algorithm was to increase the structural symmetry of unsymmetric matrices. We are exploring further the use of ordering techniques that more directly attempt to increase structural symmetry. Acknowledgments We are grateful to Michele Benzi of Los Alamos National Laboratory and Miroslav Tuma of the Czech Academy of Sciences for their assistance on the preconditioned iterative methods and Daniel Ruiz of ENSEEIHT for his help on block iterative methods. --R Orderings for incomplete factorization preconditioning of nonsymmetric problems Assignment and Matching Problems: Solution Methods with FORTRAN-Programs Experimental study of ILU preconditioners for indefinite matrices Reducing the bandwidth of sparse symmetric matrices University of Florida sparse matrix collection A supernodal approach to sparse partial pivoting The design and use of algorithms for permuting large entries to the diagonal of sparse matrices Users' guide for the Harwell-Boeing sparse matrix collection (Release 1) Making sparse Gaussian elimination scalable by static pivoting Solution of large sparse unsymmetric linear systems with a block iterative method in a multiprocessor environment Iterative methods for sparse linear systems Data structures and network algorithms --TR --CTR Laura Grigori , Xiaoye S. Li, A new scheduling algorithm for parallel sparse LU factorization with static pivoting, Proceedings of the 2002 ACM/IEEE conference on Supercomputing, p.1-18, November 16, 2002, Baltimore, Maryland Abdou Guermouche , Jean-Yves L'Excellent , Gil Utard, Impact of reordering on the memory of a multifrontal solver, Parallel Computing, v.29 n.9, p.1191-1218, September Abdou Guermouche , Jean-Yves L'excellent, Constructing memory-minimizing schedules for multifrontal methods, ACM Transactions on Mathematical Software (TOMS), v.32 n.1, p.17-32, March 2006 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 Kai Shen, Parallel sparse LU factorization on second-class message passing platforms, Proceedings of the 19th annual international conference on Supercomputing, June 20-22, 2005, Cambridge, Massachusetts Patrick R. Amestoy , Iain S. Duff , Jean-Yves L'excellent , Xiaoye S. Li, Analysis and comparison of two general sparse solvers for distributed memory computers, ACM Transactions on Mathematical Software (TOMS), v.27 n.4, p.388-421, December 2001 Kai Shen, Parallel sparse LU factorization on different message passing platforms, Journal of Parallel and Distributed Computing, v.66 n.11, p.1387-1403, November 2006 Xiren Wang , Wenjian Yu , Zeyi Wang , Xianlong Hong, An improved direct boundary element method for substrate coupling resistance extraction, Proceedings of the 15th ACM Great Lakes symposium on VLSI, April 17-19, 2005, Chicago, Illinois, USA Anshul Gupta, Recent advances in direct methods for solving unsymmetric sparse systems of linear equations, ACM Transactions on Mathematical Software (TOMS), v.28 n.3, p.301-324, September 2002 Timothy A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software (TOMS), v.30 n.2, p.165-195, June 2004 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

bipartite weighted matching;shortest path algorithms;sparse matrices;iterative methods;preconditioning;direct methods

587812

Approximation of the Determinant of Large Sparse Symmetric Positive Definite Matrices.

This paper is concerned with the problem of approximating det(A)1/n for a large sparse symmetric positive definite matrix A of order n. It is shown that an efficient solution of this problem is obtained by using a sparse approximate inverse of A. The method is explained and theoretical properties are discussed. The method is ideal for implementation on a parallel computer. Numerical experiments are described that illustrate the performance of this new method and provide a comparison with Monte Carlo--type methods from the literature.

Introduction . Throughout this paper, A denotes a real symmetric positive definite matrix of order n with eigenvalues In a number of applications, for example in lattice Quantum Chromodynamics [12], certain functions of the determinant of A, such as det(A) 1=n or ln(det(A)) are of interest. It is well-known (cf. also x2) that for large n the function A ! det(A) has poor scaling properties and can be very ill-conditioned for certain matrices A. In this paper we consider the function A few basic properties of this function are discussed in x2. In this paper we present a new method for approximating d(A) for large sparse matrices A. The method is based on replacing A by a matrix which is in a certain sense close to A \Gamma1 and for which the determinant can be computed with low computational costs. One popular method for approximating A is based on the construction of an incomplete Cholesky factorization. This incomplete factorization is often used as a preconditioner when solving linear systems with matrix A. In this paper we use another preconditioning technique, namely that of sparse approximate inverses (cf. [1, 7, 9, 11]). In Re-mark 3.10 we comment on the advantages of the use of sparse approximate inverse preconditoning for approximating d(A). Let A = LL T be the Cholesky decomposition of A. Then using techniques known from the literature a sparse approximate inverse GE of L, i.e. a lower triangular matrix GE which has a prescribed sparsity structure E and which is an approximation of L \Gamma1 , can be constructed. We then use det(GE ii as an approximation for d(A). In x3 we explain the construction of GE and discuss theoretical properties of this sparse approximate inverse. For example, such a sparse approximate inverse can be shown to exist for any symmetric positive definite A and has an interesting optimality property related to d(A). As a direct consequence of this optimality property one obtains that holds and that the approximation of d(A) by det(GE ) \Gamma2=n becomes better if a larger sparsity pattern E is used. Institut fur Geometrie und Praktische Mathematik, RWTH Aachen, Templergraben 55, D-52056 Germany. A. REUSKEN In x4 we consider the topic of error estimation. In the paper [2] bounds for the determinant of symmetric positive definite matrices are derived. These bounds, in which the Frobenius norm and an estimate of the extreme eigenvalues of the matrix involved are used, often yield rather poor estimates of the determinant (cf. experiments in [2]). In x4.1 we apply this technique to the preconditioned matrix GEAG T E and thus obtain reliable but rather pessimistic error bounds. It turns out that this error estimation technique is rather costly. In x4.2 we introduce a simple and cheap Monte Carlo technique for error estimation. In x5 we apply the new method to a few examples of large sparse symmetric positive definite matrices. 2. Preliminaries. In this section we discuss a few elementary properties of the function d. We give a comparision between the conditioning of the function d and of the fuction A ! We use the notation k \Delta k 2 for the Euclidean norm and denotes the spectral condition number of A. The trace of the matrix A is denoted by tr(A). Lemma 2.1. Let A and ffiA be symmetric positive definite matrices of order n. The following inequalities hold: Proof. The result in (2.1a) follows from Y The result in (2.1b) follows from the inequality between the geometric and arithmetic mean: Y From the Courant-Fischer characterization of eigenvalues it follows that for all i. Hence holds. Now note that Y Y APPROXIMATION OF DETERMINANTS 3 Thus the result in (2.1c) is proved. The result in (2.1c) shows that the function d(A) is well-conditioned for matrices A which have a not too large condition number (A). We now briefly discuss the difference in conditioning between the functions A ! det(A). For any symmetric positive definite matrix B of order n we have From the Courant-Fischer eigenvalue characterization we obtain for all i. Hence B is SPD B is SPD with equality for I . Thus for the condition number of the function d we have =n Note that for the diagonal matrix in the inequality in (2.2) one obtains equality for n !1. For this A and with we have equality in the second inequality in (2.1c), too. For ~ the condition number is given by ~ i.e. n times larger than the condition number in (2.2). The condition numbers for d and ~ d give an indication of the sensitivity if the perturbation kffiAk 2 is sufficiently small. Note that the bound in (2.1c) is valid for arbitrary symmetric positive definite perturbations ffiA. The bound shows that even for larger perturbations the function is well-conditioned at A if (A) is not too large. For the function ~ the effect of relatively large perturbations can be much worse than for the asymptotic case (ffiA ! 0), which is characterized by the condition number in (2.3). Consider, for example, for 2 a perturbation ~ ~ which is very large if, for example, The results in this section show that the numerical approximation of the function is a much easier task than the numerical approximation of A ! det(A). 3. Sparse approximate inverse. In this section we explain and analyze the construction of a sparse approximate inverse of the matrix A. Let A = LL T be the Cholesky factorization of A, i.e. L is lower triangular and L \Gamma1 AL I . Note that ii . We will construct a sparse lower triangular approximation G of L \Gamma1 and approximate d(A) by d(G) ii . The construction of a sparse approximate inverse that we use in this paper was introduced in [9, 10, 11] and can also be found in [1]. Some of the results derived in this section are presented in [1], too. 4 A. REUSKEN We first introduce some notation. Let E ae f(i; ng be a given sparsity pattern. By #E we denote the number of elements in E. Let SE be the set of n \Theta n matrices for which all entries are set to zero if the corresponding index is not in E: For use the representation For we define the projection Note that the matrix is symmetric positive definite. Typical choices of the sparsity pattern E (cf. x5) are such that n i is a very small number compared to n (e.g. In such a case the projected matrix P i AP T i has a small dimension. To facilitate the analysis below, we first discuss the construction of an approximate sparse inverse ME 2 SE in a general framework. For we use the representation Note that if n For given A; B 2 R n\Thetan with A symmetric positive definite we consider the following problem: determine In (3.3) we have #E equations to determine #E entries in ME . We first give two basic lemmas which will play an important role in the analysis of the sparse approximate inverse that will be defined in (3.9). Lemma 3.1. The problem (3.3) has a unique solution the ith row of ME is given by m T i with i is the ith row of B. Proof. The equations in (3.3) can be represented as (b T for all i with i is the ith row of ME . Consider an i with . For the unknown entries in m i we obtain the system of equations APPROXIMATION OF DETERMINANTS 5 which is equivalent to The matrix P i AP T i is symmetric positive definite and thus m i must satisfy Using we obtain the result in (3.4). The construction in this proof shows that the solution is unique. Below we use the Frobenius norm, denoted by k Lemma 3.2. Let A = LL T be the Cholesky factorization of A and let ME 2 SE be the unique solution of (3.3). Then ME is the unique minimizer of the functional Proof. Let e i be the ith basis vector in R n . Take M 2 SE . The ith rows of M and B are denoted by m T The minimum of the functional (3.6) is obtained if in (3.7) we minimize the functionals for all i with (3.8) can be rewritten as The unique minimum of this functional is obtained for " Using Lemma 3.1 it follows that ME is the unique minimizer of the functional (3.6). Sparse approximate inverse. We now introduce the sparse approximate inverse that will be used as an approximation for L \Gamma1 . For this we choose a lower triangular ng and we assume that (i; i) 2 E l for all i. The sparse approximate inverse is constructed in two steps: l such that ( " 2: 6 A. REUSKEN The construction of G E l in (3.9) was first introduced in [9]. A theoretical background for this factorized sparse inverse is given in [11]. The approximate inverse " (3.9a) is of the form (3.3) with I . From Lemma 3.1 it follows that in (3.9a) there is a unique solution " . Note that because E l is lower triangular and (i; i) 2 E l we have Hence it follows from Lemma 3.1 that the ith row of " denoted by g T i , is given by The ith entry of g i , i.e. e T which is strictly positive because i is symmetric positive definite. Hence diag( " contains only strictly positive entries and the second step (3.9b) is well-defined. . The sparse in (3.9a) can be computed by solving the (low dimensional) symmetric positive definite systems We now derive some interesting properties of the sparse approximate inverse as in (3.9). We start with a minimization property of " Theorem 3.3. Let A = LL T be the Cholesky factorization of A and D := l as in (3.9a) is the unique minimizer of the functional Proof. The construction of " in (3.9a) is as in (3.3) with I . Hence Lemma 3.2 is applicable with I . It follows that " l is the unique minimizer of Decompose L \GammaT as L \GammaT strictly upper triangular. Then D and R are lower and strictly upper triangular, respectively, and we obtain: F Hence the minimizers in (3.13) and (3.12) are the same. Remark 3.4. From the result in Theorem 3.3 we see that in a scaled Frobenius norm (scaling with D l is the optimal approximation of " in the set S E l , in the sense that " L is closest to the identity. A seemingly more natural minimization problem is min i.e. we directly approximate L \Gamma1 (instead of " do not use the scaling with . The minimization problem (3.14) is of the form as in Lemma 3.2 with APPROXIMATION OF DETERMINANTS 7 . Hence the unique minimizer in (3.14), denoted by ~ must satisfy (3.3) Because E l contains only indices (i; ~ must satisfy This is similar to the system of equations in (3.9a), which characterizes " l . However, in (3.16) one needs the values L ii , which in general are not available. Hence opposite to the minimization problem related to the functional (3.12) the minimization problem (3.14) is in general not solvable with acceptable computational costs. 2 The following lemma will be used in the proof of Theorem 3.7. Lemma 3.5. Let " l be as in (3.9a). Decompose " L), with D diagonal and " strictly lower triangular. Define E l ng. L is the unique minimizer of the functional and also of the functional Furthermore, for " D we have Proof. From the construction in (3.9a) it follows that is such that ( " . This is of the form (3.3) From Lemma 3.2 we obtain that " L is the unique minimizer of the functional i.e., of the functional (3.17). From the proof of Lemma 3.2, with that the minimization problem min decouples into seperate minimization problems (cf. (3.8)) for the rows of L: min l Al i g (3.20) 8 A. REUSKEN for all i with i and a T i are the ith rows of L and A, respectively. The minimization problem corresponding to (3.18) is min Y Y Al This decouples into the same minimization problems as in (3.20). Hence the functionals in (3.17) and (3.18) have the same minimizer. Using the construction of " in (3.9a) we obtain ii J D) (i;k)2E l Hence " ii holds for all i, i.e., (3.19) holds. Corollary 3.6. From (3.19) it follows that diag( " and thus, using (3.9b) we obtain for the sparse approximate inverse G E l . 2 The following theorem gives a main result in the theory of approximate inverses. It was first derived in [11]. A proof can be found in [1], too. Theorem 3.7. Let G E l be the approximate inverse in (3.9). Then G E l is the unique minimizer of the functional Proof. For G 2 S E l we use the decomposition diagonal and L . Furthermore, for L The inequality in (3.23) follows from the inequality between the arithmetic and geometric in (3.9a) we use the decomposition " L). For the approximate L). From Lemma 3.5 it follows that det(J L ) det(J " . Furthermore from Lemma 3.5 we obtain that for G L) we have I and thus equality in We conclude that G E l is the unique minimizer of the functional APPROXIMATION OF DETERMINANTS 9 in (3.22). Remark 3.8. The quantity can be seen as a nonstandard condition number (cf. [1, 9]). Properties of this quantity are given in [1] (Theorem 13.5). One elementary property is Corollary 3.9. For the approximate inverse G E l as in (3.9) we have (cf.(3.21)) i.e., Y Y Let ~ l be a lower triangular sparsity pattern that is larger than E l , i.e., E l ae ~ ng. From the optimality result in Theorem 3.7 it follows that ~ Motivated by the theoretical results in Corollary 3.9 we propose to use the sparse approximate l as in (3.9) for approximating an estimate for d(A). Some properties of this method are discussed in the following remark. Remark 3.10. We consider the method of approximating d(A) by d(G . The practical realization of this method boils down to chosing a sparsity and solving the (small) systems in (3.11). We list a few properties of this 1. The sparse approximate inverse exists for every symmetric positive definite A. Note that such an existence result does not hold for the incomplete Cholesky factorization. Furthermore, this factorization is obtained by solving low dimensional symmetric positive definite systems of the form P i AP T (cf. (3.11)), which can be realized in a stable way. 2. The systems P i AP T solved in parallel. 3. For the computation of d(G need the diagonal entries of " (cf. (3.24)). In the systems P i AP T e i we then only have to compute the last entry of " . If these systems are solved using the Cholesky lower triangular) we only need the of L i , since ("g i 4. The sparse approximate inverse has an optimality property related to the determinant: The functional G From this the inequality (3.24) and the monotonicity result (3.25) follow. A. REUSKEN 5. From(3.24) we obtain the upper bound 0 for the relative error 1. In x4 we will derive useful lower bounds for this relative error. These are a posteriori error bounds which use the matrix G E l . 2 4. A posteriori error estimation. In the previous section it has been explained how an estimate d(G E l ) \Gamma2 of d(A) can be computed. From (3.24) we have the error bound In this section we will discuss a posteriori estimators for the error d(A)=d(G In x4.1 we apply the analysis from [2] to derive an a posteriori lower bound for the quantity in (4.1). This approach results in safe, but often rather pessimistic bounds for the error. In x4.2 we propose a very simple stochastic method for error estimation. This method, although it does not yield guaranteed bounds for the error, turns out to be very useful in practice. 4.1. estimation based on bounds from [2] . In this section we show how the analysis from [2] can be used to obtain an error estimator. We first recall a main result from [2] (Theorem 2). Let A be a symmetric positive matrix of order n, F and exp l exp In [2] this result is applied to obtain computable bounds for d(A). Often these bounds yield rather poor estimates of d(A). In the present paper we approximate use the result in (4.2) for error estimation. The upper bound turns out to be satisfactory in numerical experiments (cf. x5). Therefore we restrict ourselves to the derivation of a lower bound for d(A)=d(G E l ) \Gamma2 , based on the left inequality in (4.2). Theorem 4.1. Let G E l be the approximate inverse from (3.9) and 0 ! ff 1. The following holds: ff 1, and exp Proof. The right inequality in (4.3) is already given in (4.1). We introduce the for the eigenvalues of G E l AG T l . From (3.21) we obtainn and from this it follows that ff 1 1 holds. Furthermore, APPROXIMATION OF DETERMINANTS 11 yields We now use the left inequality in (4.2) applied to the matrix G E l AG T l . Note that A simple computation yieldsn l and Substitution of (4.5) in (4.4) results inn l Using this the left inequality in (4.3) follows from the left inequality in (4.2). Note that for the lower bound in (4.3) to be computable, we need F and a strictly positive lower bound ff for the smallest eigenvalue of G E l AG T l . We now discuss methods for computing ff and . These methods are used in the numerical experiments in x5. We first discuss two methods for computing ff. The first method, which can be applied if A is an M-matrix, is based on the following lemma, where we use the Lemma 4.2. Let A be a symmetric positive definite matrix of order n with A ij 0 for all i 6= j and G E l its sparse approximate inverse (3.9). Furthermore, let z be such that Then holds. Proof. From the assumptions it follows that A is an M-matrix. In [11] (Theorem 4.1) it is proved that then G E l AG T l is an M-matrix, too. Let z Because nonnegative entries it follows that A. REUSKEN Hence Using min (G E l AG T we obtain the result (4.6). Based on this lemma we obtain the following method for computing ff. Choose apply the conjugate gradient method to the system G E l AG T This results in approximations z of z . One iterates until the stopping criterion 1 . In view of efficiency one should not take a very small tolerance j. In our experiments in x5 we use 1. Note that the CG method is applied to a system with the preconditioned matrix G E l AG T l . In situations where the preconditioning is effective one may expect that relatively few CG iterations are needed to compute z j such that kG E l AG T numerical experiments are presented in x5. As a second method for determining ff, which is applicable to any symmetric positive definite A, we propose the Lanczos method for approximating eigenvalues applied to the matrix G E l AG T l . This method yields a decreasing sequence (1) l ) of approximations (j) 1 of min (G E l AG T holds, then can be used in Theorem 4.1. However, in practice it is usually not known how to obtain reasonable values for " in (4.7). Therefore, in our experiments we use a simple heuristic for error estimation (instead of a rigorous bound as in (4.7)), based on the observed convergence behaviour of (j) It is known that for the Lanczos method the convergence to extreme eigenvalues is relatively fast. Moreover, it often occurs that the small eigenvalues of the preconditioned are well-separated from the rest of the spectrum, which has a positive effect on the convergence speed (j) In numerical experiments we indeed observe that often already after a few Lanczos iterations we have an approximation of min (G E l AG T with an estimated relative error of a few percent. However, for the ff computed in this second method we do not have a rigorous analysis which guarantees that from numerical experiments we see that this method is satisfactory. This is partly explained by the relatively fast convergence of the Lanczos method towards the smallest eigenvalue. A further explanation follows from the form of the lower bound in (4.3). For ff which is typically the case in our experiments in x5, this lower bound essentially behaves like exp(ffi ln ff) =: g(ff). Note that holds. Hence the sensitivity of the lower bound with respect to perturbations in ff is very mild. We now discuss the computation of the quantity F , which is needed in (4.3). Clearly, for computing one needs the matrices G E l and A. To avoid un-necessary storage requirements one should not compute the matrix X := G E l AG T and then determine F . A with respect to storage more efficient approach can be based on where e i is the ith basis vector in R n . For the computation of kG E l AG T APPROXIMATION OF DETERMINANTS 13 be done in parallel, one needs only sparse matrix-vector multiplications with the matrices G E l and A. Furthermore, for the computation of AG T use that (DG E l It follows from (3.10) that holds. Remark 4.3. Note that for the error estimators discussed in this section the must be available (and thus stored), whereas for the computation of the approximation d(G E l ) \Gamma2 of d(A) we do not have to store the matrix G E l (cf. Remark 3.10 item 3). Furthermore, as we will see in x5, the computation of these error estimators is relatively expensive. 2 4.2. estimation based on a Monte Carlo approach. In this section we discuss a simple error estimation method which turns out to be useful in practice. Opposite to those treated in the previous section this method does not yield (an approximation of) bounds for the error. The exact error is given by l is a sparse symmetric positive definite matrix. Fore ease of presentation we assume that the pattern E l is sufficiently large such that holds. In [11] it is proved that if A is an M-matrix or a (block) H-matrix then (4.8) is satisfied for every lower triangular pattern E l . In the numerical experiments (cf. with matrices which are not M-matrices or (block) H-matrices (4.8) turns out to be satisfied for standard choices of E l . We note that if (4.8) does not hold then the technique discussed below can still be applied if one replaces a suitable damping factor such that ae(I \Gamma !E For the exact error we obtain, using a Taylor expansion of ln(I \Gamma B) for B 2 R n\Thetan with \Gamman Hence, an error estimation can be based on estimates for the partial sums Sm := The construction of G E l is such that diag(E E l and thus tr(E 14 A. REUSKEN For S 3 we obtain Note that in S 2 and S 3 the quantity tr(E 2 F occurs which is also used in the error estimator in x4.1. In this section we use a Monte Carlo method to approximate the trace quantities in Sm . The method we use is based on the following proposition [8, 3]. Proposition 4.4. Let H be a symmetric matrix of order n with tr(H) 6= 0. Let V be the discrete random variable which takes the values 1 and \Gamma1 each with probability 0:5 and let z be a vector of n independent samples from V . Then z T Hz is an unbiased estimator of tr(H): E(z and For approximating the trace quantity in S 2 we use the following Monte Carlo algorithm 1. Generate z j 2 R n with entries which are uniformly distributed in (0; 1). 2. If (z j 3. y j Based on Proposition 4.4 and (4.10) we use as an approximation for S 2 . The corresponding error estimator is For the approximation of S 3 we replace step 3 in the algorithm above by 3. y j and we use as an estimate for S 3 . The corresponding error estimator is exp(\Gamman APPROXIMATION OF DETERMINANTS 15 Clearly, this technique can be extended to the partial sums Sm with m ? 3. However, in our applications we only use " S 3 for error estimation. It turns out that, at least in our experiments, the two leading terms in the expansion (4.9) are sufficient for a reasonable error estimation. Note that due to the truncation of the Taylor ex- pansion, the estimators E 2 and E 3 for are biased. It is shown in [3] that based on the so-called Hoeffding inequality (cf. [13]) probabilistic bounds for can be derived, where z are independent random variables as in Proposition 4.4. In this paper we do not use these bounds. Based on numerical experiments we take a fixed small value for the parameter M in the Monte Carlo algorithm above (in the experiments in x5: Remark 4.5. In the setting of this paper Proposition 4.4 is applied with is a known polynomial of degree 2 or 3. In the Monte Carlo technique for approximating Proposition 4.4 is applied with ln(A). The quantity z T ln(A)z, which can be considered as a Riemann-Stieltjes integral, is approximated using suitable quadrature rules. In [3] this quadrature is based on a Gauss-Christoffel technique where the unknown nodes and weights in the quadrature rule are determined using the Lanczos method. For a detailed explanation of this method we refer to [3]. A further alternative that could be considered for error estimation is the use of this method from [3]. In the setting here, this method could be used to compute a (rough) approximation of det(G E l AG T We did not investigate this possibility. The results in [2, 3] give an indication that this alternative is probably much more expensive than the method presented in this section. 2 5. Numerical experiments. In this section we present some results of numerical experiments with the methods introduced in x3 and x4. All experiments are done using a MATLAB implementation. We use the MATLAB notation nnz(B) for the number of nonzero entries in a matrix B. Experiment 1 (discrete 2D Laplacian). We consider the standard 5-point discrete Laplacian on a uniform square grid with m mesh points in both directions, i.e. For this symmetric positive definite matrix the eigenvalues are known: sin For the choice of the sparsity pattern E l we use a simple approach based on the nonzero structure of (powers of) the matrix A: We first describe some features of the methods for the case that we will vary m and k. Let A denote the discrete Laplacian for the case LA its lower triangular part. We then have nnz(LA 2640. For the sparse approximate inverse we obtain nnz(G E l (2) 6002. The systems P i AP T that have to be solved to determine G E l (2) (cf. (3.11)) have dimensions between 1 and 7; the mean of these dimensions is 6.7. As an approximation of obtain Y A. REUSKEN Hence 0:965. For the computation of this approximation along the lines as described in Remark 3.10, item 3, we have to compute the Cholesky factorizations n. For this approximately are needed (in the MATLAB implementation). If we compare this with the costs of one matrix-vector multiplication A x (8760 flops), denoted by MATVEC, it follows that for computing this approximation of d(A), with an error of 3.5 percent, we need work comparable to only 5 MATVEC. We will see that the arithmetic costs for error estimation are significantly higher. We first consider the methods of x4.1. The arithmetic costs are measured in terms of MATVEC. For the computation of ff as indicated in Lemma 4.2 with using the CG method with starting vector need 8 iterations. In each CG iteration we have to compute a matrix-vector multiplication G E l AG T which costs approximately 3.7 MATVEC. We obtain ff 0:0155. For the method based on the Lanczos method for approximating min (G E l AG T use the heuristic stopping criterion We then need 7 Lanczos iterations, resulting in ff direct computation results in min (G E l AG T For the computation of F we first computed the lower triangular part of l and then computed kXkF (making use of symmetry). The total costs of this are approximately MATVEC. Application of Lemma 4.1, with ff CG and ff Lanczos yields the two intervals which both contain the exact error 0.965. In both cases, the total costs for error estimation are 40-45 MATVEC, which is approximately 10 times more than the costs for computing the approximation d(G E l (2) ) \Gamma2 . We now consider the method of x4.2. We use the estimators E 2 and E 3 from (4.13), (4.15) with 6. The results are 0:973. Note that the order of magnitude of the exact error (3:5 percent) is approximated well by both In step 3 in the Monte Carlo algorithm for computing " need one matrix-vector multiplication G E l AG T MATVEC). The total arithmetic costs for E 2 are approximately 20 MATVEC. For S 3 we need two matrix-vector multiplications with l in the third step of the Monte Carlo algorithm. The total costs for E 3 are approximately 40 MATVEC. In Table 5.1 we give results for the discrete 2D Laplacian with We use the sparsity pattern E l (2). In the third column of this table we give the computed approximation of d(A) and the corresponding relative error. In the fourth column we give the total arithmetic costs for the Cholesky factorization of the matrices P i AP T item 3). In the columns 5-8 we give the results and corresponding arithmetic costs for the error estimators discussed in x4. The fifth column corresponds to the method discussed in x4.1 with ff determined using the CG method applied to G E l AG T with starting vector 1. In the stopping criterion we take The computed used as input for the lower bound in (4.3). The resulting bound for the relative error and the arithmetic costs for computing this error bound are shown in column 5. In column 6 one finds the computed error bounds if ff is determined using the Lanczos method with stopping criterion (5.2). In the last two APPROXIMATION OF DETERMINANTS 17 Table Results for 2D discrete Laplacian with costs for Thm. 4.1, Thm. 4.1, MC MC Table Results for 2D discrete Laplacian with costs for Thm. 4.1, Thm. 4.1, MC MC columns the results for the Monte Carlo estimators are given. In Table 5.2 we show the results and corresponding arithmetic costs for the method with sparsity pattern Concerning the numerical results we note the following. From the third and fourth column in Table 5.1 we see that using this method we can obtain an approximation of d(A) with relative error only a few percent and arithmetic costs only a few MATVEC. Moreover, this efficiency hardly depends on the dimension n. Comparison of the third and fourth columns of the Tables 5.1 and 5.2 shows that the approximation significantly improves if we enlarge the pattern from E l (2) to E l (4). The corresponding arithmetic costs increase by a factor of about 9. This is caused by the fact that the mean of the dimensions of the systems P i AP T from approximately 7 (E l (2)) to approximately 20. For For the other n values we have similar ratios between the number of nonzeros in the matrices LA and . Note that the matrix G E l has to be stored for the error estimation but not for the computation of the approximation d(G E l ) \Gamma2 . The error bounds in the fifth and sixth column in the Tables 5.1 and 5.2 are rather conservative and expensive. Furthermore there is some deterioration in the quality and a quite strong increase in the costs if the dimension n grows. The strong increase in the costs is mainly due to the fact that the CG and Lanczos method both need significantly more iterations if n increases. This is a well-known phenomenom (the matrix G E l AG T E l has a condition number that is proportional to n). Also note that the costs for these error estimators are (very) high compared to the costs of the computation of d(G E l ) \Gamma2 . The results in the last two columns indicate that the Monte Carlo error estimators, although less reliable, are more favourable. In Figure 5.1 we show the eigenvalues of the matrix G E l AG T l for the case (computed with the MATLAB function eig). The eigenvalues are in the A. REUSKEN interval [0:025; 1:4]. The mean of these eigenvalues is 1 can see that relatively many eigenvalues are close to 1 and only a few eigenvalues are close to zero. Fig. 5.1. Eigenvalues of the matrix G E l AG T l in Experiment 1 Experiment 2 (MATLAB random sparse matrix). The sparsity structure of the matrices considered in Experiment 1 is very regular. In this experiment we consider matrices with a pattern of nonzero entries that is very irregular. We used the MATLAB generator (sprand(n; n; 2=n)) to generate a matrix B of order n with approximately 2n nonzero entries. These are uniformly distributed random entries in (0; 1). The matrix B T B is then sparse symmetric positive semidefinite. In the generic case this matrix has many eigenvalues zero. To obtain a positive definite matrix we generated a random vector d with all entries chosen from a uniform distribution on the interval (0; 1) (d :=rand(n; 1)). As a testmatrix we used A := B T B+diag(d). We performed numerical experiments similar to those in Experiment 1 above. We only consider the case with sparsity pattern (2). The error estimator based on the CG method is not applicable because the sign condition in Lemma 4.2 is not fulfilled. For the case 900 the eigenvalues of A and of G E l AG T are shown in Figure 5.2. For A the smallest and largest eigenvalues are 0:0099 and 5:70, respectively. The picture on the right in Figure 5.2 shows that for this matrix A sparse approximate inverse preconditioning results in a very well-conditioned matrix. Related to this, one can see in Table 5.3 that for this random matrix A the approximation of d(A) based on the sparse approximate inverse is much better than for the discrete Laplacian in Experiment 1. For and respectively. For the mean of the dimensions of the systems P i AP T spectively. In all three cases the costs for a matrix-vector multiplication G E l AG E l x are approximately 4.3 MV. Furthermore, in all three cases the matrix G E l AG T l is well-conditioned and the number of Lanczos iterations needed to satisfy the stopping criterion (5.2) hardly depends on n. Due to this, for increasing n, the growth in the costs for the error estimator based on Theorem 4.1 (column 5) is much slower than in Experiment 1. As in the Tables 5.1 and 5.2, in Table 5.3 the error quantities in the columns 3, 5,6,7 are bounds or estimates for the relative error APPROXIMATION OF DETERMINANTS 19 Fig. 5.2. Eigenvalues of the matrices A and G E l AG T l in Experiment 2 Table Results for MATLAB random sparse matrices with costs for Thm. 4.1, MC MC For the values of d(A) are not given (column 2). This has to do with the fact that for these matrices with very irregular sparsity patterns the Cholesky factorization suffers from much more fill-in than for the matrices in the Experiments 1 and 3. For the matrix A in this experiment with 10000 we run into storage problems if we try to compute the Cholesky factorization using the MATLAB function chol. Experiment 3 (QCD type matrix). In this experiment we consider a complex Hermitean positive definite matrix with sparsity structure as in Experiment 1. This matrix is motivated by applications from the QCD field. In QCD simulations the determinant of the so-called Wilson fermion matrix is of interest. These matrices and some of their properties are discussed in [4, 5]. The nonzero entries in a Wilson fermion matrix are induced by a nearest neighbour coupling in a regular 4-dimensional grid. These couplings consist of 12 \Theta 12 complex matrices M xy , which have a tensor product structure M xy\Omega U xy , where P xy 2 R 4\Theta4 is a projector, U xy 2 C 3\Theta3 is from SU 3 and x and y denote nearest neighbours in the grid. These coupling matrices M xy strongly fluctuate as a function of x and y. Here we consider a (toy) problem with a matrix which has some similarities with these Wilson fermion matrices. We start with a 2-dimensional regular grid as in Experiment 1 (n grid points). For the couplings with nearest neighbours we use complex numbers with length 1. These numbers are chosen as follows. The couplings with south and west neighbours at a grid point x are exp(2iff S (x)) and exp(2iff W (x)), respectively, where ff S (x) and ff W (x) are chosen from a uniform distribution on the interval (0; 1). The couplings with the north and east neighbours are taken such that the matrix is hermitean. To make the comparison with Experiment 1 easier the matrix is scaled by the factor n, A. REUSKEN i.e. the couplings with nearest neighbours have length n. For the diagonal we take flI , where fl is chosen such that the smallest eigenvalue of the resulting matrix is approximately 1 (this can be realized by using the MATLAB function eigs for estimating the smallest eigenvalue). We performed numerical experiments as in Experiment 1 with (2). The number of nonzero entries in LA and G E l are the same as in Experiment 1. For 900 the eigenvalues of the matrices A and G E l AG T are shown in Figure 5.3. These spectra are in the intervals respectively. The results of numerical experiments are presented in Table 5.4. Note that the error Fig. 5.3. Eigenvalues of the matrices A and G E l AG T l in Experiment 3 estimator from x4.1 in which the CG method is used for computing ff can not be used for this matrix (assumptions in Lemma 4.2 are not satisfied). We did not consider the case here because then the application of the eig function for computing the smallest eigenvalue led to memory problems. Comparison of the results in Table 5.4 with those in Table 5.1 shows that when the Table Results costs for Thm. 4.1, MC MC method is applied to the QCD type of problem instead of the discrete Laplacian the performance of the method does not change very much. Finally, we note that in all measurements of the arithmetic costs we did not take into account the costs of determining the sparsity pattern E l (k) and of building the matrices --R Cambridge University Press Bounds on the trace of the inverse and the determinant of symmetric positive definite matrices Some large scale matrix computation problems Progress on lattice QCD algorithms Exploiting structure in Krylov subspace methods for the Wilson fermion matrix Matrix Computations Parallel preconditioning with sparse approximate inverses A stochastic estimator of the trace of the influence matrix for laplacian smoothing splines An alternative approach to estimating the convergence rate of the CG method On a family of two-level preconditionings of the incomplete block factorization type Factorized sparse approximate inverse precondi- tionings I : Theory Quantum Fields on a Lattice Convergence of Stochastic Processes --TR

sparse approximate inverse;determinant;preconditioning

587814

Generalized Polar Decompositions for the Approximation of the Matrix Exponential.

In this paper we describe the use of the theory of generalized polar decompositions [H. Munthe-Kaas, G. R. W. Quispel, and A. Zanna, Found. Comput. Math., 1 (2001), pp. 297--324] to approximate a matrix exponential. The algorithms presented have the property that, if $Z \in {\frak{g}}$, a Lie algebra of matrices, then the approximation for exp(Z) resides in G, the matrix Lie group of ${\frak{g}}$. This property is very relevant when solving Lie-group ODEs and is not usually fulfilled by standard approximations to the matrix exponential. We propose algorithms based on a splitting of Z into matrices having a very simple structure, usually one row and one column (or a few rows and a few columns), whose exponential is computed very cheaply to machine accuracy. The proposed methods have a complexity of ${\cal O}(\kappa n^{3})$, with constant $\kappa$ small, depending on the order and the Lie algebra ${\frak{g}}$. % The algorithms are recommended in cases where it is of fundamental importance that the approximation for the exponential resides in G, and when the order of approximation needed is not too high. We present in detail algorithms up to fourth order.

Introduction With the recent developements in the theory of Lie-group integration schemes for ordinary differential equations (Iserles, Munthe-Kaas, N-rsett & Zanna 2000), the problem of approximating the matrix exponential has lately received a renewed attention. Most Lie-group methods require a number of computation of matrix exponentials from a Lie algebra g ' R n\Thetan to a Lie group G ' GL(n; R), that usually constitutes a bottleneck in the numerical implementation of the schemes (Celledoni, Iserles, N-rsett & Orel 1999). The matrix exponentials need be approximated to the order of the underlying ODE method (hence exact computation is not an issue), however, it is of fundamental importance that such approximations resides in G. In generality, this property is not fullfilled by many standard approximations to the exponential function (Moler & van Loan 1978) unless the exponential is evaluated exactly. In some few cases (usually for small dimension) the exponential of a matrix can be evaluated exactly. This happens, for instance, for three by three skew-symmetric matrices, whose exponential can be Institutt for informatikk, University of Bergen, H-yteknologisenteret, Thorm-hlensgate 55, N-5020 Bergen, Norway. Email: anto@ii.uib.no, hans@ii.uib.no calculated exactly by means of the well known Euler-Rodriguez formula sin ff ff where (Marsden & Ratiu 1994). Exact formulas for skew-symmetric matrices and matrices in so(p; q) can be derived up to dimension eight making use of the Cayley-Hamilton theorem (Horn & Johnson 1985) with significant savings with respect to approximation techniques (Barut, Zeni & Laufer 1994, Leite & Crouch 1999). However, the algorithms are not practical for larger dimensions, for several reasons. First, they require high powers of the matrix in question (and each matrix-matrix multiplication amounts to O secondly, it is well known that the direct use of the characteristic polynomial, for large scale matrices, may lead to computational instabilities. The problem of approximating the exponential of a matrix from a Lie algebra to its corresponding Lie group has been recently considered by (Celledoni & Iserles 2000, Celledoni & Iserles 1999). In the first paper, the authors construct the approximation by first splitting the matrix X 2 g as the sum of bordered matrices. Strang-type splittings of order two are considered, so that one could apply a Yoshida technique (Yoshida 1990), based on a symmetric composition of a basic scheme whose error locally expands in odd powers of time only, to increase the order. In the second paper, the authors consider techniques based on canonical coordinates of the second kind (CCSK) (Varadarajan 1984). To follow that approach, it is necessary to choose a basis of the Lie algebra g. The choice of the basis plays a significant role in the computational complexity of the algorithms (Owren & Marthinsen 1999), and, by choosing Chevalley bases (Carter, Segal & Macdonald 1995) which entail a large number of zero structure constants, it is possible to reduce significantly the cost of the methods from O \Delta to O In this paper we consider the problem of approximating to a given order of accuracy F (t; Z) - exp(tZ) 2 G; Z 2 so that F (t; Z) 2 G, where g ' gl(R; n) and G ' GL(R;n). The techniques we introduce consist in a Lie-algebra splitting of the matrix Z by means of an iterated generalized polar decomposition induced by an appropriate involutive automorphism oe G, as discussed in (Munthe-Kaas et al. 2000b). We introduce a general technique for approximations of arbitrary high order, and discuss practical algorithms of order two, three and four. For large n, these algorithms are very competitive with standard approximations of the exponential function (for example diagonal Pad'e approximants). The paper is organized as follows. In Section 2 we discuss the background theory of the polar decomposition on Lie groups and its symmetric version. Such polar decomposition can be used to induce splitting in the Lie algebra g. As long as this splitting is practical to compute, together with the exponential of each 'splitted' part, it leads to splitting methods for the approximation of the exponential of practical interest. In Section 3 we use the theory developed in x2 to derive approximations of the exponential function for some relevant matrix Lie groups as SO(R;n), and SL(R; n). Methods of order two, three and four are discussed in greater detail, together with their computational complexity. The methods are based on splittings in bordered matrices, whose exact exponentials are very easy to compute. Section 4 is devoted to some numerical experiments where we illustrate the results derived in this paper, and finally Section 5 is devoted to some concluding remarks. Background theory It is usual in differential geometry to denote Lie-group elements with lower case letters and Lie- algebra elements with upper-case letters, whether they represent matrices, vectors or scalars (Hel- gason 1978). We adopt this convention throughout this section. Let G be a Lie group with Lie algebra g. We restrict our attention to matrix groups, i.e to the case when G ' GL(R;n). It is known that, provided G is an involutive automorphism of G, every element z 2 G sufficiently close to the identity can be decomposed in the product wg, the subgroup of elements of G fixed under oe and is the subset of anti-fixed points of oe (Lawson 1994, Munthe-Kaas et al. 2000b). The set G oe has the structure of a symmetric space (Helgason 1978) and is closed under the product as it can be easily verified by application of oe to the right-hand-side of the above relation. The decomposition (2:1) is called the polar decomposition of z in analogy with the case of real matrices with the choice of automorphism g. The automorphism oe induces an involutive automorphims doe on g in a natural manner, d dt and it defines a splitting of the algebra g into the sum of two linear spaces, Zg is a subalgebra of g, while \GammaZ g has the structure of a Lie-triple system, a set closed under the double commutator, To keep our presentation relevant to the argument matter of this paper, we refer the reader to (Munthe-Kaas et al. 2000b, Munthe-Kaas, Quispel & Zanna 2000a) and references therein for a more extensive treatement of such decompositions. However, it is of fundamental importance to note that the sets k and p possess the following properties: We denote by \Pi the canonical projection onto the subspace p and by \Pi k k the projection onto k. Then, where Assume that x and y in (2:1) are of the form and they can be expanded in series where the X i and Y i can be explicitely calculated by means of the following recurrence relations c 2' and k1 ;:::;k 2k?0 k1+\Delta\Delta\Delta+k 2k =2q (2m)! ad 2m Z (Zanna 2000). Note that Y (t) expands in odd powers of t only. The first terms in the expansions of X(t) and Y (t) are O We also consider a symmetric-type generalized polar decomposition, where, as above, X(t) 2 p and Y (t) 2 k. To compute X(t), we apply oe to both sides of (2:7) to obtain Isolating the y term in (2:8) and (2:7) and equating the result, we obtain This leads to a differential equation for X which is very similar to the one obeyed by Y in (2:5) (Zanna 2000). Using the recursions in (Zanna 2000) we obtain recursions for X(t) and Y (t). The first terms are given as and both X(t) and Y (t) expand in odd powers of t only. Generalized polar decomposition and its symmetric version for the approximation of the exponential Assume now that we wish to approximate exp(tZ) for some Z 2 g, and that oe 1 is an involutive automorphism so that the exponential of terms in as well as analytic functions of adP , are easy to compute. Then and we can approximate where X [1] and Y [1] obey the order conditions (2:4)-(2:6) to suitable order. Alternatively, we can approximate where X [1] and Y [1] obey now the order conditions (2:10)-(2:11) to given accuracy. The same mechanism can be applied to split k 1 in p 2 \Phi k 2 by means of a suitable automorphism oe 2 . The procedure can be iterated and, provided that the exponential of k m is easy to compute, we have an algorithm to approximate exp(tZ) to a given order of accuracy. In this circumstance, (3:1) will read while the analogue of (3:2) is both corresponding to the algebra splitting 3.1 On the choice of the automorphisms oe i In what follows, we will consider automorphisms oe of the form G; (3.6) where S is an idempotent matrix, i.e. S I [Munthe-Kaas and Zanna, 2000]. Clearly, and for simplicity, we will abuse notation writing oeZ in place of doeZ, given that all our computations take place in the space of matrices. all the eigenvalues of S are either +1 or \Gamma1. Thus, powers of matrices as well as powers of adP , are easy to evaluate by means of the (+1)- and (\Gamma1)-eigenspace of S (Munthe-Kaas & Zanna 2000). 3.2 Automorphisms that lead to banded matrices splittings Let Z 2 gl(n; R) be a n \Theta n matrix and consider the automorphism is the idempotent matrix . 0 It is easy to verify that Z =2 z z while \Pi k1 z In general, assume that, at the j-th step, the space consists of matrices of the form . O O w O Then, the obvious choice is O ~ . 0 where I j \Gamma1 denotes the (j \Gamma 1) \Theta (j \Gamma 1) identity matrix and ~ so that the subspace p j consists of matrices of the form O ~ Exponentials of matrices of the form (3:11) are very easy to compute: in effect, exp O ~ O exp( ~ where exp( ~ can be computed exactly either with a formula analogous to the Euler-Rodriguez formula (1:1): denote a exp( ~ I ~ ~ a T I I ~ ~ \Gammaa T Note that ~ Another alternative for the exact exponential of ~ is the one proposed in (Celledoni & Iserles exp( ~ where a j e 1 is the vector [1; finally 1)=z. The latter formula (3:13), as we shall see in the sequel, leads to significant savings in the computation and assembly of the exponentials. Moreover, given that O ~ where ~ w j;j a Next, if Z 2 g, to obtain an approximation of the exponential in G by these automorphisms, we shall require that oe i 's, defined by the above matrices S i , map g into g. Clearly, this is the case for ffl so(n; R), since oe i Z is a map from so(n) ! so(n) given that each S i is an orthogonal matrix; ffl sl(n; R), since oe i leaves the diagonal elements of Z (hence its trace) unchanged; ffl quadratic Lie algebras and the commute. This is for instance the case when J is diagonal, hence our formulas are valid for not for the symplectic algebra sp(n; R). In the latter situation, we consider different choices for the automorphisms oe i , discussed at a greater length in (Munthe-Kaas & Zanna 2000). 3.3 Splittings of order two to four, their implementation and complexity In this section we describe in more details the algorithms, the implementation and the complexity of the splittings induced by the automorphisms described above. The cases of a polar-type repre- sentation, xy, or a symmetric polar-type representation, z = xyx, are discussed separately. Algorithm 1 (Polar-type splitting, order two) Based on the iterated generalized polar decomposition (3:3). Note that the \Pi p j and \Pi k j projections need not be stored in separate matrices but can be stored in places of the rows and columns of the matrix Z. We truncate the expansions (2:6) to order two, hence at each step only the p j -part needs correction. Taking in mind (3:3), the matrices X [j] are low rank matrices with nonzero entries only on the j-th row, column row are stored in place of the corresponding Z entries. The matrix Y [n\Gamma1] is diagonal and is stored in the diagonal entries of Z. Purpose: 2nd order approximation of the splitting (3:3) overwritten with the nonzero elements of X [i] and Y [m] as: a The computation of the splitting requires at each step two matrix-vector multiplications, each amounting to O floating point operations (we count both multiplications and ad- ditions), as well as two vector updates, which are O(n operations. Hence, for large n, the cost of computing the splitting is of the order ffl 2n 3 for so(n), taking into account that b Note that both for so(p; q) and so(n) the matrix Y [n\Gamma1] is the zero matrix. Algorithm 2 (Symmetric polar-type splitting, order two) Based on the iterated generalized polar decomposition (3:4) We truncate the expansions (2:10)-(2:11) to order two. The storing of the entries is as above. Purpose: 2nd order approximation of the splitting (3:4) overwritten with the nonzero elements of X [i] and Y [m] as: % Computation of the splitting a This splitting costs only ffl n(n\Gamma1)for so(n), because of skew-symmetry. Algorithm 3 (Polar-type splitting, order three) We truncate (2:6)-(2:7) to include O terms. Note that the term [K; [P; K]] is of the form (3:15). We need include also the term of the form [P; [P; K]]. We observe that Purpose: 3rd order approximation of the splitting (3:3) overwritten with the nonzero elements of X [i] and Y [m] as: % Computation of the splitting a Analyzing the computations involved, the most costly part is constituted by the matrix-vector products in the computations in c products in the update of Z(j j). The computation of c amounting to 8n 3 in the whole process. For the update of Z(j need to compute two vector-vector products (O operations to uptdate the elements of the matrix. Thus, the whole cost of updating the matrix Z(j n) is 5n 3 . The update of z j;j requires operations per step, which give a 2n 3 contribution to the total cost of the splitting. In summary, the total cost of the splitting is ffl 5n 3 for so(p; q) and sl(n) ffl for so(n), note that d j need not be calculated as well as z Similarly, we take into account that b and that only half of the elements of Z(j need be updated. The total amounts to 2 1 It is easy to modify the splitting above to obtain order four. Note that which requires the computation of the scalar b T costing 2=3n 3 operations in the whole pro- cess. However, all the other powers ad i ~ ~ no further computation. Next can be computed with just two (one) extra matrix-vector computations for sl(n) (resp. so(n)), which contribute 4n 3 (resp. 2n 3 ) to the cost of the splitting, so that the splitting of order four costs a total of 7n 3 operations for sl(n) (resp. 4n 3 for so(n)). Algorithm 4 (Symmetric polar-type splitting, order four) We truncate (2:10)-(2:11) to include O terms. Also in this case, the term [K; [P; K]] is of the form (3:15), while the term [P; [P; K]] is computed according to (3:16). Purpose: 4th order approximation of the splitting (3:4) overwritten with the nonzero elements of X [i] and Y [m] as: a We need to compute a total of four matrix-vector products, yielding 8 operations. The update of the block Z(j costs 5n 3 operations, while the update of z(j; costs 2n 3 operations, for a total of ffl 5n 3 operations for sl(n) and so(p operations for so(n). 3.4 On higher order splittings The costs of implementing splittings following (3:3) or (3:4) depend on the type of commutation involved: commutators of the form [P; K] and [P contribute as an O \Delta term to the total complexity of the splitting, however, commutators of the form [K for easily contribute an O \Delta to the total complexity of the splittings if the special structure of the terms involved is not taken into consideration. If carefully implemented, also these terms can be computed with only matrix-vector and vector-vector products, contributing O \Delta operations to the total cost of the splitting. For example, let us consider the term which appears in the O contribution in the expansion of the Y part, both for the polar-type and symmetric polar-type splitting. One has denotes the matrix z j;j I \Gamma - . The parenthesis indicate the correct order in which the operations should be executed to obtain the right complexity (O per iteration, hence a total of O \Delta for the splitting). Many of the terms are already computed for the lower order conditions, yet the complexity arises significantly. Therefore we recommend this splitting type techniques when a moderate order of approximation is required. To construct higher order approximations with these splitting techniques, one could use our symmetric polar-type splittings, together with a Yoshida-type symmetric combination. 3.5 Assembly of the approximation F (t; Z) to the exponential For each algorithm that computes the approximation to the exponential, we distinguish two cases: when the approximation is applied to a vector v and when instead the matrix exponential exp(Z) is required. Since the matrices X [j] are never constructed explicitely and are stored as vectors, computations of the exponentials exp(X [j] ) is also never performed explicitely but it is implemented as in the case of the Householder reflections (Golub & van Loan 1989) when applied to a vector. First, let us return to (3:13). It is easy to verify that, if we denote by ff has the exact form I where I is the 2 \Theta 2 identity matrix. Similar remarks hold about the matrix D \Gamma1 . Thus, the computation of can be done in a very few flops that `do not contribute' to the total cost of the algorithm. Next, if v; k; the assembly of exp( ~ according to can be computed in 6j operations. If we let j vary between 1 and n, the total cost of the multiplications is hence 3n 2 . This is precisely the complexity of for the assembly of the exponential for polar-type splittings, that has the form as in (3:3). Algorithm 5 (Polar-type approximation) Purpose: Computing the approximant (3:3) applied to a vector v containing the nonzero elements of X [i] and Y [m] as: a old and new := [a new . In the case when the output needs be applied to a n \Theta n matrix B, we can apply the above algorithm to each column of B, for a total of 3n 3 operations. This complexity can be reduced to about 2n 3 taking into account that the vector can be calculated once and for all, depending only on the splitting of the matrix Z and not in any manner on the columns of B. Also can be computed once and stored for latter use. Algorithm 6 (Symmetric polar-type approximation) The approximation to the exponential is carried out in a manner very similar to that described above in Algorithm 5, except that, being (3:4) based on a Strang-type splitting, the assembly is also performed in reverse order. Purpose: Computing the approximant (3:4) applied to a vector v containing the nonzero elements of X [i] and Y [m] as: a old old new := [a new . a old Table 1: Complexity for a polar-type order-two approximant. Algorithm sl(n); so(p; q) so(n) 1+5 vector matrix vector matrix splitting 1 1n 3 1 1n 3 2n 3 2n 3 assembly exp 3n 2 2n 3 3n 2 2n 3 new := [a new The vectors ff; fi and fl need be calculated only once and stored for latter use in the reverse-order multiplication. The cost of the assembly is roughly twice as the cost of the assembly in Algorithm 1, hence it amounts to 5n 2 operations (we save n 2 operations omitting the computation of ff). When the result is applied to a matrix B, again we apply the same algorithm to each column of B, which yields n 3 operations. Also in this case the vector ff does not depend on B and can be computed once and for all, reducing the cost to 4n 3 operations. The same remark holds for the vectors fi and fl . It is important to mention that the matrix D might be singular or close to singular (for example when a j and b j are close to be orthogonal), hence the computation of exp( ~ according to (3:13) may be lead to instabilities. In this case, it is recommended to use (3:12) instead of (3:13). The latter choice is twice as expensive (5n 2 for polar-type assemblies and 9n 2 for symmetric assemblies for F (t; Z) applied to a vector), but deals better with the case when D is nearly singular. 4 Numerical experiments 4.1 Non-symmetric polar-type approximations to the exponential We commence comparing the polar-type order-2 splitting of Algorithm 1 combined with the assembly of the exponential in Algorithm 5 with the (1; 1)-Pad'e approximant for matrices in sl(n) and so(n), with corresponding groups SL(n) and SO(n). We choose diagonal Pad'e approximants as benchmarck because they are easy to implement, are the rational approximant with highest order of approximation at the origin and it is well known that they map quadratic Lie algebras into quadratic Lie groups (but not necessarily other Lie algebras into the corresponding Lie groups). Table 1 reports the complexity of the method 1+5. A (1; 1)-Pad'e approximant costs O floating point operations when applied to a vector (essentially the cost of LU-factorising a linear system) and O operations when applied to n \Theta n matrices (2n 3 operations come from the construction of the right-hand-side, 2 3 from the LU factorization and 2n 3 from the n forward and backward solution of triangular systems). In Figure 4.1 we compare the number of floating point operations scaled by n 3 for matrices Z up to size 500 as obtained in Matlab for our polar-type order-two algorithm (method 1+5) and the both applied to a matrix. We consider the cases when Z is in sl(n) and so(n). The costs of computing both approximations clearly converges to the theroretical estimates (which in the plot are represented by solid lines) given in Table 1 for large n. Flops/nmethod 1+5, sl(n) method 1+5, so(n) Figure 1: Floating point operations (scaled by n 3 ) versus size for the approximation of the exponential of a matrix in sl(n) and in so(n) applied to a matrix with the order-2 polar-type algorithm (method 1+5) and (1; 1)-Pad'e approximant. Table 2: Complexity for a polar-type order-three approximant. The numbers in parenthesis correspond to the coefficients for an order-four approximation. Algorithm sl(n); so(p; q) so(n) 3+5 vector matrix vector matrix splitting 5(7)n 3 5(7)n 3 2 1(4)n 3 2 1(4)n 3 assembly exp 3n 2 2n 3 3n 2 2n 3 total 5(7)n 3 7(9)n 3 2 1(4)n 3 4 1(6)n 3 In Figure 2 we compare the accuracy of the two approximations (left plot) for the exponential of a normalized so that kZk methods show a local truncation error of O revealing that the order of approximation to the exact exponential is two. The right plot shows the error in the determinant as a function of the Pad'e approximant has an error that behaves like h 3 , while our method preserves the determinant equal to one to machine accuracy. In table 2 we report the complexity of the method 3+5, which yields an approximation to the exponential of order three. The numbers in parenthesis refer to the cost of the algorithm with order four corrections. 4.2 Symmetric polar-type approximations to the exponential We commence comparing our method 2+6, yielding an approximation of order two, with the (1; 1) Pad'e approximant. Table 3 reports the complexity of the method 2+6. Clearly, in the matrix-vector case, our methods are one order of magnitude cheaper than the Pad'e approximant, and are definitively to be preferred (see Figure 3, for matrices in sl(n)). Furthermore, from exact exponential method 1+5, sl(n) |det 1| method 1+5, sl(n) Figure 2: Error in the approximation (left) and in the determinant (right) versus h for the approximation of the exponential of a traceless matrix of unit norm with the order-2 polar-type algorithm (method 1+5) and (1; 1)-Pad'e approximant. Table 3: Complexity for a symmetric polar-type order-two approximant. Algorithm sl(n); so(p; q) so(n) 2+6 vector matrix vector matrix splitting assembly exp 5n 2 4n 3 5n 2 4n 3 total our method maps the approximation in SL(n), while the Pad'e approximant does not. When comparing approximations of the matrix exponential applied to a vector, it is a must to consider Krylov subspace methods (Saad 1992). We compare the method 2+6 with a Krylov subspace method when Z is a matrix in sl(n), normalized so that kZk a vector of unit norm. The Krylov subspaces are obtained by Arnoldi iterations, whose computational cost amounts to circa 2mn 2 counting both multiplications and additions. Here m is the dimension of the subspace Km j spanfv; vg. To obtain the total cost of a Krylov method, we have to add O computations arising from the evaluation of the exponential of the Hessenberg matrix obtained with the Arnoldi iteration, plus 2nm operations arising from the multiplication of the latter with the orthogonal basis. However, when n is large and m - n, these costs are subsumed in that of the Arnoldi iteration, and the leading factor is 2mn 2 . The error, computed as and the floating point operations of both approximations for are given in Table 4. The Krylov method converges very fast: in all the three cases eight-nine iterations are sufficient to obtain almost machine accuracy, while two iterations yield an error which is of the order of method 2+6, at about two thirds (0:64; 0:68; 0:69 respectively) the cost. On the other hand, Krylov methods do not produce an SL(n) approximation to the exponential, unless the computation is performed to machine accuracy, which, in our particular example, is 3:30, 2:84 and 2:85, circa three times more costly than the 2+6 algorithm. For what the SO(n) case is concerned, it should be noted that, if Z 2 so(n), then the approximation w - exp(Z)v produced by the Krylov method has the feature that kwk independently of the number m of iterations: in this case, the Hessenberg matrix produced by the Arnoldi iterations is tridiagonal and skew-symmetric, hence its exponential orthogonal. Thus, Krylov methods are the method of choice for actions of SO(n) on R n (Munthe-Kaas & Zanna 1997). One might extrapolate that, Floating point operations method 2+6, sl(n) Figure 3: Floating point operations versus size for the approximation of the exponential of a matrix in sl(n) applied to a vector with the order-2 symmetric polar-type algorithm (method 2+6) and Table 4: Krylov subspace approximations versus the method 2+6 for the approximation of exp(Z)v. Krylov 2+6 size n error m flops error flops0.74 1 21041 7. Table 5: Complexity for a symmetric polar-type order-four approximant. Algorithm sl(n); so(p; q) so(n) 4+6 vector matrix vector matrix splitting 5n 3 5n 3 2 1n 3 2 1n 3 assembly exp 5n 2 4n 3 5n 2 4n 3 total 5n 3 9n 3 2 1 if we wish to compute the exponential exp(Z)Q, where Q 2 SO(n), one could perform only a few iterations of the Krylov method to compute w being columns of Q. Unfortunately, the approximation [w ceases to be orthogonal: although the vectors w i cease to be linearly independent and the final approximation is not in SO(n). Similar analysis yields for Stiefel manifolds, unless Krylov methods are implemented to approximate the exponential to machine accuracy. In passing, we recall that our methods based on a symmetric polar-type decomposition are time- symmetric. Hence it is possible to compose a basic scheme in a symmetric manner, following a technique introduced by Yoshida (Yoshida 1990), to obtain higher order approximations: two orders of accuracy can be obtained at three times the cost of the basic method. For instance we can use the method 2+6 as a basic algorithm to obtain an approximation of order four. Thus an approximation of order four applied to a vector can be obtained in 17n 2 operations for sl(n) (two splittings and three assemblies), compared to O operations required by the method 4+6. To conclude our gallery, we compare the method 4+6, an order-four scheme, whose complexity is described in Table 5, with a (2; 2)-Pad'e approximant, which requires 2 2 floating point operations when applied to vectors (2n 3 for the assembly and 2 for the LU factorization) and 6 2 matrices (since we have to resolve for multiple right-hand sides). The figures obtained by numerical simulations for matrices in sl(n) and SO(n) clearly agree with the theoretical asymptotic values (plotted as solid lines), as shown in Figure 4. The cost of both methods is very similar, as is the error from the exact exponential although, in the SL(n) case, the 4+6 scheme preserves the determinat to machine accuracy while the Pad'e scheme does not (see Figure 5). Conclusions In this paper we have introduced numerical algorithms for approximating the matrix exponential. The methods discussed possess the feature that, if Z 2 g, then the output is in G, the Lie group of g, a property that is fundamental in the integration of ODEs by means of Lie-group methods. The proposed methods have a complexity of O denotes the size of the matrix whose exponential we wish to approximate. Typically, for moderate order (up to order four), the constant - is less than 10 whereas the exact computation of a matrix exponential in Matlab (which employes the scaling and squaring with a Pad'e approximant) generally costs between 20n 3 and 30n 3 . Comparing methods of the same order of accuracy applied to a vector v 2 R n and to a matrix G: ffl For the case F (t; Z)v - exp(tZ)v, where v is a vector: Symmetric polar-type methods are slightly cheaper than their non-symmetric variant. For the SO(n) case, the complexity of symmetric methods is very comparable with that of diagonal Pad'e approximants of the same Flops/nmethod 4+6, sl(n) method 4+6, so(n) Figure 4: Floating point operations (scaled by n 3 ) versus size for the approximation of the exponential of a matrix in sl(n) applied to a n \Theta n matrix with the order-4 symmetric polar-type algorithm (method 4+6) and (2; 2)-Pad'e approximant. from exact exponential methods 4+6, sl(n) |det 1| methods 4+6, sl(n) Figure 5: Error in the approximation(left) and in the determinant (right) versus h for the approximation of the exponential of a traceless matrix of unit norm with the order-4 symmetric polar-type algorithm (method 4+6) and (2; 2)-Pad'e approximant. order. The complexity of the method 2+6 is O while for the rest of our methods it is O Krylov subspace methods do, however, have the complexity O if the number of iterations is independent of n. Thus, if it is important to stay on the group, we recommend Krylov methods with iteration to machine accuracy for this kind of problems. If convergence of Krylov methods is slow, our methods might be good alternatives. See (Hochbruck & Lubich 1997) for accurate bounds on the number m of iterations of Krylov methods. ffl For the case F (t; Z)B - exp(tZ)B, with B an n \Theta n matrix: Non-symmetric polar-type methods are marginally cheaper than their symmetric counterpart; however the latter should be preferred when the underlying ODE scheme is time-symmetric. The proposed methods have a complexity very comparable with that of diagonal Pad'e approximants of the same order (they require slightly less operations in the SO(n) case) in addition they map sl(n) to SL(n), a property that is not shared by Pad'e approximants. For these problems our proposed methods seem to be the best choice. It should also be noted that significant advantages arise when Z is a banded matrix. For instance, the cost of method 2+6 scales as O(nr) for F (t; Z) applied to a vector and O applied to a matrix when Z has bandwidth 2r + 1. The savings are less striking for higher order methods since commutation usually causes fill-in in the splitting. Our schemes have an implementation cost smaller than those proposed by (Celledoni & Iserles 1999), that also produce an output in G when Z 2 g. For the SO(n) case, Celledoni et al. propose an order-four scheme whose complexity is 11 1 our order-four schemes (method 3+5 with order-four corrections and method 4+6) costs 6n 3 , 6 1 operations - very comparable with the diagonal Pad'e approximant of the same order. Furthermore, the implementation of the schemes of Celledoni et al. requires a precise choice of a basis in g, hence the knowledge of the structure constants of the algebra. Our approach is instead based on the inclusion relations (2:3) and is easily expressed in very familiar linear algebra formalism. --R Lectures on Lie Groups and Lie Algebras Methods for the approximation of the matrix exponential in a Lie-algebraic setting Complexity theory for Lie-group solvers Matrix Computations Differential Geometry Matrix Analysis 'Polar and Ol'shanskii decompositions' 'Closed forms for the exponential mapping on matrix Lie groups based on Putzer's method' Introduction to Mechanics and Symmetry Numerical integration of differential equations on homogeneous manifolds Integration methods based on canonical coordinates of the second kind Lie Groups Recurrence relation for the factors in the polar decomposition on Lie groups --TR --CTR Ken'Ichi Kawanishi, On the Counting Process for a Class of Markovian Arrival Processes with an Application to a Queueing System, Queueing Systems: Theory and Applications, v.49 n.2, p.93-122, February 2005 Jean-Pierre Dedieu , Dmitry Nowicki, Symplectic methods for the approximation of the exponential map and the Newton iteration on Riemannian submanifolds, Journal of Complexity, v.21 n.4, p.487-501, August 2005

lie algebra;matrix exponential;lie-group integrator

587816

More Accurate Bidiagonal Reduction for Computing the Singular Value Decomposition.

Bidiagonal reduction is the preliminary stage for the fastest stable algorithms for computing the singular value decomposition (SVD) now available. However, the best-known error bounds on bidiagonal reduction methods on any matrix are of the form \[ A are orthogonal, $\varepsilon_M$ is machine precision, and f(m,n) is a modestly growing function of the dimensions of A.A preprocessing technique analyzed by Higham [Linear Algebra Appl., 309 (2000), pp. 153--174] uses orthogonal factorization with column pivoting to obtain the factorization \[ A=Q \left( \begin{array}{c} C^T \\ 0 \end{array} \right) P^T, \] where Q is orthogonal, C is lower triangular, and P is permutation matrix. Bidiagonal reduction is applied to the resulting matrix C.To do that reduction, a new Givens-based bidiagonalization algorithm is proposed that produces a bidiagonal matrix B that satisfies $C bounded componentwise and $\delta C$ satisfies a columnwise bound (based upon the growth of the lower right corner of C) with U and V orthogonal to nearly working precision. Once we have that reduction, there is a good menu of algorithms that obtain the singular values of the bidiagonal matrix B to relative accuracy, thus obtaining an SVD of C that can be much more accurate than that obtained from standard bidiagonal reduction procedures. The additional operations required over the standard bidiagonal reduction algorithm of Golub and Kahan [J. Soc. Indust. Appl. Math. Ser. B Numer. Anal., 2 (1965), pp. 205--224] are those for using Givens rotations instead of Householder transformations to compute the matrix V, and 2n3/3 flops to compute column norms.

Introduction . We consider the problem of reducing an m \Theta n matrix A to bidiagonal form. That is, we find orthogonal matrices U n\Thetan such that To denote B in (1.1) we use the shorthand or use the MATLAB-like form We will also use MATLAB notation for submatrices. Thus A(i: j; k: ') denotes the submatrix of A consisting of rows i through j and columns k through '. Likewise, denotes all of columns k through ' and A(i: all of rows i through j. For a matrix denote the i th singular value of X , for ng. We also let X y be the Moore-Penrose psuedoinverse of X and we let J(i; be a Givens rotation through an angle ' ij applied to columns i and j. Department of Computer Science and Engineering, The Pennsylvania State University, University Park, PA 16802-6106, e-mail: barlow@cse.psu.edu, URL: http://trantor.cse.psu.edu/~barlow. The research of Jesse L. Barlow was supported by the National Science Foundation under grants no. CCR-9201612 and CCR-9424435. Part of this work was done while the author was visiting the University of Manchester, Department of Mathematics, Manchester M13 9PL UK The reduction (1.1) is usually done as a preliminary stage for computing the singular value decomposition of A. There are now a number of very good algorithms for computing the singular value decomposition of bidiagonal matrices. We know that the "zero-shift" Q-R algorithm [13], bisection [4], and the dqds algorithm [15] can compute all of the singular values of B to relative accuracy. We also know that it is not reasonable to expect any algorithm to compute all of the singular values of a matrix to relative accuracy unless that matrix has an acyclic graph [11] or is totally sign compound [12]. Thus, it is not surprising that no algorithm can be expected to produce the bidiagonal form of a general matrix to relative accuracy in fixed precision arithmetic. The Jacobi algorithm is a more accurate method for finding the singular values of a general matrix than any algorithm that requires bidiagonal reduction. Unfortunately, the Jacobi algorithm is usually slower. For simplicity, assume that reduction followed by the Q-R algorithm can produce that SVD in about 20n 3 flops. One Jacobi sweep requires about 7n 3 flops. Thus, for Jacobi to be competitive, it must converge in about three sweeps, and that rarely happens. In this paper, we present a bidiagonal reduction method that will often preserve more of the accuracy in the singular value decomposition. The reduction is computed in two stages. In the first stage, using a Householder factorization method of Cox and Higham [10], we reduce A to a lower triangular n\Thetan . In floating point arithmetic with machine unit " M , the first stage reduction satisfies an ); where Here f(n) is a modestly sized functions and ae A is a growth factor given in [10]. A similar reduction is recommended by Demmel and Veseli'c [14] before using the Jacobi method. Thus the difference in our algorithm is the reduction of C. In the second stage, we apply a new bidiagonal reduction algorithm to C. This algorithm produces a bidiagonal matrix B such that for some n \Theta n matrices ~ and some orthogonal matrices U and V , we have where s is the smallest integer such that kC(: M kCkF . The growth ae V is bounded provided that the s \Theta s principal submatrix of the corresponding Krylov matrix is nonsingular. If that submatrix is singular, the standard backward error bounds from [17] apply. This is not as good a bound as the Jacobi method achieves [14, 27, 28], but can be much better than that achieved by the standard algo- rithm. Moreover, the algorithm can be implemented in less than 2 8 flops than the Lawson-Hanson-Chan SVD [26, pp.107-120],[9]. Using fast versions of Givens rotations, that additional overhead may be reduced to about 8=27n 3 flops. Our procedure for bidiagonal reduction has some important differences from the Golub-Kahan Householder transformation based procedure [17]. They are Givens transformations are used in the construction of the right orthogonal matrix V . (Clearly, 2 \Theta 2 Householder transformations could also be used.) ffl The matrix A is preprocessed by Householder factorization with maximal column pivoting using a new procedure due to Cox and Higham [10]. ffl The first column of V is not e 1 , the first column of the identity matrix. ffl The computation of the matrices U and V are interleaved in a different manner to preserve accuracy in the small columns. In the next section, we give our algorithm for producing the bidiagonal form. In x4 we prove the bounds (1.2)-(1.3). In x5, we give some tests and a conclusion. 2. Reduction to triangular form. Before giving the bidiagonal reduction pro- cedure, we preprocess A by reducing it to lower triangular form using a Householder transformation based procedure due to Cox and Higham [10]. It is based upon the row and column pivoting procedure of Powell and Reid [30], but uses a simpler form of pivoting. The procedure is as follows. 1. Reorder the rows of A so that 2. Using the maximal column pivoting algorithm of Businger and Golub [8], factor A into n\Thetan is lower triangular. This particular Householder factorization algorithm has very strong numerical stability properties. If we let C be the computed lower triangular factor, then Cox and Higham [10] (based on the analysis of a more complicated pivoting strategy by Powell and Reid [30]) showed that for some orthgonal matrix U 0 , and (2. ae A is a growth factor bounded by p 2. The column oriented error bound (2.2) holds for standard Householder factorization [31]. The second rowwise error bound (2.3) can only be shown for algorithms that do some kind of row and column permutations for stability. Similar results for Givens based algorithms are given by Barlow [3] and Barlow and Handy [5]. The Givens based algorithm in [3] could be substituted for the Cox-Higham algorithm. Gulliksen [21] has given a new framework for orthogonal transformations with this property. Cox and Higham demonstrate that Householder's original version of Householder transformation must be used for these bounds to hold. The bound does not hold if Parlett's [29] version of the Householder transformation is used. The columns of the matrix C satisfy In fact, any reduction of A satisfying the property (2.4) will be a suitable preprocessing step and will lead to the results given here. We now give algorithms for computing the bidiagonal reduction of C. 3. Bidiagonal Reduction Algorithms. 3.1. The Golub-Kahan Bidiagonal Reduction. The usual bidiagonal reduction algorithm is that given by Golub and Kahan [17, pp.208-210,Theorem 1], see also Golub and Reinsch [19, pp.404-405]. It is given below for the square matrix C. Since C is lower triangular, it is exactly rank deficient only if it has zero diagonals. Minor and obvious modifications to the bidiagonal reduction algorithms in this section are necessary if C has zero diagonals. Algorithm 3.1 (Golub-Kahan Bidiagonal Reduction). 1. Find an orthogonal transformation U 1 such that 2. for (a) Find an orthogonal transformation V k such that (b) Find an orthogonal transformation U k such that 3. % The bidiagonal reduction of C is given by Golub and Kahan [17] used Householder transformations to describe their algo- rithm. In that case, it requires 4n 3 +O(n 2 ) flops if the matrices U and V are kept in factored form. If U and V are accumulated, it requires 8n 3 +O(n 2 ) flops. 3.2. A Givens Based Bidiagonal Reduction Algorithm. Below is our new algorithm for the bidiagonal reduction of C. Algorithm 3.2 (New Procedure for Bidiagonal Reduction). We now present a Givens-based bidiagonal reduction procedure for an n \Theta n matrix C satisfying (2.4). In x4, we show that this new algorithm will achieve error bounds of the form (1.5)-(1.8). 1. Determine the smallest integer s such that 22 is an empty matrix and we use Algorithm 3.1, complete steps 2-4. 2. Compute the vector z given by 22 be the product of Givens rotations such that Compute Let U 1 be an orthogonal transformation such that 3. for (a) Let V k be the product of Givens rotations that satisfies (b) Find an orthogonal transformation U k such that C(k: n; k: n) / U T 4. % The bidiagonal reduction of C is given by If we use standard Givens rotations, steps 2-4 of this algorithm require 10n 3 flops. The use of fast Givens rotations as described in Hammarling [22], Gentleman [16], Bareiss [2], Barlow [3], or Anda and Park [1] would produce an algorithm with approximately the same as the Golub-Kahan Householder- based procedure. Step one requires the minimum length solution of the least squares problem 11 is already upper triangular, a procedure in Lawson and Hanson [26, pp.77- 83] would allow us to compute the orthogonal factorization of the above matrix in flops. The maximum complexity of this step is for this step never costs more than 8 Table 3.1 summarizes the complexity of the two bidiagonal reduction algorithms. Table Complexity of Bidiagonal Reduction Algorithms Compute 3.1 Algorithm 3.2(SG) Algorithm 3.2(FG) Givens rotations fast Givens rotations 4. Growth Factors and Error Bounds. 4.1. Bounding the columns of C. To show the advantages of Algorithm 3.2 for bidiagonal reduction, we consider the effects of orthogonal transformations from the right used to form V in Algorithm 3.1. Let U be the matrix from Algorithm 3.1, then consider By orthogonal equivalence, If we let ~ then has the form where and F (k) 12 is zero except for the last row. Therefore, ~ in effect, zeros out lower k \Theta (n \Gamma k) block of F . The following lemma bounds the effect of a large class of orthogonal transformations from the right. Lemma 4.1. Let F n\Thetan be partitioned according to where V 11 is nonsingular. Let Then ~ 22 where ~ Proof: Matching blocks in (4.1) and (4.2) yields Using the fact that V 11 is nonsingular, block Gaussian elimination yields (4.3). The result of Lemma 4.1 generalizes to the case given next. Lemma 4.2. Let F 2 ! m\Thetan be partitioned according to 22 0 and V has the form in (4.1). Let G = FV be partitioned Then where ~ Proof: We note that 22 V (1) If we partition V according to (4.1) then First we note that V 11 is nonsingular if and only if V (i) are nonsingular. Evaluating ~ 22 leads to ~ \GammaV (2) Thus k ~ . Now we have that 22 F 23 where the (1,3) and (2,3) blocks of C are unaffected. We then apply Lemma 4.1 to F (1) F (1) !/ to obtain (4.5). The following corollary relates the growth of the (2; 3) block to growth in Gaussian elimination. Corollary 4.3. Let F , G and V be as in Lemma 4.2. Then V is the growth factor for k steps of Gaussian elimination on V in the Euclidean norm. Proof: If we let where L is unit lower triangular and R is upper triangular then Using the fact that V we have that ~ 22 solves '' ~ ~ 22 22 and that ~ V 22 are just the result of performing k steps of Gaussian elimination on V . Thus ~ Taking norms yields 22 V is the growth factor from k steps of Gaussian elimination on V . Unfortunately, the bound on growth for Gaussian elimination for a given row ordering is no better for orthogonal matrices than it is for all matrices. The following result is proven by Barlow and Zha [7]. Proposition 4.4. Let n\Thetan be nonsingular and have the P-L-R factoriza- tion by Gaussian elimination with the row ordering P where L is lower triangular and R is upper triangular. For each have the growth factor ae (k) . Let X have the factorization where V is orthogonal and Y is upper triangular. Then Gaussian elimination with partial on V obtains the P-L-R factorization where and for each k, ae (k) X . Suppose that C 2 ! n\Thetan is a lower triangular matrix satisfying j. Note that row pivoting procedure used in the previous section assures us that a small block C 22 may be isolated. n\Thetan be given by Then X has blocks satisfying where 22 C 21 =j; (4. 22 C 22 Note that 2. We can prove some results above the Krylov matrix associated with X . We now give the following lemma about Krylov matrices for X . Lemma 4.5. Let n\Thetan be given by (4.7)-(4.10) and the matrix X 11 in (4.7) be nonsingular. Let y be such that s y Assume that y is chosen so that kzk Proof: The proof is just a verification of the formulae. First consider have that The first term of both rows satisfy the lemma, and second term may be bounded by kp (1) kp (1) The induction step is simply, where For both of the reccurances (4.11) and (4.12), we can bound the norms by where Simple substitution shows that i k - k(1 We can now prove the following lemma. Lemma 4.6. Let z n\Thetan and n\Thetan be as in Lemma 4.5 and let n\Thetan be the Krylov matrix Let K be partitioned in the form s K 11 K 12 where s is as defined in (4.7). If K 11 is nonsingular, then Proof: First we note that if two lower triangular matrices are given by s I 0 s I 0 then s I 0 above be given by Then s K 11 K 12 ~ K 22 Here and s: Thus, leading to the bound If we let s K 11 K 12 K 22 If we reconstruct this factorization, then Taking norms yields which is the desired result. It is well known that matrix V from bidiagonal reduction is the orthgonal factor of the Krylov matrix K [18, pp.472-473]. That allows us to find the following bound of the growth factor for bidiagonal reduction. A caveat to all of our results is that the s \Theta s matrix K 11 be nonsingular. Proposition 4.7. Let V be as in Lemma 4.2. Let C and s be as in (3.1), let X be as in (4.7)-(4.10), and let K be as in Lemma 4.6. Assume that K 11 is nonsingular. Then where Proof: From Proposition 4.4, since the matrix K has the orthogonal factorization where V is the set of Lanczos vector for the bidiagonal reduction of C with the first we have that where and By a classical equivalence, Since jX 22 Corollary 4.8. Let C satisfy (4.6) for some value of s. Let z be defined by s y for some y 6= 0. Let Algorithm 3.2 be applied to C and let U be the orthogonal transformations generated by that algorithm. Define ~ where Let C (k) be defined by and let ae V be defined by (4.13)-(4.15). Then Proof: Since ~ k is product of Givens rotations in the standard order, we directly apply the results in Lemma 4.2. First, let Then Since ~ taking advantage of rotations that commute, we can write ~ where Y Y Y Y Thus, - have the structure in (4.1). Using the terminology of Lemma 4.2, we have (2)and that 22 Combining the results of Lemma 4.2 and Proposition 4.7 obtains Note that the ability to factor ~ 2 is a feature of a Givens rotation based algorithm for bidiagonal reduction. 4.2. Error Bounds and Implications. The error bounds for this paper are stated in two theorems. The first one is proven in x7, the second is a consequence of the first. Theorem 4.9. Let C 2 ! n\Thetan and let n\Thetan be the bidiagonal matrix computed by Algorithm 3.2 in floating point arithmetic with machine be the contents of C after k passes thorough the main loop of Algorithm 3.2. Then there exist U; modestly growing functions where and for Theorem 4.10. Let C 2 ! n\Thetan satisfy (2.4), and let n\Thetan be the bidiagonal matrix computed by Algorithm 3.2 in floating point arithmetic with machine unit " M . Let C r be the contents of C after k passes thorough the main loop of Algorithm 3.2. Let s be the smallest integer such that and let ae V be defined by (4.13). Then there exist U; satisfying (4.16) and a modestly growing function g 5 (\Delta) such that B and C satisfy (4.17), ffiB is as in Theorem 4.9 and Proof: The proof of this result is simply a matter of bounding for each value of k. For k - s, we have that For k ? s, orthogonal equivalence yields 2: An application of (4.21) yields (4.22). The standard error bounds on bidiagonal reduction are of the form (4.17) but where ffiC only satisfies a bound of the form For the Golub-Kahan procedure, this bound is probably as good as we can expect. These lead to error bounds on the singular values of the form This is satisfactory for large singular values, but of little use for small ones. The following 4 \Theta 4 example illustrates the difference between standard bidiagonal reduction and the approach advocated here. Example 4.1. Let A be the 4 \Theta 4 matrix are small parameters. Using the MATLAB value we chose That yields a matrix that has two singular values clustered at 1, and two distinct singular values smaller than ffl. To the digits displayed, If we perform Algorithm 3.1 on A without the preprocessing in x2, we obtain The use of the bisection routine in [4] obtain the singular values The computed singular vector matrices are The invariant subspaces for the double singular value at 1 and for singular values 3 and 4 are correct. However, the individual singular vectors for singular values 3 and 4 are wrong. Algorithm 3.2 used after the reduction in x2 obtains The computed singular values to the number of digits displayed are Moreover, these corresponded to those computed by the Jacobi method to about 15 significant digits. The computed singular vector matrices were \Gamma0:408248 \Gamma0:408248 \Gamma0:553113 0:600611 \Gamma0:408248 \Gamma0:408248 \Gamma0:243588 \Gamma0:779315 \Gamma0:408248 \Gamma0:408248 0:7967 0:178704C C A Our version of the Jacobi method (coded by P.A. Yoon) obtains the slightly different singular vector matrices However, singular vectors for oe 3 and oe 4 are essentially the same, and subspace for the clustered singular value near 1 is also essentially the same. Quite recently, there have been a number of papers on the singular values and vectors of matrices under structured perturbations. We give two results below that are relevant to singular values for the perturbation given in Theorem 4.9. The first is due to Kahan [13]. Lemma 4.11. Let ~ n\Thetan , and let - 1. If- ~ ~ Lemma 4.11 says the forward errors in the matrix B make only a small relative difference in the singular values. The following result shows that the non-orthogonality of U and V in Theorem 4.9 causes only a small relative change in the singular values. This theorem is given in [24, pp.423-424,problem 18]. Lemma 4.12. Let A 2 ! m\Thetan and let n\Thetap where p - n. Then for Standard bounds on eigenvalue perturbation [18, Chapter 8] and taking square roots leads to The errors in B and non-orthogonality of the transformations U and V , make only small relative changes in the singular values. The important effect is that of the error ffiC in Theorems 4.9 and 4.10. To characterize the effect of the error ffiC in (4.17), we use a generalization of results that have been published by Barlow and Demmel [4] , Demmel and Veseli'c [14], and Gu [20]. This version was proven by Barlow and Slapni-car [6] in a work in preparation. Consider the family of symmetric matrices and the associated family of eigenproblems (4. Let (- i (i); x i (i)) be the ith eigenpair of (4.25) and define S(ffi) be the set of indices given by The set S(ffi) is the set of eigenvalues for which relative error bounds can be found. The next theorem gives such a bound. Its proof follows that of Theorem 4 in [4, p.773]. Lemma 4.13. Let (- i (i); x i (i)) be the ith eigenpair of the Hermitian matrix in (4.24). Let S(ffi) be defined by (4.26). If i 2 S(ffi) then where jx jx Proof: First assume that - i (i) is simple at the point i. Then from standard eigenvalue perturbation theory for sufficiently small - we have x x x x Thus we have jx If - i (i) is simple for all i 2 [0; ffi], then the bound (4.29) follows by integrating from 0 to ffi. In Kato[25, Theorem II.6.1,p.139], it is shown that the eigenvalues of H(i) in are real analytic, even when they are multiple. Moreover, Kato [25, p.143] goes on to point out that there are only a finite number of i where - i (i) is multiple, so that - i (i) is continuous and piecewise analytic throughout the interval [0; ffi]. Thus we can obtain (4.27)-(4.28) by integrating over each of the intervals in which - i (i) is analytic. The proof of the following proposition was given by Slapni-car [6]. Proposition 4.14. Let have the singular value decomposition n\Thetan are orthogonal and Then for each we have that where Proof: We prove this lemma is by noting that oe i of H(i) where A T (i) 0 Its associated eigenvector is The matrix EH in (4.24) is given by Thus from Lemma 4.13, oe i (i) satisfies (4.30) where A u T which is the formula for - A Proposition 4.14 can be used to bound the error in the singular values caused by both stages of the bidiagonal reduction. For the stage in x2, we have bounds of the form (1.3)-(1.4). Thus for From (2.3), the value of ae A is a growth factor bounded by y is the Moore-Penrose pseudoinverse of A. Now consider the error bound in Theorem 4.9. In context, we now consider u v i to be the ith left and right singular vectors of C rather than A as in the preceding paragraph. If we let where d i is defined in (1.8). Then for i is bounded by A matrix satisfying (2.4) will often have small singular values that have modest values of - C . As pointed out by Demmel and Veseli'c [14], only an algorithm with columnwise backward error bounds can take advantage of that fact. The error bounds on computed singular vectors of C are also better for a bidi- agonal reduction satisfying Theorem 4.9. For such bounds see the papers by Veseli'c and Demmel [14] or Barlow and Slapni-car [6]. 5. Numerical Tests. We performed two sets of numerical tests on the bidiag- onal reduction algorithms. Our tests compared the singular values to those obtained by the Jacobi method described by Demmel and Veseli'c [14]. The two test sets are as follows Example 5.1 (Test Set 1). We use the set was the Cholesky factor of the Hilbert matrix of dimension k. That is, R k was the upper triangular matrix with positive diagonals that satisfied where H k is a k \Theta k matrix whose (i; j) entry is Example 5.2 (Test Set 2). The set These are just the transposes of the matrices in Example 5.1. The reduction in x2 produces an upper triangular matrix from that. Thus the resulting bidiagonal matrices tended to be more graded. Example 5.3 (Test Set 3). We computed the SVD of L 35 from Example 5.2 using the MATLAB SVD routine to obtain We then constructed 50 matrices of the form where F was 50\Theta35 matrix generated by the MATLAB function randn (for generating matrices with normally distributed entries). Each matrix in both test sets were reduced to a matrix C using the algorithm in x2. Three separate routines were used to find the SVD of C. Plots for Bidiagonal Reduction for the SVD for Factor Hilbert Matrices FromtoFig. 5.1. Relative Error Plots from Example 5.1 Algorithm J The Jacobi method described in [14]. ffl Algorihtm G The bidiagonalization method of Algorithm 3.2 followed by the bisection routine of Demmel and Kahan [13]. ffl Algorihtm H The bidiagonalization method of Algorithm 3.1 followed by the same bisection routine. For each matrix, we calculated the two ratios 1-i-n joe G oe J 1-i-n oe J where oe J i are the ith singular values as calculated by Algorithms J, G, and H respectively. Thus we were trying to measure how well the SVDs calculated from the two bidiagonal reduction algorithms agreed with that from the Jacobi algorithm. The graphs of these values for the three test sets are given in Figures 5.1, 5.2, and 5.3. A logarithmic plot of singular values of the matrix R 90 from Example 5.1 is given in Figure 5.4. Clearly the singular values of L 90 in Example 5.2 span the same range. All three Examples had singular values that spanned a wide range. As can be seen from the figures, the maximum relative error for any singular value from either test set for Algorithm G or H is about 10 \Gamma13 or approximately 10 3 times Plots for Bidiagonal Reduction for the SVD for Factor Hilbert Matrices FromtoFig. 5.2. Relative Error Plots from Example 5.2 machine precision in MATLAB. There seems to be no measurable difference between Algorithms G and H in the quality of the singular values produced. The error analysis produced in the x4 indicates that we should expect better accuracy from Algorithm G, but does not explain why the singular values produced from both algorithms are so accurate. We suspect that matrix C produced from the reduction in x2 will almost always have good bidiagonal reduction from either algorithm, but we know of no analysis to explain why this happens for Algorithm H. 6. Conclusion. We have presented a new bidiagonal reduction algorithm that have four difference from the standard algorithm. Although it cost from 8 to 2 8 flops than the Golub-Kahan algorithm , our analysis shows that the new reduction gives a better guarantee of accurate singular values. Our numerical tests seemed to indicate that we performed the Cox-Higham Householder factorization routine as a pre-processor, both the new algorithm and the Golub-Kahan routine produced singular values that were even more accurate than any known theory predicts. Thus we strongly recommend this preprocessing step whenever it is feasible. The matrix C resulting from this reduction is usually highly graded and we suspect that there is still much to understand about the behavior of bidiagonal redcution on graded matrices. The extra cost of the new bidiagonal reduction method is far less than that of any implementation of the Jacobi method to date. It gives a reasonable guarantee of relative accuracy singular values larger than " 3=2 M and tests confirm that behaves well Plots for Bidiagonal Reduction for the SVD fortrials for Very Small Singular values Fig. 5.3. Relative Error Plots from Example 5.3 for singular values smaller than " 3=2 M . --R Fast plane rotations with dynamic scaling. Numerical solution of the weighted linear least squares problem by G- transformations Stability analysis of the G-algorithm and a note on its application to sparse least squares problems Computing accurate eigensystems of scaled diagonally dominant matrices. The direct solution of weighted and equality constrained least squares problems. Optimal perturbation bounds for the Hermitian eigenvalue prob- lem Growth factors in Gaussian elimination Linear least squares solutions by Householder transformations. An improved algorithm for computing the singular value decomposition. Stability of Householder QR factorization for weighted least squares problems. On computing accurate singular values and eigenvalues of matrices with acyclic graphs. Computing the Singular Value Decomposition with High Relative Accuracy Accurate singular values of bidiagonal matrices. Jacobi's method is more accurate than QR. Accurate singular values and differential qd algorithms. Least squares computations by Givens rotations without square roots. Calculating the singular values and pseudoinverse of a matrix. Matrix Computations Singular value decomposition and least squares solutions. Studies in Numerical Linear Algebra. Backward error analysis for the constrained and weighted linear least squares problem when using the weighted QR factorization. A note on the modifications to the Givens plane rotation. Accuracy and Stability of Numerical Algorithms. Matrix Analysis. A Short Introduction to Perturbation Theory for Linear Operators. Solving Least Squares Problems. Fast accurate eigensystem computation by Jacobi methods. Fast accurate eigenvalue methods for graded positive definite matrices Numer. Analysis of algorithms for reflectors in bisectors. On applying Householder's method to linear least squares prob- lems The Algebraic Eigenvalue Problem. --TR

singular values;bidiagonal form;orthogonal reduction;accuracy

587825

A Fully Asynchronous Multifrontal Solver Using Distributed Dynamic Scheduling.

In this paper, we analyze the main features and discuss the tuning of the algorithms for the direct solution of sparse linear systems on distributed memory computers developed in the context of a long term European research project. The algorithms use a multifrontal approach and are especially designed to cover a large class of problems. The problems can be symmetric positive definite, general symmetric, or unsymmetric matrices, both possibly rank deficient, and they can be provided by the user in several formats. The algorithms achieve high performance by exploiting parallelism coming from the sparsity in the problem and that available for dense matrices. The algorithms use a dynamic distributed task scheduling technique to accommodate numerical pivoting and to allow the migration of computational tasks to lightly loaded processors. Large computational tasks are divided into subtasks to enhance parallelism. Asynchronous communication is used throughout the solution process to efficiently overlap communication with computation.We illustrate our design choices by experimental results obtained on an SGI Origin 2000 and an IBM SP2 for test matrices provided by industrial partners in the PARASOL project.

Introduction We consider the direct solution of large sparse linear systems on distributed memory computers. The systems are of the form A is an n \Theta n symmetric positive definite, general symmetric, or unsymmetric sparse matrix that is possibly rank deficient, b is the right-hand side vector, and x is the solution vector to be computed. The work presented in this article, has been performed as Work Package 2:1 within the PARASOL Project. PARASOL is an ESPRIT IV Long Term Research Project (No 20160) for "An Integrated Environment for Parallel Sparse Matrix Solvers". The main goal of this Project, which started on January 1996 and finishes in June 1999, is to build and test a portable library for solving large sparse systems of equations on distributed memory systems. The final library will be in the public domain and will contain routines for both the direct and iterative solution of symmetric and unsymmetric systems. In the context of PARASOL, we have produced a MUltifrontal Massively Parallel Solver [27, 28] referred to as MUMPS in the remainder of this paper. Several aspects of the algorithms used in MUMPS combine to give an approach which is unique among sparse direct solvers. These include: ffl classical partial numerical pivoting during numerical factorization requiring the use of dynamic data structures, ffl the ability to automatically adapt to computer load variations during the numerical phase, ffl high performance, by exploiting the independence of computations due to sparsity and that available for dense matrices, and ffl the capability of solving a wide range of problems, including symmetric, unsymmetric, and rank-deficient systems using either LU or LDL T factorization. To address all these factors, we have designed a fully asynchronous algorithm based on a multifrontal approach with distributed dynamic scheduling of tasks. The current version of our package provides a large range of options, including the possibility of inputting the matrix in assembled format either on a single processor or distributed over the processors. Additionally, the matrix can be input in elemental format (currently only on one processor). MUMPS can also determine the rank and a null-space basis for rank-deficient matrices, and can return a Schur complement matrix. It contains classical pre- and postprocessing facilities; for example, matrix scaling, iterative refinement, and error analysis. Among the other work on distributed memory sparse direct solvers of which we are aware [7, 10, 12, 22, 23, 24], we do not know of any with the same capabilities as the MUMPS solver. Because of the difficulty of handling dynamic data structures efficiently, most distributed memory approaches do not perform numerical pivoting during the factorization phase. Instead, they are based on a static mapping of the tasks and data and do not allow task migration during numerical factorization. Numerical pivoting can clearly be avoided for symmetric positive definite matrices. For unsymmetric matrices, Duff and Koster [18, 19] have designed algorithms to permute large entries onto the diagonal and have shown that this can significantly reduce numerical pivoting. Demmel and Li [12] have shown that, if one preprocesses the matrix using the code of Duff and Koster, static pivoting (with possibly modified diagonal values) followed by iterative refinement can normally provide reasonably accurate solutions. They have observed that this preprocessing, in combination with an appropriate scaling of the input matrix, is a issue for the numerical stability of their approach. The rest of this paper is organized as follows. We first introduce some of the main terms used in a multifrontal approach in Section 2. Throughout this paper, we study the performance obtained on the set of test problems that we describe in Section 3. We discuss, in Section 4, the main parallel features of our approach. In Section 5, we give initial performance figures and we show the influence of the ordering of the variables on the performance of MUMPS. In Section 6, we describe our work on accepting the input of matrices in elemental form. Section 7 then briefly describes the main properties of the algorithms used for distributed assembled matrices. In Section 8, we comment on memory scalability issues. In Section 9, we describe and analyse the distributed dynamic scheduling strategies that will be further analysed in Section 10 where we show how we can modify the assembly tree to introduce more parallelism. We present a summary of our results in Section 11. Most results presented in this paper have been obtained on the 35 processor IBM SP2 located at GMD (Bonn, Germany). Each node of this computer is a 66 MHertz processor with 128 MBytes of physical memory and 512 MBytes of virtual memory. The SGI Cray Origin 2000 from Parallab (University of Bergen, Norway) has also been used to run some of our largest test problems. The Parallab computer consists of 64 nodes sharing 24 GBytes of physically distributed memory. Each node has two R10000 MIPS RISC 64-bit processors sharing 384 MBytes of local memory. Each processor runs at a frequency of 195 MHertz and has a peak performance of a little under 400 Mflops per second. All experiments reported in this paper use Version 4:0 of MUMPS. The software is written in Fortran 90. It requires MPI for message passing and makes use of BLAS [14, 15], LAPACK [6], BLACS [13], and ScaLAPACK [9] subroutines. On the IBM SP2, we are currently using a non-optimized portable local installation of ScaLAPACK, because the IBM optimized library PESSL V2 is not available. Multifrontal methods It is not our intention to describe the details of a multifrontal method. We rather just define terms used later in the paper and refer the reader to our earlier publications for a more detailed description, for example [3, 17, 20]. In the multifrontal method, all elimination operations take place within dense submatrices, called frontal matrices. A frontal matrix can be partitioned as shown in Figure 1. In this matrix, pivots can be chosen from within the block F 11 only. The Schur complement matrix computed and used to update later rows and columns of the overall matrix. We call this update matrix, the contribution block. fully summed rows - partly summed rows - fully summed columns partly summed columns Figure 1: Partitioning of a frontal matrix. The overall factorization of the sparse matrix using a multifrontal scheme can be described by an assembly tree, where each node corresponds to the computation of a Schur complement as just described, and each edge represents the transfer of the contribution block from the son node to the father node in the tree. This father node assembles (or sums) the contribution blocks from all its son nodes with entries from the original matrix. If the original matrix is given in assembled format, complete rows and columns of the input matrix are assembled at once, and, to facilitate this, the input matrix is ordered according to the pivot order and stored as a collection of arrowheads. That is, if the permuted matrix has entries in, for example, columns of row of column the arrowhead list associated with variable i is g. In the symmetric case, only entries from the lower triangular part of the matrix are stored. We say that we are storing the matrix in arrowhead form or by arrowheads. For unassembled matrices, complete element matrices are assembled into the frontal matrices and the input matrix need not be preprocessed. In our implementation, the assembly tree is constructed from the symmetrized pattern of the matrix and a given sparsity ordering. By symmetrized pattern, we mean the pattern of the matrix A+A T where the summation is symbolic. Note that this allows the matrix to be unsymmetric. Because of numerical pivoting, it is possible that some variables cannot be eliminated from a frontal matrix. The fully summed rows and columns that correspond to such variables are added to the contribution block that is sent to the father node. The assembly of fully summed rows and columns into the frontal matrix of the father node means that the corresponding elimination operations are delayed. This will be repeated until elimination steps on the later frontal matrices have introduced stable pivots to the delayed fully summed part. The delay of elimination steps corresponds to an a posteriori modification of the original assembly tree structure and in general introduces additional (numerical) fill-in in the factors. An important aspect of the assembly tree is that operations at a pair of nodes where neither is an ancestor of the other are independent. This gives the possibility for obtaining parallelism from the tree (so-called tree parallelism). For example, work can commence in parallel on all the leaf nodes of the tree. Fortunately, near the root node of the tree, where the tree parallelism is very poor, the frontal matrices are usually much larger and so techniques for exploiting parallelism in dense factorizations can be used (for example, blocking and use of higher Level BLAS). We call this node parallelism. We discuss further aspects of the parallelism of the multifrontal method in later sections of this paper. Our work is based on our experience of designing and implementing a multifrontal scheme on shared and virtual shared memory computers (for example, [2, 3, 4]) and on an initial prototype distributed memory multifrontal version [21]. We describe the design of our resulting distributed memory multifrontal algorithm in the rest of this paper. 3 Test problems Throughout this paper, we will use a set of test problems to illustrate the performance of our algorithms. We describe the set in this section. In Tables 1 and 2, we list our unassembled and assembled test problems, respectively. All except one come from the industrial partners of the PARASOL Project. The remaining matrix, bbmat, is from the forthcoming Rutherford-Boeing Sparse Matrix Collection [16]. For symmetric matrices, we show the number of entries in the lower triangular part of the matrix. Typical PARASOL test cases are from the following major application areas: computational fluid dynamics (CFD), structural mechanics, modelling compound devices, modelling ships and mobile offshore platforms, industrial processing of complex non-Newtonian liquids, and modelling car bodies and engine components. Some test problems are provided in both assembled format and elemental format. The suffix (rsa or rse) is used to differentiate them. For those in elemental format, the original matrix is represented as a sum of element matrices where each A i has nonzero entries only in those rows and columns that correspond to variables in the ith element. Because element matrices may overlap, the number of entries of a matrix in elemental format is usually larger than for the same matrix when assembled (compare the matrices from Det Norske Veritas of Norway in Tables 1 and 2). Typically there are about twice the number of entries in the unassembled elemental format. Real Symmetric Elemental (rse) Matrix name Order No. of elements No. of entries Origin t1.rse 97578 5328 6882780 Det Norske Veritas ship 001.rse 34920 3431 3686133 Det Norske Veritas ship 003.rse 121728 45464 9729631 Det Norske Veritas shipsec1.rse 140874 41037 8618328 Det Norske Veritas shipsec5.rse 179860 52272 11118602 Det Norske Veritas shipsec8.rse 114919 35280 7431867 Det Norske Veritas thread.rse 29736 2176 3718704 Det Norske Veritas x104.rse 108384 6019 7065546 Det Norske Veritas Table 1: Unassembled symmetric test matrices from PARASOL partner (in elemental format). In Tables 3, 4, and 5, we present statistics on the factorizations of the various test problems using MUMPS. The tables show the number of entries in the factors and the number of floating-point operations (flops) for elimination. For unsymmetric problems, we show both the estimated number, assuming no pivoting, and the actual number when numerical pivoting is used. The statistics clearly depend on the ordering used. Two classes of ordering will be considered in this paper. The first is an Approximate Minimum Degree ordering (referred to as AMD, see [1]). The second class is based on a hybrid Nested Dissection and minimum degree technique (referred to as ND). These hybrid orderings were generated using ONMETIS [26] or a combination of the graph partitioning tool SCOTCH [29] with a variant of AMD (Halo- AMD, see [30]). For matrices available in both assembled and unassembled format, we used nested dissection based orderings provided by Det Norske Veritas and denote these by MFR. Note that, in this paper, it is not our intention to compare the packages that we used to obtain the orderings; we will only discuss the influence of the type of ordering on the performance of MUMPS (in Section 5). The AMD ordering algorithms are tightly integrated within the MUMPS code; the other orderings are passed to MUMPS as an externally computed ordering. Because of this tight integration, we observe in Table 3 that the analysis time is smaller using AMD than some Real Unsymmetric Assembled (rua) Matrix name Order No. of entries Origin mixing-tank 29957 1995041 Polyflow S.A. bbmat 38744 1771722 Rutherford-Boeing (CFD) Real Symmetric Assembled (rsa) Matrix name Order No. of entries Origin oilpan 73752 1835470 INPRO b5tuer 162610 4036144 INPRO crankseg 1 52804 5333507 MacNeal-Schwendler bmw7st 1 141347 3740507 MacNeal-Schwendler ship 001.rsa 34920 2339575 Det Norske Veritas ship 003.rsa 121728 4103881 Det Norske Veritas shipsec1.rsa 140874 3977139 Det Norske Veritas shipsec5.rsa 179860 5146478 Det Norske Veritas shipsec8.rsa 114919 3384159 Det Norske Veritas thread.rsa 29736 2249892 Det Norske Veritas x104.rsa 108384 5138004 Det Norske Veritas Table 2: Assembled test matrices from PARASOL partners (except the matrix bbmat). AMD ordering Entries in Flops Time for Matrix factors (\Theta10 6 estim. actual estim. actual (seconds) mixing-tank 38.5 39.1 64.1 64.4 4.9 inv-extrusion-1 30.3 31.2 34.3 35.8 4.6 bbmat 46.0 46.2 41.3 41.6 8.1 ND ordering Entries in Flops Time for Matrix factors (\Theta10 6 estim. actual estim. actual (seconds) mixing-tank 18.9 19.6 13.0 13.2 12.8 bbmat 35.7 35.8 25.5 25.7 11.3 Table 3: Statistics for unsymmetric test problems on the IBM SP2. user-defined precomputed ordering (in this paper ND or MFR orderings). In addition, the cost of computing the external ordering is not included in these tables. AMD ordering ND ordering Entries Flops Time for Entries Flops Matrix in factors analysis in factors b5tuer 26 13 15 24 12 bmw7st 1 Table 4: Statistics for symmetric test problems on the IBM SP2. Entries Flops Matrix in factors ship 003 57 73 shipsec1 37 shipsec5 51 52 shipsec8 34 34 thread Table 5: Statistics for symmetric test problems, available in both assembled (rsa) and unassembled (rse) formats (MFR ordering). 4 Parallel implementation issues In this paper, we assume a one-to-one mapping between processes and processors in our distributed memory environment. A process will thus implicitly refer to a unique processor and, when we say for example that a task is allocated to a process, we mean that the task is also mapped onto the corresponding processor. As we did before in a shared memory environment [4], we exploit parallelism both arising from sparsity (tree parallelism) and from dense factorizations kernels (node parallelism). To avoid the limitations due to centralized scheduling, where a host process is in charge of scheduling the work of the other processes, we have chosen a distributed scheduling strategy. In our implementation, a pool of work tasks is distributed among the processes that participate in the numerical factorization. A host process is still used to perform the analysis phase (and identify the pool of work tasks), distribute the right-hand side vector, and collect the solution. Our implementation allows this host process to participate in the computations during the factorization and solution phases. This allows the user to run the code on a single processor and avoids one processor being idle during the factorization and solution phases. The code solves the system in three main steps: 1. Analysis. The host performs an approximate minimum degree ordering based on the symmetrized matrix pattern A carries out the symbolic factorization. The ordering can also be provided by the user. The host also computes a mapping of the nodes of the assembly tree to the processors. The mapping is such that it keeps communication costs during factorization and solution to a minimum and balances the memory and computation required by the processes. The computational cost is approximated by the number of floating-point operations, assuming no pivoting is performed, and the storage cost by the number of entries in the factors. After computing the mapping, the host sends symbolic information to the other processes. Using this information, each process estimates the work space required for its part of the factorization and solution. The estimated work space should be large enough to handle the computational tasks that were assigned to the process at analysis time plus possible tasks that it may receive dynamically during the factorization, assuming that no excessive amount of unexpected fill-in occurs due to numerical pivoting. 2. Factorization. The original matrix is first preprocessed (for example, converted to arrowhead format if the matrix is assembled) and distributed to the processes that will participate in the numerical factorization. Each process allocates an array for contribution blocks and factors. The numerical factorization on each frontal matrix is performed by a process determined by the analysis phase and potentially one or more other processes that are determined dynamically. The factors must be kept for the solution phase. 3. Solution. The right-hand side vector b is broadcast from the host to the other processes. They compute the solution vector x using the distributed factors computed during the factorization phase. The solution vector is then assembled on the host. 4.1 Sources of parallelism We consider the condensed assembly tree of Figure 2, where the leaves represent subtrees of the assembly tree. SUBTREES Type 2 Type 3 Type 2 Type 2 Type 1 Figure 2: Distribution of the computations of a multifrontal assembly tree over the four processors P0, P1, P2, and P3. If we only consider tree parallelism, then the transfer of the contribution block from a node in the assembly tree to its father node requires only local data movement when the nodes are assigned to the same process. Communication is required when the nodes are assigned to different processes. To reduce the amount of communication during the factorization and solution phases, the mapping computed during the analysis phase assigns a subtree of the assembly tree to a single process. In general, the mapping algorithm chooses more leaf subtrees than there are processes and, by mapping these subtrees carefully onto the processes, we achieve a good overall load balance of the computation at the bottom of the tree. We have described this in more detail in [5]. However, if we exploit only tree parallelism, the speedups are very disappointing. Obviously it depends on the problem, but typically the maximum speedup is no more than 3 to 5 as illustrated in Table 6. This poor performance is caused by the fact that the tree parallelism decreases while going towards the root of the tree. Moreover, it has been observed (see for example [4]) that often more than 75% of the computations are performed in the top three levels of the assembly tree. It is thus necessary to obtain further parallelism within the large nodes near the root of the tree. The additional parallelism will be based on parallel blocked versions of the algorithms used during the factorization of the frontal matrices. Nodes of the assembly tree that are treated by only one process will be referred to as nodes of type 1 and the parallelism of the assembly tree will be referred to as type 1 parallelism. Further parallelism is obtained by a one-dimensional (1D) block partitioning of the rows of the frontal matrix for nodes with a large contribution block (see Figure 2). Such nodes will be referred to as nodes of type 2 and the corresponding parallelism as type 2 parallelism. Finally, if the frontal matrix of the root node is large enough, we partition it in a two-dimensional (2D) block cyclic way. The parallel root node will be referred to as a node of type 3 and the corresponding parallelism as type 3 parallelism. 4.2 Type 2 parallelism During the analysis phase, a node is determined to be of type 2 if the number of rows in its contribution block is sufficiently large. If a node is of type 2, one process (called the master) holds all the fully summed rows and performs the pivoting and the factorization on this block while other processes (called slaves) perform the updates on the partly summed rows (see Figure 1). The slaves are determined dynamically during factorization and any process may be selected. To be able to assemble the original matrix entries quickly into the frontal matrix of a type 2 node, we duplicate the corresponding original matrix entries (stored as arrowheads or element matrices) onto all the processes before the factorization. This way, the master and slave processes of a type 2 node have immediate access to the entries that need to be assembled in the local part of the frontal matrix. This duplication of original data enables efficient dynamic scheduling of computational tasks, but requires some extra storage. This is studied in more detail in Section 8. (Note that for a type 1 node, the original matrix entries need only be present on the process handling this node.) At execution time, the master of a type 2 node first receives symbolic information describing the structure of the contribution blocks of its son nodes in the tree. This information is sent by the (master) processes handling the sons. Based on this information, the master determines the exact structure of its frontal matrix and decides which slave processes will participate in the factorization of the node. It then sends information to the processes handling the sons to enable them to send the entries in their contribution blocks directly to the appropriate processes involved in the type 2 node. The assemblies for this node are subsequently performed in parallel. The master and slave processes then perform the elimination operations on the frontal matrix in parallel. Macro-pipelining based on a blocked factorization of the fully summed rows is used to overlap communication with computation. The efficiency of the algorithm thus depends on both the block size used to factor the fully summed rows and on the number of rows allocated to a slave process. Further details and differences between the implementations for symmetric and unsymmetric matrices are described in [5]. 4.3 Type 3 parallelism At the root node, we must factorize a dense matrix and we can use standard codes for this. For scalability reasons, we use a 2D block cyclic distribution of the root node and we use ScaLAPACK [9] or the vendor equivalent implementation (routine PDGETRF for general matrices and routine PDPOTRF for symmetric positive definite matrices) for the actual factorization. Currently, a maximum of one root node, chosen during the analysis, is processed in parallel. The node chosen will be the largest root provided its size is larger than a computer dependent parameter (otherwise it is factorized on only one processor). One process (also called the master) holds all the indices describing the structure of the root frontal matrix. We call the root node, as determined by the analysis phase, the estimated root node. Before factorization, the structure of the frontal matrix of the estimated root node is statically mapped onto a 2D grid of processes. This mapping fully determines to which process an entry of the estimated root node is assigned. Hence, for the assembly of original matrix entries and contribution blocks, the processes holding this information can easily compute exactly the processes to which they must send data to. In the factorization phase, the original matrix entries and the part of the contribution blocks from the sons corresponding to the estimated root can be assembled as soon as they are available. The master of the root node then collects the index information for all the delayed variables (due to numerical pivoting) of its sons and builds the final structure of the root frontal matrix. This symbolic information is broadcast to all processes that participate in the factorization. The contributions corresponding to delayed variables are then sent by the sons to the appropriate processes in the 2D grid for assembly (or the contributions can be directly assembled locally if the destination is the same process). Note that, because of the requirements of ScaLAPACK, local copying of the root node is required since the leading dimension will change if there are any delayed pivots. 4.4 Parallel triangular solution The solution phase is also performed in parallel and uses asynchronous communications both for the forward elimination and the back substitution. In the case of the forward elimination, the tree is processed from the leaves to the root, similar to the factorization, while the back substitution requires a different algorithm that processes the tree from the root to the leaves. A pool of ready-to-be-activated tasks is used. We do not change the distribution of the factors as generated in the factorization phase. Hence, type 2 and 3 parallelism are also used in the solution phase. At the root node, we use ScaLAPACK routine PDGETRS for general matrices and routine PDPOTRS for symmetric positive definite matrices. 5 Basic performance and influence of ordering From earlier studies (for example [25]), we know that the ordering may seriously impact both the uniprocessor time and the parallel behaviour of the method. To illustrate this, we report in Table 6 performance obtained using only type 1 parallelism. The results show that using only type 1 parallelism does not produce good speedups. The results also show (see columns "Speedup") that we usually get better parallelism with nested dissection based orderings than with minimum degree based orderings. We thus gain by using nested dissection because of both a reduction in the number of floating-point operations (see Tables 3 and 4) and a better balanced assembly tree. We now discuss the performance obtained with MUMPS on matrices in assembled format that will be used as a reference for this paper. The performance obtained on matrices provided in elemental format is discussed in Section 6. In Tables 7 and 8, we show the performance of MUMPS using nested dissection and minimum degree orderings on the IBM SP2 and the SGI Origin 2000, respectively. Note that speedups are difficult to compute on the IBM SP2 because memory paging often occurs on a small number of processors. Hence, the better performance with nested dissection orderings on a small number of processors of the IBM SP2 is due, in part, to the reduction in the memory required by each processor (since there are less entries in the factors). To get a better idea of the true algorithmic speedups (without memory paging effects), we give, in Table 7, the uniprocessor CPU time for one processor, instead of the elapsed time. Matrix Time Speedup AMD ND AMD ND oilpan 12.6 7.3 2.91 4.45 bmw7st 1 55.6 21.3 2.55 4.87 bbmat 78.4 49.4 4.08 4.00 b5tuer 33.4 25.5 3.47 4.22 Table Influence of the ordering on the time (in seconds) and speedup for the factorization phase, using only type 1 parallelism, on 32 processors of the IBM SP2. When the memory was not large enough to run on one processor, an estimate of the Megaflop rate was used to compute the uniprocessor CPU time. (This estimate was also used, when necessary, to compute the speedups in Table 6.) On a small number of processors, there can still be a memory paging effect that may significantly increase the elapsed time. However, the speedup over the elapsed time on one processor (not given) can be considerable. Matrix Ordering Number of processors oilpan AMD 37 13.6 9.0 6.8 5.9 5.8 b5tuer AMD 116 155.5 24.1 16.8 16.1 13.1 crankseg 1 AMD 456 508.3 162.4 78.4 63.3 bmw7st 1 AMD 142 153.4 46.5 21.3 18.4 16.7 ND 104 105.7 36.7 20.2 12.9 11.7 mixing-tank AMD 495 - 288.5 70.7 64.5 61.3 ND 104 32.80 26.1 17.4 14.4 14.8 bbmat AMD 320 276.4 68.3 47.8 44.0 39.8 ND 198 106.4 76.7 35.2 34.6 30.9 Table 7: Impact of the ordering on the time (in seconds) for factorization on the IBM estimated CPU time on one processor; - means not enough memory. Table 8 also shows the elapsed time for the solution phase; we observe that the speedups for this phase are quite good. In the remainder of this paper, we will use nested dissection based orderings, unless stated otherwise. Factorization phase Matrix Ordering Number of processors bmw7st 1 AMD 85.7 56.0 28.2 18.5 15.1 14.2 ND 306.6 182.7 80.9 52.9 41.2 35.5 ND 152.1 93.8 52.5 33.0 22.1 17.0 Solution phase Matrix Ordering Number of processors crankseg 2 AMD 6.8 5.8 4.4 2.9 2.4 2.3 ND 4.3 2.7 1.8 1.5 1.1 1.8 bmw7st 1 AMD 4.2 2.4 2.3 1.9 1.4 1.6 ND 3.3 2.1 1.7 1.4 1.6 1.5 ND 8.3 4.7 2.7 2.1 1.8 2.0 ND 6.3 3.8 2.9 2.4 2.0 2.4 Table 8: Impact of the ordering on the time (in seconds) for factorization and solve phases on the SGI Origin 2000. 6 Elemental input matrix format In this section, we discuss the main algorithmic changes to handle efficiently problems that are provided in elemental format. We assume that the original matrix can be represented as a sum of element matrices where each A i has nonzero entries only in those rows and columns that correspond to variables in the ith element. A i is usually held as a dense matrix, but if the matrix A is symmetric, only the lower triangular part of each A i is stored. In a multifrontal approach, element matrices need not be assembled in more than one frontal matrix during the elimination process. This is due to the fact that the frontal matrix structure contains, by definition, all the variables adjacent to any fully summed variable of the front. As a consequence, element matrices need not be split during the assembly process. Note that, for classical fan-in and fan-out approaches [7], this property does not hold since the positions of the element matrices to be assembled are not restricted to fully summed rows and columns. The main modifications that we had to make to our algorithms for assembled matrices to accommodate unassembled matrices lie in the analysis, the distribution of the matrix, and the assembly process. We will describe them in more detail below. In the analysis phase, we exploit the elemental format of the matrix to detect supervariables. We define a supervariable as a set of variables having the same list of adjacent elements. This is illustrated in Figure 3 where the matrix is composed of two overlapping elements and has three supervariables. (Note that our definition of a supervariable differs from the usual definition, see for example [11]). Supervariables have been used successfully in a similar context to compress graphs associated with assembled matrices from structural engineering prior to a multiple minimum degree ordering [8]. For assembled matrices, however, it was observed in [1] that the use of supervariables in combination with an Approximate Minimum Degree algorithm was not more efficient. Graph size with Matrix supervariable detection OFF ON t1.rse 9655992 299194 ship 003.rse 7964306 204324 shipsec1.rse shipsec5.rse 9933236 256976 shipsec8.rse 6538480 171428 thread.rse 4440312 397410 x104.rse 10059240 246950 Table 9: Impact of supervariable detection on the length of the adjacency lists given to the ordering phase. Table 9 shows the impact of using supervariables on the size of the graph processed by the ordering phase (AMD ordering). Graph size is the length of the adjacency lists of variables/supervariables given as input to the ordering phase. Without supervariable detection, Initial graph of variables Initial matrix7426,7,8 4,5 Graph of supervariables (sum of two overlapping elements) Figure 3: Supervariable detection for matrices in elemental format. Graph size is twice the number of off-diagonal entries in the corresponding assembled matrix. The working space required by the analysis phase using the AMD ordering is dominated by the space required by the ordering phase and is Graph size plus an overhead that is a small multiple of the order of matrix. Since the ordering is performed on a single processor, the space required to compute the ordering is the most memory intensive part of the analysis phase. With supervariable detection, the complete uncompressed graph need not be built since the ordering phase can operate directly on the compressed graph. Table 9 shows that, on large graphs, compression can reduce the memory requirements of the analysis phase dramatically. Table shows the impact of using supervariables on the time for the complete analysis phase (including graph compression and ordering). We see that the reduction in time is not only due to the reduced time for ordering; significantly less time is also needed for building the much smaller adjacency graph of the supervariables. Time for analysis Matrix supervariable detection OFF ON t1.rse 4.6 (1.8) 1.5 (0.3) ship 003.rse 7.4 (2.8) 3.2 (0.7) shipsec1.rse 6.0 (2.2) 2.6 (0.6) shipsec5.rse 10.1 (4.6) 3.9 (0.8) shipsec8.rse 5.7 (2.0) 2.6 (0.5) thread.rse 2.6 (0.9) 1.2 (0.2) x104.rse 6.4 (3.5) 1.5 (0.3) Table 10: Impact of supervariable detection on the time (in seconds) for the analysis phase on the SGI Origin 2000. The time spent in the AMD ordering is in parentheses. The overall time spent in the assembly process for matrices in elemental format will differ from the overall time spent in the assembly process for the equivalent assembled matrix. Obviously, for the matrices in elemental format there is often significantly more data to assemble (usually about twice the number of entries as for the same matrix in assembled format). However, the assembly process of matrices in elemental format should be performed more efficiently than the assembly process of assembled matrices. First, because we potentially assemble at once a larger and more regular structure (a full matrix). Second, because most input data will be assembled at or near leaf nodes in the assembly tree. This has two consequences. The assemblies are performed in a more distributed way and most assemblies of original element matrices are done at type 1 nodes. (Hence, less duplication of original matrix data is necessary.) A more detailed analysis of the duplication issues linked to matrices in elemental format will be addressed in Section 8. In our experiments (not shown here), we have observed that, despite the differences in the assembly process, the performance of MUMPS for assembled and unassembled problems is very similar, provided the same ordering is used. The reason for this is that the extra amount of assemblies of original data for unassembled problems is relatively small compared to the total number of flops. The experimental results in Tables 11 and 12, obtained on the SGI Origin 2000, show the good scalability of the code for both the factorization and the solution phases on our set of unassembled matrices. Matrix Number of processors ship 003.rse 392 242 156 120 92 shipsec1.rse 174 128 shipsec5.rse 281 176 114 63 43 shipsec8.rse 187 127 68 36 thread.rse 186 120 69 46 37 x104.rse 56 34 20 Table 11: Time (in seconds) for factorization of the unassembled matrices on the SGI Origin 2000. MFR ordering is used. Matrix Number of processors t1.rse 3.5 2.1 1.1 1.2 0.8 ship 003.rse 6.9 3.6 3.3 2.5 2.0 shipsec1.rse 3.8 3.1 2.1 1.6 1.5 shipsec5.rse 5.5 4.2 2.9 2.2 1.9 shipsec8.rse 3.8 3.1 2.0 1.4 1.3 thread.rse 2.3 1.9 1.3 1.0 0.8 x104.rse 2.6 1.9 1.4 1.0 1.1 Table 12: Time (in seconds) for the solution phase of the unassembled matrices on the SGI Origin 2000. MFR ordering is used. 7 Distributed assembled matrix The distribution of the input matrix over the available processors is the main preprocessing step in the numerical factorization phase. During this step, the input matrix is organized into arrowhead format and distributed according to the mapping provided by the analysis phase. In the symmetric case, the first arrowhead of each frontal matrix is also sorted to enable efficient assembly [5]. If the assembled matrix is initially held centrally on the host, we have observed that the time to distribute the real entries of the original matrix can sometimes be comparable to the time to perform the actual factorization. For example, for matrix oilpan, the time to distribute the input matrix on 16 processors of the IBM SP2 is on average 6 seconds whereas the time to factorize the matrix is 6.8 seconds (using AMD ordering, see Table 7). Clearly, for larger problems where more arithmetic is required for the actual factorization, the time for factorization will dominate the time for redistribution. With a distributed input matrix format we can expect to reduce the time for the redistribution phase because we can parallelize the reformatting and sorting tasks, and we can use asynchronous all-to-all (instead of one-to-all) communications. Furthermore, we can expect to solve larger problems since storing the complete matrix on one processor limits the size of the problem that can be solved on a distributed memory computer. Thus, to improve both the memory and the time scalability of our approach, we should allow the input matrix to be distributed. Based on the static mapping of the tasks to processes that is computed during the analyis phase, one can a priori distribute the input data so that no further remapping is required at the beginning of the factorization. This distribution, referred to as the MUMPS mapping, will limit the communication to duplications of the original matrix corresponding to type 2 nodes (further studied in Section 8). To show the influence of the initial matrix distribution on the time for redistribution, we compare, in Figure 4, three ways for providing the input matrix: 1. Centralized mapping: the input matrix is held on one process (the host). 2. MUMPS mapping: the input matrix is distributed over the processes according to the static mapping that is computed during the analysis phase. 3. Random mapping: the input matrix is uniformly distributed over the processes in a random manner that has no correlation to the mapping computed during the analysis phase. The figure clearly shows the benefit of using asynchronous all-to-all communications (required by the MUMPS and random mappings) compared to using one-to-all communications (for the centralized mapping). It is even more interesting to observe that distributing the input matrix according to the MUMPS mapping does not significantly reduce the time for redistribution. We attribute this to the good overlapping of communication with computation (mainly data reformatting and sorting) in our redistribution algorithm. Number of Processors1357 Distribution time (seconds) Centralized matrix using MUMPS mapping Random mapping Figure 4: Impact of the initial distribution for matrix oilpan on the time for redistribution on the IBM SP2. 8 Memory scalability issues In this section, we study the memory requirements and memory scalability of our algorithms. Figure 5 illustrates how MUMPS balances the memory load over the processors. The figure shows, for two matrices, the maximum memory required on a processor and the average over all processors, as a function of the number of processors. We observe that, for varying numbers of processors, these values are quite similar. Number of Processors50150250 Size of total space (Mbytes) Maximum Average 28 Number of Processors100300500700Size of total space (Mbytes) Maximum Average Figure 5: Total memory requirement per processor (maximum and average) during factorization (ND ordering). Table 13 shows the average size per processor of the main components of the working space used during the factorization of the matrix bmw3 2. These components are: ffl Factors: the space reserved for the factors; a processor does not know after the analysis phase in which type 2 nodes it will participate, and therefore it reserves enough space to be able to participate in all type 2 nodes. ffl Stack area: the space used for stacking both the contribution blocks and the factors. ffl Initial matrix: the space required to store the initial matrix in arrowhead format. ffl Communication buffers: the space allocated for both send and receive buffers. ffl Other: the size of all the remaining workspace allocated per processor. ffl Total: the total memory required per processor. The lines ideal in Table 13 are obtained by dividing the memory requirement on one processor by the number of processors. By comparing the actual and ideal numbers, we get an idea how MUMPS scales in terms of memory for some of the components. Number of processors Factors 423 211 107 58 ideal - 211 106 53 26 Stack area 502 294 172 Initial matrix 69 34.5 17.3 8.9 5.0 4.0 3.5 ideal - 34.5 17.3 8.6 4.3 2.9 2.2 Communication buffers 0 45 34 14 6 6 5 Other 20 20 20 20 20 20 20 Total 590 394 243 135 82 69 67 ideal - 295 147 74 37 25 Table 13: Analysis of the memory used during factorization of matrix bmw3 2 (ND ordering). All sizes are in MBytes per processor. We see that, even if the total memory (sum of all the local workspaces) increases, the average memory required per processor significantly decreases up to 24 processors. We also see that the size for the factors and the stack area are much larger than ideal. Part of this difference is due to parallelism and is unavoidable. Another part, however, is due to an overestimation of the space required. The main reason for this is that the mapping of the type 2 nodes on the processors is not known at analysis and each processor can potentially participate in the elimination of any type 2 node. Therefore, each processor allocates enough space to be able to participate in all type 2 nodes. The working space that is actually used is smaller and, on a large number of processors, we could reduce the estimate for both the factors and the stack area. For example, we have successfully factorized matrix bmw3 2 on 32 processors with a stack area that is 20% smaller than reported in Table 13. The average working space used by the communication buffers also significantly decreases up to 16 processors. This is mainly due to type 2 node parallelism where contribution blocks are split among processors until a minimum granularity is reached. Therefore, when we increase the number of processors, we decrease (until reaching this minimum granularity) the size of the contribution blocks sent between processors. Note that on larger problems, the average size per processor of the communication buffers will continue to decrease for a larger number of processors. We see, as expected, that the line Other does not scale at all since it corresponds to data arrays of size O(n) that need to be allocated on each process. We see that this space significantly affects the difference between Total and ideal, especially for larger numbers of processors. However, the relative influence of this fixed size area will be smaller on large matrices from 3D simulations and therefore does not affect the asymptotic scalability of the algorithm. The imperfect scalability of the initial matrix storage comes from the duplication of the original matrix data that is linked to type 2 nodes in the assembly tree. We will study this in more detail in the remainder of this section. We want to stress, however, that from a user point of view, all numbers reported in this context should be related to the total memory used by the MUMPS package which is usually dominated, on large problems, by the size of the stack area. An alternative to the duplication of data related to type 2 nodes would be to allocate the original data associated with a frontal matrix to only the master process responsible for Matrix Number of processors oilpan Type 2 nodes 0 Total entries 1835 1845 1888 2011 2235 2521 bmw7st Total entries 3740 3759 3844 4031 4308 4793 Total entries 5758 5767 5832 6239 6548 7120 shipsec1.rsa Type 2 nodes 0 0 4 11 19 21 Total entries 3977 3977 4058 4400 4936 5337 shipsec1.rse Type 2 nodes Total entries 8618 8618 8618 8627 8636 8655 thread.rsa Type 2 nodes Total entries 2250 2342 2901 4237 6561 8343 thread.rse Type 2 nodes Total entries 3719 3719 3719 3719 3719 3719 Table 14: The amount of duplication due to type 2 nodes. "Total entries" is the sum of the number of original matrix entries over all processors (\Theta10 3 ). The number of nodes is also given. the type 2 node. During the assembly process, the master process would then be in charge of redistributing the original data to the slave processes. This strategy introduces extra communication costs during the assembly of a type 2 node and thus has not been chosen. With the approach based on duplication, the master process responsible for a type 2 node has all the flexibility to choose collaborating processes dynamically since this will not involve any data migration of the original matrix. However, the extra cost of this strategy is that, based on the decision during analysis of which nodes will be of type 2, partial duplication of the original matrix must be performed. In order to keep all the processors busy, we need to have sufficient node parallelism near the root of the assembly tree, MUMPS uses a heuristic that increases the number of type 2 nodes with the number of processors used. The influence of the number of processors on the amount of duplication is shown in Table 14. On a representative subset of our test problems, we show the total number of type 2 nodes and the sum over all processes of the number of original matrix entries and duplicates. If there is only one processor, type 2 nodes are not used and no data is duplicated. Figure 6 shows, for four matrices, the number of original matrix entries that are duplicated on all processors, relative to the total number of entries in the original matrix. Since the original data for unassembled matrices are in general assembled earlier in the assembly tree than the data for the same matrix in assembled format, the number of duplications is often relatively much smaller with unassembled matrices than with assembled matrices. Matrix thread.rse (in elemental format) is an extreme example since, even on 16 processors, type 2 node parallelism does not require any duplication (see Table 14). To conclude this section, we want to point out that the code scales well in terms of memory usage. On (virtual) shared memory computers, the total memory (sum of local workspaces over all the processors) required by MUMPS can sometimes be excessive. Therefore, we are currently investigating how we can reduce the current overestimates of the local stack areas so we can Number of Processors515Percentage BMW3_2 THREAD.RSA Figure Percentage of entries in the original matrix that are duplicated on all processors due to type 2 nodes. reduce the total memory required. A possible solution might be to limit the dynamic scheduling of a type 2 node (and corresponding data duplication) to a subset of processors. 9 Dynamic scheduling strategies To avoid the drawback of centralized scheduling on distributed memory computers, we have implemented distributed dynamic scheduling strategies. We remind the reader that type 1 nodes are statically mapped to processes at analysis time and that only type 2 tasks, which represent a large part of the computations and of the parallelism of the method, are involved in the dynamic scheduling strategy. To be able to choose dynamically the processes that will collaborate in the processing of a type 2 node, we have designed a two-phase assembly process. Let Inode be a node of type 2 and let Pmaster be the process to which Inode is initially mapped. In the first phase, the (master) processes to which the sons of Inode are mapped, send symbolic data (integer lists) to Pmaster. When the structure of the frontal matrix is determined, Pmaster decides a partitioning of the frontal matrix and chooses the slave processes. It is during this phase that Pmaster will collect information concerning the load of the other processors to help in its decision process. The slave processes are informed that a new task has been allocated to them. Pmaster then sends the description of the distribution of the frontal matrix to all collaborative processes of all sons of Inode so that they can send their contribution blocks (real values) in pieces directly to the correct processes involved in the computation of Inode. The assembly process is thus fully parallelized and the maximum size of a message sent between processes is reduced (see Section 8). A pool of tasks private to each process is used to implement dynamic scheduling. All tasks ready to be activated on a given process are stored in the pool of tasks local to the process. Each process executes the following algorithm: Algorithm 1 while ( not all nodes processed ) if local pool empty then blocking receive for a message; process the message elseif message available then receive and process message else extract work from the pool, and process it endif while Note that the algorithm gives priority to message reception. The main reasons for this choice are first that the message received might be a source of additional work and parallelism and second, the sending process might be blocked because its send buffer is full (see [5]). In the actual implementation, we use the routine MPI IPROBE to check whether a message is available. We have implemented two scheduling strategies. In the first strategy, referred to as cyclic scheduling, the master of a type 2 node does not take into account the load on the other processors and performs a simple cyclic mapping of the tasks to the processors. In the second strategy, referred to as (dynamic) flops-based scheduling, the master process uses information on the load of the other processors to allocate type 2 tasks to the least loaded processors. The load of a processor is defined here as the amount of work (flops) associated with all the active or ready-to-be-activated tasks. Each process is in charge of maintaining local information associated with its current load. With a simple remote memory access procedure, using for example the one-sided communication routine MPI GET included in MPI-2, each process has access to the load of all other processors when necessary. However, MPI-2 is not available on our target computers. To overcome this, we have designed a module based only on symmetric communication tools (MPI asynchronous send and receive). Each process is in charge of both updating and broadcasting its local load. To control the frequency of these broadcasts, an updated load is broadcast only if it is significantly different from the last load broadcast. When the initial static mapping does not balance the work well, we can expect that the dynamic flops-based scheduling will improve the performance with respect to cyclic scheduling. Tables 15 and 16 show that significant performance gains can be obtained by using dynamic flops-based scheduling. On more than 24 processors, the gains are less significant because our test problems are too small to keep all the processors busy and thus lessen the benefits of a good dynamic scheduling algorithm. We also expect that this feature will improve the behaviour of the parallel algorithm on a multi-user distributed memory computer. Another possible use of dynamic scheduling is to improve the memory usage. We have seen, in Section 8, that the size of the stack area is overestimated. Dynamic scheduling based on Matrix & Number of processors scheduling 28 cyclic 79.1 47.9 40.7 41.3 38.9 flops-based 61.1 45.6 41.9 41.7 40.4 cyclic 52.4 31.8 26.2 29.2 23.0 flops-based 29.4 27.8 25.1 25.3 22.6 Table 15: Comparison of cyclic and flops-based schedulings. Time (in seconds) for factorization on the IBM SP2 (ND ordering). Matrix & Number of processors scheduling 4 8 ship 003.rse cyclic 156.1 119.9 91.9 flops-based 140.3 110.2 83.8 shipsec5.rse cyclic 113.5 63.1 42.8 flops-based 99.9 61.3 37.0 shipsec8.rse cyclic 68.3 36.3 29.9 flops-based 65.0 35.0 25.1 Table Comparison of cyclic and flops-based schedulings. Time (in seconds) for factorization on the SGI Origin 2000 (MFR ordering). memory load, instead of computational load, could be used to address this issue. Type 2 tasks can be mapped to the least loaded processor (in terms of memory used in the stack area). The memory estimation of the size of the stack area can then be based on a static mapping of the tasks. Splitting nodes of the assembly tree During the processing of a parallel type 2 node, both in the symmetric and the unsymmetric case, the factorization of the pivot rows is performed by a single processor. Other processors can then help in the update of the rows of the contribution block using a 1D decomposition (as presented in Section 4). The elimination of the fully summed rows can represent a potential bottleneck for scalability, especially for frontal matrices with a large fully summed block near the root of the tree, where type 1 parallelism is limited. To overcome this problem, we subdivide nodes with large fully summed blocks, as illustrated in Figure 7.3571 ASSEMBLY TREE Pivot blocks Contribution blocks ASSEMBLY TREE AFTER SPLITTING2NFRONT5NPIV NPIV father son son Figure 7: Tree before and after the subdivision of a frontal matrix with a large pivot block. In this figure, we consider an initial node of size NFRONT with NPIV pivots. We replace this node by a son node of size NFRONT with NPIV son pivots, and a father node of size son , with NPIV father = NPIV \GammaNPIV son pivots. Note that by splitting a node, we increase the number of operations for factorization, because we add assembly operations. Nevertheless, we expect to benefit from splitting because we increase parallelism. We experimented with a simple algorithm that postprocesses the tree after the symbolic factorization. The algorithm considers only nodes near the root of the tree. Splitting large nodes far from the root, where sufficient tree parallelism can already be exploited, would only lead to additional assembly and communication costs. A node is considered for splitting only if its distance to the root, that is, the number edges between the root and the node, is not more than Let Inode be a node in the tree, and d(Inode) the distance of Inode to the root. For all nodes Inode such that d(Inode) dmax , we apply the following algorithm. Algorithm 2 Splitting of a node large enough then 1. Compute number of flops performed by the master of Inode. 2. Compute number of flops performed by a slave, assuming that NPROCS \Gamma 1 slaves can participate. 3. if W master ? W slave 3.1. Split Inode into nodes son and father so that NPIV son 3.2. Apply Algorithm 2 recursively to nodes son and father. endif endif Algorithm 2 is applied to a node only when NFRONT - NPIV/2 is large enough because we want to make sure that the son of the split node is of type 2. (The size of the contribution block of the son will be NFRONT - NPIV son .) A node is split only when the amount of work for the master (W master ) is large relative to the amount of work for a slave (W slave ). To reduce the amount of splitting further away from the root, we add, at step 3 of the algorithm, a relative factor to W slave . This factor depends on a machine dependent parameter and increases with the distance of the node from the root. Parameter p allows us to control the general amount of splitting. Finally, because the algorithm is recursive, we may divide the initial node into more than two new nodes. The effect of splitting is illustrated in Table 17 on both the symmetric matrix crankseg 2 and the unsymmetric matrix inv-extrusion-1. Ncut corresponds to the number of type 2 nodes cut. A value used as a flag to indicate no splitting. Flops-based dynamic scheduling is used for all runs in this section. The best time obtained for a given number of processors is indicated in bold font. We see that significant performance improvements (of up to 40% reduction in time) can be obtained by using node splitting. The best timings are generally obtained for relatively large values of p. More splitting occurs for smaller values of p, but the corresponding times do not change much. p Number of processors 28 200 Time 37.9 31.4 30.4 29.5 25.4 150 Time 41.8 31.3 31.0 28.9 27.2 100 Time 39.8 32.3 28.4 28.6 26.7 Ncut 9 11 13 14 15 50 Time 36.7 33.6 31.4 29.6 27.4 Ncut 28 p Number of processors 200 Time 25.5 16.7 13.4 12.1 12.4 150 Time 24.9 16.3 13.5 13.4 12.4 100 Time 24.9 16.2 13.7 13.1 13.6 50 Time 24.9 17.0 13.5 13.6 16.6 Table 17: Time (in seconds) for factorization and number of nodes cut for different values of parameter p on the IBM SP2. Nested dissection ordering and flops-based dynamic scheduling are used. Summary Tables and 19 show results obtained with MUMPS 4.0 using both dynamic scheduling and node splitting. Default values for the parameters controlling the efficiency of the package have been used and therefore the timings do not always correspond to the fastest possible execution time. The comparison with results presented in Tables 7, 8, and 11 summarizes well the benefits coming from the work presented in Sections 9 and 10. Matrix Number of processors oilpan 33 11.1 7.5 5.2 4.8 4.6 b5tuer 108 82.1 51.9 13.4 13.1 10.5 bmw7st 1 104 - 29.8 13.7 11.7 11.3 mixing-tank 104 30.8 21.6 16.4 14.7 14.8 bbmat 198 255.4 85.2 34.8 32.8 30.9 Table 18: Time (in seconds) for factorization using MUMPS 4.0 with default options on IBM SP2. ND ordering is used. estimated CPU time; means swapping or not enough memory. Matrix Number of processors bmw7st 1 62 36 ship 003.rse 392 237 124 108 51 shipsec1.rse 174 125 63 shipsec5.rse 281 181 103 62 37 shipsec8.rse thread.rse 186 125 70 38 24 x104.rse 56 34 19 12 11 Table 19: Time (in seconds) for factorization using MUMPS 4.0 with default options on SGI Origin 2000. ND or MFR ordering is used. The largest problem we have solved to date is a symmetric matrix of order 943695 with more than 39 million entries. The number of entries in the factors is 1:4 \Theta 10 9 and the number of operations during factorization is 5:9 \Theta 10 12 . On one processor of the SGI Origin 2000, the factorization phase required 8.9 hours and on two (non-dedicated) processors 6.2 hours were required. Because of the total amount of memory estimated and reserved by MUMPS, we could not solve it on more than 2 processors. This issue will have to be addressed to improve the scalability on globally addressable memory computers and further analysis will be performed on purely distributed memory computers with a larger number of processors. Possible solutions to this have been mentioned in the paper (limited dynamic scheduling and/or memory based dynamic scheduling) and will be developed in the future. Acknowledgements We are grateful to Jennifer Scott and John Reid for their comments on an early version of this paper. --R An approximate minimum degree ordering algorithm. Linear algebra calculations on a virtual shared memory computer. Vectorization of a multiprocessor multifrontal code. Memory management issues in sparse multifrontal methods on multiprocessors. Multifrontal parallel distributed symmetric and unsymmetric solvers. The fan-both family of column-based distributed Cholesky factorisation algorithms Compressed graphs and the minimum degree algorithm. ScaLAPACK Users' Guide. A parallel solution method for large sparse systems of equations. A supernodal approach to sparse partial pivoting. Making sparse Gaussian elimination scalable by static pivoting. Working Note 94: A Users' Guide to the BLACS v1. Algorithm 679. Algorithm 679. The Rutherford-Boeing Sparse Matrix Collection Direct Methods for Sparse Matrices. The design and use of algorithms for permuting large entries to the diagonal of sparse matrices. On algorithms for permuting large entries to the diagonal of a sparse matrix. The multifrontal solution of indefinite sparse symmetric linear systems. D'eveloppement d'une approche multifrontale pour machines 'a m'emoire distribu'ee et r'eseau h'et'erog'ene de stations de travail. Sparse Cholesky factorization on a local memory multiprocessor. Highly scalable parallel algorithms for sparse matrix factorization. Parallel algorithms for sparse linear systems. Improving the runtime and quality of nested dissection ordering. Scotch 3.1 User's guide. Hybridizing nested dissection and halo approximate minimum degree for efficient sparse matrix ordering. --TR --CTR Kai Shen, Parallel sparse LU factorization on different message passing platforms, Journal of Parallel and Distributed Computing, v.66 n.11, p.1387-1403, November 2006 Omer Meshar , Dror Irony , Sivan Toledo, An out-of-core sparse symmetric-indefinite factorization method, ACM Transactions on Mathematical Software (TOMS), v.32 n.3, p.445-471, September 2006 Patrick R. Amestoy , Iain S. Duff , Jean-Yves L'Excellent , Xiaoye S. Li, Impact of the implementation of MPI point-to-point communications on the performance of two general sparse solvers, Parallel Computing, v.29 n.7, p.833-849, July Kai Shen, Parallel sparse LU factorization on second-class message passing platforms, Proceedings of the 19th annual international conference on Supercomputing, June 20-22, 2005, Cambridge, Massachusetts Hong Zhang , Barry Smith , Michael Sternberg , Peter Zapol, SIPs: Shift-and-invert parallel spectral transformations, ACM Transactions on Mathematical Software (TOMS), v.33 n.2, p.9-es, June 2007 Mark Baertschy , Xiaoye Li, Solution of a three-body problem in quantum mechanics using sparse linear algebra on parallel computers, Proceedings of the 2001 ACM/IEEE conference on Supercomputing (CDROM), p.47-47, November 10-16, 2001, Denver, Colorado Iain S. Duff , Jennifer A. Scott, A parallel direct solver for large sparse highly unsymmetric linear systems, ACM Transactions on Mathematical Software (TOMS), v.30 n.2, p.95-117, June 2004 Adaptive grid refinement for a model of two confined and interacting atoms, Applied Numerical Mathematics, v.52 n.2-3, p.235-250, February 2005 Abdou Guermouche , Jean-Yves L'Excellent , Gil Utard, Impact of reordering on the memory of a multifrontal solver, Parallel Computing, v.29 n.9, p.1191-1218, September Vladimir Rotkin , Sivan Toledo, The design and implementation of a new out-of-core sparse cholesky factorization method, ACM Transactions on Mathematical Software (TOMS), v.30 n.1, p.19-46, March 2004 Dror Irony , Gil Shklarski , Sivan Toledo, Parallel and fully recursive multifrontal sparse Cholesky, Future Generation Computer Systems, v.20 n.3, p.425-440, April 2004 Abdou Guermouche , Jean-Yves L'excellent, Constructing memory-minimizing schedules for multifrontal methods, ACM Transactions on Mathematical Software (TOMS), v.32 n.1, p.17-32, March 2006 Olaf Schenk , Klaus Grtner, Two-level dynamic scheduling in PARDISO: improved scalability on shared memory multiprocessing systems, Parallel Computing, v.28 n.2, p.187-197, February 2002 Patrick R. Amestoy , Abdou Guermouche , Jean-Yves L'Excellent , Stphane Pralet, Hybrid scheduling for the parallel solution of linear systems, Parallel Computing, v.32 n.2, p.136-156, February 2006 Patrick R. Amestoy , Iain S. Duff , Jean-Yves L'excellent , Xiaoye S. Li, Analysis and comparison of two general sparse solvers for distributed memory computers, ACM Transactions on Mathematical Software (TOMS), v.27 n.4, p.388-421, December 2001 Xiaoye S. Li , James W. Demmel, SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems, ACM Transactions on Mathematical Software (TOMS), v.29 n.2, p.110-140, June Olaf Schenk , Klaus Grtner, Solving unsymmetric sparse systems of linear equations with PARDISO, Future Generation Computer Systems, v.20 n.3, p.475-487, April 2004 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002 Patrick R. Amestoy , Iain S. Duff , Stphane Pralet , Christof Vmel, Adapting a parallel sparse direct solver to architectures with clusters of SMPs, Parallel Computing, v.29 n.11-12, p.1645-1668, November/December Anshul Gupta, Recent advances in direct methods for solving unsymmetric sparse systems of linear equations, ACM Transactions on Mathematical Software (TOMS), v.28 n.3, p.301-324, September 2002 Timothy A. Davis, A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Transactions on Mathematical Software (TOMS), v.30 n.2, p.165-195, June 2004 Nicholas I. M. Gould , Jennifer A. Scott , Yifan Hu, A numerical evaluation of sparse direct solvers for the solution of large sparse symmetric linear systems of equations, ACM Transactions on Mathematical Software (TOMS), v.33 n.2, p.10-es, June 2007 A. N. F. Klimowicz , M. D. Mihajlovi , M. Heil, Deployment of parallel direct sparse linear solvers within a parallel finite element code, Proceedings of the 24th IASTED international conference on Parallel and distributed computing and networks, p.310-315, February 14-16, 2006, Innsbruck, Austria A. Bendali , Y. Boubendir , M. Fares, A FETI-like domain decomposition method for coupling finite elements and boundary elements in large-size problems of acoustic scattering, Computers and Structures, v.85 n.9, p.526-535, May, 2007

gaussian elimination;dynamic scheduling;multifrontal methods;asynchronous parallelism;sparse linear equations;distributed memory computation

587829

On Weighted Linear Least-Squares Problems Related to Interior Methods for Convex Quadratic Programming.

It is known that the norm of the solution to a weighted linear least-squares problem is uniformly bounded for the set of diagonally dominant symmetric positive definite weight matrices. This result is extended to weight matrices that are nonnegative linear combinations of symmetric positive semidefinite matrices. Further, results are given concerning the strong connection between the boundedness of weighted projection onto a subspace and the projection onto its complementary subspace using the inverse weight matrix. In particular, explicit bounds are given for the Euclidean norm of the projections. These results are applied to the Newton equations arising in a primal-dual interior method for convex quadratic programming and boundedness is shown for the corresponding projection operator.

Introduction . In this paper we study certain properties of the weighted linear least-squares problem where A is an m n matrix of full row rank and W is a positive denite symmetric whose matrix square root is denoted by W 1=2 . (See, e.g., Golub and Loan [14, p. 149] for a discussion on matrix square roots.) Linear least-squares problems are fundamental within linear algebra, see, e.g., Lawson and Hanson [20], Golub and Van Loan [14, Chapter 5] and Gill et al. [12, Chapter 6]. An individual problem of the form (1.1) can be converted to an unweighted problem by substituting e g. However, our interest is in sequences of weighted problems, where the weight matrix W changes and A is constant. The present paper is a continuation of the paper by Forsgren [10], in which W is assumed to be diagonally dominant. Our concern is when the weight matrix is of the form where H is a constant positive semidenite symmetric matrix and D is an arbitrary positive denite diagonal matrix. Such matrices arise in interior methods for convex quadratic programming. See Section 1.1 below for a brief motivation. The solution of (1.1) is given by the normal equations To appear in SIAM Journal on Matrix Analysis and Applications. y Optimization and Systems Theory, Department of Mathematics, Royal Institute of Technology, 44 Stockholm, Sweden (anders.forsgren@math.kth.se). Research supported by the Swedish Natural Science Research Council (NFR). z Optimization and Systems Theory, Department of Mathematics, Royal Institute of Technology, 44 Stockholm, Sweden (goran.sporre@math.kth.se). Research supported by the Swedish Natural Science Research Council (NFR). A. FORSGREN AND G. SPORRE or alternatively as the solution to the augmented system (or KKT system) r In some situations, we will prefer the KKT form (1.4), since we are interested in the case when M is a positive semidenite symmetric and singular matrix. In this situation, W 1 and (1.3) are not dened, but (1.4) is well dened. This would for example be the case in an equality-constrained weighted linear least-squares problem, see, e.g., Lawson and Hanson [20, Chapter 22]. For convenience, we will mainly use the form (1.3). mathematically, (1.3) and (1.4) are equivalent. From a computational point of view, this need not be the case. There is a large number of papers giving reasons for solving systems of one type or the other, starting with Bartels et al. [1], followed by, e.g., Du et al. [9], Bjorck [4], Gulliksson and Wedin [17], Wright [29, 31], Bjorck and Paige [5], Vavasis [26], Forsgren et al. [11], and Gill et al. [13]. The focus of the present paper is linear algebra, and we will not discuss these important computational aspects. If A has full row rank and if W+ is dened as the set of n n positive denite symmetric matrices, then for any W 2 W+ , the unique solution of (1.1) is given by In a number of applications, it is of interest to know if the solution remains in a compact set as the weight matrix changes, i.e., the question is whether sup remains bounded for a particular subset W of W+ . It should be noted that boundedness does not hold for an arbitrary subset W of W+ . Take for example let for > 0. Then W This implies that k(AWA T ) 1 AWk is not bounded when W is allowed to vary in W+ . See Stewart [24] for another example of unboundedness and related discussion. For the case where W is the set of positive denite diagonal matrices, Dikin [8] gives an explicit formula for the optimal in (1.1) as a convex combination of the basic solutions formed by satisfying m linearly independent equations. From this result, the boundedness is obvious. If A does not have full row rank, it is still possible to show boundedness, see Ben-Israel [2, p. 108]. Later, Wei [28] has also studied boundedness in absence of a full row rank assumption on A, and has furthermore given some stability results. Bobrovnikova and Vavasis [6] have given boundedness results for complex diagonal weight matrices. The geometry of the set (AWA T varies over the set of positive denite diagonal matrices has been studied by Hanke LEAST-SQUARES PROBLEMS RELATED TO QUADRATIC PROGRAMMING 3 and Neumann [18]. Based on the formula derived by Dikin [8], Forsgren [10] has given boundedness results when W is the set of positive denite diagonally dominant matrices. We show boundedness for the set of weight matrices that are arbitrary nonnegative combinations of a set of xed positive semidenite symmetric matrices, and the set of inverses of such matrices. As a special case, we then obtain the set of weight matrices of the form (1.2), which was our original interest. The boundedness is shown in the following way. In Section 2, we review results for the characterization of as W varies over the set of symmetric matrices such that AWA T is nonsingular. Section 3 establishes the boundedness when W is allowed to vary over a set of matrices that are nonnegative linear combinations of a number of xed positive semidenite matrices such that AWA T is positive denite. In Section 4, results that are needed to handle the projection using the inverse weight matrix are given. In Section 5, we combine results from the previous two sections to show boundedness for the that solves (1.4) when M is allowed to vary over the nonnegative linear combinations of a set of xed positive semidenite symmetric matrices. The research was initiated by a paper by Gonzaga and Lara [15]. The link to that paper has subsequently been superseded, but we include a discussion relating our results to the result of Gonzaga and Lara in Appendix A. 1.1. Motivation. Our interest in weighted linear least-squares problems is from interior methods for optimization, and in particular for convex quadratic program- ming. There is a vast number of papers on interior methods, and here is only given a brief motivation for the weighted linear least-squares problems that arise. Any convex quadratic programming problem can be transformed to the form minimize subject to x 0; where H is a positive semidenite symmetric n n matrix and A is an m n matrix of full row rank. For x 2 IR n , 2 IR m and s 2 IR n such that x > 0 and s > 0, an iteration of a primal-dual path-following interior method for solving (1.6) typically takes a Newton step towards the solution of the equations (1.7a) (1.7c) where is a positive barrier parameter, see, e.g., Monteiro and Adler [21, page 46]. similarly below diag(s). Strict positivity of x and s is implicitly required and typically maintained by limiting the step length. If is set equal to zero in (1.7) and the implicit requirements x > 0 and s > 0 are replaced by x 0 and s 0, the optimality conditions for (1.6) are obtained. Consequently, equations (1.7) and the implicit positivity of x and s may be viewed as a perturbation of the optimality conditions for (1.6). In a primal-dual path-following interior method, the perturbation is driven to zero to make the method converge to an optimal solution. The equations (1.7) are often referred to as the primal-dual equations. Forming the Newton equations associated with (1.7) for the corrections x, , s, and 4 A. FORSGREN AND G. SPORRE eliminating s gives x If x and s are strictly feasible, i.e., x and s are strictly positive and x satises then a comparison of (1.4) and (1.8) shows that the Newton equations (1.8) can be associated with a weighted linear least-squares problem with a positive denite weight matrix (H +X 1 S) 1 . A sequence of strictly feasible iterates fx k g 1 k=0 gives rise to a sequence of weighted linear least-squares problems, where the weight matrix changes but A is constant. In a number of convergence proofs for linear programming, a crucial step is to ensure boundedness of the step (x; s), see, e.g., Vavasis and Ye [27, Lemma 4] and Wright [30, Lemmas 7.2 and A.4]. Since linear programming is the special case of convex quadratic programming where are interested in extending this boundedness result to convex quadratic programming. Therefore, the boundedness of as X 1 S varies over the set of diagonal positive denite matrices is of interest. This boundedness property of (1.9) is shown in Section 5. 1.2. Notation. When we refer to matrix norms, and make no explicit reference to what type of norm is considered, it can be any matrix norm that is induced from a vector norm such that k(x T holds for any vector x. To denote the ith eigenvalue and the ith singular value, we use i and i respectively. For symmetric matrices A and B of equal dimension, A B means that A B is positive semidenite. Similarly, A B means that A B is positive denite. The remainder of this section is given in Forsgren [10]. It is restated here for completeness. For an m n matrix A of full row rank, we shall denote by J (A) the collection of sets of column indices associated with the nonsingular mm submatrices of A. For J 2 J (A), we denote by A J the mm nonsingular submatrix formed by the columns of A with indices in J . Associated with J 2 J (A), for a diagonal n n matrix D, we denote by D J the mm diagonal matrix formed by the elements of D that have row and column indices in J . Similarly, for a vector g of dimension n, we denote by g J the vector of dimension m with the components of g that have indices in J . The slightly dierent meanings of A J , D J and g J are used in order not to make the notation more complicated than necessary. For an example clarifying the concepts, see Forsgren [10, p. 766]. The analogous notation is used for an m n matrix A of full row rank and an n r matrix U of full row rank in that we associate J (AU) with the collection of sets of column indices corresponding to nonsingular mm submatrices of AU . Associated with J 2 J (AU), for a diagonal r r matrix D, we denote by D J the mm diagonal matrix formed by the elements of D that have row and column indices in J . Similarly, for a vector g of dimension r, we denote by g J the vector of dimension m with the components of g that have indices in J . Since column indices of AU are also column indices of U , for J 2 J (AU ), we denote by U J the n m submatrix of full column rank formed by the columns of U with indices in J . Note that each element of J (A) as well as each element of J (AU) is a collection of m indices. LEAST-SQUARES PROBLEMS RELATED TO QUADRATIC PROGRAMMING 5 2. Background. In this section, we review some fundamental results. The following theorem, which states that the solution of diagonally weighted linear least-squares problem can be expressed as a certain convex combination, is the basis for our results. As far as we know, it was originally given by Dikin [8] who used it in the convergence analysis of the interior point method for linear programming he proposed [7]. The proof of the theorem is based on the Cauchy-Binet formula and Cramer's rule. Theorem 2.1 (Dikin [8]). Let A be an m n matrix of full row rank, let g be a vector of dimension n, and let D be a positive denite diagonal n n matrix. Then, A T is the collection of sets of column indices associated with nonsingular mm submatrices of A. Proof. See, e.g., Ben-Tal and Teboulle [3, Corollary 2.1]. Theorem 2.1 implies that if the weight matrix is diagonal and positive denite, then the solution to the weighted least-squares problem (1.1) lies in the convex hull of the basic solutions formed by satisfying m linearly independent equations. Hence, this theorem provides an expression on the supremum of k(ADA T diagonal and positive denite, as the following corollary shows. Corollary 2.2. Let A be an m n matrix of full row rank, and let D+ denote the set of positive denite diagonal n n matrices. Then, sup is the collection of sets of column indices associated with nonsingular mm submatrices of A. Proof. See, e.g., Forsgren [10, Corollary 2.2]. The boundedness has been discussed by a number of authors over the years, see, e.g., Ben-Tal and Teboulle [3], O'Leary [22], Stewart [24], and Todd [25]. Theorem 2.1 can be generalized to the case where the weight matrix is an arbitrary symmetric, not necessarily diagonal, matrix such that AWA T is nonsingular. The details are given in the following theorem. Theorem 2.3 (Forsgren [10]). Let A be an m n matrix of full row rank and let W be a symmetric n n matrix such that AWA T is nonsingular. Suppose UDU T , where D is diagonal. Then, where J (AU) is the collection of sets of column indices associated with nonsingular mm submatrices of AU . Proof. See Forsgren [10, Theorem 3.1]. 3. Nonnegative combinations of positive semidenite matrices. Let A be an mn matrix of full row rank and assume that we are given an nn symmetric weight matrix W (), which depends on a vector 2 IR t for some t. If W () can be decomposed as W does not depend on and D() is diagonal, Theorem 2.3 can be applied, provided AW ()A T is nonsingular, and the 6 A. FORSGREN AND G. SPORRE matrices (AU J J involved do not depend on . If, in addition D() 0, then the linear combination of Theorem 2.3 is a convex combination. Consequently, the norm remains bounded as long as the supremum is taken over a set of values of for which In particular, we are interested in the case where a set of positive semidenite and symmetric matrices, W i , t, are given and W () is dened as W . The following two lemmas and associated corollary concern the decomposition of W (). The rst lemma concerns the set of all possible decompositions of a positive semidenite matrix W as the relation between dierent decompositions of this type. Lemma 3.1. Let W be a symmetric positive semidenite n n matrix of rank r, and let U is nonempty and compact. Further, if U and e U belong to U , then there is an r r orthogonal matrix Q such that UQ. Proof. It is possible to decompose W as is an n r matrix of full column rank, for example using a Cholesky factorization with symmetric inter- changes, see, e.g., Golub and Van Loan [14, Section 4.2.9]. Therefore, U is nonempty. If U and e U T both belong to U , then U U U Hence, U T and e U T have the same null space, which implies that the range spaces of U and e U are the same. Therefore, there is a nonsingular r r matrix M such that UM , from which it follows that e U T . Premultiplying this equation by e U T and postmultiplying it by e U gives e U is nonsingular, (3.1) gives MM I . Compactness is established by proving boundedness and closedness. Boundedness holds because kU T e is the ith unit vector. Let fU (i) g 1 be a sequence converging to U , such that U (i) 2 U for all i. From the continuity of matrix multiplication, U belongs to U , and the closedness of U follows. A consequence of this lemma is that we can decompose each W i , t, as stated in the following corollary. Corollary 3.2. For t, let W i be an nn symmetric positive semidefinite matrix of rank r i . Let is a well-dened compact subset of IR nr . Furthermore, if U and e U belong to U , then, t, there are orthogonal r i r i matrices Q i , such that U Proof. The result follows by applying Lemma 3.1 to each W i . It should be noted that U depends on the matrices W i . This dependence will be suppressed in order not to make the notation more complicated than necessary. From Corollary 3.2, we get a decomposition result for matrices that are nonnegative linear combinations of symmetric positive semidenite matrices, as is stated in the following lemma. It shows that if we are given a set of positive semidenite and symmetric matrices, t, and W () is dened as W then we can decompose W () into the form W does not depend on and D() is diagonal. Lemma 3.3. For 2 IR t , let W are symmetric positive semidenite n n matrices. Further, let U be associated with LEAST-SQUARES PROBLEMS RELATED TO QUADRATIC PROGRAMMING 7 t, according to Corollary 3.2, and for each i, let r i denote rank(W i ) and let I i be an identity matrix of dimension r i . Then W () may be decomposed as where U is any matrix in U and Proof. Corollary 3.2 shows that we may write where U is an arbitrary matrix in U and Note that D() is positive semidenite if 0. An application of Theorem 2.3 to the decomposition of Lemma 3.3 now gives the boundedness result for nonnegative combinations of positive semidenite matrices, as stated in the following proposition. Proposition 3.4. Let A be an mn matrix of full row rank. For 2 IR t , 0, let W are symmetric positive semidenite nn matrices. If W () is decomposed as W according to Lemma 3.3, then for 0 and AW ()A T 0, Furthermore, sup 0: U2U where J (AU) is the collection of sets of column indices associated with nonsingular submatrices of AU , and U is associated with t, according to Corollary 3.2. Proof. If AW ()A T 0, Theorem 2.3 immediately gives Since 0, it follows that D() 0. Consequently, det(D J ()) 0 for all J 2 J (AU ). Thus, the above expression gives sup Since this result holds for all U 2 U , it holds when taking the inmum over U 2 U . To show that the inmum is attained, let for every J that is a subset of ng such that jJ m. For a xed J , f J is continuous at every e U such that det(A e Further, at e U such that A e U J is 8 A. FORSGREN AND G. SPORRE singular, f J is a lower semi-continuous function, see, e.g., Royden [23, p. 51]. Hence, f J is lower semi-continuous everywhere. Due to the construction of f J (U ), J:jJj=m The maximum of a nite collection of lower semi-continuous functions is lower semi- continuous, see, e.g., Royden [23, p. 51], and the set U is compact by Corollary 3.2. Therefore, the inmum is attained, see, e.g., Royden [23, p. 195], and the proof is complete. Note that Proposition 3.4 as special cases includes two known cases: (i) the diagonal matrices, where W and (ii) the diagonally dominant matrices, where In both these cases, the supremum bound of (3.2) is sharp. This is because all the matrices whose nonnegative linear combinations form the weight matrices are of rank one. In that case, the minimum over U in (3.2) is not necessary since it follows from Corollary 3.2 that the columns of U are unique up to multiplication by 1. Hence, D() may be adjusted so as to give weight one to the submatrix AU J for which the maximum of the right hand side of (3.2) is achieved, and negligible weight to the other submatrices. In general, when not all matrices whose nonnegative linear combinations form the weight matrix have rank one, it is an open question if the supremum bound is sharp. 4. Inversion of the weight matrix. For a constant positive semidenite matrix H , our goal is to obtain a bound on k(A(H D is an arbitrary positive denite diagonal matrix. One major obstacle in applying Theorem 2.3 is the inverse in the weight matrix (H +D) 1 . The following proposition and its subsequent corollary and lemma provide a solution to this problem. Proposition 4.1. Suppose that an n n orthogonal matrix Q is partitioned as is an n s matrix, and 2s n. Further, let W be a symmetric nonsingular n n matrix such that Z T W 1 Z and Y T WY are nonsingular. Then and s: Proof. The orthogonality of Q ensures that Y T I . This gives and hence LEAST-SQUARES PROBLEMS RELATED TO QUADRATIC PROGRAMMING 9 proving the rst part of the proposition. are nonsingular, we may write (Z I (Z T W 1 (4.2a) (4.2b) The orthogonality of Q ensures that s: We also have I (Z T W 1 (Z combination of (4.2a), (4.3) and (4.4) gives s: An analogous argument applied to (4.2b), taking into account that 2s n gives (4.6a) s: (4.6b) The second part of the proposition follows by a combination of (4.1), (4.5) and (4.6). In particular, Proposition 4.1 gives the equivalence between the Euclidean norms of a projection and the projection onto the complementary space using the inverse weight matrix, given that the matrices used to represent the spaces are orthogonal. This is shown in the following corollary. Corollary 4.2. Suppose that an n n orthogonal matrix Q is partitioned as is an nm matrix. Further, let W be a symmetric nonsingular matrix such that Z T W 1 Z and Y T WY are nonsingular. Then Further, let W+ denote the set of n n positive denite symmetric matrices, and let W W+ . Then, sup k(Z Proof. If m n=2, the rst statement follows by letting Proposition 4.1. The second statement is a direct consequence of the rst one. If m < n=2, we may similarly apply Proposition 4.1 after interchanging the roles of Y and Z, and W and As noted above, Corollary 4.2 states the equality between the Euclidean norms of two projections, given that the matrices describing the spaces onto which we project are orthogonal. The following lemma relates the Euclidean norms of the projections when the matrices are not orthogonal. A. FORSGREN AND G. SPORRE Lemma 4.3. Let A be an m n matrix of full row rank, and let N be a matrix whose columns form a basis for the null space of A. Further, let W be a symmetric nonsingular n n matrix such that N T W 1 N and A T WA are nonsingular. Then k(N Proof. Let be an orthogonal matrix such that the columns of Z form a basis for the null space of A. Then, there are nonsingular matrices RZ and R Y such that a matrix norm which is induced from a vector norm is submultiplicative, see, e.g., Horn and Johnson [19, Thm. 5.6.2], this giveskRZ k k(N k(Z Z k; Y k: (4.7b) If the Euclidean norm is used, the bounds in (4.7) can be expressed in terms of singular values of A and N since Y and Z are orthogonal matrices, i.e. (4.8a) (4.8b) A combination of Corollary 4.2, (4.7), and (4.8) gives the stated result. If the weight matrix is allowed to vary over some subset of the positive denite symmetric matrices, it follows from Lemma 4.3 that the norm of the projection onto a subspace is bounded if and only if the norm of the projection onto the orthogonal complement is bounded when using inverses of the weight matrices. This is made precise in the following corollary. Corollary 4.4. Let W+ denote the set of n n positive denite symmetric matrices, and let W W+ . Let A be an m n matrix of full row rank and let N be a matrix whose columns form a basis for the null space of A. Then sup only if sup k(N In particular, sup k(N sup sup k(N Proof. The second statement follows by multiplying the inequalities in Lemma 4.3 by k(N and then taking the supremum of the three expressions. The rst statement of the corollary then follows from the equivalence of matrix norms that are induced from vector norms, see, e.g., Horn and Johnson [19, Thm. 5.6.18]. 5. Inversion and nonnegative combination. Let A be an m n matrix of full row rank, and let Z be a matrix whose columns form an orthonormal basis for the null space of A. Further, let are LEAST-SQUARES PROBLEMS RELATED TO QUADRATIC PROGRAMMING 11 given symmetric positive semidenite n n matrices. In Section 3 the weight matrix was assumed to be the nonnegative combination of symmetric positive semidenite matrices. This section concerns weight matrices that are the inverse of such combi- nations, i.e., where the weight matrix is the inverse of M(). Further, if the problem is originally posed as the KKT-system, cf. (1.4), it makes sense to study the problem under the assumption that Z T M()Z 0, since in our situation, Z T M()Z 0 if and only if the matrix of (5.1) is nonsingular, see Gould [16, Lemma 3.4]. Note that Z T M()Z 0 is a weaker assumption than M() 0, which is necessary if the least-squares formulation is to be valid. A combination of Proposition 3.4 and Lemma 4.3 shows that () remains bounded under the abovementioned assumptions. This is stated in the following theorem, which is the main result of this paper. Theorem 5.1. Let A be an m n matrix of full row rank and let g be an n- vector. Further, let Z be a matrix whose columns form an orthonormal basis for the null space of A. For 2 IR t , 0, let are symmetric positive semidenite nn matrices. Further, let r() and () satisfy Then, sup 0: In particular, if Z T M()Z 0, then Finally, if M() is decomposed according to Lemma 3.3, then sup 0: where J (Z T U) is the collection of sets of column indices associated with nonsingular submatrices of Z T U , and U is associated with M i , t, according to Corollary 3.2. Proof. For I 0. Therefore, is well-dened. By Lemma 4.3 it follows that (5. A. FORSGREN AND G. SPORRE For such that Z T M()Z 0, the matrix in the system of equations dening () and r() is nonsingular, see Gould [16, Lemma 3.4]. Then, the implicit function theorem implies that lim !0 Therefore, letting (5.3). Taking the supremum over such that 0 and Z T M()Z 0, and using Proposition 3.4 gives (5.4), from which (5.2) follows upon observing that all norms on a real nite-dimensional vector space are equivalent, see, e.g., Horn and Johnson [19, As a consequence of Theorem 5.1, we are now able to prove the boundedness of the projection operator for the application of primal-dual interior methods to convex quadratic programming described in Section 1.1. Corollary 5.2. Let H be a positive semidenite symmetric nn matrix, let A be an m n matrix of full row rank, and let D+ denote the space of positive denite diagonal n n matrices. Then, sup Proof. If M() 0, then () of Theorem 5.1 satises Theorem 5.1 implies that () is bounded. This holds for any vector g, and hence sup 0:M()0 The stated result follows by applying (5.6) with and letting For convenience in notation, it has been assumed that all variables of the convex quadratic program are subject to bounds. It can be observed that the analogous results hold when some variables are not subject to bounds. In this situation, M of may be partitioned as where H is symmetric and positive semidenite and D 11 is diagonal and positive denite. Let A be partitioned conformally with M as . Then, (1.4) has a unique solution as long as there is no nonzero p 2 such that A 2 Gould [16, Lemma 3.4]. Hence, under this additional assumption, Theorem 5.1 can be applied to bound k()k as D 11 varies over the set of positive denite diagonal matrices. 6. Summary . It has been shown that results concerning the boundedness of for A of full row rank and W diagonal, or diagonally dominant, and symmetric positive denite can be extended to a more general case where W is a nonnegative linear combination of a set of symmetric positive semidenite matrices such that AWA T 0. Further, boundedness has been shown for the projection onto the null space of A using as weight matrix the inverse of a nonnegative linear combination of a number of symmetric positive semidenite matrices. This result LEAST-SQUARES PROBLEMS RELATED TO QUADRATIC PROGRAMMING 13 has been used to show boundedness of a projection operator arising in a primal-dual interior method for convex quadratic programming. The main tools for deriving these results have been the explicit formula for the solution of a weighted linear least-squares problem given by Dikin [8], and the relation between a projection onto a subspace with a certain weight matrix and the projection onto the orthogonal complement using the inverse weight matrix. An interesting question that is left open is whether the explicit bounds that are given are sharp or not. In the case where all the matrices whose nonnegative linear combination form the weight matrix are of rank one, the bounds are sharp. In the general case, this is an open question. On a higher level, an interesting question is whether the results of this paper can be utilized to give new complexity bounds for quadratic programming, analogous to the case of linear programming, see, e.g., Vavasis and Ye [27, Section 9]. Appendix A. Relationship to partitioned orthogonal matrices. In this appendix we review a result by Gonzaga and Lara [15] concerning diagonally weighted projections onto orthogonally complementary subspaces, and combine this result with a result concerning singular values of submatrices of orthogonal matrices. It was in fact these results which lead to the more general results relating weighted projection onto a subspace and the projection onto its complementary subspace using the inverse weight matrix, as described in Section 4. Gonzaga and Lara [15] state that if Y is an n m orthogonal matrix and Z is a matrix whose columns form an orthonormal basis for the null space of Y T , then sup where D+ is the set of positive denite diagonal nn matrices. They use a geometric approach to prove this result. We note that Corollary 4.2, specialized to the case of diagonal positive denite weight matrices, allows us to state the same result. Fur- thermore, we obtain an explicit expression for the supremum by Corollary 2.2. The following corollary summarizes this result. Corollary A.1. Suppose that an n n orthogonal matrix Q is partitioned as is an n m matrix. Let D+ denote the set of diagonal positive denite n n matrices. Then, sup ~ where J (Z T ) is the collection of sets of column indices associated with nonsingular (n m) (n m) submatrices of Z T and J (Y T ) is the collection of sets of column indices associated with nonsingular mm submatrices of Y T . Proof. Since D 2 D+ if and only if D 1 2 D+ , Corollary 4.2 shows that sup The explicit expressions for the two suprema follow from Corollary 2.2. Hence, in our setting, we would rather state the result of Gonzaga and Lara [15] in the equivalent form sup 14 A. FORSGREN AND G. SPORRE with the expressions for the suprema stated in Corollary A.1. Note that an implication of Corollary A.1 is that if an nn orthogonal matrix Q is partitioned as , where Y has m columns, there is a certain relationship between the smallest singular value of all nonsingular (n m)(n m) submatrices of Z and the smallest singular value of all nonsingular mm submatrices of Y . This is in fact a consequence of a more general result, namely that if Q is partitioned further as where Z 1 is (n m) (n m), then all singular values of Z 1 and Y 2 that are less than one are identical. This in turn is a consequence of properties of singular values of submatrices of orthogonal matrices that can be obtained by the CS-decomposition of an orthogonal matrix, see, e.g., Golub and Van Loan [14, Section 2.6.4]. This result relating the singular values of Z 1 and Y 2 of (A.1) implies the existence of J and ~ J , that are complementary subsets of which the maxima in Corollary A.1 are achieved. This observation lead us to the result that for any positive denite diagonal D. Subsequently, this result was superseded by the more general analysis presented in Section 4. Acknowledgement . We thank the two anonymous referees for their constructive and insightful comments, which signicantly improved the presentation. --R Numerical techniques in mathematical programming A Volume A geometric property of the least squares solution of linear equations A norm bound for projections with complex weights Iterative solution of problems of linear and quadratic programming The factorization of sparse symmetric inde On linear least-squares problems with diagonally dominant weight matrices Stability of symmetric ill-conditioned systems arising in interior methods for constrained optimization Numerical Linear Algebra and Optimization On the stability of the Cholesky factorization for symmetric quasi-de nite systems Matrix Computations A note on properties of condition numbers On practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem The geometry of the set of scaled projections Matrix Analysis Interior path-following primal-dual algorithms On bounds for scaled projections and pseudoinverses Real Analysis On scaled projections and pseudoinverses A Dantzig-Wolfe-like variant of Karmarkar's interior-point linear programming algorithm Stable numerical algorithms for equilibrium systems A primal-dual interior point method whose running time depends only on the constraint matrix Upper bound and stability of scaled pseudoinverses Stability of linear equations solvers in interior-point methods --TR

quadra- tic programming;weighted least-squares problem;interior method;unconstrained linear least-squares problem

587833

Stability of Structured Hamiltonian Eigensolvers.

Various applications give rise to eigenvalue problems for which the matrices are Hamiltonian or skew-Hamiltonian and also symmetric or skew-symmetric. We define structured backward errors that are useful for testing the stability of numerical methods for the solution of these four classes of structured eigenproblems. We introduce the symplectic quasi-QR factorization and show that for three of the classes it enables the structured backward error to be efficiently computed. We also give a detailed rounding error analysis of some recently developed Jacobi-like algorithms of Fassbender, Mackey, and Mackey [Linear Algebra Appl., to appear] for these eigenproblems. Based on the direct solution of 4 4, and in one case 8 8, structured subproblems these algorithms produce a complete basis of symplectic orthogonal eigenvectors for the two symmetric cases and a symplectic orthogonal basis for all the real invariant subspaces for the two skew-symmetric cases. We prove that, when the rotations are implemented using suitable formulae, the algorithms are strongly backward stable and we show that the QR algorithm does not have this desirable property.

Introduction . This work concerns real structured Hamiltonian and skew- Hamiltonian eigenvalue problems where the matrices are either symmetric or skew- symmetric. We are interested in algorithms that are strongly backward stable for these problems. In general, a numerical algorithm is called backward stable if the computed solution is the true solution for slightly perturbed initial data. If, in addition, this perturbed initial problem has the same structure as the given problem, then the algorithm is said to be strongly backward stable. There are three reasons for our interest in strongly backward stable algorithms. First, such algorithms preserve the algebraic structure of the problem and hence force the eigenvalues to lie in a certain region of the complex plane or to occur in particular kinds of pairings. Because of rounding errors, algorithms that do not respect the structure of the problem can cause eigenvalues to leave the required region [26]. Second, by taking advantage of the structure, storage and computation can be lowered. Finally, structure-preserving algorithms may compute eigenpairs that are more accurate than the ones provided by a general algorithm. Structured Hamiltonian eigenvalue problems appear in many scientific and engineering applications. For instance, symmetric skew-Hamiltonian eigenproblems arise in quantum mechanical problems with time reversal symmetry [9], [23]. In response theory, the study of closed shell Hartree-Fock wave functions yields a linear response eigenvalue equation with a symmetric Hamiltonian [21]. Also, total least squares problems with symmetric constraints lead to the solution of a symmetric Hamiltonian problem [17]. # Received by the editors February 23, 2000; accepted for publication (in revised form) by V. Mehrmann November 24, 2000; published electronically May 3, 2001. http://www.siam.org/journals/simax/23-1/36800.html Department of Mathematics, University of Manchester, Manchester, M13 9PL, England (ftisseur@ma.man.ac.uk, http://www.ma.man.ac.uk/-ftisseur). This work was supported by Engineering and Physical Sciences Research Council grant GR/L76532. The motivation for this work comes from recently developed Jacobi algorithms for structured Hamiltonian eigenproblems [10]. These algorithms are structure-preserving, inherently parallelizable, and hence attractive for solving large-scale eigenvalue prob- lems. Our first contribution is to define and show how to compute structured backward errors for structured Hamiltonian eigenproblems. These backward errors are useful for testing the stability of numerical algorithms. Our second contribution concerns the stability of these new Jacobi-like algorithms. We give a unified description of the algorithms for the four classes of structured Hamiltonian eigenproblems. This provides a framework for a detailed rounding error analysis and enables us to show that the algorithms are strongly backward stable when the rotations are implemented using suitable formulae. The organization of the paper is as follows. In section 2 we recap the necessary background concerning structured Hamiltonians. In section 3 we derive computable structured backward errors for structured Hamiltonian eigenproblems. In section 4, we describe the structure-preserving QR-like algorithms proposed in [5] for structured Hamiltonian eigenproblems. We give a unified description of the new Jacobi-like algorithms and detail the Jacobi-like update for each of the four classes of structured Hamiltonian. In section 5 we give the rounding error analysis and in section 6 we use our computable backward errors to confirm empirically the strong stability of the algorithms. 2. Preliminaries. A matrix P # R 2n-2n is symplectic if P T -In In I n is the n - n identity matrix. A matrix H # R 2n-2n is Hamiltonian if Hamiltonian matrices have the form where E, F, G # R n-n and F G. We denote the set of real Hamiltonian matrices by H 2n . A matrix S # R 2n-2n is skew-Hamiltonian if Skew-Hamiltonian matrices have the form where E, F, G # R n-n and F are skew-symmetric. We denote the set of real skew-Hamiltonian matrices by SH 2n . Note that if H # H 2n , then P SH 2n , where P is an arbitrary symplectic matrix. Thus symplectic similarities preserve Hamiltonian and skew-Hamiltonian structure. Also, symmetric and skew-symmetric structures are preserved by orthogonal similarity transformations. Therefore structure-preserving algorithms for symmetric or skew-symmetric Hamiltonian or skew-Hamiltonian eigenproblems have to use real symplectic orthogonal trans- formations, that is, matrices U # R 2n-2n satisfying U T As in [10], we denote by SpO(2n) the group of real symplectic orthogonal matrices. Any U # SpO(2n) can be written as I and In Tables 2.1 and 2.2, we summarize the structure of Hamiltonian and skew- Hamiltonian matrices that are either symmetric or skew-symmetric, their eigenvalue STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 105 Table Properties of structured Hamiltonian matrices H # H 2n . Symmetric real, Skew-symmetric pure imaginary, pairs #, - Table Properties of structured skew-Hamiltonian matrices S # SH 2n . Symmetric real, double # D 0 Skew-symmetric pure imaginary, double, properties, and their symplectic orthogonal canonical form. We use D # R n-n to denote a diagonal matrix and B # R n-n to denote a block-diagonal matrix that is the direct sum of 1 - 1 zero blocks and 2 - 2 blocks of the form [ 0 -b 0 ]. These canonical forms are consequences of results in [19]. Next, we show that the eigenvectors of skew-symmetric Hamiltonian matrices can be chosen to have structure. This property is important when defining and deriving structured backward errors. Lemma 2.1. The eigenvectors of a skew-symmetric Hamiltonian matrix H can be chosen to have the form [ z -iz ] with z # C n . Proof. Let HU be the canonical form of H with symplectic orthogonal. The matrix -iI I iI ] is unitary and diagonalizes the canonical form of H: # . Hence is an eigenvector basis for H and this shows that the eigenvectors can be taken to have the form [ z -iz ] with z # C n . Note that an eigenvector of a skew-symmetric Hamiltonian matrix does not necessarily have the form [ z -iz ]. For instance, consider 106 FRANC-OISE TISSEUR Table t: Number of parameters defining H. Hamiltonian Skew-Hamiltonian is an eigenvector of H, corresponding to the eigenvalue -id, that is not of the form [ z 3. Structured backward error. We begin by developing structured backward errors that can be used to test the strong stability of algorithms for our classes of Hamiltonian eigenproblems. 3.1. Definition. For notational convenience, the symbol H denotes from now on both Hamiltonian and skew-Hamiltonian matrices. Let (#x, #) be an approximate eigenpair for the structured Hamiltonian eigenvalue problem R 2n-2n . A natural definition of the normwise backward error of an approximate eigenpair is where we measure the perturbation in a relative sense and # denotes any vector norm and the corresponding subordinate matrix norm. Deif [8] derived the explicit expression for the 2-norm #x -H#x is the residual. This shows that the normwise relative backward error is a scaled residual. The componentwise backward error is a more stringent measure of the backward error in which the components of the perturbation #H are measured individually: #x, |#H| #|H| # . Here inequalities between matrices hold componentwise. Geurts [12] showed that 1#i#2n The componentwise backward error provides a more meaningful measure of the stability than the normwise version when the elements in H vary widely in magnitude. However, this measure is not entirely appropriate for our problems as it does not respect any structure (other than sparsity) in H. Bunch [2] and Van Dooren [25] have also discussed other situations when it is desirable to preserve structure in definitions of backward errors. The four classes of structured Hamiltonian matrices we are dealing with are defined by t # real parameters that make up E and F (see Table 3.1). We write this dependence as extend the notion of componentwise backward error to allow dependence of the perturbations on STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 107 a set of parameters and they define structured componentwise backward errors. Following their idea and notation we define the structured relative normwise backward error by -(#x, implies that #H has the same structure as H. The structured relative componentwise backward error #x, #) is defined as in (3.1) but with the constraint #H#F #H#F replaced by |#H| #|H|. In our case, the dependence of the data on the t parameters is linear. We naturally require (#x, #) to have any properties forced upon the exact eigenpairs, otherwise the backward error will be infinite. In the next subsections, we give algorithms for computing these backward errors. We start by describing a general approach that was used in [13] in the context of structured linear systems and extend it to the case where the approximate solution lies in the complex plane. 3.2. A general approach for the computation of -(# x, #). Let and -+i#. By equating real and imaginary parts, the constraint #x in (3.1) becomes # u or equivalently #H [ # u . Applying the vec operator (which stacks the columns of a matrix into one long vector), we obtain where# denotes the Kronecker product. We refer to Lancaster and Tismenetsky [18, Chap. 12] for properties of the vec operator and the Kronecker product. By linearity we have -t of full rank and where #p is the t-vector of parameters defining #H. There exists a diagonal matrix D 1 depending on the structure of H (symmetric/skew- symmetric Hamiltonian/skew-Hamiltonian) such that # u . Using (3.4) we can rewrite (3.3) as Y BD using (3.5), -(#x, y This shows that the structured backward error is given in terms of the minimal 2-norm solution to an underdetermined system. If the underdetermined system is consistent, then the minimal 2-norm solution is given in terms of the pseudo-inverse by In this case -(#x, When H is a symmetric structured Hamiltonian, we can assume that # and # x are real. Therefore and from (3.2) we have [ Applying the vec operation gives # u I 2n # vec(#-I As -I - H is also a symmetric structured Hamiltonian, we have by linearity that vec(#-I #- is the t-vector of parameters defining #- lies in the range of Y BD Therefore, the underdetermined system in (3.6) is consistent for symmetric Hamiltonians and for symmetric skew- Hamiltonians. For a skew-symmetric Hamiltonian, we can again prove consistency for pure imaginary approximate eigenvalues and approximate eigenvectors of the form in Lemma 2.1. We have not been able to prove that the underdetermined system is consistent for the skew-symmetric skew-Hamiltonian case. As the dependence on the parameters is linear, in the definition of the structured relative componentwise backward error #x, #), we have the equivalence |#H| #|H| #p| #|p|. q. Then the smallest # satisfying |#p| #|p| is #q# . The minimal #-norm solution of Y BD 2 can be approximated by minimizing in the 2-norm. We have #). By looking at each problem individually, it is possible to reduce the size of the underdetermined system. Nevertheless, solution of the system by standard techniques still takes O(n 3 ) operations. In the next section, we show that by using a symplectic quasi-QR factorization of the approximate eigenvector and residual (or some appropriate parts) we can derive expressions for -(#x, #) that are cheaper to compute for all the structured Hamiltonians of interest except for skew-symmetric skew-Hamiltonians. First, we define a symplectic quasi-QR factorization. 3.3. Symplectic quasi-QR factorization. We define the symplectic quasi-QR factorization of an 2n -m matrix A by where Q is real symplectic orthogonal, T 1 # R n-m is upper trapezoidal, and T 2 # R n-m is strictly upper trapezoidal. Such a symplectic quasi-QR factorization has also been discussed by Bunse-Gerstner [3, Cor. 4.5(ii)].Before giving an algorithm to STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 109 compute this symplectic quasi-QR factorization, we need to describe two types of elementary orthogonal symplectic matrices that can be used to zero selected components of a vector. A symplectic Householder matrix H # R 2n-2n is a direct sum of n-n Householder matrices: where diag # I k-1 , I n-k+1 -v T v I n otherwise, and v is determined such that for a given x # R n , P (k, A symplectic Givens rotation G(k, # R 2n-2n is a Givens rotation where the rotation is performed in the plane (k, k G(k, #) has the form where # is chosen such that for a given x # R 2n , G(k, We use a combination of these orthogonal transformations to compute our symplectic quasi-QR factorization: symplectic Householder matrices are used to zero large portions of a vector and symplectic Givens are used to zero single entries. Algorithm 3.1 (symplectic quasi-QR factorization). Given a matrix with A 1 , A 2 # R n-m , this algorithm computes the symplectic quasi-QR factorization (3.8). For End Determine G End We illustrate the procedure for a generic 6 - 4 matrix: -# -# G3 -# 3.4. Symmetric Hamiltonian eigenproblems. Let be the residual vector and be the symplectic quasi-QR factorization (3.8) with Q symplectic orthogonal and . 0 e n+1,2 We have Q which is equivalent to e 110 e 22e n+1,2# still a symmetric Hamiltonian matrix. Equation (3.10) defines the first column of # H. As |e 11 e 22 /e 11 , # - h E) T STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 111 and # be such that e 11 e 22 -0 - e 11 where the -'s are arbitrary real coe#cients. Then, any symmetric Hamiltonian of the F F -# #x. The Frobenius norm of #H is minimized by setting the -'s to zero in the definition of # F . We obtain the following lemma. Lemma 3.2. The backward error of an approximate eigenpair of a symmetric Hamiltonian eigenproblem is given by -(#x, |e 11 | is the quasi-triangular factor in the symplectic quasi-QR factorization of [#x r] with #I -H)#x. We also have -(#x, where e 2 is the second unit vector. 3.5. Skew-symmetric Hamiltonian eigenproblems. For skew-symmetric Hamiltonian eigenproblems the technique developed in section 3.4 needs to be modified as in this case r, # x are complex vectors and we want to define a real skew-symmetric Hamiltonian perturbation so that (H #x. In the definition of the structured backward error (3.1), we now assume that # is pure imaginary and that # x has the form [ (see Lemma 2.1). Taking the plus sign in # x, the equation (H #x can be written as E)#z. Multiplying (3.12) by -i gives (3.11). Hence, we carry out the analysis with (3.11) only. Setting in (3.11) and equating real and imaginary parts yields which is equivalent to Using we show that w and s are orthogonal: For the other choice of sign with , the equation #x is equivalent to and we can show that w T We can now carry on the analysis as in section 3.4. Let be the symplectic quasi-QR factorization of [w s]. As w T we have that e obtain #H by solving the underdetermined system e 110 #e 22e n+1,2# Lemma 3.3. The backward error of an approximate eigenpair (#x, #) of a skew-symmetric Hamiltonian eigenproblem with pure imaginary and x of the form is given by -(#x, |e 11 | is the quasi-triangular factor in the symplectic quasi-QR factorization of [w s] with if We also have -(#x, where e 2 is the second unit vector. 3.6. Symmetric skew-Hamiltonian eigenproblems. The analysis for symmetric skew-Hamiltonian eigenproblems is similar to that in section 3.4. The only di#erence comes from noting that F STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 113 using v T #I-E)#x 1 . Instead of computing a symplectic quasi-QR factorization of [#x r], we compute a symplectic quasi-QR factorization of # x r] in order to introduce one more zero in the triangular factor R. We summarize the result in the next lemma. Lemma 3.4. The backward error of an approximate eigenpair of a symmetric skew-Hamiltonian eigenproblem is given by -(#x, #) =|e 11 | x r] is the quasi-triangular factor in the symplectic quasi-QR factorization of [J x r] with #I -H)#x. We also have -(#x, #H#F . 3.7. Comments. Lemmas 3.2-3.4 provide an explicit formula for the backward error that can be computed in O(n 2 ) operations. For skew-symmetric skew-Hamiltonian matrices H, the eigenvectors are complex with no particular structure. The constraint (H #x in (3.1) can be written in the form #H[#x), is the residual. We were unable to explicitly construct matrices #H satisfying this constraint via a symplectic QR factorization of [#x), #x), #(r), #(r)]. Thus, in this case, we have to use the approach described in section 3.2 to compute -(#x, #), which has the drawback that it requires O(n 3 ) operations. 4. Algorithms for Hamiltonian eigenproblems. A simple but ine#cient approach to solve structured Hamiltonian eigenproblems is to use the (symmetric or unsymmetric as appropriate) QR algorithm on the 2n - 2n structured Hamiltonian matrix. This approach is computationally expensive and uses 4n 2 storage locations. Moreover, the QR algorithm does not use symplectic orthogonal transformations and is therefore not structure-preserving. Benner, Merhmann, and Xu's method [1] for computing the eigenvalues and invariant subspaces of a real Hamiltonian matrix uses the relationship between the eigenvalues and invariant subspaces of H and an extended 4n - 4n Hamiltonian ma- trix. Their algorithm is structure-preserving for the extended Hamiltonian matrix but is not structure-preserving for H. Therefore, it is not strongly backward stable in the sense of this paper. 4.1. QR-like algorithms. Bunse-Gerstner, Byers, and Mehrmann [5] provide a chart of numerical methods for structured eigenvalue problems, most of them based on QR-like algorithms. In this section, we describe their recommended algorithms for our structured Hamiltonian eigenproblems. In the limited case where rank(F Byer's Hamiltonian QR algorithm [6] based on symplectic orthogonal transformations yields a strongly backward stable algorithm. For symmetric Hamiltonian eigenproblems, the quaternion QR algorithm [4] is suggested. The quaternion QR algorithm is an extension of the Francis QR algorithm for complex or real matrices to quaternion matrices. This algorithm uses exclusively quaternion unitary similarity transformations so that it is backward stable. Compared with the standard QR algorithm for symmetric matrices, this algorithm cuts the storage and work requirements approximately in half. However, its implementation requires quaternion arithmetic and it is not clear whether it is strongly backward stable. A skew-symmetric Hamiltonian H is first reduced via symplectic orthogonal transformations to block antidiagonal form [ 0 -T the blocks are symmetric tridi- agonal. The complete solution is obtained via the symmetric QR algorithm applied to T . The whole algorithm is strongly backward stable as it uses only real symplectic orthogonal transformations that are known to be backward stable. For symmetric skew-Hamiltonian problems, the use of the "X-trick" is suggested: # with X =# 2 I I -iI iI # . The eigenvalues of H are computed from the eigenvalue of the Hermitian matrices using the Hermitian QR algorithm for instance. One drawback of this approach is that it uses complex arithmetic and does not provide a real symplectic orthogonal eigenvector basis. Hence the algorithm does not preserve the "realness" of the original matrix. Finally, for the skew-symmetric skew-Hamiltonian case, H is reduced to block-diagonal form via a finite sequence of symplectic orthogonal transformations. The blocks are themselves tridiagonal and skew-symmetric. Then Paardekooper's Jacobi algorithm [22] or the algorithm in [11] for skew-symmetric tridiagonal matrices can be used to obtain the complete solution. The whole algorithm is strongly backward stable. 4.2. Jacobi-like algorithms. Byers [7] adapted the nonsymmetric Jacobi algorithm [24] to the special structure of Hamiltonian matrices. The Hamiltonian Jacobi algorithm based on symplectic Givens rotations and symplectic double Jacobi rotations of the form J# I 2n , where J is a 2 - 2 Jacobi rotation, preserves the Hamiltonian structure. This Jacobi algorithm, when it converges, builds a Hamiltonian Schur decomposition [7, Thm. 1]. For symmetric H, this Jacobi algorithm converges to the canonical form [ D0 -D ] and is strongly backward stable. For skew-symmetric Hamiltonian H, this Jacobi algorithm does not converge as the symplectic orthogonal canonical form for H is not Hamiltonian triangular. Recently, Fa-bender, Mackey, and Mackey [10] developed Jacobi algorithms for structured Hamiltonian eigenproblems that preserve the structure and produce a complete basis of symplectic orthogonal eigenvectors for the two symmetric cases and a symplectic orthogonal basis for all the real invariant subspaces for the two skew-symmetric cases. These Jacobi algorithms are based on the direct solution of 4 - 4, and in one case 8 - 8, subproblems using appropriate transformations. The algorithms work entirely in real arithmetic. Note that "realness" of the initial matrix can be viewed as additional structure that these Jacobi algorithms preserve. We give a unified description of these Jacobi-like algorithms for the four classes of structured Hamiltonian eigenproblems under consideration. Let H # R 2n-2n be a structured Hamiltonian matrix (see Table 2.1 and 2.2). These Jacobi methods attempt to reduce the quantity (o#-diagonal norm) STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 115 where S is a set of indices depending on the structure of the problem using a sequence of symplectic orthogonal transformations H # SHS T with S # R 2n-2n . The aim is that H converges to its canonical form. In the following, we note A i,j,i+n,j+n the restriction to the (i, n) plane of A. Algorithm 4.1. Given a structured Hamiltonian matrix H # R 2n-2n and a tolerance tol > 0, this algorithm overwrites H with its approximate canonical form orthogonal and o#(PHP T while Choose (i, Compute a symplectic orthogonal S such that (SHS T ) i,j,i+n,j+n is in canonical form. preserving structure preserving structure Note that the pair (i, uniquely determines a 4 - 4 principal submatrix that also inherits the Hamiltonian or skew-Hamiltonian structure together with the symmetry or skew-symmetry property. There are many ways of choosing the indices (i, j) but this choice does not a#ect the rest of the analysis. We refer to n(n - 1)/2 updates as a sweep. Each sweep must be complete, that is, every part of the matrix must be reached. We see immediately that any complete sweep of the (1, 1) block of H consisting of 2-2 principal submatrices generates a corresponding complete sweep of H. For each 4 - 4 target submatrix, a symplectic orthogonal matrix that directly computes the corresponding canonical form is constructed and embedded into the in the same way that the 4 - 4 target has been extracted. For skew-symmetric skew-Hamiltonians, the 4-4 based Jacobi algorithm does not converge. The aim of these Jacobi algorithms is to move the weight to the diagonal of either the diagonal blocks or o#-diagonal blocks. That cannot be done for a skew-symmetric skew-Hamiltonian because these diagonals are zero. There is no safe place where the norm of the target submatrix can be kept. However, if an 8 - 8 skew-symmetric skew-Hamiltonian problem is solved instead, the 2 - 2 diagonal blocks of H become a safe place for the norm of target submatrices and the resulting 8 - 8 based Jacobi algorithm is expected to converge. The complete sweep is defined by partitioning blocks along the rightmost and lower edges when n is odd. Hence, in this case we must also be able to directly solve subproblems. Immediately, we see that the di#cult part in deriving these algorithms is to define the appropriate symplectic orthogonal transformation S that computes the canonical form of the restriction to the (i, n) plane of H. Fa-bender, Mackey, and Mackey [10] show that by using a quaternion representation of the 4 - 4 symplectic orthogonal group, as well as 4 - 4 Hamiltonian and skew-Hamiltonian matrices in the tensor square of the quaternion algebra, we can define and construct 4 - 4 symplectic orthogonal matrices R that do the job. These transformations are based on rotations of the subspace of pure quaternions. We need to give all the required transformations in a form suitable for rounding error analysis and also to facilitate the description of the structure preserving Jacobi algorithms. We start by defining two types of quaternion rotations. This enables us to encode the formulas in [10] into one. Let e s #= e 1 be a standard basis vector of R 4 and p # R 4 such that p #= 0, e T (p is a pure quaternion), and p/#p# 2 #= e s . Let s # . We define the left quaternion rotation by # . QL is symplectic orthogonal and not di#cult to compute. We have x and the other components of x are just permutations of the coordinates of p. We define the right quaternion rotation by # . The matrix QR is orthogonal. It is symplectic when s #= 3 and x R 4 be nonzero. Following [10], we define the 4 - 4 symplectic orthogonal Givens rotation associated with p by # . We now have all the tools needed to define the symplectic orthogonal transformations that directly compute the canonical form for each of the 4 - 4 structured Hamiltonian eigenproblems of interest. We refer to [10] for more details about how these transformations have been derived. 4.2.1. Symmetric Hamiltonian. Let H # R 4-4 be a symmetric Hamiltonian matrix. The canonical form of H is obtained in two steps: first H is reduced to 2 - 2 block diagonal form and then the complete diagonalization is obtained by using a double Jacobi rotation. For the first step we consider the singular value decomposition of the 3-3 matrix Let u 1 and v 1 be the left and right singular vectors corresponding to the largest singular value # 1 and let v1 ]. We have A T STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 117 so that e T v, the vector x in (4.3) is such that which implies that the right quaternion rotation QR (v, 2) is symplectic orthogonal. As shown in [10], the product diagonalizes H, that is, QHQ E). Complete diagonalization is obtained by using a double Jacobi rotation is chosen such that sin # sin # diagonalizes In summary, the symplectic orthogonal transformation S used in Algorithm 4.1 is equal to the identity matrix except in the (i, plane, where the (i, j, n j)-restriction matrix is given by 4.2.2. Skew-symmetric Hamiltonian. Let H # R 4-4 be a skew-symmetric Hamiltonian matrix and let p # R 4 be defined from the elements of H by It is easy to verify that for 4.2.3. Symmetric skew-Hamiltonian. Let H # R 4-4 be a symmetric skew- Hamiltonian matrix and let p # R 4 be defined from the elements of H by diagonalizes H and 4.2.4. Skew-symmetric skew-Hamiltonian. For the convergence of the Jacobi algorithm to be possible we need to solve an 8 - 8 subproblem. The matrix H # R 8-8 is block diagonalized with three 4 - 4 symplectic Givens rotations of the form (4.6) and one symplectic Givens rotation of the form (3.9). Let G be the product of these rotations. We have where tridiagonal and skew-symmetric. The complete 2 - 2 block- diagonalization is obtained by directly transforming its real Schur form as follows. In [20], Mackey showed that the transformation directly computes the real Schur form of E, that is, (Q# I 2 )G is the symplectic orthogonal transformation that computes the real Schur form of the 8 - 8 skew-symmetric skew- Hamiltonian H: # . When n is odd, we have to solve a 6-6 subproblem for each complete sweep of the Jacobi algorithm. As for the 8-8 case, the 6-6 skew-symmetric skew-Hamiltonian H is first reduced to the form (4.7), where tridiagonal and skew-symmetric. This is done by using just one 4 - 4 symplectic Givens rotation followed by one symplectic Givens rotation. Let and . Then computes directly the real Schur form of Moreover, we have e T Q) and # with 5. Error analysis of the Jacobi algorithms. In floating point arithmetic, Algorithm 4.1 computes an approximate canonical form T such that where P is symplectic orthogonal, and an approximate basis of symplectic orthogonal eigenvectors P . We want to derive bounds for #H# - I#, and 5.1. Preliminaries. We use the standard model for floating point arithmetic [16] where u is the unit roundo#. We assume that (5.1) holds also for the square roots operation. To keep track of the higher terms in u we make use of the following result [16, Lem. 3.1]. Lemma 5.1. If |# i | # u and # We define where p denotes a small integer constant whose value is unimportant. In the following, computed quantities will be denoted by hats. First, we consider the construction of a 4 - 4 Givens rotation and left and right quaternion rotations. Lemma 5.2. Let a 4 - 4 Givens rotation constructed according to (4.6) with p # R 4 . Then the computed G satisfies | # G-G| # 5 |G|. Proof. This result is a straightforward extension of Lemma 18.6 in [16] concerning Givens rotations. The rounding error properties of right and left quaternion rotations require more attention. When p s < 0, the computation of #p# therefore the computation of QL (p, s) or QR (p, s) is a#ected by cancellation. This problem can be overcome by using another formula as shown in the next lemma. Lemma 5.3. Let 4 - 4 left and right quaternion rotations constructed according to where and with p # R 4 given. Then the computed QL and | # QL -QL | # Proof. It is straightforward to verify that the expressions for QL (p, s) and QR (p, s) in (5.2) and (5.3) agree with the definitions in (4.4) and (4.5). We have f As p s # 0, there exists # 5 such that f using the same argument we have We also have f l ## ## Using [16, Lem. 3.3] we have and Hence, we certainly have In the following we use the term elementary symplectic orthogonal matrix to describe any double Givens rotation, 4-4 Givens rotation, or left or right quaternion rotation that is embedded as a principal submatrix of the identity matrix I # R 2n-2n . We have proved that any computed elementary symplectic orthogonal matrix used by the Jacobi algorithm satisfies a bound of the form | # Lemma 5.4. Let x # R 2n-2n and consider the computation of Px, where P is a computed elementary symplectic orthogonal matrix satisfying (5.5). The computed y satisfies where P is the exact elementary symplectic orthogonal matrix. Proof. The vector y di#ers from x only in elements We have We obtain similar results for y n+i , and y n+j . Hence, As Finally, we define and note that #x# Now, we consider the pre- and postmultiplication of a matrix H by an approximate elementary symplectic orthogonal matrix Lemma 5.5. Let H # R 2n-2n and P # R 2n-2n be any elementary symplectic orthogonal matrix such that f l(P ) satisfies (5.5). Then, f STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 121 Proof. Let h i be the ith column of H. By Lemma 5.4 we have The same result holds for h j , h n+i , and h n+j and the other columns of H are un- changed. Hence, f Similarly, B)P T with # B#F . Then, with with # As a consequence of Lemma 5.5, if H k+1 is the matrix obtained after one Jacobi update with S k (which is the product up to six elementary symplectic orthogonal matrices), we have is the exact transformation for Up to now, we made no assumption on H. If H is a structured Hamiltonian matrix, the (i, j)-restriction of RHR T is in canonical form. For instance, if H is a skew-symmetric Hamiltonian matrix, in a computer implementation the diagonal elements of H are not computed but are set to zero. Also, h ij , h i,j+n and by skew-symmetry h ji , h j+n,i are set to zero. But by forcing these elements to be zero, we are making the error smaller so the bounds still hold. Because of the structure of the problem, both storage and the flop count can be reduced by a factor of four. Any structured Hamiltonian matrix needs less than n 2 +n storage locations. If only the t parameters defining H are computed, the structure in the error is preserved and #H has the same structure as H. It is easy to see that the bounds in Lemma 5.6 are still valid with the property that #H has the same structure as H. Theorem 5.6. Algorithm 4.1 for structured Hamiltonians H compute a canonical T such that where #H has the same structure as H and #H#F # k #H#F , where k is the number of symplectic orthogonal transformations S i applied for each Jacobi update. The computed basis of symplectic orthogonal eigenvectors satisfies Proof. From (5.6), one Jacobi update of H satisfies For the second update we have 122 FRANC-OISE TISSEUR Continuing in this fashion, we find that, after k updates, 1 . S T k with #H k #F # In a similar way, using the first part of Lemma 5.5 we have After k updates, readily. Theorem 5.6 shows that the computed eigenvalues are the exact eigenvalues of a nearby structured Hamiltonian matrix and that the computed basis of eigenvectors is orthogonal and symplectic up to machine precision. This proves the strong backward stability of the Jacobi algorithms. 6. Numerical experiments. To illustrate our results we present some numerical examples. All computations were carried out in MATLAB, which has unit roundo# For symmetric Hamiltonians, symmetric skew-Hamiltonians, and skew-symmetric Hamiltonians with approximate eigenvector # x of the form [ z -iz ], computing -(#x, #) in involves a symplectic quasi-QR factorization of a 2n - 2 matrix, which can be done in order n 2 flops, a cost negligible compared with the O(n 3 ) cost of the whole eigendecomposition. For skew-symmetric Hamiltonians with approximate eigenvector x not of the form -iz ], and for skew-symmetric skew-Hamiltonians, the computation of -(#x, #) requires O(n 3 ) flops as we have to find the minimal 2-norm solution of a large underdetermined system in (3.6). Thus, in this case, -(#x, #) is not a quantity we would compute routinely in the course of solving a problem. Note that in our implementation of the Jacobi-like algorithm for skew-symmetric Hamiltonians we choose the approximate eigenvectors to be the columns of P [ I -iI I where P is the accumulation of the symplectic orthogonal transformations used by the algorithm to build the canonical form. In this case, the approximate eigenvectors x are guaranteed to be of the form [ z To test the strong stability of numerical algorithms for solving structured Hamiltonian eigenproblems, we applied the direct search maximization routine mdsmax of the MATLAB Test Matrix Toolbox [15] to the function 1#i#2n are the computed eigenpairs. In this way we carried out a search for problems on which the algorithms performs unstably. As expected from the theory, we could not generate examples for which the structured backward error for the Jacobi-like algorithms is large: -(#x, #) < nu#H#F in all our tests. The symmetric QR algorithm does not use symplectic orthogonal transformations and is therefore not structure-preserving. To our surprise, we could not generate examples of symmetric Hamiltonian and symmetric skew-Hamiltonian matrices for which STABILITY OF STRUCTURED HAMILTONIAN EIGENSOLVERS 123 Table Backward error of the eigenpair for of the 4-4 skew-symmetric Hamiltonian defined by (6.1). #) -max (#x, #) Jacobi-like algorithm Table Backward errors of the approximation of the eigenvalue 0 for a 30-30 random skew-symmetric skew-Hamiltonian matrix. | #) any of the eigenpairs computed by the symmetric QR algorithm has a large backward error. However, the QR algorithm does not compute a symplectic orthogonal basis of eigenvectors and also, it is easy to generate examples for which the -# structure for symmetric Hamiltonians and eigenvalue multiplicity 2 structure for symmetric skew-Hamiltonians is not preserved. If we generalize the definition of the structured backward error of a single eigenpair to a set of k eigenpairs, the symmetric QR algorithm is likely to produce sets of eigenpairs with an infinite structured backward error. The QR-like algorithm for symmetric skew-Hamiltonians is likely to provide eigenvectors that are complex instead of real, yielding an infinite structured backward error in (3.14). The good backward stability of individual eigenpairs computed by the QR algorithm does not hold for the skew-symmetric Hamiltonian case. For instance, we considered the skew-symmetric Hamiltonian eigenproblem # , with whose eigenvalues are distinct: In Table 6.1, we give the normwise, componentwise, and structured normwise backward error of the eigenpair for computed by the unsymmetric QR algorithm and the skew-symmetric Jacobi algorithm. The QR algorithm does not use symplectic orthogonal transformations and the computed eigenvectors do not have the structure -iz ]. Therefore, for the computation of - max (#x, #), we use the general formula (3.7). In the skew-symmetric skew-Hamiltonian case, when n is odd, 0 is an eigenvalue of multiplicity two and is not always well approximated with the unsymmetric QR algorithm. We generated a random 15 - 15 E and F . We give in Table 6.2 the backward errors associated with the approximation of the eigenvalue 0 for both the QR algorithm and Jacobi algorithm. 7. Conclusion. The first contribution of this work is to extend existing definitions of backward errors in a way appropriate to structured Hamiltonian eigen- problems. We provided computable formulae that are inexpensive to evaluate except for skew-symmetric skew-Hamiltonians. Our numerical experiments showed that for symmetric structured Hamiltonian eigenproblems, the symmetric QR algorithm computes eigenpairs with a small structured backward error but the algebraic properties of the problem are not preserved. Our second contribution is a detailed rounding error analysis of the new Jacobi algorithms of Fa-bender, Mackey, and Mackey [10] for structured Hamiltonian eigen- problems. These algorithms are structure-preserving, inherently parallelizable, and hence attractive for solving large-scale eigenvalue problems. We proved their strong stability when the left and right quaternion rotations are implemented using our formulae (5.2), (5.3). Jacobi algorithms are easy to implement and o#er a good alternative to QR algorithms, namely, the unsymmetric QR algorithm, which we showed to be not strongly backward stable for skew-symmetric Hamiltonian and skew-Hamiltonian eigenproblems, and the algorithm for symmetric skew-Hamiltonians based on applying the QR algorithm to (4.1), which does not respect the "realness" of the problem. Acknowledgments . I thank Nil Mackey for pointing out the open question concerning the strong stability of the Jacobi algorithms for structured Hamiltonian eigenproblems and for her suggestion in fixing the cancellation problem when computing the quaternion rotations. I also thank Steve Mackey for his helpful comments on an earlier manuscript. --R A new method for computing the stable invariant subspace of a real Hamiltonian matrix The weak and strong stability of algorithms in numerical linear algebra Matrix factorizations for symplectic QR-like methods A quaternion QR algorithm A chart of numerical methods for structured eigenvalue problems A Hamiltonian QR algorithm IEEE Trans. A relative backward perturbation theorem for the eigenvalue problem Hamilton and Jacobi come full circle: Jacobi algorithms for structured Hamiltonian eigenproblems Accurately counting singular values of bidiagonal matrices and eigenvalues of skew-symmetric tridiagonal matrices A contribution to the theory of condition Backward error and condition of structured linear systems Structured backward error and condition of generalized eigenvalue problems The Test Matrix Toolbox for Matlab (version 3.0) Accuracy and Stability of Numerical Algorithms Oxford University Press The Theory of Matrices Canonical forms for Hamiltonian and symplectic matrices and pencils Hamilton and Jacobi meet again: Quaternions and the eigenvalue problem Solution of the large matrix equations which occur in response theory An eigenvalue algorithm for skew-symmetric matrices A Jacobi-like algorithm for computing the Schur decomposition of a non-Hermitian matrix Structured linear algebra problems in digital signal processing A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix --TR

quaternion rotation;structure-preserving;symplectic;jacobi algorithm;backward error;skew-Hamiltonian;hamiltonian;symmetric;rounding error;skew-symmetric

587834

Structured Pseudospectra for Polynomial Eigenvalue Problems, with Applications.

Pseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrix equation and thereby factorizes the quadratic $\lambda$-matrix. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques.

Introduction . Pseudospectra are an established tool for gaining insight into the sensitivity of the eigenvalues of a matrix to perturbations. Their use is widespread with applications in areas such as fluid mechanics, Markov chains, and control theory. Most of the existing work is for the standard eigenproblem, although attention has also been given to matrix pencils [4], [23], [33], [40], [46]. The literature on pseudospectra is large and growing. We refer to Trefethen [41], [42], [43] for thorough surveys of pseudospectra and their computation for a single matrix; see also the Web site [3]. In this work we investigate pseudospectra for polynomial matrices (or #-matrices) 0: m. We first define the #-pseudospectrum and obtain a computationally useful characterization. We examine the relation between the backward error of an approximate eigenpair of the polynomial eigenvalue problem associated with (1.1), the #-pseudospectrum, and the stability radius. We consider both unstructured perturbations and structured perturbations of a type commonly used in control theory. Existing methods for the computation of pseudospectra in the case standard and generalized eigenvalue problems) do not generalize straightforwardly to matrix polynomials. We develop two techniques that allow e#cient computation for m > 1. A transfer function approach employs the generalized Schur decomposition of the mn - mn companion form pencil. For the quadratic case an alternative # Received by the editors May 1, 2000; accepted for publication (in revised form) by M. Chu February 7, 2001; published electronically June 8, 2001. This work was supported by Engineering and Physical Sciences Research Council grant GR/L76532. http://www.siam.org/journals/simax/23-1/37145.html Department of Mathematics, University of Manchester, Manchester, M13 9PL, England (ftisseur@ma.man.ac.uk, http://www.ma.man.ac.uk/-ftisseur/, higham@ma.man.ac.uk, http://www.ma.man.ac.uk/-higham/). The work of the second author was supported by a Royal Society Leverhulme Trust Senior Research Fellowship. solvent approach computes a solvent of the associated quadratic matrix equation thereby factorizes the quadratic #-matrix; it works all the time with n - n matrices once the solvent has been obtained. We give a detailed comparison of these approaches and also outline techniques that can be e#ciently used when n is so large as to preclude factorizations. In the last section, we illustrate our theory and techniques on applications from vibrating systems, control theory, acoustics, and fluid mechanics. 2. Pseudospectra. 2.1. Definition. The polynomial eigenvalue problem is to find the solutions (x, #) of where P (#) is of the form (1.1). If x #= 0 then # is called an eigenvalue and x the corresponding right eigenvector; y #= 0 is a left eigenvector if y # P of eigenvalues of P is denoted by #(P ). When Am is nonsingular P has mn finite eigenvalues, while if Am is singular P has infinite eigenvalues. Good references for the theory of #-matrices are [8], [20], [21], [37]. Throughout this paper we assume that P has only finite eigenvalues (and pseu- doeigenvalues); how to deal with infinite eigenvalues is described in [16]. For notational convenience, we introduce We define the #-pseudospectrum of P by with Here the # k are nonnegative parameters that allow freedom in how perturbations are measured-for example, in an absolute sense (# k # 1) or a relative sense By setting # unperturbed. The norm, here and throughout, is any subordinate matrix norm. Occasionally, we will specialize to the norm # p subordinate to the H-older vector p-norm. When reduces to the standard definition of #-pseudospectrum of a single matrix: with #A# . It is well known [43] that (2.4) is equivalent to In the following lemma, we provide a generalization of this equivalence for the #- pseudospectrum of P . Lemma 2.1. STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 189 Proof. Let S denote the set on the right-hand side of the claimed equality. We first show that # (P ) implies # S. If # is an eigenvalue of P this is immediate, so we can assume that # is not an eigenvalue of P and hence that P (#) is nonsingular. Since is singular, we have so that # S. Now let # S. Again we can assume that with #, so that #x# = 1. Then there exists a matrix H with Lem. 6.3]). Let y y and We now apportion E between the A k by defining where for complex z we define Then and #A k # k #, 0: m. Hence # (P ). The characterization of the #-pseudospectrum in Lemma 2.1 will be the basis of our algorithms for computing pseudospectra. We note that for is the root neighborhood of the polynomial P introduced by Mosier [28], that is, the set of all polynomials obtained by elementwise perturbations of P of size at most #. This set is also investigated by Toh and Trefethen [38], who call it the #-pseudozero set. 2.2. Connection with backward error. A natural definition of the normwise backward error of an approximate eigenpair (x, #) of (2.1) is and the backward error for an approximate eigenvalue # is given by #) := min #(x, #). 190 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM By comparing the definitions (2.3) and (2.6) it is clear that the #-pseudospectrum can be expressed in terms of the backward error of # as The following lemma gives an explicit expression for #(x, #) and #). This lemma generalizes results given in [36] for the 2-norm and earlier in [5], [10] for the generalized eigenvalue problem. Lemma 2.2. The normwise backward error #(x, #) is given for x #= 0 by p(|#x# If # is not an eigenvalue of P then #) =p(|#P (#) Proof. It is straightforward to show that the right-hand side of (2.8) is a lower bound for #(x, #). That the lower bound is attained is proved using a construction for #A k similar to that in the proof of Lemma 2.1. The expression (2.9) follows on using the equality, for nonsingular C # C n-n , min x#=0 #Cx#x#C We observe that the expressions (2.7) and (2.9) lead to another proof of Lemma 2.1. 2.3. Structured perturbations. We now suppose that P (#) is subject to structured perturbations that can be expressed as with . The matrices D and E are fixed and assumed to be of full rank, and they define the structure of the perturbations; # is an arbitrary matrix whose elements are the free parameters. Note that #A 0 , . , #Am in (2.10) are linear functions of the parameters in #, but that not all linear functions can be represented in this form. We choose this particular structure for the perturbations because it is one commonly used in control theory [17], [18], [30] and it leads to more tractable formulae than a fully general approach. Note, for instance, that the system (which leads to a polynomial eigenvalue problem with may be interpreted as a closed loop system with unknown static linear output feedback #; see Figure 2.1. Note that unstructured perturbations are represented by the special case of (2.10) with For notational convenience, we introduce Corresponding to (2.10) we have the following definition of structured backward error for an approximate eigenpair (x, #): STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 191 Fig. 2.1. Closed loop system with unknown static linear output feedback #. and the backward error for an approximate eigenvalue is #(x, #; D,E). In the next result we use a superscript "+" to denote the pseudo-inverse [9]. Lemma 2.3. The structured backward error #(x, #; D,E) in the Frobenius norm is given by #F if the system is consistent; otherwise # F (x, #; D,E) is infinite. Proof. It is immediate that # F (x, #; D,E) is the Frobenius norm of the minimum Frobenius norm solution to (2.13). The result follows from the fact that is the solution of minimum Frobenius norm to the consistent system sect. 3.4.8]. To gain some insight into the expression (2.12) we consider the case of unstructured but weighted perturbations, as in (2.11) but with The system (2.13) is now trivially consistent and (2.12) gives using the fact that #ab . The expression (2.14) di#ers from that for #(x, #) in (2.8) for the 2-norm only by having the 2-norm of the vector rather than the 1-norm in the denominator. Lemma 2.4. If # is not an eigenvalue of P (#) then the structured backward error #; D,E) is given by (2. 192 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM Proof. We have min The companion form of P (#P (#) is given by where . 0 I -A I I I Am and #Am # . As # . Then, using the identity det(I both AB and BA are defined [47, p. 54], det(P (#P using [45, Lem. 1], we have But it is easily verified that D, STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 193 so that We define the structured #-pseudospectrum by Analogously to the unstructured case, # so from Lemma 2.4 we have which is a generalization of a result of Hinrichsen and Kelb [17, Lem. 2.2] for the #-pseudospectrum of a single matrix. 2.4. Connection between backward error and stability radius. In many mathematical models (e.g., those of a dynamical system) it is required for stability that a matrix has all its eigenvalues in a given open subset C g # of the complex plane. Various stability radii have been defined that measure the ability of a matrix to preserve its stability under perturbations. We partition the complex plane C into two disjoint subsets C g and C b , with # an open set. Consider perturbations of the form in (2.10). Following Pappas and Hinrichsen [30] and Genin and Van Dooren [7], we define the complex structured stability radius of the #-matrix P with respect to the perturbation structure (D, E) and the partition by r Let #C b be the boundary of C b . By continuity, we have r #C s-t { #(P (#) +D#E(#C b # } #; D,E). Thus we have expressed the stability radius as an infimum of the eigenvalue backward error. Using Lemma 2.4 we obtain the following result. Lemma 2.5. If # is not an eigenvalue of P then r and for unstructured perturbations and the p-norm we have r The result for the unstructured case in the second part of this lemma is also obtained by Pappas and Hinrichsen [30, Cor. 2.4] and Genin and Van Dooren [7, Thm. 2]. 194 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM 3. Computation of pseudospectra. In this section, we consider the computation of # (P ), concentrating mainly on the 2-norm. We develop methods for unstructured perturbations and show how they can be extended to structured perturbations of the form in (2.10). Lemma 2.1 shows that the boundary of # (P ) comprises points z for which the scaled resolvent norm p(|z|)#P (z) Hence, as for pseudospectra of a single matrix, we can obtain a graphical representation of the pseudospectra of a polynomial eigenvalue problem by evaluating the scaled resolvent norm on a grid of points z in the complex plane and sending the results to a contour plotter. We refer to Trefethen [42] for a survey of the state of the art in computation of pseudospectra of a single matrix. The region of interest in the complex plane will usually be determined by the underlying application or by prior knowledge of the spectrum of P . In the absence of such information we can select a region guaranteed to enclose the spectrum. If Am is nonsingular (so that all eigenvalues are finite) then by applying the result (A)| #A#" to the companion form (2.16) we deduce that for any p-norm. Alternatively, we could bound max j |# j (P )| by the largest absolute value of a point in the numerical range of P [24], but computation of this number is itself a nontrivial problem. For much more on bounding the eigenvalues of matrix polynomials see [15]. For the 2-norm, #P (z) denotes the smallest singular value. If the grid is # and # min is computed using the Golub-Reinsch SVD algorithm then the whole computation requires roughly which is prohibitively expensive for matrices of large dimension and a fine grid. Using the fact that # min (P (z)) is the square root of # min (P (z) # P (z)), we can approximate with the power iteration or Lanczos iteration applied to P (z) In the case of a single matrix, Lui [25] introduced the idea of using the Schur form of A in order to speed up the computation of # min Unfortunately, for matrix polynomials of degree m # 2 no analogue of the Schur form exists (that is, at most two general matrices can be simultaneously reduced to triangular form). We therefore look for other ways to e#ciently evaluate or approximate #P (z) -1 # for many di#erent z. 3.1. Transfer function approach. The idea of writing pseudospectra in terms of transfer functions is not new. Simoncini and Gallopoulos [34] used a transfer function framework to rewrite most of the techniques used to approximate #-pseudospectra of large matrices, yielding interesting comparisons as well as better understanding of the techniques. Hinrichsen and Kelb [17] investigated structured pseudospectra of a single matrix with perturbations of the form in (2.10), and they expressed the structured #-pseudospectrum in terms of a transfer function. Consider the equation STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 195 It can be rewritten as wm where F and G are defined in (2.16). Hence -I # u. Since this equation holds for all u, it follows that -I # . This equality can also be deduced from the theory of #-matrices [21, Thm. 14.2.1]. We have thus expressed the resolvent in terms of a transfer function. In control theory, P (z) -1 corresponds to the transfer function of the linear time-invariant multivariate system described by I Several algorithms have been proposed in the literature [22], [27] to compute transfer functions at a large number of frequencies, most of them assuming that G = I. Our objective is to e#ciently compute the norm of the transfer function, rather than to compute the transfer function itself. For structured perturbations we see from (2.17) that the transfer function P (z) is replaced by E(z)P (z) -D # . All the methods described below for the dense case are directly applicable with obvious changes. We would like a factorization of F - zG that enables e#cient evaluation or application of di#erent z. There are various possibilities, including, when G is nonsingular, 196 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM is a Schur decomposition, with W unitary and T upper tri- angular. However this approach is numerically unstable when G is ill conditioned. A numerically stable reduction is obtained by computing the generalized Schur decom- position where W and Z are unitary and T and S are upper triangular. Then -I # . Hence once the generalized Schur decomposition has been computed, we can compute x at a cost of O((mn) 2 ) flops, since T - zS is triangular of dimension mn. For the 2-norm we can therefore e#ciently approximate #P (z) using inverse iteration or the inverse Lanczos iteration, that is, the power method or the Lanczos method applied to P (z) The cost of the computation breaks into two parts: the cost of the initial transformations and the cost of the computations at each of the # 2 grid points. Assuming that (3.3) is computed using the QZ algorithm [9, Sec. 7.7] and the average number of power method or Lanczos iterations per grid point is k, the total cost is about For the important special case eigenvalue problem), this cost is Comparing with (3.1) we see that this method is a significant improvement over the SVD-based approach for a su#ciently fine grid and a small degree m. For the 2-norm note that, because of the two outer factors in (3.4), we cannot discard the unitary matrices Z and W , unlike in the analogous expression for the resolvent of a single matrix in the standard eigenproblem. For the 1- and #-norms we can e#ciently estimate #P (z) using the algorithm of Higham and Tisseur [14], which requires only the ability to multiply matrices by P (z) -1 and P (z) -# . An alternative to the generalized Schur decomposition is the generalized Hessenberg-triangular form, which di#ers from (3.3) in that one of T and S is upper Hessenberg. The Hessenberg form is cheaper to compute but more expensive to work with. It leads to a smaller overall flop count when k# 2 > # 25mn. 3.2. Factorizing the quadratic polynomial. The transfer function-based method of the previous section has the drawback that it factorizes matrices of dimension m times those of the original polynomial matrix. We now describe another method, particular to the quadratic case, that does not increase the size of the problem. Suppose we can find a matrix S such that A 2 S 2 +A 1 S+A that is, a solvent of the quadratic matrix equation A 2 If we compute the Schur decomposition STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 197 and the generalized Schur decomposition then so a vector can be premultiplied by Q(z) -1 or its conjugate transpose in O(n 2 ) flops for any z. Moreover, for the 2-norm we can drop the outer Q and W # factors in (3.7), by unitary invariance, and hence we do not need to form W . For the 2-norm, the total cost of this method is where c S is the cost of computing a solvent and we have assumed that we precompute Z. Comparing this flop count with (3.5) we see that the cost per grid point of the solvent approach is much lower. The success of this method depends on two things: the existence of solvents and being able to compute one at a reasonable cost. Some su#cient conditions for the existence of a solvent are summarized in [13]. In particular, for an overdamped problem, one for which A 2 and A 1 are Hermitian positive definite, A 0 is Hermitian positive semidefinite, and a solvent is guaranteed to exist. Various methods are available for computing solvents [12], [13]. One of the most generally useful is Newton's method, optionally with exact line searches, which requires a generalized Sylvester equation in n-n matrices to be solved on each iteration, at a total cost of about 56n 3 flops per iteration. If Newton's method converges within 8 iterations or so, so that c S # 448n 3 flops, this approach is certainly competitive in cost with the transfer function approach. When there is a gap between the n largest and n smallest eigenvalues ordered by modulus, as is the case for overdamped problems [20, Sec. 7.6], Bernoulli iteration is an e#cient way of computing the dominant or minimal solvent S [13]. If t iterations are needed for convergence to the dominant or minimal solvent then the cost of Bernoulli iteration is about c flops. Bernoulli iteration converges only linearly, but convergence is fast if the eigenvalue gap is large. A third approach to computing a solvent is to use a Schur method from [13], based on the following theorem. Let F and G be defined as in (2.16), so that -A # . Theorem 3.1 (Higham and Kim [13]). All solvents of Q(X) are of the form is a generalized Schur decomposition with Q and Z unitary and T and S upper tri- angular, and where all matrices are partitioned as block 2 - 2 matrices with n - n blocks. The method consists of computing the generalized Schur decomposition (3.9) by the QZ algorithm and then forming 11 . The generalized Schur decomposition may need to be reordered in order to obtain a nonsingular Z 11 . Note that the 198 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM unitary factor Q does not need to be formed. For this method, c where the constant r depends on the amount of reordering required. From (3.8), the total cost is now which is much more favorable than the cost (3.5) of the transfer function method. For higher degree polynomials we can generalize this approach by attempting to linear factors by recursively computing solvents. However, for degrees greater than 2 classes of problem for which a factorization into linear factors exists are less easily identified and the cost of Newton's method (for example) is much higher than for 3.3. Large-scale computation. All the methods described above are intended for small- to medium-scale problems for which Schur and other reductions are pos- sible. For large, possibly sparse, problems, di#erent techniques are necessary. These techniques can be classified into two categories: those that project to reduce the size of the problem and then compute the pseudospectra of the reduced problem, and those that approximate the norm of the resolvent directly. 3.3.1. Projection approach. For a single matrix, A, Toh and Trefethen [39] and Wright and Trefethen [48] approximate the resolvent norm by the Arnoldi method; that is, they approximate or by # min ( where Hm is the square Hessenberg matrix of dimension m # n obtained from the Arnoldi process and Hm is the matrix Hm augmented by an extra row. Simoncini and Gallopoulos [34] show that a better but more costly approximation is obtained by approximating #(A- zI) is the orthonormal basis generated during the Arnoldi process. These techniques are not applicable to the polynomial eigenvalue problem of degree larger than one because of the lack of a Schur form for the Arnoldi method to approximate. A way of approximating #P (z) -1 # for all z is through a projection of P (z) a lower dimensional subspace. Let V k be an n - k matrix with orthonormal columns. We can apply one of the techniques described in the previous sections to compute pseudospectra of the projected polynomial eigenvalue problem A possible choice for V k is an orthonormal basis of k selected linearly independent eigenvectors of P (#). In this case, P (#) is the matrix representation of the projection of P (#) onto the subspace spanned by the selected eigenvectors. The eigenvectors can be chosen to correspond to parts of the spectrum of interest and can be computed using the Arnoldi process on the companion form pencil (F, G) or directly on P (#) with the Jacobi-Davidson method or its variants [26], [35]. In the latter case, the during the Davidson process. 3.3.2. Direct approach. This approach consists of approximating #P (z) at each grid point z. Techniques analogous to those used for single matrices can be applied, such as the Lanczos method applied to P (z) # P (z) or its inverse. We refer the reader to [42] for more details and further references. 4. Applications and numerical experiments. We give a selection of applications of pseudospectra for polynomial eigenvalue problems, using them to illustrate the performance of our methods for computing pseudospectra. All our examples are for 2-norm pseudospectra. STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 199 4.1. The wing problem. The first example is based on a quadratic polynomial A 0 from [6, Sec. 10.11], with numerical values modified as in [20, Sec. 5.3]. The eigenproblem for Q(#) arose from the analysis of the oscillations of a wing in an airstream. The matrices are 17.6 1.28 2.89 1.28 0.824 0.413 7.66 2.45 2.1 # . The left plot in Figure 4.1 shows the boundaries of #-pseudospectra with perturbations measured in the absolute sense . The eigenvalues are plotted as dots. Another way of approximating a pseudospectrum is by random perturbations of the original matrices [41]. We generated 200 triples of complex random normal perturbation matrices (#A 1 , 1: 3. In the right plot of Figure 4.1 are superimposed as small dots the eigenvalues of the perturbed polynomials # 2 The solid curve marks the boundary of the #-pseudospectrum for pictures show that the pair of complex eigenvalues are much more sensitive to perturbations than the other two complex pairs. The eigenvalues of Q(#) are the same as those of the linearized problem A- #I, where -A # . Figure 4.2 shows boundaries of #-pseudospectra for this matrix, for the same # as in Figure 4.1. Clearly, the #-pseudospectra of the linearized problem (4.1) do not give useful information about the behavior of the eigensystem of Q(#) under perturbations. This emphasizes the importance of defining and computing pseudospectra for the quadratic eigenvalue problem in its original form. 4.2. Mass-spring system. We now consider the connected damped mass-spring system illustrated in Figure 4.3. The ith mass of weight m i is connected to the (i+1)st mass by a spring and a damper with constants k i and d i , respectively. The ith mass is also connected to the ground by a spring and a damper with constants # i and # i , respectively. The vibration of this system is governed by a second-order di#erential equation d dt where the mass matrix diagonal, and the damping matrix C and sti#ness matrix K are symmetric tridiagonal. The di#erential equation leads to the quadratic eigenvalue problem In our experiments, we took all the springs (respectively, dampers) to have the same constant except the first and last, for which the constant is 2# (respectively, 2# ), and we took m i # 1. Then 200 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM Fig. 4.1. Wing problem. Left: # (Q), for # [10 approximation to #- pseudospectrum with Fig. 4.2. Wing problem. # (A), for A in (4.1) with # [10 and the quadratic eigenvalue problem is overdamped. We take an of freedom mass-spring system over a 100 - 100 grid. A plot of the pseudospectra is given in Figure 4.4. For this problem we compare all the methods described. In the solvent approach exact line searches were used in Newton's method and no reordering was used in the generalized Schur method. The solvents from the Bernoulli and Schur methods were refined by one step of Newton's method. The Bernoulli iteration converged in 12 iterations while only 6 iterations were necessary for Newton's method. The Lanczos inverse iteration converged after 3 iterations on average. In Table 4.1 we give the estimated flop counts, using the formulae from section 3, together with execution times. The computations were performed in MATLAB 6, which is an excellent environment for investigating pseudospectra. While the precise times are not important, the con- STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 201 d i-1 d n-1 Fig. 4.3. An n degree of freedom damped mass-spring system. Fig. 4.4. Pseudospectra of a 250 degree of freedom damped mass-spring system on a 100 - 100 grid. clusion is clear: in this example, the three solvent-based methods are much faster than the SVD and transfer function methods. (The high speed of the SVD method relative to its flop count is attributable to MATLAB's very e#cient svd function.) 4.3. Acoustic problem. Acoustic problems with damping can give rise to large quadratic eigenvalue problems (4.2), where, again, M is the mass matrix, C is the damping matrix, and K the sti#ness matrix. We give in Figure 4.5 the sparsity pattern of the three matrices M , C, and K of order 107 arising from a model of a speaker box [1]. These matrices are symmetric and the sparsity patterns of M and K are identical. There is a large variation in the norms: #M# We plot in Figure 4.6 pseudospectra with perturbations measured in both an absolute sense a relative sense together with pseudospectra of the corresponding standard eigenvalue problem of the form (4.1). The eigenvalues are all pure imaginary and are marked by dots on the plot. The two first plots are similar, both showing that the most sensitive Table Comparison in terms of flops and execution time of di#erent techniques. Method Estimated cost in flops Execution time Golub-Reinsch SVD 26747n 3 102 min Transfer function 3408n 3 106 min Solvent: Schur 1677n 3 37 min Matrices M and K Matrix C Fig. 4.5. Sparsity patterns of the three 107 - 107 matrices M,C, and K of an acoustic problem. eigenvalues are located at the extremities of the spectrum; the contour lines di#er mainly around the zero eigenvalue. The last plot is very di#erent; clearly it is the eigenvalues close to zero that are the most sensitive to perturbations of the standard eigenproblem form. We mention that for this problem we have been unable to compute a solvent. 4.4. Closed loop system. In multi-input and multioutput systems in control theory the location of the eigenvalues of matrix polynomials determine the stability of the system. Figure 4.7 shows a closed-loop system with feedback with gains 1 and The associated matrix polynomial is given by # . We are interested in the values of # for which P (z) has all its eigenvalues inside the unit circle. By direct calculation with det(P (z)), using the Routh array, for example, it can be shown that P (z) has all its eigenvalues inside the unit circle if and only if # < 0.875. The matrix P (z) can be viewed as a perturbed matrix polynomial with structured perturbations: STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 203 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 Fig. 4.6. Acoustic problem, Perturbations measured in an absolute sense (top left) and relative sense (top right). Pseudospectra of the equivalent standard eigenvalue problem are shown at the bottom. where I zI z 2 I # . We show in Figure 4.8 the structured pseudospectra as defined by (2.17). The dashed lines mark the unit circle. Since the outermost contour has value just touches the unit circle, this picture confirms the value for the maximal # that we obtained analytically. 4.5. The Orr-Sommerfeld equation. The Orr-Sommerfeld equation is a linearization of the incompressible Navier-Stokes equations in which the perturbations in velocity and pressure are assumed to take the form y, 204 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM z z Fig. 4.7. Closed-loop system with feedback gains 1 and 1 10.60.20.20.60.80.36Fig. 4.8. Structured pseudospectra of a closed-loop system with one-parameter feedback. where # is a wavenumber and # is a radian frequency. For a given Reynolds number R, the Orr-Sommerfeld equation may be written We consider plane Poiseuille flow between walls at in the streamwise x direction, for which the boundary conditions are For a given real value of R, the boundary conditions will be satisfied only for certain combinations of values of # and #. Two cases are of interest. Case 1. Temporal stability. If # is fixed and real, then (4.3) is linear in the parameter # and corresponds to a generalized eigenvalue problem. The perturbations STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 205 are periodic in x and grow or decay in time depending on the sign of the imaginary part of #. This case has been studied with the help of pseudospectra by Reddy, Schmid, and Henningson [32]. Case 2. Spatial stability. For most real flows, the perturbations are periodic in time, which means that # is real. Then the sign of the imaginary part of # determines whether the perturbations will grow or decay in space. In this case, the parameter is #, which appears to the fourth power in (4.3), so we obtain a quartic polynomial eigenvalue problem. Bridges and Morris [2] calculated the spectrum of (4.3) using a finite Chebyshev series expansion of # combined with the Lanczos tau method and they computed the spectrum of the quartic polynomial by two methods: the QR algorithm applied to the corresponding standard eigenvalue problem in companion form, and Bernoulli iteration applied to determine a minimal solvent and hence to obtain the n eigenvalues of minimal modulus. For our estimation of the pseudospectra of the Orr-Sommerfeld equation we use a Chebyshev spectral discretization that combines an expansion in Chebyshev polynomials and collocation at the Chebyshev points with explicit enforcement of the boundary conditions. We are interested in the eigenvalues # that are the closest to the real axis, and we need Im(#) > 0 for stability. The linear eigenvalue problem (Case 1) has been solved by Orszag [29]. The critical neutral point corresponding to # and # both real for minimum R was found at the frequency our calculations we set R and # to these values and we computed the modes #, taking which gives matrices of order 1. The first few modes are plotted in Figure 4.9. For the first mode we obtained which compares favorably with the result of Orszag. Figure 4.10 shows the pseudospectra in a region around the first few modes on a 100-100 grid, with # since A 4 is the identity matrix and is not subject to uncertainty. The plot shows that the first mode is very sensitive. Interest- ingly, the second and subsequent modes are almost as sensitive, with perturbations of in the matrix coe#cients being su#cient to move all these modes across the real axis, making the flow unstable. The pseudospectra thus give a guide to the accuracy with which computations must be carried out for the numerical approximations to the modes to correctly determine the location of the modes. For more on the interpretation of pseudospectra for this problem, see [32] and [44]. Again, for comparison we computed the pseudospectra of the corresponding standard eigenvalue problem. The picture was qualitatively similar, but the contour levels were several orders of magnitude smaller, thus not revealing the true sensitivity of the problem. 206 FRANC-OISE TISSEUR AND NICHOLAS J. HIGHAM Fig. 4.9. The first few modes of the spectrum of the Orr-Sommerfeld equation for 109.39.058.88.99.159.59.5Fig. 4.10. Pseudospectra of the Orr-Sommerfeld equation for STRUCTURED PSEUDOSPECTRA FOR POLYNOMIAL EIGENPROBLEMS 207 --R http://www. Elementary Matrices and Some Applications to Dynamics and Di Stability Radii of Polynomial Matrices Matrix Polynomials Matrix Computations Structured backward error and condition of generalized eigenvalue problems Accuracy and Stability of Numerical Algorithms Solving a Quadratic Matrix Equation by Newton's Method with Exact Line Searches Numerical analysis of a quadratic matrix equation A block algorithm for matrix 1-norm estimation Bounds for Eigenvalues of Matrix Polynomials More on Pseudospectra for Polynomial Eigenvalue Problems and Applications in Control Theory Spectral value sets: A graphical tool for robustness analysis Real and complex stability radii: A survey Numerical solution of matrix polynomial equations by Newton's method The Theory of Matrices Numerical range of matrix polynomials Computation of pseudospectra by continuation Locking and restarting quadratic eigenvalue solvers A determinant identity and its application in evaluating frequency response matrices Root neighborhoods of a polynomial Accurate solution of the Orr-Sommerfeld stability equation Robust stability of linear systems described by higher order Generalized Inverse of Matrices and Its Applications Pseudospectra of the Orr-Sommerfeld operator Transfer functions and resolvent norm approximation of large matrices Backward error and condition of polynomial eigenvalue problems The quadratic eigenvalue problem Pseudozeros of polynomials and pseudospectra of companion matrices Calculation of pseudospectra by the Arnoldi iteration Portraits Spectraux de Matrices: Un Outil d'Analyse de la Stabilit-e Pseudospectra of matrices Computation of pseudospectra Spectra and pseudospectra Hydrodynamic stability without eigenvalues On stability radii of generalized eigenvalue problems Pseudospectra for matrix pencils and stability of equilibria The Algebraic Eigenvalue Problem --TR --CTR Kui Du, Note on structured indefinite perturbations to Hermitian matrices, Journal of Computational and Applied Mathematics, v.202 n.2, p.258-265, May, 2007 Graillat, A note on structured pseudospectra, Journal of Computational and Applied Mathematics, v.191 n.1, p.68-76, 15 June 2006 Kui Du , Yimin Wei, Structured pseudospectra and structured sensitivity of eigenvalues, Journal of Computational and Applied Mathematics, v.197 n.2, p.502-519, 15 December 2006 Kirk Green , Thomas Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, v.196 n.2, p.567-578, 15 November 2006

structured perturbations;matrix polynomial;solvent;orr-sommerfeld equation;stability radius;pseudospectrum;quadratic matrix equation;backward error;transfer function;lambda-matrix;polynomial eigenvalue problem

587858

A Multilevel Dual Reordering Strategy for Robust Incomplete LU Factorization of Indefinite Matrices.

A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph-based strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILU threshold preconditioner.

Introduction This paper is concerned with reordering strategies used in developing robust preconditioners based on incomplete LU (ILU) factorization of the coefficient matrix of sparse linear system of the form where A is an unstructured matrix of order n. In particular, we are interested in ILU preconditioning techniques for which A is an indefinite matrix; i.e., a matrix with an indefinite symmetric part. Indefinite matrices arise frequently from finite element discretizations of coupled partial differential equations in computational fluid dynamics and from other applications. Technical Report 285-99, Department of Computer Science, University of Kentucky, Lexington, KY, 1999. This work was supported in part by the University of Kentucky Center for Computational Sciences and in part by the University of Kentucky College of Engineering. y E-mail: jzhang@cs.uky.edu. URL: http://www.cs.uky.edu/~jzhang. ILU preconditioning techniques have been successful for solving many nonsymmetric and indefinite matrices, despite the fact that their existence in these applications is not guaranteed. However, their failure rates are still too high for them to be used as blackbox library software for solving general sparse matrices of practical interests [9]. In fact, the lack of robustness of preconditioned iterative methods is currently the major impediment for them to gain acceptance in industrial applications, in spite of their intrinsic advantage for large scale problems. For indefinite matrices, there are at least two reasons that make ILU factorization approaches problematic [9]. The first problem is due to small or zero pivots [23]. Pivots in an indefinite matrix can be arbitrarily small. This may lead to unstable and inaccurate factorizations. In such cases, the size of the elements in the LU factors may be very large and these large size elements lead to inaccurate factorization. The second problem is due to unstable triangular solves [18]. The incomplete factors of an indefinite matrix are usually not diagonally dominant. An indication of unstable triangular solves is when kL are extremely large while the offdiagonal elements of L and U are reasonably bounded. Such problems are usually caused by very small pivots. They may sometimes happen without a small pivot. A statistic, condest, was introduced by Chow and Saad [9] to measure the stability of the triangular solves. It is defined to be k(LU) \Gamma1 ek1 , where e is a vector of all ones. This statistic is useful when its value is very large, e.g., in the order of 10 15 . Small pivots are usually related small or zero diagonal elements. It can be argued that by restricting the magnitude of the diagonal elements, we may be able to alleviate, if not eliminate, these two problems of ILU factorizations to a certain degree. Such restrictions can be seen in the form of full or partial pivoting strategies in Gaussian elimination. In ILU factorization, column pivoting strategy has been implemented with Saad's ILUT, resulting in an ILUTP techniques [32]. However, ILUTP has not always been helpful in dealing with nonsymmetric matrices [3, 9]. As Chow and Saad pointed [9], a poor pivoting sequence can occasionally trap a factorization into a zero pivot, even if the factorization would have succeeded without pivoting. In addition, existing pivoting strategies for incomplete factorization cannot guarantee that a nonzero pivot can always be found, unlike the case with Gaussian elimination [9]. Another obvious strategy of dealing with small pivots is to replace them by a larger value. The ILU factorization can continue and the resulting preconditioner may be well conditioned. In such a way, the ILU factorization is said to be stabilized. However, this strategy alters the values of the matrix and the resulting preconditioner may be inaccurate. Thus, the choice of the replacing value for the small pivots is critical for a good performance and a good choice is usually problem dependent [23]. Too large a value will result in a stable but inaccurate factorization; too small a value will result in an unstable factorization. A similar strategy is to factor a shifted matrix A + ffI , where ff is a positive scalar so that A+ ffI is well conditioned [27, 44]. Such a strategy too obviously has a tradeoff between stable and accurate factorization. For more studies on the stability of ILU factorizations, we refer to [19, 29, 42, 13, 45]. It is also possible to reorder the rows of the matrix so that their diagonal dominance in a certain sense is in decreasing order. In this way, small pivots are in the last rows of the matrix and may not be used in an ILU factorization. This strategy also has some problems since the values of the pivots are modified in an unpredictable way, small pivots may still affect the ILU factorization. In addition, the effect of standard reordering schemes applied to general nonsymmetric sparse matrices is still an unsettled issue [17, 24, 43]. This paper follows the above idea of putting the rows with small diagonal elements to the last few rows. However, these small diagonal elements will never be used in the ILU factorization. Instead, these rows form the rows of a Schur complement matrix and the values of the diagonal elements are modified in a systematic way. This process is continued for a few times until all small diagonal elements are removed; or until the last Schur complement matrix is small enough that a complete pivoting strategy can be implemented inexpensively. With this reordering strategy, we can expect to obtain a stable and accurate ILU factorization. We also implement a graph based reordering strategy (minimum degree algorithm) to reduce the fill-in amount during the stable ILU factorization. This paper is organized as follows. The next section introduces a dual reordering strategy based on both the values and the graph of the matrix. Section 3 discusses a partial ILU factorization technique to construct the Schur complement matrix implicitly. Section 4 gives analyses on the values of the diagonal elements of the Schur complement matrix and shows how the stability of the LU factor affects the quality of a preconditioner. Section 5 outlines the multilevel dual reordering algorithm. Section 6 contains numerical experiments. Concluding remarks are included in Section 7. Reordering Strategy Most reordering strategies are originally developed for the direct solution of sparse matrices based on Gaussian elimination. They are mainly used to reduce the fill-in elements in the Gaussian elimination process or to extract parallelism from LU factorizations [15, 22]. They have also been used in ILU preconditioning techniques for almost the same reasons [16, 20, 30]. Various reordering strategies were first studied for preconditioned conjugate gradient methods, i.e., for the cases where the matrix is symmetric positive definite [1, 4, 5, 10, 11, 26, 31]. They were then extended for treating nonsymmetric problems [2, 7, 12, 14]. Most of these strategies are based on the adjacency graph but not on the values of the matrices. They are robust for general sparse matrices only if used with suitable pivoting strategies, which are based on the values of the matrices, to prevent unstable factorizations. Hence, reordering strategies based on matrix values are needed to yield robust stable ILU factorizations. Such an observation has largely been overlooked in ILU techniques for some time, partly because the early ILU techniques were mainly developed to solve sparse matrices arising from finite difference discretizations of partial differential equations [28]. In such cases, the diagonal elements of the matrices usually have nonzero values. In this paper, we introduce a dual reordering strategy for robust ILU factorization for solving general sparse indefinite matrices. To this end, we first introduce a strategy to determine the row diagonal dominance of a matrix. 1 We actually compute a certain measure to determine the relative strength of the diagonal element with respect to a certain norm of the row in question. Algorithm 2.1 is an example of computing a diagonal dominance measure for each row of the matrix and was originally introduced in [41] as a diagonal threshold strategy in a multilevel ILU factorization. 1 The reference to row diagonal dominance is due to the assumption that our matrix is stored in a row oriented format, such as in the compressed sparse row format [34]. The proposed strategy works equally well if the matrix is stored in a column oriented format with the reference to column diagonal dominance. Algorithm 2.1 Computing a measure for each row of a matrix. 1. For do 2. r 3. If r i 6= 0, then 4. ~ t 5. End if 6. End do 7. 8. For do 9. 10. End do In Line 2 of the Algorithm 2.1 the set defined as i.e., the nonzero row pattern for the row i. A row with a small absolute diagonal value will have a small t i measure. A row with a zero diagonal value will have an exact zero t i measure. denote the adjacency graph of the matrix A, where is the set of vertices and K is the set of edges. Let (v denote an edge from vertex v j to vertex v k . Since a node in the adjacency graph of a matrix corresponds to a row of the matrix, we will use the term node and row of a matrix interchangeably. Given a diagonal threshold tolerance ffl ? 0, we divide the nodes of A into two parts, V 1 and V 2 , such that It is obvious that For convenience, we assume that a symmetric permutation is performed so that the nodes in V 1 are listed first, followed by the nodes in V 2 . Since the nodes in V 1 are "good" for ILU factorization in terms of stability, we may further improve the quality of the ILU factorization by implementing a graph based reordering strategy. The following minimum degree reordering algorithm is just one example of such graph based reordering strategies to reduce the fill-in elements in the ILU factorization We denote by deg(v i ) the degree of the node v i , which equals the number of nonzero elements of the ith row minus one; i.e., The set of the degrees of the rows of the matrix A can be conveniently computed when Algorithm 2.1 is run to compute the diagonal dominance measure of A. For example, in Line 2 of Algorithm 2.1, the number of nonzero elements of the ith row will be counted. After the first reordering based on the threshold tolerance ffl, we perform a second reordering based on the degrees of the nodes. But the second reordering is only performed with respect to the nodes in V 1 . To be more precise, we reorder the nodes in V 1 in a minimum degree fashion; i.e., the nodes with smaller degrees are listed first, those with larger degrees are listed last. After the two steps of reorderings, we have are the permutation matrices corresponding to the threshold tolerance reordering and the minimum degree reordering, respectively. We use P g here to emphasize that it is just a graph based reordering strategy, and is not necessarily restricted to the minimum degree reordering. Other graph based reordering strategies such as the Cuthill-McKee or reverse Cuthill-McKee algorithms [25] may be used to replace the minimum degree strategy. But their meaning may be slightly changed since not all neighboring nodes of a node in V 1 belong to V 1 , some of them may be in V 2 . For simplicity, we use A to denote both the original and the permuted matrices in the sequel so that the permutation matrices will no longer appear explicitly. We also refer to the two reordering strategies as threshold reordering and graph reordering for short. 3 Partial ILU Factorization An incomplete LU factorization process with a double dropping strategy (ILUT) is first applied to the upper part (D F ) of the reordered matrix A in (2). The ILUT algorithm uses two parameters and - to control the amount of fill-in elements caused by the Gaussian elimination process and is described in detail in [32]. ILUT builds the preconditioning matrix row by row. For each row of the LU factors, ILUT first drops all computed elements whose absolute values are smaller than - times the average nonzero absolute values of the current row. After an (incomplete) row is computed, ILUT performs a search with respect to the computed current row such that the largest p elements in absolute values are kept, the rest nonzero elements are dropped again. Thus the resulting ILUT factorization has at most p elements in each row of the L and U parts. The use of a double dropping strategy ensures that the memory requirement be met. It is easy to see that the total storage cost for ILUT is bounded by 2pn for a matrix of order n. The ILUT process is continued to the second part of the matrix A in (2) with respect to the C) submatrix. However, the elimination process is only performed with respect to the columns in E, and linear combinations for columns in C are performed accordingly. In other words, the elements corresponding to the C submatrix are not eliminated. Such a process is called a partial Gaussian elimination or a partial LU factorization in [38]. Note that, due to the partial Gaussian elimination, all rows in the (E C) submatrix can be processed independently (in parallel). This is because all nodes in the E submatrix that are to be eliminated use only the computed (I)LU factorization of the (D F ) part. Note also that the diagonal values of the rows of the C submatrix are never used as pivots. It can be shown [38] that such a partial Gaussian elimination process modifies C into the (incomplete) Schur complement of A. In exact arithmetic, C would be changed into where LU is the standard LU factorization of the D submatrix. Hence, this method constructs the Schur complement indirectly, in contrast to some alternative methods, e.g., the BILUM preconditioner in [37], in which the Schur complement is constructed explicitly by matrix-matrix multiplications. The partial ILU factorization process just described yields a block LU factorization of the matrix A of the form ' D F where I and 0 are generic identity and zero matrices, respectively. If the factorization is exact and if we can solve the Schur complement matrix A 1 , the solution of the original linear system (1) can be found by a backward substitution. This process is similar to the sparse direct solution method based on one step cyclic reduction technique [22]. The partial ILU factorization process is the backbone of a domain based multilevel ILU pre-conditioning technique (BILUTM) described in [38]. Such an ILU factorization with a suitable block independent set ordering yields a preconditioner (BILUTM) that is highly robust and possesses high degree of parallelism. However, in this paper, the parallelism due to block independent set ordering is not our concern, we restrict our attention to the robustness of multilevel ILU factorization resulting from removing small pivots. We can heuristically argue that the ILU factorization resulting from applying the above partial ILU factorization to the reordered matrix is likely to be more stable than that would be generated by applying ILUT directly to the original matrix. This is because the factorization is essentially performed with respect to the nodes in V 1 that have a relatively good diagonal dominance. The partial ILU factorization with respect to the nodes in V 2 never needs to divide any pivot elements. So there is no reason that large size elements should be produced. As remarked previously, if we can solve the Schur complement matrix A 1 in (3) to a certain degree, we can develop a two level preconditioner for the matrix A. An alternative is based on the observation that A 1 is another sparse matrix and we can apply the same procedures to A 1 that have been applied to A to yield an even smaller Schur complement A 2 . This is the philosophy of multilevel ILU preconditioning techniques developed in [33, 37, 38]. However, for this moment, we only discuss the possible construction of a two level preconditioner. A two level preconditioner. The easiest way to construct a two level preconditioner is to apply the ILUT factorization technique to the matrix A 1 . One question will be naturally asked: is the ILUT factorization more stable when applied to A 1 than when applied to A? Notice that since the nodes with good diagonal dominance have all been factored out, we tend to think that the nodes of A 1 are not good for a stable ILUT factorization. This may not always be true, since the measure of diagonal dominance computed in Algorithm 2.1 is relatively to a certain norm of the row in question. We need to examine relative changes in size of the diagonal value when a node is considered as a node in A and when it is considered as a node in A 1 . 4 Analyses Diagonal submatrix D. For the easy of analysis, unless otherwise indicated explicitly, we assume that the partial LU factorization described above is exact; i.e., no dropping strategy is enforced. We also assume that, in the reordered matrix, the D submatrix is diagonal. Such a reordering can be achieved by an independent set search as in a multielimination strategy of Saad [33, 39]. Thus, the factorization (4) is reduced to ED '' D F We now assume that all indices are local to individual submatrices. In other words, when we say the ith row of the matrix F , we mean the ith row of the submatrix F , not the ith row of the matrix A, original or permuted. For convenience we assume that D is of dimension m and A 1 is of dimension m. We also use the notations: It can be shown [22, 38] that, with the partial LU factorization without dropping, an arbitrary element of the Schur complement matrix A 1 is: Since we assume that the nodes with large diagonal dominance measure are in V 1 and the nodes in have small or zero diagonal dominance measure, we are interested in knowing how the diagonal value of a node of A may change when it becomes a node in A 1 . row i F e ik rowk l l Figure 1: An illustration of the partial LU factorization to eliminate e ik in the E submatrix. The following proposition is obvious from Equation (6) and from Figure 1. Proposition 4.1 If either the jth column of the submatrix F or the ith row of the submatrix E is a zero vector, then s Definition 4.2 A node v i of the vertex set V is said to be independent to a subset V I of V if and only if An immediate consequence of the independentness is the following corollary that is first proved in [39]. Corollary 4.3 If a node v i in V 2 is independent to all the nodes in V 1 , then s i.e., the values of the ith row of C will not be modified in the partial LU factorization. We now modify our threshold tolerance reordering strategy slightly to a diagonal threshold strategy, similar to that discussed in [39]. We assume that the node v i is in V 1 if ja ii j - ffl and D is still a diagonal matrix. With such a modification, we have jd m. Denote by the size of the largest elements in absolute value of A. Proposition 4.4 The size of the elements of the Schur complement matrix A 1 is bounded by M(1+ mM=ffl). Proof. Starting from Equation (6): +mM=ffl):Proposition 4.4 shows that the size of the elements of the Schur complement matrix cannot grow uncontrollably if ffl is large enough. This result indicates that our first level (I)LU factorization is stable. As we hinted previously, we will be interested in recursively applying our strategy to the successive Schur complement matrices. We may assume that the matrix A is presparsified so that small nondiagonal elements are removed. To be more specific, for the parameter - used in the ILUT fac- torization, we assume min 1-i;j-n fja ij jg - for all nonzero elements of A, except for possibly the diagonal elements. With some additional assumptions, we can have a lower bound on the variation of the diagonal values of the Schur complement matrix A 1 . Proposition 4.5 Suppose ja ij j - for all nonzero elements of the matrix A, and suppose that either c ik f ki =d k - 0 or c ik f ki =d k - 0 holds for all 1 - k - m. Then card where are the index sets of the nonzero elements of the ith row of the E submatrix and the ith column of the F submatrix, respectively. card(V ) denotes the cardinality of a set V . Proof. If either c ik f ki =d k - 0 or c ik f ki =d k - 0 holds for all 1 - k - m, we have The kth term in the right-hand side sum of (7) is nonzero if and only if both e ik and f ki are nonzero. This happens if and only if k 2 Nz(E Note that jc ik j -; jf ki j - and jd k m. It follows that card :It is implicitly assumed that ffl ! M . In practice, ffl is small so that the set V 1 may be large enough to avoid constructing a large Schur complement matrix. Denote card By the motivation of the diagonal threshold strategy, the value of jc ii j is zero or very small. Thus the size of js ii j can be considered as being close to \Delta i . Corollary 4.6 Under the conditions of the Proposition 4.5, if c Corollary 4.6 shows that if the ith diagonal element of A 1 is zero in A and if the set Nz(E i is nonempty, then the size of the ith diagonal element is nonzero in the Schur complement. Thus, under these conditions, a zero pivot is removed. In fact, the cardinality of Nz(E seems to be the key factor to remove zero diagonal elements. It is difficult to derive more useful bounds for general sparse matrices. If certain conditions are given to restrict the class of matrices under consideration, it is possible to obtain more realistic bounds to characterize the size of the elements of the Schur complement matrix, especially the size of its diagonal elements. General submatrix D. For general submatrix D corresponding to the factorization (4), it is easy to see that, if the jth column of the submatrix F is zero, the jth column of the submatrix L \Gamma1 F is zero. Hence, Proposition 4.1 carries over to the general case. At this moment, we are unable to show results analogous to Propositions 4.4 and 4.5 for general submatrix D. However, it can be argued heuristically that, if D is not a diagonal matrix, the cardinality of the set Nz(E i likely to be larger than that of Nz(E Size of k(LU) \Gamma1 k. Let us consider the quality of preconditioning in a nonstandard way. Denote by the error (residual) matrix of the ILU factorization. At each iteration, the preconditioning step solves for - w the system where r is the residual of the current iterate. In a certain sense, we can consider - w as an approximate to the correction term of the current iterate. The quality of the preconditioning step (9) can be judged by comparing (9) with the exact or perfect preconditioning step If Equation (10) could be solved to yield the exact correction term w, the preconditioned iterative method would converge in one step. Of course, solving the Equation (10) is as hard as solving the original system (1). However, we can measure the relative difference in the correction term when approximating the Equation (10) by the Equation (9). This difference may tell us how good the preconditioning step (9) approximates the exact preconditioning step (10). The following proposition is motivated by the work of Kershaw [23]. Proposition 4.7 Suppose the matrix A and the factor LU from the incomplete LU factorization are nonsingular, the following inequality holds: wk kwk for any consistent norm k \Delta k. Proof. It is obvious that r 6= 0, otherwise the iteration would have converged. The nonsingularity of A implies that w 6= 0. Note that - Equation (10), we have It follows that, for any consistent norm, The desired result (11) follows immediately by dividing kwk on both sides. 2 It is well known that the size of the error matrix E directly affects the convergence rate of the preconditioned iterative methods [16]. Proposition 4.7 shows that the quality of a preconditioning step is directly related to the size of both (LU) \Gamma1 and R. A high quality preconditioner must be accurate; i.e., it must have an error matrix that is small in size. A high quality preconditioner must also have a stable factorization and stable triangular solves; i.e., the size of (LU) \Gamma1 must be small. Since the condition estimate, condest is a lower bound for k(LU) \Gamma1 k1 , it should provide some information about the quality of the preconditioner and may be used to measure the stability of the LU factorization and of the triangular solves. 5 Multilevel Dual Reordering and ILU Factorization Based on our previous analyses, the size of a diagonal element of the matrix A 1 is likely to be larger than that of the same element in A. 2 We can apply Algorithm 2.1 to A 1 and repeat on A 1 the procedures that were applied to A. This process may be repeated for a few times until 2 This is obviously false for an M-matrix. However, there will be no Schur complement matrix at all if A is an all small diagonal elements are modified to large values, or until the last Schur complement matrix is small enough that an ILU factorization with a complete pivoting strategy can be implemented inexpensively. Since the number of small or zero pivots in the last Schur complement matrix is small, a third strategy is to replace them by a large value. This will not introduce too much error to the overall factorization. Given a maximum level L and denote A the multilevel dual reordering strategy and ILU factorization can be formulated as Algorithm 5.1. Algorithm 5.1 Multilevel dual reordering and ILU factorization. 1. Given the parameters -; p; ffl; L 2. For 3. Run Algorithm 2.1 with ffl to find permutation matrices and P jg 4. Perform matrix permutation A 5. If no small pivot has been found, then 6. Apply ILUT(p; -) to A j and exit 7. Else 8. Apply a partial ILU factorization to A j 9. to yield a Schur complement matrix A j+1 10. End if 11. End do 12. Apply ILUTP or a stabilized ILUT to AL if AL exists The ILU preconditioner constructed by Algorithm 5.1 is structurally similar to the BILUTM preconditioner in [38]. The difference is that we do not construct a block independent set for the D j submatrix. Instead, we set up a diagonal measure constraint and employ a graph reordering scheme to reduce fill-in. The emphasis of this paper is on solving indefinite matrices by removing small pivots. It can be seen, if L levels of reduction are performed, the resulting ILU preconditioner has the following The application of the preconditioner can be done by a level by level forward elimination, followed by a level by level backward substitution. There are also permutations and inverse permutations to be performed, specific procedures depend on implementations. For detailed descriptions, we refer to [37, 38]. 6 Numerical Experiments Standard implementations of multilevel preconditioning methods have been described in detail in [33, 37, 38]. We used full GMRES as the accelerator [35]. We tested three preconditioners: standard ILUT of [32], a column pivoting variant ILUTP [32], and the multilevel dual reordering preconditioner designed in this paper, abbreviated as MDRILU (multilevel dual reordering ILU factorization). All preconditioners used a safeguard (stabilization) procedure by replacing a zero pivot with (0:0001+-)r i , where r i was computed as the average nonzero values of the row in question. They were used as right preconditioners for GMRES [34]. The main parameters used in all three preconditioners are the pair (p; -) in the double dropping strategy. ILUTP needs another parameter 0 - oe - 1 to control the actual pivoting. A nondiagonal element a ij is a candidate for a permutation only when oeja It is suggested that reasonable values of oe are between 0:5 and 0:01, with 0:5 being the best in many cases [34, p. 295]. MDRILU also needs another parameter ffl to enforce the diagonal threshold reordering as in Algorithm 5.1. The maximum possible level number in MDRILU was levels of dual reorderings the Schur complement A 10 is not empty, a stabilized ILUT factorization was employed to factor A 10 . 3 For all linear systems, the right-hand side was generated by assuming that the solution is a vector of all ones. The initial guess was a vector of some random numbers. The iteration was terminated when the 2-norm of the residual was reduced by a factor of 10 7 . We also set an upper bound of 100 for the full GMRES iteration. A symbol"-" indicates lack of convergence. In all tables with numerical results, "iter" shows the number of preconditioned GMRES iter- ations; "spar" shows the sparsity ratio which is the ratio between the number of nonzero elements of the preconditioner to that of the original matrix; "prec" shows the CPU time in seconds spent in constructing the preconditioners; is the condition estimate of the preconditioners as introduced in Section 1. Since these ILU preconditioners approach direct solvers as robustness with respect to the memory cost (sparsity ratio). We remark that our codes were not optimized and they computed and outputed information such as the number of zero diagonals, smallest pivots, ets. Consequently, the CPU times reported in this paper only have relative meaning. Note that the solution time at each iteration is mainly the cost of the matrix (both A and the preconditioner) vector products and is thus proportional to the product of the iteration count and the sparsity ratio, i.e., solution time The numerical experiments were conducted on a Power-Challenge XL Silicon Graphics workstation equipped with 512 MB of main memory, one 190 MHz R10000 processor, and 1 MB secondary cache. We used Fortran 77 programming language in 64 bit arithmetic computation. Test matrices. Three test matrices were selected from different applications. Table 1 contains simple descriptions of the test matrices. They have been used in several other papers [6, 9, 39, 46]. None of the three matrices has a zero diagonal. Matrix order nonzeros description buckling problem for container model simulation WIGTO966 3 864 238 252 Euler equation model Table 1: Simple descriptions of the test matrices. 3 We found stabilized ILUT was better than ILUTP for solving the last system. We did not implement an ILUT factorization with a full pivoting strategy. WIGTO966 matrices. The WIGTO966 matrix 4 was supplied by L. Wigton from Boeing Com- pany. It is solvable by ILUT with large values of p [6]. This matrix was also used to compare BILUM with ILUT in [36], and BILUTM with ILUT in [38], and to test point and block preconditioning techniques in [8, 9]. Since ILUT requires very large amount of fill-in to converge, the WIGTO966 matrix is ideal to test alternative preconditioners and to show the least memory that is required for convergence. For example, BILUM (with GMRES(10)) was shown to be 6 times faster than ILUT with only one-third of the memory required by ILUT [36]. BILUTM (with GMRES(50)) converged almost 5 times faster and used just about one-fifth of the memory required by ILUT [38]. Table 2 lists results from several runs to compare MDRILU and ILUT. It shows that MDRILU could converge with low sparsity ratios, as low as 0:94. The threshold parameter ffl was in a fixed range when the other parameters p and - changed. For all the values of p and - tested in Table 2, ILUT did not converge. We found that there was no very small pivot, the size of the smallest pivot in all tests in Table was 1.19e-5. But the condition estimates for ILUT were very large, the smallest condest value is 1.1e+82, indicating unstable triangular solves had resulted during the factorization and solution processes. MDRILU ILUT iter prec spar cond iter prec spar cond 50 1.0e-3 0.38 27 4.92 2.17 4.4e+4 - 9.9e+116 50 1.0e-4 0.38 25 7.48 2.55 2.7e+4 - 2.7e+91 Table 2: Comparison of MDRILU and ILUT for solving the WIGTO966 matrix. We further compared ILUTP and ILUT and list the results in Table 3. We see that ILUTP is more robust than ILUT for solving the WIGTO966 matrix. ILUT required high sparsity ratios to converge. For those cases, ILUTP was able to converge with fewer iterations. When we chose failed to converge, but ILUTP converged in 49 iteration with a sparsity ratio 3:06. Notice that both ILUTP and ILUT did not converge with MDRILU could converge with these parameters. We point out that the condition estimates of ILUTP are much smaller than those of ILUT. This implies that ILUTP did stabilize the ILU factorization process with a column pivoting strategy, although there was no very small pivot in the factorization. The results of Table 3 also show that the additional cost of implementing ILUTP is not high in this test. However, as far as solving the WIGTO966 matrix is concerned, computing an MDRILU preconditioner is much cheaper than computing either an ILUT or an ILUTP preconditioner. RAEFSKY4 matrices. The RAEFSKY4 matrix 5 was supplied by H. Simon from Lawrence Berkeley National Laboratory (originally created by A. Raefsky from Centric Engineering). This is 4 The WIGTO966 matrix is available from the author. 5 The RAEFSKY4 matrix is available online from the University of Florida Sparse Matrix Collection at http://www.cise.ufl.edu/~davis/sparse. ILUTP ILUT iter prec spar cond iter prec spar cond 100 1.0e-4 0.50 34 22.90 3.08 2.3e+5 - 3.0e+69 300 1.0e-3 0.10 9 44.98 7.39 1.2e+4 74 51.52 7.91 1.3e+8 Table 3: Comparison of ILUTP and ILUT for solving the WIGTO966 matrix. probably the hardest one in the total of 6 RAEFSKY matrices. Figure 2 shows the convergence history of three preconditioners with 1.0e-4. The other parameters were for MDRILU and oe = 0:03 for ILUTP. We see that both ILUT and ILUTP did not have much convergence in 100 iterations; MDRILU converged in 13 iterations. 2-norm residual iterations RAEFSKY4 Matrix DRILUDRILU (dashed line) ILUTP (dashdot line) ILUT (solid line) Figure 2: Convergence history of preconditioned GMRES for solving the RAEFSKY4 matrix. In Figure 3 we plotted the iteration counts (left part) and the values of condition estimate (right part) of the MDRILU preconditioner with different values of the threshold parameter ffl, keeping We found that the iteration count and the condition estimate were linked to each other. A large value of condition estimate is usually accompanied by a large iteration count of MDRILU. We also see that the convergence rates of MDRILU are not very sensitive to the choice of the value of ffl. For 0:38 - ffl - 0:78, MDRILU gave very similar performance. UTM5940 matrix. The UTM5940 matrix 6 is the largest matrix from the TOKAMAK collection and was provided by P. Brown of Lawrence Livermore National Laboratory. Table 4 contains a few runs with MDRILU and ILUT with different sparsity ratios. It is clear that MDRILU is more efficient than ILUT when the sparsity ratios are low. The results are also consistent with other test results, 6 The UTM5940 matrix is available from online the MatrixMarket of the National Institute of Standards and Tech- RAEFSKY4 Matrix iterations epsilon 0.850015002500condest epsilon RAEFSKY4 Matrix Figure 3: Iteration counts (left) and condition estimates (right) of MDRILU with different values of ffl for solving the RAEFSKY4 matrix. indicating that MDRILU is able to solve this problem with less storage cost than ILUT. If sufficient memory space is available, ILUT may be efficient in certain cases. Note that if both MDRILU and ILUT converge with similar iteration counts, MDRILU is more expensive to construct than ILUT. MDRILU ILUT iter prec spar cond iter prec spar cond 50 1.0e-4 0.30 42 7.49 6.26 2.2e+7 86 3.67 5.72 1.3e+7 Table 4: Comparison of MDRILU and ILUT for solving the UTM5940 matrix. Figure 4 shows the convergence history of MDRILU with different values of dropping tolerance - to solve the UTM5940 matrix, keeping We note that the number of iterations did not change very much when - changed from 1.0e-2 to 1.0e-5 and the sparsity ratio changed from 2:67 to 4:15. It seems that MDRILU worked quite well with a relatively strict dropping tolerance. FIDAP matrices. The FIDAP matrices 7 were extracted from the test problems provided in the FIDAP package [21]. They were generated by I. Hasbani of Fluid Dynamics International and B. Rackner of Minnesota Supercomputer Center. The matrices were resulted from modeling the incompressible Navier-Stokes equations and were generated using whatever solution method was specified in the input decks. However, if the penalty method was used, there is usually a corresponding 7 All FIDAP matrices are available online from the MatrixMarket of the National Institute of Standards and Tech- 2-norm residual iterations Matrix solid line: dashed line: dashdot line: dotted line: Figure 4: Convergence history of MDRILU with different values of dropping tolerance - for solving the UTM5940 matrix. FIDAPM matrix, which was constructed using a fully coupled solution method (mixed u-p formula- tion). The penalty method gives very ill conditioned matrices, whereas the mixed u-p method gives indefinite, larger systems (they include pressure variables). Many of these matrices contain small or zero diagonal values. 8 The zero diagonals are due to the incompressibility condition of the Navier-Stokes equations [9]. The substantial amount of zero diagonals makes these matrices indefinite. It is remarked in [6] that the FIDAP matrices are difficult to solve with ILU preconditioning techniques, which require high level of fill-in to be effective and the performance of the preconditioners is unstable with respect to the amount of fill-in. Many of them cannot be solved by the standard BILUM preconditioner and in some cases, even the construction of BILUM failed due to the occurrence of very ill conditioned blocks. Nevertheless, some of them may be solved by the enhanced version of BILUM using singular value decomposition based regularized inverse technique and variable block size [40]. The details of all of the largest 31 FIDAP matrices (n ? 2000) are listed in Table 5 and the corresponding test results are given in Table 6. The second column of Table 6 lists the number of zero diagonals of the given matrix. In our tests, we first set 0:5; 0:3; 0:1; 0:01. If none of these ffl values showed any promise, we increased the p value or decreased the - value. If for a given pair of (p; - ), MDRILU with a certain value of ffl converged or showed some convergence, we adjusted the value of ffl to get improved convergence rates if possible. However, there was no effort made to find the best parameters. We stopped refining the parameters when we found the iteration count was reasonable and the sparsity ratio was not high, or the computations took too much time in case of large matrices. Once MDRILU was tested, the same pair (p; -) was used to test ILUTP and ILUT. For ILUTP, we varied the value of oe analogously to what we did to choose the value of ffl. Table 6 shows that MDRILU can solve 27 out of the 31 largest FIDAP matrices. To the best of 8 The FIDAP matrices have structural zeros added on the offdiagonals to make them structurally symmetric. Structural zeros were also added to the diagonals. Matrix order nonzeros description developing flow in a vertical channel impingment cooling flow over multiple steps in a channel flow in lid-driven wedge FIDAP015 6 867 96 421 spin up of a liquid in an annulus turbulent flow over a backward-facing step developing pipe flow, turbulent attenuation of a surface disturbance coating convection two merging liquids with an interior interface turbulent flow in axisymmetric U-bend species deposition on a heated plate FIDAP035 19 716 218 308 turbulent flow in a heated channel FIDAP036 3 079 53 851 chemical vapor deposition flow of plastic in a profile extrusion die flow past a cylinder in free stream natural convection in a square enclosure developing flow in a vertical channel impingment cooling flow over multiple heat sources in a channel FIDAPM11 22 294 623 554 3D steady flow, head exchanger FIDAPM15 9 287 98 519 spin up of a liquid in an annulus turbulent flow is axisymmetric U-bend radiation heat transfer in a square cavity FIDAPM37 9 152 765 944 flow of plastic in a profile extrusion die Table 5: Description of the largest 31 FIDAP matrices. our knowledge, this is the first time that so many FIDAP matrices were solved by a single iterative technique. (20 were solved in [40], in [46], 9 in [39], and 8 in [9].) In Table 6 the term "unstable" means that convergence was not reached in 100 iterations and the condition estimate was greater than Similarly the term "inaccurate" means that convergence was not reached, but the condition estimate did not exceed 10 15 . They are categorized according to Chow and Saad's arguments [9]. We remark that the results of "inaccurate" or "unstable" in Table 6 do not indicate that ILUT or ILUTP can or cannot solve the given matrices with different parameters. The results only mean that they did not converge with the parameters that made MDRILU converge. It is worth pointing out that, in several tests, we observed that ILUTP encountered zero pivots when ILUT did not. Although we allowed 10 levels of maximum dual reorderings to be performed, there were very few cases that 10 levels of reorderings were actually needed. In most cases, 3 to 4 levels of dual reorderings were performed for the FIDAP matrices. In many cases, the first Schur complement MDRILU ILUTP ILUT Matrix zero-d p - ffl iter spar iter spar iter spar unstable unstable unstable unstable FIDAP026 457 20 1.0e-4 0.30 84 0.77 unstable unstable FIDAP036 504 20 1.0e-4 0.10 23 1.75 83 1.91 unstable FIDAPM07 432 300 1.0e-4 0.20 78 6.66 80 7.71 inaccurate FIDAPM08 780 20 1.0e-4 0.10 25 1.70 78 2.22 unstable unstable unstable 43 7.51 21 7.37 unstable 28 14.38 11 7.47 13 7.46 43 3.61 unstable unstable Table Solving the FIDAP matrices by MDRILU, ILUTP and ILUT. matrix did not have any zero diagonal, even if the original matrix A did have many zero diagonals. We listed in Table 7 those matrices that did have zero diagonals in their Schur complement matrices. For all the FIDAP matrices solved by MDRILU, only the FIDAP026 matrix had 12 zero diagonals in the last Schur complement A 5 . The test results show that the multilevel dual reordering strategy does have the effect of removing small and zero pivots from ILU factorizations. Remarks. Ironically, the four matrices, FIDAP011, FIDAP015, FIDAP018, and FIDAP035, that were not solved by MDRILU do not have any zero diagonals. They may be solved by ILUT with small values of - . Some of them may even be solved by GMRES without preconditioning if enough iterations are allowed. We think this is because these matrices are very nonsymmetric and the preconditioned matrices were worse conditioned than the original matrices, causing GMRES iteration to converge extremely slowly. One of our strong feeling in these numerical experiments is that, in general, MDRILU does not seem to work well when - is very small. Large values of p usually improve convergence. This observation can be seen in Figure 5 which depicts the convergence history of Matrix A 0 A 1 A 3 A 4 A 5 Table 7: Number of zero diagonals in the Schur complement matrices. MDRILU for solving the largest FIDAP matrix, FIDAPM11. We used tested two values of 1.0e-3. It is clear that more accurate (in terms of dropping tolerance) ILU factorization does not help and sometimes hampers convergence. Good values for the parameter ffl are between 0:1 and 0:5. For most problems, the performance of MDRILU is not very sensitive to the choice of ffl, as long as it is in the range of 0:1 and 0:5. 2-norm residual iterations FIDAPM11 Matrix solid line: dashed line: Figure 5: Convergence history of MDRILU with different values of dropping tolerance - for solving the FIDAPM11 matrix. 7 Conclusion We have proposed a multilevel dual reordering strategy for constructing robust ILU preconditioners for solving general sparse indefinite matrices. This reordering strategy is combined with a partial ILU factorization procedure to construct recursive Schur complement matrices. The preconditioner is a multilevel ILU preconditioner. However, the constructed preconditioner (MDRILU) is different from all existing multilevel preconditioners in a fundamental concept [37, 47]. MDRILU never intends to utilize any traditional multilevel property, it uses the Schur complement approach solely for the purpose of removing small pivots. We conducted analyses on simplified model problems to find out how the size of the small diagonal elements and other elements is modified when these elements become the elements of the Schur complement matrix. We gave an upper bound for the size of general elements of the Schur complement matrix to show that their size will not grow uncontrollably if a suitable threshold reordering based on the diagonal dominance measure is implemented. We also showed that under certain conditions, a zero or very small diagonal element is likely to be modified to favor a stable ILU factorization by the Schur complement procedure. We further studied the quality of a preconditioning step. We showed that the quality of a preconditioning step is directly related to the size of both (LU) \Gamma1 and R (the error matrix). Hence, a high quality preconditioner must have a stable ILU factorization and stable triangular solves, as well as a small size error matrix. In other words, both accuracy and stability affect the quality of a preconditioner. We performed numerical experiments to compare MDRILU with two popular ILU precondi- tioners. Our numerical results show that MDRILU is much more robust than both ILUT and ILUTP for solving most indefinite matrices under current consideration. The most valuable advantage of MDRILU is that it can construct a sparse high quality preconditioner with low storage cost. The preconditioners computed by MDRILU are more stable than those computed by ILUT and ILUTP, thanks to the ability of MDRILU to remove (not replace) the small diagonal values. Both analytic and numerical results strongly support our conclusion that the multilevel dual reordering strategy developed in this paper is a very useful strategy to construct robust ILU preconditioners for solving general sparse indefinite matrices. Due to the time and space limit, we have not tested other graph reordering algorithms in the multilevel dual reordering algorithm. Some of the popular reordering strategies such as Cuthill-McKee and reverse Cuthill-McKee algorithms may be useful in such applications to further improve quality of the ILU preconditioner. However, we fell the robustness of MDRILU is mainly a result of using threshold tolerance reordering strategy and partial ILU factorization to remove small pivots. The difference arising from using different graph algorithm may be significant in terms of the number of iterations. But such a difference is unlikely to alter the stability problem in a systematic manner in the ILU factorization. --R Comparison of fast iterative methods for symmetric systems. Incomplete factorization methods for fully implicit simulation of enhanced oil recovery. Orderings for incomplete factorization preconditioning of nonsymmetric problems. An incomplete-factorization preconditioning using red-black ordering Parallel elliptic preconditioners: Fourier analysis and performance on the Connection machine. On preconditioned Krylov subspace methods for discrete convection-diffusion problems An object-oriented framework for block preconditioning Experimental study of ILU preconditioners for indefinite matrices. Weighted graph based ordering techniques for preconditioned conjugate gradient methods. Ordering methods for preconditioned conjugate gradient methods applied to unstructured grid problems. SOR as a preconditioner. Stability and spectral properties of incomplete factorization. On parallelism and convergence of incomplete LU factorizations. Direct Methods for Sparse Matrices. The effect of reordering on preconditioned conjugate gradients. The effect of reordering on the preconditioned GMRES algorithm for solving the compressible Navier-Stokes equations A stability analysis of incomplete LU factorization. Relaxed and stabilized incomplete factorization for nonselfadjoint linear systems. Ordering techniques for the preconditioned conjugate gradient method on parallel computers. FIDAP: Examples Manual Computer Solution of Large Sparse Positive Definite Systems. On the problem of unstable pivots in the incomplete LU-conjugate gradient method Conjugate gradient methods and ILU preconditioning of non-symmetric matrix systems with arbitrary sparsity patterns Comparative analysis of the Cuthill-McKee and the reverse Cuthill-McKee ordering algorithms for sparse matrices Ordering strategies for modified block incomplete factorizations. An incomplete factorization technique for positive definite linear systems. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix On the stability of the incomplete LU-factorization and characterizations of H-matrices Ordering methods for approximate factorization preconditioning. Orderings for conjugate gradient preconditionings. ILUT: a dual threshold incomplete LU preconditioner. ILUM: a multi-elimination ILU preconditioner for general sparse matrices Iterative Methods for Sparse Linear Systems. GMRES: a generalized minimal residual method for solving non-symmetric linear systems Domain decomposition and multi-level type techniques for general sparse linear systems BILUM: block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems. BILUTM: a domain-based multi-level block ILUT preconditioner for general sparse matrices Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems A multi-level preconditioner with applications to the numerical simulation of coating problems On the stability of the incomplete Cholesky decomposition for a singular perturbed problem Incomplete LU preconditioners for conjugate-gradient-type iterative methods Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from PDE-problems Stabilized incomplete LU-decompositions as preconditionings for the Tchebycheff iteration Preconditioned Krylov subspace methods for solving nonsymmetric matrices from CFD applications. A grid based multilevel incomplete LU factorization preconditioning technique for general sparse matrices. --TR --CTR Wang , Jun Zhang, A new stabilization strategy for incomplete LU preconditioning of indefinite matrices, Applied Mathematics and Computation, v.144 n.1, p.75-87, 20 November 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, Jeonghwa Lee , Jun Zhang , Cai-Cheng Lu, Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems, Journal of Computational Physics, v.185 n.1, p.158-175, 10 February 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 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 Michele Benzi, Preconditioning techniques for large linear systems: a survey, Journal of Computational Physics, v.182 n.2, p.418-477, November 2002

sparse matrices;multilevel incomplete LU preconditioner;incomplete LU factorization;reordering strategies

587861

An Implicitly Restarted Symplectic Lanczos Method for the Symplectic Eigenvalue Problem.

An implicitly restarted symplectic Lanczos method for the symplectic eigenvalue problem is presented. The Lanczos vectors are constructed to form a symplectic basis. The inherent numerical difficulties of the symplectic Lanczos method are addressed by inexpensive implicit restarts. The method is used to compute some eigenvalues and eigenvectors of large and sparse symplectic operators.

Introduction . We consider the numerical solution of the real symplectic eigenvalue problem where M 2 IR 2n\Theta2n is large and possibly sparse. A matrix M is called symplectic iff or equivalently, M T and I n is the n \Theta n identity matrix. The symplectic matrices form a group under multiplication. The eigenvalues of symplectic matrices occur in reciprocal pairs: If - is an eigenvalue of M with right eigenvector x, then - \Gamma1 is an eigenvalue of M with left eigenvector (Jx) T . The computation of eigenvalues and eigenvectors of such matrices is an important task in applications like the discrete linear-quadratic regulator problem, discrete Kalman filtering, or the solution of discrete-time algebraic Riccati equations. See, e.g., [21, 22, 28] for applications and further references. Symplectic matrices also occur when solving linear Hamiltonian difference systems [6]. In order to develop fast, efficient, and reliable methods, the symplectic structure of the problem should be preserved and exploited. Then important properties of symplectic matrices (e.g., eigenvalues occurring in reciprocal pairs) will be preserved and not destroyed by rounding errors. Different structure-preserving methods for solving have been proposed. In [25], Lin introduces the S which can be used to compute the eigenvalues of a symplectic matrix by a structure-preserving method similar to Van Loan's square-reduced method for the Hamiltonian eigenvalue problem [38]. Flaschka, Mehrmann, and Zywietz show in [14] how to construct structure-preserving methods based on the SR method [10, 11, 26]. Patel [34, 33] and Mehrmann [27] developed structure-preserving algorithms for the symplectic generalized eigenproblem submitted in July 1998 y Universit?t Bremen, Fachbereich 3 - Mathematik und Informatik, Zentrum f?r Technomathe- matik, 28357 Bremen, FRG. E-mail: benner@math.uni-bremen.de z corresponding author, Universit?t Bremen, Fachbereich 3 - Mathematik und Informatik, Zentrum f?r Technomathematik, 28357 Bremen, FRG. E-mail: heike@math.uni-bremen.de Benner and Fa-bender Recently, Banse and Bunse-Gerstner [2, 3] presented a new condensed form for symplectic matrices. The 2n \Theta 2n condensed matrix is symplectic, contains nonzero entries, and is determined by 4n \Gamma 1 parameters. This condensed form, called symplectic butterfly form, can be depicted as a symplectic matrix of the following @ @@ @7 5 Once the reduction of a symplectic matrix to butterfly form is achieved, the SR algorithm [10, 11, 26] is a suitable tool for computing the eigenvalues/eigenvectors of a symplectic matrix. The SR algorithm preserves the butterfly form in its iterations and can be rewritten in a parameterized form that works with the 4n\Gamma1 parameters instead of the (2n) 2 matrix elements in each iteration. Hence, the symplectic structure, which will be destroyed in the numerical process due to roundoff errors, can be restored in each iteration for this condensed form. An analysis of the butterfly SR algorithm can be found in [2, 4, 5]. In [2, 3] an elimination process for computing the butterfly form of a symplectic matrix is given which uses elementary unitary symplectic transformations as well as non-unitary symplectic transformations. Unfortunately, this approach is not suitable when dealing with large and sparse symplectic matrices as an elimination process can not make full use of the sparsity. Hence, symplectic Lanczos methods which create the symplectic butterfly form if no breakdown occurs are derived in [2, 4]. Given and a symplectic matrix M 2 IR 2n\Theta2n , these Lanczos algorithms produce a matrix S which satisfies a recursion of the form MS is a butterfly matrix of order 2k \Theta 2k, and the columns of S 2n;2k are orthogonal with respect to the indefinite inner product defined by J (1.3). The latter property will be called J-orthogonality throughout this paper. The residual r k+1 depends on v k+1 and w Such a symplectic Lanczos method will suffer from the well-known numerical difficulties inherent to any Lanczos method for unsymmetric matrices. In [2], a symplectic look-ahead Lanczos algorithm is presented which overcomes breakdown by giving up the strict butterfly form. Un- fortunately, so far there do not exist eigenvalue methods that can make use of that special reduced form. Standard eigenvalue methods as QR or SR algorithms have to be employed resulting in a full symplectic matrix after only a few iteration steps. A different approach to deal with the numerical difficulties of the Lanczos process is to modify the starting vectors by an implicitly restarted Lanczos process (see the fundamental work in [9, 35]); for the unsymmetric eigenproblem the implicitly restarted Arnoldi method has been implemented very successfully, see [24]). The problems are addressed by fixing the number of steps in the Lanczos process at a prescribed value k which depends upon the required number of approximate eigenvalues. J-orthogonality of the k Lanczos vectors is secured by re-J-orthogonalizing these vectors when necessary. The purpose of the implicit restart is to determine initial vectors such that the associated residual vectors are tiny. Given (1.4), an implicit Lanczos restart computes the Lanczos factorization An implicitly restarted symplectic Lanczos method 3 which corresponds to the starting vector (where p(M) 2 IR 2n\Theta2n is a polynomial) without having to explicitly restart the Lanczos process with the vector - v 1 . Such an implicit restarting mechanism is derived here analogous to the technique introduced in [4, 18, 35]. Section 2 reviews the symplectic butterfly form and some of its properties that will be helpful for analyzing the symplectic Lanczos method which reduces a symplectic matrix to butterfly form. This symplectic Lanczos method is presented in Section 3. Further, that section is concerned with finding conditions for the symplectic Lanczos method terminating prematurely such that an invariant subspace associated with certain desired eigenvalues is obtained. We will also consider the important question of determining stopping criteria. The implicitly restarted symplectic Lanczos method itself is derived in Section 4. Numerical properties of the proposed algorithm are discussed in Section 5. In Section 6, we present some preliminary numerical examples. 2. The Symplectic Butterfly Form. A symplectic matrix is called a butterfly matrix if B 11 and B 21 are diagonal, and B 12 and B 22 are tridiag- onal. Banse and Bunse-Gerstner [2, 3] showed that for every symplectic matrix M , there exist numerous symplectic matrices S such that MS is a symplectic butterfly matrix. In [2], an elimination process for computing the butterfly form of a symplectic matrix is presented (see also [4]). In [4], an unreduced butterfly matrix is introduced in which the lower right tridiagonal matrix is unreduced, that is, the subdiagonal elements of B 22 are nonzero. Using the definition of a symplectic matrix, one easily verifies that if B is an unreduced butterfly matrix, then B 21 is nonsingular. This allows the decomposition of B into two simpler symplectic matrices: I T I @@ @ is tridiagonal and symmetric. Hence parameters that determine the symplectic matrix can be read off directly. The unreduced butterfly matrices play a role analogous to that of unreduced Hessenberg matrices in the standard QR theory [2, 4, 5]. We will frequently make use of the decomposition (2.1) and will denote it by a a (2. 4 Benner and Fa-bender . d2 . dn . b2 d2 . . a2d2 . an Remark 2.1. (See [4].) a) Any unreduced butterfly matrix is similar to an unreduced butterfly matrix with b) We will have deflation if d j. Then the eigenproblem can be split into two smaller ones with unreduced symplectic butterfly matrices. Eigenvalues and eigenvectors of symplectic butterfly matrices can be computed efficiently by the SR algorithm [7], which is a QR like algorithm in which the QR decomposition is replaced by the SR decomposition. Almost every matrix A 2 IR 2n\Theta2n can be decomposed into a product A = SR where S is symplectic and R is J - triangular, that is R where all submatrices R ij 2 IR n\Thetan are upper triangular, and R 21 is strictly upper triangular [12]. In the following a matrix D 2 IR 2n\Theta2n will be called trivial if it is both symplectic and J-triangular. D is trivial if and only if it has the form where C and F are diagonal matrices, C nonsingular. If the SR decomposition A = SR exists, then other SR decompositions of A can be built from it by passing trivial factors back and forth between S and R. That is, if D is a trivial matrix, ~ R is another SR decomposition of A. If A is nonsingular, then this is the only way to create other SR decompositions. In other words, the SR decomposition is unique up to trivial factors. The SR algorithm is an iterative algorithm that performs an SR decomposition at each iteration. If B is the current iterate, then a spectral transformation function An implicitly restarted symplectic Lanczos method 5 q is chosen (such that q(B) 2 IR 2n\Theta2n ) and the SR decomposition of q(B) is formed, if possible: Then the symplectic factor S is used to perform a similarity transformation on B to yield the next iterate, which we will call b B: If is an unreduced symplectic butterfly matrix, then so is B in (2.5) [2, 3]. If rank is an unreduced symplectic butterfly matrix, then b in (2.5) is of the form (see [4]) @ @@ @ -z-z-z-z- where is a symplectic butterfly matrix and ffl the eigenvalues of are just the - shifts that are eigenvalues of B. An algorithm for explicitly computing S and R is presented in [8]. As with explicit QR steps, the expense of explicit SR steps comes from the fact that q(B) has to be computed explicitly. A preferred alternative is the implicit SR step, an analogue to the Francis QR step [15, 17, 20]. As the implicit SR step is analogous to the implicit QR step, this technique will not be discussed here (see [4, 5] for details). A natural way to choose the spectral transformation function q is to choose a polynomial these choices make use of the symmetries of the spectrum of symplectic matrices. But, as explained in [5], a better choice is a Laurent polynomial to drive the SR step. For example, instead of p 4 (-) we will use 2: This reduces the size of the bulges that are introduced, thereby decreasing he number of computations required per iteration. Moreover, the use of Laurent polynomials improves the convergence and stability properties of the algorithm by effectively treating each reciprocal pair of eigenvalues as a unit. Using a generalized Rayleigh-quotient strategy, the butterfly SR algorithm is typically cubic convergent [5]. The right eigenvectors of unreduced butterfly matrices have the following property which will be helpful when analyzing the symplectic Lanczos method introduced in the next section. 6 Benner and Fa-bender Lemma 2.2. Suppose that B 2 IR 2n\Theta2n is an unreduced butterfly matrix as in (2.4). If In order to proof this lemma we need the following definition. Let Pn be the permutation matrix If the dimension of Pn is clear from the context, we leave off the subscript. Proof. The proof is by induction on the size of B. The entries of the eigenvector x will be denoted by x Suppose that 2. The second and fourth row of a Since B is unreduced, we know that a 2 6= 0 and d 2 6= 0. If x then from (2.9) we obtain while (2.8) gives b 2 Using (2.10) we obtain x The third row of a Using since B is unreduced, we obtain x which contradicts the assumption x 6= 0. Assume that the lemma is true for matrices of order 2(n\Gamma1). Let B 2n;2n 2 IR 2n\Theta2n be an unreduced butterfly matrix. For simplicity we will consider the permuted equation B 2n;2n Partition 2n\Gamma2 an an c n5 ; ~ x ~ is an unreduced butterfly matrix and y 2 This implies since an 6= 0 as B 2n;2n is unreduced. Further we have Hence, using (2.11) we get ~ x -y. Using ~ further obtain from (2.11) y This is a contradiction, because by induction hypothesis e T Remark 2.3. If y be the right eigenvector of B to - y, then (Jy) that e T 2n y 6= 0, hence the nth component of the left eigenvector of B corresponding to - is 6= 0. An implicitly restarted symplectic Lanczos method 7 3. A Symplectic Lanczos Method for Symplectic Matrices. In this sec- tion, we review the symplectic Lanczos method to compute the butterfly form (2.4) for a symplectic matrix M derived in [4]. The usual unsymmetric Lanczos algorithm generates two sequences of vectors. Due to the symplectic structure of M it is easily seen that one of the two sequences can be eliminated here and thus work and storage can essentially be halved. (This property is valid for a broader class of matrices, see [16].) Further, this section is concerned with finding conditions for the symplectic Lanczos method terminating prematurely such that an invariant subspace associated with certain desired eigenvalues is obtained. Finally we will consider the important question of determining stopping criteria. In order to simplify the notation we use in the following permuted versions of M and B as in the previous section. Let with the permutation matrix P as in (2.7). 3.1. The symplectic Lanczos factorization. We want to compute a symplectic matrix S such that S transforms the symplectic matrix M to a symplectic butterfly matrix B; in the permuted version MS = SB yields Equivalently, as , we can consider where a a . dn 0 . The structure preserving Lanczos method generates a sequence of permuted symplectic matrices satisfying 8 Benner and Fa-bender T is a permuted 2k \Theta 2k symplectic butterfly matrix. The vector r k+1 := d k+1 (b k+1 is the residual vector and is JP - orthogonal to the columns of S 2n;2k P , the Lanczos vectors. The matrix B 2k;2k P is the -orthogonal projection of MP onto the range of S 2n;2k Here J 2k;2k P denotes a permuted 2k \Theta 2k matrix J of the form (1.3). Equation (3.5) defines a length 2k Lanczos factorization of MP . If the residual vector r k+1 is the zero vector, then equation (3.5) is called a truncated Lanczos factorization when k ! n. Note that r n+1 must vanish since (S 2n;2n and the columns of S 2n;2n form a JP -orthogonal basis for IR 2n . In this case the symplectic Lanczos method computes a reduction to permuted butterfly form. The symplectic Lanczos factorization is, up to multiplication by a trivial matrix, specified by the starting vector v 1 (see [4, Theorem 4.1]). wn ]. For a given v 1 , a Lanczos method constructs the matrix SP columnwise from the equations From this we obtain the algorithm given in Table 3.1 (for a more detailed discussion see [4]). method Choose an initial vector e do (update of wm ) set e am e wm (computation of c m ) (update of v m+1 ) e Table Symplectic Lanczos Method Remark 3.1. Using the derived formulae for w k+1 , the residual term r can be expressed as An implicitly restarted symplectic Lanczos method 9 There is still some freedom in the choice of the parameters that occur in this algorithm. Essentially, the parameters b m can be chosen freely. Here we set b Likewise a different choice of the parameters am ; dm is possible. Note that M \Gamma1 since M is symplectic. Thus M \Gamma1 just a matrix-vector-product with the transpose of MP . Hence, only one matrix-vector product is required for each computed Lanczos vector wm or v m . Thus an efficient implementation of this algorithm requires 6n nz is the number of nonzero elements in MP and 2k is the number of Lanczos vectors computed (that is, the loop is executed k times). The algorithm as given in Table 3.1 computes an odd number of Lanczos vectors, for a practical implementation one has to omit the computation of the last vector v k+1 (or one has to compute an additional vector w In the symplectic Lanczos method as given above we have to divide by parameters that may be zero or close to zero. If such a case occurs for the normalization parameter dm+1 , the corresponding vector e v m+1 is zero or close to the zero vector. In this case, a (good approximation to a) JP -orthogonal invariant subspace of MP or equivalently, a symplectic invariant subspace of M is detected. By redefining e v m+1 to be any vector satisfying m, the algorithm can be continued. The resulting butterfly matrix is no longer unreduced; the eigenproblem decouples into two smaller subproblems. In case e wm is zero (or close to zero), an invariant subspace of MP with dimension is found (or a good approximation to such a subspace). In this case the parameter am will be zero (or close to zero). From Table 3.1 we further obtain that in this case is a real eigenvalue of MP (and hence of M) with corresponding Due to the symmetry of the spectrum of M , we also have that 1=b m is an eigenvalue of M . Computing an eigenvector y of MP corresponding to 1=b m , we can try to augment the (2m \Gamma 1)-dimensional invariant subspace to an MP -invariant subspace of even dimension. If this is possible, the space can be made JP -orthogonal by JP -orthogonalizing y against f and normalizing such that y T JP Thus if either v m+1 or wm+1 vanishes, the breakdown is benign. If v m+1 6= 0 and wm+1 6= 0 but then the breakdown is serious. No reduction of the symplectic matrix to a symplectic butterfly matrix with v 1 as first column of the transformation matrix exists. A convergence analysis for the symplectic Lanczos algorithm analogous to the one for the unsymmetric Lanczos algorithm presented by Ye [39] can be given. Moreover, an error analysis of the symplectic Lanczos algorithm in finite-precision arithmetic analogous to the analysis for the unsymmetric Lanczos algorithm presented by Bai [1] can also be derived. These results will be presented in [13]. As to be expected, the computed Lanczos vectors loose J(JP )-orthogonality when some Ritz values begin to converge. 3.2. Truncated symplectic Lanczos factorizations. This section is concerned with finding conditions for the symplectic Lanczos method terminating prema- turely. This is a welcome event since in this case we have found an invariant symplectic (Following [17], we define each floating point arithmetic operation together with the associated integer indexing as a flop.) Benner and Fa-bender subspace S 2n;2k and the eigenvalues of B 2k;2k are a subset of those of M . We will first discuss the conditions under which the residual vector of the symplectic Lanczos factorization will vanish at some step k. Then we will show how the residual vector and the starting vector are related. Finally a result indicating when a particular starting vector generates an exact truncated factorization is given. First the conditions under which the residual vector of the symplectic Lanczos factorization will vanish at some step k will be discussed. From the derivation of the algorithm it is immediately clear that if no breakdown occurs, then where K(X; v; vg. Further it is easy to see that If dim K(MP Hence, there exist real scalars ff such that Using the definition of a k+1 as given in Table 3.1 and the above expression we obtain because of J-orthogonality, a 0: As e This implies that an invariant subspace of MP with dimension 2k If dim K(MP g. Hence a for properly chosen ff and from the algorithm in Table 3.1 An implicitly restarted symplectic Lanczos method 11 Therefore e v This implies that the residual vector of the symplectic Lanczos factorization will vanish at the first step k such that the dimension of K(M; is equal to 2k and hence is guaranteed to vanish for some k - n. Next we will discuss the relation between the residual term and the starting vector. If dim K(M; and Cn is a generalized companion matrix of the form . 1 (see [2, proof of Satz 3.6]). Thus, Define the residual in (3.7) by Note that where We will now show that f k+1 is up to scaling the residual of the length 2k symplectic Lanczos iteration with starting vector v 1 . Together with (3.9) this reveals the relation between residual and starting vectors. Since det (C J-orthogonal columns (that is, (S 2n;2k ) T JnS is a J-triangular matrix. Then . The diagonal elements of R are nonzero if and only if the columns of are linear independent. Choosing 12 Benner and Fa-bender assures that (\GammaJ k (S 2n;2k multiplying (3.7) from the right by is an unreduced butterfly matrix (see [2, proof of Satz 3.6]) with the same characteristic polynomial as C k . Equation (3.10) is a valid symplectic Lanczos recursion with starting vector v residual vector f k+1 =r 2k;2k . By (3.9) and due to the essential uniqueness of the symplectic Lanczos recursion any symplectic Lanczos recursion with starting vector v 1 yields a residual vector that can be expressed as a polynomial in M times the starting vector v 1 . Remark 3.2. From (3.8) it follows that if dim K(M; then we can choose c 1 ; :::; c 2k such that f This shows that if the Krylov subspace forms an 2k-dimensional M-invariant subspace, the residual of the symplectic Lanczos recursion will be zero after k Lanczos steps such that the columns of S 2n;2k span a symplectic basis for the subspace K(M; 1). The final result of this section will give necessary and sufficient conditions for a particular starting vector to generate an exact truncated factorization in a similar way as stated for the Arnoldi method in [35]. This is desirable since then the columns of S 2n;2k form a basis for an invariant symplectic subspace of M and the eigenvalues of B 2k;2k are a subset of those of M . Here, - will denote the Lanczos vectors after permuting them back, i.e., - v Theorem 3.3. Let MS 2n;2k be the symplectic Lanczos factorization after k steps, with B 2k;2k unreduced. Then Jordan matrix of order 2k. Proof. If d XJ be the Jordan canonical form of B 2k;2k and put X. Then Suppose now that it follows that Hence by (3.6) dim K(M; unreduced, dim K(M; k. Hence dim K(M; and therefore, d A similar result may be formulated in terms of Schur vectors or symplectic Schur vectors (see, e.g., [28, 29] for the real symplectic Schur decomposition of a symplectic matrix). These theorems provide the motivation for the implicit restart developed in the next section. Theorem 3.3 suggests that one might find an invariant subspace by iteratively replacing the starting vector with a linear combination of approximate eigenvectors corresponding to eigenvalues of interest. Such approximations are readily available through the Lanczos factorization. 3.3. Stopping Criteria. Now assume that we have performed k steps of the symplectic Lanczos method and thus obtained the identity (after permuting back) MS An implicitly restarted symplectic Lanczos method 13 If the norm of the residual vector is small, the 2k eigenvalues of B 2k;2k are approximations to the eigenvalues of M . Numerical experiments indicate that the norm of the residual rarely becomes small by itself. Nevertheless, some eigenvalues of B 2k;2k may be good approximations to eigenvalues of M . Let - be an eigenvalue of B 2k;2k with the corresponding eigenvector y. Then the vector The vector x is referred to as Ritz vector and - as Ritz value of M . If the last component of the eigenvector y is sufficiently small, the right-hand side of (3.11) is small and the pair f-; xg is a good approximation to an eigenvalue-eigenvector pair of M . Note that by Lemma 2.2 je T is unreduced. The pair (-; x) is exact for the nearby problem A small jjEjj is not sufficient for the Ritz pair f-; xg being a good approximation to an eigenvalue-eigenvector pair of M . The advantage of using the Ritz estimate jd w k+1 jj is to avoid the explicit formation of the residual deciding about the numerical accuracy of an approximate eigenpair. It is well-known that for non-normal matrices the norm of the residual of an approximate eigenvector is not by itself sufficient information to bound the error in the approximate eigenvalue. It is sufficient however to give a bound on the distance to the nearest matrix to which the given approximation is exact. In the following, we will give a computable expression for the error. Assume that B 2k;2k is diagonalizable Since MS 2k , it follows that MS or 2k Y: Thus k. The last equation can be re-written as Using Theorem 2' of [19] we obtain that (- is an eigen-triplet of where jjJx k+i jj g: 14 Benner and Fa-bender Furthermore, when jjEjj is small enough, then is an eigenvalue of M and Consequently, the symplectic Lanczos algorithm should be continued until both jjEjj is small and cond(- j )jjEjj is below a given threshold for accuracy. 4. An Implicitly Restarted Symplectic Lanczos Method. In the previous sections we have briefly mentioned two algorithms for computing approximations to the eigenvalues of a symplectic matrix M . The symplectic Lanczos algorithm is appropriate when the matrix M is large and sparse. If only a small subset of the eigenvalues is desired, the length k symplectic Lanczos factorization may suffice. The analysis in the last chapter suggests that a strategy for finding 2k eigenvalues in a length k factorization is to find an appropriate starting vector that forces the residual r k+1 to vanish. The SR algorithm, on the other hand, computes approximations to all eigenvalues and eigenvectors of M . From Theorem 4.1 in [4] (an implicit Q-theorem for the SR case) we know that in exact arithmetic, when using the same starting vector, the SR algorithm and the length n Lanczos factorization generate the same symplectic butterfly matrices (up to multiplication by a trivial matrix). Forcing the residual for the symplectic Lanczos algorithm to zero has the effect of deflating a subdiagonal element during the SR algorithm: by Remark 3.1 r from the symplectic Lanczos process we have d Hence a zero residual implies a zero d k+1 such that deflation occurs for the corresponding butterfly matrix. Our goal in this section will be to construct a starting vector that is a member of the invariant subspace of interest. Our approach is to implicitly restart the symplectic Lanczos factorization. This was first introduced by Sorensen [35] in the context of unsymmetric matrices and the Arnoldi process. The scheme is called implicit because the updating of the starting vector is accomplished with an implicit shifted SR mechanism on This allows to update the starting vector by working with a symplectic matrix in IR 2j \Theta2j rather than in IR 2n\Theta2n which is significantly cheaper. The iteration starts by extending a length k symplectic Lanczos factorization by steps. Next, 2p shifts are applied to B 2(k+p);2(k+p) using double or quadruple SR steps. The last 2p columns of the factorization are discarded resulting in a length k factorization. The iteration is defined by repeating this process until convergence. For simplicity let us first assume that and that a 2n \Theta 2(k P is known such that as in (3.5). Let - be a real shift and Then (using will be a permuted butterfly matrix and SP is an upper triangular matrix with two additional subdiagonals. With this we can re-express (4.1) as MP (S 2n;2k+2 An implicitly restarted symplectic Lanczos method 15 P SP this yields The above equation fails to be a symplectic Lanczos factorization since the columns of the matrix d k+2 (b k+2 v k+2 2k+2 SP are nonzero. Let ij be the (i; j)th entry of SP . The residual term in (4.2) is Rewriting (4.2) as where Z is blocked as6 6 6 6 4 dk+1e T dk+1e T dk+2 bk+2s 2k+2;2k e T dk+2ak+2s 2k+2;2k e T we obtain as a new Lanczos identity r where d a Here, - a k+1 , - b k+1 , - d k+1 denote parameters of - are parameters of B 2k+2;2k+2 P . In addition, - w k+1 are the last two column vectors from - are the two last column vectors of S 2n;2k+2 As the space spanned by the columns of S orthogonal, and SP is a permuted symplectic matrix, the space spanned by the columns of - is J-orthogonal. Thus (4.3) is a valid symplectic Lanczos factorization. The new starting vector is - ae 2 IR. This can be seen as follows: first note that for unreduced butterfly matrices B 2k+2;2k+2 we have q 2 (B 2k+2;2k+2 Hence, from q 2 (B 2k+2;2k+2 we obtain q 2 (B 2k+2;2k+2 is an upper triangular matrix. As q 2 (B 2k+2;2k+2 Using (4.3) it follows that ae S 2n;2k+2 ae S 2n;2k+2 =ae (MP S 2n;2k+2 =ae (MP S 2n;2k+2 Benner and Fa-bender as r k+2 e T using again (4.3) we get \Gammaae as e T Note that in the symplectic Lanczos process the vectors v j of S 2n;2k P satisfy the condition and the parameters b j are chosen to be one. This is no longer true for the odd numbered column vectors of SP generated by the SR decomposition and the parameters - b j from - P and thus for the new Lanczos factorization (4.3). Both properties could be forced using trivial factors. Numerical tests indicate that there is no obvious advantage in doing so. Using standard polynomials as shift polynomials instead of Laurent polynomials as above results in the following situation: In p 2 (B 2k+2;2k+2 is an upper triangular matrix with four (!) additional subdiagonals. Therefore, the residual term in (4.2) has five nonzero entries. Hence not the last two, but the last four columns of (4.2) have to be discarded in order to obtain a new valid Lanczos factorization. That is, we would have to discard wanted information which is avoided by using Laurent polynomials. This technique can be extended to the quadruple shift case using Laurent polynomials as the shift polynomials as discussed in Section 2. The implicit restart can be summarized as given in Table 4.1. In the course of the iteration we have to choose shifts in order to apply 2p shifts: choosing a real shift - k implies that - \Gamma1 k is also a shift due to the symplectic structure of the problem. Hence, - \Gamma1 k is not added to \Delta as the use of the Laurent polynomial q 2 guarantees that - \Gamma1 k is used as a shift once - k 2 \Delta. In case of a complex shift - k , implies that - k is also a shift not added to \Delta. For complex shifts - k , - k in \Delta. Numerous choices are possible for the selection of the p shifts. One possibility is the case of choosing p "exact" shifts with respect to B 2(k+p);2(k+p) . That is, first the eigenvalues of B 2(k+p);2(k+p) are computed (by the SR algorithm), then p unwanted eigenvalues are selected. One choice for this selection might be: sort the eigenvalues by decreasing magnitude. There will be k eigenvalues with modulus greater than or equal to 1 Select the 2p eigenvalues with modulus closest to 1 as shifts. If - k+1 is complex with then we either have to choose 2p shifts or just 2p as - k+1 belongs to a quadruple pair of eigenvalues of B 2(k+p);2(k+p) P and in order to preserve the symplectic structure either - k and - k+1 have to be chosen or none. An implicitly restarted symplectic Lanczos method 17 restarted symplectic Lanczos method perform k steps of the symplectic Lanczos algorithm to compute S 2n;2k obtain the residual vector r k+1 while jjr k+1 jj ? tol perform p additional steps of the symplectic Lanczos method to compute S 2n;2(k+p) select p shifts - i compute - via implicitly shifted SR steps set S 2n;2k obtain the new residual vector r k+1 while Table k-step restarted symplectic Lanczos method A different possibility of choosing the shifts is to keep those eigenvalues that are good approximations to eigenvalues of M . That is, eigenvalues for which (3.11) is small. Again we have to make sure that our set of shifts is complete in the sense described above. Choosing eigenvalues of B 2(k+p);2(k+p) P as shifts has an important consequence for the next iterate. Assume for simplicity that B 2(k+p);2(k+p) P is diagonalizable. Let be a disjoint partition of the spectrum of B 2(k+p);2(k+p) . Selecting the exact shifts - in the implicit restart, following the rules mentioned above yields a matrix g. This follows from (2.6). Moreover, the new starting vector has been implicitly replaced by the sum of 2k approximate eigenvectors: ae ae properly chosen. The last equation follows since q(B 2(k+p);2(k+p) )e 1 has no component along an eigenvector of associated with Hence It should be mentioned that the k-step restarted symplectic Lanczos method as in Table 4.1 with exact shifts builds a J-orthogonal basis for a number of generalized Krylov subspaces simultaneously. The subspace of length 2(k +p) generated during a restart using exact shifts contains all the Krylov subspaces of dimension 2k generated from each of the desired Ritz vectors, for a detailed discussion see [13]. A similar Benner and Fa-bender observation for Sorensen's restarted Arnoldi method with exact shifts was made by Morgan in [30]. For a discussion of this observation see [30] or [23]. Morgan infers 'the method works on approximations to all of the desired eigenpairs at the same time, without favoring one over the other' [30, p. 1220,l. 7-8 from the bottom]. This remark can also be applied to the method presented here. In the above discussion we have assumed that the permuted SR decomposition exists. Unfortunately, this is not always true. During the bulge-chase in the implicit SR step, it may happen that a diagonal element a j of B 1 (2.2) is zero (or almost zero). In that case no reduction to symplectic butterfly form with the corresponding first column - does exist. In the next section we will prove that a serious breakdown in the symplectic Lanczos algorithm is equivalent to such a breakdown of the SR decomposition. Moreover, it may happen that a subdiagonal element d j of the (2; 2)-block of B 2 (2.3) is zero (or almost zero) such that The matrix - P is split, an invariant subspace of dimension j is found. If shifts have been applied, then the iteration is halted. Otherwise we continue similar to the procedure described by Sorensen in [35, Remark 3]. As the iteration progresses, some of the Ritz values may converge to eigenvalues of long before the entire set of wanted eigenvalues have. These converged Ritz values may be part of the wanted or unwanted portion of the spectrum. In either case it is desirable to deflate the converged Ritz values and corresponding Ritz vectors from the unconverged portion of the factorization. If the converged Ritz value is wanted then it is necessary to keep it in the subsequent factorizations; if it is unwanted then it must be removed from the current and the subsequent factorizations. Lehoucq and Sorensen develop in [23, 36] locking and purging techniques to accomplish this in the context of unsymmetric matrices and the restarted Arnoldi method. These ideas can be carried over to the situation here. 5. Numerical Properties of the Implicitly Restarted Symplectic Lanczos Method. 5.1. Stability Issues. It is well known that for general Lanczos-like methods the stability of the overall process is improved when the norm of the Lanczos vectors is chosen to be equal to 1 [32, 37]. Thus, Banse proposes in [2] to modify the prerequisite our symplectic Lanczos method to \Gammaoe and An implicitly restarted symplectic Lanczos method 19 For the resulting algorithm and a discussion of it we refer to [2]. It is easy to see that BP SP is no longer a permuted symplectic matrix, but it still has the desired form of a butterfly matrix. Unfortunately, an SR step does not preserve the structure of and thus, this modified version of the symplectic Lanczos method can not be used in connection with our restart approaches. some form of reorthogonalization any Lanczos algorithm is numerically unstable. Hence we re-J P -orthogonalize each Lanczos vector as soon as it is computed against the previous ones via wm where for defines the indefinite inner product implied by J n This re-J P -orthogonalization is costly, it requires 16n(m \Gamma 1) flops for the vector wm and 16nm flops for v m+1 . Thus, if 2k Lanczos vectors are computed, the re-J P -orthogonalization adds a computational cost of the order of flops to the overall cost of the symplectic Lanczos method. For standard Lanczos algorithms, different reorthogonalization techniques have been studied (for references see, e.g., [17]). Those ideas can be used to design analogous re-J P -orthogonalizations for the symplectic Lanczos method. It should be noted that if k is small, the cost for re-J P -orthogonalization is not too expensive. Another important issue is the numerical stability of the SR step employed in the restart. During the SR step on the 2k \Theta 2k symplectic butterfly matrix, all but are orthogonal. These are known to be numerically stable. For the nonorthogonal symplectic transformations that have to be used, we choose among all possible transformations the ones with optimal (smallest possible) condition number (see [8]). 5.2. Breakdowns in the SR Factorization. If there is a starting vector - aeq(M)v 1 for which the explicitly restarted symplectic Lanczos method breaks down, then it is impossible to reduce the symplectic matrix M to symplectic butterfly form with a transformation matrix whose first column is - v 1 . Thus, in this situation the SR decomposition of q(B) can not exist. As will be shown in this section, this is the only way that breakdowns in the SR decomposition can occur. In the SR step, most of the transformations used are orthogonal symplectic transformations; their computation can not break down. The only source of breakdown can be one of the symplectic Gaussian eliminations L j . For simplicity, we will discuss the double shift case. Only the following elementary elimination matrices are used in the implicit SR step: elementary symplectic Givens matrices [31] where 20 Benner and Fa-bender elementary symplectic Householder transformation and elementary symplectic Gaussian elimination matrices [8] where Assume that k steps of the symplectic Lanczos algorithm are performed, then from (3.5) Now an implicit restart is to be performed using an implicit double shift SR step. In the first step of the implicit SR step, a symplectic Householder matrix H 1 is computed such that H 1 is applied to B 2k;2k introducing a small bulge in the butterfly form: additional elements are found in the positions (2; 1), (1; 2), (n 1). The remaining implicit transformations perform a bulge-chasing sweep down the subdiagonal to restore the butterfly form. An algorithm for this is given in [2] or [4]; it can be summarized for the situation here as in Table 5.1, where ~ G j and G j both denote symplectic Givens transformation matrices acting in the same planes but with different rotation angles. compute G '+1 such that (G '+1 B 2k;2k ) compute L '+1 such that (L '+1 B 2k;2k ) compute ~ G '+1 such that (B 2k;2k ~ G '+1 compute H '+1 such that (B 2k;2k H '+1 Table Reduction to butterfly form - double shift case. An implicitly restarted symplectic Lanczos method 21 Suppose that the first exist and that we have computed ~ In order to simplify the notation, we switch to the permuted version and rewrite the permuted symplectic matrix b SP as I 2n\Gamma2j \Gamma2 making use of the fact that the accumulated transformations affect only the rows 1 to j and j. The leading (2j principal submatrix of is given by e x x x x x where the hatted quantities denote unspecified entries that would change if the SR update could be continued. Next, the (2j should be annihilated by a permuted symplectic Gaussian elimination. This elimination will fail to exist if the SR decomposition of q(B 2k;2k ) does not exist. As will be needed later, - a implies that - y This follows as e P is From e we obtain - a j x x If - a (otherwise the last Gaussian transformation did not exist). Next we show that this breakdown in the SR decomposition implies a breakdown in the Lanczos process started with the starting vector - 22 Benner and Fa-bender For this we have to consider (5.1) multiplied from the right by b SP . From the derivations in the last section we know that the starting vector of that recursion is given by - As the trailing (2n submatrix of b SP is the identity, we can just as well consider multiplied from the right by SP P SP corresponds to the matrix in (5.2) (no butterfly w j+1 ]. The columns of - are JP -orthogonal The starting vector of the recursion (5.3) is given by - Deleting the last four columns of - P in the same way as in the implicit restart we obtain a valid symplectic Lanczos factorization of length 2. In order to show that a breakdown in the SR decomposition of q(B) implies a breakdown in the above symplectic Lanczos recursion, we need to show From (5.2) and (5.3) we obtain and Further we do know from the symplectic Lanczos algorithm all of these quantities are already known. Now consider x3 Obviously, x Using (5.6) we obtain 2. Hence x Using (5.5) end (5.4) will see that x z3 An implicitly restarted symplectic Lanczos method 23 As - a From (5.3) we obtain Hence using (5.4) yields Similar, it follows that z This argumentation has shown that an SR breakdown implies a serious Lanczos breakdown. The opposite implication follows from the uniqueness of the Lanczos factorization. The result is summarized in the following theorem. Theorem 5.1. Suppose the symplectic butterfly matrix B 2k;2k corresponding to (3.5) is unreduced and let - 2 IR. Let L j be the jth symplectic Gauss transformation required in the SR step on (B If the first symplectic Gauss transformations of this SR step exist, then L j fails to exist if and only if - v T j as in (4.3). 6. Numerical Experiments. Some examples to demonstrate the properties of the (implicitly restarted) symplectic Lanczos method are presented. The computational results are quite promising but certainly preliminary. All computations were done using Matlab Version 5.1 on a Sun Ultra 1 with IEEE double-precision arithmetic and machine precision Our code implements exactly the algorithm as given in Table 4.1. In order to detect convergence in the restart process, the rather crude criterion was used. This ad hoc stopping rule allowed the iteration to halt quite early. Usually, the eigenvalues largest in modulus (and their reciprocals) of the wanted part of the spectrum are much better approximated than the ones of smaller modulus. In a black-box implementation of the algorithm this stopping criterion has to be replaced with a more rigorous one to ensure that all eigenvalues are approximated to the desired accuracy (see the discussion in Section 3.3). Benign breakdown in the symplectic Lanczos process was detected by the criterion while a serious breakdown was detected by Our implementation intends to compute the k eigenvalues of M largest in modulus and their reciprocals. In the implicit restart, we used exact shifts where we chose the shifts to be the 2p eigenvalues of B 2k+p;2k+p closest to the unit circle. Our observations have been the following. Benner and Fa-bender ffl Re-J-orthogonalization is necessary; otherwise J-orthogonality of the computed Lanczos vectors is lost after a few steps, and ghost eigenvalues (see, e.g., [17]) appear. That is, multiple eigenvalues of B 2k;2k correspond to simple eigenvalues of M . ffl The implicit restart is more accurate than the explicit one. ffl The leading end of the 'wanted' Ritz values (that is, the eigenvalues largest in modulus and their reciprocals) converge faster than the tail end (closest to cut off of the sort). The same behavior was observed in [35] for the implicitly restarted Arnoldi method. In order to obtain faster convergence, it seems advisable (similar to the implementation of Sorensen's implicitly restarted Arnoldi method in Matlab's eigs) to increase the dimension of the computed Lanczos factorization. That is, instead of computing S 2n;2k as a basis for the restart, one should compute a slightly larger factorization, e.g. dimension 2(k instead of dimension 2k. When 2' eigenvalues have converged, a subspace of dimension 2(k computed as a basis for the restart, followed by p additional Lanczos steps to obtain a factorization of length k Using implicit SR steps this factorization is reduced to one of length k If the symplectic Lanczos method would be implemented following this approach, the convergence check could be done using only the k Ritz values of largest modulus (and their reciprocals) or those that yield the smallest Ritz residual jd where the y j are the eigenvectors of B 2k;2k . ffl It is fairly difficult to find a good choice for k and p. Not for every possible choice of k, there exists an invariant subspace of dimension 2k associated to the k eigenvalues - i largest in modulus and their reciprocals. If - k is complex and - then we can not choose the 2p eigenvalues with modulus closest to the unit circle as shifts as this would tear a quadruple of eigenvalues apart resulting in a shift polynomial q such that q(B 2(k+p);2(k+p) we can do is to choose the 2p \Gamma 2 eigenvalues with modulus closest to 1 as shifts. In order to get a full set of 2p shifts we add as the last shift the real eigenvalue pair with largest Ritz residual. Depending on how good that real eigenvalue approximates an eigenvalue of M , this strategy worked, but the resulting subspace is no longer the subspace corresponding to the k eigenvalues largest in modulus and their reciprocals. If the real eigenvalue has converged to an eigenvalue of M , it is unlikely to remove that eigenvalue just by restarting, it will keep coming back. Only a purging technique like the one discussed by Lehoucq and Sorensen [23, 36] will be able to remove this eigenvalue. Moreover, there is no guarantee that there is a real eigenvalue of P that can be used here. Hence, in a black-box implementation one should either try to compute an invariant subspace of dimension or of dimension 2(k 1). As this is not known a priori, the algorithm should adapt k during the iteration process appropriately. This is no problem, if as suggested above, one always computes a slightly larger Lanczos factorization than requested. Example 6.1. The first test performed concerned the loss of J-orthogonality of the computed Lanczos vectors during the symplectic Lanczos method and the ghost An implicitly restarted symplectic Lanczos method 25 eigenvalue problem (see, e.g. [17]). To demonstrate the effects of re-J-orthogonali- zation, a 100 \Theta 100 symplectic matrix with eigenvalues 200; 100; 50; was used. A symplectic block-diagonal matrix with these eigenvalues on the block-diagonal was constructed and a similarity transformation with a randomly generated orthogonal symplectic matrix was performed to obtain a symplectic matrix M . As expected, when using a random starting vector M 's eigenvalues largest in modulus (and the corresponding reciprocals) tend to emerge right from the start, e.g., the eigenvalues of B 10;10 are 199:99997; 100:06771; 48:71752; 26:85083; 8:32399 and their reciprocals. Without any form of re-J-orthogonalization, the J-orthogo- nality of the Lanczos vectors is lost after a few iterations as indicated in Figure 6.1. number of Lanczos steps 100,2k JS 100,2k ||Fig. 6.1. loss of J-orthogonality after k symplectic Lanczos steps The loss of J-orthogonality in the Lanczos vectors results, as in the standard Lanczos algorithm, in ghost eigenvalues. That is, multiple eigenvalues of B 2k;2k correspond to simple eigenvalues of M . For example, using no re-J-orthogonalization, after 17 iterations the 6 eigenvalues largest in modulus of B 34;34 are Using complete re-J-orthogonalization, this effect is avoided: 200; 100; 49:99992; 47:02461; 45:93018; 42:31199: The second test performed concerned the question whether an implicit restart is more accurate than an explicit one. After nine steps of the symplectic Lanczos method (with a random starting vector) the resulting butterfly had the eigenvalues (using the Matlab function eig) 200:000000000000 99:999999841718 13:344815062428 3:679215125563 \Sigma5:750883779240i 26 Benner and Fa-bender and their reciprocals. Removing the 4 complex eigenvalues from B 18;18 using an implicit restart as described in Section 4, we obtain a symplectic butterfly matrix impl whose eigenvalues are 200:000000000000 99:999999841719 13:344815062428 and their reciprocals. From (2.6) it follows that these have to be the 14 real eigenvalues of B 18;18 which have not been removed. As can be seen, we lost one digit during the implicit restart (indicated by the 'underbar' under the 'lost' digits in the above table). Performing an explicit restart with the explicitly computed new starting vector butterfly expl whose eigenvalues are 200:000000000000 99:999999841793 and their reciprocals. This time we lost up to nine digits. The last set of tests performed on this matrix concerned the k-step restarted symplectic Lanczos method as given in Table 4.1. As M has only one quadruple of complex eigenvalues, and these eigenvalues are smallest in magnitude there is no problem in choosing k - n. For every such choice there exists an invariant symplectic subspace corresponding to the k eigenvalues largest in magnitude and their reciprocals. In the tests reported here, a random starting vector was used. Figure 6.2 shows a plot of jjr k+1 jj versus the number of iterations performed. Iteration Step 1 refers to the norm of the residual after the first k Lanczos steps, no restart is performed. The three lines in Figure 6.2 present three different choice for k and p: Convergence was achieved for all three examples (and many more, not shown here). Obviously, the choice results in faster convergence than the choice 8. Convergence is by no means monotonic, during the major part of the iteration the norm of the residual is changing quite dramatically. But once a certain stage is achieved, the norm of the residual converges. Although convergence quite fast, this does not imply that convergence is as fast for other choices of k and p. The third line in Figure 6.2 demonstrates that the convergence for does need twice as many iteration steps as for Example 6.2. Symplectic matrix pencils that appear in discrete-time linear-quadratic optimal control problems are typically of the form - I \GammaBB T (Note: For F 6= I , L and N are not symplectic, but L \Gamma -N is a symplectic matrix pencil.) Assuming that L and N are nonsingular (that is, F is nonsingular), solving this generalized eigenproblem is equivalent to solving the eigenproblem for the symplectic matrix - I \GammaBB T If one is interested in computing a few of the eigenvalues of L \Gamma -N , one can use the An implicitly restarted symplectic Lanczos method 27 number of iterations norm(r Fig. 6.2. k-step restarted symplectic Lanczos method, different choices of k and p restarted symplectic Lanczos algorithm on In each step of the symplectic Lanczos algorithm, one has to compute matrix-vector products of the form Mx and Making use of the special form of L and N this can be done without explicitly inverting us consider the computation of First compute Next one has to solve the linear system analogous to x and z, then from Ny = z we obtain In order to solve y we compute the LU decomposition of F and solve the linear system F T y using backward and forward substitution. Hence, the explicit inversion of N or F is avoided. In case F is a sparse matrix, sparse solvers can be employed. In particular, if the control system comes from some sort of discretization scheme, F is often banded which can be used here by computing an initial band LU factorization of F in order to minimize the cost for the computation of y 2 . Note that in most applications, such that the computational cost for C T Cx 1 and significantly cheaper than a matrix-vector product with an n \Theta n matrix. In case of single-input the corresponding operations come down to two dot products of length n each. Using Matlab's sparse matrix routine sprandn sparse normally distributed random matrices F; B; C (here, n) of different dimensions and with different densities of the nonzero entries were generated. Here an example of dimension presented, where the density of the different matrices was chosen to be matrix - nonzero entries 28 Benner and Fa-bender Matlab computed the norm of the corresponding matrix to be - 5:3 \Theta In the first set of tests k was chosen to be 5, and we tested As can be seen in Figure 6.3, for the first 3 iterations, the norm of the residual decreases for both choice of p, but then increases quite a bit. During the first step, the eigenvalues of B 10;10 are approximating the 5 eigenvalues of L \Gamma -N largest in modulus and their reciprocals. In step 4, a 'wrong' choice of the shifts is done in both cases. The extended matrices B 20;20 and B 30;30 both still approximate the 5 eigenvalues of L \Gamma -N largest in modulus, but there is a new real eigenvalue coming in, which is not a good approximation to an eigenvalue of L \Gamma -N . But, due to the way the shifts are chosen here, this new eigenvalue is kept, while an already good approximated eigenvalue - a little smaller in magnitude - is shifted away, resulting in a dramatic increase of jjr k+1 jj. Modifying the choice of the shifts such that the good approximation is kept, while the new real eigenvalue is shifted away, the problem is resolved, the 'good' eigenvalues are kept and convergence occurs in a few steps (the 'o'-line in Figure 6.3). Using a slightly larger Lanczos factorization as a basis for the restart, e.g., a factorization of length k + 3 instead of length k and using a locking technique to decouple converged approximate eigenvalues and associated invariant subspaces from the active part of the iteration, this problem is avoided. number of iterations norm(r modified Fig. 6.3. k-step restarted symplectic Lanczos method, different choices of the shifts Figure 6.4 displays the behavior of the k-step restarted symplectic Lanczos method for different choices of k and p, where k is quite small. Convergence is achieved in any case. So far, in the tests presented, k was always chosen such that there exists a deflating subspace of L \Gamma -N corresponding to the k eigenvalues largest in modulus and their reciprocals. For there is no such deflating subspace (there is one for and one for Figure 6.5 for a convergence plot. The eigenvalues of B 2(k+p);2(k+p) in the first iteration steps approximate the k eigenvalues of largest modulus and their reciprocals (where 5 - j - p) quite well. Our choice of shifts is to select the 2p eigenvalues with modulus closest to 1, but as - k+1 is complex with 1, we can only choose shifts that way. The last shift is chosen An implicitly restarted symplectic Lanczos method 29 number of iterations norm(r Fig. 6.4. k-step restarted symplectic Lanczos method, different choices of k and p according to the strategy explained above. This eigenvalue keeps coming back before it is annihilated. A better idea to resolve the problem is to adapt k appropriately. number of iterations norm(r Fig. 6.5. k-step restarted symplectic Lanczos method, different choices of k and p 7. Concluding Remarks. We have investigated a symplectic Lanczos method for symplectic matrices. Employing the technique of implicitly restarting the method using double or quadruple shifts as zeros of the driving Laurent polynomials, this results in an efficient method to compute a few extremal eigenvalues of symplectic matrices and the associated eigenvectors or invariant subspaces. The residual of the Lanczos recursion can be made to zero by choosing proper shifts. It is an open problem how these shifts should be chosen in an optimal way. The preliminary numerical tests reported here show that for exact shifts, good performance is already achieved. Before implementing the symplectic Lanczos process in a black-box algorithm, some more details need consideration: in particular, techniques for locking of con- Benner and Fa-bender verged Ritz values as well as purging of converged, but unwanted Ritz values, needs to be derived in a similar way as it has been done for the implicitly restarted Arnoldi method. --R analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem Symplektische Eigenwertverfahren zur L-osung zeitdiskreter optimaler Steuerungs- probleme A condensed form for the solution of the symplectic eigenvalue problem The symplectic eigenvalue problem SR and SZ algorithms for the symplectic (butterfly) eigenproblem Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions Matrix factorization for symplectic QR-like methods A symplectic QR-like algorithm for the solution of the real algebraic Riccati equation An implicitly restarted Lanczos method for large symmetric eigenvalue problems Sur quelques Algorithmes de recherche de valeurs propres Numerical linear algorithms and group theory On some algebraic problems in connection with general eigenvalue algorithms Symplectic Methods for Symplectic Eigenproblems An analysis of structure preserving methods for symplectic eigenvalue problems The QR transformation Part I and Part II Matrix Computations Model reduction of state space systems via an implicitly restarted Lanczos method Residual bounds on approximate eigensystems of nonnormal matrices On some algorithms for the solution of the complete eigenvalue problem The Algebraic Riccati Equation Invariant subspace methods for the numerical solution of Riccati equations Deflation techniques for an implicitly restarted Arnoldi itera- tion Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods A new method for computing the closed loop eigenvalues of a discrete-time algebraic Riccati equation Canonical forms for Hamiltonian and symplectic matrices and pencils On restarting the Arnoldi method for large nonsymmetric eigenvalue problems A Schur decomposition for Hamiltonian matrices Computation of the stable deflating subspace of a symplectic pencil using structure preserving orthogonal transformations Implicit application of polynomial filters in a k-step Arnoldi method Analysis of the look ahead Lanczos algorithm A symplectic method for approximating all the eigenvalues of a Hamiltonian matrix A convergence analysis for nonsymmetric Lanczos algorithms --TR

implicit restarting;symplectic Lanczos method;symplectic matrix;eigenvalues

587865

Differences in the Effects of Rounding Errors in Krylov Solvers for Symmetric Indefinite Linear Systems.

The three-term Lanczos process for a symmetric matrix leads to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving a reduced system in one way or another. This leads to well-known methods: MINRES (minimal residual), GMRES (generalized minimal residual), and SYMMLQ (symmetric LQ). We will discuss in what way and to what extent these approaches differ in their sensitivity to rounding errors.In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods, and we will not consider the errors in the Lanczos process itself. We will show that the method of solution may lead, under certain circumstances, to large additional errors, which are not corrected by continuing the iteration process.Our findings are supported and illustrated by numerical examples.

Introduction We will consider iterative methods for the construction of approximate solutions, starting with for the linear system A an n by n symmetric matrix, in the k-dimensional Krylov subspace with r With the standard 3-term Lanczos process, we generate an orthonormal basis v 1 process can be recast in matrix formulation as in which V j is defined as the n by j matrix with columns v 1 tridiagonal matrix. Mathematical Institute, Utrecht University, Budapestlaan 6, Utrecht, the Netherlands. y Institute of Mathematics, Medical University of L-ubeck, Wallstra-e 40, 23560 L-ubeck, Germany. E-mail: sleijpen@math.uu.nl, vorst@math.uu.nl, modersitzki@informatik.mu-luebeck.de This assumption does not mean a loss of generality, since the case x0 6= 0 can be reduced to this by a simple shift Paige [11] has shown that in finite precision arithmetic, the Lanczos process can be implemented so that the computed V k+1 and T k satisfy with, under mild conditions for k, (u is the machine precision, m 1 denotes the maximum number of nonzeros in any row of A). we obtain the convenient expression Popular Krylov subspace methods for symmetric linear systems can be derived with formula (1) as a starting point: MINRES, GMRES, 2 and SYMMLQ. The matrix T k can be interpreted as the restriction of A with respect to the Krylov subspace, and the main idea behind these Krylov solution methods is that the given system replaced by a smaller system with T k over the Krylov subspace. This reduced system is solved - implicitly or explicitly - in a convenient way and the solution is transformed with V k to a solution in the original n-dimensional space. The main differences between the methods are due to a different way of solution of the reduced system and to differences in the backtransformation to an approximate solution of the original system. We will describe these differences in relevant detail in coming sections. Of course, these methods have been derived assuming exact arithmetic, for instance, the generating formulas are all based on an exact orthogonal basis for the Krylov subspace. In reality, however, we have to compute this basis, as well as all other quantities in the methods, and then it is of importance to know how the generating formulas behave in finite precision arithmetic. The errors in the underlying Lanczos process have been analysed by Paige [11, 12]. It has been proven by Greenbaum and Strakos [8], that rounding errors in the Lanczos process may have a delaying effect on the convergence of iterative solvers, but do not prevent eventual convergence in general. Usually, this type of error analysis is on a worst case scenario, and as a consequence the error bounds are pessimistic. In particular, the error bounds cannot very well be used to explain differences between these methods, so as we observe them in practical situations. In this paper, we propose a different way of analysing these methods, different in the way that we do not attempt to derive sharper upper bounds, but that we try to derive upper bounds for relevant differences between these processes in finite precision arithmetic. This will not help us to understand why any of these methods converges in finite precision, but it will give us some insight in answering practical questions such as: - When and why is MINRES less accurate than SYMMLQ? This question was already posed in the original publication [14], but the answer in [14, p.625] is largely speculative. - Is MINRES suspect for ill-conditioned systems, because of the minimal residual approach (see [14, p.619])? Although hints are given for the reasons of inaccuracies in MINRES, for MINRES, it is also stated in [14, p. 625] that it is not as accurate as SYMMLQ for the reason 2 GMRES has been designed in combination with Arnoldi's method for unsymmetric systems, but for symmetric systems Arnoldi's method and Lanczos' method lead, in exact arithmetic, to the same relation (1) that the minimal residual method is suspect. In [3, p. 43] an explicit relation is suggested between MINRES and working with A 2 , and it is argued that for that reason sensitivity to rounding errors of the solution depends on - 2 (A) 2 (it is even stated: 'the squared condition number of A 2 ', implying - 2 which seems to be a mistake). - Why and when is SYMMLQ slower than for instance MINRES or GMRES? - Why does MINRES sometimes lead to rather large residuals, whereas the error in the approximation is significantly smaller? See, for instance observations on this, made in [14, p.626]. Most important, understanding the differences between these methods will help us in making a choice. We will now briefly characterize the different methods in our investigation: 1. MINRES [14]: determine x is minimal. This minimization leads to a small system with T k , and the tridiagonal structure of T k is exploited to get a short recurrence relation for x k . The advantage of this is that only three vectors from the Krylov subspace have to be saved (in fact, MINRES works with transformed basis vectors; this will be explained in Section 2.3). For the implementation of MINRES that we have used, see the Appendix. 2. GMRES [16]: This method also minimizes, for y k 2 R k , the residual kb \Gamma Ax k k 2 . GMRES was designed for unsymmetric matrices, for which the orthogonalisation of the Krylov basis is done with Arnoldi's method. This leads to a small upper Hessenberg system that has to be solved. However, when A is symmetric, then, in exact arithmetic, the Arnoldi method is equivalent to the Lanczos method (see also [7, p.41]). Although GMRES is commonly presented with an Arnoldi basis, there are various implementations of it that differ in finite precision, for instance, with Modified Gram-Schmidt, Classical Gram-Schmidt, Householder, and other variants. We view Lanczos as one way to obtain an orthogonal basis, and therefore stick to the name GMRES rather than to introduce a new and possibly confusing acronym. Due to the way of solution in GMRES, all the basis vectors v j have to be stored, also when A is symmetric. 3. SYMMLQ [14]: determine x such that the error x Euclidean length. It may come as a surprise that can be minimized without knowing x, but this can be accomplished by restricting the choice of x k to AK k (A; r 0 ). Conjugate Gradient approximations can, if they exist, be computed with little effort from the SYMMLQ information. In the SYMMLQ implementation suggested in [14] this is used to terminate iterations either at a SYMMLQ iterate or a Conjugate Gradient iterate, depending on which one is best. For the implementation of SYMMLQ that we have used, see the Appendix. Note that these methods can be carried out with exactly the same basis vectors v j and tridiagonal matrix T j . Most of our bounds on perturbations in the solutions at the kth iteration step will be expressed as bounds for corresponding perturbations to the residual in the kth step, relative to the norm of an initial residual. Since all these iteration methods construct their search spaces from residual vector information (that is, they all start with kr 0 k 2 ), and since we make at least errors in the order of u kbk 2 in the computation of the residuals, we may not expect perturbations of order less than u- 2 (A)kbk 2 in the iteratively computed solutions. So our bounds can only be expected to show up in the computed residuals, if the errors are larger than the error induced by the computation of the residuals itself. Notations: Quantities associated with n dimensional spaces will be represented in bold face, like A, and v j . Vectors and matrices on low dimensional subspaces are denoted in normal mode: T , y. Constants will be denoted by Greek symbols, with the exception that we will use u to denote the relative machine precision. The absolute value of a matrix refers to elementwise absolute values, that is 2 Differences in round-off error behaviour between MINRES and GMRES 2.1 The basic formulas for GMRES and MINRES in exact arithmetic We will first describe the generic formulas for the iterative methods MINRES and GMRES, and we will assume exact arithmetic in the derivation of these formulas. Without loss of generality, we may assume that x The aim is to minimize kb \Gamma Axk 2 over the Krylov subspace, and since we see that for minimizing must be the linear least squares solution of the overdetermined system In GMRES this system is solved with Givens rotations, which leads to an upper triangular reduction of in which R k is k by k upper triangular with bandwidth 3, and Q k is a orthonormal columns. Using (6), y k can be solved from and since x (R (R The parentheses have been included in order to indicate the order of computation. In the original publication [16], GMRES was proposed for unsymmetric A, in combination with Arnoldi's method for an orthonormal basis for the Krylov subspace. However, when A is symmetric then Arnoldi's method is equivalent to Lanczos' method, so that (8) describes GMRES for symmetric A. The well-known disadvantage of this approach is that we have to store all columns of V k for the computation of x k . MINRES follows essentially the same approach as GMRES for the minimization of the residual, but it exploits the banded structure of R k , in order to get short recurrences for x k , and in order to save on memory storage. Indeed, the computations in the generating formula (8) can be reordered as z k For the computation of W k , it is easy to see that the last column of W k is obtained from the last two columns of W k\Gamma1 and v k . This makes it possible to update x to x k with a short recurrence, since z k follows from the kth Givens rotation applied to the vector (z T This interpretation leads to MINRES. We see that MINRES and GMRES both use V k , R k , T k , Q k , and z k , for the computation of x k . Of course, we are not dictated to compute these items in exactly the same way for the two methods, but there is no reason to compute them differently. Therefore, we will compare implementations of GMRES and MINRES that are based on exactly the same items in floating point finite arithmetic. From now on we will study in what way MINRES and GMRES differ in finite precision arithmetic, given exactly the same set V k , R k , T k , Q k , and z k (computed in finite precision too) for the two different methods. Hence, the differences in finite precision between GMRES and MINRES are only caused by a different order of computation of the namely ffl for GMRES: x ffl for MINRES: x z k . Of course, we could have tried to get upper bounds for all errors made in each process, but this would most likely not reveal the differences between the two methods. If we want to study the differences between the two methods then we have to concentrate on the two generating formulas. 2.2 Error analysis for GMRES In order to understand the difference between GMRES and MINRES, we have to study the computational errors in V k . We will indicate actual computation in floating point finite precision arithmetic by fl, and the result will be denote by a b. Then, according to [5, p. 89], in floating point arithmetic the computed solution b y (R This implies that b k so that apart from second order terms in u Here is the exact value based on the computed R k and z k . Then we make also errors in the computation of x k , that is we compute b x y k ). With the error bounds for the matrix vector product [10, p.76], we obtain with Hence, the error \Deltax that can be attributed to differences between MINRES and GMRES, has two components This error leads to a contribution \Deltar k to the residual, that is \Deltar k is that part of r k that can be attributed to differences between MINRES and GMRES (ignoring O(u 2 ) \Deltar Note that in finite precision we have that AV , and that, because of (3), the leads to a contribution of O(u 2 ) in \Deltar k . This is also the case in forthcoming situations where we replace AV k by V k+1 T k in the derivation of upper bounds for error contributions. Using the bound in (10) and the bound for \Delta 2 , we get (skipping higher order terms in u) k 3 Here we have used that k jR k from [21, Th. 4.2]; see Lemma 5.1 for details). The factor - 2 denotes the condition number with respect to the Euclidean norm. 3 Note that we could bound kV k+1 k 2 by which is, because of the local orthogonality of the v j , a crude overestimate. According to [15, p. 267 (bottom)], it may be more realistic to replace this factor m, where m denotes the number of times that a Ritz value of T k has converged to an eigenvalue of A. When solving a linear system, this value of m is usually very modest, 2 or 3 say. Finally, we note that R T It has been shown in [6] that the matrix T k that has been obtained in finite precision arithmetic, may interpreted as the exact Lanczos matrix obtained from a matrix e A in which eigenvalues of A are replaced by multiplets. Each multiplet contains eigenvalues that differ by O(u) 1 4 from an original eigenvalue of A. 4 With e k we denote the orthogonal matrix that generates T k , in exact arithmetic, from e A. Hence, e A T e A e 3 We also have used that the computed Q k are orthogonal matrices, with errors in the order of u, i.e., O(u). These O(u)-errors lead to O(u) 2 -errors in (13). 4 This order of difference is pessimistic; factors proportional to (u) 1 2 , or even u, are more likely, but have not been proved [7, Sect.4.4.2]. so that oe min (R T A T e and oe (R T A T e which implies - 2 (R k (ignoring errors proportional to mild orders of u). This finally results in the upper bound for the error in the residual due to the difference between GMRES and MINRES: Note that, even if there were only rounding errors in the matrix-vector multiplication, then the perturbation \Deltax to A \Gamma1 b would have been (in norm) in the order of u This corresponds to an error kA\Deltaxk 2 - u- 2 (A)kbk 2 in the residual. Therefore, the stability of GMRES cannot essentially be improved. 2.3 Error analysis for MINRES The differences in finite precision between MINRES and GMRES are reflected by z k . We will first analyze the floating point errors introduced by the computation of the columns of k . The jth row w j;: of W k satisfies w which means that in floating point finite precision arithmetic we obtain the solution b w j;: of a perturbed system: with Note that the perturbation term \Delta R j depends on j. This gives b w when we combine the relations for c with We may replace c k in (18), because this leads only to O(u 2 ) errors. Finally, we make errors in the computation of x k because of finite precision errors in the multiplication of c with The errors made in c k and the error term are the only errors that can be held responsible for the difference between MINRES and GMRES. Added together, they lead to the \Deltax k related to MINRES: and this leads to the following contribution to the MINRES residual: \Deltar If we use the bound (18) for \Delta W , and use for other quantities bounds similar as for GMRES, then we obtain 3 3 Here we have also used the fact that and, with kV k k F - k, the expression can be further bounded. This finally results in the following upper bound for the error contribution in the residual due to the differences in the implementation between MINRES and GMRES: 3k We see that the different implementation for MINRES leads to a relative error in the residual that is proportional to the squared condition number of A, whereas for the GMRES implementation the difference led to a relative error proportional to the condition number only. This means that if we plot the residuals for MINRES and GMRES then we may expect to see differences, more specifically, the difference between the computed residuals for the two methods may be expected to be in the order of the square of the condition number. As soon as the computed residual of GMRES gets below u difference may be visible. 2.4 Discussion In Fig. 1, we have plotted the residuals obtained for GMRES and MINRES. Our analysis suggests that there may be a difference between both in the order of the square of the condition number times machine precision relative to kbk 2 . Of course, the computed residuals reflect all errors made in both processes, and if all these errors together lead to perturbations in the same order for MINRES and GMRES, then we will not see much difference. However, as we see, all the errors in GMRES lead to something proportional to the condition number, and now the effect of the square of the condition number is clearly visible in the error in the residual for MINRES. Our analysis implies that one has to be careful with MINRES when solving linear systems with an ill-conditioned matrix A, specially when eigenvector components in the solution, corresponding to small eigenvalues, are important. The residual norm reduction kr k k 2 =kbk 2 for the exact (but unknown) MINRES residual can be computed efficiently as a product ae k j js 1 of the sines s k of the Givens rotations. In MINRES (as well as GMRES) this value ae k is used to measure the reduction of the residual norm: in practical computations, a residual norm is not often computed explicitly Convergence history MINRES A=Q'*diag(D)*Q, Q Givens log10(|r|) (solid line), log10(|rho|) (dotted line) -2Convergence history MINRES , A=Q'*diag(D)*Q, Q Givens log10(|r|) (solid line), log10(|rho|) (dotted line) -2Convergence history GMRES A=Q'*diag(D)*Q, Q Givens log10(|r|) (solid line), log10(|rho|) (dotted line) -2Convergence history GMRES , A=Q'*diag(D)*Q, Q Givens log10(|r|) (solid line), log10(|rho|) (dotted line) Figure 1. MINRES (top) and GMRES (bottom): solid line dotted line of the estimated residual norm reduction ae k . The pictures show the results for a positive definite system (the left pictures) and for a non-definite system (the right pictures). For both examples -2 To be more specific: at the left and G the Givens rotation in the (1; 30)-plane over an angle of at the right diagonal G the same Givens rotation as for the left example; in both examples (and others to come) b is the vector with all coordinates equal to 1, and the relative machine precision as the kth floating point approximate. Therefore, it is of interest to know how much the computed ae k may differ from the exact residual norm reduction. The errors made in the computation of ae k are of order u and can be neglected. Since the computation of ae k and of b x k are based on the same inexact Lanczos process, (22) implies that The situation for GMRES is much better: the difference between ae k and the true residual reduction for GMRES can be bounded by the quantity in the right hand side of (14). In fact, as observed at the end of x2.2, except for the moderate constant (3 about the most accurate computation that can be expected. 2.5 Diagonal matrices Numerical Analysts often carry out experiments for (unpreconditioned) iterative solvers with diagonal matrices, because, at least in exact arithmetic, the convergence behaviour depends on the distribution of the eigenvalues and the structure of the matrix plays no role in Krylov solvers. However, the behaviour of these methods for diagonal systems may be quite different in finite precision, as we will show now, and, in particular for MINRES, experiments with diagonal matrices may give a too optimistic view on the behaviour of the method. Rotating the matrix from diagonal to non-diagonal (i.e., diagonal and Q orthogonal, instead of A = D) has hardly any influence on the errors in the GMRES residuals (no results shown here). This is not the case for MINRES: experimental results (cf. Fig. 2) indicate that the errors in the MINRES residuals for diagonal matrices are of order (A), as for GMRES. This can be understood as follows. If we neglect O(u 2 ) terms, then, according to (15), the error, due to the inversion of R k , in the jth coordinate of the MINRES-x k is given by k When A is diagonal with (j; j)-entry - j , the error in the jth coordinate of the MINRES residual is equal to (use (1) and (6)) k k Therefore, in view of (16), and including the error term for the multiplication with c (cf. (19)), we have for MINRES applied to a diagonal matrix: which is the same upper bound as for the errors in the GMRES residuals in (14). The perturbation matrix \Delta R j depends on the row index j. Since, in general, \Delta R j will be different for each coordinate j, (23) cannot be expected to be correct for non-diagonal matrices. In fact, if orthogonal matrix, then errors of order in the jth coordinate of x k can be transferred by Q to an mth coordinate and may not be damped by a small value j- m j. More precisely, if \Gamma is the maximum size of the off-diagonal elements of A that "couple" small diagonal elements of A to large ones, then the error in the MINRES residual will be of order \Gamma (R we recover the bound (22). 2.6 The errors in the approximations In exact arithmetic we have that . Assuming that, in finite precision, this also gives about the right order of magnitude, then the errors related to differences between MINRES and GMRES, for the approximate solutions in (11) and (20) can be bounded by essentially the same upper bound: . (3 Convergence history MINRES with A=diag(D) log10(|r|) (solid line), log10(|rho|) (dotted line) -2Convergence history MINRES with A=diag(D) log10(|r|) (solid line), log10(|rho|) (dotted line) Figure 2. MINRES: solid line dotted line (\Delta \Delta \Delta) log 10 of the estimated residual norm reduction ae k . The pictures show the results for a positive definite diagonal system (the left picture) and for a non-definite diagonal system (the right picture). Except for the Givens rotation, the matrices in these examples are equal to the matrices of the examples in Fig. 1: here This may come as a surprise since the bound for the error contribution to the residual for MINRES is proportional to Based upon our observations for numerical experiments, we think that this can be explained as follows. The error in the GMRES approximation has mainly large components in the direction of the small singular vectors of A. These components are relatively reduced by multiplication with A, and then have less effect to the norm of the residual. On the other hand the errors in the MINRES approximation are more or less of the same magnitude over the spectrum of singular values of A and multiplication with A will make error components associated with larger singular values more dominating in the residual. We will support our viewpoint by a numerical example. The results in Fig. 3 are obtained with a positive definite matrix with two tiny eigenvalues. For b we took a random perturbation of Ay in the order of 0:01: This example mimics the situation where the right-hand side vector is affected by errors from measurements. The solution x of the equation has huge components in the direction of the two singular vectors with smallest singular value. In the other directions x is equal to y plus a perturbation of less than one percent. The coordinates of the vector y in our example form a parabola, which makes the effects easier visible. The convergence history of GMRES and of MINRES (not shown here) for this example with is comparable to the ones in the left pictures of Fig. 1, but, because of a higher condition number, the final stagnation of the residual norm in the present example takes place on a higher level (- 3 Fig. 3 shows the solution x k as computed at the 80th step of GMRES (top pictures) and of MINRES (bottom pictures); the right pictures show the component of x k orthogonal to the two singular vectors with smallest singular value, while the left pictures show the complete x k . Note that kx k k . The curve of the projected GMRES solution (top-right picture) is a slightly perturbed parabola indeed (the irregularities are due to the perturbation p). The computational errors from the GMRES process are not visible in this picture: these errors are -0.050.050.150.25x_{GMRES} proj on span(V(3:n)) sing. vectors V with increasing sing. values -0.50.51.5x_{MINRES} proj on span(V(3:n)) sing. vectors V with increasing sing. values Figure 3. The pictures show the solution x of computed with 80 steps of GMRES (top pictures) and of MINRES (bottom pictures). The ith coordinate of xk (along the vertical axis) is plotted against i (along the horizontal axis). sin 0:01. The right pictures show the component of xk orthogonal to the two singular vectors with smallest singular value, while the left pictures show the complete xk . mainly in the direction of the two small singular vectors. In contrast, the irregularities in the MINRES curve (bottom-right) are almost purely the effect of rounding errors in the MINRES process. In SYMMLQ we minimize the norm of x which means that y k is the solution of the normal equations This system can be further simplified by exploiting the Lanczos relations (1): A stable way of solving this set of normal equations is based on an L e Q decomposition of T T and this is equivalent to the transpose of the Q k R k decomposition of T k (see (6)), which is constructed for GMRES and MINRES: This leads to from which the basic generating formula for SYMMLQ is obtained: with k . We will further assume that x The actual implementation of SYMMLQ [14] is based on an update procedure for V k+1 Q k , and on a three term recurrence relation for kr Note that SYMMLQ can be carried out with exactly the same computed values for V k+1 , Q k , R k , and r 0 , as for GMRES and MINRES. In fact, there is no good reason for using different values for each of the algorithms. Therefore, differences because of round-off, between the three methods, must be attributed to the additional rounding errors made in the evaluation of the right-hand side of (25). The largest factor in the upper bound for these additional rounding errors in the construction of the SYMMLQ approximation x k is caused by the inversion of L k . The multiplication with and the assembly of x k , leads to a factor k k in the upper bound (similar as for MINRES and GMRES). In order to simplify the much more complicated analysis for SYMMLQ, we have chosen to study only the effect of the errors introduced by the inversion of L k . The resulting error \Deltax k is written as where g k represents the exact solution and b g k is the value obtained in finite precision arithmetic. We likewise the coordinates of bg k =kr 0 k 2 are denoted by These coordinates can be written as manipulation leads to where From (25) it follows that Hence, the error in the SYMMLQ residual r ME k can be written as The first term can be treated as in GMRES: We define By combining (29), (27), and the definition for t k , we conclude that and because of the orthogonality of v k and v k+1 , we have that The computed residual reduction k b t k k 2 is usually used for monitoring the convergence, in a stopping criterion. In actual computations with SYMMLQ, no residual vectors are computed. Expression (30) can now be bounded realistically byp 3 Here we have used that k jL k Hence 3 A straight-forward estimate is 3 which is much larger than the first term in (33). Experiments indicate that k b t towards 0 (even below the value u- 2 (A)). Below, we will explain why this is to be expected (cf. (49)). Fig. 4 illustrates that the upper bound in (33), with k b t Accuracy. From (33) it follows that 3 is the SYMMLQ residual with respect to the computed SYMMLQ approximate and r k is the SYMMLQ residual for the exact SYMMLQ approximate (for the finite precision Lanczos). Apparently, assuming that kr k increases, SYMMLQ is rather accurate since, for any method, errors in the order u should be expected anyway. Convergence history SYMMLQ A=Q'*diag(D)*Q, Q Givens log10(|r|) (solid line), log10(|rho|) (dotted line) -55Convergence history SYMMLQ , A=Q'*diag(D)*Q, Q Givens log10(|r|) (solid line), log10(|rho|) (dotted line) Figure 4. SYMMLQ: solid line dotted line (\Delta \Delta \Delta) log 10 of the estimated residual norm reduction k b tk k2 . The pictures show the results for the positive definite system (the left picture) and for the non-definite system (the right picture) of Fig. 1. Both systems have condition number Convergence. It is not clear yet whether the convergence of SYMMLQ is insensitive to rounding errors. This would follow from (33) if both t k and b t k would approach 0. It is unlikely that will be (much) larger than k b t k k 2 , that is, it is unlikely that the inexact process converges faster than the process in exact arithmetic. Therefore, when it is observed that k b t k k 2 is small (of order u- 2 (A)), it may be concluded that the speed of convergence has not been affected seriously by rounding errors. In experiments, we see that b t k approaches zero if k increases. For practical applications, assuming that kt k k 2 . k b t k k 2 , it is useful to know that the computable value k b t k k 2 informs us on the accuracy of the computed approximate and on a possible loss of speed of convergence. However, it is of interest to know in advance whether the computed residual reduction will decrease to 0. Moreover, we would like to know whether course, it is impossible to prove that SYMMLQ will converge for any symmetric problem: one can easily construct examples for which kr k k 2 will be of order 1 for any k ! n. But, as we will analyse in the next subsection, the interesting quantities can be bounded in terms of the MINRES residual. That result will be used in order to show that the will be relatively unimportant as soon as MINRES has converged to some degree. 3.1 A relation between SYMMLQ and MINRES residual norms In this section we will assume exact arithmetic, in particular the Lanczos process is assumed to be exact. The residuals r MR k and r ME k denote the residuals of MINRES and SYMMLQ, respectively. The norm of the residual b \Gamma Ax b , with x b the best approximate of x in K k can be bounded in terms of the norm of the residual r MR kr MR This follows from the observation that r MR k where x MR k is from the same subspace from which the best approximate x b has been selected, and furthermore that kb \Gamma Ax b k 2 - Unfortunately, SYMMLQ selects its approximation x k from a different subspace, namely AK k (A; r 0 ). This makes a comparison less straight forward. The following lemma will be used for bounding the SYMMLQ error in terms of the MINRES error. Its proof uses the fact that r MR connects K k+1 spanned by r MR k and AK k (A; r 0 ). Lemma 3.1 For each z 2 K k+1 kr MR 2: (36) Proof. For simplicity we will assume that x By construction x ME z in the space AK k (A; r 0 ). Hence implies that By construction we have that x ME as a consequence: From Pythogoras' theorem, with (37), we conclude that and (36) follows by combining this result with (38). Unfortunately, a combination of (36) with k and the obvious estimate jff k j kr MR , from (37) does not lead to a useful result. An interesting result follows from an upper bound for jff k j that can be obtained from a relation between two consecutive MINRES residuals and a Lanczos basis vector. This result is formulated in the next theorem. Theorem 3.2 Proof. We use the relation r MR where r CG k is the kth Conjugate Gradient residual. The scalars s and c represent the Givens transformation used in the kth step of MINRES. This relation is a special case of the slightly more general relation between GMRES and FOM residuals, formulated in [2, 22]. For symmetric A, GMRES is equivalent with MINRES, and FOM is equivalent with CG. Since r CG r MR kr MR kr MR and . Moreover, since r MR k . Therefore, with e ME kr MR r MR kr MR kr MR r MR kr MR kr MR and hence kr MR kr MR A combination of (42) and (36) with k+1 leads to kr MR kr MR With and using the minimal residual property kr MR we obtain the following recursive upper bound from (43): kr MR A simple induction argument shows that fi k - k+1 , and the definition of fi k implies kr MR which completes the proof. For our analysis of the additional errors in SYMMLQ, we also need a slightly more general result, formulated in the next theorem. Theorem 3.3 Let y. For the best approximation y ME k of y in AK k (A; r 0 ), and for y MR is the best approximation of c in AK k (A; r 0 ), with - k as in (39), we have kr MR i-k kr MR Proof. The proof comes along the same lines as the proof of Theorem 3.2. Replace the quantities x and x MR k by y and y MR k . Since the y quantities fulfill the same orthogonality relations, (36) is valid also in the y quantities. This is also the case for the upper bound for jff k j kr MR Hence, with e ME j , we have kr MR kr MR If we define b we find that kr MR which implies (45). For the relations between SYMMLQ and MINRES we have assumed exact arithmetic, that is we have assumed an exact Lanczos process as well as an exact solve of the systems with L k . However, we can exclude the influence of the Lanczos process by applying Theorem 3.2 right away to a system with a Lanczos matrix Tm and initial residual kr 0 k 2 e 1 . In this setting, we have, for k ! m, that ([2, 22]) kr MR with s j the sine in the jth Givens rotation for the QR decomposition of T is the estimated reduction of the norms of the MINRES residuals. From relation (44) in combination with (31) we conclude that Note that inequality (47) is correct for any symmetric tri-diagonal extension e Tm of T (47) holds with e Tm instead of Tm . It has been shown in [6] that there is an extension e Tm of which any eigenvalue is in a O(u) 1 4 -neighborhood of some eigenvalue of A, and therefore in fairly good precision. This leads to our upper bound In x3.1.1, we will show that The upper bound in (49) contains a square of the condition number. However, in the interesting situation where ae k decreases towards 0, the effect of the condition number squared will be annihilated eventually. Remark 3.4 Except for the constants 'k the estimates (48) and (49), respectively, appear to be sharp (see Fig. 5). Although the maximal values of the ratio of kt Fig. 5 exhibit slowly growing behavior, the growth is not of order k 3 . In the proof of (49) (cf. x3.1.1), upper bounds as in (48) are used in a consecutive number of steps. In view of the irregular convergence of SYMMLQ, the upper bound (48) will be sharp for at most a few steps. By exploiting this observation, one can show that a growth of order k 2 , or even less, will be more likely. versus MINRES log10 of quotient of the estimates of the residual norms: SYMMLQ / MINRES -22(e_k^t(L+Delta)\e1)./rho_k, |Delta|<eps*|L|, eps=2.958e-13 log10 of perturbations in SYMMLQ Figure 5. Results for the non-definite matrix with condition number (as in the right pictures) of Fig. 1 and Fig. 4. The left picture shows log 10 of the ratio k b tk k2 =ae k of the estimated residual norm reduction of SYMMLQ with the one of MINRES, the right picture models k b tk \Gamma tk k2 =ae k : it shows the log 10 of e T 3.1.1 SYMMLQ recurrences In this section we derive the upper bound (49). Suppose that the jth recurrence for the fl i 's is perturbed by a relatively small ffi and all other recurrence relation are exact: The resulting perturbed quantities are labeled as e. Then For is a multiple of the SYMMLQ residual for the Tm -system (m ? as in the proof of inequality (48), Theorem 3.2 could be applied for estimating k e t . For the situation where j 6= 1, Theorem 3.3 can be used. To be more precise, with we have (in the notation of Theorem 3.3), for y and ae k with c j the cosine in the jth Givens rotation. Therefore, by Theorem 3.3, ae k For this specific situation, the estimate for fi k in the last paragraph of the proof of Theorem 3.2 can be improved. It can be shown that fi j - 1 if fi k - k\Gammaj . Therefore, the - k+1 in (54) can be replaced by - k\Gammaj . A combination of (51) with (54) gives Using the definition of M j and the recurrence relations for the fl j , we can express \Gamma' jj Therefore, from (48), we have that Hence (cf. (50)) and, with (55), this gives gives Because the recurrences are linear, the effect of a number of perturbations is the cumulation of the effects of single perturbations. If each recurrence relation is perturbed as in (50) then the estimate (49) appears as a cumulation of bounds as in (57). The vector b t k in (49) represents the result of these successive perturbations due to finite precision arithmetic. Finally, we will explain that the effect of rounding errors in solving L can be described as the result of successively perturbed recurrence relations (50), with First we note that the efl k 's resulting from the perturbation are the same as those resulting from the perturbation which means that a perturbation to the second term in the jth recurrence relation can also be interpreted as a similar perturbation to the first term in the (j \Gamma 1)st recurrence relation. Now we consider perturbations that are introduced in each recurrence relation due to finite precision arithmetic errors. Let b actually computed and this can be rewritten, with different - and - 0 , as Since the perturbation to the second term in this jth recurrence relation can be interpreted as a similar perturbation to the first term in the (j \Gamma 1)st recurrence relation (which was already perturbed with a factor (1 + 3-)), we have that the computed b fl j can be interpreted as the result of perturbing each leading term with a factor (1 4 Discussion and Conclusions In Krylov subspace methods there are two main effects of floating point finite precision arithmetic errors. One effect is that the generated basis for the Krylov subspace deviates from the exact one. This may lead to a loss of orthogonality of the Lanczos basis vectors, but the main effect on the iterative solution process is a delay in convergence rather than mis-convergence. In fact, what happens is that we try to find an approximated solution in a subspace that is not as optimal, with respect to its dimension, as it could have been. The other effect is that the determination of the approximation itself is perturbed with rounding errors, and this is, in our view a serious point of concern; it has been the main theme of this study. In our study we have restricted ourselves to symmetric indefinite linear systems b. Before we review our main results, it should be noted that we should expect upper bounds for relative errors in approximations for x that contain at least the condition number of A, simply because we can in general not compute Ax k exactly. We have studied the effects of perturbations to the computed solution through their effect on the residual, because the residual (or its norm) is often the only information that we get from the process. This residual information is often obtained in a cheap way from some update procedure, and it is not uncommon that the updated residual may take values far beyond machine precision (relative to the initial residual). Our analysis shows that there are limits on the reduction of the true residual because of errors in the approximated solution. In view of the fact that we may expect at least a linear factor - 2 (A), when working with Euclidean norms, GMRES (x2.2) and SYMMLQ (x3) lead to acceptable approximate solutions. When these methods converge then the relative error in the approximate solution is, apart from modest factors, bounded by u - 2 (A). SYMMLQ is attractive since it minimizes the norm of the error, but it does so with respect to A times the Krylov subspace, which may lead to a delay in convergence with respect to GMRES (or MINRES), by a number of iterations that is necessary to gain a reduction by in the residual, see Theorem 3.2. For ill-conditioned systems this may be considerable. As has been pointed out in [14], the Conjugate Gradient iterates can be constructed with little effort from SYMMLQ information if the they exist. For indefinite systems the Conjugate Gradient iterates are well-defined for at least every other iteration step, and they can be used to terminate the iteration if this is advantageous. However, the Conjugate Gradient process has no minimization property (as for the positive definite case) when the matrix is indefinite and so there is no guarantee that any of these iterates will be sufficiently close to the desired solution before SYMMLQ converges. For indefinite symmetric systems we see that MINRES may lead to large perturbation errors: for MINRES the upper bound contains a factor This means that if the condition number is large, then the methods of choice are GMRES or SYMMLQ. Note that for the symmetric case, GMRES can be based on the three-term recurrence relation, which means that the only drawback is the necessity to store all the Lanczos vectors. If storage is at premium then SYMMLQ is the method of choice. If the given system is well-conditioned, and if we are not interested in very accurate solu- tions, then MINRES may be an attractive choice. Of course, one may combine any of the discussed methods with a variation on iterative refinement: after stopping the iteration at some approximation x k , we compute the residual possible in higher precision, and we continue to solve solution z j of this system is used to correct x . The procedure could be repeated and eventually this leads to approximations for x so that the relative error in the residual is in the order of machine precision (for more details on this, see [20]). However, if we would use MINRES then, after restart, we have to carry out at least a number of iterations for the reduction by a factor equal to the condition number, in order to arrive at something of the same quality as GMRES, which may make the method much less effective than GMRES. For situations where u, MINRES may be even incapable of getting at a sufficient reduction for the iterative refinement procedure to converge. It is common practice, among numerical analysts, to test the convergence behavior of Krylov subspace solvers for symmetric systems with well-chosen diagonal matrices. This gives often a quite good impression of what to expect for non-diagonal matrices with the same spectrum. However, as we have shown in our x2.5, for MINRES this may lead to a too optimistic picture, since floating point error perturbations with MINRES lead to errors in the residual (and the approximated solution) that are a factor smaller as for non-diagonal matrices. --R Templates for the solution of linear sys- tems:building blocks for iterative methods A theoretical comparison of the Arnoldi and GMRES algorithms A survey of preconditioned iterative methods Polynomial Based Iteration Methods for Symmetric Linear Systems Matrix Computations Behavior of slightly perturbed Lanczos and conjugate-gradient recurrences Iterative Methods for Solving Linear Systems Methods of conjugate gradients for solving linear systems Accuracy and Stability of Numerical Algorithms analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix Accuracy and effectiveness of the Lanczos algorithm for the symmetric eigenproblem Approximate solutions and eigenvalue bounds from Krylov subspaces Solutions of sparse indefinite systems of linear equations The symmetric eigenvalue problem GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems Reliable updated residuals in hybrid Bi-CG methods Relaxiationsmethoden bester Strategie zur L-osung linearer Gleichungssysteme Efficient High Accuracy Solutions with GMRES(m) The superlinear convergence behaviour of GMRES --TR

GMRES;MINRES;linear systems;iterative methods;SYMMLQ;stability

587872

Polynomial Instances of the Positive Semidefinite and Euclidean Distance Matrix Completion Problems.

Given an undirected graph G=(V,E) with node set V=[1,n], a subset $S\subseteq V$, and a rational vector $a\in {\rm\bf Q}^{S\cup E}$, the positive semidefinite matrix completion problem consists of determining whether there exists a real symmetric n n positive semidefinite matrix X=(xij) satisfying xii=ai ($i\in S$) and xij=aij ($ij\in E$). Similarly, the Euclidean distance matrix completion problem asks for the existence of a Euclidean distance matrix completing a partially defined given matrix. It is not known whether these problems belong to NP. We show here that they can be solved in polynomial time when restricted to the graphs having a fixed minimum fill-in, the minimum fill-in of graph G being the minimum number of edges needed to be added to G in order to obtain a chordal graph. A simple combinatorial algorithm permits us to construct a completion in polynomial time in the chordal case. We also show that the completion problem is polynomially solvable for a class of graphs including wheels of fixed length (assuming all diagonal entries are specified). The running time of our algorithms is polynomially bounded in terms of n and the bitlength of the input a. We also observe that the matrix completion problem can be solved in polynomial time in the real number model for the class of graphs containing no homeomorph K4.

Introduction . 1.1. The matrix completion problem. This paper is concerned with the completion problem for positive semidefinite and Euclidean distance matrices. The positive semidefinite matrix completion problem (P) is defined as follows: Given a graph E), a subset S ' V and a rational vector a 2 Q S[E , determine whether there exists a real matrix (The notation X - 0 means that X is a symmetric positive semidefinite matrix or, for short, a psd matrix.) In words, problem (P) asks whether a partially specified matrix can be completed to a psd matrix; the terminology of graphs being used as a convenient tool for encoding the positions of the specified entries. When problem (P) has a positive answer, one says that a is completable to a psd matrix; a matrix X satisfying (1.1) is called a psd completion of a and a positive definite (pd) completion when X is positive definite. We let (P s ) denote problem (P) when diagonal entries are specified. If one looks for a pd completion then one can assume without loss of generality that all diagonal entries are specified (cf. Lemma 2.5); this is however not obviously so if one looks for a psd completion (although this can be shown to be true when restricting the problem to the class of chordal graphs; cf. the proof of Theorem 3.5). i;j=1 is called a Euclidean distance matrix (a distance matrix, for short) if there exist vectors CWI, P.O. Box 94079, 1090 GB Amsterdam, The Netherlands (monique@cwi.nl). M. LAURENT (Here, kuk denotes the Euclidean norm of vector u 2 R k .) A set of vectors u i satisfying (1.2) is called a realization of Y . Note that all diagonal entries of a distance matrix are equal to zero. The Euclidean distance matrix completion problem (D) is defined as follows: Given a graph E) and a rational vector d 2 determine whether there exists a real matrix Y is a distance matrix and y Hence problem (D) asks whether a partially specified matrix can be completed to a distance matrix. As will be recalled in Section 2.3, psd matrices and distance matrices are closely related and, thus, their associated completion problems can often be treated in an analogous manner. These matrix completion problems have many applications, e.g., to multidimensional scaling problems in statistics (cf. [28]), to the molecule conformation problem in chemistry (cf. [10], [17]), to moment problems in analysis (cf. [5]). 1.2. An excursion to semidefinite programming. The psd matrix completion problem is obviously an instance of the general semidefinite programming feasibility problem Given integral n \Theta n symmetric matrices Q determine whether there exist real numbers z The complexity status of problem (F) is a fundamental open question in the theory of semidefinite programming; this is true for both the Turing machine model and the real number model, the two most popular models of computation used in complexity theory. In particular, it is not known whether there exists an algorithm solving (F) whose running time is polynomial in the size L of the data, that is, the total space needed to store the entries of the matrices Q The Turing machine model (also called rational number model, or bit model; cf. [12]) works on rational numbers and, more precisely, on their binary representations; in par- ticular, the running time of an elementary operation (+; \Gamma; \Theta; \Xi) depends on the length of the binary representations of the rational numbers involved. Hence, the size L of the data of problem (F) in this model can be defined as mn 2 L 0 , where L 0 is the maximum number of bits needed to encode an entry of a matrix Q i . On the other hand, the real number model (introduced in [9]) works with real numbers and it assumes that exact real arithmetic can be performed; in particular, an elementary operation (+; \Gamma; \Theta; \Xi) between any two real numbers takes unit time. Hence, the size L of the data of (F) in this model is equal to mn 2 . Semidefinite programming (SDP) deals with the decision problem (F) and its optimization version: s.t. . SDP can be seen as a generalization of linear programming (LP), obtained by replacing the nonnegativity constraints of the vector variable in LP by the semidefinite- ness of the matrix variable in SDP. Information about SDP can be found in the handbook [40]; cf. also the survey [38], and [3, 16] with an emphasis on applications to discrete optimization. A standard result in LP is that every feasible linear system: Ax - b with rational coefficients has a solution whose size is polynomially bounded in terms of the size of A and b (cf. [36], corollary 3.2b). This implies that the problem of testing feasibility of an LP program belongs to NP in the bit model (this fact is obvious for the real number model). Moreover, any LP optimization problem can be solved in polynomial time in the bit model using the ellipsoid algorithm of Khachiyan [22] or the interior-point method of Karmarkar [21]; it is an open question whether LP can be solved in polynomial time in the real number model (cf. p. 60 in [41]). The feasibility problem (F) belongs to NP in the real number model (since one can test in polynomial time whether a matrix is psd, for instance, using Gaussian elimination; in fact, for a rational matrix the running time is polynomial in its bitlength (cf. p. 295 in [15])). However, it is not known whether problem (F) belongs to NP in the bit model. Indeed, in contrast with LP, it is not true that if a solution exists then one exists which is rational and has a polynomially bounded size. Consider, for instance, the following 2 is the unique real for which X - 0; hence, this is an instance where there is a real solution but no rational solution. Consider now the following matrix (taken from and thus any rational solution has exponential bitlength. More examples of 'ill-conditioned' sdp's can be found in [33]. However, Ramana [33] has developed an exact duality theory for SDP which enables him to show the following results: Problem (F) belongs to NP " co-NP in the real number model. In the bit model, (F) belongs to NP if and only if it belongs to co-NP; hence, is not NP-complete nor co-NP complete unless NP=co-NP. Algorithms have been found that permit to solve the optimization problem (1.5) approximatively in polynomial time; they are based on the ellipsoid method (cf. [15]) and interior-point methods (cf. [31], [3]). 4 M. LAURENT More precisely, set K := fz S(K; ffl) := fy j 9z 2 K with points that are in the ffl-neighborhood of K') and S(K; \Gammaffl) ('the points that are at distance at least ffl from the border of K'). let L denote the maximum bit size of the entries of the matrices . Assume that we know a constant R ? 0 such that either with kzk - R. Then, the ellipsoid based algorithm, given rational ffl ? 0, either finds or asserts that S(K; Its running time is polynomial in n; m;L and log ffl and this algorithm is polynomial in the bit model. Assume that we know a constant R ? 0 such that kzk - R for all z 2 K and a point z 2 K for which 'strictly feasible'). There is an interior-point algorithm which finds y 2 K strictly feasible such that c T y - time polynomial in n; m; L; log ffl; log R and the bitlength of z . Note, however, that no polynomial bound has been established for the bitlengths of the intermediate numbers occurring in the algorithm. Khachiyan and Porkolab have shown that problem (F) and its analogue in rational numbers can be solved in polynomial time in the bit model for a fixed number m of variables. Theorem 1.1. (i) [32] Problem (F) can be solved in polynomial time for any fixed m. (ii) [23] The following problem can be solved in polynomial time for any fixed m: Given n \Theta n integral symmetric matrices Q rational numbers satisfying (1.4) or determine that no such numbers exist. The result from Theorem 1.1 (ii) extends to the context of semidefinite programming the result of Lenstra [29] on the polynomial solvability of integer LP in fixed dimension. 1.3. Back to the matrix completion problem. As the matrix completion problem is a special instance of SDP, it can be solved approximatively in polynomial time; specific interior-point algorithms for finding approximate psd and distance matrix completions have been developed, e.g., in [19],[10],[2],[30]. However, such algorithms are not garanteed to find exact completions in polynomial time. This motivates our study in this paper of some classes of matrix completion problems that can be solved exactly in polynomial time. As mentioned earlier, one of the difficulties in the complexity analysis of SDP arises from the fact that a rational SDP problem might have no rational solution (recall the example from (1.6)). This raises the following question in the context of matrix completion: If a rational partial matrix has a psd completion, does a rational completion always exist ? We do not know of a counterexample to this statement. On the other hand, we will show that the answer is positive, e.g., when the graph of specified entries is chordal or has minimum fill-in 1 (cf. Lemma 4.2). (Note that the answer is obviously positive if a pd completion exists.) Motivated by the above discussion, let us define for each of the problems (P) and (D) its rational analogue (P Q ) and (D Q ). Problem (P Q ) is defined as follows: Given a graph E), a subset S ' V and a rational vector a 2 Q S[E , find a rational matrix X satisfying (1.1) or determine that no such matrix exists. diagonal entries are specified), we denote the problem as (P Q Problem (D Q ) is defined as follows: Given a graph E) and a rational vector d 2 rational matrix Y satisfying (1.3) or determine that no such matrix exists. The complexity of the problems (P), (D), (P Q ), and (D Q ) is not known; in particular, it is not known whether they belong to NP in the bit model (they do trivially in the real number model). In this paper, we present some instances of graphs for which the completion problems can be solved in polynomial time. All our complexity results apply for the bit model (unless otherwise specified, as in Section 5.3). Recall that a graph is said to be chordal if it does not contain a circuit of length - 4 as an induced subgraph. Then, the minimum fill-in of graph G is defined as the minimum number of edges needed to be added to G in order to obtain a chordal graph. Note that computing the minimum fill-in of a graph is an NP-hard problem [42]. The following is the main result of Sections 3 and 4. Theorem 1.2. For any integer m - 0, problems (P), (P Q ), (D) and (D Q ) can be solved in polynomial time (in the bit model) when restricted to the class of graphs whose minimum fill-in is equal to m. The essential ingredients in the proof of Theorem 1.2 are the subcase case), Theorem 1.1, and the link (exposed in Section 2.3) between psd matrices and distance matrices. In the chordal case, a simple combinatorial algorithm permits to solve the completion problem in polynomial time. The psd matrix completion problem for chordal graphs has been extensively studied in the literature (cf. the survey of Johnson [18] for detailed references). In some sense, this problem has been solved by Grone, Johnson, S'a and Wolkowicz [14] who, building upon a result of Dym and Gohberg [11], have characterized when a vector a indexed by the nodes and edges of a chordal graph admits a psd completion; cf. Theorem 3.1. From this follows the polynomial time solvability of problem (P s ) for chordal graphs. In fact, the result from Theorem 3.1 is proved in [14] in a constructive manner and, thus, yields an algorithm permitting to solve problem (P Q s ) for chordal graphs. This algorithm has a polynomial running time in the real number model; however, it has to be modified in order to achieve a polynomial running time in the bit model. To summarize, the result from Theorem 1.2 also holds in the real number model for chordal graphs would hold for all graphs having fixed minimum fill-in m - 1 if the result from Theorem 1.1 would remain valid in the real number model 1 . We present in Section 5.1 another class of graphs for which the matrix completion problem (P s ) can be solved in polynomial time (in the bit model). This class contains (generalized) circuits and wheels having a fixed length (and fatness); these graphs arise naturally when considering the polar approach to the psd matrix completion problem. Then, Section 5.2 contains a brief description of this polar approach, together with some open questions and remarks. In the final Section 5.3, we consider the matrix completion problem for the class of graphs containing no homeomorph of K 4 (it contains circuits). Then a condition characterizing existence of a psd or distance matrix completion exists which permits to obtain a simple combinatorial algorithm solving the existence and construction problems in polynomial time in the real number model. claims to have a proof of this fact. 6 M. LAURENT 2. Preliminaries. We recall here some basic facts about Schur complements and Euclidean distance matrices that will be needed in the paper and we make a few observations about psd completions. 2.1. Schur complements. For a symmetric matrix M , set In(M) := (p; q; r), where denotes the number of positive (resp. negative, zero) eigenvalues of M . When M - 0, a maximal nonsingular principal submatrix of M is a nonsingular principal submatrix of M of largest possible order, thus equal to the rank of M . Lemma 2.1. Let be a symmetric matrix, where A is nonsingular. Then, the matrix known as the Schur complement of A in M . In particular, A is a maximal nonsingular principal submatrix of M , then As a direct application, we have the following results which will be used at several occasions in the paper. Lemma 2.2. Let X be a symmetric matrix having the following block decomposition: where T , R, Z, A, S, D are rational matrices of suitable orders; all entries of X being specified except those of Z that have to be determined in order to obtain X - 0. Assume that R A 0: In the case when n - 1 and A 6= 0, let A 0 be a maximal nonsingular principal submatrix of A, let denote the corresponding block decompositions of A and X. Then, X - 0 if we set Z when Proof. The result follows using Lemma 2.1 after noting that the Schur complement of A 0 in X is given by@ T R T Indeed, the Schur complement in A is equal to 0 since A - 0 and A 0 is a maximal nonsingular principal submatrix of A; as implies that Lemma 2.3. Let X be a symmetric matrix of the form R A where A - 0 and T is a symmetric matrix of order ' whose diagonal entries are all equal to some scalar N . Let A 0 be a maximal nonsingular principal submatrix of A and let denote the corresponding block decompositions of A and X. Then, X - 0 if and only if (i) In particular, X is pd if and only if A and are pd. Moreover, T large enough (namely, for N greater or equal to the largest eigenvalue of R T diagonal entries and as off-diagonal entries those of T ). 2.2. Some observations about positive semidefinite completions. Given a graph E), a subset S ' V , a vector a 2 Q S[E , and a scalar N ? 0, let a denote the extension of a obtained by setting a i := N for all Obviously, Lemma 2.4. a is completable to a psd matrix if and only if a N is completable to a psd matrix for some N ? 0 (and then for all N 0 - N ). Therefore, if one can "guess" a value N to assign to the unspecified diagonal entries, then one can reduce the problem to the case when all diagonal entries are specified. This can be done when the graph G of specified off-diagonal entries is chordal as we see later or if we look for a pd completion as the next result shows. Lemma 2.5. Given a 2 Q its restriction to the subgraph induced by S. Then, a has a pd completion if and only if b has a pd completion. Proof. Apply Lemma 2.3. This result does not extend to psd completions (which contradicts a claim from [14] (psd case in Prop. 1)). Indeed, the following partial matrix@ has no psd completion while its lower principal 2 \Theta 2 submatrix is psd. 8 M. LAURENT A final observation is that if a partial matrix contains a fully specified row, then the completion problem can be reduced to considering a matrix of smaller order. Indeed, suppose that is a partial symmetric matrix whose first row is fully specified. If a A is not completable. If a then A is completable if and only if its first row is identically zero and its lower principal submatrix of order If a 11 ? 0 then one can reduce to a problem of order considering the Schur complement of a 11 in A. 2.3. Euclidean distance matrices. The following connection (2.4) between psd and distance matrices has been established by Schoenberg [35]. Let square symmetric matrix with zeros on its main diagonal and whose rows and columns are indexed by a set V , and let i 0 be a given element of V . Then, ' i 0 (Y ) denotes the square whose rows and columns are indexed by set V nfi 0 g and whose entries are given by Then, one can easily verify that Y is a distance matrix (Indeed, a set of vectors u realization of the matrix Y if and only if ' i 0 (Y ) is the Gram matrix of the vectors u which means that its (i; j)-th entry is equal to establishes a linear bijection between the set of distance matrices of order jV j and the set of psd matrices of order (2.4) has a direct consequence for the corresponding matrix completion problems. Let E) be a graph and assume that i 0 2 V is a universal node, i.e., that i 0 is adjacent to all other nodes of G. Then, an algorithm permitting to solve the psd matrix completion problem for graph Gni 0 can be used for solving the distance matrix completion problem for graph G and vice versa. Indeed, Y is a distance matrix completion of d 2 R E completion of ' i 0 (d) (For the definition of ' i 0 (d) use (2.3) restricted to the pairs ij with or i 6= j with ij edge of G.) For more information about connections between the two problems, see [20],[26]. 3. The Matrix Completion Problem for Chordal Graphs. We consider here the matrix completion problems for chordal graphs. First, we recall results from [14] and [4] yielding a good characterization for the existence of a completion; then, we see how they can be used for constructing a completion in polynomial time. 3.1. Characterizing existence of a completion. Let E) be a graph and let a 2 Q V [E be a vector; in the distance matrix case, the entries of a indexed by V (corresponding to the diagonal entries of a matrix completion) are assumed to be equal to zero. If K ' V is a clique in G (i.e., any two distinct nodes in K are joined by an edge in G), the entries a ij of vector a are well-defined for all nodes denote the jKj \Theta jKj symmetric matrix whose rows and columns are indexed by K and with ij-th entry a ij for Obviously, if a is completable to a psd matrix, then a satisfies: for every maximal clique K in G: Similarly, if a is a completable to a distance matrix, then a satisfies: a(K) is a distance matrix for every maximal clique K in G: The conditions (3.1) and (3.2) are not sufficient in general for ensuring the existence of a completion. For instance, if E) is a circuit and a 2 Q V [E has all its entries equal to 1 except one entry on an edge equal to \Gamma1, then a satisfies (3.1) but a is not completable to a psd matrix. However, if G is a chordal graph, then (3.1) and (3.2) suffice for ensuring the existence of a completion. Theorem 3.1. Let E) be a chordal graph and let a 2 R V [E . If a satisfies (3.1), then a is completable to a psd matrix [14]; if a satisfies (3.2), then a is completable to a distance matrix [4]; moreover, if a is rational valued, then a admits a rational completion. As the maximal cliques in a chordal graph can be enumerated in polynomial time [37] (cf. below) and as one can check positive semidefiniteness of a rational matrix in polynomial time (cf. [15], p. 295), one can verify whether (3.1) holds in polynomial time when G is chordal; in view of (2.4), one can also verify whether (3.2) holds in polynomial time when G is chordal. This implies: Theorem 3.2. Problems (P s ) and (D) can be solved in polynomial time for chordal graphs. The proof given in [14, 4] for Theorem 3.1 is constructive; thus, it provides an algorithm for constructing a completion and, as we see below, a variant of it can be shown to have a polynomial running time. The proof is based on the following properties of chordal graphs. E) be a graph. Then, G is chordal if and only if it has a perfect elimination ordering; moreover, such an ordering can be found in polynomial time [34]. An ordering v of the nodes of a graph E) being called a perfect elimination ordering if, for every 1, the set of nodes v k with k ? j that are adjacent to v j induces a clique in G. For the clique consisting of node v j together with the nodes are adjacent to v j ; then the cliques K maximal cliques of a chordal graph G. Hence, if G is chordal and not a clique, then one can find (in polynomial time) an edge which the graph H := G (obtained by adding e to G) is chordal. (Indeed, let i be the largest index in [1; n] for which there exists j ? i such that v i and v j are not adjacent in G; then we can choose for e the pair ij as v remains a perfect elimination ordering for H.) Moreover, if G is chordal then, for any e 62 E, there exists a unique maximal clique in containing edge e [14] (easy to check). Therefore, if G is complete and not a clique, we can order the missing edges in G as in such a way that the graph G q := chordal for every q be the unique maximal clique in G q containing edge e q . Given a 2 Q V [E satisfying (3.1), set G 0 := G and x 0 := a. We execute the following step for Find z q 2 Q for which the vector x q := This can be done in view of Lemma 2.2 (case applied to the matrix X := x q (K q ) and one can choose for z q the rational value given by (2.2). Then, the final vector M. LAURENT rational psd completion of a. This shows Theorem 3.1 in the psd case (the Euclidean distance matrix case being similar). As mentioned earlier, the preprocessing step (find the suitable ordering e the missing edges and the cliques K q ) can be done in polynomial time. Then, one can construct the values z yielding a psd completion of a in p - n 2 steps. Therefore, the algorithm is polynomial in the real number model. In order to show polynomiality in the bit model, one has to verify that the encoding sizes of z polynomially bounded in terms of n and the encoding size of a. This is, however, not clear. Indeed, both R 0 and S 0 in the definition of z q via (2.2) may involve some previously defined z h for (in fact, the same may hold for A 0 ); then, we have a quadratic dependence between z q and the previously defined z which may cause a problem when trying to prove that the encoding size of z q remains polynomially bounded. However, as we see below, the above algorithm can be modified to obtain a polynomial running time. The basic idea is that, instead of adding the missing edges one at a time, one adds them by 'packets' consisting of edges sharing a common endnode. Then, in view of Lemma 2.2, one can specify simultaneously all the entries on these edges, which permits to achieve a linear dependency among the z q 's. 3.2. Constructing a psd completion in polynomial time. Let E) be a chordal graph and let perfect elimination ordering of its nodes. For and let denote the elements set F ' := fi ' denote the graph with node set V and edge set Hence, we have a sequence of graphs: where each G ' is chordal remains a perfect elimination ordering of its nodes) and GL is the complete graph. We now show that G ' has only one maximal clique which is not a clique in G '\Gamma1 . Lemma 3.3. For there is a unique maximal clique K ' in G ' which is not a clique in G '\Gamma1 . Moreover, J(i is a clique in G and the set K ' n J(i ' ) is a clique in G. Proof. Let K be a maximal clique in G ' which is not a clique in G first show that J(i ' ) ' K. For this, assume that By maximality of K, there exists an element i 2 K such that i and j 0 are not adjacent in G ' . Then, since the set [i ' ; n] is a clique in G ' . Therefore, the pairs ij and ii ' are edges of G ' and, thus, of G. Since the ordering of the nodes is a perfect elimination ordering for G, this implies that i ' and j must be adjacent in G, yielding a contradiction. Suppose now that K;K 0 are two distinct maximal cliques in G ' such that and exist nodes i 2 K n K 0 that are not adjacent in G ' . Given a node j 2 J(i ' ), one can easily verify that (i; i is an induced circuit in G '\Gamma1 , which contradicts the fact that G '\Gamma1 is chordal and, thus, shows unicity of the clique K ' . It is obvious that K ' n fi ' g is a clique in G '\Gamma1 . We now verify that K ' n J(i ' ) is a clique in G. For this, note first that i ' is adjacent to every node of K ' G ' and, thus, in G. Suppose now that x 6= y are two nodes in K ' n (J(i that are not adjacent in G. Then, as xy is an edge of G '\Gamma1 , we have: y. As i ' is adjacent to both x and y in G this implies that x and y must be adjacent in G, yielding a contradiction. We now describe the modified algorithm. Let E) be a chordal graph and let a Setting x 0 := a, we execute the following step for Find z ' 2 Q F ' for which the vector x ' := Then, the final vector rational psd completion of a. For instance, we can choose for z ' the value given by relation (2.2), applying Lemma 2.2 to the matrix X := x ' (K ' ). (Indeed, in view of Lemma 3.3, can be verified by induction.) We verify that the encoding sizes of z are polynomially bounded in terms of n and the encoding size of a. For this, we note that z are determined by a recurrence of the form: are matrices of (appropriate) orders - n. A crucial observation is that all entries of R ' and A ' belong to the set, denoted as A, of entries of a (as K ' n J(i ' ) is a clique in G, by Lemma 3.3), while the entries of S ' belong to the set A [ Z Z denotes the set of entries of (z For r 2 Q, let hri denote the encoding size of r, i.e., the number of bits needed to encode r in binary notation and, for a vector i). One can verify that, for two vectors a denote the maximum encoding length of the entries of vector a and, for We derive from (3.6) that for all ' (setting S 0 := 0). This implies that As L - n, we obtain that all encoding sizes of z are polynomially bounded in terms of n and the encoding size of a. (We also use here the fact that the entries of A \Gamma1 are polynomially bounded in the input size; cf. chap. 1.3 in [15].) Thus, we have shown: Theorem 3.4. Problem (P Q s ) can be solved in polynomial time for chordal graphs. We finally indicate how to solve the general problem when some diagonal entries are unspecified. Theorem 3.5. Problems (P) and (P Q ) can be solved in polynomial time for chordal graphs. Proof. Let E) be a chordal graph, let S ' V and let a 2 Q S[E satisfying: for each maximal clique K ' S (else, we can conclude that a is not completable). M. LAURENT Following Lemma 2.4, we search for a scalar N ? 0 such that a is completable if and only if its extension a N 2 Q V [E (assigning value N to the unspecified diagonal entries) is completable or, equivalently, a N (K) - 0 for all maximal cliques K in G. Note that each matrix a N (K) has the same form as matrix X from Lemma 2.3. Therefore, such N exists if and only if the linear condition (i) from Lemma 2.3 holds for each clique K and an explicit value for N can be constructed as indicated in Lemma 2.3. Once N has been determined, we proceed with completing a N by applying the algorithm presented above. To conclude note that the algorithm presented in this section outputs a pd completion if one exists. 3.3. Constructing a distance matrix completion. The distance matrix completion problem for chordal graphs can be solved in an analogous manner. Namely, let E) be a chordal graph, let be the sequence of chordal graphs from (3.4), let K ' be the cliques constructed in Lemma 3.3, and let a 2 Q Setting a 0 := a, we execute the following step for Find z ' 2 Q F ' for which the vector x ' := (a x ' (K ' ) is a distance matrix. Then, the final vector provides a distance matrix completion of a. The above step can be performed as follows. If then we let z ' be defined by z ' (j) := x is a given element of J(i ' ). Otherwise, let is a universal node in G[K ' ], the subgraph of G induced by Therefore, in view of relation (2.5), we can find z ' satisfying (3.7) by applying Lemma 2.2. The polynomial running time of the above algorithm follows from the polynomial running time of the corresponding algorithm in the psd case. Thus, we have shown: Theorem 3.6. Problem (D Q ) can be solved in polynomial time for chordal graphs. 4. The Matrix Completion Problem for Graphs with Fixed Minimum Fill- In. In this section we describe an algorithm permitting to solve problems (P), (P Q and (D Q ) in polynomial time for the graphs having minimum fill-in m, where m - 1 is a given integer. This algorithm is based on Theorems 1.1, 3.1, 3.2, 3.4 and 3.6. E) be a graph with minimum fill-in m, let S ' V and let a 2 Q S[E be given. (Again we assume that a in the distance matrix case.) We first execute the following step. Step 0. Find edges which the graph H := chordal and find the maximal cliques K (Such edges exist since G has and they can be found in polynomial time, simply by enumeration as m is fixed. The maximal cliques in H can also be enumerated in polynomial time since H is chordal and, moreover, p - n.) Then, we perform step x in order to solve problem (x) for x=P,P Q ,D,D Q . Step P. Determine whether there exist real numbers z for which the vector defined by x i := a x eh := z h Step D. Determine whether there exist real numbers z for which the vector x 2 defined by x are distance matrices. Then, a has a completion if and only if the answer in step P or D is positive. Step P Q . Find rational numbers z holds or determine that no such numbers exist; if they exist, find a rational psd completion of x. Step D Q . Find rational numbers z holds or determine that no such numbers exist; if they exist, find a rational distance matrix completion of x. Steps P and P Q can be executed in the following manner. Let M denote the block diagonal matrix with the p matrices x(K 1 diagonal blocks (and zeros else- Hence, M has order jK holds if and only if M - 0. Clearly, the matrix M can be written under the form: are symmetric matrices with (0,1)-entries and Q 0 is a symmetric matrix whose nonzero entries belong to the set of entries of a. Therefore, in view of Theorem 1.1, one can determine the existence of z satisfying (4.1) in polynomial time. Then, finding a rational psd completion of x in step P Q can be done in polynomial time in view of Theorem 3.4. In the distance matrix case, we use the following construction for distance matrices. For a be a square symmetric matrix whose rows and columns are indexed by set V a and let i a be a given element of V a . We construct a new matrix D, denoted as whose rows and columns are indexed by set and whose entries are given by ae D a (i; D a (i; i a Lemma 4.1. D is a distance matrix if and only if D are distance matrices. Proof. The 'only if' part is obvious. Conversely, assume that D are distance matrices; we show that D := D 1 is a distance matrix. For a 2 [1; p], let u a a ) be vectors providing a realization of D a ; we can assume without loss of generality that u a i a we construct a sequence of vectors w i 2 R n1+:::+np (i 2 setting w i := (0 denotes the zero vector in R n ). One can easily verify that the vectors w i provide a realization of D. Steps D and D Q can be performed as follows. Let M := x(K 1 the matrix indexed by K constructed as indicated in relation (4.3). Clearly, M can be written under the form: 14 M. LAURENT are symmetric matrices with entries in f0; 1g and Q 0 is a symmetric matrix whose nonzero entries are sums of at most two entries of a. Let i 0 be a given element of K Hence, (4.2) holds if and only if matrix M is a distance matrix (by Lemma 4.1) or, equivalently, if and only if ' i 0 (M) is positive semidefinite (by relation (2.4)). Therefore, in view of Theorems 3.2 and 3.6, steps D and D Q can be executed in polynomial time. This completes the proof of Theorem 1.2. Lemma 4.2. When the minimum fill-in m is equal to 1, existence of a completion implies existence of a rational one. Proof. To see it, suppose first that all diagonal entries are specified; then, steps P and P Q can be executed in an elementary manner. Indeed, each matrix x(K i has at most one unspecified entry z 1 . Hence, the set of scalars z 1 for which x(K i an interval of the form I to see from Lemma 2.2). Therefore, (4.1) holds if and only if z 1 2 I and v := min i there is a completion (i.e., if u - v), then one can find one with z 1 rational. This is obvious if this follows from the fact (easy to verify) that Suppose now some diagonal entries are unspecified. If there is a completion with value z 2 at the unspecified diagonal entries, then we can assume that z 2 is rational (replacing if necessary z 2 by a larger rational number). Then, by the above discussion, the off-diagonal unspecified entry z 1 can also be chosen to be rational. 5. Further Results and Open Questions. We present in Section 5.1 another class of graphs for which the completion problem can be solved in polynomial time (in the bit model). Then, we discuss in Section 5.2 some open questions arising when considering a polar approach to the positive semidefinite completion problem. Finally, we describe in Section 5.3 a simple combinatorial algorithm permitting to solve the completion problem in polynomial time (in the real number model) for the class of graphs containing no homeomorph of K 4 . 5.1. Another class of polynomial instances. We present here another class of graphs for which the positive semidefinite matrix completion problem (P s ) can be solved in polynomial time. Given two integers p;q be the class consisting of the E) satisfying the following properties: There exist two disjoint subsets is disjoint from E, the graph is chordal, and H has q maximal cliques that are not cliques in G. We show: Theorem 5.1. Given integers p; q - 1, the positive semidefinite completion problem can be solved in polynomial time (in the bit model) over the class G p;q . Examples of graphs belonging to class G p;q arise from circuits, wheels and some gen- eralizations. A generalized circuit of length n is defined in the following manner: its node set is U with two nodes being adjacent if and only if generalized wheel of length n is obtained by adding a set U 0 (the center of the wheel) of pairwise adjacent nodes to a generalized circuit of length n and making each node in U 0 adjacent to each node in U Call a generalized circuit or wheel p-fat if min(jU p-fat generalized circuit or wheel of length q belongs to G p;q . We will see in Section 5.2 that generalized circuits and wheels arise as basic objects when studying the matrix completion problem on graphs of small order. figwheel.eps Fig. 5.1. (a) The wheel of length 4; (b) a 2-fat generalized wheel of length 4 The proof of Theorem 5.1 is based on the following result of Barvinok [8], which shows that one can test feasibility of a system of quadratic equations in polynomial time for any fixed number of equations 2 . Theorem 5.2. For be a quadratic polynomial in x 2 R n , where A i is an n \Theta n symmetric matrix, b i 2 R n and c i 2 R. One can test feasibility of the system: f i in polynomial time (in the bit model) for any given m. Proof of Theorem 5.1. Let E) be a graph in class G p;q and let a 2 R V [E be given. We are also given the sets V 1 and V 2 for which, say, adding to G all edges in F := fij creates a chordal graph H . We show that deciding whether a can be completed to a psd matrix amounts to testing the feasibility of a system of m quadratic polynomials where m depends only on p and q. As H is chordal, a is completable to a psd matrix if and only if there exists a matrix Z of order for which x := (a; Z) 2 R V [E[F satisfies: x(K) - 0 for each maximal clique K in H . We assume that for each maximal clique K of H contained in G (else, we can conclude that a is not completable). Consider now a maximal clique K of H which is not contained in G. Then, x(K) has the following form: setting the submatrix of Z with row indices in the notation of Lemma 2.2, we obtain that x(K) - 0 if and only if the following matrix is positive semidefinite (we have assumed that A - 0). We can apply again a Schur decomposition to matrix MK in order to reformulate the condition on Z. Setting TK := In [8] Barvinok considers the homogeneous case, where each equation is of the form: f i for some symmetric matrix A i . However, the general nonhomogeneous case can be derived from it (Barvinok, personal communication, 1998). M. LAURENT we have that . Let D 0 0 be a largest nonsingular submatrix of D 0 and let denote the corresponding block decompositions of D 0 and MK . Taking the Schur complement of D 0 0 in MK , we obtain that MK - 0 if and only if Let YK := Z[V denote the column submatrix of Z with column indices in and set Then, Therefore, the condition: x(K) - 0 can be rewritten as the sytem: ae are matrices depending on input data a. We can reformulate condition (1K) as an equation by introducing a new square matrix SK of order namely, rewrite (1K) as the columns of matrix Z, and let s K denote the columns of matrix SK for each clique K. Then, condition (1'K) can be expressed as a system of equations of the form: where f is a quadratic polynomial; similarly for condition (2K). The total number of quadratic equations obtained in this manner depends only on p and q. Therefore, in view of Theorem 5.2, one can check feasibility of this system in polynomial time when p and q are fixed. p;q denote the subclass of G p;q consisting of the graphs G for which every maximal clique of H (the chordal extension of G) which is not a clique of G is not contained in Then, the Euclidean distance matrix completion problem can be solved in figX.eps Fig. 5.2. The matrix completion problem for generalized circuits of length 4 polynomial time over the class G 0 p;q for any fixed p and q. The proof is similar to that of Theorem 5.1, since we can get back to the psd case using relation (2.4) (a matrix and its image under ' i 0 having same pattern of unknown entries if i 0 belongs to V n In particular, the Euclidean distance matrix completion problem can be solved in polynomial time for generalized circuits of length 4 and fixed fatness, or for generalized wheels (with a nonempty center) of fixed length and fatness. The complexity of the psd completion problem for generalized wheels and circuits is not known; in fact, in view of the remark made at the end of Section 2.2, it suffices to consider circuits. In view of Theorem 5.1, the problem is polynomial if we fix the length and the fatness of the circuit. It would be particularly interesting to determine the complexity of the completion problem for generalized circuits of length 4 and unrestricted fatness. This problem can be reformulated as follows: Determine whether and how one can fill the unspecified entries in the blocks marked '?' of the matrix X shown in Figure 5.2, so as to obtain X - 0 (all entries are assumed to be specified in the grey blocks). Indeed, as will be seen in Section 5.2, these graphs constitute in some sense the next case to consider after chordal graphs. 5.2. A polar approach to the completion problem. Given a graph consider the cone CG consisting of the matrices lies on an extremal ray of the cone CG (i.e., define the order of G as the maximum rank of an extremal matrix X 2 CG . It is shown in [1] that a 2 R V [E is completable to a psd matrix if and only if a satisfies: a for every extremal matrix One might suspect that the psd matrix completion problem is somewhat easier to solve for graphs having a small order since the extremal matrices in CG have then a small rank. Indeed, the graphs of order 1 are precisely the chordal graphs, for which the problem is polynomially solvable. On the other hand, a circuit of length n has order which is the highest possible order for a graph on n nodes. Moreover, if i 0 is a universal node in a graph G, then both graphs G and G n i 0 have the same order, which corroborates the observation made at the end of Section 2.2. A natural question concerns the complexity of the problem for graphs of order 2. The graphs of order 2 have been characterized in [27]. It is shown there that, up to a simple graph operation (clique-sum), they belong to two basic classes G 1 and G 2 . All the graphs in G 1 have minimum fill-in at most 3; hence, the problem is polynomially solvable for them (by Theorem 1.2). The graphs in class G 2 are the generalized wheels of length 4 (and unrestricted fatness). Hence, if the psd matrix completion problem is polynomially solvable for generalized wheels of length 4, then the same holds for all graphs of order 2. 5.3. The matrix completion problem for graphs with no homeomorph of K 4 . We now discuss the matrix completion problem for the class H consisting of the graphs M. LAURENT figK4.eps Fig. 5.3. A homeomorph of K 4 containing no homeomorph of K 4 as a subgraph; a homeomorph of K 4 being obtained from K 4 by repacing its edges by paths. (Graphs in H are also known as series parallel graphs.) Clearly, H contains all circuits. The case of circuits is certainly interesting to understand since circuits are the most simple nonchordal graphs. Similarly to the chordal case, a condition characterizing existence of a psd completion is known for the graphs in H. Namely, the following is shown in [24] (using a result of [7]). Given a graph E) in H and a 2 R V [E satisfying: a then a has a psd completion if and only if the scalars x e := 1 - arccos a e (e 2 E) satisfy the inequalities: G; jF j odd. Proposition 5.3. [6] Given x 2 [0; test in polynomial time whether x satisfies the linear system (5.2). Proof. Consider the graph ~ E) consists of the pairs ij; is easy to see that x satisfies (5.2) if and only if z(P for every path P from i to i 0 in ~ G and every . The result now follows as one can compute shortest paths in polynomial time. Therefore, problem (P s ) is polynomial time solvable in the real number model for graphs in H. It is not clear how to extend this result to the bit model since the scalars - arccos a e are in general irrational and, thus, one encounters problems of numerical stability when trying to check whether (5.2) holds. Moreover, there is a simple combinatorial algorithm (already briefly mentioned in [25]) permitting to construct a psd completion in polynomial time in the real number model. E) be a graph in H and let a 2 R V [E be given satisfying a The algorithm performs the following steps. 1. Set x e := 1 - arccos a e for e 2 E and test whether x satisfies (5.2). If not, one can conclude that a has no psd completion. Otherwise, go to 2. 2. Find a set F of edges disjoint from E for which the graph H := and contains no homeomorph of K 4 . 3. Find an extension y 2 [0; 1] E[F of x satisfying the linear system (5.2) with respect to graph H . 4. Set b e := cos(-y e ) for e is completable to a psd matrix (since y satisfies (5.2) and H has no homeomorph of K 4 ) and one can compute a psd completion X of b with the algorithm of Section 3.2 (since H is chordal). Then, X is a completion of a. All steps can be executed in polynomial time. This follows from earlier results for steps 1 and 4; for step 2 use a result of [39] and, for step 3, one can use an argument similar to the proof of Proposition 5.3. Namely, given x 2 [0; satisfying (5.2), in order to extend x to [0; 1] E[feg in such a way that (5.2) remains valid with respect to G+ e, one has to find a scalar ff 2 [0; We have: ;. With the notation of the proof of Proposition 5.3, one finds that is an ab path in ~ is an ab \Gamma path in ~ Hence one can compute ff in polytime. One can then determine the extension y of x to H by iteratively applying this procedure. The distance matrix completion problem for graphs in H can be treated in a similar manner. Indeed, given E) in H and a 2 R E a e for e 2 E. Then, a is completable to a distance matrix if and only if x satisfies the linear inequalities: f2Cne circuits C in G and all e 2 C: (Cf. [26].) Again one can test in polynomial time whether x - 0 satisfies (5.3) (simply, test for each edge is an ab \Gamma path in G)). An algorithm analogous to the one exposed in the psd case permits to construct a distance matrix completion. Therefore, we have shown: Theorem 5.4. One can construct a real psd (distance matrix) completion or decide that none exists in polynomial time in the real number model for the graphs containing no homeomorph of K 4 . It is an open question whether the above result extends to the bit model of computa- tion, even for the simplest case of circuits. Acknowledgements . We are grateful to A. Barvinok for providing us insight about Theorem 5.2, to L. Porkolab for bringing [23] to our attention, and to A. Schrijver for discussions about Section 3. We also thank the referees for their careful reading and for their suggestions which helped us improving the presentation of the paper. --R Positive semidefinite matrices with a given sparsity pattern. Solving Euclidean distance matrix completion problems via semidefinite programming. Interior point methods in semidefinite programming with applications in combinatorial optimization. The Euclidean distance matrix completion problem. On the matrix completion method for multidimensional moment problems. The real positive definite completion problem for a simple cycle. Feasibility testing for systems of real quadratic equations. On a theory of computation and complexity over the real num- bers: NP-completeness Distance Geometry and Molecular Conformation. Extensions of band matrices with band inverses. Computers and Intractability: A Guide to the Theory of NP- Completeness Algorithmic Theory and Perfect Graphs. An interior-point method for semidefinite programming The molecule problem: exploiting structure in global optimization. Matrix completion problems: a survey. An interior-point method for approximate positive semidefinite completions Connections between the real positive semidefinite and distance matrix completion problems. A new polynomial-time algorithm for linear programming A polynomial algorithm in linear programming. Computing integral points in convex semi-algebraic sets The real positive semidefinite completion problem for series-parallel graphs Cuts, matrix completions and graph rigidity. A connection between positive semidefinite and Euclidean distance matrix completion problems. On the order of a graph and its deficiency in chordality. Theory of multidimensional scaling. Interior Point Polynomial Algorithms in Convex Program- ming: Theory and Algorithms On the complexity of semidefinite programs. An exact duality theory for semidefinite programming and its complexity implica- tions Algorithmic aspects of vertex elimination on graphs. Remarks to M. Theory of Linear and Integer Programming. Decomposition by clique separators. programming. Steiner trees Handbook of Semidefinite Program- ming: Theory Computing the minimum fill-in is NP-complete --TR --CTR Henry Wolkowicz , Miguel F. Anjos, Semidefinite programming for discrete optimization and matrix completion problems, Discrete Applied Mathematics, v.123 n.1-3, p.513-577, 15 November 2002

real number model;chordal graph;positive semidefinite matrix;polynomial algorithm;euclidean distance matrix;order of a graph;minimum fill-in;bit model;matrix completion

587873

Preconditioners for Nondefinite Hermitian Toeplitz Systems.

This paper is concerned with the construction of circulant preconditioners for Toeplitz systems arising from a piecewise continuous generating function with sign changes.If the generating function is given, we prove that for any $\varepsilon >0$, only ${\cal O} (\log N)$ eigenvalues of our preconditioned Toeplitz systems of size N N are not contained in $[-1-\varepsilon, -1+\varepsilon] \cup [1 -\varepsilon, 1+\varepsilon]$. The result can be modified for trigonometric preconditioners. We also suggest circulant preconditioners for the case that the generating function is not explicitly known and show that only ${\cal O} (\log N)$ absolute values of the eigenvalues of the preconditioned Toeplitz systems are not contained in a positive interval on the real axis.Using the above results, we conclude that the preconditioned minimal residual method requires only ${\cal O} (N \log ^2 N)$ arithmetical operations to achieve a solution of prescribed precision if the spectral condition numbers of the Toeplitz systems increase at most polynomial in N. We present various numerical tests.

Introduction Let L 2- be the space of 2-periodic Lebesgue integrable real-valued functions and let C 2- be the subspace of 2-periodic real-valued continuous functions with norm The Fourier coefficients of f 2 L 2- are given by a \Gamma- Research supported in part by the Hong Kong-German Joint Research Collaboration Grant from the Deutscher Akademischer Austauschdienst and the Hong Kong Research Grants Council. and the sequence fAN (f)g 1 N=1 of (N; N)-Toeplitz matrices generated by f is defined by real-valued the matrices AN (f) are Hermitian. We are interested in the iterative solution of Toeplitz systems where the generating function f 2 L 2- . To be more precise, we are looking for good preconditioning strategies so that Krylov space methods applied to the preconditioned system converge in a few number of iteration steps. Note that by the Toeplitz structure of AN each iteration step requires only O(N log N) arithmetical operations by using fast Fourier transforms. Preconditioning techiques for Toeplitz systems have been well-studied in the past 10 years. However, most of the papers in this area are concerned with the case where the generating function f is either positive or nonnegative, see for instance [4, 3, 18, 6, 16, 9] and the references therein. In this paper, we consider f that has sign changes. The method we propose here will also work for generating functions that are positive or nonnegative. Up to now iterative methods for Toeplitz systems with generating functions having different signs were only considered in [18, 20] and in connection with non-Hermitian systems in [7, 5]. In [7], we have constructed circulant preconditioners for non-Hermitian Toeplitz matrices with known generating function of the form where p is an arbitrary trigonometric polynomial and h is a function from the Wiener class with jhj ? 0. We proved that the preconditioned matrices have singular values properly clustered at 1. Then, if the spectral condition number of AN (f) fulfills - 2 (AN the conjugate gradient method (CG) applied to the normal equation requires only O(log N) iteration steps to produce a solution of fixed precision. However, in general nothing can be said about the eigenvalues of the preconditioned matrix. In this paper, we consider real-valued functions f 2 L 2- of the form where Y is a trigonometric polynomial with a finite number of zeros of even order 2s j and where h 2 L 2- is a piecewise continuous function with simple discontinuities at there exist h(- j \Sigma 0))=2. Further, we assume that In particular, we are interested in the Heavyside function h. A similar setting was also considered in [18]. S. Serra Capizzano suggested the application of band-Toeplitz preconditioners AN (p s ) in combination with CG applied to the normal equation. He proved, beyond a more general result which can not directly be used for precon- ditioning, that at most o(N) eigenvalues of the preconditioned matrix AN (p s absolute values not contained in a positive interval on the real axis. A result with o(N) outlyers was also obtained in [19], where the application of preconditioned GMRES was examined. In the following, we construct circulant preconditioners for the minimal residual method (MIN- RES). Note that preconditioned MINRES avoids the transformation of the original system to the normal equation but requires Hermitian positive definite preconditioners. Then, the preconditioned matrices are again Hermitian, so that the absolute values of their eigenvalues coincide with their singular values. If the generating function is given, we prove that for " ? 0, only O(log N) singular values of the preconditioned matrices are not contained in [1 \Gamma We also construct circulant preconditioners for the case that the generating function of the Toeplitz matrices is not explicitly known. For this, we use positive reproducing kernels with special properties previously applied by the authors in [16, 9] and show that O(log N) singular values of the preconditioned matrices are not contained in a positive interval on the real axis. Then, if in addition - 2 (AN preconditioned MINRES converges in at most O(log N) iteration steps. In summary, the proposed algorithm requires only O(N log 2 N) arithmetical operations. This paper is organized as follows: In Section 2, we introduce circulant preconditioners for (1.1) under the assumption that the generating function of the sequence of Toeplitz matrices is known and prove clustering results for the eigenvalues of the preconditioned matrices. Section 3 deals with the construction of preconditioners if the generating function of the Toeplitz matrices is not explicitly known. In Section 4, we modify the results of Section 2 with respect to trigonometric preconditioners. The convergence of MINRES applied to our preconditioned Toeplitz systems is considered in Section 5. Finally, we present numerical results in Section 6. Circulant preconditioners involving generating functions First we introduce some basic notation. By RN (M) we denote arbitrary (N; N)-matrices of rank at most M . Let M N (g) be the circulant (N; N)-matrix diag F where F N denotes the N-th Fourier matrix e \Gamma2-ijk=N and where F is the transposed complex conjugate matrix of F . For a trigonometric polynomial k=\Gamman 1 the matrices AN (q) and M N (q) are related by (see [14]). For a function g with a finite number of zeros we define the set I N (g) by I N (g) := and the points xN;l (g) (l xN;l (g) := where ~ l 2 1g is the next higher index to l so that ~ l 2 I N (g). For N large enough we can simply choose ~ l By M N;g (f) we denote the circulant matrix diag (f(x N;l If g has m zeros, then we have by construction that Assume now that the sequence fAN (f)g 1 N=1 of nonsingular Toeplitz matrices is generated by a known piecewise continuous function f 2 L 2- of the form (1.2) - (1.4). Then we suggest the Hermitian positive definite circulant matrix M N;f (jf j) as preconditioner for MINRES. We examine the distribution of the eigenvalues of M N;f (jf . The following theorem is Lemma 10 of [22] written with respect to our notation. Theorem 2.1 Let h 2 L 2- be a piecewise continuous function having only simple discontinuities at By FN we denote the Fej'er kernel FN (t) := e ikt cos kt (2.4) sin and by FN h the cyclic convolution of FN and h. Then, for any " ? 0, there exist constants independent of N so that the number -("; AN ) of eigenvalues of AN absolute value exceeding " can be estimated by In other words, we have by Theorem 2.1 that where V N is a matrix of spectral norm - " and where Using Theorem 2.1, we can prove the following lemma. Lemma 2.2 Let be given by (1:2) - (1.4). Then, for any " ? 0 and sufficiently large N , the number of singular values of M N;f 2 which are not contained in the interval [1 \Gamma Proof. By (2.6) and since the eigenvalues of M N;f (jhj) are restricted from below by h \Gamma , it remains to show that for any " ? 0 and sufficiently large N , except for O(log N) eigenvalues, all eigenvalues of M N;f (jhj) \Gamma1 M N (FN h) have absolute values in [1 \Gamma Indeed we will prove that there are only O(1) outlyers. For this we follow mainly the lines of proof of Gibb's ph-anomenon. Without loss of generality we assume that h 2 L 2- has only one jump at First we examine FN g, where g is given by By (2.4) and since g has Fourier series sin we obtain Z xFN (t) sin and further by (2.5) Z x/ sin Ntsin t! 2 Z x/ sin sin N t' 2 Z Nx0 sin t xand by partial integration and definition of g where si (y) := y Rsin t dt. We are interested in the behavior of Here dxe denotes the smallest integer - x. It is well known that lim . Thus, if so that " for all N - The same holds if we approach 0 from the left, i.e. if we consider 2-l=N for Next we have by definition of g and h that is a continuous function. Since FN is a reproducing kernel, for any " ? 0, there exists ~ so that for all l 2 Assume that l = we obtain by (2.7) and (2.8) that for any " ? 0 there exists and consequently, since jh denote the number of zeros of f which are equal to one of the points 2-l=N 1). Then the set contains at least absolute values of eigenvalues of M N;f (jhj) \Gamma1 M N (FN h) and we conclude by (2.9) that except for O(1) eigenvalues and sufficiently large N , all eigenvalues of have absolute values contained in [1 \Gamma This completes the proof. Remark 2.3 In a similar way as above we can prove that for any " ? 0 and N sufficiently large, the number of eigenvalues of AN (h) with absolute values not in the interval is O(log N ). Note that the property that at most o(N) eigenvalues of AN (h) have absolute values not contained in [h simply from the fact that the singular values of AN (h) are distributed as jhj [13, 19]. Theorem 2.4 Let be given by (1:2) - (1.4). Then, for any " ? 0 and sufficiently large N , except for O(log N) singular values, all singular values of contained in [1 \Gamma Proof. The polynomial p s in (1.3) can be rewritten as where Y and - p(t) is the complex conjugate of p(t). By straightforward computation it is easy to check that where only the first s columns (rows) of R c (r) N are nonzero columns (rows). p jhj the eigenvalues of M N;f (jf coincide with the eigenvalues of BN Now we obtain by (2.10), (2.1) and (2.3) that BN By Lemma 2.2, for any " ? 0 and N sufficiently large, except for O(log N) singular values, all singular values of M N;f are contained in [1 \Gamma "; 1+ "]. Now the assertion follows by (2.12) and Weyl's interlacing theorem [12, p. 184]. 3 Circulant preconditioners involving positive kernels In many applications we only know the entries a k (f) of the Toeplitz matrices AN (f ), but not the generating function itself. In this case, we use even positive reproducing kernels . These are trigonometric polynomials of the form c N;k cos kt; c satisfying KN - 0,2- \Gamma- KN and the reproducing property lim Since (KN \Gamma- a k (f) c N;k e ikx ; the cyclic convolution of KN and f is determined by the first N Fourier coefficients of f . As preconditioner which can be constructed from the entries of AN (f) without explicit knowledge of f we suggest the circulant matrix M N;KN \Lambdaf (jK N f j). In order to obtain a suitable distribution of the eigenvalues of the preconditioned matrices, we need kernels with a special property which is related to the order of the zeros of p s . The generalized Jackson kernels J m;N of degree - are defined by determined by (3.1). Here btc denotes the largest integer - t. In particular, we have that i.e. there exist positive constants c 1 ; c 2 so that c 1 N 1\Gamma2m - m;N - c 2 N 1\Gamma2m . See [10, pp. possibility for the construction of the Fourier coefficients of J m;N is prescribed in [9]. The B-spline kernels B m;N of degree - are defined by sinc where Mm denotes the centered cardinal B-spline of order m and sinc t := sin t See [16, 8]. Since cos kt the Fourier coefficients of B m;N are given by values of centered cardinal B-splines. Note that is just the Fej'er kernel FN . The above kernels have the following important property: Theorem 3.1 be given by (1:2) - (1:4). Assume that for all t j (j 2 exists a neighborhood so that f is a monotone function in this neighborhood and moreover f(t m;N be given by (3.2) or (3.3), where Then there exist so that for N !1, except for O(1) points, all points of the set Proof. 1. First we consider the upper bound. Since p s and KN are nonnegative, we obtain Z \Gamma- Z \Gamma- In [16, 9], we proved that m - oe implies that for all x 2 I N (p s ) ' I N (f ), there exists a (KN p s )(x) Thus, since jh(x)j - h \Gamma for (KN p s )(x) 2. Next we deal with the lower bound. 2.1. Let x 2 I N (f) be not in the neighborhood of t j (j exist independent of N so that since KN is a reproducing kernel and by using the same arguments as in the proof of Lemma 2.2 if x is in the neighborhood of some we obtain that, for any " ? 0 there exists N("), so that except for at most a constant number of points, all considered points x 2 I N (f) satisfy and thus 2.2. It remains to consider the points For simplicity we assume that i.e. p s has only a zero of order 2s at lim For any fixed 0 (KN Z \Gammab \Gammab Z \Gamma- Z Z \Gammab b+x and since f is bounded (KN Z \Gammab b+x By definition of KN we see that for any fixed 0 ! ~ b - Z KN (t) dt - const N \Gamma2m+1 ; (3.5) so that we get for small x (e.g. x ! b=2) (KN Z \Gammab 2.2.1. Assume that h has no jump at that h(t) - h \Gamma or h(t) - \Gammah \Gamma for t 2 [\Gamma"; "]. We restrict our attention to the case h - h \Gamma . monotone increasing on (0; -), we obtain for x(N) 2 (0; ") " I N (f) and N sufficiently large that Z \Gamma" Z Z with a positive constant c independent of N . On the other hand, we have by definition of s and since by assumption . Then we obtain by (3.6) with that for N large enough (KN f)(x(N)) const with a positive constant const independent of N . The proof for x(N) 2 (\Gamma"; follows the same lines. 2.2.2. Finally, we assume that h has a jump at Without loss of generality let h(0 by assumption on f , there exists that Z We consider points of the form with lim in case of Jackson kernels and fl := 1 in case of B-spline kernels. Then we have for t 2 and consequently for sufficiently small " 1 and y, since sin is odd and monotone increasing on (0; -=2) that Further, by definition of the B-spline kernels m;N (t) := N sinc and similarly as in (3.9) we see that By assumption h does not change the sign in (0; " 1 ). Then we obtain by (3.8), monotonicity of p s in (0; -) and m - s Z Z y where K 0 m;N g. Set there exist Z wl l=r\Gammak wl Z and further by (3.5) and since lim Z Straightforward computation yields const Z' sin u Hence we get for N large enough that Z const and by (3.10) that Z const (3.11) with positive constants const independent of N . Now we consider x(N) 2 I N (f) with y k (N) - x(N) ! y k+1 (N ). Z Z Z Z and since f is by assumption monotone increasing on [\Gamma" Z Z Z Z Z Z and by (3.5) and since f is bounded Z Z By assumption R const R and since f(y k (N) - const N \Gamma2s and m - s by (3.12), (3.11) that for N large enough Z const with a nonnegative constant const independent of N . Finally, we use (3.6) with again to finish the proof. To show our main result we also need the following lemma. Lemma 3.2 Let A 2 C N;N be a Hermitian positive definite matrix having in [a be a Hermitian matrix with singular values in [b 1. Then at least N \Gamma 4n of A B are contained in [\Gammaa Proof. 1. Assume first that n i.e. A has only eigenvalues in [a the j-th eigenvalue of the matrix B. We consider the eigenvalues of to t 2 R. By Weyl's interlacing theorem (see [12, p. 184]) we obtain for t - 0 that and for t ! 0 that we obtain by (3.13) and (3.14) that - j . On the other hand, we see by (3.13) and (3.14) that - j . Thus, since - j is a continuous function in t 2 R, there exists This implies that t j is an eigenvalue of AB. Consequently, every corresponds to an eigenvalue of AB. (Eigenvalues are called with multiplicities.) The examination of the same lines. In summary, N \Gamma n 2 eigenvalues of AB are contained in [\Gammaa 2. Let n 1 eigenvalues of A be outside [a since A is positive definite, the matrix can be splitted as A 1=2 where ~ A 1=2 is Hermitian with all eigenvalues in [a 1=2 of rank n 1 . The eigenvalues of AB coincide with the eigenvalues of A 1=2 BA 1=2 . Hence it remains to show that at most 4n 1 singular values of A 1=2 BA 1=2 are not contained in we have A 1=2 BA A 1=2 A 1=2 BA 1=2 ~ A 1=2 A 1=2 By 1. all but n 2 singular values of ~ A 1=2 A 1=2 are contained in [a and Weyl's interlacing theorem yield the assertion. Theorem 3.3 Let be given by (1:2) - (1:4). Assume that for all t j (j 2 exists a neighborhood so that f is a monotone function in this neighborhood and moreover f(t m;N be given by (3.2) or (3.3), where By ff; fi we denote the constants from Theorem 3:1. Then, for any " ? 0 and sufficiently large N , except for O(log N) singular values, all singular values of M N (jK N f are contained in [ff \Gamma Proof. Let BN (f) be defined by (2.11). Then we obtain by (2.12) that The distribution of the eigenvalues of M N;f 2 is known by Lemma 2.2. It remains to examine the eigenvalues of the Hermitian positive definite matrix These eigenvalues coincide with the reciprocal eigenvalues of M N;f (jf By definition of M N;g and since KN is a reproducing kernel, except for O(1) eigenvalues, all eigenvalues of M N;f (jf are given by j(K N f)(2-l=N)j=jf(2-l=N)j (l 2 I N (f )). Thus, by Theorem 3.1, for N !1 only O(1) eigenvalues of are not contained in [ff; fi]. Consequently, by (3.18), Lemma 2.2, Lemma 3.2 and Weyl's interlacing theorem at most O(log N) singular values of 2 are not contained in [ff \Gamma 4 Trigonometric preconditioners In addition to Section 2, we suppose that the Toeplitz matrices AN 2 R N;N are symmetric, i.e. the generating function f 2 L 2- is even. This suggests the application of so-called trigonometric preconditioners. Note that in the symmetric case the multiplication of a vector with AN can be realized using fast trigonometric transforms instead of fast Fourier transforms (see [14]). In this way complex arithmetic can be completely avoided in the iterative solution of (1.1). This is one of the reasons to look for preconditioners which can be diagonalized by trigonometric matrices corresponding to fast trigonometric transforms instead of the Fourier matrix F N . In practice, four discrete sine transforms (DST I - IV) and four discrete cosine transforms (DCT I - IV) were used (see [21]). Any of these eight trigonometric transforms can be realized with O(N log N) arithmetical operations. Likewise, we can define preconditioners with respect to any of these transforms. In this paper, we restrict our attention to the so-called discrete cosine transform of type II (DCT-II) and discrete sine transform of type II (DST-II), which are determined by the following transform matrices: where ffl N 1). We propose the preconditioners diag (jf(~x N;l diag (jf(~x N;l )j) N where ~ xN;l := l- l- ~ l- and where ~ l 2 1g is the next higher index to l such that jf(~x N;l )j ? 0. See [15]. Then we can prove in a completely similar way as in Section 2 that for any " ? 0 and sufficiently large N except for O(log N) singular values, all singular values of are contained in [1 \Gamma 5 Convergence of preconditioned MINRES In order to prescribe the convergence behavior of preconditioned MINRES with our preconditioners of the previous sections, we have to estimate the smaller outlyers for increasing N . Lemma 5.1 Let f 2 L 2- be defined by (1:2)-(1:4). Assume that - 2 (AN (ff ? 0). Then the smallest absolute values of the eigenvalues of M N;f (jf behave for N !1 as O(N \Gammaff ). Proof. Since and both kM N;f (jf j)k 2 and kM N;KN \Lambdaf (jK N f j)k 2 are restricted from above, it remains to show that there exists a constant c ? 0 independent of N so that kAN The above inequality follows immediately from the fact that the singular values of AN (f) are distributed as jf j (see [13, 19]). We want to combine our knowledge of the distribution of the eigenvalues of our preconditioned matrices with results concerning the convergence of MINRES. Theorem 5.2 Let A 2 C N;N be a Hermitian matrix with p and q isolated large and small singular values, respectively: Let for the solution of iteration steps to achieve precision - , i.e. jjr (k) jj 2 the k-th iterate. The theorem can be proved by using the same technique as in [1, pp. 569 - 573]. Namely, based on the known estimate jjr k denotes the space of polynomials of degree - k with p k are the eigenvalues of A, we choose p k as product of the linear polynomials passing through the outlyers and the modified Chebyshev polynomials The above summand p ln 2 can be further reduced if we use polynomials of higher degree for the larger outlyers. Note that a similar estimate can be given for the CG method applied to the normal equation A b. Here we need iteration steps to archive precision jje (0) jj A Note that the latter method requires two matrix-vector multiplications in each iteration step. By Theorem 2.4, Theorem 3.3 and Lemma 5.1 our preconditioned MINRES with both preconditioners produces a solution of (1.1) of prescribed precision in O(log N) iteration steps and with O(N log 2 N) arithmetical operations. The same holds for preconditioned CG applied to the normal equation. 6 Numerical results In this section, we test our circulant and trigonometric preconditioners in connection with different iterative methods on a SGI O2 work station. As transform length we use right-hand side b of (1.1) the vector consisting of N entries "1" and as start vector the zero vector. We begin with a comparison of MINRES applied to N g and CGNE (Craig's method) (cf. [17, p. 239]) applied to For both algorithms we have used MATLAB implementations of B. Fischer. See also [11]. In particular, his implementation of preconditioned MINRES avoids the splitting (6.2). In order to make the following computations with MINRES and CGNE comparable, we have stopped both computations if Example 1. We begin with Hermitian Toeplitz matrices AN (f) arising from the generating function Table 1 presents the number of iterations for circulant preconditioners. The first row of the table contains the exponent n of the transform length . According to Theorem 2.4 and Theorem 5.2, the preconditioners M N (jf j; F N ) lead to very good results. As expected, the preconditioners M N;KN \Lambdaf (jK N f j; F N ) with the Fej'er kernels are not suitable for (1.1) (cf. also [16]), while the preconditioners with do their job. Further, CGNE needs half the number of iterations but twice the number of matrix-vector multiplications per iteration than MINRES. See also Section 5. method MINRES I N 23 71 277 * 43 CGNE I N 11 37 164 * Table 1: Example 2. Next, we consider the symmetric Toeplitz matrices AN (f) arising from the generating function with Tables 2 presents the number of iterations for trigonometric preconditioners. The results are similar to those of Example 1, except that CGNE requires nearly the same number of iterations as MINRES. method MINRES I N 9 17 MINRES M N;f (jf j; C II MINRES M N;f (jf j; S II MINRES M N;FN \Lambdaf (jF N f j; C II CGNE I Table 2: f 2 Acknowledgment . The authors wish to thank B. Fischer for for the MATLAB implementations of PMINRES and CGNE. --R Iterative Solution Methods. preconditioning for Toeplitz matrices. Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions. Conjugate gradient methods of Toeplitz systems. Preconditioners for non-Hermitian Toeplitz systems Circulant preconditioners from B-splines The best circulant preconditioners for Hermitian Toeplitz matrices. Constructive Approximation. Polynomial Based Iteration Methods for Symmetric Linear Systems. Matrix Analysis. On the distribution of singular values of Toeplitz matrices. Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems. Preconditioners for ill-conditioned Toeplitz matrices Preconditioners for ill-conditioned Toeplitz matrices constructed from positive kernels Iterative Methods for Sparse Linear Systems. Preconditioning strategies for Hermitian Toeplitz systems with nondefinite generating functions. A unifying approach to some old and new theorems on distribution and clustering. Fast algorithms for the discrete W transform and for the discrete Fourier transform. Circulant preconditioners for Toeplitz matrices with piecewise continuous generating functions. --TR

circulant matrices;preconditioners;nondefinite Toeplitz matrices;krylov space methods

587883

Methods for Large Scale Total Least Squares Problems.

The solution of the total least squares (TLS) problems, $\min_{E,f}\|(E,f)\|_F$ subject to (A+E)x=b+f, can in the generic case be obtained from the right singular vector corresponding to the smallest singular value $\sigma_{n+1}$ of (A, b). When A is large and sparse (or structured) a method based on Rayleigh quotient iteration (RQI) has been suggested by Bjrck. In this method the problem is reduced to the solution of a sequence of symmetric, positive definite linear systems of the form $(A^TA-\bar\sigma^2I)z=g$, where $\bar\sigma$ is an approximation to $\sigma_{n+1}$. These linear systems are then solved by a {\em preconditioned} conjugate gradient method (PCGTLS). For TLS problems where A is large and sparse a (possibly incomplete) Cholesky factor of ATA can usually be computed, and this provides a very efficient preconditioner. The resulting method can be used to solve a much wider range of problems than it is possible to solve by using Lanczos-type algorithms directly for the singular value problem. In this paper the RQI-PCGTLS method is further developed, and the choice of initial approximation and termination criteria are discussed. Numerical results confirm that the given algorithm achieves rapid convergence and good accuracy.}

Introduction . The estimation of parameters in linear models is a fundamental problem in many scientific and engineering applications. A statistical model that is often realistic is to assume that the parameters x to be determined satisfy a linear relation where A 2 R m\Thetan , and b 2 R m , are known and (E; f) is an error matrix with rows which are independently and identically distributed with zero mean and the same variance. (To satisfy this assumption the data (A; b) may need to be premultiplied by appropriate scaling matrices, see Golub and Van Loan [10].) In statistics this model is known as the "errors-in-variables model". The estimate of the true but unknown parameter vector x in the model (1.1) is obtained from the solution of the total least squares (TLS) problem min subject to Department of Mathematics, University of Link-oping, S-581 83 Link-oping, Sweden. e-mail: akbjo@math.liu.se, pontus.matstoms@vti.se. The work of these authors was supported by the Swedish Research Council for Engineering Sciences, TFR. y Department of Informatics, University of Bergen, N-5020 Bergen, Norway, email: pinar@ii.uib.no denotes the Frobenius matrix norm. If a minimizing pair (E; f) has been found for the problem (1.2) then any x satisfying said to solve the TLS problem. Due to recent advances in data collection techniques LS or TLS problems where A is large and sparse (or structured) frequently arise, e.g., in signal and image processing applications. For the solution of the LS problem both direct methods based on sparse matrix factorizations and iterative methods are well developed, see [2]. An excellent treatment of theoretical and computational aspects of the TLS problem is given in Van Huffel and Vandewalle [25]. Solving the TLS problem requires the computation of the smallest singular value and the corresponding right singular vector of (A; b). When A is large and sparse this is a much more difficult problem than that of computing the LS solution. For example, it is usually not feasible to compute the SVD or any other two-sided orthogonal factorization of A since the factors typically are not sparse. Iterative algorithms for computing the singular subspace of a matrix associated with its smallest singular values, with applications to TLS problems with slowly varying data, have previously been studied by Van Huffel [24]. In [27, 3] a new class of methods based on a Rayleigh quotient iteration was developed for the efficient solution of large scale TLS problems. Related methods for Toeplitz systems were studied by Kamm and Nagy [14]. In this paper the methods in [3] are further developed and numerical results given. Similar algorithms for solving large scale multidimensional TLS problems will be considered in a forthcoming paper [4]. In Section 2 we recall how the solution to the TLS problem can be expressed in terms of the smallest singular value and corresponding right singular vector of the compound matrix (A; b). We discuss the conditioning of the LS and TLS problems and illustrate how the TLS problem can rapidly become intractable. Section 3 first reviews a Newton iteration for solving a secular equation. For this method to converge to the TLS solution strict conditions on the initial approximation have to be satisfied. We then derive the Rayleigh quotient method, which ultimately achieves cubic convergence. The choice of initial estimates and termination criteria are discussed. A preconditioned conjugate gradient method is developed in Section 4 for the efficient solution of the resulting sequence of sparse symmetric linear systems. Finally, in Section 5, numerical results are given which confirm the rapid convergence and numerical stability of this class of methods. Preliminaries. 2.1 The TLS problem. The TLS problem (1.2) is equivalent to finding a perturbation matrix (E; f) having minimal Frobenius norm, which lowers the rank of the matrix (A; b). Hence it can be analyzed in terms of the singular value decomposition are the singular values of (A; b). Note that by the minmax characterization of singular values it follows that the singular values oe 0 i of A interlace those of (A; b), i.e., We assume in the following that A has full rank, that is, oe that . Then the minimum is attained for the rank one perturbation for which k(E; f)k solution is then obtained from the right singular vector z x TLS provided that i 6= 0. If the TLS problem is called nongeneric, and there is no solution. This case cannot occur if oe and in the following we always assume that this condition holds. From the characterization (2.2) it follows that n+1 and the system of nonlinear equations A T A A T b x x Putting n+1 the first block row of this system of equations can be written which can be viewed as "the normal equations" for the TLS problem. Note that from our assumption that oe 0 n ? oe n+1 it follows that A T A \Gamma oe 2 n+1 I is positive definite. 2.2 Conditioning of the TLS problem. For the evaluation of accuracy and stability of the algorithms to be presented we need to know the sensitivity of the TLS problem to perturbations in data. We first recall that if x LS 6= 0 the condition number for the LS problem is (see [2, Sec. 1.4]) n . Note that the condition number depends on both A and b, and that for large residual problems the second term may dominate. Condition number for problem LS and TLS beta kappa kappa LS kappa TLS Figure 2.1: Condition numbers -LS and - TLS as function of Equation (2.4) shows that the TLS problem is always worse conditioned than the LS problem. From (2.3), multiplying from the left with This inequality is wek, but shows that kx TLS k 2 will be large when kr LS k 2 AE oe 0 n . Golub and Van Loan [10] showed that an approximate condition number for the TLS problem is the TLS condition number can be much greater than -(A). The relation between the two condition numbers (2.5) and (2.7) depend on the relation between the kr LS k 2 and oe n+1 , which is quite intricate. (For a study of this relation in another context see Paige and Strako-s [17].) As an illustration we consider the following small overdetermined system@ Trivially, the LS solution is If we take in (2.8) oe independent of fi, and hence does not reflect the illconditioning of A. The TLS solution is of similar size as the LS solution as long as jfij - oe 0 2 . However, when jfij AE oe 0 2 then from (2.6) it follows that kx TLS k 2 is large. In Fig. 2.1 the two condition numbers are plotted as a function of jfij. We note that - LS increases proportionally to jfij because of the second term in (2.5). For the condition number - TLS grows proportionally to jfij 2 . It can be verified that kx TLS k 2 also grows proportionally to jfij 2 . 3 Newton and Rayleigh Quotient methods. 3.1 A Newton method. Equation (2.3) constitutes a system of (n equations in x and -. One way to proceed (see [14]) is to eliminate x to obtain the rational secular equation for method applied to (3.1) leads to the iteration This iteration will converge monotonically at a rate that is asymptotically quad- ratic. The convergence of this method can be improved by using a rational interpolation similar to that in [6] to solve the secular equation. However, in any case, - will converge to oe 2 n+1 and x (k) to the TLS solution only if the initial approximation satisfies In general it is hard to verify this assumption. For the special case of a Toeplitz TLS problem Kamm and Nagy [14] use a bisection algorithm based on a fast algorithm for factorizing Toeplitz matrices to find an initial starting value satisfying (3.4). 3.2 The Rayleigh quotient method. The main drawback of the Newton method above is that unless (3.4) is satisfied it will converge to the wrong singular value. A different Newton method is obtained by applying Newton's method to the full system As remarked in [20] this is closely related to inverse itera- tion, which is one of the most widely used methods for refining eigenvalues and eigenvectors. Rayleigh quotient iteration (RQI) is inverse iteration with a shift equal to the Rayleigh quotient. RQI has cubic convergence for the symmetric eigenvalue problem, see [18, Sec.4-7], and is superior to the standard Newton method applied to (3.5). For the eigenvalue problem (2.3) the Rayleigh quotient equals Let x (k) be the current approximation and ae k the corresponding Rayleigh quotient. Then the next approximation x (k+1) in RQI and the scaling factor fi k are obtained from the symmetric linear system where If J (k) is positive definite the solution can be obtained by block Gaussian elimi- nation, ' \Gamma(z where It follows that x In [2] a reformulation was made to express the solution in terms of the residual vectors of (3.5) ' where r This uses the following formulas to compute The RQI iteration is defined by equations (3.10)-(3.13). 3.3 Initial estimate and global convergence. Parlett and Kahan [19] have shown that for almost all initial vectors the Rayleigh quotient iteration converges to some singular value and vector pair. However, in general we cannot say to which singular vector RQI will converge. If the LS solution is known, a suitable starting approximation for - may be Conditions to ensure that RQI will converge to the TLS solution from the starting approximation (ae(x LS ); x LS ) are in general difficult to verify and often not satisfied in practice. However, in contrast to the simple Newton iteration in Section 3.1, the method may converge to the TLS solution even when The Rayleigh quotient ae(x LS ) will be a large overestimate of oe 2 n+1 when the residual norm kr LS k 2 is large and kx LS k 2 does not reflect the illconditioning of A. Note that it is typical for illconditioned least squares problems that the right-hand side is such that kx LS k 2 is not large! For example, least squares problems arising from ill-posed problems usually satisfy a so called Picard condition, which guarantees that the right-hand side has this property, see [11, Sec. 1.2.3]. Szyld [23] suggested that one or more steps of inverse iteration could be applied initially before switching to RQI, in order to ensure convergence to the smallest eigenvalue. Inverse iteration for oe 2 n+1 corresponds to taking oe in the RQI algorithm. Starting from x = x LS the first step of inverse iteration simplifies as follows. Using (3.9) and (3.10) with ae z and the new approximation becomes Several steps of inverse iteration may be needed to ensure convergence of RQI to the smallest singular value. However, since inverse iteration only converges lin- early, taking more than one step will usually just hold up the rapid convergence of RQI. We therefore recommend in general steps as the default value. To illustrate the situation consider again the small 3 \Theta 2 system (2.8) with . This has the LS solution x does not reflect the illconditioning of A the initial Rayleigh quotient approximation equals By the interlacing property we have that oe 3 - oe 0 2 . Since jfij AE oe 0 2 it is clear that the Rayleigh quotient fails to approximate oe 2 3 . This is illustrated in Figure 3.1, where ae(x LS ) 1=2 and oe 3 are plotted as function of jfij. It is easily verified, however, that after one step of inverse iteration ae(x INV ) will be close to oe 0 2 -5 beta sqrt(r LS Figure 3.1: Rayleigh quotient approximation and oe 3 for 3.4 Termination criteria for RQI. The RQI algorithm for the TLS problem is defined by (3.10)-(3.13). When should the RQI iteration be terminated? We suggest two different criteria. The first is based on the key fact in the proof of global convergence that the normalized residual norm always decreases, fl k+1 - fl k , for all k. Thus, if an increase in the norm occurs this must be caused by roundoff, and then it makes no sense to continue the iterations. This suggests that we terminate the iterations with x k+1 when A second criterion is based on the observation that since the condition number for computing oe n+1 equals 1, we can expect to obtain oe n+1 to full machine precision. Since convergence of RQI is cubic a criterion could be to stop when the change in the approximation to oe n+1 is of the order of oe 1 u 1=p , where similar criterion with used by Kamm and Nagy [14] for terminating the Newton iteration.) However, as will be evident from the numerical results in Section 5, full accuracy in x TLS in general requires one more iteration after oe n+1 has converged. Therefore we recommend to stop when either (3.16) or is satisfies, where u is the machine unit and C a suitable constant. We summarize below the RQI algorithm with one step of inverse iteration (cf. Algorithm 3.1. Rayleigh Quotient Iteration. solve A T solve solve 3.5 Rounding errors and stability. If the RQI iteration converges then f (k) , g (k) , and fi k will tend to zero. Consider the rounding errors which occur in the evaluation of the residuals (3.11). Let ~ where u is the unit roundoff; see [13, Chap. 3]. Then the computed residual vector satisfies - Obviously convergence will cease when the residuals (3.11) are dominated by roundoff. Assume that we perform one iteration from the exact solution, x TLS , r TLS , and n+1 . Then the first correction to the current approximation is obtained by solving the linear system in (3.13), which now becomes For the correction this gives the estimate This estimate is consistent with the condition estimate for the TLS problem. We note that the equations (3.18) are of similar form to those that appear in the corrected semi-normal equations for the LS problem; see [1], [2, Sec. 6.6.5]. A detailed roundoff error analysis similar to that done for the LS problem would become very complex and is not attempted here. It seems reasonable to conjecture that if if oe 0 2 will suffice to solve the linear equations for the correction w (k) using the Cholesky factorization of I). Methods for the solution of the linear systems are considered in more detail in Section 4. 4 Solving the linear systems. In the RQI method formulated in the previous section the main work consists of solving in each step two linear systems of the form Here oe is an approximation to oe n+1 and varies from step to step. Provided that the system (4.1) is symmetric and positive definite. 4.1 Direct linear solvers. then the system (4.1) can be solved by computing the (sparse) Cholesky factorization of the matrix A T A \Gamma oe 2 I. Note that A T A only has to be formed once and the symbolic phase of the factorization does not have to be repeated. However, it is a big disadvantage that a new numerical factorization has to be computed at each step of the RQI algorithm. For greater accuracy and stability in solving LS problems it is often preferred to use a QR factorization instead of a Cholesky factorization. However, since in the TLS normal equations the term oe 2 I is subtracted from A T A, this is not straightforward. The Cholesky factor of the matrix A T A \Gamma oe 2 I can be obtained from the QR factorization of the matrix A ioeI , where i is the imaginary unit. This is a downdating problem for the QR factorization and can be performed using stabilized hyperbolic rotations, see [2, pp. 143-144], or hyperbolic Householder transformations, see [22]. However, in the sparse case this is not an attractive alternative, since it would require nontrivial modifications of existing software for sparse QR factorization. 4.2 Iterated deregularization. To solve the TLS normal equations using only a single factorization of A T A we can adapt an iterated regularization scheme due to Riley and analyzed by Golub [9]. In this scheme, we solve the TLS normal equations by the iteration A T Affi If lim k!1 x b. This iteration will converge with linear rate equal to ae n provided that ae ! 1. This iteration may be implemented very efficiently if the QR decomposition of A is available. We do not pursue this method further, since it has no advantage over the preconditioned conjugate gradient method developed in [3]. 4.3 A preconditioned conjugate gradient algorithm. Performing the change of variables is a given nonsingular matrix, and multiplying from the left with S \GammaT the system (4.1) becomes This system is symmetric positive definite provided that oe ! oe 0 n , and hence the conjugate gradient method can be applied. We can use for S the same preconditioners as have been developed for the LS problem; for a survey see [2, Ch. 7]. In the following we consider a special choice of preconditioner, the complete Cholesky factor R of A T A (or R from a QR decomposition of A). Unless A is huge this is often a feasible choice, since efficient software for sparse Cholesky and sparse QR factorization are readily available [2, Ch. 7]. Using AR I, the preconditioned system (4.2) simplifies to (Note that although A and A T have disappeared from this system of equations matrix-vector multiplications with these matrices are used to compute the right-hand side f !) In the inverse iteration step used in the initialization, the solution obtained by two triangular solves. The standard conjugate gradient method applied to the system (4.2) can be formulated in terms of the original variables w. The resulting algorithm is a slightly simplified version of the algorithm PCGTLS given in [3] and can be Algorithm 4.1. PCGTLS Preconditioned gradient method for solving using the Cholesky factor R of A T A as preconditioner. Initialize: w ks . For ks (j+1) k 2fi Denote the original and the preconditioned matrix by I and e respectively. Then a simple calculation shows that for the condition number of the transformed system is reduced by a factor of -(A), !/ The spectrum of ~ C will be clustered close to 1. In particular in the limit when the eigenvalues of e C will lie in the interval (Note the relation to the condition number - TLS !) Hence, unless oe 0 can expect this choice of preconditioner to work very well for solving the shifted system (4.1). The I is positive definite if oe ! oe 0 n . In this case in PCGTLS, and the division in computing ff k can always be carried out. If n then the system (4.2) is not positive definite and a division by zero can occur. This can be avoided by including a test to ensure that equivalently kp the CG iterations are considered to have failed. The RQI step is then repeated with a new smaller value of oe 2 e.g., The accuracy of TLS solutions computed by Rayleigh Quotient Iteration will basically depend on the accuracy residuals and the stability of the method used to solve the linear systems (4.1). We note that the cg method CGLS1 for the LS problem, which is related to PCGTLS, has been shown to have very good numerical stability properties, see [5]. 4.4 Termination criteria in PCGTLS. The RQI iteration, using PCGTLS as an inner iteration for solving the linear systems, is an inexact Newton method for solving a system of nonlinear equa- tions. Such methods have been studied by Dembo, Eisenstat, and Steihaug [7], who consider the problem of how to terminate the iterative solver so that the rate of convergence of the outer Newton method is preserved. Consider the iteration where r k is the residual error. In [7] it is shown that maintaining a convergence order of 1 requires that when k !1, the residuals satisfy inequalities is a forcing sequence. In practice the above asymptotic result turns out to be of little practical use in our context. Once the asymptotic cubic convergence is realized, the ultimate accuracy possible in double precision already has been achieved. A more prac- tical, ad hoc termination criterion for the PCGTLS iterations will be described together with the numerical results reported below. Remark. In the second linear system to be solved in RQI, the right-hand side converges to x TLS . Hence it is tempting to use the value of u obtained from the last RQI to initialize PCGTLS in the next step. However, our experience is that this slows down the convergence compared to initializing u to zero. 5 Numerical results. 5.1 Accuracy and termination criteria. Numerical tests were performed in Matlab on a SUN SPARC station 10 using double precision with unit roundoff . For the initial testing we used contrived test problems [A; similar to those in [5] and generated in the following way. 1 Let e where Y; Z are random orthogonal matrices and Further, let Ax: This ensures that the norm of the solution does not reflect the illconditioning of A. We then add random perturbations Note that since oe there is a perturbation E to A with which makes A rank deficient. Therefore it is not realistic to consider perturbations with To test the termination criteria for the inner iterations iterations log0||x Figure 5.1: Errors problem PS(30,15), with systems solved by PCGTLS with iterations. iterations log0||x Figure 5.2: Errors systems solved by PCGTLS with we used problem P (30; 15), oe 0 The linear systems arising in RQI were solved using PCGTLS with the Cholesky factor of A T A as preconditioning. The criterion (4.6) shows that the linear systems should be solved more and more accurately as the RQI method converges. The rate of convergence depends on the ratio oe n+1 =oe 0 n , see (4.4), and is usually very rapid. We have used a very simple strategy where in the kth step of RQI These test problems are neither large nor sparse! iterations are performed, where - 0 is a parameter to be chosen. In Figure 5.1 we show results for 2. The plots for are almost indistinguishable, whereas delay in convergence. Indeed, for this problem taking iterations in PCGTLS suffices to give the same result as using an exact (direct) solver. Since no refactorizations are performed the object should be to minimize the total number of PCGTLS iter- ations. Based on these considerations and the test results we recommend taking should work well for problems where the ratio oe n+1 =oe 0 n is smaller. Rarely more than 2-3 RQI iterations will be needed. In Figure 5.2 we show results for problem PS(30,15), and different error levels Here respectively, were needed to achieve an accuracy of about 10 \Gamma11 in x TLS . Since oe 0 this is equal to the best limiting accuracy that can be expected. Note also that the error in oe n+1 converges to machine precision, usually in one less iteration, which supports the use of the criterion (3.17) to terminate RQI. 5.2 Improvement from inverse iteration. We now show the improvement resulting from including an initial step of inverse iteration. In Figure 5.3 we show results for the problem considered above. For the first two error levels only one RQI iteration now suffices. For the highest error level oe n+1 converges in two iterations and x TLS in three. -22 iterations log0||x Figure 5.3: Errors One step of inverse iteration. Linear systems solved by PCGTLS with k +1 iteration. We now consider the second test problem in [14], which is defined where A 2 R n\Thetan\Gamma1 . Here e is a vector with entries generated randomly from a normal distribution with mean 0:0 and variance 1:0, and scaled so that jjejj . For 0:01 the condition numbers in (2.5)-(2.7) are respectively. This problem has features similar to those of the small illcondi- tioned example discussed previously in Section 2.2, although here the norm of the solution x LS is large. -22 iterations log0||x Figure 5.4: Second test problem with 0:001. RQI without/with one step of inverse iteration Applying the RQI algorithm we obtained the results shown in Figure 5.4. The initial approximation ae(x LS ) is here far outside the interval [oe n+1 ; oe 0 n ). Thus the matrix A T A \Gamma oe 2 I is initially not positive definite and we cannot guarantee the existence of the Cholesky factor. However, the Algorithm PCGTLS still does not break down, and as shown in Figure 5.4 the limiting accuracy is obtained after five RQI iterations. This surprisingly good performance of RQI can be explained by the fact that even though x LS does not approximate x TLS well, the angle between them is small; the cosine equals 0:98453. Performing one step of inverse iteration before applying the RQI algorithm gives much improved convergence. The one initial step of inverse iteration here suffices to give an initial approximation in the interval [oe n+1 ; oe 0 n ). This can be compared with 12-23 steps of bisection needed to achieve such a starting approximation, see [14]! Three RQI iterations now give the solution x TLS with an error close to the limiting accuracy, see Fig. 5.4. We note that in both cases we obtained oe n+1 to full machine precision. Also, the relative error norm of in the TLS solution was consistent with the condition 5.3 A problem in signal restoration. The Toeplitz matrix used in this example comes from an application in signal restoration, see [14, Example 3]. Specifically, an n \Theta (n \Gamma 2!) convolution matrix T is constructed to have entries in the first column given by exp and zero otherwise. Entries in the first row given by t zero otherwise, where 8. A Toeplitz matrix T and right-hand side vector g is then constructed as e, where E is a random Toeplitz matrix with the same structure as T , and e is a random vector. The entries in E and e are generated randomly from a normal distribution with mean 0.0 and variance 1.0, and scaled so that In [14] problems with convergence were reported. However, these are due to the choice of right-hand side - 1 , which was taken to be a vector of all ones. For the unperturbed problem this vector is orthogonal to the space spanned by the left singular vector corresponding to the smallest singular value. Therefore the magnitude of the component in this direction of the initial vector x LS will be very small, of the order fl. Also, although A is quite well conditioned the least squares residual is large. The TLS problem is therefore close to a nongeneric problem and thus very illconditoned. Because of the extreme illconditioning for this right-hand side, the behavior of any solution method becomes very sensitive to the particular random perturbation added. We have therefore instead chosen a right-hand side - g 2 given by - m. For this the TLS problem is much better conditioned, see Table 5.1. Convergence is now obtained in just two iterations, see Figure 5.5. Table 5.1: Condition numbers for test problem 3 for right-hand sides - g i , -2iterations log0||x Figure 5.5: Third test problem; RQI with one step of inverse iteration, 6 Summary . We have developed an algorithm for solving large scale TLS problems based on Rayleigh quotient iteration for computing the right singular vector of defining the solution. The main work in this method consists of solving a sequence of linear systems with matrix A T A \Gamma oe 2 I, where oe is the current approximation to the smallest singular value of oe n+1 of (A; b). For large and sparse TLS problems these linear systems can be solved by a preconditioned conjugate gradient method. An efficient preconditioner is given by a (possibly incomplete) Cholesky factorization of A T A or QR factorization of A. Termination criteria for the inner and outer iterations have been given. We conjecture that the described method almost always computes the TLS solution with an accuracy compatible with a backward stable method. Although a detailed error analysis is not given this conjecture is supported by numerical results. Methods for solving the TLS problem are by necessity more complex than those for the (linear) LS problem. Our algorithm contains several ad hoc choices. On the limited set of test problems we have tried it has only failed for almost singular problems, for which the total least squares model is not relevant and should not be used. In our method the perturbation E is a rank one matrix which in general is dense. Sometimes it is desired to find a perturbation E that preserves the sparsity structure of A. A Newton method for this more difficult problem has been developed by Rosen, Park, and Glick [21]. However, the complexity of this algorithm limits it to fairly small sized problems. Recently a method, which has the potential to be applied to large sparse problems has been given by Yalamov and Yun Yuan [26]. Their algorithm only converges with linear rate, which may suffice to obtain a low accuracy solution. --R Improving the accuracy of computed singular values. Numerical methods for solving least squares problems An analysis of the total least squares problem A survey of condition number estimation for triangular matrices Accuracy and Stability of Numerical Algorithms A total least squares method for Toeplitz systems of equations The minimum eigenvalue of a symmetric positive-definite Toeplitz matrix and rational Hermitian interpolation Sparse QR factorization in MATLAB The Symmetric Eigenvalue Problem On the convergence of a practical QR algorithm Total least norm formulation and solution for structured problems stability and pivoting Criteria for combining inverse and Rayleigh quotient iteration Iterative algorithms for computing the singular subspace of a matrix associated with its smallest singular values The Total Least Squares Problem: Computational Aspects and Analysis A successive least squares method for structured total least squares. Iterative Methods for Least Squares and Total Least Squares Problems --TR --CTR Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization, Applied Numerical Mathematics, v.49 n.1, p.39-61, April 2004

rayleigh quotient iteration;singular values;conjugate gradient method;total least squares

587895

The Recursive Inverse Eigenvalue Problem.

The recursive inverse eigenvalue problem for matrices is studied, where for each leading principal submatrix an eigenvalue and associated left and right eigenvectors are assigned. Existence and uniqueness results as well as explicit formulas are proven, and applications to nonnegative matrices, Z-matrices, M-matrices, symmetric matrices, Stieltjes matrices, and inverse M-matrices are considered.

Introduction Inverse eigenvalue problems are a very important subclass of inverse problems that arise in the context of mathematical modeling and parameter identification. They have been studied extensively in the last 20 years, see e.g. [3, 5, 7, 11, 12, 14] and the references therein. In particular, the inverse eigenvalue problem for non-negative matrices is still a topic of very active research, since a necessary and sufficient condition for the existence of a nonnegative matrix with a prescribed spectrum is still an open problem, see [4, 11]. In this paper we study inverse eigenvalue problems in a recursive matter, that allows to extend already constructed solutions if further data become available, as is frequently the case in inverse eigenvalue problems, e.g. [3]. We investigate the following recursive inverse eigenvalue problem of order n: Let F be a field, let s l 1;1 l 2;1 l 2;2 l n;1 r 1;1 r 1;2 r 2;2 r 1;n be vectors with elements in F . Construct a matrix A 2 F n;n such l T where Ahii denotes the i-th leading principal submatrix of A. In the sequel we shall use the notation RIEP(n) for "the recursive inverse eigenvalue problem of order n". In Section 2 we study the existence and uniqueness of solutions for RIEP(n) in the general case. Our main result gives a recursive characterization of the solution for RIEP(n). We also obtain a nonrecursive necessary and sufficient condition for unique solvability as well as an explicit formula for the solution in case of uniqueness. The results of Section 2 are applied in the subsequent sections to special cases. In Section 3 we discuss nonnegative solutions for RIEP(n) over the field IR of real numbers. We also introduce a nonrecursive sufficient condition for the existence of a nonnegative solution for RIEP(n). Uniqueness of nonnegative solutions for RIEP(n) is discussed in Section 4. In Section 5 we study Z-matrix and M-matrix solutions for RIEP(n) over IR. In Section 6 we consider real symmetric solutions for RIEP(n) over IR. In Section 7 we consider positive semidefinite real symmetric solutions for RIEP(n) over IR. In Section 8 we combine the results of the previous two sections to obtain analogous results for Stieltjes matrices. Finally, in Section 9 we investigate inverse M-matrix solutions for RIEP(n). Existence and uniqueness results In this section we study the existence and uniqueness of solutions for RIEP(n) in the general case. For this purpose we introduce some further notation. For the vectors l ~ l l i;1 r 1;i The case is easy to verify. Proposition 1 If l solves RIEP(1). If either l 1;1 6= 0 or r 1;1 6= 0 then is the unique solution for RIEP(1). For we have the following recursive characterization of the solution for RIEP(n). Theorem 2 Let n - 2. There exists a solution for RIEP(n) if and only if there exists a solution B for RIEP(n-1) such that l and There exists a unique solution for RIEP(n) if and only if there exists a unique solution for RIEP(n-1) and l n;n r n;n 6= 0. Proof. Let A be an n \Theta n matrix. Partition A as where B is an (n-1) \Theta (n-1) matrix. Clearly, A solves RIEP(n) if and only if B solves RIEP(n-1) and It thus follows that there exists a solution for RIEP(n) if and only if there exists a solution B for RIEP(n-1) such that the equations (4)-(7) (with unknown x, y and z) are solvable. We now show that these equations are solvable if and only if (1) and (2) hold. Distinguish between four cases: 1. r Here (4) is equivalent to (2), (5) is equivalent to l n;n and (6) then follows from (4). For every y 2 F n\Gamma1 we can find z 2 F such that (7) holds. 2. l Here (5) is equivalent to (1), (4) is equivalent to r n;n and (7) then follows from (5). For every x 2 F n\Gamma1 we can find z 2 F such that (6) holds. 3. l Here (4) is equivalent to (2) and (5) is equivalent to (1). For any x 2 F n\Gamma1 with x T ~ r we have (6), and for any y 2 F n\Gamma1 with we have (7), where z can be chosen arbitrarily. 4. l n;n 6= 0; r n;n 6= 0. Here (4)-(7) have a unique solution, given by (8), and l n;n r n;n It follows that (4)-(7) are solvable if and only if (1) and (2) hold. To prove the uniqueness assertion, note that it follows from our proof that if either l a solution is not unique, since at least one of the vectors x, y and z can be chosen arbitrarily. If both l n;n 6= 0 and r n;n 6= 0 then every solution B for RIEP(n-1) defines a unique solution A for RIEP(n). The uniqueness claim follows. This result is recursive and allows to derive a recursive algorithm to compute the solution, but we do not get explicit nonrecursive conditions that characterize the existence of solutions. In order to get a necessary and sufficient condition for unique solvability as well as an explicit formula for the solution in case of uniqueness, we define the n \Theta n matrix R n to be the matrix whose columns are appended at the bottom to obtain n-vectors. Similarly, we define the n \Theta n matrix L n to be the matrix whose rows are l appended at the right to obtain n-vectors. That is, we have l 2;1 l 2;2 r 2;2 . r n\Gamma1;n We denote Also, we denote by ffi the Hadamard (or elementwise) product of matrices. Proposition 3 A solution A for RIEP(n) satisfies Proof. We prove our claim by induction on n. For the claim follows easily. Assume that the assertion holds for Partition A as in (3). We have l n;n R By the inductive assumption we have L Also, by (4) we have B~r by (5) we have ~ l T n , and by (7) we have ~ l T n;n . It thus follows that In general, the converse of Proposition 3 does not hold, that is, a matrix A satisfying does not necessarily form a solution for RIEP(n), as is demonstrated by Example 5 below. Theorem 4 There is a unique solution for RIEP(n) if and only if l 1;1 6= 0 or r 1;1 6= 0 and l i;i r i;i 6= 0; Furthermore, the unique solution is given by Proof. The uniqueness claim follows from Proposition 1 and Theorem 2. The fact that the unique solution for RIEP(n) is given by (14) follows immediately from Proposition 3. In the case that the solution is not unique, that is, whenever l or whenever l i;i or r i;i vanish for some i ? 1, the matrices L n and R n defined in (11) are not invertible. Therefore, in this case (14) is invalid. We conclude this section by an example showing that, in general, a revised form of (14), with inverses replaced by generalized inverses, does not provide a solution for RIEP(n). Example 5 Let and let hi "0 We have be the Moore-Penrose inverses of L and R respectively, see [1]. We have Since Ah2i does not have an eigenvalue 2, A is not a solution for RIEP(3). Note that we still have L n AR In this section we have characterized solvability of RIEP(n) over a general field F in terms of recursive conditions. We have also given a necessary and sufficient condition for unique solvability and an explicit formula for the unique solution. In the following sections we shall discuss the special cases of nonnegative matrices, Z-matrices, M-matrices, real symmetric matrices, positive semidefinite matrices, Stieltjes matrices and inverse M-matrices. 3 Existence of nonnegative solutions In this section we apply the results of Section 2 to nonnegative solutions for RIEP(n) over the field IR of real numbers. A matrix A 2 IR n;n is said to be nonnegative [positive] if all elements of A are nonnegative [positive]. In this case we In order to state our results we define a vector over IR to be unisign if its nonzero components have the same sign. Theorem 6 Let n - 2. There exists a nonnegative solution for RIEP(n) if and only if we have l i or r i is a unisign nonzero vector =) s and there exists a nonnegative solution B for RIEP(n-1) sn ~ rn rn;n rn;n l n;n l n;n and l n;n r n;n 6= 0 =) s n ( l n;n r n;n l n;n r n;n There exists a positive solution for RIEP(n) if and only if there exists a positive solution B for RIEP(n-1) such that (15)-(18) hold with strict inequalities and every nonzero unisign vector l i or r i has no zero components. Proof. Let A 2 IR n;n . As in the proof of Theorem 2, partition A as in (3), and so A solves RIEP(n) if and only if B solves RIEP(n-1) and (4)-(7) hold. Therefore, if A is a nonnegative solution for RIEP(n) then we have (16)-(18). Also, it follows from the nonnegativity of A that (15) holds. Conversely, assume that (15) holds and that B forms a nonnegative solution for RIEP(n-1) satisfying (16)-(18). We show that in this case we can find nonnegative solutions x, y and z for (4)-(7). Distinguish between four cases: 1. r Here x is given by (8), y can be chosen arbitrarily, and z should be chosen such that (7) holds. It follows from (17) that x is nonnegative. If s n - 0 then we choose so we have a nonnegative solution for (4)-(7). If s n ! 0 then, by (15), l n is not unisign and hence ~ l T l n;n has at least one negative component. It follows that we can find a positive vector y such that ~ l T l n;n by (7) we have l n;n , it follows that z ? 0, and so again we have a nonnegative solution for (4)-(7). 2. l Here y is given by (9), x can be chosen arbitrarily, and z should be chosen such that (6) holds. The proof follows as in the previous case. 3. l should be chosen such that x T ~ and z can be chosen arbitrarily. In order to obtain a nonnegative solution we can choose x, y and z to be zero. 4. l n;n 6= 0; r n;n 6= 0. Here x is given by (8), y is given by (9), and z is given by (10). It follows from (17), (16) and (18) that x, y and z are nonnegative. Assume now that A is a positive solution for RIEP(n). It is easy to verify that in this case (15)-(18) should hold with strict inequalities. Also, for every nonzero unisign vector l i [r i ], the vector l T has no zero components, implying that l i , [r i ] has no zero components. Conversely, assume that (15) holds with a strict inequality, that every nonzero unisign vector l i or r i has no zero components, and that B forms a positive solution for RIEP(n-1) satisfying (16)-(18) with strict inequalities. We show that in this case we can find positive solutions x, y and z for (4)-(7). Note that in Case 1 above, the vector x now becomes positive. Also, since the inequality in (15) is now strict, we have either s n ? 0, in which case we can choose positive y sufficiently small such that z is positive, or s n - 0, in which case y can be chosen positive as before and the resulting z is positive. The same arguments hold for Case 2. In Case 4, it follows from the strict inequalities (17)-(18) that x, y and z are positive. Finally, in Case 3, since l n and r n both have at least one zero component, it follows that both vectors are not unisign. Hence, we can find positive x and y such that x T ~ r We assign any positive number to z to find a positive solution A for RIEP(n). By the Perron-Frobenius theory, see e.g. [8, 2], the largest absolute value ae(A) of an eigenvalue of a nonnegative n \Theta n matrix A is itself an eigenvalue of A, the so called Perron root of A, and it has an associated nonnegative eigenvector. Furthermore, if A is irreducible, that is, if either there exists no permutation matrix P such that P T where B and D are square, then ae(A) is a simple eigenvalue of A with an associated positive eigenvector. If A is not necessarily irreducible then we have the following, see e.g. [2]. Theorem 7 If B is a principal submatrix of a nonnegative square matrix A then ae(B) - ae(A). Furthermore, ae(A) is an eigenvalue of some proper principal submatrix of A if and only if A is reducible. Note that if we require that the s i are the Perron roots of the principal submatrices Theorem 7, we have If, furthermore, all the leading principal submatrices of A are required to be irreducible, then Condition (19) is not sufficient to guarantee that a nonnegative solution A for RIEP(n) necessarily has s Perron roots of Ahii, demonstrated by the following example. Example 8 Let and let hi "0 The nonnegative matrix 24 In order to see cases in which s are the Perron roots of Ahii, respectively, we prove Proposition 9 If the vector l n or r n is positive then for a nonnegative solution A for RIEP(n) we have Proof. The claim follows immediately from the known fact that a positive eigenvector of a nonnegative matrix corresponds to the spectral radius, see e.g. Theorem 2.1.11 in [2, p. 128]. ng we have either l i ? 0 or r i ? 0 then for every nonnegative solution A for RIEP(n) we have Lemma 11 Assume that there exists a nonnegative solution A for RIEP(n) such that ae(Ahn-1i) ! s n . If r n 6= 0 or l n 6= 0 then Proof. Since r n 6= 0 or l n 6= 0 it follows that s n is an eigenvalue of A. Assume that ae(A). It follows that the nonnegative matrix A has at least two eigenvalues larger than or equal to s n . By [6, p. 473], see also [10, Corollary 1], it follows that ae(Ahn-1i) - s n , which is a contradiction. Therefore, we have s Corollary 12 If for every ng we have either r i 6= 0 or l i 6= 0, and if holds then for every nonnegative solution A for RIEP(n) we have Proof. Note that Our result follows using Lemma 11 repeatedly. Lemma 13 Assume that r n - 0 and r n;n 6= 0 or that l n - 0 and l n;n 6= 0. Then for every nonnegative solution A for RIEP(n) we have g. Proof. Without loss of generality, we consider the case where r n - 0 and r n;n 6= 0. If r n is positive then, by Proposition 9, we have since by the Perron-Frobenius theory we have ae(Ahn-1i) - ae(A), the result follows. Other- wise, r n has some zero components. Let ff be the set of indices i such that r i;n ? 0 and let ff c be the complement of ff in ng. Note that since r n is a nonnegative eigenvector of the nonnegative matrix A it follows that the submatrix A[ff c jff] of A, with rows indexed by ff c and columns indexed by ff, is a zero matrix. It follows that A is a reducible matrix and ae(A[ffjff])g. Note that the subvector r n [ff] of r n indexed by ff is a positive eigenvector of A[ffjff] associated with the eigenvalue s n . It thus follows that it follows that A[ff c jff c ] is a submatrix of Ahn-1i. Thus, by the Perron-Frobenius theory we have ae(A[ff c jff c ]) - ae(Ahn-1i) - ae(A). Hence, it follows that g. Corollary 14 Assume that for every ng we have either r i - 0 and r i;i 6= 0 or l i - 0 and l i;i 6= 0. Then for every nonnegative solution A for RIEP(n) we have g. Proof. Note that Our result follows using repeatedly. Corollary 15 Assume that for every r i;i 6= 0 or l i - 0 and l i;i 6= 0. If (19) holds then for every nonnegative solution A we have Another interesting consequence of Theorem 4 is the following relationship between the matrix elements and the eigenvectors associated with the Perron roots of the leading principal submatrices of a nonnegative matrix. Corollary 2. Let A 2 IR n;n be a nonnegative matrix, let s be the Perron roots and associated left and right eigenvectors of Ahii, respectively, and assume that (20) holds. Let S defined as in (11) and (12). Then Proof. Since (20) holds, it follows that s i is not an eigenvalue of Ahi-1i, Therefore, it follows from (1) and (2) that l i;i r i;i 6= 0. Also, since l 1 and r 1 are eigenvectors of Ah1i, we have l 1;1 r 1;1 6= 0. It now follows from Theorem 4 that Ahii is the unique solution for RIEP(i), and is given by (21). While Theorem 6 provides a recursive characterization for nonnegative solvability of RIEP(n), in general nonrecursive necessary and sufficient conditions for the existence of nonnegative solution are not known. We now present a nonrecursive sufficient condition. Corollary 17 Assume that the vectors l are all positive and that the numbers s are all positive. Let r j;i r r j;i r l l i;j l i\Gamma1;j l i;j l i\Gamma1;j If we have and then there exists a (unique) nonnegative solution A for RIEP(n). Furthermore, if all the inequalities (22)-(24) hold with strict inequality then there exists a (unique) positive solution A for RIEP(n). Proof. We prove our assertion by induction on n. The case the inductive assumption we can find a nonnegative solution B for RIEP(n-1). Note that Therefore, it follows from (22) that and so (16) holds. Similarly we prove that (17) holds. To prove that holds note that by (25) we have B~r n - Bm r Similarly, we have ~ l T . Hence, it follows that ~ l T . By (24) applied to Theorem 6, there exists a nonnegative solution for RIEP(n). The proof of the positive case is similar. The conditions in Corollary 17 are not necessary as is demonstrated by the following example. Example hi "1 We have m r 2. Note that both (22) and (23) do not hold for 3. Nevertheless, the unique solution for RIEP(3) is the nonnegative matrix 2 4 Uniqueness of nonnegative solutions When considering uniqueness of nonnegative solutions for RIEP(n), observe that it is possible that RIEP(n) does not have a unique solution but does have a unique nonnegative solution, as is demonstrated by the following example. Example 19 Let and let hi "0 "1 By Theorem 2, there is no unique solution for RIEP(2). Indeed, the solutions for RIEP(2) are all matrices of the form a \Gammaa Clearly, the zero matrix is the only nonnegative solution for RIEP(2). Observe that, unlike in Theorem 2, the existence of a unique nonnegative solution for RIEP(n) does not necessarily imply the existence of a unique nonnegative solution for RIEP(n-1), as is demonstrated by the following example. Example 20 Let and let hi "0 Observe that all matrices of the form a a solve RIEP(2), and hence there is no unique nonnegative solution for RIEP(2). However, the only nonnegative solution for RIEP(3) is the matrix6 4 We remark that one can easily produce a similar example with nonnegative vectors In order to introduce necessary conditions and sufficient conditions for uniqueness of nonnegative solutions for RIEP(n) we prove Lemma and assume that B forms a nonnegative solution for satisfying (15)-(18). Then there exist unique nonnegative vectors x, y and z such that the matrix solves RIEP(n) if and only if either l n;n r n;n 6= 0, or s l n is a unisign vector with no zero components, or is a unisign vector with no zero components. Proof. We follow the proof of Theorem 6. Consider the four cases in that proof. In Case 1, the vector x is uniquely determined and any nonnegative assignment for y is valid as long as l n;n nonnegative vector sufficiently small will do. If s as is shown in the proof of Theorem 6, we can find a positive y such that z ? 0, and by continuity arguments there exist infinitely many such vectors y. If s a unique such y exists if and only if there exists a unique nonnegative vector y such that ~ l T l n;n l n has a nonpositive component then every vector y whose corresponding component is positive and all other components are zero solves the problem. On the other hand, if ~ l n ? 0, which is equivalent to saying that l n is a unisign vector with no zero components, then the only nonnegative vector y that solves the problem is Similarly, we prove that, in case 2, a unique nonnegative solution exists if and only if s is a unisign vector with no zero components. We do not have uniqueness in Case 3 since then z can be chosen arbitrarily. Finally, there is always uniqueness in Case 4. Lemma 21 yields sufficient conditions and necessary conditions for uniqueness of nonnegative solutions for RIEP(n). First, observe that if s l n is a unisign vector with no zero components, or if s is a unisign vector with no zero components, then the zero matrix is the only nonnegative solution of the problem. A less trivial sufficient condition is the following. Corollary 22 Let n - 2, and let A be a nonnegative solution for RIEP(n). If Ahn-1i forms a unique nonnegative solution for RIEP(n-1) and if l n;n r n;n 6= 0, then A is the unique nonnegative solution for RIEP(n). Necessary conditions are given by the following Corollary 23 Let n - 2. If there exists a unique nonnegative solution for RIEP(n) then either l n;n r n;n 6= 0, or s l n is a unisign vector with no zero components, or s is a unisign vector with no zero components. The condition l n;n r n;n 6= 0 is not sufficient for the uniqueness of a nonnegative solution for RIEP(n), as is shown in the following example. Example and let hi "0 Although we have l n;n r n;n 6= 0, all matrices of the a a 0 a a 07solve RIEP(3), and hence there is no unique nonnegative solution for RIEP(3). 5 The Z-matrix and M-matrix case A real square matrix A is said to be a Z-matrix if it has nonpositive off-diagonal elements. Note that A can be written as ff is a real number and B is a nonnegative matrix. If we further have that ff - ae(B) then we say that A is an M-matrix. In this section we discuss Z-matrix and M-matrix solutions for RIEP(n) over the field IR of real numbers. The proofs of the results are very similar to the proofs of the corresponding results in Sections 3 and 4 and, thus, are omitted in most cases. Theorem 25 Let n - 2. There exists a Z-matrix solution for RIEP(n) if and only if there exists a Z-matrix solution B for RIEP(n-1) sn ~ rn rn;n rn;n and 8 l n;n l n;n Furthermore, if l n or r n is positive then a Z-matrix solution for RIEP(n) is an M-matrix if and only if s n - 0. Proof. The proof of the first part of the theorem is similar to the proof of Theorem 6, observing that here the vectors x and y are required to be nonnegative and that the sign of z is immaterial. The proof of the second part of the Theorem follows, similarly to Proposition 9, from the known fact that a positive eigenvector of a Z-matrix corresponds to the least real eigenvalue. Theorem 26 Let n - 2. Let A 2 IR n;n be a Z-matrix, let s be the least real eigenvalues and the corresponding left and right eigenvectors of Ahii, respectively, and assume that defined as in (11) and (12). Then For the numbers M r and m l , defined in Corollary 17, we have Theorem 27 Assume that the vectors l are all positive and that the numbers s are all positive. If we have and then there exists a (unique) M-matrix solution A for RIEP(n). Theorem 28 Let n - 2, let A be a Z-matrix solution for RIEP(n) and assume that Ahn-1i forms a unique Z-matrix solution for RIEP(n-1). Then A is the unique Z-matrix solution for RIEP(n) if and only if l n;n r n;n 6= 0. Here too, unlike in Theorem 2, the existence of a unique Z-matrix solution for RIEP(n) does not necessarily imply the existence of a unique Z-matrix solution for RIEP(n-1), as is demonstrated by the following example. Example 29 Let s hi "0 "1 Observe that all matrices of the form a \Gammaa solve RIEP(2), and hence there is no unique Z-matrix solution for RIEP(2). However, it is easy to verify that the zero matrix is the only Z-matrix solution for RIEP(3). 6 The real symmetric case The inverse eigenvalue problem for real symmetric matrices is well studied, see e.g. [3]. In this section we consider symmetric solutions for RIEP(n) over the field IR of real numbers. We obtain the following consequence of Theorem 2, characterizing the real symmetric case. Theorem 2. There exists a symmetric solution for RIEP(n) if and only if there exists a symmetric solution B for RIEP(n-1) such that the implications (1) and (2) hold, and l n;n r n;n 6= 0 =) (s n I ~ l n l n;n ~ r n;n Furthermore, if there exists a unique symmetric solution for RIEP(n) then l n;n 6= or r n;n 6= 0. Proof. Let A 2 IR n;n . Partition A as in (3), and so A solves RIEP(n) if and only solves RIEP(n-1) and (4)-(7) hold. It was shown in the proof of Theorem 2 that (4)-(7) are solvable if and only if (1) and (2) hold. Therefore, all we have to show that if B is symmetric then we can find solutions x, y and z for (4)-(7) such that only if (26) holds. We go along the four cases discussed in Theorem 2. In Case 1, the vector x is uniquely determined and the vector y can be chosen arbitrarily. Therefore, in this case we set y = x, and z is then uniquely determined. In Case 2, the vector y is uniquely determined and the vector x can be chosen arbitrarily. Thus, in this case we set y, and z is then uniquely determined. In Case 3, we can choose any x and y as long as x T ~ r In particular, we can choose Furthermore, z can be chosen arbitrarily. Finally, in Case 4, we have only if (26) holds. Note that this is the only case in which, under the requirement that the vectors x, y and z are uniquely determined. We remark that, unlike in Theorem 2, the existence of a unique symmetric solution for RIEP(n) does not necessarily imply the existence of a unique symmetric solution for RIEP(n-1), as is demonstrated by the following example. Example and let hi "1 l 4 =6 6 610 \Gamma17 7 7: It is easy to verify that all symmetric matrices of the form61 1 a a a b7 solve RIEP(3), while the unique solution for This example also shows that there may exist a unique solution for RIEP(n) even if l Naturally, although not necessarily, one may expect in the symmetric case to have the condition Indeed, in this case we have the following corollary of Theorems 2 and 30. Corollary assume that (27) holds. The following are equivalent (i) There exists a symmetric solution for RIEP(n). (ii) There exists a solution for RIEP(n). (iii) There exists a symmetric solution B for RIEP(n-1) such that (1) holds. (iv) There exists a solution B for RIEP(n-1) such that (1) holds. Proof. Note that since (27) holds, we always have (26). We now prove the equivalence between the four statements of the theorem. (i) =) (ii) is trivial. (ii) =) (iv) by Theorem 2. (iv) =) (iii). Since (27) holds, it follows that B+B Talso solves RIEP(n-1). (iii) =) (i). Since B is symmetric and since we have (27), the implications (1) and (2) are identical. Our claim now follows by Theorem 30. For uniqueness we have Theorem 33 Let n - 2 and assume that (27) holds. The following are equivalent (i) There exists a unique symmetric solution for RIEP(n). (ii) There exists a unique solution for RIEP(n). (iii) We have l i;i 6= 0; Proof. In view of (27), the equivalence of (ii) and (iii) follows from Theorem 4. To see that (i) and (iii) are equivalent note that, by the construction in Theorem 30, for every symmetric solution B for RIEP(n-1) there exists a solution A for RIEP(n) such that Furthermore, A is uniquely determined if and only if l n;n 6= 0. Therefore, it follows that there exists a unique symmetric solution for RIEP(n) if and only if there exists a unique symmetric solution for and l n;n 6= 0. Our assertion now follows by induction on n. We conclude this section remarking that a similar discussion can be carried over for complex Hermitian matrices. 7 The positive semidefinite case In view of the discussion of the previous section, it would be interesting to find conditions for the existence of a positive (semi)definite real symmetric solution for RIEP(n). Clearly, a necessary condition is nonnegativity of the numbers s i n. Nevertheless, this condition is not sufficient even if a real symmetric solution exists, as is demonstrated by the following example. Example 34 Let and let hi "1 The unique solution for RIEP(3) is the symmetric matrix6 4 which is not positive semidefinite. The following necessary and sufficient condition follows immediately from Theorem 4. Theorem assume that (27) holds. Assume, further, that r i;i 6= n. Then the unique solution for RIEP(n) is positive semidefinite [positive definite] if and only if S n ffi (R T positive semidefinite [positive definite]. Remark 36 By Theorem 33, in the case that r we do not have uniqueness of symmetric solutions for RIEP(n). Hence, if there exists a symmetric solution for RIEP(n) then there exist at least two different such solutions A and B. Note that A a symmetric solution for RIEP(n) for every real number c. It thus follows that in this a case it is impossible to have all solutions for RIEP(n) positive semidefinite. Therefore, in this case we are looking for conditions for the existence of some positive semidefinite solution for RIEP(n). The following necessary condition follows immediately from Proposition 3. Theorem 37 Let n - 2 and assume that (27) holds. If there exists a positive semidefinite real symmetric solution for RIEP(n) then S n ffi (R T positive semidefinite. In order to find sufficient conditions for the existence of a positive semidefinite solution for RIEP(n), we denote by oe(A) the least eigenvalue of a real symmetric matrix A. Lemma 38 Let n - 2 and assume that (27) holds. Assume that there exists a symmetric solution A for RIEP(n) such that oe(Ahn-1i) ? s n . If r n 6= 0 then Proof. Since r n 6= 0 it follows that s n is an eigenvalue of A. Assume that It follows that A has at least two eigenvalues smaller than or equal to s n . By the Cauchy Interlacing Theorem for Hermitian matrices, e.g. [8, Theorem 4.3.8, p. 185], it follows that oe(Ahn-1i) - s n , which is a contradiction. Therefore, we have Corollary assume that (27) holds. If r i 6= 0 for all i, then every real symmetric solution A for RIEP(n) is positive semidefinite. If s n ? 0 then every real symmetric solution for RIEP(n) is positive definite. Proof. Note that Using Lemma 38 repeatedly we finally obtain implying our claim. Remark 40 In view of Remark 36, it follows from Corollary 39 that if r i 6= 0 for all i and if has a unique (positive semidefinite) solution. The converse of Corollary 39 is, in general, not true. That is, even if every real symmetric solution for RIEP(n) is positive semidefinite we do not necessarily have as is demonstrated by the following example. Example and let hi "1 The unique solution for RIEP(3) is the positive definite matrix Nevertheless, we do not have s 1 - s 2 . We conclude this section with a conjecture motivated by Theorems 35 and 37. One direction of the conjecture is proven in Theorem 37. Conjecture 42 Let n - 2 and assume that (27) holds. Then there exists a positive semidefinite [positive definite] real symmetric solution for RIEP(n) if and only if S n ffi (R T positive semidefinite [positive definite]. 8 The Stieltjes matrix case In this section we combine the results of the previous two sections to obtain analogous results for Stieltjes matrices, that is, symmetric M-matrices. The following theorem follows immediately from Theorems 30 and 25. Theorem 43 Let n - 2. There exists a symmetric Z-matrix solution for RIEP(n) if and only if there exists a symmetric Z-matrix solution B for satisfying 8 sn ~ rn rn;n rn;n l n;n l n;n and l n;n r n;n 6= 0 =) (s n I ~ l n l n;n ~ r n;n Furthermore, if l n or r n is positive then a symmetric Z-matrix solution for RIEP(n) is a Stieltjes matrix if and only if s n - 0. Corollary 44 Let n - 2, and assume that the vectors l i , are all positive and that (27) holds. There exists a symmetric Z-matrix solution A for RIEP(n) if and only if there exists a symmetric Z-matrix solution B for satisfying s n ~ . The solution A is a Stieltjes matrix if and only if s n - 0. The following nonrecursive sufficient condition from Theorem 27. Theorem assume that the vectors l i , are all positive, that (27) holds, and that the numbers s are all positive. If we have then there exists a (unique) Stieltjes matrix solution A for RIEP(n). Proof. By Theorem 27 there exists a unique M-matrix solution A for RIEP(n). Since A T also solves the problem, it follows that A = A T and the result follows. 9 The inverse M-matrix case It is well known that for a nonsingular M-matrix A we have A ingly, a nonnegative matrix A is called inverse M-matrix if it is invertible and A \Gamma1 is an M-matrix. An overview of characterizations of nonnegative matrices that are inverse M-matrices can be found in [9].In this section we discuss, as a final special case, inverse M-matrix solutions for RIEP(n). The following theorem follows immediately from two results of [9]. Theorem 46 Let A 2 IR n;n be partitioned as in (3). Then A is an inverse M-matrix if and only if B is an inverse M-matrix and and for the diagonal entries: (31) Proof. By Corollary 3 in [9], if A is an inverse M-matrix then B is an inverse M-matrix. By Theorem 8 in [9], if B is an inverse M-matrix then A is an inverse M-matrix if and only if (28)-(31) hold. Our claim follows. The next result gives necessary and sufficient recursive conditions for the existence of an inverse M-matrix solution for RIEP(n). Theorem 2. There exists an inverse M-matrix solution for RIEP(n) if and only if s n ? 0 and there exists an inverse M-matrix solution B for satisfying 8 N~rn rn;n l n;n l n;n r n;n 6= 0 =) l n;n r n;n and, except for the diagonal entries, l n;n r n;n 6= 0 =) s n l n;n r n;n l n;n r n;n Proof. As in the proof of Theorem 2, partition A as in (3). If A is an inverse M - matrix solution for RIEP(n) then, as is well known, its eigenvalues lie in the open right half plane, and so the real eigenvalue s n must be positive. Furthermore, by Theorem 46, B is an inverse M-matrix and (28)-(31) hold. Finally, we have (4)-(7). Distinguish between four cases: 1. r Here x is given by (8), and so it follows from (29) that l n;n Theorem 2 we have B~r implying that N ~ 2. l Here y is given by (9), and so it follows from (28) that N~rn rn;n Theorem 2 we have ~ l T 3. l Similarly to the previous cases prove that N ~ 4. l n;n 6= 0; r n;n 6= 0. Here x is given by (8), y is given by (9), and z is given by (10). It follows from (28) that N~rn rn;n - 0, and from (29) that ~ l T l n;n - 0. It follows from (30) that l n;n r n;n l n;n r n;n l n;n r n;n 0: now follows that ~ l T l n;nrn;n ! 1. Finally, it follows from (31) that, except for the diagonal entries, l n;n r n;n r n;n l n;n l n;n r n;n We have thus proven that if A is an inverse M-matrix solution for RIEP(n) then is an inverse M-matrix solution B for RIEP(n-1) satisfying (32)-(35). Conversely, assume that s n ? 0 and B is an inverse M-matrix solution B for satisfying (32)-(35). We show that x, y and z can be chosen such that (28)-(31) hold, and so by Theorem 46, A is an inverse M-matrix. Here too we distinguish between four cases: 1. r Here x is given by (8), and by (33) we obtain (29). Note that y can be chosen arbitrarily, and and z should be chosen such that (7) holds. If we choose It follows that so we also have (30). Finally, since is an M-matrix, it follows that (31) holds (except for the diagonal entries). 2. l Here y is given by (9), and by (32) we obtain (28). The vector x can be chosen arbitrarily, so we choose The proof follows as in the previous case. 3. l should be chosen such that x T ~ and z can be chosen arbitrarily. We choose and the proof follows. 4. l n;n 6= 0; r n;n 6= 0. Here x is given by (8), y is given by (9), and z is given by (10). By (32) and (33) we obtain (28) and (29) respectively. Finally, similarly to the corresponding case in the proof of the other direction, (34) implies (30) and (35) implies (31). Note that Conditions (32)-(33) imply immediately Conditions (16)-(17) by multiplying the inequality by the nonnegative matrix B. This is not surprising, since an inverse M-matrix is a nonnegative matrix. The converse, however, does not hold in general. The following example shows that although (16)-(17) is satisfied, (32)-(33) do not hold. Example 48 Let and let hi 0:5257 0:8507 0:3859 0:91267 The unique solution for RIEP(3) is the nonnegative matrix A =6 4 which is not an inverse M-matrix since A 1:6429 \Gamma1:5714 0:4286 \Gamma1:5714 2:2857 \Gamma0:7143 0:4286 \Gamma0:7143 0:2857 Indeed, the unique nonnegative solution for RIEP(2) satisfies (16), as 2:8673 8:2024 However, B does not satisfy (32), since the vector \Gamma1:3688 2:2816 is not nonnegative. --R Generalized Matrix Inverses: Theory and Applications Nonnegative Matrices in Mathematical Sci- ences A survey of matrix inverse eigenvalue problems The spectra of non-negative matrices via symbolic dynamics On an inverse problem for nonnegative and eventually nonnegative matrices On some inverse problems in matrix theory Matrix Analysis Inverse eigenvalue problems for matrices A note on an inverse problem for nonnegative matrices Nonnegative matrices whose inverses are M-matrices Note on an inverse characteristic value problem --TR --CTR Fan-Liang Li , Xi-Yan Hu , Lei Zhang, Left and right inverse eigenpairs problem of skew-centrosymmetric matrices, Applied Mathematics and Computation, v.177 n.1, p.105-110, 1 June 2006

m-matrices;recursive solution;stieltjes matrices;inverse eigenvalue problem;inverse M-matrices;hermitian matrices;nonnegative matrices;z-matrices

587916

Extremal Properties for Dissections of Convex 3-Polytopes.

A dissection of a convex d-polytope is a partition of the polytope into d-simplices whose vertices are among the vertices of the polytope. Triangulations are dissections that have the additional property that the set of all its simplices forms a simplicial complex. The size of a dissection is the number of d-simplices it contains. This paper compares triangulations of maximal size with dissections of maximal size. We also exhibit lower and upper bounds for the size of dissections of a 3-polytope and analyze extremal size triangulations for specific nonsimplicial polytopes: prisms, antiprisms, Archimedean solids, and combinatorial d-cubes.

Introduction . Let A be a point conguration in R d with its convex hull having dimension d. A set of d-simplices with vertices in A is a dissection of A if no pair of simplices has an interior point in common and their union equals conv(A). A dissection is a triangulation of A if in addition any pair of simplices intersects at a common face (possibly empty). The size of a dissection is the number of d-simplices it contains. We say that a dissection is mismatching when it is not a triangulation (i.e. it is not a simplicial complex). In this paper we study mismatching dissections of maximal possible size for a convex polytope and compare them with maximal triangulations. This investigation is related to the study of Hilbert bases and the hierarchy of covering properties for polyhedral cones which is relevant in Algebraic Geometry and Integer Programming (see [5, 10, 24]). Maximal dissections are relevant also in the enumeration of interior lattice points and its applications (see [2, 15] and references there). It was rst shown by Lagarias and Ziegler that dissections of maximal size turn out to be, in general, larger than maximal triangulations, but their example uses interior points [16]. Similar investigations were undertaken for mismatching minimal dissections and minimal triangulations of convex polytopes [4]. In this paper we augment previous results by showing that it is possible to have simultaneously, in the same 3-polytope, that the size of a mismatching minimal (maximal) dissection is smaller (larger) than any minimal (maximal) triangulation. In addition, we show that the gap between the size of a mismatching maximal dissection and a maximal triangulation can grow linearly on the number of vertices and that this occurs already for a family of simplicial convex 3-polytopes. A natural question is how dierent are the upper and lower bounds for the size of mismatching dissections versus those bounds known for triangulations (see [21]). We prove lower and upper bounds on their size with respect to the number of vertices for dimension three and exhibit examples showing that our technique of proof fails already in dimension four. Here is the rst Dept. of Mathematics, Univ. of California-Davis (deloera@math.ucdavis.edu). The research of this author partially supported by NSF grant DMS-0073815. y Depto. de Matematicas, Estad. y Comput., Univ. de Cantabria (santos@matesco.unican.es). The research of this author was supported partially by grant PB97-0358 of the Spanish Direccion General de Investigacion Cientca y Tecnica. z Dept. of Information Science, Univ. of Tokyo (fumi@is.s.u-tokyo.ac.jp). summary of results: Theorem 1.1. 1. There exists an innite family of convex simplicial 3-polytopes with increasing number of vertices whose mismatching maximal dissections are larger than their maximal triangulations. This gap is linear in the number of vertices (Corollary 2.2). 2. (a) There exists a lattice 3-polytope with 8 vertices containing no other lattice point other than its vertices whose maximal dissection is larger than its maximal triangulations. (b) There exists a 3-polytope with 8 vertices for which, simultaneously, its minimal dissection is smaller than minimal triangulations and maximal dissection is larger than maximal triangulations. (Proposition 2.3) 3. If D is a mismatching dissection of a 3-polytope with n vertices, then the size of D is at least n 2. In addition, the size of D is bounded above by n 2 (Proposition 3.2). A consequence of our third point is that the result of [4], stating a linear gap between the size of minimal dissections and minimal triangulations, is best possible. The results are discussed in Sections 2 and 3. The last section presents a study of maximal and minimal triangulations for combinatorial d-cubes, three-dimensional prisms and anti-prisms, as well as other Archimedean polytopes. The following theorem and table summarize the main results: Theorem 1.2. 1. There is a constant c > 1 such that for every d 3 the maximal triangulation among all possible combinatorial d-cubes has size at least c d d! (Proposition 4.1). 2. For a three-dimensional m-prism, in any of its possible coordinatizations, the size of a minimal triangulation is 2m 5+ d me. For an m-antiprism, in any of its possible coordinatizations, the size of a minimal triangulation is 3m 5 (Proposition 4.3). The size of a maximal triangulation of an m-prism depends on the coordinatization, and in certain natural cases it is (m (Proposition 4.4). 3. The following table species sizes of the minimal and maximal triangulations for some Platonic and Archimidean solids. These results were obtained via integer programming calculations using the approach described in [8]. All computations used the canonical symmetric coordinatizations for these polytopes [6]. The number of vertices is indicated in parenthesis (Remark 4.5): Icosahedron (12) 15 20 Dodecahedron (20) 23 36 Cuboctahedron Icosidodecahedron Truncated Truncated Octahedron Truncated Cube (24) 25 48 Small Rhombicuboctahedron Pentakis Dodecahedron (32) 54 ? Rhombododecahedron Table Sizes of extremal triangulations of Platonic and Archimidean solids. 2. Maximal dissections of 3-polytopes. We introduce some important de- nitions and conventions: We denote by Qm a convex m-gon with m an even positive be two edges parallel to Qm , orthogonal to each other, on opposite sides of the plane containing Qm , and such that the four segments v i intersect the interior of Qm . We suppose that v 1 v 2 and u 1 u 2 are not parallel to any diagonal or edge of Qm . The convex hull Pm of these points has m+ 4 vertices and it is a simplicial polytope. We will call north (respectively south) vertex of Qm the one which maximizes (respectively minimizes) the scalar product with the vector v 2 v 1 . Similarly, we will call east (west) the vertex which maximizes (minimizes) the scalar product with u 2 u 1 . We denote these four vertices n, s, e and w, respectively. See Figure 2.1. e s Fig. 2.1. North, South, East, and West vertices. We say that a directed path of edges inside Qm is monotone in the direction v 1 (respectively when the vertices of the path appear in the path following the same order given by the scalar product with formulation is that any line orthogonal to v 1 v 2 cuts the path in at most one point. We remark that by our choice of v 1 v 2 and u 1 u 2 all vertices of Qm are ordered by the values of their scalar products with v 2 v 1 and also with respect to u 2 u 1 . In the same way, a sequence of vertices of Qm is ordered in the direction of v 1 v 2 (respectively if the order is the same as the one provided by using the values of the scalar products of the points with the vector Consider the two orderings induced by the directions of v 1 v 2 and u 1 u 2 on the set of vertices of Qm . Let us call horizontal (respectively vertical) any edge joining two consecutive vertices in the direction of v 1 v 2 (respectively of u 1 u 2 ). As an example, if Qm is regular then the vertical edges in Qm form a zig-zag path as shown in Figure 2.2. Our examples in this section will be based on the following observation and are inspired by a similar analysis of maximal dissections of dilated empty lattice tetrahedra in R 3 by Lagarias and Ziegler [16]: Let Rm be the convex hull of the m+2 vertices consisting of the m-gon Qm and . Rm is exactly one half of the polytope Pm . Consider a triangulation T 0 of Qm and a path of edges of T 0 monotone with respect to the direction u 1 u 2 . Observe that divides T 0 in two regions, which we will call the \north" and the \south". Then, the following three families of tetrahedra form a triangulation T of Rm : the edges of joined to the edge the southern triangles of T 0 joined to v 1 ; and the northern triangles of T 0 joined to v 2 (see Figure 2.3). Moreover, all the triangulations of Rm are obtained in this way: Any triangulation T e Fig. 2.2. The minimal monotone path (middle) and the maximal monotone path made by the vertical edges (right) in the direction u 1 u 2 . s e s e s e Fig. 2.3. Three types of tetrahedra in Rm . of Rm induces a triangulation T 0 of Qm . The link of v 1 v 2 in T is a monotone path of edges contained in T 0 and it divides T 0 in two regions, joined respectively to v 1 and Using the Cayley trick, one can also think of the triangulations of Rm as the ne mixed subdivisions of the Minkowski sum Qm (see [13] and references within). The size of a triangulation of Rm equals is the number of edges in the path . There is a unique minimal path in Qm of length one (Figure 2.2, middle) and a unique maximal path of length m 1 (Figure 2.2, right). Hence the minimal and maximal triangulations of Rm have, respectively, m 1 and 2m 3 tetrahedra. The maximal triangulation is unique, but the minimal one is not: after choosing the diagonal in the rest of the polygon Qm can be triangulated in many ways. From the above discussion regarding Rm we see that we could independently triangulate each of the two halves of Pm with any number of tetrahedra from m 1 to 2m 3. Hence, Pm has dissections of sizes going from 2m 2 to 4m 6. Among the triangulations of Pm , we will call halving triangulations those that triangulate the two halves of Pm . Equivalently, the halving triangulations are those which do not contain any of the four edges Proposition 2.1. Let Pm be as described above, with Qm being a regular m-gon. No triangulation of Pm has more than 7m+ 1 tetrahedra. On the other hand, there are mismatching dissections of Pm with 4m 6 tetrahedra. Proof. Let T be a triangulation of Pm . It is an easy application of Euler's formulas for the 3-ball and 2-sphere that the number of tetrahedra in a triangulation of any 3-ball without interior vertices equals the number of vertices plus interior edges minus three (such formula appears for instance in [9]). Hence our task is to prove that T has at most 5minterior edges. For this, we classify the interior edges according to how many vertices of Qm they are incident to. There are only four edges not incident to any vertex of Qm (the edges contains at most m 3 edges incident to two vertices of Qm (i.e. diagonals of Qm ), since in any family of more than m 3 such edges there are pairs which cross each other. Thus, it su-ces to prove that T contains at most 3m1 edges incident to just one vertex of Qm , i.e. of the form v i p or u i p with Let p be any vertex of Qm . If p equals w or e then the edges pv 1 and pv 2 are both in the boundary of Pm ; for any other p, exactly one of pv 1 and pv 2 is on the boundary and the other one is interior. Moreover, we claim that if pv i is an interior edge in a triangulation T , then the triangle pv 1 v 2 appears in T . This is so because there is a plane containing pv i and having v 3 i as the unique vertex on one side. At the same time the link of pv i is a cycle going around the edge. Hence, v 3 i must appear in the link of pv i . It follows from the above claim that the number of interior edges of the form pv i in T equals the number of vertices of Qm other than w and e in the link of In a similar way, the number of interior edges of the form pu i in T equals the number of vertices of Qm other than n and s in the link of u 1 u 2 . In other words, if we call in the index and of the vertices are reversed, because in this way u is monotone with respect to with respect to v 1 v 2 ), then the number of interior edges in T incident to exactly one vertex of Qm equals jvertices( v )j Our goal is to bound this number. As an example, Figure 2.4 shows the intersection of Qm with a certain triangulation of Pm 12). The link of v 1 v 2 in this triangulation is the chain of vertices and edges wabu 1 nu 2 ce (the star of v 1 v 2 is marked in thick and grey in the gure). u consists of the chains wab and ce and the isolated vertex n. In turn, the link of u 1 u 2 is the chain nv 1 s and v consists of the isolated vertices n and s. s e a c Fig. 2.4. Illustration of the proof of Proposition 2.1. Observe that v has at most three connected components, because it is obtained by removing from link T the parts of it incident to v 1 and v 2 , if any. Each component is monotone in the direction of v 1 v 2 and the projections of any two components to a line parallel to v 1 v 2 do not overlap. The sequence of vertices of Qm ordered in the direction of v 1 v 2 , can have a pair of consecutive vertices contained in only where there is a horizontal edge in v or in the at most two discontinuities of v . This is true because Qm is a regular m-gon. We denote n hor the number of horizontal edges in v and n 0 hor this number plus the number of discontinuities in v (hence n 0 hor n hor non-horizontal edge of v produces a jump of at least two in the v 1 v 2 -ordering of the vertices of Pm , hence we have hor hor Analogously, and with the obvious similar meaning for n vert and n 0 vert , jvertices( vert vert can be completed to a triangulation of Qm , and exactly four non- interior edges of Qm are horizontal or vertical, we have n hor hor vert m+ 5. Hence, hor vert 3: Thus, there are at most 3m1 interior edges in T of the form pv i or pu i and at most 5minterior edges in total, as desired. Corollary 2.2. The polytope Pm described above has the following properties: It is a simplicial 3-polytope with m+ 4 vertices. Its maximal dissection has at least 4m 6 tetrahedra. Its maximal triangulation has at most 7m+ 1 tetrahedra. In particular, the gap between sizes of the maximal dissection and maximal triangulation is linear on the number of vertices. Three remarks are in order: First, the size of the maximal triangulation for Pm may depend on the coordinates or, more specically on which diagonals of Qm intersect the tetrahedron v 1 concerning the size of the minimal triangulation of Pm , we can easily describe a triangulation of Pm with only m+ 5 tetrahedra: let the vertices n, s, e and w be as dened above (see Figure 2.1) and let us call northeast, northwest, southeast and southwest the edges in the arcs ne, nw, se and sw in the boundary of Qm . Then, the triangulation consists of the ve tetrahedra (shown in the left part of Figure 2.5) together with the edges respectively, to the northeast, northwest, southeast and southwest edges of Qm . The right part of Figure 2.5 shows the result of slicing through the triangulation by the plane containing the polygon Qm . Finally, although the corollary above states a dierence between maximal dissections and maximal triangulations only for Pm with m > 14, experimentally we have observed there is a gap already for 8. Now we discuss two other interesting examples. The following proposition constitutes the proof of Theorem 1.1 (2). Proposition 2.3. 1. Consider the following eight points in R 3 The vertices of a square in the plane The vertices of a horizontal edge above the square, and The vertices of a horizontal edge below the square. These eight points are the vertices of a polytope P whose only integer points are precisely its eight vertices and with the following properties: (a) Its (unique) maximal dissection has 12 tetrahedra. All of them are uni- modular, i.e. they have volume 1=6. (b) Its (several) maximal triangulations have 11 tetrahedra. e s e s Fig. 2.5. For the triangulation of Pm with its ve central tetrahedra (left) and the intersection of the triangulation with the polygon Qm (right) are shown. The four interior vertices are the intersection points of the edges with the plane containing Qm . 2. For the 3-polytope with vertices the sizes of its (unique) minimal dissection and (several) minimal triangulations are 6 and 7 respectively, and the sizes of its (several) maximal triangulations and (unique) maximal dissection are 9 and 10 respectively. Proof. The polytopes constructed are quite similar to P 4 constructed earlier except that Q 4 is non-regular (in part 2) and the segments u 1 u 2 and v 1 v 2 are longer and are not orthogonal, thus ending with dierent polytopes. The polytopes are shown in Figure 2.6. Figure 2.7 describes a maximal dissection of each of them, in ve parallel slices. Observe that both polytopes have four vertices in the plane and another four in the plane y = 1. Hence, the rst and last slices in parts (a) and (b) of Figure 2.7 completely describe the polytope. e s e s Fig. 2.6. The two polytopes in Proposition 2.3. (1) The vertices in the planes quadrangles whose only integer points are the four vertices. This proves that the eight points are in convex position and that the polytope P contains no integer point other than its vertices. Let us now prove the assertions on maximal dissections and triangulations of (a) Consider the paths of length three are monotone respectively in the directions orthogonal to v 1 v 2 and u 1 u 2 . Using them, u2 s e (a) (b) y y y y y Fig. 2.7. Five 2-dimensional slices of the maximal dissections of the polytopes in Proposition 2.3. The rst and last slices are two facets of the polytopes containing all the vertices. we can construct two triangulations of size ve of the polytopes conv(nsewv 1 they do not ll P completely. There is space left for the tetrahedra swv 1 u 1 and env 2 u 2 . This gives a dissection of P with twelve tetrahedra. All the tetrahedra are unimodular, so no bigger dissection is possible. (b) A triangulation of size 11 can be obtained using the same idea as above, but with paths v and u of lengths three and two respectively, which can be taken from the same triangulation of the square nswe. To prove that no triangulation has bigger size, it su-ces to show that P does not have any unimodular triangulation. This means all tetrahedra have volume 1=6. We start by recalling a well-known fact (see Corollary 4.5 in [25]). A lattice tetrahedron has volume 1=6 if and only if each of its vertices v lies in a consecutive lattice plane parallel to the supporting plane of the opposite facet to v. Two parallel planes are said to be consecutive if their equations are ax+ by Suppose that T is a unimodular triangulation of P . We will rst prove that the triangle u 1 u 2 e is in T . The triangular facet u 1 lying in the hyperplane has to be joined to a vertex in the plane x 1. The two possibilities are e and v 1 . With the same argument, if the tetrahedron u 1 u 2 sv 1 is in which lies in the hyperplane 2x will be joined to a vertex in 2x 2, and the only one is e. This nishes the proof that u 1 is a triangle in T . Now, u 1 u 2 e is in the plane x must be joined to a vertex in i.e. to w. Hence u 1 u 2 ew is in T and, in particular, T uses the edge ew. P is symmetric under the rotation of order two on the axis g. Applying this symmetry to the previous arguments we conclude that T uses the edge ns too. But this is impossible since the edges ns and ew cross each other. (2) This polytope almost ts the description of P 4 , except for the fact that the edges intersect the boundary and not the interior of the planar quadrangle nsew. With the general techniques we have described, it is easy to construct halving dissections of this polytope with sizes from 6 to 10. Combinatorially, the polytope is a 4-antiprism. Hence, Proposition 4.3 shows that its minimal triangulation has 7 tetrahedra. The rest of the assertions in the statement were proved using the integer programming approach proposed in [8], which we describe in Remark 4.5. We have also veried them by enumerating all triangulations [19, 29]. It is interesting to observe that if we perturb the coordinates a little so that the planar quadrilateral becomes a tetrahedron with the right orientation and without changing the face lattice of the polytope, then the following becomes a triangulation with ten 3. Bounds for the size of a dissection. Let D be a dissection of a d-polytope . Say two (d 1)-simplices S 1 and S 2 of D intersect improperly in a (d 1)- hyperplane H if both lie in H , are not identical, and they intersect with non-empty relative interior. Consider the following auxiliary take as nodes the (d 1)- simplices of a dissection, and say that two (d 1)-simplices are adjacent if they intersect improperly in certain hyperplane. A mismatched region is the subset of R d that is the union of (d 1)-simplices over a connected component of size larger than one in such a graph. Later, in Proposition 3.4 we will show some of the complications that can occur in higher dimensions. Dene the simplicial complex of a dissection as all the simplices of the dissection together with their faces, where only faces that are identical (in R d ) are identied. This construction corresponds intuitively to an in ation of the dissection where for each mismatched region we move the two groups of (d 1)-simplices slightly apart leaving the relative boundary of the mismatched region joined. Clearly, the simplicial complex of a dissection may be not homeomorphic to a ball. The deformed d-simplices intersect properly, and the mismatched regions become holes. The numbers of vertices and d-simplices do not change. Lemma 3.1. All mismatched regions for a dissection of a convex 3-polytope P are convex polygons with all vertices among the vertices of P . Distinct mismatched regions have disjoint relative interiors. Proof. Let Q be a mismatched region and H the plane containing it. Since a mismatched region is a union of overlapping triangles, it is a polygon in H with a connected interior. If two triangles forming the mismatched region have interior points in common, they should be facets of tetrahedra in dierent sides of H . Otherwise, the two tetrahedra would have interior points in common, contradicting the denition of dissection. Triangles which are facets of tetrahedra in one side of H cover Q. Triangles coming from the other side of H also cover Q. take triangles coming from one side. As mentioned above, they have no interior points in common. Their vertices are among the vertices of the tetrahedra in the dissection, thus among the vertices of the polytope P . Hence, the vertices of the triangles are in convex position, thus the triangles are forming a triangulation of a convex polygon in H whose vertices are among the vertices of P . For the second claim, suppose there were distinct mismatched regions having an interior point in common. Then their intersection should be an interior segment for Let Q be one of the mismatched regions. It is triangulated in two dierent ways each coming from the tetrahedra in one side of the hyperplane. The triangles in either triangulation cannot intersect improperly with the interior segment. Thus the two triangulations of Q have an interior diagonal edge in common. This means the triangles in Q consists of more than one connected components of the auxiliary graph, contradicting the denition of mismatched region. Proposition 3.2. 1. The size of a mismatching dissection D of a convex 3-polytope with n vertices is at least n 2. 2. The size of a dissection of a 3-polytope with n vertices is bounded from above by n 2 . Proof. (1) Do an in ation of each mismatched region. This produces as many holes as mismatched regions, say m of them. Each hole is bounded by two triangulations of a polygon. This is guaranteed by the previous lemma. Denote by k i the number of vertices of the polygon associated to the i-th mismatched region. In each of the holes introduce an auxiliary interior point. The point can be used to triangulate the interior of the holes by lling in the holes with the coning of the vertex with the triangles it sees. We now have a triangulated ball. Denote by jDj the size of the original dissection. The triangulated ball has then total. The number of interior edges of this triangulation is the number of interior edges in the dissection, denoted by e i (D), plus the new additions, for each hole of length k i we added k i interior edges. In a triangulation T of a 3-ball with n boundary vertices and n 0 interior vertices, the number of tetrahedra jT j is related to the number of interior edges e i of T by the formula: jT The proof is a simple application of Euler's formula for triangulated 2-spheres and 3-balls and we omit the easy details. Thus, we have the following equation: 3: This can be rewritten as Taking into account that e i (D) (because diagonals in a polygon are interior edges of the dissection), we get an inequality 3: Finally note that in a mismatching dissection we have m 1 and k i 4. This gives the desired lower bound. (2) Now we look at the proof of the upper bound on dissections. Given a 3- dissection, we add tetrahedra of volume zero to complete to a triangulation with at simplices that has the same number of vertices. One can also think we are lling in the holes created by an in ation with (deformed) tetrahedra. The lemma states that mismatched regions were of the shape of convex polygons. The 2-simplices forming a mismatched region were divided into two groups (those becoming apart by an in ation). The two groups formed dierent triangulations of a convex polygon, and they had no interior edges in common. In this situation, we can make a sequence of ips (see [17]) between the two triangulations with the property that any edge once disappeared does not appear again (see Figure 3.1). We add one abstract, volume zero tetrahedron for each ip, and obtain an abstract triangulation of a 3-ball. The triangulation with at simplices we created is a triangulated 3-ball with n vertices. By adding a new point in a fourth dimension, and coning from the boundary 2-simplices to the point, we obtain a triangulated 3-sphere containing the original 3- ball in its boundary. From the upper bound theorem for spheres (for an introduction Fig. 3.1. Filling in holes with tetrahedra according to ips. to this topic see [30]) its size is bounded from above by the number of facets of a cyclic 4-polytope minus 2n 4, the number of 2-simplices in the boundary of D. The 4-dimensional cyclic polytope with vertices is well-known to have (n 2)=2 facets (see [11, page 63]), which completes the proof after a trivial algebraic calculation. Open Problem 3.3. What is the correct upper bound theorem for dissections of d-dimensional polytopes with d 4? In our proof of Proposition 3.2 we built a triangulated PL-ball from a three-dimensional dissection, using the ip connectivity of triangulations of a convex n-gon. Unfortunately the same cannot be applied in higher dimensions as the ip connectivity of triangulations of d-polytopes is known to be false for convex polytopes in general [22]. But even worse, the easy property we used from Lemma 3.1 that mismatched regions are convex polyhedra fails in dimension d 4. Proposition 3.4. The mismatched regions of a dissection of a convex 4-polytope can be non-convex polyhedra. Proof. The key idea is as follows: suppose we have a 3-dimensional convex polytope P and two triangulations T 1 and T 2 of it with the following properties: removing from P the tetrahedra that T 1 and T 2 have in common, the rest is a non-convex polyhedron P 0 such that the triangulations T 0 2 of it obtained from T 1 and do not have any interior 2-simplex in common (actually, something weaker would su-ce: that their common interior triangles, if any, do not divide the interior of the polytope). In these conditions, we can construct the dissection we want as a bipyramid over to one of the apices and T 2 to the other one. The bipyramid over the non-convex polyhedron P 0 will be a mismatched region of the dissection. For a concrete example, start with Schonhardt's polyhedron whose vertices are labeled in the lower face and 4; 5; 6 in the top face. This is a non-convex polyhedron made, for example, by twisting the three vertices on the top of a triangular prism. Add two antipodal points 7 and 8 close to the \top" triangular facets (those not breaking the quadrilaterals see Figure 3.2). For example, take as coordinates for the points Let P 0 be this non-convex polyhedron and let T 0 1468g. 1 cones vertex 7 to the rest of the boundary of P 0 , and T 0 vertex 8. Any common interior triangle of T 0 would use the edge 78. But the link of 78 in contains only the points 1, 2 and 3, and the link in T 0 contains only 4, 5 and 6. Let P be the convex hull of the eight points, and let T 1 and T 2 be obtained from 2 by adding the three tetrahedra 1245, 2356 and 1346. Fig. 3.2. The mismatched region of a four-dimensional dissection. 4. Optimal dissections for specic polytopes. The regular cube has been widely studied for its smallest dissections [12, 14]. This receives the name of simplexity of the cube. In contrast, because of the type of simplices inside a regular d-cube, a simple volume argument shows that the maximal size of a dissection is d!, the same as for triangulations. On the other hand, we know that the size of the maximal triangulation of a combinatorial cube can be larger than that: For example, the combinatorial 3-cube obtained as the prism over a trapezoid (vertices on a parabola for instance) has triangulations of size 7. Figure 4.1 shows a triangulation with 7 simplices for those coordinatizations where the edges AB and GH are not coplanar. The tetrahedron ABGH splits the polytope into two non-convex parts, each of which can be triangulated with three simplices. To see this, suppose that our polytope is a very small perturbation of a regular 3-cube. In the regular cube, ABGH becomes a diagonal plane which divides the cube into two triangular prisms ABCDGH and ABEFGH . In the non-regular cube, the diagonals AH and BG, respectively, become non-convex. Any pair of triangulations of the two prisms, each using the corresponding diagonal, together with tetrahedron ABGH give a triangulation of the perturbed cube with 7 tetrahedra. The boundary triangulation is shown in the at diagram. It is worth noticing that for the regular cube the boundary triangulation we showed does not extend to a triangulation of the interior. A F G A F G Fig. 4.1. A triangulation of a combinatorial 3-cube into seven tetrahedra. One can then ask, what is the general growth for the size of a maximal dissection of a combinatorial cube? To answer this question, at least partially, we use the above construction and we adapt an idea of M. Haiman, originally devised to produce small triangulations of regular cubes [12]. The idea is that from triangulations of a d 1 - cube and a d 2 -cube of sizes s 1 and s 2 respectively we can get triangulations of the rst subdividing it into s 1 s 2 copies of the product of two simplices of dimensions d 1 and d 2 and then triangulating each such piece. We recall that any triangulation of the Cartesian product of a d 1 -simplex and a d 2 -simplex has d1+d2 maximal simplices. Hence, in total we have a triangulation of the (d 1 maximal simplices. Recursively, if one starts with a triangulation of size s of the d-cube, one obtains triangulations for the rd-cube of size (rd)!( s In Haiman's context one wants s to be small, but here we want it to be big. More precisely, denote by f(d) the function max C: d-cube (max T of C jT j) and call Haiman's argument shows that if f(d 1 dierently, that g(d 1 . The value on the right hand side is the weighted geometric mean of g(d 1 ) and g(d 2 ). In particular, if both g(d 1 ) and g(d 2 ) are 1 and one of them is > 1 then g(d 1 We have constructed above a triangulation of size 7 for the Klee-Minty 3-cube, which proves g(3) 3 Haiman's idea we can now construct \large" triangulations of certain 4-cubes and 5-cubes, which prove respectively that equal to one and two respectively). Finally, since any d > 5 can be expressed as a sum of 3's and 4's, we have g(d) minfg(3); g(4)g 1:039 for any d > 5. Hence: Proposition 4.1. For the family of combinatorial d-cubes with d > 2 the function admits the lower bound f(d) c d d! where c 1:031. Exactly as in Haiman's paper, the constant c can be improved (asymptotically) if one starts with larger triangulations for the smaller dimensional cubes. Using computer calculations (see Remark 4.5), we obtained a maximal triangulation for the Klee-Minty 4-cube with 38 maximal simplices, which shows that g(d) 4 1:122 for every d divisible by 4 (see [1] for a complete study of this family of cubes). We omit listing the triangulation here but it is available from the authors by request. Open Problem 4.2. Is the sequence g(d) bounded? In other words, is there an upper bound of type c d d! for the function f(d)? Observe that the same question for minimal triangulations of the regular d-cube (whether there is a lower bound of type c d d! for some c > 0) is open as well. See [26] for the best lower bound known. We continue our discussion with the study of optimal triangulations for three-dimensional prisms and antiprisms. We will call an m-prism any 3-polytope with the combinatorial type of the product of a convex m-gon with a line segment. An m- antiprism will be any 3-polytope whose faces are two convex m-gons and 2m triangles, each m-gon being adjacent to half of the triangles. Vertices of the two m-gons are connected with a band of alternately up and down pointing triangles. Each such polyhedron has a regular coordinatization in which all the faces are regular polygons, and a realization space which is the set of all possible coordinatiza- tions that yield the same combinatorial information [20]. Our rst result is valid in the whole realization space. Proposition 4.3. For any three-dimensional m-prism, in any of its possible coordinatizations, the number of tetrahedra in a minimal triangulation is 2m 5+d m e. For any three-dimensional m-antiprism, in any of its possible coordinatizations, the number of tetrahedra in a minimal triangulation is 3m 5. Proof. In what follows we use the word cap to refer to the m-gon facets appearing in a prism or antiprism. We begin our discussion proving that any triangulation of the prism or antiprism has at least the size we state, and then we will construct triangulations with exactly that size. We rst prove that every triangulation of the m-prism requires at least 2m 5 We call a tetrahedron of the m-prism mixed if it has two vertices on the top cap and two vertices on the bottom cap of the prism, otherwise we say that the tetrahedron is top-supported when it has three vertices on the top (respectively bottom-supported). For example, Figure 4.2 shows a triangulation of the regular 12- prism, in three slices. Parts (a) and (c) represent, respectively, the bottom and top caps. Part (b) is the intersection of the prism with the parallel plane at equal distance to both caps. In this intermediate slice, bottom or top supported tetrahedra appear as triangles, while mixed tetrahedra appear as quadrilaterals. (b) (a) (c) Fig. 4.2. A minimal triangulation of the regular 12-prism. Because all triangulations of an m-gon have m 2 triangles there are always exactly 2m 4 tetrahedra that are bottom or top supported. In the rest, we show there are at least d m mixed tetrahedra. Each mixed tetrahedra marks an edge of the top, namely the edge it uses from the top cap. Of course, several mixed tetrahedra could mark the same top edge. Group together top-supported tetrahedra that have the same bottom vertex. This grouping breaks the triangulated top m-gon into polygonal regions. Note that every edge between two of these regions must be marked. For example, in part (c) of Figure 4.2 the top cap is divided into 6 regions by 5 marked edges (the thick edges in the Figure). Let r equal the number of regions under the equivalence relation we set. There are r 1 interior edges separating the r regions, and all of them are marked. Some boundary edges of the top cap may be marked too (none of them is marked in the example of Figure 4.2). We can estimate the marked edges in another way: There are m edges on the boundary of the top, which appear partitioned among some of the regions (it could be the case some region does not contain any boundary edge of the m-gon). We claim that no more than two boundary edges per region will be unmarked (). This follows because a boundary edge is not marked only when the top supported tetrahedron that contains it has the point in the bottom cap that is directly under one of the vertices of the edge. In a region, at most two boundary edges can satisfy this. Hence we get at least m 2r marked edges on the boundary of the top and at least (r marked edges in total. Thus the number of mixed tetrahedra is at least the maximum of r 1 and m r 1. In conclusion, we get that, indeed, the number of mixed tetrahedra is bounded below by d m e 1. Note that we only use the combinatorics and convexity of the prism in our arguments. We will show that minimal triangulations achieve this lower bound, but then, observe that if m is even, in a minimal triangulation we must have no boundary edge can be marked, as is the case in Figure 4.2. If m is odd, then we must have r 2 f(m 1)=2; (m+1)=2g and at most one boundary edge can be marked. The proof that any triangulation of an m-antiprism includes at least 3m 5 tetrahedra is similar. There are 2m 4 top-supported and bottom-supported tetrahedra in any triangulation and there are r 1 marked edges between the regions in the top. The only dierence is that, instead of claim (), one has at most one unmarked boundary edge per region. Thus there are at least m r marked edges in the boundary of the top, and in total at least (r marked edges in the top. Hence there exist at least (2m 4)+(m in any triangulation. For an m-antiprism we can easily create a triangulation of size 3m 5 by choosing any triangulation of the bottom m-gon and then coning a chosen vertex v of the top m-gon to the m 2 triangles in that triangulation and to the 2m 3 triangular facets of the m-antiprism which do not contain v. This construction is exhibited in Figure 4.3. Parts (a) and (c) show the bottom and top caps triangulated (each with its 5 marked edges) and part (b) an intermediate slice with the 5 mixed tetrahedra appearing as quadrilaterals. (c) (b) (a) Fig. 4.3. A minimal triangulation of the regular 6-antiprism. For an m-prism, let u i and v i , denote the top and bottom vertices respectively, so that the vertices of each cap are labeled consecutively and u always an edge of the prism. If m is even we can chop o the vertices u i for odd i and v j for even j, so that the prism is decomposed into m tetrahedra and an ( m)-antiprism. The antiprism can be triangulated into 3m5 tetrahedra, which gives a triangulation of the prism into 5m5 tetrahedra, as desired. Actually, this is how the triangulation of Figure 4.2 can be obtained from that of Figure 4.3. If m is odd we do the same, except that we chop o only the vertices and vertex is chopped in the edge um v m ). This produces m 1 tetrahedra and an ( m+1 )-antiprism. We triangulate the antiprism into 3m+3 tetrahedra and this gives a triangulation of the m-prism into 5m+15 tetrahedra. We have seen that the coordinates are not important when calculating minimal triangulations of the three-dimensional prisms and antiprisms. On the other hand, the dierence in size of the maximal triangulation can be quite dramatic. Below we prove that in certain coordinatizations it is roughly m 2and show experimental data indicating that for the regular prism it is close to m 2Proposition 4.4. Let Am be a prism of order m, with all its side edges parallel. 1. The size of a maximal triangulation of Am is bounded as 2. The upper bound is achieved if the two caps (m-gon facets) are parallel and there is a direction in which the whole prism projects onto one of its side quadrangular facets. (For a concrete example, let one of the m-gon facets have vertices on a parabola and let Am be the product of it with a segment). Proof. Let the vertices of the prism be labeled um and v so that the u i 's and the v j 's form the two caps, vertices in each cap are labeled consecutively and u i v i is always a side edge. For the upper bound in part (1), we have to prove that a triangulation of Am has at most m 2 +m 62m diagonals. The possible diagonals are the edges is not in f1; 0; 1g modulo m. This gives exactly twice the number we want. But for any i and j the diagonals u one of them can appear in each triangulation. We now prove that the upper bound is achieved if Am is in the conditions of part (2). In fact, the condition on Am that we will need is that for any 1 i < j k < l m, the point v j sees the triangle v i u k u l from the same side as v k and v l (i.e. \from above" if we call top cap the one containing the v i 's). With this we can construct a triangulation with First cone the vertex v 1 to any triangulation of the bottom cap (this gives m 2 tetrahedra). The m 2 upper boundary facets of this cone are visible from v 2 , and we cone them to it (again m 2 tetrahedra). The new m 2 upper facets are visible from v 3 and we cone them to it (m 2 tetrahedra more). Now, one of the upper facets of the triangulation is of the upper cap, but the other m 3 are visible from v 4 , so we cone them and introduce m 4 tetrahedra. Continuing the process, we will introduce coning the vertices which gives a total of The triangulation we have constructed is the placing triangulation [17] associated to any ordering of the vertices nishing with dierent description of the same triangulation is that it cones the bottom cap to v 1 , the top cap to um , and its mixed tetrahedra are all the possible 1. This gives We nally prove the lower bound stated in part (1). Without loss of general- ity, we can assume that our prism has its two caps parallel (if not, do a projective transformation keeping the side edges parallel). Then, Am can be divided into two prisms in the conditions of part (2) of sizes k and l with 2: take any two side edges of Am which posses parallel supporting planes and cut Am along the plane containing both edges. By part (2), we can triangulate the two subprisms with respectively, taking care that the two triangulations use the same diagonal in the dividing plane. This gives a triangulation of Am with This expression achieves its minimum when k and l are as similar as possible, i.e. Plugging these values in the expression gives a triangulation of size l Based on an integer programming approach we can compute maximal triangulations of specic polytopes (see remark at the end of the article). Our computations with regular prisms up to show that the size of their maximal triangulations achieve the lower bound stated in part (1) of Proposition 4.4 (see Table 4.1). In other words, that the procedure of dividing them into two prisms of sizes b m and d m in the conditions of part (2) of Proposition 4.4 and triangulating the subprisms independently yields maximal triangulations. We have also computed maximal sizes of triangulations for the regular m-antiprisms up to which turn out to follow the formula . A construction of a triangulation of this size for every m can be made as follows: Let the vertices of the regular m-antiprism be labeled um and v so they are forming the vertices of the two caps consecutively in this order and v i u i and u i v i+1 are side edges. We let v . The triangulation is made by placing the vertices in any ordering nishing with . The tetrahedra used are the bottom-supported tetrahedra with apex v 1 , top-supported tetrahedra with apex u d me and the mixed tetrahedra We conjecture that these formulas for regular base prisms and antiprisms actually give the sizes of their maximal triangulations for every m, but we do not have a proof. Prism (regular base) 43 50 Antiprism (regular base) 4 8 12 17 22 28 34 41 48 56 Table Sizes of maximal triangulations of prisms and antiprisms. Remark 4.5. How can one nd minimal and maximal triangulations in specic instances? The approach we followed for computing Tables 1.1 and 4.1 and some of the results in Proposition 2.3 is the one proposed in [8], based on the solution of an integer programming problem. We think of the triangulations of a polytope as the vertices of the following high-dimensional polytope: Let A be a d-dimensional polytope with n vertices. Let N be the number of d-simplices in A. We dene PA as the convex hull in R N of the set of incidence vectors of all triangulations of A. For a triangulation T the incidence vector v T has coordinates (v T and (v T . The polytope PA is the universal polytope dened in general by Billera, Filliman and Sturmfels [3] although it appeared in the case of polygons in [7]. In [8], it was shown that the vertices of PA are precisely the integral points inside a polyhedron that has a simple description in terms of the oriented matroid of A (see [8] for information on oriented matroids). The concrete integer programming problems were solved using C-plex Linear Solver TM . The program to generate the linear constraints is a small C ++ program written by Samuel Peterson and the rst author. Source code, brief instructions, and data les are available via ftp at http://www.math.ucdavis.edu/~deloera. An alternative implementation by A. Tajima is also available [27, 28]. He used his program to corroborate some of these results. It should be mentioned that a simple variation of the ideas in [8] provides enough equations for an integer program whose feasible vertices are precisely the 0=1-vectors of dissections. The incidence vectors of dissections of conv(A), for a point set A, are just the 0=1 solutions to the system of equations hx; v T are the incidence vectors for every regular triangulation T of the Gale transform A (regular triangulations in the Gale transform are the same as chambers in A). Generating all these equations is as hard as enumerating all the chambers of A. Nevertheless, it is enough to use those equations coming from placing triangulations (see [23, Section 3.2]), which gives a total of about n d+1 equations if A has n points and dimension d. Acknowledgments . We are grateful to Alexander Below and Jurgen Richter- Gebert for their help and ideas in the proofs of Proposition 3.2 and 3.4. Alexander Below made Figure 3.2 using the package Cinderella. The authors thank Akira Tajima and Jorg Rambau for corroborating many of the computational results. We thank Samuel Peterson for his help with our calculations. Finally, we thank Hiroshi Imai, Bernd Sturmfels, and Akira Tajima for their support of this project. --R Deformed products and maximal shadows of polytopes An algorithmic theory of lattice points in polyhedra Constructions and complexity of secondary poly- topes Minimal simplicial dissections and triangulations of convex 3-polytopes New York Triangulations (tilings) and certain block triangular matrices Tetrahedrizing point sets in three di- mensions binary covers of rational polyhedral cones A simple and relatively e-cient triangulation of the n-cube The Cayley trick Simplexity of the cube Triangulations of integral polytopes and Ehrhart polynomials Subdivisions and triangulations of polytopes Triangulations for the cube TOPCOM: a program for computing all triangulations of a point set On triangulations of the convex hull of n points A point con Triangulations of oriented matroids Optimality and integer programming formulations of triangulations in general di- mension Optimizing geometric triangulations by using integer programming Enumerating triangulations for products of two simplices and for arbitrary con --TR --CTR Mike Develin, Note: maximal triangulations of a regular prism, Journal of Combinatorial Theory Series A, v.106 n.1, p.159-164, April 2004 Jesus A. De Loera , Elisha Peterson , Francis Edward Su, A polytopal generalization of Sperner's lemma, Journal of Combinatorial Theory Series A, v.100 n.1, p.1-26, October 2002

archimedean solid;antiprism;lattice polytope;mismatched region;combinatorial d-cube;prism;dissection;triangulation

587931

Scheduling Unrelated Machines by Randomized Rounding.

We present a new class of randomized approximation algorithms for unrelated parallel machine scheduling problems with the average weighted completion time objective. The key idea is to assign jobs randomly to machines with probabilities derived from an optimal solution to a linear programming (LP) relaxation in time-indexed variables. Our main results are a $(2+\varepsilon)$-approximation algorithm for the model with individual job release dates and a $(3/2+\varepsilon)$-approximation algorithm if all jobs are released simultaneously. We obtain corresponding bounds on the quality of the LP relaxation. It is an interesting implication for identical parallel machine scheduling that jobs are randomly assigned to machines, in which each machine is equally likely. In addition, in this case the algorithm has running time O(n log n) and performance guarantee 2. Moreover, the approximation result for identical parallel machine scheduling applies to the on-line setting in which jobs arrive over time as well, with no difference in performance guarantee.

Introduction It is by now well-known that randomization can help in the design of algorithms, cf., e. g., [27, 26]. One way of guiding randomness is the use of linear programs (LPs). In this paper, we give LP-based approximation algorithms for problems which are particularly well-known for the difficulties to obtain good lower bounds: machine (or processor) scheduling problems. Because of the random choices involved, our algorithms are rather randomized approximation algorithms. A randomized r-approximation algorithm is a polynomial-time algorithm that produces a feasible solution whose expected value is within a factor of r of the optimum; r is also called the (expected) performance guarantee of the algorithm. Actually, most often we compare the output of an algorithm with a lower bound given by an optimum solution to a certain LP relaxation. Hence, at the same time we obtain an analysis of the quality of the respective LP. All our off-line algorithms can be derandomized with no difference in performance guarantee, but at the cost of increased (but still polynomial) running times. We consider the following model. We are given a set J of n jobs (or tasks) and m unrelated parallel machines. Each job j has a positive integral processing requirement which depends on the machine i job j will be processed on. Each job j must be processed for the respective amount of time on one of the m machines, and may be assigned to any of them. Every machine can process at most one job at a time. In preemptive schedules, a job may repeatedly be interrupted and continued later on another (or the same) machine. In nonpreemptive schedules, a job must be processed in an uninterrupted fashion. Each job j has an integral release date r j ? 0 before which Parts of this paper appeared in a preliminary form in [36, 35] # M.I.T., Sloan School of Management and Operations Research Center, E53-361, Fachbereich Mathematik, MA 6-1, Technische Universitat Berlin, Strae des 17. Juni 136, D-10623 Berlin, Germany, skutella@math.tu- berlin.de it cannot be processed. We denote the completion time of job j in a schedule S by C S or C j , if no confusion is possible. We seek to minimize the total weighted completion time: a weight w associated with each job j and the goal is to find a schedule S that minimizes j2J w j C j . In scheduling, it is quite convenient to refer to the respective problems using the standard classification scheme of Graham, Lawler, Lenstra, and Rinnooy Kan [17]. The nonpreemptive problem R j r just described, is strongly NP-hard [23]. Scheduling to minimize the total weighted completion time (or, equivalently, the average weighted completion time) has recently achieved a great deal of attention, partly because of its importance as a fundamental problem in scheduling, and also because of new applications, for instance, in compiler optimization [5] or in parallel computing [3]. There has been significant progress in the design of approximation algorithms for this kind of problems which led to the development of the first constant worst-case bounds in a number of settings. This progress essentially follows on the one hand from the use of preemptive schedules to construct nonpreemptive ones [31, 4, 7, 14, 16]. On the other hand, one solves an LP relaxation and then a schedule is constructed by list scheduling in a natural order dictated by the LP solution [31, 19, 34, 18, 25, 14, 8, 28, 37]. In this paper, we utilize a different idea: random assignments of jobs to machines. To be more precise, we exploit an LP relaxation in time-indexed variables for the problem R j r and we then show that a certain variant of randomized rounding leads to a e)-approximation algorithm, for any e ? 0. In the absence of nontrivial release dates, the performance guarantee can be improved to 3=2 e. At the same moment, the corresponding LP is a respectively, i. e., the true optimum is always within this factor of the optimal value of the LP relaxation. Our algorithm improves upon a 16=3-approximation algorithm of Hall, Shmoys, and Wein [19] that is also based on time-indexed variables which have a different meaning, how- ever. In contrast to their approach, our algorithm does not rely on Shmoys and Tardos' rounding technique for the generalized assignment problem [39]. We rather exploit the LP by interpreting LP values as probabilities with which jobs are assigned to machines. For an introduction to and the application of randomized rounding to other combinatorial optimization problems, the reader is referred to [33, 26]. Using a different approach, the second author has subsequently developed slightly improved approximation results for the problems under consideration. For the problem R he gives a 3=2-approximation algorithm [41] that is based on a convex quadratic programming relaxation in assignment variables, which is inspired by the time-indexed LP relaxation presented in this paper. Only recently, this approach has been generalized to the problem with release dates for which it yields performance guarantee 2 [42]. For the special case of identical parallel machines, i. e., for each job j and all machines i we have Chakrabarti et al. [4] obtained a (2:89 + e)-approximation by refining an online greedy framework of Hall et al. [19]. The former best known LP-based algorithm, however, relies on an LP relaxation solely in completion time variables which is weaker than the one we propose. It has performance guarantee (4 \Gamma 1=m) (see [18] for the details). For the LP we use here, an optimum solution can greedily be obtained by a certain preemptive schedule on a virtual single machine which is m times as fast as any of the original machines. The idea of using a preemptive relaxation on such a virtual machine was employed before by Chekuri, Motwani, Natarajan, and Stein [7]. They show that any preemptive schedule on such a machine can be converted into a nonpreemptive schedule on the identical parallel machines such that, for each job j, its completion time in the nonpreemptive schedule is at most (3 \Gamma 1=m) times its preemptive completion time. For the problem to minimize the average completion time, they refine this to a 2:85-approximation algorithm. For the algorithm we propose delivers in time O(nlogn) a solution that is expected to be within a factor of 2 of the optimum. Since the LP relaxation we use is even a relaxation of the corresponding preemptive problem, our algorithm is also a 2-approximation for which improves upon a 3-approximation algorithm due to Hall, Schulz, Shmoys, and Wein [18]. In particular, our result implies that the value of an optimal nonpreemptive schedule is at most a factor 2 the value of an optimal preemptive schedule. For the problem without release dates, our algorithm achieves performance guarantee 3=2. Since an optimum solution to the LP relaxation can be obtained greedily, our algorithm also works in the corresponding online setting where jobs continually arrive to be processed and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards; the algorithm achieves competitive ratio 2 for both the nonpreemptive and the preemptive variant of this setting. Recently, Skutella and Woeginger [43] developed a polynomial-time approximation scheme for the problem which improves upon the previously best known (1 2)=2-approximation algorithm due to Kawaguchi and Kyan [22]. Subsequently, Chekuri, Karger, Khanna, Skutella, and Stein [6] gave polynomial-time approximation schemes for the problem its preemptive variant also for the corresponding problems on a constant number of unrelated machines, Rm j r On the other hand, it has been shown by Hoogeveen, Schuurman, and Woeginger [20] that the problems R j r are MAXSNP-hard and therefore do not have a polynomial time approximation scheme, unless P=NP. The rest of the paper is organized as follows. In Section 2, we start with the discussion of our main result: the algorithm with performance guarantee 2 in the general context of unrelated parallel machine scheduling. In the next section, we give combinatorial approximation algorithms for identical parallel machine scheduling. We also show how to use these algorithms in an online setting. Then, in Section 4, we discuss the derandomization of the previously given randomized algorithms. Finally, in Section 5 we give the technical details of turning the pseudo-polynomial algorithm of Section 2 into a polynomial-time algorithm with performance guarantee We conclude by pointing out some open problems in Section 6. Unrelated Parallel Machine Scheduling with Release Dates In this section, we consider the problem R j r As in [30, 19, 18, 42], we will actually discuss a slightly more general problem in which the release date of every job j may also depend on the machine. The release date of job j on machine i is thus denoted by r i j . Machine-dependent release dates are relevant to model network scheduling in which parallel machines are connected by a network, each job is located at a given machine at time 0, and cannot be started on another machine until sufficient time elapses to allow the job to be transmitted to its new machine. This model has been introduced in [9, 1]. The problem R j r is well-known to be strongly NP-hard; in fact, already P2 strongly NP-hard, see [2, 23]. The first nontrivial approximation algorithm for this problem was given by Phillips, Stein, and Wein [30]. It has performance guarantee O(log 2 n). Subsequently, Hall et al. [19] gave a 16=3-approximation algorithm which relies on a time-indexed LP relaxation whose optimum value serves as a surrogate for the true optimum in their estimations. We use a somewhat similar LP relaxation, but whereas Hall et al. invoke the deterministic rounding technique of Shmoys and Tardos [39] to construct a feasible schedule we randomly round LP solutions to feasible schedules. be the time horizon, and introduce for every job j 2 J, every machine m, and every point which represents the time job j is processed on machine i within the time interval (t; t Equivalently, one can say that a y i jt =p i j -fraction of job j is being processed on machine i within the time interval (t; t 1]. The LP, which is an extension of a single machine LP relaxation of Dyer and Wolsey [10], is as follows: minimize subject to for all Equations (1) ensure that the whole processing requirement of every job is satisfied. The machine capacity constraints (2) simply express that each machine can process at most one job at a time. Now, for (3), consider an arbitrary feasible schedule S where job j is being continuously processed between time C S on machine h. Then, the expression for C LP in (3) corresponds to the real completion time C S j of j if we assign the values to the LP variables y i jt as defined above, i. e., y wise. The right-hand side of (4) equals the processing time p h j of job j in the schedule S, and is therefore a lower bound on its completion time C S . Finally, constraints (5) ensure that no job is processed before its release date. Hence, (LP R ) is a relaxation of the scheduling problem R j r In fact, note that even the corresponding mixed-integer program, where the y-variables are forced to be binary, is only a relaxation. The following algorithm takes an optimum LP solution, and then constructs a feasible schedule by using a kind of randomized rounding. Algorithm LP ROUNDING Compute an optimum solution y to (LP R ). Assign each job j to a machine-time pair (i; t) independently at random with probability y i jt draw t j from the chosen time interval (t; t +1] independently at random with uniform distribution. Schedule on each machine i the jobs that were assigned to it nonpreemptively as early as possible in order of nondecreasing t j . In the last step ties can be broken arbitrarily; they occur with probability zero. For the analysis of the algorithm it will be sufficient to assume that the random decisions for different jobs are pairwise independent. Remark 2.1. The reader might wonder whether the seemingly artificial random choice of the t j 's in Algorithm LP ROUNDING is really necessary. Indeed, it is not, which also implies that we could work with a discrete probability space: The following results are still true if we simply set t is assigned to a machine-time pair (i; t); ties are broken randomly or even arbitrarily. We mainly chose this presentation for the sake of giving a different interpretation in terms of so-called a-points in Section 3. The following lemma illuminates the intuition in Algorithm LP ROUNDING by relating the implications of the second step to the solution y of (LP R ). For the analysis of the algorithm, however, we will only make use of the second part of the Lemma. Its first part is a generalization of a result due to Goemans [14] for the single machine case. Lemma 2.2. Let y be the optimum solution to (LP R ) in the first step of Algorithm LP ROUNDING. Then, for each J the following holds: a) The expected value of t j is equal to m y jt b) The expected processing time of job j in the schedule constructed by Algorithm LP ROUNDING is equal to Proof. First, we fix a machine-time pair (i; t) job j has been assigned to. Then, the expected processing time of j under this condition is p i j . Moreover, the conditional expectation of t j is equal to t . By adding these conditional expectations over all machines and time intervals, weighted by the corresponding probabilities y jt , we get the claimed results. Note that the lemma remains true if we start with an arbitrary, not necessarily optimal solution y to (LP R ) in the first step of Algorithm LP ROUNDING. This is also true for the following results. The optimality of the LP solution will only be needed to get a lower bound on the value of an optimal schedule. Lemma 2.3. The expected completion time of each job j in the schedule constructed by Algorithm LP ROUNDING can be bounded by this bound is even true if t j is set to t in the second step of the algorithm and ties are broken arbitrarily, see Remark 2.1. In the absence of nontrivial release dates the following stronger bound holds: this bound also holds if t j is set to t in the second step of the algorithm and ties are broken uniformly at random. Proof. We consider an arbitrary, but fixed job j 2 J. To analyze the expected completion time of job j, we first also consider a fixed assignment of j to a machine-time pair (i; t). Then, the expected starting time of job j under these conditions precisely is the conditional expected idle time plus the conditional expected amount of processing of jobs that come before j on machine i. Observe that there is no idle time on machine i between the maximum release date of jobs on machine i which start no later than j and the starting time of job j. It follows from the ordering of jobs and constraints (5) that this maximum release date and therefore the idle time of machine i before the starting time of j is bounded from above by t. In the absence of nontrivial release dates there is no need for idle time at all. On the other hand, we get the following bound on the conditional expected processing time on machine i before the start of j: y ik' y ik' The last inequality follows from the machine capacity constraints (2). However, if t j is set to t in the second step of the algorithm and ties are broken arbitrarily, we have to replace E[t by 1 on the right-hand side and get a weaker bound of t +1. Putting the observations together we get an upper bound of 2 2 ) for the conditional expectation of the starting time of j. In the absence of nontrivial release dates it can be bounded by t . Unconditioning the expectation by the formula of total expectation together with Lemma 2.2 b) yields the result. Theorem 2.4. For instances of R j r the expected value of the schedule constructed by Algorithm LP ROUNDING is bounded by twice the value of an optimal solution. Proof. By Lemma 2.3 and constraints (3) the expected completion time of each job is bounded by twice its LP completion time C LP . Since the optimal LP value is a lower bound on the value of an optimal schedule and the weights are nonnegative, the result follows by linearity of expectations. Note that Theorem 2.4 still holds if we use the weaker LP relaxation where constraints (4) are missing. How- ever, this is not true for the following result. Theorem 2.5. For instances of R the expected value of the schedule constructed by Algorithm LP ROUNDING is bounded by 3=2 times the value of an optimal solution. Proof. The result follows from Lemma 2.3 and the LP constraints (3) and (4). Independently, the result in Theorem 2.5 has also been obtained by Fabian A. Chudak (communicated to us by David B. Shmoys, March 1997) after reading a preliminary version of the paper on hand which only contained the bound of 2 for R j r Theorem 2.4. In the absence of nontrivial release dates, Algorithm LP ROUNDING can be improved and simplified: Corollary 2.6. For instances of R the approximation result of Theorem 2.5 also holds for the following improved and simplified variant of Algorithm LP ROUNDING: In the second step we assign each job j independently at random with probability T to machine i. In the last step we apply Smith's ratio rule [44] on each machine, i. e., we schedule the jobs that have been assigned to machine i in order of nonincreasing ratios w Proof. Notice that the random assignment of jobs to machines remains unchanged in the described variant of Algorithm LP ROUNDING. However, for a fixed assignment of jobs to machines, sequencing the jobs according to Smith's ratio rule on each machine is optimal. In particular, it improves upon the random sequence used in the final step of Algorithm LP ROUNDING. In the analysis of Algorithm LP ROUNDING we have always compared the value of the computed solution to the optimal LP value which is itself a lower bound on the value of an optimal solution. Therefore we can state the following result on the quality of the LP relaxation: Corollary 2.7. The linear program (LP R ) is a 2-relaxation for R j r (even without constraints (4)) and a2 -relaxation for R We show in the following section that (LP R ) without constraints (4) is not better than a 2-relaxation, even for instances of P . On the other hand, the relaxation can be strengthened by adding the constraints These constraints ensure that in each time period no job can use the capacity of more than one machine. Unfor- tunately, we do not know how to use these constraints to get provably stronger results on the quality of the LP relaxation and better performance guarantees for Algorithm LP ROUNDING. Notice that the results in Theorem 2.4 and Theorem 2.5 do not directly lead to approximation algorithms for the considered scheduling problems. The reason is that we cannot solve (LP R ) in polynomial time due to the exponentially many variables. However, we can overcome this drawback by introducing new variables which are not associated with exponentially many time intervals of length 1, but rather with a polynomial number of intervals of geometrically increasing size. In order to get polynomial-time approximation algorithms in this way, we have to pay for with a slightly worse performance guarantee. For any constant e ? 0 we get approximation algorithms with performance guarantee 2+ e and 3=2+ e for the scheduling problems under consideration. We elaborate on this in Section 5. It is shown in [40] that the ideas and techniques presented in this section and Section 5 can be modified to construct approximation algorithms for the corresponding preemptive scheduling problems. Notice that, although the LP relaxation (LP R ) allows preemptions of jobs, it is not a relaxation of R j r it is shown in [40, Example 2.10.8.] that the right-hand side of (3) can in fact overestimate the actual completion time of a job in the preemptive schedule corresponding to a solution of (LP R ). However, one can construct an LP relaxation for the preemptive scheduling problem by replacing (3) with a slightly weaker constraint. This leads to a (3 approximation algorithm for R j r e)-approximation algorithm for R j pmtn j w j C j . These results can again be slightly improved by using convex quadratic programming relaxations, see [42]. Scheduling with Release Dates We now consider the special case of m identical parallel machines. The processing requirement and the release date of job j no longer depend on the machine job j is processed on and are thus denoted by p j and r j , respectively. As mentioned above, already the problem P2 In this setting, Algorithm LP ROUNDING can be turned into a purely combinatorial algorithm. Taking up an idea that has been used earlier, e. g., by Chekuri et al. [7], we reduce the identical parallel machine instance to a single machine instance. However, the single machine is assumed to be m times as fast as each of the original machines, i. e., the virtual processing time of job j on this virtual single machine is p 0 without loss of generality that p j is a multiple of m). Its weight and its release date remain the same. The crucial idea for our algorithm is to assign jobs uniformly at random to machines. Then, on each machine, we sequence the assigned jobs in order of random a-points with respect to a preemptive schedule on the fast single machine. For 1, the a-point C S j (a) of job j with respect to a given preemptive schedule S on the fast single machine is the first point in time when an a-fraction of job j has been completed, i. e., when j has been processed for a time units. In particular, C S j and for j (0) to be the starting time of job j. Slightly varying notions of a-points were considered in [31, 19], but their full potential was only revealed when Chekuri et al. [7] as well as Goemans [14] chose the parameter a at random. The following algorithm may be seen as an extension of their single machine techniques to identical parallel machines. Algorithm: RANDOM ASSIGNMENT Construct a preemptive schedule S on the virtual single machine by scheduling at any point in time among the available jobs the one with largest w j =p 0 ratio. For each job j 2 J, draw a j independently at random and uniformly distributed from [0; 1] and assign j uniformly at random to one of the m machines. Schedule on each machine i the jobs that were assigned to it nonpreemptively as early as possible in nondecreasing order of C S Notice that in the first step whenever a job is released, the job being processed (if any) will be preempted if the released job has a larger ratio. An illustration of Algorithm RANDOM ASSIGNMENT can be found in the Appendix . The running time of this algorithm is dominated by the effort to compute the preemptive schedule in the first step. Goemans observed that this can be done in O(nlogn) time using a priority queue [14]. In the following we will show that Algorithm RANDOM ASSIGNMENT can be interpreted as a reformulation of Algorithm LP ROUNDING for the special case of identical parallel machines. One crucial insight for the analysis is that the above preemptive schedule on the virtual single machine corresponds to an optimum solution to an LP relaxation which is equivalent to (LP R ). We introduce a variable y jt for every job j and every time period (t; t +1] that is set to 1=m if job j is being processed on one of the m machines in this period and to 0 otherwise. Notice that in contrast to the unrelated parallel machine case we do not need machine dependent variables since there is no necessity to distinguish between the identical parallel machines. We can express the new variables y jt in the old variables y i jt by setting This leads to the following simplified LP (ignoring constraints (4) of (LP R )): minimize subject to y jt for all For the special case was introduced by Dyer and Wolsey [10]. They also indicated that it follows from the work of Posner [32] that the program can be solved in O(nlogn) time. Goemans [13] showed (also for the case m= 1) that the preemptive schedule that is constructed in the first step of Algorithm RANDOM ASSIGNMENT defines an optimum solution to (LP P ). This result as well as its proof can be easily generalized to an arbitrary number of identical parallel machines: Lemma 3.1. For instances of the problems the relaxation (LP P ) can be solved in O(nlogn) time and the preemptive schedule on the fast single machine in the first step of Algorithm RANDOM ASSIGNMENT corresponds to an optimum solution. Theorem 3.2. Algorithm RANDOM ASSIGNMENT is a randomized 2-approximation algorithm for Proof. We show that Algorithm RANDOM ASSIGNMENT can be interpreted as a special case of Algorithm LP ROUNDING. The result then follows from its polynomial running time and Theorem 2.4. Lemma 3.1 implies that in the first step of Algorithm RANDOM ASSIGNMENT we simply compute an optimum solution to the LP relaxation (LP P ) which is equivalent to (LP R ) without constraints (4). In particular, the corresponding solution to (LP R ) is symmetric with regard to the m machines. Therefore, in Algorithm LP ROUNDING each job is assigned uniformly at random to one of the machines. The symmetry also yields that for each job j the choice of t j is not correlated with the choice of i in Algorithm LP ROUNDING. It can easily be seen that the probability distribution of the random variable t j in Algorithm LP ROUNDING exactly equals the probability distribution of C S Algorithm RANDOM ASSIGNMENT. For this, observe that the probability that C S equals the fraction y jt =p 0 j of job j that is being processed in this time interval. Moreover, since a j is uniformly distributed in (0;1] each point in (t; t + 1] is equally likely to be obtained for C S Therefore, the random choice of C S Algorithm RANDOM ASSIGNMENT is an alternative way of choosing t j as it is done in Algorithm LP ROUNDING. Consequently, the two algorithms coincide for the identical parallel machine case. In particular, the expected completion time of each job is bounded from above by twice its LP completion time and the result follows by linearity of expectations. At this point, let us briefly compare the approximation results of this section for the single machine case related results. If we only work with one a for all jobs instead of individual and independent a j 's and if we draw a uniformly from [0; 1], then RANDOM ASSIGNMENT precisely becomes Goemans' randomized 2-approximation algorithm RANDOM a for 1jr j j w j C j [14]. Goemans, Queyranne, Schulz, Skutella, and Wang have improved this result to performance guarantee 1:6853 by using job-dependent a j 's as in Algorithm RANDOM ASSIGNMENT together with a nonuniform choice of the a j 's [15]. The same idea can also be applied in the parallel machine setting to get a performance guarantee better than 2 for Algorithm RANDOM ASSIGNMENT. This improvement, however, depends on m. We refer the reader to the single machine case for details. A comprehensive treatment and a detailed overview of the concept of a-points for machine scheduling problems can be found in [40, Chapter 2]. We have already argued in the last section that (LP R ) and thus (LP P ) is a 2-relaxation of the scheduling problem under consideration: Corollary 3.3. The relaxation (LP P ) is a 2-relaxation of the scheduling problem and this bound is tight, even for Proof. The positive result follows from Corollary 2.7. For the tightness, consider an instance with m machines and one job of length m and unit weight. The optimum LP completion time is (m+ 1)=2, whereas the optimum completion time is m. When m goes to infinity, the ratio of the two values converges to 2. Our approximation result for identical parallel machine scheduling can be directly generalized to the corresponding preemptive scheduling problem. In preemptive schedules a job may repeatedly be interrupted and continued later on another (or the same) machine. It follows from the work of McNaughton [24] that already is NP-hard since there always exists an optimal nonpreemptive schedule and the corresponding nonpreemptive problem is NP-hard. We make use of the following observation: Lemma 3.4. The linear program (LP P ) is also a relaxation of the preemptive problem Proof. Since all release dates and processing times are integral, there exists an optimal preemptive schedule where preemptions only occur at integral points in time. Take such an optimal schedule S and construct the corresponding feasible solution to (LP P ) by setting y being processed on one of the m machines within the time interval (t; t +1] and y It is an easy observation that C LP j and equality holds if and only if job j is continuously processed in the time interval (C S Thus, the value of the constructed solution to (LP P ) is a lower bound on the value of an optimal schedule. This observation leads to the following results which generalize Theorem 3.2 and Corollary 3.3. Corollary 3.5. The value of the (nonpreemptive) schedule constructed by Algorithm RANDOM ASSIGNMENT is not worse than twice the value of an optimum preemptive schedule. Moreover, the relaxation (LP P ) is a 2- relaxation of the scheduling problem and this bound is tight. The 2-approximation algorithm in Corollary 3.5 improves upon a performance guarantee of 3 due to Hall, Schulz, Shmoys, and Wein [18]. Another consequence of our considerations is the following result on the power of preemption: Corollary 3.6. For identical parallel machine scheduling with release dates so as to minimize the weighted sum of completion times, the value of an optimal nonpreemptive schedule is at most twice as large as the value of an optimal preemptive one. Moreover, the techniques in Algorithm LP ROUNDING can be used to convert an arbitrary preemptive schedule into a nonpreemptive one such that the value increases at most by a factor of 2: for a given preemptive schedule, construct the corresponding solution to (LP P ) or (LP R ), respectively. The value of this feasible solution to the LP relaxation is a lower bound on the value of the given preemptive schedule. Using Algorithm LP ROUNDING, the solution to (LP R ) can be turned into a nonpreemptive schedule whose expected value is bounded by twice the value of the LP solution, and thus by twice the value of the preemptive schedule we started with. This improves upon a bound of 7=3 due to Phillips et al. [29]. Algorithm RANDOM ASSIGNMENT can easily be turned into an online algorithm. There are several different online paradigms that have been studied in the area of scheduling, see [38] for a survey. We consider the setting where jobs continually arrive over time and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards. In particular, the characteristics of a job, i. e., its processing time and its weight become only known at its release date. In order to apply Algorithm RANDOM ASSIGNMENT in the online setting, note that for each job j its random variable a j can be drawn immediately when the job is released since there is no interdependency with any other decisions of the randomized algorithm. The same holds for the random machine assignments. Moreover, the preemptive schedule in the first step can be constructed until time t without the need of any knowledge of jobs that are released afterwards. Furthermore, it follows from the analysis in the proof of Lemma 2.3 that we get the same performance guarantee if job j is not started before time t j (respectively C S Thus, in the online variant of Algorithm RANDOM ASSIGNMENT we schedule the jobs as early as possible in order of nondecreasing C S with the additional constraint that no job j may start before time C S The following result improves upon the competitive ratio 2:89 Corollary 3.7. The online variant of Algorithm RANDOM ASSIGNMENT achieves competitive ratio 2. The perhaps most appealing aspect of Algorithm RANDOM ASSIGNMENT is that the assignment of jobs to machines does not depend on job characteristics; any job is put with probability 1=m to any of the machines. This technique also proves useful for the problem without (nontrivial) release dates: Theorem 3.8. Assigning jobs independently and uniformly at random to the machines and then applying Smith's ratio rule on each machine is a 3=2-approximation algorithm for P There exist instances for which this bound is asymptotically tight. Proof. First notice that the described algorithm exactly coincides with Algorithm RANDOM ASSIGNMENT (LP ROUNDING, respectively). Because of the negative result in Corollary 3.3, we cannot derive the bound 3=2 by comparing the expected value of the computed solution to the optimal value of (LP P ). Remember that we used a stronger relaxation including constraints (4) in order to derive this bound in the unrelated parallel machine setting. However, as a result of Lemma 2.3 we get since the second term on the right-hand side of (6) is equal to p j for the case of identical parallel machines. Since both are lower bounds on the value of an optimal solution, the result follows. In order to show that the performance guarantee 3=2 is tight, we consider instances with m identical parallel machines and m jobs of unit length and weight. We get an optimal schedule with value m by assigning one job to each machine. On the other hand we can show that the expected completion time of a job in the schedule constructed by random machine assignment is 3=2 \Gamma 1=2m which converges to 3=2 for increasing m. Since the jobs, we can without loss of generality schedule on each machine the jobs that were assigned to it in a random order. Consider a fixed job j and the machine i it has been assigned to. The probability that a job k 6= j was assigned to the same machine is 1=m. In this case k is processed before j on the machine with probability 1=2. We therefore get E[C j 2m . Quite interestingly, the derandomized variant of this algorithm precisely coincides with the WSPT-rule for which Kawaguchi and Kyan proved performance guarantee (1 2)=2 1:21 [22]: list the jobs according to nonincreasing ratios w j =p j and schedule the next job whenever a machine becomes available. Details for the derandomization are given in Section 4. While the proof given by Kawaguchi and Kyan is somewhat complicated, our simpler randomized analysis yields performance guarantee 3=2 for their algorithm. However, this weaker result also follows from the work of Eastman, Even, and Isaacs [11] who gave a combinatorial lower bound for which coincides with the lower bound given by (LP P ). The latter observation is due to Uma and Wein [48] and Williamson [50]. Derandomization Up to now we have presented randomized algorithms that compute a feasible schedule the expected value of which can be bounded from above in terms of the optimum solution to the scheduling problem under consideration. This means that our algorithms will perform well on average; however, we cannot give a firm guarantee for the performance of any single execution. From a theoretical point of view it is perhaps more desirable to have (deterministic) algorithms that obtain a certain performance in all cases. One of the most important techniques for derandomization is the method of conditional probabilities. This method is implicitly contained in a paper of Erdos and Selfridge [12] and has been developed in a more general context by Spencer [45]. The idea is to consider the random decisions in a randomized algorithm one after another and to always choose the most promising alternative. This is done by assuming that all of the remaining decisions will be made randomly. Thus, an alternative is said to be most promising if the corresponding conditional expectation for the value of the solution is as small as possible. The randomized algorithms in this paper can be derandomized by the method of conditional probabilities. We demonstrate this technique for the most general problem R j r Algorithm LP ROUNDING. Making use of Remark 2.1 and Lemma 2.3 we consider the variant of this algorithm where we set t being assigned to the machine-time pair (i; t) (ties are broken by prefering jobs with smaller indices). Thus, we have to construct a deterministic assignment of jobs to machine-time pairs. Our analysis of Algorithm LP ROUNDING in the proof of Lemma 2.3 does not give a precise expression for the expected value of the computed solution but only an upper bound. Hence, for the sake of a more accessible derandomization, we modify Algorithm LP ROUNDING by replacing its last step with the following variant: 3') Schedule on each machine i the jobs that were assigned to it nonpreemptively in nondecreasing order of t j , where ties are broken by preferring jobs with smaller indices. At the starting time of job j the amount of idle time on its machine has to be exactly t j . for each job j that has been assigned to machine i and t j 6 t k if job k is scheduled after job j, Step 3' defines a feasible schedule. In the proof of Lemma 2.3 we have bounded the idle time before the start of job j on its machine from above by t j . Thus, the analysis still works for the modified Algorithm LP ROUNDING. The main advantage of the modification of Step 3 is that we can now give precise expressions for the expectations and conditional expectations of completion times. Let y be the optimum solution we started with in the first step of Algorithm LP ROUNDING. Using the same arguments as in the proof of Lemma 2.3 we get the following expected completion time of job j in the schedule constructed by our modified Algorithm LP ROUNDING y ikt Moreover, we are also interested in the conditional expectation of j's completion time if some of the jobs have already been assigned to a machine-time pair. Let K ' J be such a subset of jobs. For each job k 2 K the 0=1- variable x ikt for t ? r ik indicates whether k has been assigned to the machine-time pair (i; t) not enables us to give the following expressions for the conditional expectation of j's completion time. If j 62 K we get y ikt and, if j 2 K, we get where (i; t) is the machine-time pair job j has been assigned to, i. e., x 1. The following lemma is the most important part of the derandomization of Algorithm LP ROUNDING. Lemma 4.1. Let y be the optimum solution we started with in the first step of Algorithm LP ROUNDING, K ' J, and x a fixed assignment of the jobs in K to machine-time pairs. Furthermore let j 2 J nK. Then, there exists an assignment of j to a machine-time pair (i; t) (i. e., x i t such that 6 EK;x Proof. Using the formula of total expectation, the conditional expectation on the right-hand side of (11) can be written as a convex combination of conditional expectations E K[f jg;x \Theta over all possible assignments of job j to machine-time pairs (i; t) with coefficients y i jt We therefore get a derandomized version of Algorithm LP ROUNDING if we replace the second step by 0; x:=0; for all j 2 J do i) for all possible assignments of job j to machine-time pairs (i; t) (i. e., x i \Theta ii) determine the machine-time pair (i; t) that minimizes the conditional expectation in i); set K := K[f jg; x Notice that we have replaced Step 3 of Algorithm LP ROUNDING by 3' only to give a more accessible analysis of its derandomization. Since the value of the schedule constructed in Step 3 is always at least as good as the one constructed in Step 3', the following theorem can be formulated for Algorithm LP ROUNDING with the original Step 3. Theorem 4.2. If we replace Step 2 in Algorithm LP ROUNDING with 2' we get a deterministic algorithm whose performance guarantee is at least as good as the expected performance guarantee of the randomized algorithm. Moreover, the running time of this algorithm is polynomially bounded in the number of variables of the LP relaxation Proof. The result follows by an inductive use of Lemma 4.1. The computation of (9) and (10) is polynomially bounded in the number of variables. Therefore, the running time of each of the n iterations in Step 2' is polynomially bounded in this number. The same derandomization also works for the polynomial time algorithms that are based on interval-indexed LP relaxations described in Section 5. Since these LP relaxations only contain a polynomial number of variables, the running time of the derandomized algorithms is also bounded by a polynomial in the input size of the scheduling problem. Notice that, in contrast to the situation for the randomized algorithms, we can no longer give job-by-job bounds for the derandomized algorithms. An interesting application of the method of conditional probabilities is the derandomization of Algorithm RANDOM ASSIGNMENT in the absence of release dates. We have already discussed this matter at the end of Section 3. It essentially follows from the considerations above that the derandomized version of this algorithm always assigns a job to the machine with the smallest load so far if we consider the jobs in order of nonincreasing . Thus, the resulting algorithm coincides with the WSPT-rule analyzed by Kawaguchi and Kyan [22]. 5 Interval-Indexed LP Relaxations As mentioned earlier, our LP-based algorithms for unrelated parallel machine scheduling suffer from the exponential number of variables in the corresponding LP relaxation (LP R ). However, we can overcome this drawback by using new variables which are not associated with exponentially many time intervals of length 1, but rather with a polynomial number of intervals of geometrically increasing size. This idea was earlier introduced by Hall et al. [19]. We show how Algorithm LP ROUNDING can be turned into a polynomial time algorithm for R j r at the cost of an increase in the performance guarantee to 2 e. The same technique can be used to derive a e)-approximation algorithm for R For a given h ? 0, L is chosen to be the smallest integer such that (1 Consequently, L is polynomially bounded in the input size of the considered scheduling problem. Let I \Theta and for 1 6 ' 6 L let I . We denote with jI ' j the length of the '-th interval, i. e., jI ' To simplify notation we define (1 +h) '\Gamma1 to be 1 with the following interpretation: y is the time job j is processed on machine i within time interval I ' , or, equivalently: (y i j' \Delta jI ' j)=p i j is the fraction of job j that is being processed on machine i within I ' . Consider the following linear program in these interval-indexed variables: minimize subject to for all Consider a feasible schedule and assign the values to the variables y i j' as defined above. This solution is clearly feasible: Constraints (12) are satisfied since a job j consumes units if it is processed on machine constraints (13) are satisfied since the total amount of processing on machine i of jobs that are processed within the interval I ' cannot exceed its size. Finally, if job j is continuously being processed between C machine h, then the right-hand side of equation (14) is a lower bound on the real completion time. Thus, (LP h R ) is a relaxation of the scheduling problem R j r Since (LP R ) is of polynomial size, an optimum solution can be computed in polynomial time. We rewrite Algorithm LP ROUNDING for the new LP: Algorithm: LP ROUNDING Compute an optimum solution y to (LP h Assign each job j to a machine-interval pair (i; I ' ) independently at random with probability from the chosen time interval I ' independently at random with uniform distribution. On each machine i schedule the jobs that were assigned to it in order of nondecreasing t j . The following lemma is a reformulation of Lemma 2.2 b) for the new situation and can be proved analogously. Lemma 5.1. The expected processing time of each job j 2 J in the schedule constructed by Algorithm LP ROUNDING is equal to m Theorem 5.2. The expected completion time of each job j in the schedule constructed by Algorithm LP ROUNDING is at most 2 \Delta (1 +h) \Delta C LP . Proof. We argue almost exactly as in the proof of Lemma 2.3, but use Lemma 5.1 instead of Lemma 2.2 b). We consider an arbitrary, but fixed job j 2 J. We also consider a fixed assignment of j to machine i and time interval I ' . Again, the conditional expectation of j's starting time equals the expected idle time plus the expected processing time on machine i before j is started. With similar arguments as in the proof of Lemma 2.3, we can bound the sum of the idle time plus the processing time by 2 This, together with Lemma 5.1 and (14) yields the theorem. For any given e ? 0 we can choose e)-approximation algorithm for the problem R j r R ) is a 6 Concluding Remarks and Open Problems In this paper, we have developed LP-based approximation algorithms for different scheduling problems and in doing so we have also gained some insight of the quality of the employed time-indexed LPs. A number of open problems arises from this and related research, and in the following wrap up we distinguish between the off-line and the on-line setting. Our central off-line result is the (2+e)-approximation for R j r there exist instances which show that the underlying LP relaxation ((LP R ) without inequalities (4)) is indeed not better than a 2-relaxation. However, it is open whether the quality of (LP R ) (with (4) and/or (7)) is better than 2 and therefore also whether it can be used to derive an approximation algorithm with performance guarantee strictly less than 2. On the negative side, In other words, the best known approximation algorithm for R j r performance guarantee 2 (we proved 2+ e here and [42] gets rid of the e using a convex quadratic relaxation), but the only known limit to its approximation is the non-existence of a polynomial-time approximation scheme, unless NP. The situation for R j j w j C j is similar. (LP R ) is a 3=2-relaxation, the quality of (LP R ) together with (7) is unknown, the 3=2-approximation given in [41] (improving upon the (3=2+e)-approximation in Section 2) is best known, and again there cannot be a PTAS, unless As far as identical parallel machines are concerned, one important property of our 2-approximation algorithm for is that it runs in time O(nlogn). The running time of the recent PTAS is O [6]. The other important feature of the O(nlogn) algorithm is that it is capable of working in an on-line context as well, which brings us to the second set of open problems. If jobs arrive over time and if the performance of algorithms is measured in terms of their competitiveness to optimal off-line algorithms, it is theoretically of the utmost importance to distinguish between deterministic and randomized algorithms. For identical parallel machine scheduling to minimize total weighted completion time, there is a significant gap between the best-known deterministic lower bound and the competitive ratio of the best-known deterministic algorithm. The lower bound of 2 follows from the fact that for on-line single machine scheduling to minimize total completion time no deterministic algorithm can have competitive ratio less than 2 [21, 46]. A (4 e)-competitive algorithm emerges from a more general framework [19, 18]. For randomized algorithms, our understanding seems slightly better. The best-known randomized lower bound of e=(e \Gamma 1) is again inherited from the single machine case [47, 49], and there is a randomized 2-competitive algorithm given in the paper in hand. Acknowledgements . The authors are grateful to Chandra S. Chekuri, Michel X. Goemans, and David B. Shmoys for helpful comments on an earlier version of this paper [36]. --R Competitive distributed job scheduling Resource scheduling for parallel database and scientific applications Improved scheduling algorithms for minsum criteria Approximation schemes for minimizing average weighted completion time with release dates. Approximation techniques for average completion time scheduling Approximation algorithms for precedence-constrained scheduling problems on parallel machines that run at different speeds Deterministic load balancing in computer networks Formulating the single machine sequencing problem with release dates as a mixed integer program Bounds for the optimal scheduling of n jobs on m processors A supermodular relaxation for scheduling with release dates Single machine scheduling with release dates. RINNOOY KAN Scheduling to minimize average completion time: Off-line and on-line approximation algorithms Scheduling to minimize average completion time: Off-line and on-line algorithms Optimal on-line algorithms for single-machine schedul- ing Worst case bound of an LRF schedule for the mean weighted flow-time problem RINNOOY KAN Management Science Randomized approximation algorithms in combinatorial opti- mization Randomized Algorithms Approximation bounds for a general class of precedence constrained parallel machine scheduling problems Improved bounds on relaxations of a parallel machine scheduling problem Task scheduling in networks A sequencing problem with release dates and clustered jobs A technique for provably good algorithms and algorithmic proofs Scheduling to minimize total weighted completion time: Performance guarantees of LP-based heuristics and lower bounds New approximations and LP lower bounds An approximation algorithm for the generalized assignment problem Approximation and Randomization in Scheduling A PTAS for minimizing the weighted sum of job completion times on parallel machines Various optimizers for single-stage production Ten Lectures on the Probabilistic Method Cited as personal communication in How low can't you go? On the relationship between combinatorial and LP-based approaches to NP-hard scheduling problems PhD thesis Cited as personal communication in --TR --CTR Feng Lu , Dan C. Marinescu, An R || Cmax Quantum Scheduling Algorithm, Quantum Information Processing, v.6 n.3, p.159-178, June 2007 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 Martin Skutella, Convex quadratic and semidefinite programming relaxations in scheduling, Journal of the ACM (JACM), v.48 n.2, p.206-242, March 2001

on-line algorithm;scheduling;linear programming relaxation;randomized rounding;approximation algorithm

587939

On Lower Bounds for Selecting the Median.

We present a reformulation of the 2n+o(n) lower bound of Bent and John [Proceedings of the 17th Annual ACM Symposium on Theory of Computing, 1985, pp. 213--216] for the number of comparisons needed for selecting the median of n elements. Our reformulation uses a weight function. Apart from giving a more intuitive proof for the lower bound, the new formulation opens up possibilities for improving it. We use the new formulation to show that any pair-forming median finding algorithm, i.e., a median finding algorithm that starts by comparing $\lfloor n/2\rfloor$ disjoint pairs of elements must perform, in the worst case, at least 2.01 n comparisons. This provides strong evidence that selecting the median requires at least cn+o(n) comparisons for some c> 2.

Introduction . Sorting and selection problems have received extensive attention by computer scientists and mathematicians for a long time. Comparison based algorithms for solving these problems work by performing pairwise comparisons between the elements until the relative order of all elements is known, in the case of or until the i-th largest element among the n input elements is found, in the case of selection. Sorting in a comparison based computational model is quite well understood. Any deterministic algorithm can be modeled by a decision tree in which all internal nodes represent a comparison between two elements; every leaf represents a result of the computation. Since there must be at least as many leaves in the decision tree as there are possible re-orderings of n elements, all algorithms that sort n elements use at least dlog n!e n log n n log e comparisons in the worst case. (All logarithms in this paper are base 2 logarithms.) The best known sorting method, called merge insertion by Knuth [9], is due to Lester Ford Jr. and Selmer Johnson [7]. It sorts n elements using at most n log n 1:33n Thus, the gap between the upper and lower bounds is very narrow in that the error in the second order term is bounded by 0:11n. The problem of nding the median is the special case of selecting the i-th largest in an ordered set of n elements, when Although much eort has been put into nding the exact number of required comparisons, there is still an annoying gap between the best upper and lower bounds currently known. Knowing how to sort, we could select the median by rst sorting, and then selecting the middle-most element; it is quite evident that we could do better, but how much better? This question received a somewhat surprising answer when Blum et al. [3] showed, in 1973, how to determine the median in linear time using at most 5:43n comparisons. This result was improved upon in 1976 when Schnhage, Paterson, and Pippinger [13] presented an algorithm that uses only 3n School of Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel. E-mail: dorit@checkpoint.com and zwick@post.tau.ac.il. y Department of Numerical Analysis and Computing Science, Royal Institute of Technology, 100 44 Stockholm, Sweden. E-mail: fjohanh,staffanug@nada.kth.se. D. DOR AND J. H ASTAD AND S. ULFBERG AND U. ZWICK main invention was the use of factories which mass-produce certain partial orders that can be easily merged with each other. This remained the best algorithm for almost 20 years, until Dor and Zwick [5] pushed down the number of comparisons a little bit further to 2:95n + o(n) by adding green factories that recycle debris from the merging process used in the algorithm of [13]. The rst non-trivial lower bound for the problem was also presented, in 1973, by Blum et al. [3] using an adversary argument. Their 1:5n lower bound was subsequently improved to 1:75n + o(n) by Pratt and Yao [12] in 1973. Then Yap [14], and later Munro and Poblete [10], improved it to 38 43 n+O(1), respectively. The proofs of these last two bounds are long and complicated. In 1979, Fussenegger and Gabow [8] proved a 1:5n + o(n) lower bound for the median using a new proof technique. Bent and John [2] used the same basic ideas when they gave, in 1985, a short proof that improved the lower bound to 2n which is currently the best available. Thus, the uncertainty in the coe-cient of n is larger for nding the median than it is for sorting, even though the linear term is the second order term in the case of sorting. Since our methods are based on the proof by Bent and John, let us describe it in some detail. Given the decision tree of a comparison based algorithm, they invented a method to prune it that yields a collection of pruned trees. Then, lower bounds for the number of pruned trees and for their number of leaves are obtained. A nal argument saying that the leaves of the pruned trees are almost disjoint then gives a lower bound for the size of the decision tree. In Section 2 we reformulate the proof by Bent and John by assigning weights to each node in the decision tree. The weight of a node v corresponds to the total number of leaves in subtrees with root v in all pruned trees where v occurs in the proof by Bent and John. The weight of the root is approximately 2 2n ; we show that every node v in the decision tree has a child whose weight is at least half the weight of v, and that the weights of all the leaves are small. When the proof is formulated in this way, it becomes more transparent, and one can more easily study individual comparisons, to rule out some as being bad from the algorithm's point of view. For many problems, such as nding the maximal or the minimal element of an ordered set, and nding the maximal and minimal element of an ordered set, there are optimal algorithms that start by making bn=2c pairwise comparisons between singleton elements. We refer to algorithms that start in this way as being pair- forming. It has been discussed whether there are optimal pair-forming algorithms for all partial orders, and in particular this question was posed as an open problem by Aigner [1]. Some examples were then found by Chen [4], showing that pair-forming algorithms are not always optimal. It is interesting to note that the algorithms in [5] and [13] are both pair-forming. It is still an open problem whether there are optimal pair-forming algorithms for nding the median. In Section 3 we use our new approach to prove that any pair-forming algorithm uses at least 2:01227n comparisons to nd the median. Dor and Zwick [6] have recently been able to extend the ideas described here and obtain a (2+)n lower bound, for some tiny > 0, on the number of comparisons performed, in the worst case, by any median selection algorithm. ON LOWER BOUNDS FOR SELECTING THE MEDIAN 3 2. Bent and John revisited. Bent and John [2] proved that 2n + o(n) comparisons are required for selecting the median. Their result, in fact, is more general and provides a lower bound for the number of comparisons required for selecting the i-th largest element, for any 1 i n. We concentrate here on median selection although our results, like those of Bent and John, can be extended to general i. Although the proof given by Bent and John is relatively short and simple, we here present a reformulation. There are two reasons for this: the rst is that the proof gets more transparent; the second is that this formulation makes it easier to study the eect of individual comparisons. Theorem 2.1 (Bent and John [2]). Finding the median requires 2n comparisons Proof. Any deterministic algorithm for nding the median can be represented by a decision tree T , in which each internal node v is labeled by a comparison a : b. The two children of such a node, v a<b and v a>b , represent the outcomes a < b and a > b, respectively. We assume that decision trees do not contain redundant comparisions between elements whose relative order has already been established. We consider a universe U containing n elements. For every node v in T and subset C of U we make the following denitions: every comparison a : b above v with b 2 C had outcome a > b every comparison a : b above v with b 2 C had outcome a < b Before we proceed with the proof that selecting the median requires 2n parisons, we present a proof of a somewhat weaker result. We assume that U contains show that selecting the two middlemost elements requires comparisons. The proof in this case is slightly simpler, yet it demonstrates the main ideas used in the proof of the theorem. We dene a weight function on the nodes of T . This weight function satises the following three properties: (i) the weight of the root is 2 2n+o(n) . (ii) each internal node v has a child whose weight is at least half the weight of v. (iii) the weight of each leaf is small. For every node v in the decision tree, we keep track of subsets A of size m which may contain the m largest elements with respect to the comparisons already made. Let A(v) contain all such sets which are called upper half compatible with v. The As are assigned weights which estimate how far from a solution the algorithm is, assuming that the elements in A are the m largest. The weight of every A 2 A(v) is dened as and the weight of a node v is dened as The superscript 1 in w 1 (A) is used as we shall shortly have to dene a second weight function w 2 (B). 4 D. DOR AND J. H ASTAD AND S. ULFBERG AND U. ZWICK case w 1 a 2 A b 2 A 1 a 2 A b 2 a 2 a 2 Table The weight of a set A 2 A(v) in the children of a node v, relative to its weight in v. In the root r of T , all subsets of size m of U are upper half compatible with r so that . Also, each A 2 A(r) has weight 2 2m , and we nd, as promised, that 2m Consider the weight w 1 (A) of a set A 2 A(v) at a node v labeled by the comparison b. What are the weights of A in v's children? This depends on which of the elements a and b belongs to A (and on which of them is minimal in A or maximal in A). The four possible cases are considered in Table 2.1. The weights given there are relative to the weight w 1 of A at v. A zero indicates that A is no longer compatible with this child and thus does not contribute to its weight. The weight w 1 va<b (A), when example, is 1 (A), and is w 1 (A), otherwise. As can be seen, v always has at least one child in which the weight of A is at least half its weight at v. Furthermore, in each one of the four cases, w 1 (A). Each leaf v of the decision tree corresponds to a state of the algorithm in which the two middlemost elements were found. There is therefore only one set A left in A(v). Since we have identied the minimum element in A and the maximum element in A, we get that w 1 4. So, if we follow a path from the root of the tree and repeatedly descend to the child with the largest weight, we will, when we eventually reach a leaf, have performed at least 2n We now prove that selecting the median also requires at least 2n isons. To make the median well dened we assume that 1. The problem that arises in the above argument is that the weights of the leaves in T , when the selection of the median, and not the two middlemost elements, is considered, are not necessarily small enough: it is possible to know the median without knowing any relations between elements in A (which now contains m 1 elements); this is remedied as follows. In a node v where the algorithm is close to determining the minimum element in A, we essentially force it to determine the largest element in A instead. This is done by moving an element a 0 out of A and creating a set g. This set is lower half compatible with v and the median is the maximum element in B. By a suitable choice of a 0 , most of max v ( is in max v (B). A set B is lower half compatible with v may contain the m smallest elements in U . We keep track of Bs in the multiset B(v). For the root r of T , we let A(r) contain all subsets of size m of U as before, and let B(r) be empty. We exchange some As for Bs as the algorithm proceeds. The ON LOWER BOUNDS FOR SELECTING THE MEDIAN 5 case a 2 a 2 Table The weight of a set B 2 B(v) in the children of a node v, relative to its weight in v. weight of a set B is dened as The weight of B estimates how far the algorithm is from a solution, assuming that the elements in B are the m smallest elements. The weight of a node v is now dened to be In the beginning of an algorithm (in the upper part of the decision tree), the weight of a node is still the sum of the weights of all As, and therefore We now dene A(v) and B(v) for the rest of T more exactly. For any node v in T , except the root, simply copy A(v) and B(v) from the parent node and remove all sets that are not upper or lower half compatible with v, respectively. We ensure that the weight of every leaf is small by doing the following: If, for some A 2 A(v) we have ne, we select an element a 0 2 min v (A) which has been compared to the fewest number of elements in A; we then remove the set A from A(v) and add the set to B(v). Note that at the root, jmin r and that this quantity decreases by at most one for each comparison until a leaf is reached. In a leaf v the median is known; thus, A(v) is empty. Lemma 2.2. Let A(v) and B(v) be dened by the rules described above. Then, every internal node v (labeled a : b) in T has a child with at least half the weight of v, i.e., w(v a<b ) w(v)=2 or w(v a>b ) w(v)=2. Proof. Table 2.1 gives the weights of a set A 2 A(v) at v's children, relative to the weight of A at v. Similarly, Table 2.2 gives the weights of a set B 2 B(v) in v's children, relative to the weight w 2 v (v) of B at v. As w 1 v (B), for every A 2 A(v) and B 2 B(v), all that remains to be checked is that the weight does not decrease when a lower half compatible set B replaces an upper half compatible set A. This is covered by Lemma 2.3. Lemma 2.3. If A is removed from A(v) and B is added in its place to B(v), and if fewer than 4n comparisons have been performed on the path from the root to v, then (A). Proof. A set A 2 A(v) is replaced by a set only when ne. The element a 0 , in such a case, is an element of min v (A) that has been compared to the fewest number of elements in A. If a 0 was compared to at least 2 n elements in A, we get that each element of min v (A) was compared to at 6 D. DOR AND J. H ASTAD AND S. ULFBERG AND U. ZWICK least 2 n elements in A, and at least 4n comparisons have been performed on the path from the root to v, a contradiction. We get therefore that a 0 was compared to fewer than 2 n elements of A and thus jmax v (B)j > jmax v ( n. As a consequence, we get that 4 A)j and thus 2 4 as required. We now know that the weight of the root is large, and that the weight does not decrease too fast; what remains to be shown is that the weights of the leaves are relatively small. This is established in the following lemma. Lemma 2.4. For a leaf v (in which the median is known), w(v) 2m2 4 Proof. Clearly, the only sets compatible with a leaf of T are the set A containing the m largest elements, and the set B containing the m smallest elements. Since we get that w 2 Since there are exactly m elements that can be removed from B to obtain a corresponding A, there can be at most m copies of B in B(v). Let T be a comparison tree that corresponds to a median nding algorithm. If the height of T is at least 4n, we are done. Otherwise, by starting at the root and repeatedly descending to a child whose weight is at least half the weight of its parent, we trace a path whose length is at least 2n + o(n) and Theorem 2.1 follows. Let us see how the current formalism gives room for improvement that did not exist in the original proof. The 2n lower bound is obtained by showing that each node v in a decision tree T that corresponds to a median nding algorithm has a child whose weight is at least half the weight of v. Consider the nodes v along the path obtained by starting at the root of T and repeatedly descending to the child with the larger weight, until a leaf is reached. If we could show that su-ciently many nodes on this path have weights strictly larger than half the weights of their parents, we would obtain an improved lower bound for median selection. If w(v i then the length of this path, and therefore the depth of T , is at least 2n 3. An improved lower bound for pair-forming algorithms. Let v be a node of a comparison tree. An element x is a singleton at v if it was not compared above v with any other element. Two elements x and y form a pair at v if the elements x and y were compared to each other above v, but neither of them was compared to any other element. A pair-forming algorithm is an algorithm that starts by constructing By concentrating on comparisons that involve elements that are part of pairs, we obtain a better lower bound for pair-forming algorithms. Theorem 3.1. A pair-forming algorithm for nding the median must perform, in the worst case, at least 2:00691n Proof. It is easy to see that a comparison involving two singletons can be delayed until just before one of them is to be compared for the second time. We can therefore restrict our attention to comparison trees in which the partial order corresponding to each node contains at most two pairs. Allowing only one pair is not enough as algorithms should be allowed to construct two pairs fa; bg and fa compare an element from fa; bg with an element from fa g. We focus our attention on nodes in the decision tree in which an element of a pair is compared for the second time and in which the number of non-singletons is at most m, for some < 1. If v is a node in which the number of non-singletons is at ON LOWER BOUNDS FOR SELECTING THE MEDIAN 7 r a r a r a r r a r a r a r A A Fig. 3.1. The six possible ways that a, b, and c may be divided between A and A. Note that c is not necessarily a singleton element; it may be part of a larger partial order. most m, for some < 1, then B(v) is empty and thus we do not have to consider Table 2.2 for the rest of the section. Recall that A(v) denotes the collection of subsets of U size m that are upper half compatible with v. If H;L U are subsets of U , of arbitrary size, we let Ag: We let wH=L (v) be the contribution of the sets of AH=L (v) to the weight of v, i.e., For brevity, we write A (v) for A fh1 ;:::;h r g=fl1 ;:::;l s g (v) and w (v) for w fh1 ;:::;h r g=fl1 ;:::;l s g (v). Before proceeding, we describe the intuition that lies behind the rest of the proof. Consider Table 2.1 from the last section. If, in a node v of the decision tree, the two cases a 2 A; b 2 A and a 2 are not equally likely, or more precisely, if the contributions w a=b (v) and w b=a (v) of these two cases to the total weight of v are not equal, there must be at least one child of v whose weight is greater than half the weight of v. The di-culty in improving the lower bound of Bent and John lies therefore at nodes in which the the contributions of the two cases a 2 A; b 2 A and a 2 are almost equal. This fact is not so easily seen when looking at the original proof given in [2]. Suppose now that v is a node in which an element a of a pair fa; bg is compared with an arbitrary element c and that the number of non-singletons in v is at most m. We assume, without loss of generality, that a > b. The weights of a set A 2 A(v) in v's children depend on which of the elements a, b, and c belongs to A, and on whether c is minimal in A or maximal in A. The six possible ways of dividing the elements a, b, and c between A and A are shown in Figure 3.1. The weights of the set A in v's children, relative to the weight w 1 of A at v, in each one of these six cases are given in Table 3.1. Table 3.1 is similar to Table 2.1 of the previous section, with c playing the role of b. There is one important dierence, however. If a; b; c 2 A, as in the rst row of Table 3.1, then the weight of A in v a>c is equal to the weight of A in v. The weight is not halved, as may be the case in the rst row of Table 2.1. If the contribution w abc= (v) of the case a; b; c 2 A to the weight of v is not negligible, there must again be at least one child of v whose weight is greater than half the weight of v. The improved lower bound is obtained by showing that if the contributions of the cases a 2 A, b 2 A and a 2 are roughly equal, and if most elements in the partial order are singletons, then the contribution of the case a; b; c 2 A is non- negligible. The larger the number of singletons in the partial order, the larger is the relative contribution of the weight w abc= (v) to the weight w(v) of v. Thus, whenever 8 D. DOR AND J. H ASTAD AND S. ULFBERG AND U. ZWICK case w 1 a 2 A b 2 a 2 A b 2 A c 2 a 2 A b 2 a 2 A 11 Table The weight of a set A 2 A(v) in the children of a node v, relative to its weight in v, when the element a of a pair a > b is compared with an arbitrary element c. an element of a pair is compared for the second time, we make a small gain. The above intuition is made precise in the following lemma: Lemma 3.2. If v is a node in which an element a of a pair a > b is compared with an element c, and if the number of singletons in v is at least m+ 2 p (w a=c (v) w c=a (v) Proof. Both inequalities follow easily by considering the entries in Table 3.1. To obtain the second inequality, for example, note that w(v a>c As w c=ab w a=c (v), the second inequality follows. It is worth pointing out that in Table 3.1 and in Lemma 3.2, we only need to assume that a > b; we do not use the stronger condition that a > b is a pair. This stronger condition is crucial however in the sequel, especially in Lemma 3.4. To make use of Lemma 3.2 we need bounds on the relative contributions of the dierent cases. The following lemma is a useful tool for determining such bounds. Lemma 3.3. Let E) be a bipartite graph. Let - 1 and - 2 be the minimal degree of the vertices of V 1 and V 2 , respectively. Let 1 and 2 be the maximal degree of the vertices of V 1 and V 2 , respectively. Assume that a positive weight function w is dened on the vertices of G such that w(v 1 and (v r Proof. Let denote the two vertices connected by the edge e. We then have ON LOWER BOUNDS FOR SELECTING THE MEDIAN 9 The other inequality follows by exchanging the roles of V 1 and V 2 . Using Lemma 3.3 we obtain the following basic inequalities. Lemma 3.4. If v is a node in which a > b is a pair and the number of non- singletons in v is at most m, then2 Each one of these inequalities relates a weight, such as w abc= (v), to a weight, such as w ac=b (v), obtained by moving one of the elements of the pair a > b from A to A. In each inequality we 'lose' a factor of 1 . When the elements a and b are joined together a factor of 2 is introduced. When the elements a and b are separated, a factor of 1is introduced. Proof. We present a proof of the inequality w abc= (v) 1 (v). The proof of all the other inequalities is almost identical. Construct a bipartite graph E) whose vertex sets are (v). Dene an edge A ac=b (v) if and only if there is a singleton d 2 A 1 such that A Suppose that is such an edge. As a 62 min v other elements are extremal with respect to A 1 if and only if they are extremal with respect to A 2 (note that b 2 min v A 2 )), we get that w 1 For every set A of size m, the number of singletons in A is at least (1 )m and at most m. We get therefore that the minimal degrees of the vertices of V 1 and V 2 are and the maximal degrees of V 1 and V 2 are 1 ; 2 m. The inequality w abc= (v) 1 therefore follows from Lemma 3.3. Using these basic inequalities we obtain: Lemma 3.5. If v is a node in which a > b is a pair and the number of non- singletons is at most m, for some < 1, then Proof. We present the proof of the rst inequality. The proof of the other two inequalities is similar. Using inequalities from Lemma 3.4 we get that and the rst inequality follows. D. DOR AND J. H ASTAD AND S. ULFBERG AND U. ZWICK We are now ready to show that if v is a node in which an element of a pair is compared for the second time, then v has a child whose weight is greater than half the weight of v. Combining Lemma 3.2 and Lemma 3.5, we get that2 (w(v a<c w(v) As a consequence, we get that The coe-cient of w(v), on the right hand side, is minimized when the two expressions whose maximum is taken are equal. This happens when . Plugging this value of into the two expressions, we get that where It is easy to check that f 1 () > 0 for < 1. A pair-forming comparison is a comparison in which two singletons are compared to form a pair. A pair-touching comparison is a comparison in which an element of a pair is compared for the second time. In a pair-forming algorithm, the number of singletons is decreased only by pair-forming comparisons. Each pair-forming comparison decreases the number of singletons by exactly two. As explained above, pair-forming comparisons can always be delayed so that a pair-forming comparison immediately followed by a comparison that touches the pair fa; bg, or by a pair-forming comparison a then by a comparison that touches both pairs g. Consider again the path traced from the root by repeatedly descending to the child with the larger weight. As a consequence of the above discussion, we get that when the i-th pair-touching comparison along this path is performed, the number of non-singletons in the partial order is at most 4i. It follows therefore from the remark made at the end of the previous section that the depth of the comparison tree corresponding to any pair-forming algorithm is at least log ON LOWER BOUNDS FOR SELECTING THE MEDIAN 11 r a r a r a r r a r a r a r r a r a r a r r c r d r c r d r c r d r c r d r c r d r c r d r c r d A A Fig. 3.2. The nine possible ways that a, b, c, and d may be divided between A and A. case w 1 Table The weight of a set A 2 A(v) in the children of a node v, relative to its weight in v, when the element a of a pair a > b is compared with an element of a pair c > d. This completes the proof of Theorem 3.1. The worst case in the proof above is obtained when the algorithm converts all the elements into quartets . A quartet is a partial order obtained by comparing elements contained in two disjoint pairs. In the proof above, we analyzed cases in which an element a of a pair a > b is compared with an arbitrary element c. If the element c is also part of a pair, a tighter analysis is possible. By performing this anaylsis we can improve Theorem 3.1. Theorem 3.6. A pair-forming algorithm for nding the median must perform, in the worst case, at least 2:01227n Proof. Consider comparisons in which the element from a pair a > b is compared with an element of a pair c > d. The nine possible ways of dividing the elements a, b, c, and d among A and A are depicted in Figure 3.2. We may assume, without loss of generality, that the element a is compared with either c or with d. Let v be a node of the comparison tree in which a > b and c > d are pairs and which one of the comparions a : c or a : d is performed. Let A 2 A(v). The weights of a set A in v's children, relative to the weight w 1 of A at v, in each one of these nine cases are given in Table 3.2. The two possible comparisons a : c and a : d are considered separately. The following equalities are easily veried. Lemma 3.7. If a > b and c > d are pairs in v then 12 D. DOR AND J. H ASTAD AND S. ULFBERG AND U. ZWICK The following inequalities are analogous to the inequalities of Lemma 3.4. Lemma 3.8. If a > b and c > d are pairs in v and if the number of non-singletons in v is at most m, for some < 1, then2 Consider rst the comparison a : c. By examining Table 3.2 and using the equalities of Lemma 3.7, we get that w(va<c)+w(va>c )= w abcd= (v) Minimizing this expression, subject to the equalities of Lemma 3.7, the inequalities of Lemma 3.8, and the fact that the weights of the nine cases sum up to w(v), amounts to solving a linear program. By solving this linear program we get that where It seems intuitively clear that the comparison a : d is a bad comparison from the algorithm's point of view. The adversary will most likely answer with a > d. Indeed, by solving the corresponding linear program, we get that w(va>d As for every 0 1, we may disregard the comparison a : d from any further consideration. It is easy to verify that (1+f 1 As a result, we get a lower bound of This completes the proof of Theorem 3.6. ON LOWER BOUNDS FOR SELECTING THE MEDIAN 13 4. Concluding remarks. We presented a reformulation of the 2n + o(n) lower bound of Bent and John for the number of comparisons needed for selecting the median of n elements. Using this new formulation we obtained an improved lower bound for pair-forming median nding algorithms. As mentioned, Dor and Zwick [6] have recently extended the ideas described here and obtained a (2+)n lower bound for general median nding algorithms, for some tiny > 0. We believe that the lower bound for pair-forming algorithms obtained here can be substantially improved. Such an improvement seems to require, however, some new ideas. Obtaining an improved lower bound for pair-forming algorithms may be an important step towards obtaining a lower bound for general algorithms which is signicantly better than the lower bound of Bent and John [2]. Paterson [11] conjectures that the number of comparisons required for selecting the median is about (log 4=3 2)n 2:41n. --R Producing posets. Finding the median requires 2n comparisons. Time bounds for selection. Partial Order Productions. Selecting the median. Median selection requires (2 A tournament problem. A counting approach to lower bounds for selection problems. The Art of Computer Programming A lower bound for determining the median. Progress in selection. On lower bounds for computing the i-th largest element New lower bounds for medians and related problems. --TR --CTR Krzysztof C. Kiwiel, On Floyd and Rivest's SELECT algorithm, Theoretical Computer Science, v.347 n.1-2, p.214-238, November 2005

median selection;lower bounds;comparison algorithms

587941

The Maximum Edge-Disjoint Paths Problem in Bidirected Trees.

A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Given a set of directed paths in a bidirected tree, the goal of the maximum edge-disjoint paths problem is to select a maximum-cardinality subset of the paths such that the selected paths are edge-disjoint. This problem can be solved optimally in polynomial time for bidirected trees of constant degree but is APX-hard for bidirected trees of arbitrary degree. For every fixed $\varepsilon >0$, a polynomial-time $(5/3+\varepsilon)$-approximation algorithm is presented.

Introduction . Research on disjoint paths problems in graphs has a long history [12]. In recent years, edge-disjoint paths problems have been brought into the focus of attention by advances in the field of communication networks. Many modern network architectures establish a virtual circuit between sender and receiver in order to achieve guaranteed quality of service. When a connection request is accepted, the network must allocate sufficient resources on all links along a path from the sender to the receiver. Edge-disjoint paths problems are at the heart of the arising resource allocation problems. We study the maximum edge-disjoint paths problem (MEDP) for bidirected tree networks. A bidirected tree is the directed graph obtained from an undirected tree by replacing each undirected edge by two directed edges with opposite directions. Bidirected tree networks have been studied intensively because they are a good model for optical networks with pairs of unidirectional fiber links between adjacent nodes [26, MEDP in bidirected trees is defined as follows. Given a bidirected tree E) and a set P of simple, directed paths in T , the goal is to find a subset P that the paths in P 0 are edge-disjoint and the cardinality of P 0 is maximized. We say that an algorithm is a ae-approximation algorithm for MEDP if it always outputs a subset of edge-disjoint paths whose cardinality is at least a (1=ae)-fraction of the cardinality of an optimal solution. The conflict graph of a set of directed paths in a bidirected tree is an undirected graph with a vertex for each path and an edge between two vertices if the corresponding paths intersect (i.e., if they share an edge). One can view MEDP in bidirected trees as a maximum independent set problem in the conflict graph. We assume that the given tree is rooted at an arbitrary node. For a node v, we let p(v) denote the parent of v. The level of a node is then defined as its distance to the root node. The root has level zero. We say that a path touches a node if it begins at that node, passes through that node, or ends at that node. The level of a path is the minimum of the levels of all nodes it touches. The unique node on a path whose level is equal to the level of the path is the least common ancestor (lca) of the path. A preliminary version of this article has appeared in the Proceedings of the 9th Annual International Symposium on Algorithms and Computation ISAAC'98, LNCS 1533, pages 179-188, 1998. y Institut f?r Informatik, TU M-unchen, 80290 M-unchen, Germany (erlebach@in.tum.de). z IDSIA Lugano, Corso Elvezia 36, 6900 Lugano, Switzerland (klaus@idsia.ch). We denote a path that begins at node u and ends at node v by (u; v) and its lca by lca(u; v). 1.1. Results. First, in x2, we determine the complexity of MEDP in bidirected trees: MEDP can be solved optimally in polynomial time in bidirected trees of constant degree and in bidirected stars, but it is MAX SNP-hard in bidirected trees of arbitrary degree. The main result of this paper is summarized by the following theorem Theorem 1.1. For every fixed " ? 0, there is a polynomial-time approximation algorithm for the maximum edge-disjoint paths problem in bidirected trees with approximation ratio 5=3 The description of the algorithm and a proof that the claimed approximation ratio is indeed achieved appear in x3. In x4, we discuss how our results can be generalized to the weighted version of the problem and to the maximum path coloring problem. 1.2. Related work. Path coloring in bidirected trees. Previous work on bidirected trees has focused on the path coloring problem: Given a set of directed paths in a bidirected tree, assign colors to the paths such that paths receive different colors if they share an edge. The goal is to minimize the total number of colors used. This problem is NP-hard even for binary trees [8, 24]. The best known approximation algorithms [11, 10] use at most d(5=3)Le colors, where L is the maximum load (the load of an edge is the number of paths using that edge) and thus a lower bound on the optimal solution. Previous algorithms had used (15=8)L colors [26] and (7=4)L colors [18, 25] in the worst case. For the special case of all-to-all path coloring, it was shown that the optimal number of colors is equal to the maximum load [14]. Multicommodity flow in trees. Garg et al. [13] studied the integral multicommodity flow problem in undirected trees, which is a generalization of MEDP in undirected trees. They showed that the problem with unit edge capacities (equivalent to MEDP in undirected trees) can be solved optimally in polynomial time. For undirected trees with edge capacities one or two, they proved the problem MAX SNP-hard. They also presented a 2-approximation algorithm for integral multicommodity flow in trees. It works by considering the demands in order of non-increasing levels of their lcas and by satisfying them greedily. This approximation algorithm can be adapted to MEDP in bidirected trees, where it also gives a 2-approximation. The main idea that leads to our improved approximation algorithm for MEDP in bidirected trees is to consider all paths with the same lca simultaneously instead of one by one. Online algorithms for MEDP in trees. MEDP has also been studied in the on-line scenario, where the paths are given to the algorithm one by one. The algorithm must accept or reject each path without knowledge about future requests. Preemption is not allowed. It is easy to see that no deterministic algorithm can have a competitive ratio better than the diameter of the tree in this case. Awerbuch et al. gave a randomized algorithm with competitive ratio O(log n) for undirected trees with n nodes [2]. Their algorithm works also for bidirected trees. An improved randomized algorithm with competitive ratio O(log d) for undirected trees with diameter d was given in [3]. MEDP for other topologies. If MEDP is studied for arbitrary graphs, the algorithm must solve both a routing problem and a selection problem. For arbitrary directed graphs with m edges, MEDP was recently shown to be NP-hard to approximate within m 1=2\Gamma" [16]. Approximation algorithms with approximation ratio O( p m) are known for the unweighted case [20, 28] and for the weighted case [22]. Better approximation ratios can be achieved for restricted classes of graphs. For a class EDGE-DISJOINT PATHS IN BIDIRECTED TREES 3 of planar graphs containing two-dimensional mesh networks, an O(1)-approximation algorithm has been devised in [21]. 2. Complexity results. MEDP in bidirected trees is NP-hard in general. This can be proved by a reduction from 3D-matching that is similar to the reduction used by Garg et al. to prove the NP-hardness of integral multicommodity flow in undirected trees with edge capacities one and two [13]. We omit the details, because the modification is straightforward. If we reduce from the bounded variant of the 3D-matching problem [19], the reduction is an L-reduction and an AP-reduction, implying that MEDP in bidirected trees is MAX SNP-hard [27] and APX -hard [7]. This shows that there is no polynomial-time approximation scheme for the problem Nevertheless, MEDP can be solved optimally in polynomial time if the input is restricted in certain ways. First, consider the case that the maximum degree of the given tree is bounded by a constant. The optimal solution can be computed by dynamic programming in this case. We process the nodes of the tree in order of non-increasing levels. At every node v, we record for each possible subset S of edge-disjoint paths touching v and its parent (note that jSj - 2) the maximum number of paths contained in the subtree rooted at v that can be accepted in addition to the paths in S. Node v is processed only when these values are known for all its children. We can then enumerate all possible edge-disjoint subsets of paths touching v. For each such subset, we can look up the corresponding values stored at children of v and update the values stored at v accordingly. Note that there are only polynomially many subsets to consider at each node. When the root node has been processed, the optimal solution can easily be constructed. Another special case that can be solved optimally in polynomial time is the case that the given bidirected tree T is a star, i.e., it contains only one node with degree greater than one. MEDP in bidirected stars can be reduced to the maximum matching problem in a bipartite graph as follows. First, we can assume without loss of generality that every given path uses exactly two edges of the star; if a path uses only one edge, we can add a new node to the star and extend the path by one edge without changing the set of solutions. Now, observe that every path uses exactly one edge directed towards the center and one edge directed away from the center of the star. Construct a bipartite graph G by including a vertex for every edge of the star and by adding an edge between two vertices u and v in G for every path in T that uses the edges corresponding to u and v. Two paths in T are edge-disjoint if and only if the corresponding edges in G do not share an endpoint. Sets of edge-disjoint paths in T correspond to matchings in G. A maximum matching in G can be computed in polynomial time [17]. The latter result can actually be generalized from stars to spiders. A spider is a bidirected tree in which at most one node (the center) has degree greater than two. MEDP in a bidirected spider can be solved in polynomial time using an algorithm for the maximum-weight bipartite matching problem as a subroutine. The bipartite graph G is constructed as above from the paths touching the center of the spider, and the weight of an edge e in G specifies how many fewer paths not touching the center of the spider can be accepted if the path corresponding to e is accepted. The details are left to the reader. 3. Approximating the optimal solution. Fix any " ? 0. Let an instance of the maximum edge-disjoint paths problem be given by a bidirected tree T and a set 4 T. ERLEBACH AND K. JANSEN P of directed paths in T . Denote by P an arbitrary optimal solution for the given instance. The algorithm proceeds in two passes. In the first pass, it processes the nodes of T in order of non-increasing levels (i.e., bottom-up). Assume that the algorithm is about to process node v. Let P v denote the subset of all paths (u; w) 2 P with that do not intersect any of the paths that have been accepted by the algorithm at a previous node and that do not use any edges that have been reserved or fixed by the algorithm (see below). For the sake of simplicity, we can assume without loss of generality that we have u otherwise, we could add an additional child to v for each path in P v starting or ending at v and make the path start or end at this new child instead. Every path uses exactly two edges incident to v, and we refer to these two edges as the top edges of p. We say that two paths are equivalent if they use the same two edges incident to v, i.e., if their top edges are the same. For a set Q of paths with the same lca, this defines a partition of Q into different equivalence classes of paths in the natural way. While the algorithm processes node v, it tries to determine for the paths in P v whether they should be included in the solution (these paths are called accepted) or not (these paths are called rejected ). Sometimes, however, the algorithm cannot make this decision right away. In these cases the algorithm will leave some paths in an intermediate state and resolve them later on. The possibilities for paths in such intermediate states are (i) undetermined paths, (ii) groups of deferred paths, (iii) groups of exclusive paths, and (iv) groups of 2-exclusive paths. We refer to undetermined paths and to paths in groups of exclusive paths and in groups of 2-exclusive paths as unresolved paths and to paths in groups of deferred paths as deferred paths. The status of unresolved paths is resolved at later nodes during the first pass. The second pass of the algorithm proceeds top-down and accepts one path from each group of deferred paths. 3.1. Paths in intermediate states. In the following we give explanations regarding the possible groups of paths in intermediate states. First, the algorithm will sometimes leave a single path p of P v in an undetermined state. If P v has only one equivalence class of paths, accepting a path might cause the algorithm to miss the chance of accepting two paths of smaller level (than v) later on. Hence, the algorithm could at best achieve a 2-approximation. Therefore, instead of accepting or rejecting the paths in P v right away, the algorithm picks one of them and makes it an undetermined path. All other paths in P v , if any, are rejected, and the undetermined path will be accepted or rejected at a later node. A second situation in which the algorithm does not accept or reject all paths in right away is sketched in Fig. 3.1. (Here and in the following, pairs of oppositely directed edges are drawn as undirected edges in all figures.) In this situation, the algorithm decides to accept one of several intersecting paths from P v , but it defers the decision which one of them to accept. The intersecting paths are called a group of deferred paths. All paths in a group of deferred paths use the same edge incident to v and to a child c of v. In the figure, this is the edge (c; v). (The case that the deferred paths share the edge (v; c) is symmetrical.) Furthermore, each deferred path uses also an edge (v; c 0 ) connecting v and a child c 0 6= c, and not all of the deferred EDGE-DISJOINT PATHS IN BIDIRECTED TREES 5 e c Fig. 3.1. A group of deferred paths. c c Fig. 3.2. Possible configuration of a group of exclusive paths (left-hand side), and situation in which both exclusive paths are blocked (right-hand side). paths use the same such edge. If the algorithm decides to create a new group of deferred paths, it marks the edge (c; v) as reserved (assuring that no path accepted at a node processed after v can use the edge), but leaves all edges (v; c 0 ) for children available. The reserved edge is indicated by a dashed arrow in Fig. 3.1. The motivation for introducing groups of deferred paths is as follows: first, the reserved edge blocks at most one path of smaller level that could be accepted in an optimal solution; second, no matter which path using the edge (p(v); v) is accepted at a node processed after v, that path uses at most one of the edges (v; c 0 ), and as there is still at least one deferred path that does not use that particular edge (v; c 0 ), the algorithm can pick such a deferred path in the second pass. When processing later nodes during the first pass, the algorithm actually treats the group of deferred paths like a single accepted path that uses only the reserved edge of the deferred paths. A group of exclusive paths is sketched in Fig. 3.2 (left-hand side). Such a group consists of one path q (called the lower path) contained in the subtree rooted at a child c of v and one path p (called the higher path) with lca v that intersects q. At most one of the two paths can be accepted, but if the algorithm picks the wrong one this choice can cause the algorithm to accept only one path while the optimal solution would accept the other path and one or two additional paths. Hence, the algorithm defers the decision which path to accept until a later node. For now, it only marks 6 T. ERLEBACH AND K. JANSEN Fig. 3.3. Group of 2-exclusive paths consisting of a pair of independent groups of exclusive paths. the top edge of path q that is intersected by p as fixed. (Fixed edges are indicated by dotted arrows in our figures.) Obviously, a group of exclusive paths has the following property. Property (E). If at most one path touching v but not using the fixed edge is accepted at a later node, either p or q can still be accepted. Only when two paths touching v are accepted at a later node, they can block p and q from being accepted. The right-hand side of Fig. 3.2 shows how two paths accepted at a later node can block both exclusive paths. While processing later nodes, the algorithm will try to avoid this whenever possible. The last types of unresolved paths are sketched in Figures 3.3 and 3.4. These groups of 2-exclusive paths consist of a set of four paths at most two of which can be accepted. More precisely, the first possibility for a group of 2-exclusive paths is to consist of two independent groups of exclusive paths (Fig. 3.3), i.e., of two groups of exclusive paths such that the fixed edge of one group is directed towards the root and the fixed edge of the other group is directed towards the leaves. Furthermore, the two groups must either be contained in disjoint subtrees (as shown in Fig. 3.3), or only their lower paths are contained in disjoint subtrees and their higher paths do not intersect each other. A pair of independent groups of exclusive paths has two fixed edges: the fixed edges of both groups. The second possibility for a group of 2-exclusive paths is to consist of a group of exclusive paths contained in a subtree rooted at a child of v and two paths with lca v that intersect the exclusive paths (but not their fixed edge) in a way such that accepting p 1 and p 2 would block both of the exclusive paths from being accepted (Fig. 3.4). Two edges are marked fixed, namely the top edge of the higher exclusive path intersected by a path with lca v and the top edge of the lower exclusive path intersected by a path with lca v. It is not difficult to show by case analysis that a group of 2-exclusive paths has the following property. Property (2E). If at most one path touching v but not using a fixed edge is accepted at a later node, two paths from the group of 2-exclusive paths can still be accepted. If two paths touching v but not using a fixed edge are accepted at a later node, at least one path from the group of 2-exclusive paths can still be accepted. While processing later nodes, the algorithm will try to avoid accepting two paths touching v such that only one path from the group of 2-exclusive paths can be accepted EDGE-DISJOINT PATHS IN BIDIRECTED TREES 7 vFig. 3.4. Further configurations of groups of 2-exclusive paths. 3.2. Invariants. In x3.4 we will present the details of how the algorithm proceeds during the first pass. At the same time, we will show that the approximation ratio achieved by the algorithm is 5=3 ". In order to establish this, we will prove by induction that the following invariants can be maintained. These invariants hold before the first node of T is processed, and they hold again each time an additional node of T has been processed. A node v is called a root of a processed subtree if the node v has already been processed but its parent has not. Invariant A. For every root v of a processed subtree, all paths in that subtree are accepted, rejected, or deferred except if one of the following cases occurs: (i) The subtree contains one undetermined path. All other paths contained in the subtree are accepted, rejected, or deferred. No edge in the subtree is marked fixed. (ii) The subtree contains one group of exclusive paths. All other paths contained in the subtree are accepted, rejected, or deferred. The only edge marked fixed in the subtree is the one from the group of exclusive paths. (iii) The subtree contains one group of 2-exclusive paths. All other paths contained in the subtree are accepted, rejected, or deferred. The only edges marked fixed in the subtree are the two from the group of 2-exclusive paths. All accepted paths are edge-disjoint and do not contain any reserved edges. Every unresolved path is edge-disjoint from all accepted paths and does not contain any reserved edges. Every deferred path contains exactly one reserved edge: the reserved edge of the group of deferred paths to which the path belongs. If a deferred path p intersects an accepted or unresolved path q, then the level of q is smaller than that of p. Invariant B. Let A be the set of all paths that have already been accepted by the algorithm. Let F be the set of all paths in P whose lca has not yet been processed and which are not blocked by any of the accepted paths, by reserved edges, or by fixed edges. Let d be the number of groups of deferred paths that are contained in processed subtrees. Let U be the set of all undetermined paths. Let X be the union of all groups of exclusive paths and groups of 2-exclusive paths. Then there is a subset O ' F [U [X of edge-disjoint paths satisfying the following conditions: (a) jP j - (5=3 (b) For every group of exclusive paths, O contains one path from that group; for every group of 2-exclusive paths, O contains two paths from that group. Intuitively, the set O represents a subset of P containing edge-disjoint paths that could still be accepted by the algorithm and that has the following property: if the algorithm accepts at least a 1=(5=3+ ")-fraction of the paths in O (in addition to the 8 T. ERLEBACH AND K. JANSEN paths it has already accepted), its output is a (5=3 + ")-approximation of the optimal solution. Observe that the invariants are satisfied initially with While it will be easy to see from the description of the algorithm that Invariant A is indeed maintained throughout the first pass, special care must be taken to prove that Invariant B is maintained as well. 3.3. The second pass. If the invariants are satisfied after the root node is processed, we have jOj. At this time, there can still be one undetermined path (which can, but need not be contained in O: therefore, jOj 2 f0; 1g in this case) one group of exclusive paths (from which O contains exactly one path, or one group of 2-exclusive paths (from which O contains two edge-disjopint paths, 2). If there is an undetermined path, the algorithm accepts it. If there is a group of exclusive paths, the algorithm accepts one of them arbitrarily. If there is a group of 2-exclusive paths, the algorithm accepts two edge-disjoint paths of them arbitrarily. The algorithm accepts at least jOj additional paths in this way, and the resulting set A 0 of accepted paths satisfies jA and, therefore, jP j - (5=3 In the second pass, the algorithm processes the nodes of the tree in reverse order, i.e., according to non-decreasing levels (top-down). At each node v that is the lca of at least one group of deferred paths, it accepts one path from each of the groups of deferred paths such that these paths are edge-disjoint from all previously accepted paths and from each other. This can always be done due to the definition of groups of deferred paths. Hence, the number of paths accepted by the algorithm increases by d in the second pass, and the set A 00 of paths that are accepted by the algorithm in the end satisfies jA 00 Theorem 1.1. 3.4. Details of the first pass. Assume that the algorithm is about to process node v. Recall that P v ' P is the set of all paths with lca v that do not intersect any previously accepted path nor any fixed or reserved edge. Let U v be the set of undetermined paths contained in subtrees rooted at children of v. Let X v be the set of all paths in groups of exclusive paths and groups of 2-exclusive paths contained in subtrees rooted at children of v. In the following, we explain how the algorithm processes node v and determines which of the paths in P v [U v [X v should be accepted, rejected, deferred, or left (or put) in an unresolved state. Observe that for a given set of paths with lca v the problem of determining a maximum-cardinality subset of edge-disjoint paths is equivalent to solving MEDP in a star and can thus be done in polynomial time by computing a maximum matching in a bipartite graph (cf. x2). Whenever we use an expression like compute a maximum number of edge-disjoint paths in S ' P v in the following, we imply that the computation should be carried out by employing this reduction to maximum matching. We will use the following property of bipartite graphs: for 2, the fact that a maximum matching in a bipartite graph G has cardinality s implies that there are s vertices in G such that every edge is incident to at least one of these s vertices. (The property holds for arbitrary values of s and is known as the K-onig theorem [23]; see, e.g., the book by Berge [4, pp. 132-133].) Observe that each child of the current node v is the root of a processed subtree, which can, by Invariant A, contain at most one of the following: one undetermined path, or one group of exclusive paths, or one group of 2-exclusive paths. Let k be the number of children of v that have an undetermined path in their subtree, let ' be EDGE-DISJOINT PATHS IN BIDIRECTED TREES 9 the number of children of v that have a group of exclusive paths, and let m be the number of children of v that have a group of 2-exclusive paths. We use the expression subtrees with exclusive paths to refer to all subtrees rooted at children of v with either a group of exclusive paths or with a group of 2-exclusive paths. Note that one main difficulty lies in determining which of the paths in U v [ should be accepted and which should be rejected. If k bounded by a constant, all possible combinations of accepting and rejecting paths in U v [X v can be tried out in polynomial time, but if k is large, the algorithm must proceed in a different way in order to make sufficiently good decisions. The exact threshold for determining when k considered large and, consequently, the running-time of the algorithm depend on the constant ". Let F , U , X , A and d denote the quantities defined in x3.2 at the instant just before the algorithm processes node v. Let F 0 , U 0 , X 0 , A 0 and d 0 denote the respective quantities right after node v is processed. Furthermore, denote by a v the number of paths that are newly accepted while processing v and by d v the number of groups of deferred paths that are newly created while processing v. We can assume that there is a set O ' F [U [X of edge-disjoint paths satisfying Conditions (a) and (b) of Invariant B before v is processed. In every single case of the following case analysis, we show how to construct a set O 0 that satisfies Invariant B after v is processed. O 0 is obtained from O by replacing paths, removing paths, or inserting paths as required. In particular, O 0 must be a set of edge-disjoint paths satisfying O 0 ' F 0 [U 0 [X 0 . Therefore, all paths intersecting a newly accepted path or the reserved edge of a newly created group of deferred paths must be removed from O. Note that at most two such paths can have smaller level than v, because all such paths of smaller level must use the edge (v; p(v)) or (p(v); v). Paths that are rejected by the algorithm must be removed or replaced in O. If a new group of exclusive paths or group of 2-exclusive paths is created, O 0 must contain one or two paths, respectively, from that group so that Condition (b) of Invariant B is maintained. Furthermore, we must ensure that jO 0 j is smaller than jOj by at most ( 5 As the value jAj+d increases by a v while v is processed (i.e., we have jA this implies that Condition (a) of Invariant B holds also after v is processed, i.e., Case 1. 2="g. The algorithm can try out all combinations of accepting or rejecting unresolved paths in the subtrees rooted at children of v: for undetermined paths there are two possibilities (accepting or rejecting the path), for groups of exclusive paths there are two possibilities (accepting the lower path or accepting the higher path), and for groups of 2-exclusive paths there are either four possibilities (in the case of a pair of independent groups of exclusive paths as shown in Fig. 3.3 on page accepting the lower or higher path in one group and the lower or higher path in the other group) or two relevant possibilities (in the cases shown in Fig. 3.4 on page 7: accepting the lower or higher path of the group of exclusive paths contained in the group of 2-exclusive paths and the edge-disjoint path among the remaining two paths; note that accepting no path of the group of exclusive paths and only the remaining two paths blocks more paths from F than any of the other two possibilities, hence we do not need to consider this third possibility) of accepting two edge-disjoint paths of the group. Hence, the number of possible combinations is bounded from above by 2 k+' 4 O(1). For each combination fl, the algorithm computes a maximum number s fl of edge-disjoint paths in P v not intersecting any of the u fl paths from U v [X v that are (tentatively) accepted for this combination. Let s be the maximum of u taken over all combinations fl. Note that s is the cardinality of a maximum-cardinality subset of edge-disjoint paths in and the algorithm does nothing and proceeds with the next node. Otherwise, we distinguish the following cases. Case 1.1. Case 1.1.1. has only one equivalence class of paths, pick one of them, say p, arbitrarily and make it an undetermined path. (Hence, and U Reject all other paths in P v . If O contains a path p 0 6= p from P v , replace p 0 by p in O to obtain O 0 (in order to ensure O O We have a j. Obviously, the invariants are satisfied. If P v has more than one equivalence class of paths, there must be an edge e incident to v that is shared by all paths in P v (as a consequence of the K-onig theorem). Make P v a group of deferred paths with reserved edge e. We have a O can contain at most one path intersecting edge e: either a path from P v or a path of smaller level. It suffices to remove this path from O in order to obtain a valid set O 0 , and we get and the invariants are satisfied. Case 1.1.2. There is one child c of v that has an undetermined path p with lca w in its subtree, possibly the algorithm does nothing and leaves p in its undetermined state. If P v 6= ;, all paths in P v must intersect p in the same edge, say in the edge (u; w) with (The case that they intersect p in an edge (w; u) is symmetrical.) The algorithm picks an arbitrary path q from P v and makes fp; qg a group of exclusive paths with fixed edge (u; w). (Hence, other paths in P v are rejected, and we have a must ensure that O 0 contains p or q in order to satisfy Condition (b) of Invariant B. If O does not contain any path from P v [ U v , by Property (E) either or q can be inserted into O after removing at most one path of smaller level. If O contains a path p 0 from P v [ U v already, this path can be replaced by p or q if and the invariants are satisfied. Case 1.1.3. 1. There is one child of v that has a group of exclusive paths in its subtree. As any path from P v could be combined with a path from the group of exclusive paths to obtain two edge-disjoint paths and because we have assumed must have Hence, the algorithm does nothing at node v and leaves the group of exclusive paths in its intermediate state. Case 1.2. 2. Observe that k 2. In many of the subcases of Case 1.2, the algorithm will yield a v 2. If O contains at most one path from removing that path and at most two paths of smaller level is clearly sufficient to obtain a valid set O 0 in such subcases. Therefore, we do not repeat this argument in every relevant subcase; instead, we discuss only the case that O contains two paths from Case 1.2.1. There is a subtree rooted at a child of v that contains a group of 2-exclusive paths. We must have any path in P v could be combined with two paths from X v to form a set of three edge-disjoint paths. Hence, the algorithm does nothing at node v and leaves the group of 2-exclusive paths in its unresolved state. Case 1.2.2. There are two children of v whose subtrees contain a group of exclusive paths. Note that this case, as any path from P v could be combined with one exclusive path from each subtree to obtain a set of three edge-disjoint paths. EDGE-DISJOINT PATHS IN BIDIRECTED TREES 11 e Fig. 3.5. Case 1.2.3.1: Pv contains two edge-disjoint paths (left-hand side); Case 1.2.3.2 (a): The fixed edge and e have the same direction (right-hand side). If the fixed edges of both groups of exclusive paths point in the same direction (i.e., are both directed to the root or to the leaves), the algorithm accepts the lower paths of both groups of exclusive paths. The higher paths are rejected, and no edge is marked fixed anymore. We have a 0, and at most three paths must be removed from O to obtain a valid set O 0 : the two paths from the groups of exclusive paths that are contained in O, and at most one path of smaller level using the edge between v and p(v) whose direction is opposite to the direction of the formerly fixed edges. If the fixed edges of the groups of exclusive paths point in different directions (i.e., one is directed towards the root and one towards the leaves), the groups represent a pair of independent groups of exclusive paths, and the algorithm can create a new group of 2-exclusive paths. Note that O contains two paths from the new group of 2-exclusive paths already, because it contained one path from each of the two groups of exclusive paths in X v due to Condition (b) of Invariant B. Therefore, we can set O and the invariants are satisfied. Case 1.2.3. There is one child of v that has a group of exclusive paths in its subtree and one child of v that has an undetermined path in its subtree. All paths in P v must intersect the undetermined path, because otherwise a path from P v could be combined with the undetermined path and an exclusive path to obtain a set of three edge-disjoint paths. Case 1.2.3.1. There are two edge-disjoint paths in P v . In this case, the situation must be as shown on the left-hand side of Fig. 3.5: the two edge-disjoint paths from must intersect the group of exclusive paths in a way that blocks all exclusive paths from being accepted, and there cannot be any other kinds of paths in P v . The algorithm accepts the lower path from the group of exclusive paths and the undetermined path, and it rejects all other paths in marked fixed anymore. We have a Note that any combination of two edge-disjoint paths from P v [U v [X v blocks at least three of the four top edges of the paths accepted by the algorithm. Hence, if O contains two paths from it can contain at most one path of smaller level intersecting the paths accepted by the algorithm, and it suffices to remove at most three paths from O to obtain a valid e e Fig. 3.6. Cases 1.2.3.2 (b) and 1.2.3.2 (c): The fixed edge and e have different directions. Case 1.2.3.2. All paths in P v intersect the same edge e of the undetermined path. Case 1.2.3.2 (a). The direction of e is the same as that of the fixed edge of the group of exclusive paths (see the right-hand side of Fig. 3.5). The algorithm accepts the undetermined path and the lower path from the group of exclusive paths. All other paths in are rejected, and no edge is marked fixed anymore. We have a O contains two paths from P v [ use the fixed edge and the edge e, and at most one further path from O can be blocked by the paths accepted by the algorithm (because such a path must use the edge between v and p(v) in the direction opposite to the direction of e). Thus, it suffices to remove at most three paths from O to obtain a valid set O 0 . Case 1.2.3.2 (b). The direction of e is different from that of the fixed edge, and there is a path that does not intersect the higher exclusive path (see the left-hand side of Fig. 3.6). The algorithm uses X v , p and the undetermined path together to create a new group of 2-exclusive paths consisting of a pair of independent groups of exclusive paths. All other paths is P v are rejected by the algorithm. In addition to the fixed edge of the old group of exclusive paths, the edge e is marked fixed. Note that O contains one path from X v due to Condition (b) of Invariant B. If O contains the undetermined path or the path p, let O O contains a path other than p from P v , replace this path either by p or by the undetermined path (one of these must be possible). If O does not contain a path from P v [ U v but contains a path p 0 using the edge between v and p(v) in the direction given by edge e, replace p 0 either by p or by the undetermined path (one of the two must be possible). If O does not contain a path from P v [ U v and no path using the edge between v and p(v) in the direction given by edge e, add either p or the undetermined path to O. In any case, the invariants are satisfied. In particular, jO Case 1.2.3.2 (c). The direction of e is different from that of the fixed edge, and all paths in P v intersect the higher exclusive path (see the right-hand side of Fig. 3.6). The algorithm accepts the undetermined path and the lower path from the group of exclusive paths, and it rejects all other paths from marked fixed anymore. We have a O contains two paths from must contain at least one of the two paths accepted by the algorithm, and the other path in O uses a top edge of the other path accepted by the algorithm. O contains at most one path of smaller level intersecting the paths accepted by the algorithm, and it suffices to remove at most three paths from O in order to obtain a valid set O 0 . EDGE-DISJOINT PATHS IN BIDIRECTED TREES 13epe pp c c" c' pp p' c c" c' Fig. 3.7. Case 1.2.4.1: Pv contains two edge-disjoint paths that block the exclusive paths. Case 1.2.4. There is one child c of v that has a group of exclusive paths in its subtree. Denote the higher and the lower path in the group of exclusive paths by p and q, respectively. Assume without loss of generality that the fixed edge e 0 of the group of exclusive paths is directed towards the root of the tree (as shown in Fig. 3.7). Note that We distinguish further cases regarding the maximum number of edge-disjoint paths in P v . Case 1.2.4.1. There are two edge-disjoint paths p 1 and p 2 in P v . As must intersect the exclusive paths in a way that blocks all of them from being accepted. See Fig. 3.7. Let p 1 intersect p, and let p 2 intersect q. Let c 0 6= c be the child of v such that p 1 uses the edges (c; v) and (v; c 0 ), and let c 00 6= c be the child of v such that p 2 uses the edges (c 00 ; v) and (v; c). Note that c the top edge of p intersected by p 1 be e 1 , and let the top edge of q intersected by p 2 be e 2 . As contains only two edge-disjoint paths, every path p must either intersect edge e 1 , or intersect edge e 2 , or intersect both p 1 and p 2 . (The latter case is possible only if c 0 6= c 00 and if all paths in P v that intersect e 1 use the edges (c; v) and (v; c 0 ) and all paths in P v that intersect e 2 use the edges (c 00 ; v) and (v; c); in that case, p 0 must use as shown on the right-hand side of Fig. 3.7.) Case 1.2.4.1 (a). All paths in P v that intersect e 1 use the edges (c; v) and (v; c 0 ), and all paths in P v that intersect e 2 use the edges First, assume that all paths in P v intersect either e 1 or e 2 . Note that there are exactly two equivalence classes of paths in P v in this case. See Fig. 3.7 (left-hand side). The algorithm uses the group of exclusive paths and one representative from each of the two equivalence classes of paths in P v to create a group of 2-exclusive paths. All other paths in P v are rejected. The fixed edge e 0 of the group of exclusive paths is no longer marked fixed, instead the edges e 1 and e 2 are marked fixed. If O contains two paths from P v [X v , one of them must be from X v due to Condition (b) of Invariant B and the other can be replaced by a path in the new group of 2-exclusive paths. Otherwise, it is possible to remove the path from X v and at most one additional path from O such that the resulting set contains no path from P v [ X v , at most one 14 T. ERLEBACH AND K. JANSEN path of smaller level touching v, and no path of smaller level intersecting a fixed edge of the new group of 2-exclusive paths. By Property (2E), two paths from the new group of 2-exclusive paths can then be inserted into that set to obtain O 0 . We have and the invariants are satisfied. Now, assume that there is a path p 0 2 P v that intersects neither e 1 nor e 2 . As noted above, we must have c 0 6= c 00 in this case, and p 0 must use the edges Fig. 3.7 (right-hand side). The algorithm accepts the lower path from the group of exclusive paths and the path p 0 , and it rejects all other paths in No edge is marked fixed anymore. We have a Note that any combination of two edge-disjoint paths from blocks at least three of the four top edges of the paths accepted by the algorithm. Hence, if O contains two paths from can contain at most one path of smaller level intersecting the paths accepted by the algorithm, and it suffices to remove at most three paths from O to obtain a valid set O 0 . Case 1.2.4.1 (b). There are at least two equivalence classes of paths in P v intersecting the higher path of the group of exclusive paths. The algorithm accepts the lower path of the group of exclusive paths and makes the paths in P v intersecting the higher path a group of deferred paths. All other paths in P v [X v are rejected, and no edge is marked fixed anymore. The reserved edge of the group of deferred paths is the top edge shared by all these paths. If O contains two paths from P v [ X v , note that one of the two paths must be from X v (due to Condition (b) of Invariant B) and that these two paths also block the top edges of the lower path of the group of exclusive paths. Hence, O cannot contain any path of smaller level intersecting the lower path, and it can contain at most one path of smaller level intersecting the reserved edge of the newly deferred paths. It suffices to remove at most three paths from O to obtain a valid set O 0 . Case 1.2.4.1 (c). There is only one equivalence class of paths in P v intersecting the higher path of the group of exclusive paths, and there are at least two equivalence classes of paths in P v intersecting the lower path of the group of exclusive paths. The algorithm accepts the higher path of the group of exclusive paths and makes the paths in P v intersecting the lower path a group of deferred paths. All other paths in are rejected, and no edge is marked fixed anymore. The reserved edge of the group of deferred paths is the top edge shared by all these paths. If O contains two paths from note that one of the two paths must be from X v (due to Condition (b) of Invariant B) and that these two paths also block edge e 1 . Hence, O cannot contain any path of smaller level intersecting e 1 , and it can contain at most one path of smaller level intersecting the reserved edge of the newly deferred paths or the top edge of the higher path that is directed towards the leaves, because all such paths must use the edge (p(v); v). It suffices to remove at most three paths from O to obtain a valid set O 0 . Case 1.2.4.2. P v does not contain two edge-disjoint paths. Let e be an edge incident to v such that all paths in P v use edge e. Case 1.2.4.2 (a). has at least two different equivalence classes of paths. The algorithm makes all paths in P v a new group of deferred paths with reserved edge e and accepts q, the lower path of the group of exclusive paths. Path p is rejected, and no edge in this subtree is marked fixed anymore. We have a O contains two paths from P v [ X v , these paths block two of the three top edges blocked by the algorithm: the fixed edge e 0 of the group of exclusive paths and edge e. O can contain at most one path of smaller level Fig. 3.8. Case 1.2.5 (a): All sets of two edge-disjoint paths use the same four top edges (left- hand side); Case 1.2.5 (b): there is only one equivalence class of paths using edge e 1 , but more than one class using edge e2 (right-hand side). that intersects the path accepted by the algorithm or the reserved edge of the new group of deferred paths, and it suffices to remove at most three paths from O to obtain a valid set O 0 . Case 1.2.4.2 (b). e 6= (v; c 0 ) for all children c 0 6= c of v, or P v has only one equivalence class of paths. If there is a path p that does not intersect q, the algorithm accepts p 0 and q. If all paths in P v intersect q, the algorithm accepts p and an arbitrary path from P v . In both cases, all other paths in P v [X v are rejected, and no edge in this subtree is marked fixed anymore. We have d 2. Assume that O contains two paths from P v [ X v . We will show that it suffices to remove at most three paths from O to obtain a valid set O 0 . If the algorithm has accepted p, O must also contain p and a path from P v , thus blocking at least three of the four top edges of the paths accepted by the algorithm. At most one further path in O can be blocked by the paths accepted by the algorithm. Now assume that the algorithm has accepted q. Observe that the two paths from that are in O must also use the edges e 0 and e, thus blocking two of the four top edges of paths accepted by the algorithm. If e and e 0 have the same direction, O can contain at most one path of smaller level intersecting the paths accepted by the algorithm, because such a path must use the edge (p(v); v). If P v has only one equivalence class of paths, the paths from P v [X v that are in O block three of the four top edges of paths accepted by the algorithm, and again it suffices to remove at most one path of smaller level from O. Finally, consider the case that P v has more than one equivalence class of paths and that e = (v; c). Since edge e blocks more paths of smaller level than the top edge of q that is directed towards the leaves, the two paths from P v [X v that are in O do in fact block at least as many paths of smaller level as three of the four top edges of the paths accepted by the algorithm. Case 1.2.5. there must be two edges incident to v such that all paths in P v use at least one of these two edges (by the K-onig theorem). Let e 1 and e 2 be two such edges. Case 1.2.5 (a). All possible sets of two edge-disjoint paths from P v use the same four edges incident to v. See the left-hand side of Fig. 3.8 for an example. The algorithm picks two arbitrary edge-disjoint paths from P v , accepts them, and rejects all other paths from P v . We have a O contains two paths from P v , removing these two paths is sufficient to obtain a valid set O 0 , because they use the same top edges as the paths accepted by the algorithm and O cannot contain any further path intersecting the paths accepted by the algorithm. In the following, let D be the set of paths in P v that intersect all other paths from P v . In other words, a path p 2 P v is in D if P v does not contain a path q that is edge-disjoint from p. Note that if Case 1.2.5 (a) does not apply, it follows that either the paths in P v n D using edge e 1 or those using edge e 2 must have more than one Fig. 3.9. Case 1.2.5 (c): Configurations in which two groups of deferred paths can be created. equivalence class of paths. Case 1.2.5 (b). There is only one equivalence class C of paths in P v n D using more than one equivalence class of paths in P v n D using edge e 2 and not intersecting a path from C. See the right-hand side of Fig. 3.8. (The case with e 1 and e 2 exchanged is symmetrical. Furthermore, note that the case that there is only one equivalence class C of paths in P v n D using edge e 1 and only one equivalence class of paths in P v n D using edge e 2 and not intersecting a path from C satisfies the condition of Case 1.2.5 (a).) The algorithm picks a path p from C arbitrarily, accepts p, and makes the paths using edge e 2 and not intersecting p a group of deferred paths with reserved edge e 2 . All other paths in P v are rejected. We have a O contains two paths from P v , these paths must also use both top edges of p and the newly reserved edge, and thus removing these two paths from O is sufficient to obtain a valid set O 0 . Case 1.2.5 (c). There is more than one equivalence class of paths in P v n D using there is more than one equivalence class of paths in P v n D using edge e 2 , and Case 1.2.5 (a) does not apply. The algorithm makes the paths in P v n D using e 1 a group of deferred paths with reserved edge e 1 and the paths in P v n D using e 2 a group of deferred paths with reserved edge e 2 . All other paths in P v are rejected. Note that no matter which paths of smaller level are accepted by the algorithm later on, there are still two paths, one in each of the two groups of newly deferred paths, that are edge-disjoint from these paths of smaller level and from each other. (Otherwise, Case 1.2.5 (a) would apply.) We have a 2. If O contains two paths from P v , these paths use e 1 and e 2 as well, and removing these two paths from O is sufficient to obtain a valid set O 0 , because O cannot contain any further path intersecting a reserved edge of the newly deferred paths. Case 1.2.6. There is one child of v that has an undetermined path p in its subtree. Let P 0 denote the set of paths in P v that do not intersect p. We begin by making some simple observations. First, P 0 v must not contain two edge-disjoint paths. Hence, there must be an edge e incident to v that is shared by all paths in P 0 v . Second, implies that the maximum number of edge-disjoint paths in P v is at most two. So there must be two edges e 1 and e 2 incident to v such that every path in P v uses at least one of these two edges. Let the lca of the undetermined path be v 0 , and let c be the child of v whose subtree contains the undetermined path (possibly of v 0 such that the undetermined path uses the edges (v 1 a number of subcases regarding the number of equivalence classes in P 0 v . Case 1.2.6 (a). P 0 v is empty. Let P 1 and P 2 denote the sets of paths in P v that intersect p in the edge (v in the edge (v respectively. Note that the algorithm accepts an arbitrary path from P i if P i has only one equivalence class of paths and creates a new group of deferred paths from P i otherwise. The undetermined path p is rejected. We have EDGE-DISJOINT PATHS IN BIDIRECTED TREES 17 v' v' v' v" v" Fig. 3.10. Case 1.2.7: v has two children with undetermined paths in their subtrees. 2. If O contains two paths from P v [ U v , removing these two paths is sufficient, because they block at least as many paths of smaller level as the newly accepted paths or newly reserved edges. Case 1.2.6 (b). P 0 v has one equivalence class of paths. The algorithm accepts an arbitrary path from P 0 v and the undetermined path p. All other paths in P v are rejected. We have a Assume that O contains two paths from P v [U v . If O contains p, O must also contain a path from P 0 v , and it suffices to remove these two paths from O to obtain a valid set O 0 . If O does not contain p but contains a path from P 0 must also contain a path from P v that intersects p; these two paths block at least three of the four top edges blocked by the algorithm, and it suffices to remove these two paths and at most one path of smaller level. Finally, if O contains neither p nor a path from P 0 must contain two paths from P v that intersect p in different top edges and at least one of which intersects also a top edge of the paths in suffices to remove at most three paths from O to obtain a valid set O 0 . Case 1.2.6 (c). P 0 v has more than one equivalence class of paths. Let e be the edge incident to v that is shared by all paths in P 0 v . The algorithm accepts the undetermined path p and creates a new group of deferred paths from the paths in . All other paths in P v are rejected. We have a 1. Assume that O contains two paths from P v [ U v . If O contains p, O must also contain a path from v , and it suffices to remove these two paths from O to obtain a valid set O 0 . If O does not contain p but contains a path from P 0 must contain a path from P v that intersects these two paths block at least two of the three top edges blocked by the algorithm, and it suffices to remove these two paths and at most one path of smaller level. Finally, if O contains neither p nor a path from P 0 must contain two paths from P v that intersect p in different top edges; again, these two paths block at least two of the three top edges blocked by the algorithm, and it suffices to remove at most three paths from O to obtain a valid set O 0 . Case 1.2.7. Two children of v have undetermined paths in their subtrees. Denote the undetermined paths by p and q. See Fig. 3.10. As every path in P v must intersect at least one undetermined path. In addition, if there are two paths in P v that intersect one undetermined path in different top edges, at least one of them must also intersect the other undetermined path. Let P 1 and P 2 denote the sets of paths in P v that intersect p and q, respectively. Note that Case 1.2.7 (a). There are edge-disjoint paths p 1 and p 2 in P v such that p 1 intersects p in a top edge e 1 but does not intersect q, and p 2 intersects q in a top edge e 2 but does not intersect p, and such that e 1 and e 2 have different directions (i.e., one is directed towards the root, and the other is directed towards the leaves). The algorithm makes p, q, p 1 and p 2 a group of 2-exclusive paths consisting of a pair of independent groups of exclusive paths and rejects all other paths from P v . The edges e 1 and e 2 are marked fixed. If O contains two paths from the new group of 2-exclusive paths already, let O Otherwise, it is possible to replace paths in O by paths from the new group of 2-exclusive paths to obtain O 0 . In any case, jO Case 1.2.7 (b). If the condition for Case 1.2.7 (a) does not hold, the algorithm accepts p and q and rejects all paths from P v . We have a that O contains two paths from P v [ U v . If O contains p and q, it suffices to remove these two paths. If O contains only one of p and q, say p, it must contain a path from P v that intersects q, and these two paths block three of the four top edges blocked by the algorithm. If O contains neither p nor q, it must contain two paths from P v . If at least one of these two paths in O intersects both p and q, these two paths again block at least three of the four top edges blocked by the algorithm. If both paths in O intersect only one of p and q, it must be the case that one of them intersects p in an edge e 1 and one of them intersects q in an edge e 2 . If e 1 and e 2 have the same direction, O can contain at most one path of smaller level intersecting a path accepted by the algorithm. If e 1 and e 2 have different directions, the condition of Case 1.2.7 (a) applies. Case 1.3. s - 3. The algorithm accepts the s paths and rejects all other paths from this subtree is marked fixed anymore. As s is the maximum number of edge-disjoint paths in can contain at most s paths from P v [U v [X v . Furthermore, O can contain at most two paths from F using the edges (v; p(v)) or (p(v); v), and these are the only two further paths in O that could possibly be blocked by the s paths accepted by the algorithm. Hence, a valid set O 0 can be obtained from O by deleting at most s paths. As s the invariants are maintained. Case 2. 2="g. In this case, the algorithm cannot try out all possibilities of accepting or rejecting unresolved paths in polynomial time. Instead, it calculates only four candidate sets of edge-disjoint paths from chooses the largest of them. For obtaining two of the four sets, we employ a method of removing paths from an arbitrary set S of edge-disjoint paths in P v such that ' exclusive paths from X v can be accepted in addition to the paths remaining in S. The resulting set of edge-disjoint paths in S [X v has cardinality jSj where r is the number of paths that were removed from S. The details of the method and a proof that will be presented later in Lemma 3.1. With this tool we are ready to describe the candidate sets S 1 be the subset of paths in P v that do not intersect any undetermined path in U v . 1. Compute a maximum number s 1 of edge-disjoint paths in P 0 v . S 1 is obtained by taking these paths, all k undetermined paths, and as many additional edge-disjoint paths from X v as possible. We have undetermined paths and at least m paths from groups of 2-exclusive paths in X v due to Property (2E). 2. S 2 is obtained from S 1 by removing r of the s 1 paths in S such that ' +2m exclusive paths can be accepted. S 2 contains ' +2m exclusive paths, and according to Lemma 3.1 only r - m)=3 of the s 1 paths in S were removed to obtain S 2 . As S 2 still contains the k undetermined paths, we have In addition, we have jS EDGE-DISJOINT PATHS IN BIDIRECTED TREES 19 because S 2 contains all k undetermined paths from U v and ' exclusive paths. 3. S 3 is obtained by first computing a maximum number s 3 of edge-disjoint paths in P v and then adding as many edge-disjoint paths from X v [ U v as possible. We have jS 3 j - s 3 +m, because S 3 contains at least m paths from groups of 2-exclusive paths in X v due to Property (2E). 4. S 4 is obtained from S 3 by removing r of the s 3 paths in S 3 " P v from S 3 such that '+2m exclusive paths can be accepted, in the same way as S 2 is obtained from S 1 . according to Lemma 3.1, we have jS 4 j - m+(2=3)(s 3 +'+m). The algorithm accepts the paths in that set S i with maximum cardinality and rejects all other paths from P v [U v [X v . We have a Note that a v - jS 2 j - maxf3; 2="g and that this implies 2 - "a v . Let O be the number of paths from P v that are contained in O v and that intersect at least one of the k undetermined paths. Observe that O v can contain at most k \Gamma b 0 =2 undetermined paths from U v . Note that the maximum number of edge-disjoint paths in P v is s 3 and that the maximum number of edge-disjoint paths in P 0 v is s 1 . Using jO and using jO With this upper bound on jO v j and the lower bounds on the cardinalities of the four sets S i , we can now prove that at least one of the sets S i satisfies jO it suffices to remove at most jO paths from O in order to obtain a valid set O 0 , this implies that the invariants are maintained. If we have jO the following cases. Case 2.1. ff ? 3=2. If ' we have a v - jS 3 use (3.1) and a v - jS 4 j to bound the ratio between jO a v a v Case 2.2. we have a v - jS 3 a use (3.1) and a v - jS 2 j to bound the ratio between jO a v a v 20 T. ERLEBACH AND K. JANSEN c e a d Fig. 3.11. Set of edge-disjoint paths in Pv . Case 2.3. ff - 4=3. From (3.1) we get jO 2m, and we have a We have shown that jO holds in all subcases of Case 2. To complete the description of Case 2, we still have to explain the method for removing paths from S 1 and S 3 in order to obtain S 2 and S 4 , respectively. The method takes an arbitrary set S of edge-disjoint paths in P v and removes paths from S to obtain a set S 0 such that every subtree with exclusive paths is touched by at most one path in S 0 . The motivation for this is that S can cause all paths from a group of exclusive paths to be blocked only if two paths from S intersect the corresponding subtree (Property (E)). Similarly, if only one path from a group of 2-exclusive paths can be accepted, S must contain two paths from P v that intersect the corresponding subtree (Property (2E)). The method proceeds as follows. Consider a graph G with the paths in S as its vertices and an edge between two paths if they touch the same child of v. G has maximum degree two and consists of a collection of chains and cycles. Note that every edge of G corresponds to a child of v that is touched by two paths in S. We are interested in the maximal parts of chains and cycles that consist entirely of edges corresponding to children of v that are the roots of subtrees with exclusive paths. There are the following possibilities for such parts: (i) A cycle such that all paths on the cycle have both endpoints in a subtree with exclusive paths. (ii) A chain such that the paths at both ends have only one endpoint in a subtree with exclusive paths, while the internal paths have both endpoints in subtrees with exclusive paths. (iii) A chain such that the path at one end has only one endpoint in a subtree with exclusive paths, while all other paths have both endpoints in a subtree with exclusive paths. (iv) A chain such that all its paths have both endpoints in a subtree with exclusive paths. Note that every such maximal part of a cycle or chain has length (number of paths) at least two, because it contains at least one edge. The method for removing paths proceeds as follows. Cycles of even length and chains are handled by removing every other path from S, starting with the second path for chains. Cycles of odd length are handled by removing two consecutive paths in one place and every other path from the rest of the cycle. Consider the example depicted in Fig. 3.11. The node v has eight children, named a to h, and six of them (c to h) are roots of subtrees with exclusive paths (indicated EDGE-DISJOINT PATHS IN BIDIRECTED TREES 21 e-a g-h a-d d-c h-f f-g c-b Fig. 3.12. Graph G representing the structure of the paths. by an exclamation mark). A set S of edge-disjoint paths in P v is sketched. The graph G obtained from this set is shown in Fig. 3.12, and the label of a vertex in G is u-w if the corresponding path begins in the subtree rooted at u and ends in the subtree rooted at w. With respect to (i)-(iv) above, G contains a cycle of type (i) with length three (containing the paths f -g, g-h, and h-f) and a chain of type (ii) with length three (containing the paths a-d, d-c, and c-b). According to the rules given above, three paths would be removed from S: two paths, say f -g and g-h, from the cycle, and the path d-c from the chain of length three. It is easy to see that this process always ensures that in the end S contains, for each subtree with exclusive paths, at most one path with an endpoint in that subtree. Hence, due to Properties (E) and (2E), S can be filled up with edge-disjoint exclusive paths until it contains all exclusive paths. Lemma 3.1. Let v be a node with '+m children with exclusive paths. Let S ' P v be a set of edge-disjoint paths. Let S 0 ' S be the set of paths obtained from S by removing paths according to the method described above. Let Proof. Let a be the number of cycles of type (i), and let a i , 1 - i - a, be the length of the ith cycle. Denote the number of chains of type (ii) by b and their lengths by b i , 1 - i - b. Denote the number of chains of type (iii) by c and their lengths by c i , Denote the number of chains of type (iv) by d and their lengths by d d. Note that a As the number of paths contained in the union of all these chains and cycles is at most s, we have P a Furthermore, considering the number of children with exclusive paths covered by each chain or cycle, we obtain P a in the latter inequality and adding up the two inequalities, we obtain P a Taking into account that P a \Sigma a i\Upsilon c and that da i =2e - (2=3)a i for a i - 2, the lemma follows. In the example displayed in Fig. 3.11, we had sufficient to remove 3 . 3.5. Running-time of the algorithm. The running-time of our algorithm is polynomial in the size of the input for fixed " ? 0, but exponential in 1=". Let a bidirected tree E) with n nodes and a set P containing h directed paths in T (each path specified by its endpoints) be given. For arbitrary " ? 0, we claim that our approximation algorithm can be implemented to run in time O The details of the implementation as well as experimental results will be reported in [9]. 22 T. ERLEBACH AND K. JANSEN Note that we can choose n) and still achieve running-time polynomial in the size of the input. The resulting algorithm achieves approximation ratio therefore, asymptotic approximation ratio 5=3. (If the optimal solution contains many paths, n must also be large, and the approximation ratio gets arbitrarily close to 5=3.) 4. Generalizations. There are several generalizations of MEDP. First, it is meaningful to consider the weighted version of the problem, where each path has a certain weight and the goal is to maximize the total weight of the accepted paths. The weighted version of MEDP can still be solved optimally in polynomial time in bidirected stars and spiders (by reduction to maximum-weight matching in a bipartite graph) and in bidirected trees of bounded degree (by a minor modification of the dynamic programming procedure given in x2). Another generalization of MEDP is the MaxPC problem. For a given bidirected tree set P of directed paths in T , and number W of colors, the maximum path coloring (MaxPC) problem is to compute a subset P 0 ' P and a W -coloring of P 0 . The goal is to maximize the cardinality of P 0 . The MaxPC problem is equivalent to finding a maximum (induced) W -colorable subgraph in the conflict graph of the given paths. Studying MaxPC is motivated by the admission control problem in all-optical WDM (wavelength-division multiplexing) networks without wavelength converters: every wavelength (color) can be used to establish a set of connections provided that the paths corresponding to the connections are edge-disjoint, and the number of available wavelengths is limited [5]. The weighted variant of MaxPC is interesting as well. MaxPC and weighted MaxPC can both be solved optimally in polynomial time for bidirected stars by using an algorithm for (the weighted version of) the capacitated b-matching problem [15, pp. 257-259]. If the number W of colors and the maximum degree of the bidirected tree are both bounded by constants, MaxPC and weighted MaxPC can be solved optimally in polynomial time by dynamic programming (similar to the procedure in x2). MaxPC is NP-hard for arbitrary W in bidirected binary trees (because path coloring is NP-hard) and for bidirected trees of arbitrary degree (because it is equivalent to MEDP in this case). In order to obtain approximation algorithms for MaxPC with arbitrary number W of colors, a technique due to Awerbuch et al. [1] can be employed. It allows reducing the problem with W colors to MEDP with only a small increase in the approximation ratio. The technique works for MaxPC in arbitrary graphs G; we discuss it here only for trees. Let an instance of MaxPC be given by a bidirected tree set P of paths in T , and a number W of colors. An approximation algorithm A for arbitrary number W of colors is obtained from an approximation algorithm A 1 for one color (i.e., for the maximum edge-disjoint paths problem) by running W copies of A 1 , giving as input to the ith copy the bidirected tree T and the set of paths that have not been accepted by the first copies of A 1 (see Fig. 4.1). The output of A is the union of the W sets of paths output by the copies of A 1 , and the paths in the ith set are assigned color i. In [1] it is shown that the algorithm A obtained using this technique has approximation ratio at most ae +1 if A 1 has approximation ratio ae, even if different colors are associated with different network topologies. For identical networks, which we have in our application, the approximation ratio achieved by A can even be bounded by aeW which is smaller than 1=(1 \Gamma e \Gamma1=ae ) for all W . This bound is mentioned in the journal version of [1] and can be viewed as an adaptation of a similar result in [6]. It can be proved easily using the fact that if A has selected p k paths after Algorithm A Input: bidirected tree T , set P of paths, number W of colors Output: disjoint subsets P 1 ,. ,P W of P (each P i is edge-disjoint) begin to W do begin Fig. 4.1. Reduction from many colors to one color. running k copies of A 1 , there is still a set of at least (jP among the remaining paths and the next copy of A 1 accepts at least a (1=ae)-fraction of this number. The reduction works also for the weighted case. Since we have an optimal algorithm for MEDP in bidirected trees of bounded degree and (5=3 ")-approximation algorithms for MEDP in arbitrary bidirected trees, we can employ the above technique and obtain approximation algorithms with bidirected trees of bounded degree and with ratio approximately 2:22 for MaxPC in arbitrary bidirected trees. Acknowledgments . The authors are grateful to Stefano Leonardi for pointing out the reduction from MaxPC with arbitrary number of colors to MaxPC with one color and to Adi Ros'en for informing them about the improved analysis for the ratio obtained by this reduction in the case of identical networks for all colors and for supplying a preliminary draft of the journal version of [1]. --R Competitive non Graphs and Hypergraphs Special issue on Dense Wavelength Division Multiplexing Techniques for High Capacity and Multiple Access Communication Systems Location of bank accounts to optimize float: An analytic study of exact and approximate algorithms Structure in approximation classes Call scheduling in trees Optimal wavelength routing on directed fiber trees An optimal greedy algorithm for wavelength allocation in directed tree networks Colouring paths in directed symmetric trees with applications to WDM routing Efficient wavelength routing on directed fiber trees Maximum bounded 3-dimensional matching is MAX SNP-complete Approximation algorithms for disjoint paths problems Approximating disjoint-path problems using greedy algorithms and packing integer programs A note on optical routing on trees Improved access to optical bandwidth in trees Efficient access to optical bandwidth Computational Complexity Improved approximations for edge-disjoint paths --TR --CTR Thomas Erlebach , Klaus Jansen, Implementation of approximation algorithms for weighted and unweighted edge-disjoint paths in bidirected trees, Journal of Experimental Algorithmics (JEA), 7, p.6, 2002 R. Sai Anand , Thomas Erlebach , Alexander Hall , Stamatis Stefanakos, Call control with k rejections, Journal of Computer and System Sciences, v.67 n.4, p.707-722, December Thomas Erlebach , Klaus Jansen, Conversion of coloring algorithms into maximum weight independent set algorithms, Discrete Applied Mathematics, v.148 n.1, p.107-125,

approximation algorithms;bidirected trees;edge-disjoint paths

587948

Improved Algorithms and Analysis for Secretary Problems and Generalizations.

In the classical secretary problem, n objects from an ordered set arrive in random order, and one has to accept k of them so that the final decision about each object is made only on the basis of its rank relative to the ones already seen. Variants of the problem depend on the goal: either maximize the probability of accepting the best k objects, or minimize the expectation of the sum of the ranks (or powers of ranks) of the accepted objects. The problem and its generalizations are at the core of tasks with a large data set, in which it may be impractical to backtrack and select previous choices.Optimal algorithms for the special case of are well known. Partial solutions for the first variant with general k are also known. In contrast, an explicit solution for the second variant with general k has not been known. It seems that the fact that the expected sum of powers of the ranks of selected items is bounded as n tends to infinity has been known to follow from standard results. We derive our results by obtaining explicit algorithms. For each $z \geq 1$, the resulting expected sum of the zth powers of the ranks of the selected objects is at most $k^{z the best possible value at all is kz O(kz). Our methods are very intuitive and apply to some generalizations. We also derive a lower bound on the trade-off between the probability of selecting the best object and the expected rank of the selected object.

Introduction In the classical secretary problem, n items or options are presented one by one in random order (i.e., all n! possible orders being equally likely). If we could observe them all, we could rank them totally with no ties, from best (rank 1) to worst (rank n). However, when the ith object appears, we can observe only its rank relative to the previous the relative rank is equal to one plus the number of the predecessors of i which are preferred to i. We must accept or reject each object, irrevocably, on the basis of its rank relative to the objects already seen, and we are required to select k objects. The problem has two main variants. In the first, the goal is to maximize the probability of obtaining the best k objects. In the second, the goal is to minimize the expectation of the sum of the ranks of the selected objects or, more generally, for a given positive integer z, minimize the expectation of the sum of the zth powers of the ranks. Solutions to the classical problem apply also in variety of more general situations. Examples include (i) the case where objects are drawn from some probability distribution; the interesting feature of this variant is that the decisions of the algorithms may be based not only on the relative rank of the item but also on an absolute "grade" that the item receives, (ii) the number of objects is not known in advance, (iii) objects arrive at random times, (iv) some limited back-tracking is allowed: objects that were rejected may be recalled, (v) the acceptance algorithm has limited memory, and also combinations of these situations. In addition to providing intuition and upper and lower bounds for the above important generalizations of the problem, solutions to the classical problem also provide in many cases very good approximations, or even exact solutions (see [4, 13, 14] for survey and also [8]). Our methods can also be directly extended to apply for these generalizations. The obvious application to choosing a best applicant for a job gives the problem its common name, although the problem (and our results) has a number of other applications in computer science. For any problem with a very large data set, it may be impractical to backtrack and select previous choices. For example, in the context of data mining, selecting records with best fit to requirements, or retrieving images from digital libraries. In such applications limited backtracking may be possible, and in fact this is one of the generalizations mentioned above. Another important application is when one needs to choose an appropriate sample from a population for the purpose of some study. In other applications the items may be jobs for scheduling, opportunities for investment, objects for fellowships, etc. 1.1 Background and Intuition The problem has been extensively studied in the probability and statistics literature (see [4, 13, 14] for surveys and also [10]). The case of k = 1. Let us first review the case of one object has to be selected. Since the observer cannot go back and choose a previously presented object which, in retrospect, turns out to be the best, it clearly has to balance the risk of stopping too soon and accepting an apparently desirable object when an even better one might still arrive, against the risk of waiting for too long and then find that the best item had been rejected earlier. It is easy to see that the optimal probability of selecting the best item does not tend to zero as n tends to infinity; consider the following stopping rule: reject the first half of the objects and then select the first relatively best one (if any). This rule chooses the best object whenever the latter is among the second half of the objects while the second best object is among the first half. Hence, for every n, this rule succeeds with probability greater than 1=4. Indeed, it has been established ([7, 5, 2]) (see below) that there exists an optimal rule that has the following form: reject the first objects and then select the first relatively best one or, if none has been chosen through the end, accept the last object. When n tends to infinity, the optimal value of r tends to n=e, and the probability of selecting the best is approximately 1=e. (Lind- ley showed the above using backward induction [7]. Later, Gilbert and Mosteller provided a slightly more accurate bound for r [5]. Dynkin established the result as an application of the theory of Markov stopping times [2].) It is not as easy to see that the optimal expected rank of the selected object tends to a finite limit as n tends to infinity. Observe that the above algorithm (for maximizing the probability of selecting the best object) yields an expected rank of n=(2e) for the selected item; the argument is as follows. With probability 1=e, the best item is among the first n=e items, and in this case the algorithm selects the last item. The conditional expectation of the rank of the last object in this case is approximately n=2. Thus, the expected rank for the selected object in this algorithm tends to infinity with n. Indeed, in this paper we show that, surprisingly, the two goals are in fact in conflict (see Section 1.2). It can be proven by backward induction that there exists an optimal policy for minimizing the expected rank of selected item that has the following form: accept an object if and only if its rank relative to the previously seen objects exceeds a certain threshold (depending on the number of objects seen so far). Note that while the optimal algorithm for maximizing the probability of selecting the best has to remember only the best object seen so far, the threshold algorithm has to remember all the previous objects. (See [11] for solutions where the observer is allowed to remember only one of the previously presented items.) This fact suggests that minimizing the expected rank is harder. Thus, not surprisingly, finding an approximate solution for the dynamic programming recurrence for this problem seems significantly harder than in the case of the first variant of the prob- lem, i.e., when the goal is to maximize the probability of selecting the best. Chow, Moriguti, Robbins, and Samuels, [1] showed that the optimal expected rank of the selected object is approximately 3:8695. The question of whether higher powers of the rank of the selected object tend to finite limits as n tends to infinity was resolved in [11]. It has also been shown that if the order of arrivals is determined by an adversary, then no algorithm can yield an expected rank better than n=2 [12]. The case of a general k. There has been much interest in the case where more than one object has to be selected. It is not hard to see that for every fixed k, the maximum probability of selecting the best k objects does not tend to zero as n tends to infinity. The proof is as follows. Partition the sequence of n objects into k disjoint intervals, each containing n=k consecutive items. Apply the algorithm for maximizing the probability of selecting the best object to each set independently. The resulting algorithm selects the best item in each interval with probability e \Gammak . The probability that the best k objects belong to distinct intervals tends to k!=k k as n tends to infinity. For this first variant of the problem, the case of was considered in [9]; Vanderbei [16], and independently Glasser, Holzager, and Barron [6], considered the problem for general k. They showed that there is an optimal policy with the following threshold form: accept an object with a given relative rank if and only the number of observations exceeds a critical number that depends on the number of items selected so in addition, an object which is worse than any of the already rejected objects need not be considered. Notice that this means that not all previously seen items have to be remembered, but only those that were already selected and the best among all those that were already rejected. This property is analogous to what happened in the case, where the goal was to maximize the probability of selecting the best item. Both papers derive recursive relations using backward induction. General solutions to their recurrences are not known, but the authors give explicit solutions (i.e., critical values and probability) for the case of [16] also presents certain asymptotic results as n tends to infinity and k is fixed and also as both k and n tend to infinity so that (2k \Gamma n)= remains finite. In analogy to the case of bounding the optimal expected sum of ranks of k selected items appears to be considerably harder than minimizing the probability of selecting the best k items. Also, here it is not obvious to see whether or not this sum tends to a finite limit when n tends to infinity. Backward induction gives recurrences that seem even harder to solve than those derived for the case of maximizing the probability of selecting the best k. Such equations were presented by Henke [8], but he was unable to approximate their general solutions. Thus, the question of whether the expected sum of ranks of selected items tends to infinity with n has been open. There has not been any explicit solution for obtaining a bounded expected sum. Thus the sec- ond, possibly more realistic, variant of the secretary problem has remained open. 1.2 Our Results In this paper we present a family of explicit algorithms for the secretary problem such that for each positive integer z, the family includes an algorithm for accepting items, where for all values of n and k, the resulting expected sum of the zth powers of the ranks of the accepted items is at most where C(z) is a constant. 2 kg. Clearly, the sum of ranks of the zth powers of the best k objects is k z+1 =(z Thus, the sum achieved by our algorithms is not only bounded by a value independent of n, but also differs from the best possible sum only by a relatively small amount. For every fixed k, this expected sum is bounded by a constant. Thus we resolve the above open questions regarding the expected sum of ranks and, in general, zth powers of ranks, of the selected objects. Our approach is very different from the dynamic programming approach taken in most of the papers mentioned above. In addition to being more successful in obtaining explicit solution to this classical prob- lem, it can more easily be used to obtain explicit solutions for numerous generalizations, because it does not require a completely new derivation for each objective function. We remark that our approach does not partition the items into k groups and select one item in each. Such a method is suboptimal since with constant probability, a constant fraction of the best k items appear in groups where they are not the only ones from the best k. Therefore, this method rejects a constant fraction of the best k with constant prob- ability, and so the expected value of the sum of the ranks obtained by such an algorithm is greater by at least a constant factor than the optimal. Since the expected sums achieved by our algorithms depend only on k and z and, in addition, the probability of our algorithms to select an object does not decrease with its rank, it will follow that the probabilities of our algorithms to actually select the best objects depend only on k and z, and hence for fixed k and z, do not tend to zero when n tends to infin- ity. In particular, this means that for our algorithms will select the best possible object with probability bounded away from zero. In contrast, for any algorithm for the problem, if the order of arrival of items is the worst possible (i.e., generated by an oblivious adversary), then the algorithm yields an expected sum of at least kn z 2 \Gamma(z+1) for the zth powers of the ranks of selected items. Our lower bound holds also for randomized algorithms. Finally, in Section 1.1 we observed that an optimal algorithm for maximizing the probability of selecting the best object results in an unbounded expected rank of the selected object. As a second part of this work we show that this fact is not a coincidence: the two goals are in fact in conflict. No algorithm can simultaneously optimize the expected rank and the probability of selecting the best. We derive a lower bound on the trade-off between the probability of accepting the best object and the expected rank of the accepted item. Due to lack of space, most proofs are omitted or only sketched. 2. The Algorithms In this section we describe a family of algorithms for the secretary problem, such that for each positive integer z, the family includes an algorithm for accepting objects, where the resulting expected sum of the zth powers of the ranks of accepted objects is In addition, it will follow that the algorithm accepts the best k objects with positive probability that depends only on k and z. Let z be the positive integer that we are given. Denote For the convenience of exposition, we assume without loss of generality that n is a power of 2. We partition the sequence [1; . ; n] (corresponding to the objects in the order of arrival) into consecutive intervals I i m), so that I fng if In other words, the first are [1; n 4 ]; . ; each containing a half of the remaining elements. The mth interval contains the last element. Note that jI Let us refer to the first intervals as the opening ones, and let the rest be the closing ones. Note that since p - 64, the last five intervals are closing. For an opening I i , the expected number of those of the top k objects in I i is (The latter is not necessarily an integer.) Further- more, for any d - (i.e., d is in one of the opening intervals), the expected number of those of the top k objects among the first d to arrive is d \Delta k n . Let Observe that pm 0 We will refer to p i as the minimum number of acceptances required for I i m). Observe that On the other hand, Intuitively, during each interval the algorithm attempts to accept the expected number of top k objects that arrive during this interval, and in addition to make up for the number of objects that should have been accepted prior to the beginning of this interval but have not. Note that since p during such intervals the algorithm only attempts to make up for the number of objects that should have been accepted beforehand and have not. Let us explain this slightly more formally. During each execution of the algorithm, at the beginning of each interval, the algorithm computes a threshold for acceptance, with the goal that by the time the processing of the last object of this interval is com- pleted, the number of accepted objects will be at least the minimumnumber of acceptances required prior to this time. In particular, recall that for denotes the minimum number of acceptances required for I i . Given a "prefix" of an execution prior to the beginning of I i 1), be the number of items accepted in I j . Let D Roughly speaking, D i\Gamma1 is the difference between the minimumnumber of acceptances required prior to the beginning of I i and the number of items that were actually accepted during the given prefix. Note that Given a prefix of an execution prior to the beginning of I i , let ae We refer to A i computed at the beginning of I i as the acceptance threshold for I i in this execution. Loosely stated, given a prefix of execution of the algorithm prior to the beginning of I i , A i is the number of objects the algorithm has to accept during I i in order to meet the minimum number required by the end of I i . The algorithm will aim at accepting at least A i objects during I i . To ensure that it accepts that many, it attempts to accept a little more. In particular, during each opening interval I i , the algorithm attempts to accept an expected number of A i +6(z +1) p A i log k. As we will see, this ensures that the algorithm accepts at least A i objects during this interval with probability of at least k \Gamma5(z+1) . During each closing interval I i , the algorithm attempts to accept an expected number of 32(z This ensures that the algorithm accepts at least A i objects during this interval with probability of at least 2 \Gamma5(z+1)(a i +1) . We make the distinction between opening and closing intervals in order to restrict the expected rank of the accepted objects. If I i is closing, then A i may be much smaller than p A i log k. Let ae A i log k if I i is opening closing. In order to accept an expected number of B i objects during interval I i , the algorithm will accept the dth item if it is one of the approximately ones among the first d. Since the order of arrival of the items is random, the rank of the dth object relative to the first d ones is distributed uniformly in the set f1; . ; dg. Therefore, the dth object will be accepted with probability of B since jI e, the expected number of objects accepted during I i is indeed B i . If at some point during the execution of the algo- rithm, the number of slots that still have to be filled equals the number of items that have not been processed yet, all the remaining items will be accepted regardless of rank. Analogously, if by the time the dth item arrives all slots have already been filled, this item will not be accepted. Finally, the algorithm does not accept any of the first dn=(8 k)e items except in executions during which the number of slots becomes equal to the number of items before dn=(8 k)e items have been pro- cessed. Roughly speaking, this modification will allow to bound the expected rank of the dth item in terms of its rank relative to the first d items. The above leads to our algorithm, which we call Select. Algorithm Select: The algorithm processes the items, one at a time, in their order of arrival. At the beginning of each interval I i , the algorithm computes A i as described above. When the dth item (d 2 I i ) arrives, the algorithm proceeds as follows. (i) If all slots have already been filled then the object is rejected. (ii) Otherwise, if d ? dn=(8 k)e, then (a) If the dth item is accepted if it is one of the top items among the first d. (b) If the algorithm accepts the dth item if it is one of the top b32(z items among the first d. (iii) Otherwise, if the number of slots that still have to be filled equals the number of items left (i.e., 1), the dth item is accepted. We refer to acceptances under (3) , i.e., when the number of slots that still have to be filled equals the number of items that remained to be seen, as manda- tory, and to all other acceptances as elective. For example, if the dth item arrives during I 1 , and the latter is opening, then the item is accepted electively if and only if it is one of the approximately k=2 log k=2 log top objects among the first d. In general, if the dth object arrives during an opening I i , then the object is accepted electively if and only if it is one of the approximately top objects among the first d. 3. Analysis of Algorithm Select Very loosely stated, the proof proceeds as follows. In Section 3.1 we show that for Observe that this implies that for high probability, A i is approximately p i , i.e., In Section 3.2 we show that if the dth object arrives during an opening I i , then the conditional expectation of the zth power of its rank, given that it is accepted electively, is not greater than 2 iz 1 z+1 A z c 4 (z)2 iz A z\Gamma0:5 log k, for some constant c 4 (z) (depend- ing on z); if I i is closing, this conditional expectation is not greater than c 6 (z)2 iz A z c 6 (z). In Section 3.3 these results of Sections 3.1 and 3.2 are combined and it is established that if the dth object arrives during an opening I i , then its conditional expected zth power of rank, given that it is accepted electively, is at most k z for some constant c(z). If I i is closing, that conditional expected zth power of rank is at most c 0 (z)k z , for some constant c 0 (z), if approximately otherwise. From this it will follow that the expected sum of the zth powers of ranks of the elec- tively accepted objects is 1 In addition we use the result of Section 3.1 to show that the expected sum of the zth powers of ranks of mandatorily accepted objects is O(k z+0:5 log k). Thus the expected sum of the zth powers of ranks of the accepted objects is 1 In addition, from the fact that the expected sum of the zth powers of ranks of the accepted objects is bounded by a value that depends only on k and z, it will also follow that the algorithm accepts the top k objects with probability that depends only on k and z. 3.1 Bounding the A i s In this section we show that for high probability, A i is very close to p i . To this end we distinguish between 'smooth' and `nonsmooth' executions (see below). 3.1.1 Smooth Prefixes. Denote by E i the prefix of an execution E prior to the end of I i . Note that Em is E. We say that computed in E i is - jI j j. Denote by ME i the event in which E i is smooth. In this section we show that for an opening interval I i , in executions whose prefix prior to the end of the 1th interval is smooth, the probability that exponentially with j (Part 1 of Lemma 3.3). For a closing I i , in executions whose prefix prior to the end of the i\Gamma1th interval is smooth, the probability that A i exponentially both with j and with i (Part 2 of Lemma 3.3). Part 1 and Part 2 of Lemma 3.3 will follow, respec- tively, from Lemmas 3.1 and 3.2 that show that in executions whose prefix prior to the end of the ith interval is smooth, in I i the algorithm accepts A i objects with high probability (where A i is computed for the prefix of the execution). Intuitively, the restriction to smooth executions is necessary since at most objects can be selected in I i . Lemma 3.1 For every any value a i of A i , Sketch of Proof: Note that D i ? 0 only if the number of objects accepted in I i is less than a i . Loosely stated, the algorithm accepts the dth object electively if it is one of the top A i log objects among the first d. Since the objects arrive in a random order, the rank of the dth object within the set of first d is distributed uniformly and hence it will be accepted electively with probability not less than b(a a i log c=d. Moreover, the rank of the dth object within the set of the first d is independent of the arrival order of the first d \Gamma 1, and hence is independent of whether or not any previous object in this interval, say the th one, is one of the top objects among the first d 1 . The rest of the proof follows from computing the expected number of accepted candidates and Chernoff inequality. Analogously, Lemma 3.2 If n - 16, then for every Lemma 3.3 (i) For (ii) If n - 16, then for Sketch of Proof: We outline the proof for Part (1). Recall that the minimum number of acceptances required for an opening interval I i is Thus if A i ? k2 \Gammai , then D are positive. These events are dependent and their probabilities are conditioned on however, it can be shown that both the dependency and the conditioning are working in our favour. Lemma 3.1 thus implies that each of the underlying events fD q ? 0g with probability less than k \Gamma5(z+1) . Hence, 3.1.2 Nonsmooth Executions. Lemma 3.3 implies that in smooth executions, with high probability, A i is very close to p i . To complete the proof that A i is close to p i , we now show that nonsmooth executions are rare. In particular, Part (1) of Lemma 3.3 is used to show: Lemma 3.4 If Analogously, Lemma 3.5 If n - 16, k - 1 The case of k - n=2 is excluded (Lemma 3.5) and thus handled separately later (Section 3.3). 3.2 Expected zth powers of Ranks Let us denote by R d the random variable of the rank of the dth object. We define the arrival rank of the dth object as its rank within the set of the first d objects, i.e., one plus the number of better objects seen so far. Denote by S d the random variable of the arrival rank. Denote by NA d the event in which the dth object is accepted electively. Lemma 3.6 There exist constants c 2 (z), c 3 (z) and c (z) such that for all d - n k and s, E(R z d d s z d d Combining the result of Lemma 3.6 with the fact that given that the object is accepted electively during an opening interval I i and A distributed uniformly in the set f1; 2; . ; b(a a i log k)2 i d=ncg, we will get: Lemma 3.7 There exist constants c 4 (z) and c 5 (z) such that for all opening intervals I i (i.e., every value a i of A i , if the dth object arrives during I i and d - n E(R z r d Analogously, Lemma 3.8 There exists a constant c 6 (z), such that for all closing intervals I i (i.e., a i of A i , if the dth object arrives during I i , and d - , then E(R z 3.3 Expected Sum of Ranks In this section we show that the expected sum of the zth powers of ranks of the k accepted objects isz (Theorem 3.1). This will follow by adding up the expected sum of the zth powers of ranks of electively accepted objects (Lemmas 3.13), and the expected sum of the zth powers of ranks of mandatorily accepted objects (Lemma 3.15). 3.3.1 Elective Acceptances. Denote by SUMZ i the sum of the zth powers of ranks of objects that are accepted electively during I i . Lemma 3.9 There exists a constant c 7 (z) such that for all opening intervals I i and for all values a i of A i , a z+1 Lemma 3.10 There exists a constant c 8 (z) such that for all closing intervals I i , for all acceptance thresholds a i computed for I i , Lemma 3.9 is combined with Part 1 of Lemma 3.3 and with Lemma 3.4 to show: Lemma 3.11 There exists a constant c 9 (z) such that for all opening intervals I i , Analogously, Lemma 3.12 If n - 16, then there exists a constant such that for any closing interval I i , The following lemma completes the proof of the upper bound on the sum of the ranks of the electively accepted objects. It sums up the expected sum of ranks of electively accepted objects over all intervals. Lemma 3.13 3.3.2 Mandatory Acceptances. This section bounds the expected sum of mandatorily accepted objects. We first observe: Lemma 3.14 If the dth object is mandatorily accepted in execution E during I i , then :ME i+1 Denote by SUMDZ i the sum of the zth powers of ranks of objects that are accepted mandatorily during I i . Lemmas 3.4 and 3.5 of Section 3.1.2 imply that, for each I i , the probability that a prefix of execution prior to the end of I i is not smooth, is at most c(z)n \Gamma2:5(z+1) log n, where c(z) is a constant. (The case of k - 1 2 n is handled without the use of Lemma 3.5, since this lemma excludes it.) Clearly, this bound applies also for the probability that objects will be mandatorily accepted in I i . We combine this bound with the facts that the rank of an object never exceeds n, and the number of accepted objects is at most k - n, to show: Lemma 3.15 There exist constants c 21 (z) and c 22 (z) such that Lemmas 3.13 and 3.15 imply: Theorem 3.1 The expected sum of ranks of accepted objects is at Corollary 3.1 Algorithm Select accepts the best k objects with positive probability that depends only on k and z. 4. Trade-Off between Small Expected Rank and Large Probability of Accepting the Best Theorem 4.1 Let p 0 be the maximum possible probability of selecting the best object. There is a c ? 0 so that for all ffl ? 0 and all sufficiently large n, if A is an algorithm that selects one of n objects, and the probability pA that A selects the best one is greater than then the expected rank of the selected object is at least c=ffl. Proof: Suppose that contrary to our assertion there is an algorithm A that selects the best object with probability of at least p yet the expected value of the rank of the selected object is less than c=ffl. Starting from A, we construct another algorithm R so that R selects the best object with a probability Denote by OPT the following algorithm: Let n=e objects pass, and then accept the first object that is better than anyone seen so far. If no object was accepted by the time the last object arrives, accept the last object. For n sufficiently large, this algorithm accepts the best object with the highest possible prob- ability, and hence with probability p 0 [7]. 3 In better approximation to r than ne \Gamma1 although the difference is never more than 1 [5]. We ignore this difference for the sake of simplicity. We define R by modifying A. The definition will depend on parameters c 1 ? d ? 0. We will assume that d is a sufficiently large absolute constant and c 1 is sufficiently large with respect to d. R will accept an object if at least one of the following conditions is (i) A accepts the object after time n=d and by time and the object is better than anybody else seen (ii) OPT accepts the object whereas A accepted earlier somebody who, at the time of acceptance, was known not to be the best one (that is there was a better one before); (iii) OPT accepts the object and A has already accepted somebody by time n=d; (iv) the object comes after time it is better than anybody else seen before and R has not yet accepted anybody based on the rules (1), (2), (v) the object is the nth object and R has not accepted yet any object. Notation: Denote by BA, BR, and BOPT the events in which A, R and OPT, repectively, accept the best object. Denote by B1, B2, and B3 the events in which the best object appears in the intervals spectively. Denote by IA1, IA2 and IA3 the events in which A makes a selection in the intervals [1; n=d], We distinguish between two cases. Case I: ProbfIA1g - 4.1 Proof: Suppose that A made a selection by time n=d. According to rule (3), in this case R will accept an object that arrives after time n=d if and only if OPT accepts this object. By choosing d sufficiently large, we have that objects are accepted by OPT only after time n=d. Thus, if A made a selection by time n=d, R will accept the object if and only if OPT accepts it. Thus, The second inequality follows since the probability that OPT accepts the best object is independent of the order of arrival of the first n=d objects, and hence independent of whether or not A makes a selection by time n=d. On the other hand, Thus, by choosing d to be sufficiently large the claim follows. 4.2 Proof: The claim follows immediately from the fact that if A picks the best object between n=d and t 0 , then this object must be the best seen so far, and hence by rule (1), R picks the same object. 4.3 Proof: If IA3 holds then neither A nor R have accepted anybody till time t 0 . Let X be the event when A chooses no later than R. By the definition of R we have that if X " IA3 holds then either A accepts an object that already at the moment of acceptance is known not to be the best, or A and R accept the same object. Thus, To complete the proof, it suffices to show that Suppose that IA3 " :X holds and R accepts an object at some time t ? t 0 . By definition, A has not accepted anybody yet, and the object accepted by R at t is better than anyone else seen earlier. Thus, if a better object than the one accepted by R arrives after time t, this means that the best object arrives after time t. Since the objects arrive in a random order, the rank of each dth arriving object within the set of first d is distributed uniformly. Hence, the probability that the best object will arrive after time t is at most (n \Gamma t)=n - c 1 ffln. Notice that this probability is independent of the ordering of the first t objects, and hence is independent of the fact that R has accepted the tth object. Therefore the probability that the object accepted by R is indeed the best object is at least 1 \Gamma c 1 ffln, while the probability that A accepts the best one later is smaller than ffln. Thus, for any fixed choice of t and fixed order of the first t objects (with the property IA3 " :X), the probability of BR is larger than BA, and hence Now we can complete the proof of Case I: ProbfBRg The second inequality follows from Claims 4.1, 4.2 and 4.3. The fourth inequality follows from (i) by the theorem assumption and (ii) ProbfIA1g - 3ffl=p 0 by Case I assumption. Case II: ProbfIA1g ! 3ffl=p 0 . Denote by BR1, BR2, and BR3 the events when R picks the best object and its selections are in the interval respectively. Denote by BA1, BA2, and BA3 the corresponding events for A. Since by the assumption of this case ProbfIA1g ! If A picks the best object between n=d and t 0 , then this object must be the best seen so far, and hence by rule (1), R picks the same object. Thus By choosing d sufficiently large, we have that objects are accepted by OPT only after time n=d. Observe that in that case, if the second best comes by time n=d and the best comes after time t 0 , then R accepts the best object. The probability that the second best object arrives by time n=d is 1=d, and the conditional probability that the best object comes after given that the second best comes by time n=d, is at least c 1 ffl. It thus follows: For bounding ProbfBA3g, we first use the assumption that the expected rank of the object selected by A is less than c=ffl, to show: Proof: Each of the 1=(10dc 1 ffl) objects with a rank smaller than 1=(10dc 1 ffl) arrives after time t probability of at most c 1 ffl. Therefore, with probability of at least 1 \Gamma 1=(10d), all objects that arrive after time t 0 are of rank larger than 1=(10dc 1 ffl). Hence, if the probability of IA3 had been greater than 1=(2d), then the expected value of the rank would have been larger than c 0 =ffl for some absolute constant the c of the theorem to be equal to c 0 , and we get a contradiction to the assumption that the expected rank of the selected object is at most c=ffl. Recall that B3 denotes the event in which the best object arrives in interval IA3g. But B3 is independent of the order of arrival of the first t 0 objects and hence independent on whether or not A has accepted an object by time t 0 . Thus, Claim 4.4 implies that ProbfIA3g \Delta ProbfB3 Equations (1) to (4) imply (The last inequality follows from our assumption that c 1 is sufficiently large with respect to d.) Therefore Acknowledgements We are indebted to James Aspnes, Eugene Dynkin, John Preater, Yossi Rinott, Mike Saks, Steve Samuels, and Robert Vanderbei for helpful references. --R "secretary problem" The optimum choice of the instant for stopping a Markov process. Who solved the secretary prob- lem? Statistical Science <Volume>4</Volume> The secretary problem and its ex- tensions: A review Recognizing the maximum of a sequence. The d Choice secretary problem. Dynamic programming and decision theory. Sequentialle Auswahlprobleme bei Unsicherheit. A generalization of the best choice problem. On multiple choice secretary prob- lems The finite memory secretary problem. Optimal counter strategies for the secretary problem. Secretary problems. Secretary problems as a source of benchmark sounds. Amortized efficiency of list updates and paging rules. The optimal choice of a sub-set of a population --TR --CTR Andrei Broder , Michael Mitzenmacher, Optimal plans for aggregation, Proceedings of the twenty-first annual symposium on Principles of distributed computing, July 21-24, 2002, Monterey, California Robert Kleinberg, A multiple-choice secretary algorithm with applications to online auctions, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia Mohammad Taghi Hajiaghayi , Robert Kleinberg , David C. Parkes, Adaptive limited-supply online auctions, Proceedings of the 5th ACM conference on Electronic commerce, May 17-20, 2004, New York, NY, USA

optimal stopping;expected rank maximization;dynamic programming

587953

Global Price Updates Help.

Periodic global updates of dual variables have been shown to yield a substantial speed advantage in implementations of push-relabel algorithms for the maximum flow and minimum cost flow problems. In this paper, we show that in the context of the bipartite matching and assignment problems, global updates yield a theoretical improvement as well. For bipartite matching, a push-relabel algorithm that uses global updates runs in $O\big(\sqrt n m\frac{\log(n^2/m)}{\log n}\big)$ time (matching the best bound known) and performs worse by a factor of $\sqrt n$ without the updates. A similar result holds for the assignment problem, for which an algorithm that assumes integer costs in the range $[\,-C,\ldots, C\,]$ and that runs in time $O(\sqrt n m\log(nC))$ (matching the best cost-scaling bound known) is presented.

Introduction . The push-relabel method [10, 13] is the best currently known way for solving the maximum flow problem [1, 2, 18]. This method extends to the minimum cost flow problem using cost scaling [10, 14], and an implementation of this technique has proven very competitive on a wide class of problems [11]. In both contexts, the idea of periodic global updates of node distances or prices has been critical to obtaining the best running times in practice. Several algorithms for the bipartite matching problem run in O( and Karp [15] first proposed an algorithm that achieves this bound. Karzanov [16] and Even and Tarjan [5] proved that the blocking flow algorithm of Dinitz [4] runs in this time when applied to the bipartite matching problem. Two phase algorithms based on a combination of the push-relabel method [13] and the augmenting path method [7] were proposed in [12, 19]. Feder and Motwani [6] give a "graph compression" technique that combines with the algorithm of Dinitz to yield an O( log n ) algorithm. This is the best time bound known for the problem. The most relevant theoretical results on the assignment problem are as follows. The best currently known strongly polynomial time bound of O(n(m log n)) is achieved by the classical Hungarian method of Kuhn [17]. Under the assumption that the input costs are integers in the range [ Gabow and Tarjan [9] use cost scaling and blocking flow techniques to obtain an O( nm log(nC)) time algorithm. An algorithm using an idea similar to global updates with the same running time appeared in [8]. Two-phase algorithms with the same running time appeared in [12, 19]. The first phase of these algorithms is based on the push-relabel method and the second phase is based on the successive augmentation approach. We show that algorithms based on the push-relabel method with global updates match the best bounds for the bipartite matching and assignment problems. Our results are based on new selection strategies: the minimum distance strategy in the bipartite matching case and the minimum price change in the assignment problem case. We also prove that the algorithms perform significantly worse without global updates. Similar results can be obtained for maximum and minimum cost flows in networks with unit capacities. Our results are a step toward a theoretical justification of the use of global update heuristics in practice. This paper is organized as follows. Section 2 gives definitions relevant to bipartite matching and maximum flow. Section 3 outlines the push-relabel method for maximum flow and shows its application to bipartite matching. In Section 4, we present the time bound for the bipartite matching algorithm with global updates, and in Section 5 we show that without global updates, the algorithm performs poorly. Section 6 gives definitions relevant to the assignment problem and minimum cost flow. In Section 7, we describe the cost-scaling push-relabel method for minimum cost flow and apply the method to the assignment problem. Sections 8 and 9 gen- denote the number of nodes and edges, respectively. eralize the bipartite matching results to the assignment problem. In Section 10, we give our conclusions and suggest directions for further research. 2. Bipartite Matching and Maximum Flow E) be an undirected bipartite graph, let A matching in G is a subset of edges M ' E that have no node in common. The cardinality of the matching is jM j. The bipartite matching problem is to find a maximum cardinality matching. The conventions we assume for the maximum flow problem are as follows: Let V; E) be a digraph with an integer-valued capacity u(a) associated with each arc 2 a 2 E. We assume that a (where a R denotes the reverse of arc a). A pseudoflow is a satisfying the following for each a 2 E: The antisymmetry constraints are for notational convenience only, and we will often take advantage of this fact by mentioning only those arcs with nonnegative flow; in every case, the antisymmetry constraints are satisfied simply by setting the reverse arc's flow to the appropriate value. For a pseudoflow f and a node v, the excess flow into v, denoted e f (v); is defined by v). A preflow is a pseudoflow with the property that the excess of every node except s is nonnegative. A node v 6= t with e f (v) ? 0 is called active. A flow is a pseudoflow f such that, for each node v 2 V , e f that a preflow f is a flow if and only if there are no active nodes. The maximum flow problem is to find a flow maximizing e f (t). 3. The Push-Relabel Method for Bipartite Matching We reduce the bipartite matching problem to the maximum flow problem in a standard way. For brevity, we mention only the "forward" arcs in the flow network; to each such arc we give unit capacity. The "reverse" arcs have capacity zero. Given an instance of the bipartite matching problem, we construct an instance \Delta of the maximum flow problem by setting ffl for each node v 2 X placing arc (s; v) in E; ffl for each node v 2 Y placing arc (v; t) in E; Sometimes we refer to an arc a by its endpoints, e.g., (v; w). This is ambiguous if there are multiple arcs from v to w. An alternative is to refer to v as the tail of a and to w as the head of a, which is precise but inconvenient. Given Matching Instance Bipartite Matching Instance Corresponding Maximum Flow Instance (Reverse arcs not shown) Figure 1. Reduction from Bipartite Matching to Maximum Flow ffl for each edge fv; wg 2 E with placing arc (v; w) in E A graph obtained by this reduction is called a matching network. Note that if G is a matching network, then for any integral pseudoflow f and for any arc a 2 E, u(a); f(a) 2 f0; 1g. Indeed, any integral flow in G can be interpreted conveniently as a matching in G: the matching is exactly the edges corresponding to those arcs a 2 X \Theta Y with 1. It is a well-known fact [7] that a maximum flow in G corresponds to a maximum matching in G. For a given pseudoflow f , the residual capacity of an arc a 2 E is The set of residual arcs E f contains the arcs a 2 E with f(a) ! u(a). The residual graph is the graph induced by the residual arcs. A distance labeling is a function d . We say a distance labeling d is valid with respect to a pseudoflow f if Those residual arcs (v; w) with the property that are called admissible arcs. We begin with a high-level description of the generic push-relabel algorithm for maximum flow specialized to the case of matching networks. The algorithm starts with the zero flow, then sets f(s; . For an initial distance labeling, the algorithm sets the algorithm applies push and relabel operations in any order until the current pseudoflow is a flow. The push and relabel operations, described below, preserve the properties that the current pseudoflow f is a preflow and that the current distance labeling d is valid with respect to f . push(v; w). send a unit of flow from v to w. end. relabel(v). replace d(v) by min (v;w)2Ef f end. Figure 2. The push and relabel operations The push operation applies to an admissible arc (v; w) whose tail node v is active. It consists of "pushing" a unit of flow along the arc, i.e., increasing f(v; w) by one, increasing e f (w) by one, and decreasing e f (v) by one. The relabel operation applies to an active node v that is not the tail of any admissible arc. It consists of changing v's distance label so that v is the tail of at least one admissible arc, i.e., setting d(v) to the largest value that preserves the validity of the distance labeling. See Figure 2. Our analysis of the push-relabel method is based on the following facts. See [13] for details; note that arcs in a matching network have unit capacities and thus push(v; w) saturates the arc (v; w). (2) Distance labels do not decrease during the computation. (3) relabel(v) increases d(v). (4) The number of relabel operations during the computation is O(n) per node. (5) The work involved in relabel operations is O(nm). If a node v is relabeled t times during a computation segment, then the number of pushes from v is at most (t degree(v). (7) The number of push operations during the computation is O(nm). The above lemma implies that any push-relabel algorithm runs in O(nm) time given that the work involved in selecting the next operation to apply does not exceed the work involved in applying these operations. This can be easily achieved using simple data structures described in [13]. 4. Global Updates and the Minimum Distance Discharge Algorithm In this section, we specify an ordering of the push and relabel operations that yields certain desirable properties. We also introduce the idea of a global distance update and show that the algorithm resulting from our operation ordering and global update strategy runs in O( time. For any nodes v; w, let dw (v) denote the breadth-first-search distance from v to w in the residual graph of the current preflow. If w is unreachable from v in the residual graph, dw (v) is infinite. Setting for every node v is called a global update operation. Such an operation can be accomplished with O(m) work that amounts to two breadth-first-search computations. The ordering of operations we use is called Minimum Distance Discharge, and it consists of repeatedly choosing an active node whose distance label is minimum among all active nodes and, if there is an admissible arc leaving that node, pushing a unit of flow along the admissible arc, otherwise relabeling the node. For convenience, we denote by \Gamma(f; d) (or simply \Gamma) the minimum distance label of an active node with respect to the pseudoflow f and the distance labeling d. We let \Gamma max denote the maximum value reached by \Gamma during the algorithm so far. attains a new maximum, we perform a global update operation. Our analysis hinges on a parameter k in the range 2 - k - n, to be chosen later. We divide the execution of the algorithm into four stages: In the first two stages, excesses are moved to t; in the final two stages, excesses that cannot reach the sink return to s. We analyze the first stage of each pair using the following lemma. Lemma 4.1. The Minimum Distance Discharge algorithm uses O work during the period beginning when \Gamma first exceeds ending when \Gamma first exceeds j. Proof: The number of relabelings that can occur when \Gamma max lies in the interval [i; j] is at most 1). Thus the relabelings and pushes require O work. The observations that a global update requires O(m) work and during the period there are O(j \Gamma i) global updates complete the proof. Lemma 4.1 allows us to account for the periods when \Gamma n+k]. The algorithm expends O(km) work during these periods. To study the behavior of the algorithm during the remainder of its execution, we introduce a combinatorial lemma that is a special case of a well-known decomposition theorem [7] (see also [5]). Lemma 4.2. Any integral pseudoflow f 0 in the residual graph of an integral preflow f in a matching network can be decomposed into cycles and simple paths that are pairwise node-disjoint except at the endpoints of the paths. Each path takes one of the following forms: ffl from s to t; ffl from a node v with e f (v) ? 0 to a node w with e f+f 0 (w can be t); ffl from a node v with e f (v) ? 0 to s. Lemma 4.2 allows us to show that when \Gamma max is outside the intervals covered by Lemma 4.1, the amount of excess the algorithm must process is small. Lemma 4.3. If \Gamma(f; d) - k ? 2, the total excess that can reach the sink is at most n=(k \Gamma 1). Proof: Let f be a maximum flow in G, and let f is a pseudoflow in G f , and therefore can be decomposed into paths as in Lemma 4.2. Because \Gamma - k and d is a valid distance labeling with respect to f , any path from an active node to t in G f must contain at least nodes. In particular, the excess-to-sink paths of Lemma 4.2 contain at least k nodes each, and are node-disjoint except for their endpoints. Since G contains only n+2 nodes, there can be no more than n=(k \Gamma 1) such paths. Since f is a maximum flow, the amount of excess that can reach the sink in G f is no more than n=(k \Gamma 1). The proof of the next lemma is similar. Lemma 4.4. If \Gamma(f; d) - n + k, the total excess at nodes in V is at most n=(k \Gamma 1). Lemma 4.3 and Lemma 4.4 show that outside the intervals covered by Lemma 4.1, the total excess processed by the algorithm is at most 2n=(k \Gamma 1). To complete the bound on the work expended by the algorithm outside these intervals, we use the following lemma and the fact that at most O(m) work takes place between consecutive global updates to deduce that O time suffices to process the excess outside the intervals covered by Lemma 4.1. Lemma 4.5. Between any two consecutive global update operations, at least one unit of excess reaches the source or the sink. Proof: For every node v, at least one of d s (v), d t (v) is finite. Therefore, immediately after a global update operation, at least one admissible arc leaves every node, by the definition of a global update. Hence the first unit of excess processed by the algorithm immediately after a global update arrives at t or at s before any relabeling occurs. The time bound for the Minimum Distance Discharge algorithm is O . Choosing n ) to balance the two terms, we see that the Minimum Distance Discharge algorithm with global updates runs in O( Feder and Motwani [6] give an algorithm that runs in o( time and produces a "com- pressed" version G ) of a bipartite graph in which all adjacency information is preserved, but that has asymptotically fewer edges if the original graph E) is dense. This graph consists of all the original nodes of X and Y , as well as a set of additional nodes W . If an edge fx; yg appears in E, either fx; yg 2 E or G contains a length-two path from x to y through some node of W . It is possible to show that an analogue to Lemma 4.2 holds in such a graph; the paths in the decomposition may not be node-disjoint at nodes of W , but remain so at nodes of X and Y , and this is enough to show that the Minimum Distance Discharge algorithm with graph compression runs in O log n time. This bound matches the bound of Feder and Motwani for Dinitz's algorithm. 1. Initialization establishes jX j units of excess, one at each node of X ; 2. Nodes of X are relabeled one-by-one, so all v 2 X have 3. While e f (t) ! jY j, 3.1. a unit of excess moves from some node v 2 X to some node w 2 Y with 3.2. w is relabeled so that 3.3. The unit of excess moves from w to t, increasing e f (t) by one. 4. A single node, x 1 with e f relabeled so that d(x 1 2. 5. ' / 1. 6. While ' - n, Remark: All nodes v 2 V now have with the exception of the one node which has d(x ' are at nodes of X ; 6.1. All nodes with excess, except the single node x ' , are relabeled one-by-one so that all 6.2. While some node y 2 Y has 6.2.1. A unit of excess is pushed from a node in X to 6.2.2. y is relabeled so 6.2.3. The unit of excess at y is pushed to a node x 2 X with 6.2.4. x is relabeled so that if some node in Y still has distance label ', otherwise 6.3. ' 7. Excesses are pushed one-by-one from nodes in X (labeled Figure 3. The Minimum Distance Discharge execution on bad examples. 5. Minimum Distance Discharge Algorithm without Global Updates In this section we describe a family of graphs on which the Minimum Distance Discharge algorithm without global updates (for values of m between \Theta(n) and This shows that the updates improve the worst-case running time of the algorithm. Given ~ n and ~ we construct a graph G as follows: G is the complete bipartite graph with ~ ~ It is straightforward to verify that this graph has m+ O( ~ edges. Note that jX j ? jY j. Figure 3 describes an execution of the Minimum Distance Discharge algorithm on G, the matching network derived from G, that time. With more complicated analysis, it is possible to show that every execution of the Minimum Distance Discharge algorithm on G It is straightforward to verify that in the execution outlined, all processing takes place at active nodes with minimum distance labels among the active nodes. Another important fact is that during the execution, no relabeling changes a distance label by more than two. Hence the execution uses \Theta(nm) work in the course of its \Theta(n 2 ) relabelings. 6. Minimum Cost Circulation and Assignment Problems Given a weight function c and a set of edges M , we define the weight of M to be the sum of weights of edges in M . The assignment problem is to find a maximum cardinality matching of minimum weight. We assume that the costs are integers in the range where C - 1. (Note that we can always make the costs nonnegative by adding an appropriate number to all arc costs.) For the minimum cost circulation problem, we adopt the following framework. We are given a graph E), with an integer-valued capacity function as before. In addition to the capacity function, we are given an integer-valued cost c(a) for each arc a 2 E. We assume c(a) = \Gammac(a R ) for every arc a. A circulation is a pseudoflow f with the property that e f for every node v 2 V . (The absence of a distinguished source and sink accounts for the difference in nomenclature between a circulation and a flow.) The cost of a pseudoflow f is given by f(a)?0 c(a)f(a). The minimum cost circulation problem is to find a circulation of minimum cost. 7. The Push-Relabel Method for the Assignment Problem We reduce the assignment problem to the minimum cost circulation problem as follows. As in the unweighted case, we mention only "forward" arcs, each of which we give unit capacity. The "reverse" arcs have zero capacity and obey cost antisymmetry. Given an instance \Delta of the assignment problem, we construct an instance \Delta of the minimum cost circulation problem by ffl creating special nodes s and t, and setting ffl for each node v 2 X placing arc (s; v) in E and defining c(s; ffl for each node v 2 Y placing arc (v; t) in E and defining c(v; ffl for each edge fv; wg 2 E with placing arc (v; w) in E and defining c(v; c(v; w); ffl placing n=2 arcs (t; s) in E and defining c(t; If G is obtained by this reduction, we can interpret an integral circulation in G as a matching in G just as we did in the bipartite matching case. Further, it is straightforward to verify that a minimum cost circulation in G corresponds to a maximum matching of minimum weight in G. Given Assignment Instance Assignment Problem Instance Corresponding Minimum Cost Circulation Instance Given Costs Large Negative Costs Zero Costs Figure 4. Reduction from Assignment to Minimum Cost Circulation A price function is a function R. For a given price function p, the reduced cost of an arc (v; w) is c p (v; ftg. Note that all arcs in E have one endpoint in U and one endpoint in its complement. U to be the set of arcs whose tail node is in U . For a constant ffl - 0, a pseudoflow f is said to be ffl-optimal with respect to a price function if, for every residual arc a 2 E f , we have ae a A pseudoflow f is ffl-optimal if f is ffl-optimal with respect to some price function p. If the arc costs are integers and ffl ! 1=n, any ffl-optimal circulation is optimal. For a given f and p, an arc a 2 E f is admissible iff ae a 2 E U and c p (a) ! ffl or The admissible graph is the graph induced by the admissible arcs. Our asymmetric definitions of ffl-optimality and admissibility are natural in the context of the assignment problem. They have the benefit that the complementary slackness conditions are violated on O(n) arcs (corresponding to the matched arcs). For the symmetric definition, complementary slackness can be violated on \Omega\Gamma m) arcs. procedure min-cost(V; E; u; c); while ffl - 1=n do end. Figure 5. The cost scaling algorithm. procedure refine(ffl; f; p); while f is not a circulation apply a push or a relabel operation; return(ffl; f; p); end. Figure 6. The generic refine subroutine. First we give a high-level description of the successive approximation algorithm (see Figure 5). The algorithm starts with the beginning of every iteration, the algorithm divides ffl by a constant factor ff and saturates all arcs a with c p (a) ! 0. The iteration modifies f and p so that f is a circulation that is (ffl=ff)-optimal with respect to p. When ffl ! 1=n, f is optimal and the algorithm terminates. The number of iterations of the algorithm is dlog ff (nC)e. Reducing ffl is the task of the subroutine refine. The input to refine is ffl, f , and p such that (except in the first iteration) circulation f is ffl-optimal with respect to p. The output from refine is ffl circulation f , and a price function p such that f is ffl 0 -optimal with respect to p. At the first iteration, the zero flow is not C-optimal with respect to the zero price function, but because every simple path in the residual graph has length of at least \GammanC, standard results about refine remain true. The generic refine subroutine (described in Figure 6) begins by decreasing the value of ffl, and setting f to saturate all residual arcs with negative reduced cost. This converts f into an ffl-optimal pseudoflow (indeed, into a 0-optimal pseudoflow). Then the subroutine converts f into an ffl-optimal circulation by applying a sequence of push and relabel operations, each of which preserves ffl-optimality. The generic algorithm does not specify the order in which these operations are applied. Next, we describe the push and relabel operations push(v; w). send a unit of flow from v to w. end. relabel(v). then replace p(v) by else replace p(v) by max (u;v)2Ef f p(u) end. Figure 7. The push and relabel operations for the unit-capacity case. As in the maximum flow case, a push operation applies to an admissible arc (v; w) whose tail node v is active, and consists of pushing one unit of flow from v to w. A relabel operation applies to an active node v. The operation sets p(v) to the smallest value allowed by the ffl-optimality constraints, namely max (v;w)2Ef otherwise. The analysis of cost scaling push-relabel algorithms is based on the following facts [12, 14]. During a scaling iteration (1) no node price increases; (2) every relabeling decreases a node price by at least ffl; (3) for any v 2 V , p(v) decreases by O(nffl). 8. Global Updates and the Minimum Change Discharge Algorithm In this section, we generalize the ideas of minimum distance discharge and global updates to the context of the minimum cost circulation problem and analyze the algorithm that embodies these generalizations. We analyze a single execution of refine, and to simplify our notation, we make some assumptions that do not affect the results. We assume that the price function is identically zero at the beginning of the iteration. Our analysis goes through without this assumption, but the required condition can be achieved at no increased asymptotic cost by replacing the arc costs with their reduced costs and setting the node prices to zero in the first step of refine. Under the assumption that each iteration begins with the zero price function, the price change of a node v during an iteration is \Gammap(v). By analogy to the matching case, we define denote the maximum value attained by \Gamma(f; p) so far in this iteration. The minimum change discharge strategy consists of repeatedly choosing a node v with applying a push or relabel operation at v. In the weighted context, a global update takes the form of setting each node price so that there is a path in GA from every excess to some deficit (a node v with e f (v) ! 0) and every node reachable in GA from a node with excess lies on such a path. This amounts to a modified shortest-paths computation, and can be done in O(m) time using ideas from Dial's work [3]. We perform a global update every time \Gamma max has increased by at least ffl since the last global update. We developed global updates from an implementation heuristic for the minimum cost circulation problem [11], but in retrospect, they prove similar in the assignment context to the one-processor Hungarian Search technique developed in [8]. We use essentially the same argument as for the unweighted case to analyze the part of the algorithm's execution when \Gamma max is small. Lemma 8.1. The Minimum Change Discharge algorithm uses O during the period beginning when \Gamma first exceeds ending when \Gamma first exceeds j. Proof: Similar to Lemma 4.1. large, the argument we used in the unweighted case does not generalize because it is not true that \Gammap(v) gives a bound on the breadth-first-search distance from v to a deficit in the residual graph. Let E(f) denote the total excess in pseudoflow f , i.e., The following lemma is analogous to Lemma 4.2. Lemma 8.2. Given a matching network G and a circulation g, any pseudoflow f in G g can be decomposed into ffl cycles and ffl paths, each from a node u with e f (u) ! 0 to a node v with e f (v) ? 0, where all the elements of the decomposition are pairwise node-disjoint except at the endpoints of the paths, and each element carries one unit of flow. We denote a path from node u to node v in such a decomposition by (u / v). The following lemma is similar in spirit to those in [8] and [12], although the single-phase push-relabel framework of our algorithm changes the structure of the proof. Lemma 8.3. At any point during refine, E(f) \Theta \Gamma max - ffl. Proof: Let c denote the (reduced) arc cost function at the beginning of this execution of refine, and let E) denote the residual graph at the same instant. For simplicity in the following analysis, we view a pseudoflow as an entity in this graph G. Let f , p be the current pseudoflow and price function at the most recent point during the execution of refine when . Then we have E(f) We will complete our proof by showing that and then deriving an upper bound on this quantity. By the definition of the reduced costs, Letting P be a decomposition of f into paths and cycles according to Lemma 8.2 and noting that cycles make no contribution to the sum, we can rewrite this expression as Since nodes u with e f are never relabeled, for such a node, and we have Because the decomposition P must account for all of f 's excesses and deficits, we can rewrite Now we derive an upper bound on c p (f) \Gamma c(f ). It is straightforward to verify that for any matching network G and integral circulation g, G g has exactly n arcs and so from the fact that the execution of refine begins with the residual graph of an (ffffl)-optimal circulation, we deduce that there are at most n negative-cost arcs in E. Because each of these arcs has cost at least \Gammaffffl, we have c(f) - \Gammaffnffl. Hence c Now consider c ffl-optimality of f with respect to p says that a R 2 Now by Lemma 8.2, f can be decomposed into cycles and paths from deficits to excesses. Let P denote this decomposition, and observe that c p the interior of a path P , i.e., the path minus its endpoints and initial and final arcs, and let @(P ) denote the set containing the initial and final arcs of P . If P is a cycle, -(P can write The total number of arcs in the cycles and path interiors is at most n+2, by node-disjointness. Also, the total excess is never more than n, so the initial and final arcs of the paths number no more than 2n. And because each arc carrying positive flow has reduced cost at most ffl, we have Therefore, c p (f) \Gamma c(f) - Now to complete our time bound, we use the following lemma. Lemma 8.4. Between any two consecutive global update operations, at least one unit of excess reaches a deficit. Proof: This lemma is a simple consequence of the ffl-optimality of f with respect to p. In particular, the definition of ffl-optimality implies that no push operation can move a unit of excess from a node to another node with higher price change, and indeed, two consecutive push operations on any given unit of excess suffice to move the excess to some node with strictly lower price change. By the definition of a global update operation, these properties suffice to ensure that a unit of excess reaches some deficit immediately after a global update, and before any relabeling occurs. Lemma 8.3 shows that when \Gamma the total excess remaining is O(n=k). Lemma 8.4 shows that O(m) work suffices to cancel each unit of excess remaining. As in the unweighted case, the total work in an execution of refine is O(mk choosing gives a O( time bound on an execution of refine. The overall time bound follows from the O(log(nC)) bound on the number of scaling iterations. Graph compression methods [6] do not apply to graphs with weights because the compressed graph preserves only adjacency information and cannot encode arbitrary edge weights. Hence the Feder-Motwani techniques do not apply in the assignment problem context. 9. Minimum Change Discharge Algorithm without Global Updates We present a family of assignment instances on which we show refine without global updates performs\Omega\Gamma nm) work in the first scaling iteration, under the minimum distance discharge selection rule. Hence this family of matching networks suffices to show that global updates account for an asymptotic difference in running time. The family of assignment instances on which we show refine without global updates takes structurally the same as the family of bad examples we used in the unweighted case, except that they are have two additional nodes and one additional edge. The costs of the edges present in the unweighted example are zero, and there are two extra nodes connected only to each other, sharing an edge with cost ff. At the beginning of the first scaling iteration, ff. The execution starts by setting 1. From this point on, the execution of refine restricted to the nodes and arcs present in the unweighted example parallels the execution of the maximum flow algorithm detailed in Section 5. 10. Conclusions and Open Questions We have given algorithms that achieve the best time bounds known for bipartite matching, namely O log n , and for the assignment problem in the cost scaling context, namely O ( nm log(nC)). We have also given examples to show that without global updates, the algorithms perform worse. Hence we conclude that global updates can be a useful tool in theoretical development of algorithms. We have shown a family of assignment instances on which refine performs poorly, but our proof seems to hinge on details of the reduction, and so it applies only in the first scaling iter- ation. An interesting open question is the existence of a family of instances of the assignment problem on which refine uses\Omega\Gamma nm) time in every scaling iteration. --R Goldberg's Algorithm for the Maximum Flow in Perspective: a Computational Study. Implementing Goldberg's Max-Flow Algorithm - A Computational In- vestigation Algorithm 360: Shortest Path Forest with Topological Ordering. Algorithm for Solution of a Problem of Maximum Flow in Networks with Power Estimation. Network Flow and Testing Graph Connectivity. Clique Partitions Faster Scaling Algorithms for Network Problems. Efficient Graph Algorithms for Sequential and Parallel Computers. An Efficient Implementation of a Scaling Minimum-Cost Flow Algorithm A New Approach to the Maximum Flow Problem. Finding Minimum-Cost Circulations by Successive Approxima- tion O nakhozhdenii maksimal'nogo potoka v setyakh spetsial'nogo vida i nekotorykh prilozheniyakh. The Hungarian Method for the Assignment Problem. Implementations of Goldberg-Tarjan Maximum Flow Algo- rithm New Scaling Algorithms for the Assignment and Minimum Cycle Mean Problems. --TR

push-relabel algorithm;dual update;assignment problem;cost scaling;zero-one flow;bipartite matching

587956

Realizing Interval Graphs with Size and Distance Constraints.

We study the following problem: given an interval graph, does it have a realization which satisfies additional constraints on the distances between interval endpoints? This problem arises in numerous applications in which topological information on intersection of pairs of intervals is accompanied by additional metric information on their order, distance, or size. An important application is physical mapping, a central challenge in the human genome project. Our results are (1) a polynomial algorithm for the problem on interval graphs which admit a unique clique order (UCO graphs). This class of graphs properly contains all prime interval graphs. (2) In case all constraints are upper and lower bounds on individual interval lengths, the problem on UCO graphs is linearly equivalent to deciding if a system of difference inequalities is feasible. (3) Even if all the constraints are prescribed lengths of individual intervals, the problem is NP-complete. Hence, problems (1) and (2) are also NP-complete on arbitrary interval graphs.

Introduction . A graph G(V; E) is an interval graph if one can assign to each vertex v an interval I v on the real line, so that two intervals have a non-empty intersection if and only if their vertices are adjacent. The set of intervals fI v g v2V is called a realization of G. The problems which we study here are concerned with the existence of an interval realization to a graph, subject to various types of distance (or difference) constraints on interval endpoints. These are inequalities of the form x y and constant C xy . Specifically, we study the following problems (we defer further definitions to section 2): Distance-Constrained Interval Graph (DCIG): INSTANCE: A graph E) and a system S of distance constraints on the variables QUESTION: Does G have a closed interval realization whose endpoints satisfy S? That is, is there a set of intervals f[l which form a realization of G and their endpoints satisfy S? A special case is DCIG in which all constraints are lower and upper bounds on interval lengths: Bounded Interval Graph Recognition (BIG): INSTANCE: A graph QUESTION: Is there a closed interval realization of G such that for each vertex v: In the following problem, each interval must have a prescribed Measured Interval Graph Recognition (MIG INSTANCE: A graph E) and a length function QUESTION: Is there a closed interval realization of G, in which for every We shall prove here that even MIG , the most restricted problem of the three, is strongly NP- complete. Unlike the situation with interval graphs, the fact that the intervals must be closed causes some loss in generality. In contrast, we show that when the interval graph admits a unique consecutive clique order (up to complete reversal), DCIG is polynomial, and hence, so are the other two problems. The class of graphs satisfying this property (which we call UCO graphs) properly contains the class of prime interval graphs, and is recognizable in linear time. Our solution is based on reducing the problem to a system of difference constraints. We also prove that we cannot do better, by showing that the problem of solving a system of difference constraints and the problem BIG on UCO graphs are linearly equivalent. Interval graphs have been intensively studied, due to their central role in many applications (cf. [33, 17, 11]). They arise in many practical problems which require the construction of a time line where each particular event or phenomenon corresponds to an interval representing its duration. Among the applications are planning [3], scheduling [22, 31], archaeology [26], temporal reasoning [2], medical diagnosis [29], and circuit design [36]. There are also non-temporal applications in genetics [6] and behavioral psychology [9]. In the Human Genome Project, a central problem which bears directly on interval graphs is the physical mapping of DNA [8, 25]: It calls for the reconstruction of a map (a realization) for a collection of DNA segments, based on information on the pairwise intersections of segments. In the applications above, size and distance constraints on the intervals may occur naturally: The lengths of events (intervals) may be known precisely, or may have upper and lower bounds. The order or distance between two events may be known. This is often the case in scheduling problems and temporal reasoning. In physical mapping, certain experiments provide information on the sizes of the DNA segments [21]. Our goal here is to study how to combine those additional constraints with precise intersection data. Green and Xu (cf. [20]) developed and implemented a program (called SEGMAP) for construction of physical maps of DNA, which utilizes intersection and size data. The intersection data is obtained by experimentally testing whether each of the segments contain a sequence of DNA (called STS) which appears in a unique, unknown location along the chromosome. Hence, two segments which contain a common STS must intersect. Their algorithm works in two phases: the first phase ignores the size data. It obtains a partition of the STSs into groups, and a linear order on the groups. The second phase uses the partial order of phase 1 together with the size data to obtain the map using linear programming algorithms. Our results in section 3 imply that faster algorithms (utilizing network flow techniques) can be used under certain conditions on the data. However, the results in section 5 imply that the general problem tackled by SEGMAP is intractable (unless P=NP) even with perfect data. Recognizing interval graphs (i.e., deciding if a graph has an interval realization) can be done in linear time [7, 28, 23]. Surprisingly, much less is known about the realization problem when the input contains additional constraints on the realization. The special case of MIG where all intervals have equal length corresponds to recognizing unit interval graphs [33], which can be done in linear time [10]. The special case of DCIG where all distance constraints have the form r is the problem of seriation with side constraints [27, 19] (also called interval graph with order constraints), which can also be solved in linear time [32]. When DCIG is further restricted to the special case where for each pair u; v where (u; v) 62 E, we have either the constraint r The problem is equivalent to recognizing an interval order, which can be done in linear time [4]. Fishburn and Graham [12] discussed a special case of BIG where all intervals have the same pair p and q of upper and lower bounds. For each p and q, they characterized the resulting class of interval graphs (and interval orders), in terms of the family of minimal forbidden induced subgraphs (respectively, suborders). They proved that such a family is finite if and only if p q is rational. In this case, for integer p and q, their characterization yields an exponential time n O(pq) algorithm for identification of such graphs (orders), where n is the number of vertices. Isaak [24] studied a variant of BIG in which the input is an interval order, there are upper and lower integer bounds on individual interval lengths, and the question is whether there exist a realization in which all endpoints are integers. Using Bellman's notion of a distance graph, Isaak gave an O(min(n 3 log time algorithm for that problem, where C is the sum of bounds on lengths. He also posed the more general problem of BIG, which we answer here. We generalize distance graphs to handle both strict and weak inequalities on endpoints, in order to solve DCIG on a particular class of graphs. There have been other studies on the realization of a set of intervals based on partial information on their intersection, length and order. Those are different from our problems here inasmuch the information on intersection is incomplete, i.e., the underlying interval graph is not completely known. Among these are studies on interval sandwich [18], interval satisfiability [19, 37, 32], on interval graphs and orders which have realizations with at most k different lengths [11, chapter 9], on the smallest interval orders whose representation requires at least k different lengths [11, chapter 10], and on the number of distinct interval graphs and orders on n vertices which have a realization with k given lengths [35]. The paper is organized as follows: Section 2 contains some preliminaries and background. Section 3 studies problem DCIG on UCO graphs, and proves its linear equivalence to solving systems of difference constraints. This implies in particular an O(min(n 3 log time algorithm for all three problems on UCO graphs. In section 4 we sketch a simple proof that DCIG is strongly NP-complete. Section 5 proves the stronger result that MIG is strongly NP-complete. The reduction (performed in two steps) is rather involved, but we feel it gives insight on the interplay between the topological side of the problem (i.e., intersection, open or closed intervals) and its metric aspect (i.e., the intervals sizes). 2. Preliminaries. A graph E) is called an intersection graph of a family of sets is called an interval graph if it is an intersection graph of a family of intervals on the real line. In that case, S is called a realization of G. Depending on the convention, each interval may be either closed or open, with no loss of generality. For simplicity, we sometimes use the same names for the intervals and for the corresponding vertices. For an interval I denote its left and right endpoints by l(I) and r(I), respectively. The length of I, denoted jIj, is r(I) \Gamma l(I). If G has a realization in which all the intervals are of equal length, then it is called a unit interval graph. be the maximal cliques in a graph vng. The clique matrix of G is the n \Theta k zero-one matrix . If the columns in C(G) can be permuted so that the ones in each row are consecutive, then we say that has the consecutive ones property, and we call such a permutation of the columns a consecutive (clique) order. According to Gilmore and Hoffman [16], G is an interval graph if and only if C(G) has the consecutive ones property. For two non-intersecting intervals x; y where x is completely to the left of y, we write xOEy or, equivalently, y-x. Let be a partial order. Call ! an interval order if there exists a set of v2V such that v ! u if and only if I v OEI u . S is called a realization for P . Call E) the incomparability graph of P , if for each u; only if u and v are incomparable in Hence, G is an interval graph if and only if it is the incomparability graph of some interval order. In this case we will say that the graph G admits the order !. For a vertex in the graph (v)[fvg. For a vertex set U ' V denote N [U . A set M ' V is called a module in E) if for each x; y 2 M , and for each u Surely, V is a module, and for each v 2 V , fvg is a module. Such modules are called trivial. If all modules in G are trivial, then G is called prime. For a subset X ae V define Xg. For a module M in the graph G, the graph G is said to be obtained from G by contracting M to v. We usually denote by n and m the number of vertices and edges, respectively, in the graph. 3. Distance Constraints in UCO graphs. We call an interval graph uniquely clique-orderable (UCO for short) if it has a unique consecutive clique order, up to complete reversal, in every realization. An interval graph G is UCO if and only if the only non-trivial modules in it are cliques [34]. Note that G is UCO if and only if the interval order admitted by G is unique, up to complete reversal, because an interval order of the vertices of G uniquely determines a linear order of the maximal cliques in G, and vice versa. Denote this order by OE G . Note also that the class of UCO graphs properly contains the class of prime interval graphs. UCO graphs can be recognized in linear time by applying the PQ-tree algorithm of Booth and Lueker [7], and noting that G is UCO if and only if the final tree consists of a single internal Q-node and the leaves. This procedure also computes OE G in O(m + n) time. In this section we study the problem DCIG when the input graph is UCO. We show how to reduce this problem, in linear time, to the problem of deciding whether a system of difference constraints is feasible. Hence, DCIG, BIG and MIG are all polynomial on UCO graphs. We also prove that for BIG and DCIG, we cannot do any better, since deciding the feasibility of a system of difference constraints can be reduced in linear time to an instance of BIG with a UCO graph. 3.1. A Polynomial Algorithm for DCIG on UCO Graphs. Let be an instance of DCIG, where E) is UCO and A is a set of difference inequalities on the interval endpoints. Construct two systems T and - T of difference constraints on the variables fl v ; r v g v2V , as follows: Both systems include all inequalities in A. In addition, for each x; y contains an inequality r x ! l y , and - T contains an inequality r y ! l x . If xy 2 E then both T and - T contain an inequality r x - l y (and r y - l x ). With these definitions we prove: Lemma 3.1. P has a realization if and only if either T or - T has a feasible solution. Proof. If r v g v2V is a feasible solution to T or to - T , then X is a solution to A, and realizes G. On the other hand, let r v ]g v2V be a realization of G, whose endpoints satisfy A. Then the order of the intervals f[ ~ l v ; ~ r v ]g v2V on the real line is either OE G , or its reversal. Therefore, f ~ l v ; ~ r v g v2V is a feasible solution to either T or - T . Hence, we can solve our problem by deciding whether system T or - T is feasible. We shall prove now that a system S of weak and strict difference constraints on n variables is reducible in linear time to a system S 0 which consists of weak difference constraints, with numbers only O(n) times larger. (Standard transformation techniques [14] would give numbers O(2 L ) times larger for binary input length L.) Assume all constants in S to be integral, and fix ffl - 1 n . Define S 0 to include every weak inequality in S, and a weak inequality strict inequality x \Gamma y ! c in S. Note that the number of variables and number of inequalities in the two systems is the same, and the constants in (after multiplying by an appropriate factor to restore integrality) are larger than the constants in S by a factor of \Theta(n). Lemma 3.2. S has a feasible solution if and only if S 0 has one. Proof. The 'if' direction is trivial, since a feasible solution to S 0 also satisfies S. To prove the 'only if', we generalize the notion of a distance graph (cf. [1, p. 103]), to handle strict and weak inequalities: For a system T of difference constraints, construct a directed weighted graph weights and arc labels, as follows: For every constraint x \Gamma y - C xy or add an arc (y; x) to D(T ) with weight C xy and label the arc - or !, respectively. D(T ) is called the distance graph of the system T . The weight of a path (or a cycle) in this graph is the sum of the weights of its arcs. Bellman has shown that when all inequalities in T are weak, T is feasible if and only if D(T ) contains no negative cycle ([5], see also [1, p. 103]). Suppose S 0 is not feasible. Then D(S 0 ) must contain a negative-weight cycle c. Let w(c) and w 0 (c) be the total weight of c in D(S) and D(S 0 ), respectively. Distinguish two cases: ffl All arcs in c have labels -. Then (y;x)2c (y;x)2c (1) Hence, S is infeasible. ffl c contains an arc marked !. Since the weight of each arc in c differs from the weight of the corresponding arc in c 0 by no more than ffl, we get: Since the weights in D(S) are integral, it follows that w(c) - 0. Since the cycle c in D(S) contains an arc marked !, the inequality (1) is strict, namely, w(c) ? 0, so S is infeasible. Corollary 3.3. A system T is feasible if and only if the weight of every cycle in its distance graph D(T ) is either positive, or it is zero and the cycle consists of - arcs only. We now show that addition of identical strict inequalities to the equivalent systems S and S 0 above maintains the equivalence between them. (We will need this property in section 5.3): For constants define the following systems 2 and S 3 on the set of variables Lemma 3.4. Let are integers and ffl ! 1 n . S has a feasible solution if and only if S 0 has one. Proof. The proof is by induction on the size of I 3 . For I this is lemma 3.2. Suppose both S and S 0 have feasible solutions, and consider adding a single strict inequality E: x \Gamma y ! C to both systems. This implies adding an arc labeled ! with to both distance graphs D(S) and D(S 0 ). By corollary 3.3, it suffices to prove that there exist a cycle of non-positive weight passing through e in D(S [E) if and only if such a cycle exists in D(S 0 [E). But for every simple path p from x to y, w S 0 is an integer. Hence, dwS 0 since C is integral, wS By lemma 3.1, and lemma 3.2, solving an instance of DCIG linearly reduces into determining if at least one of two systems of difference constraints is feasible. Using the distance graph reformulation, the feasibility of such a system with M weak inequalities on N variables, with sum of absolute values of arc weights C, can be decided in O(min(NM; log In our instance (G; there are n vertices, so Corollary 3.5. Deciding if a UCO graph with difference constraints has a realization can be done in O(min(n 3 log nC)) time. Note that the algorithms of [30, 13] for deciding the feasibility of a system also produce a feasible solution if one exists. This enables construction of a realization (if one exists) in O(min(n 3 log nC)) time. 3.2. Reducing a System of Difference Constraints to BIG on UCO graphs. Given a system of weak difference constraints, we shall show how to reduce it, in linear time, to an equivalent instance of BIG, in which the graph is UCO. According to lemma 3.2, the assumption that all constraints are weak can be made without loss of generality. Let P be the following system of weak difference constraints in the variables xN g: a new system P 0 of difference constraints on the same variable set X: Note that the choice of C guarantees that c 0 can be rewritten as: where all right hand side terms are larger than one. We call a solution f~x i g N to P 0 monotone if for each Lemma 3.6. P has a solution if and only if P 0 has a monotone solution. Moreover, if - is a feasible solution to P for which is minimal, then - is a monotone feasible solution to P 0 . Proof. Suppose P 0 has a monotone solution x N . Let ~ x for each Therefore, the i-th inequality in P is satisfied by f~x i g N Hence, P has a feasible solution. Let ~ x xN be a solution of P , for which is minimal. (P defines an intersection of closed half-spaces, which is a closed set, therefore there is a solution attaining this minimal value). By [5], \Delta is the sum of arc weights along some simple path in the distance graph By P we get, for each 1 - is a feasible solution of . For each 1 and X 0 is monotone. For the above system P , define to be the following BIG instance (compare figure 1): ffl G is the intersection graph of the set of intervals A defined as follows: i=0 where a ffl The length constraints are as follows: - For integral i: U (b i a 1 a 2 a 3 a 4 a 0 a 5 a 1 a 2 a 3 a 4 a 0 a 5 1Fig. 1. The graph G used in the reduction (top) and a realization for it (bottom) Lemma 3.7. G is UCO. Proof. Let G 0 be the intersection graph of A [ B. It is easy to see that G 0 is prime, and hence, it has a unique clique order [34]. Moreover, G 0 has exactly 2N cliques, each one containing (among other vertices) a unique and distinct b i . The set of maximal cliques in G is fN [b x namely, each clique is distinguished by a single b i . Since G 0 is UCO, its unique clique order determines a unique linear order on fb x jx 2 Bg, and hence, also on the maximal cliques of G. Hence, G is UCO. Theorem 3.8. P has a feasible solution if and only if J has a realization. Proof. Only if: Suppose f~x i g N is a feasible solution to P , for which is minimal. By lemma 3.6, fx 0 iC is a monotone solution to P 0 . Choose arbitrary 1. Define the following set R [ T [ S of intervals: is monotone, the intersection graph of R[T [S is isomorphic to G. The length bounds on vertices of T and R are trivially satisfied. If i as required. If satisfying the length bounds on the vertices of S. Suppose J has a realization. Let fy i g N be the points in a realization of J which correspond to the intervals fb i g N (which have length zero). W.l.o.g. y N ? y 1 , because otherwise we can reverse the realization. Since G is UCO, the order of the intervals in J is identical to the order of the intervals A[B [W in the definition of G. Therefore y due to the length constraint on be the interval corresponding to w in the realization. Define a system P 00 of difference constraints as follows: i . It follows that fy i g N is a monotone solution to P 00 . A proof similar to lemma 3.2 implies that P 0 and P 00 are equivalent, so P 0 is feasible. We would like to show that P 0 has a monotone solution. Let Q 0 be the system of constraints have only monotone solutions. According to lemma 3.4, adding Q 0 to both P 0 and P 00 maintains the equivalence between them. But a monotone solution of P 00 realizes has a monotone solution and according to lemma 3.6 P is feasible. Corollary 3.9. The problem of deciding whether there exists a feasible solution to a system of difference constraints is linearly reducible to the problem BIG on a UCO graph. 4. DCIG is NP-complete. We will now show, that although DCIG is polynomial when restricted to UCO graphs, it is NP-complete in general. A stronger result will be proven in the next section, but we include a sketch of this proof as it is much more transparent. Theorem 4.1. DCIG is strongly NP-complete. Proof. We show a pseudo-polynomial reduction from the problem 3-PARTITION which is known to be strongly NP-complete (see, e.g., [15]). An instance of 3-PARTITION is a set X of 2 ), such that k. The question is to determine whether there exists a partition of X into k subsets (which have to be triplets) so that for each 1 - j - k: xng be an instance of 3-PARTITION. Define an instance of DCIG, I = (G; S) where G is the empty graph on the vertices fv j g n , and S consists of the following three types of constraints: We shall see that I is satisfiable if and only if X is a "yes" instance (see figure 2). Assume for now that all intervals in X must be open. Suppose there exists a partition X Examine the set of intervals 1-j-k;1-i-3 where I a j I v i ). The intervals in T are disjoint, and their endpoints trivially satisfy S, hence, T is a realization of I. a 1 a k a 0 Fig. 2. The v i 's can be squeezed between the a i 's if and only if a 3-partition exists Conversely, suppose fI a i is a realization of I. For each g. According to the constraints, l(a the I j 's do not intersect each other, and therefore the sets X j are disjoint. Moreover, every x i is a member of some X j . Therefore is a partition of X. For each 1 - j - n: Since G is empty all the I v i 's are disjoint, hence, We assumed here that all intervals in the realization are open. To form a closed realization, it suffices to modify the reduction by allowing an interval of length 1 (instead of length 1) for each 'gap' interval [r(a sufficiently small. (If each a are integers, then Since 3-partition is strongly NP-complete, and the reduction is pseudo-polynomial, our problem is strongly NP-complete. 5. Recognizing Measured Interval Graphs is NP-Complete. In this section we prove the NP-completeness of the problem MIG , introduced in section 1. The main part in this proof is a hardness result for the following, slightly more general problem, in which we specify in advance for each interval whether it should be closed or open: Recognizing a Measured Interval Graph with Specified Endpoints (MIG): INSTANCE: A graph non-negative length L(v) for every v 2 V , and a closedg. QUESTION: Is there a realization of G, in which the length of I v is exactly L(v), and I v is open if and only if We shall denote such an instance by When P is a "yes" instance, we say that P is a measured interval graph (with endpoint specification). We shall first prove that MIG is NP-complete, and then reduce MIG to MIG . The issue of endpoint specification seems unnatural at first sight. It is well known that for interval graphs in general the endpoint specification can be arbitrary, namely, a graph is interval if and only if it has a realization for any possible specification of endpoints. This is not the case in the presence of length constraints. For example: a K 1;3 graph with length 1 assigned to all vertices has no realization if all intervals are open (or all closed), but it has a realization precisely if the degree-3 vertex and two of the others are closed, as in figure 3. a c d a c d Fig. 3. The K 1;3 graph shown (right) has a realization (left) if all intervals but c are closed. We shall often use the following implicit formulation for the problem, by representing G and OE using intervals, and using L to modify their MIG: Implicit formulation: INSTANCE: A pair (T; L) where x2V is a set of intervals, and is a length function. QUESTION: Is there a set of intervals only if I x " I y closed if and only if I x is closed. This formulation is sometimes more convenient as it suggests a possible realization. We need the following notations and definitions: Definition 5.1. OE) be a measured interval graph with endpoints specification, and let U ' V be a set of its vertices. Define the measured interval graph PU induced by P on U , to be (GU GU is the subgraph of G induced on U , and LU , OE U are the restrictions of L and OE, respectively, to U . Call two MIG instances there is a graph isomorphism f between G and G 0 , and for each namely, the length and closure properties of intervals are preserved by f . In this case denote P Definition 5.2. OE) be an instance of MIG. Let be a realization of P . Define realization of Pg. 5.1. Basic Structures. We now describe three "gadgets" which are building blocks in our NP-completeness construction, and prove some of their properties. The structure of these gadgets assures us that their realization has very few degrees of freedom. To formalize this we introduce the following notion: Definition 5.3. Two realizations of the same interval graph are isometric if they are identical up to reversal and an additive shift. Namely, there exists a function j for all j. Let OE) be an instance of MIG. We call U ' V (G) rigid in P if in any two realizations of P , the sets of intervals realizing U are isometric. In particular, all endpoints are located at fixed distances from the leftmost endpoint, including the rightmost one. Thus in every realization U has the same length. If V (G) is rigid in P , we call P rigid. Note that the fact that U is rigid in P does not imply that PU is rigid. For example, the instance P defined implicitly by the intervals in figure 3 is rigid, and in particular fb; c; dg is rigid in P , but P fb;c;dg is not rigid. 5.1.1. The Switch. We first define the switch, a gadget which will be used as a toggle in larger structures. For the parameter real value a - 1, define the MIG instance (compare figure 4): G is the graph on the five vertices assigns lengths 0; 1 respectively, and OE(v) specifies v 3 to be open, and all the other vertices to be closed. A realization fI of a Switch(a) will be called straight if I 1 is to the left of I 5 . Otherwise, it will be called reversed. We say that such a realization is located at I 3 . For a straight realization U of a Switch(a) located at by \GammaU the reverse realization located at Hence, \GammaU is a "mirror image" of U , covering the same interval [x; x along the real line. Lemma 5.4. Switch(a) is rigid. In particular, Proof. Let S be a straight realization, as in the top left of figure 4. Suppose S 0 is another realization, such that both leftmost endpoints, I 1 and I 0 are identical. The intersection graph of a Switch(a) is prime, Fig. 4. The Switch (bottom), a straight realization (top left) and a reversed realization(top right). hence, I 3 is between I 1 and I 5 , and l(I 5 therefore all inequalities hold as equalities. In particular, a. Note that lemma 5.4 implies that a realization of a straight Switch(a) located at (x; x+1) is unique. The same is true for a reversed Switch. Lemma 5.5. OE) be an instance of MIG, and let be a realization for it. Let be a module such that PU Proof. Let vertices numbered in the same order as in the definition of a Switch. According to lemma 5.4, I(v 1 ) and I(v 5 ) are one unit apart, but both of them intersect I(x) and I(y). Therefore I(x)"I(y) contains the unit length interval between I(v 1 ) and I(v 5 ), yielding jI(x)"I(y)j - 1. 5.1.2. The Fetters. Our second gadget binds two Switches and imposes a prescribed distance between them. For positive real parameters, d; r and two sets of vertices, U 1 and U 2 , define a five-parameter instance of MIG, F etters(d; r or in short, F etters = (G; L; OE), as follows: are modules in G, and each of them induces a Switch. More precisely, there exist constants a 1 ; a 2 such that F etters U1 ffl The graph ~ E), constructed from G by contracting U respectively, is as follows (compare figure 5): long long long long long long (v long long long long (v short long 2. long long endFig. 5. The F etters (bottom) and a realization of it (top). The distance between the Switches v 1 and v 2 is fixed. ffl The lengths for the remaining intervals are: 2. long 2. 2. ffl OE specifies v tot to be open, and all the other intervals outside U to be closed. When there is no confusion, we shall use the vertex and the corresponding interval in the realization interchangeably. For example, l(v short i ) is the position of the left endpoint of the interval corresponding to v short i in the realization, jv short its length, etc. Call a realization of F etters straight if v 1 is to the left of v 2 . Otherwise call the realization reversed. A realization of F etters is said to be located at the interval corresponding to v tot . The F etters instance fixes the distance between its two Switches. To formalize this notion we need the following definition: OE) be a MIG instance. Let M;M 0 ' V (G) be modules in G where For a realization of P , in which I and I 0 are the intervals corresponding to the middle vertices in M and M 0 , respectively, define Dist(M; Lemma 5.7. ~ is rigid in the F etters. In particular, in every realization of the F etters, Proof. Recall that ~ G is the graph constructed from G by contracting U 1 and U 2 into v 1 and v 2 , respectively. It is easy to see that ~ G is prime. Prime interval graphs have an interval order which is unique, up to complete reversal [34]. Hence, let us refer to the order in figure 5, where w.l.o.g. tot is between v end 1 and v end according to the length constraints: l(v end long By lemma 5.5, r(v long long long yielding l(v end long In a similar way we prove v short long and the result follows. By lemma 5.7, for a realization of the F etters(d; r which is straight (or reversed) and located at 1), the only degrees of freedom are reversals of the Switches. 5.1.3. The Frame. We now construct an element which divides an interval into sub-intervals of prescribed lengths. Each sub-interval is characterized by a distinct set of intervals which contain it. This element will be used as a frame, into which the moving and toggling elements will fit, and have the desired degrees of freedom. be a sequence of real positive numbers, whose sum is s. to be an instance consists of 3r+3 vertices, where V and g. ffl The edges in G are: k such that j is odd and ji \Gamma jj - 1. are odd and ji \Gamma jj - 2. ffl The lengths are: ffl OE specifies ff i to be open if i is odd, and fl 4 to be open, but all other intervals to be closed. A realization of a F rame is said to be straight if fl 1 is to the left of fl 3 . Otherwise it is called reversed. Such a realization is located at the interval corresponding to fl 4 . adjacent to all ff; fi Fig. 6. A graph of a F rame (top) and its realization (bottom). The F rame structure is rigid and divides an interval into smaller intervals of prescribed lengths and positions. is rigid in a F rame. Proof. Let G 0 be the subgraph of G induced on V ff [ V fi . It is easy to see that G 0 is prime, and hence, has a unique clique order [34]. Moreover, G 0 has exactly k maximal cliques, each one containing (among other vertices) a unique and distinct ff i . The set of maximal cliques in G is fN [ff i namely, each clique is distinguished by a single ff i . Since G 0 is UCO, its unique clique order determines a unique linear order on V ff , and hence, also on the maximal cliques of G. Hence, G is UCO. Let S and S 0 be two realizations of the same F rame. Suppose are their leftmost endpoints, respectively. The F rame graph is UCO, hence, the order of the ff-intervals is identical in both S and S 0 . Moreover, all the ff-intervals are disjoint, and must be between 1 and fl 0 3 , which are at distance exactly s. Thus, the position of all ff endpoints is uniquely determined. It is easy to see that also all fi-intervals except must have identical position in both realizations. By lemma 5.8, for any straight (or reversed) realization of a F rame(x located at (x; x+ s), the positions of all intervals except are uniquely determined. In the sequel, when we use a realization of such a F rame to implicitly define a MIG instance, we shall assume that fi 1 and fi k are contained in [x; x+s], so the realization has the shortest possible length. In addition, when we use any gadget in the implicit definition, and we describe its intervals by saying that "the gadget is located at we mean that "a straight realization of the gadget is located at 5.2. The Reduction. The realization of a MIG instance is a polynomial witness for a "yes" instance, hence, MIG is in NP. We describe a reduction from 3-Coloring, which is NP-complete (see, e.g., [15]). Let E) be an instance of 3-Coloring. We construct an instance P= (T; L) of MIG (in implicit form), and prove that P is a "yes" instance if and only if G is 3-colorable. The general plan is as follows: We construct measured interval sub-instances for each vertex and for each edge of G. The sub-instance of a vertex is designed so that it can be realized only in three possible ways, which will correspond to its color. The sub-instance for each edge will prevent the vertices at its endpoints from having the same color. 5.2.1. The Vertex Sub-instance. Let 1. Define the following set of intervals (compare figures 7 and 8): 1), located at (0; 11). , and each of ffi 1 and ffi 2 is a Switch(3), located at (1; 2) and (7; 8), respectively. The superscripts match the vertex numbers in each Switch. long long 2 g, such that i [ located at (\Gamma2M; 2M ). Note that the intersection graph of ! [ i is prime. For each interval I 2 S let jIj. The sub-instance of each vertex is isomorphic to (S; L), and the F rames of the n vertices are layed out contiguously as follows: For an interval J and a real number 1g. For each Denote be the measured interval graph defined implicitly by S. \Gamma2M 2M A frame Fig. 7. This is the set of intervals S. ffi can be positioned in the frame. The distance between ffi 1 and ffi 2 is enforced by the i. Fig. 8. This is a sketch of the structure of a vertex. The Switches ffi can be positioned in the frame !. The distance between enforced by the i. A realization of a vertex sub-instance is called straight (respectively, reversed) if the realization of its ! is straight (respectively, reversed). Lemma 5.9. Let [ i2V S(i) 0 be a realization of P V , with . Then either every S(i) 0 is straight, or every S(i) 0 is reversed. Proof. It suffices to prove that l(fl 1 (i) follows from the identity of the zero-length intersecting intervals and the disjointness of are both, by lemma 5.8, at distance 11 from the former pair, respectively. be a straight realization of P V . Define the function (2) Call Col the coloring defined by S 0 . We now show that each vertex subgraph can be realized in exactly three distinct colors. This is also demonstrated in figure 9. Lemma 5.10. For each Color 0: Color 1: Color 2: Fig. 9. The three possible positions of in a vertex sub-instance, which will correspond to the three possible colors of the vertex. Proof. According to lemma 5.8, in each straight S(i) 0 , the positions of the intervals in ff(i) 0 relative to l(fl 1 (i) 0 ), are fixed. Assume w.l.o.g. For each J 2 for each But according to lemma 5.7: Therefore, and 5.2.2. The Edge Sub-instance. Let the edges of G be an edge in E, where j. For each edge e k we construct an edge sub-instance, that forces the colors of the vertices i and j to be different. moving frame fixed frame Y Z Fig. 10. This is a functional sketch of the edge sub-instance. The two Switches D can only reverse in their relative fixed positions inside the moving frame. Their distances from the corresponding Switches in the vertices respectively, are fixed. The moving frame itself can have different positions along the fixed frame. We first give an overview of this construction (compare Figure 10): Each edge is assigned a fixed rame w which contains two Switches, D 1 and D 2 , which are the heart of its sub-instance. The F rames of the edges are layed out contiguously to the right of the vertex F rames. The sub-instance is a collection of intervals fA designed so that: 1. D 1 and are kept at a fixed distance (this is done by the Y intervals). 2. D 2 and are kept at a fixed distance (this is done by Z). 3. D 1 and D 2 are restricted to be in one of four possible relative positions, allowing the four possible color differences between the vertices i and j (this is done by W ). 4. D 1 and D 2 together can undergo a translation, allowing the six possible color combinations of the vertices i and j, as demonstrated in figure 12 (this is done by A 0 and w). We now describe the construction in detail (compare Figure 11): Define the following set of intervals. Let 11n. For readability, we omit the parameter k whenever possible. is a F rame(5; 12; 1), located at (0; 1), located at (7; 16) Base(k). located at (8; is a Switch(4) located at (11; 12) Base(k). long long 2 g, such that Y [ is the located at long long 2 g, such that Z [ is the located at The length function on the sub-instance (X; L) is defined so that the Y -s in which we set: L(Y short long long This change, together with the +1 in the first parameter of the F etters of Y , forces a +1 shift on the location of ffi 1 (i). This shift will be crucial in forcing the vertices i and j to have different colors. Note that the intersection graph of X(k) \Gamma D(k) is prime. Note also that the left and right ends of the F etters sub-instances Y and Z are positioned way beyond the contiguous F rames, in all edge sub-instances and vertex sub-instances. This allows every F etters to move independently, and no vertex or edge subgraph is a module. J 0 be a set of intervals with the same intersection graph as of X, which satisfy the corrected length constraints, where X(i) straight (respectively, reversed) realization if the frame w(i) 0 is straight (respectively, reversed). A proof similar to that of lemma 5.9 implies: Lemma 5.11. For each 1 only if X(j) 0 is straight. The complete constructed instance is P= (S[ X;L), where the interval lengths L are implicit in each of the two types of subgraphs, and the only exception is the corrected length in the Y (k)-s. Due to this exception simple super-imposition of Sand X does not give a realization. Lemma 5.12. If S 0 is a realization of P for which S 0 is straight, then X 0 is straight. Proof. Recall that c 1 (1) and fl 3 (n) are the leftmost and the rightmost zero length intervals in the leftmost edge sub-instance, and the rightmost vertex sub-instance, respectively. Suppose, to the contrary, that X 0 is not straight. The zero-length intersecting intervals c 1 (1) 0 and fl 3 (n) 0 must be identical. W.l.o.g. these two intervals intersect, in contrary to our constructed interval graph. d Z Z Z Z tot Z long 2 Z Z long 1 Z Z \Gamma6k \Gammad Y Y Y tot Y longY Y longY shortY \Gamma6k \GammaD1 Base(k Y Fig. 1. The whole edgeFig. 11. The edge sub-instance: A moving frame can be positioned inside the fixed frame. The Switches D 1 and D 2 are positioned inside the moving frame. Each of D 1 and D 2 is connected to its vertex sub-instance via F etters. Lemma 5.13. then for every realization S 0 of P: Col(i) 6= Col(j). Proof. Assume w.l.o.g that the realization is straight, and that l(c 1 Base(k). Again, we omit the parameter k whenever possible. Surely The first inequality follows since C 0 1 and A 0 must intersect, the second inequality follows since A 0 0 and a 0should intersect, and the last equality follows since w 0 is rigid (lemma 5.8). Since Length(W we conclude that W straight. According to lemma 5.8 the relative positions of all the intervals in the F rame (except B 0 are fixed relative to l(C 0 The realization for each of D 1 and D 2 can be either straight or reversed, giving rise to four possible combinations of positions (Any of these combinations fixes the positions of with respect to l(C 0 1 )). In particular: Due to lemma 5.7, and the realization being straight: Therefore: Corollary 5.14. If P is a "yes" instance, then G is 3-colorable. Proof. If P is a measured interval graph, then it has a straight realization (since the realization can reversed completely). Define the coloring as described in (2). By lemma 5.13, Col is a proper 3-Coloring of G. Let us now prove the converse: Lemma 5.15. If G is 3-colorable, then P admits a realization. Proof. Let be a proper 3-coloring of G. We build a realization S 0 for the instance P as follows: 1. For the vertex position its Switches 2 as follows (compare figure 9): ffl If ffl If ffl If The rest of the intervals in the vertex subgraph are positioned accordingly (cf. lemma 5.10). 2. For the edge e the directions of the Switches D 1 (k) 0 and in the realization are determined by y, thus fixing the distance between D 3 . The absolute position of these Switches is determined according to Col(i) and Col(j), as follows (compare figure 12): and S[ X have the same intersection graph, all interval lengths match the prescribed lengths, and their endpoints meet the specification. From lemma 5.15 and corollary 5.14 we can finally conclude: Theorem 5.16. MIG is NP-complete. In fact, the same reduction implies strong NP-completeness, as 3-Coloring is strongly NP-complete and the reduction is also pseudo-polynomial. 5.3. Closing the Open Intervals. We have proved that recognizing a measured interval graph with specified endpoints is NP-complete. We now show that this problem is hard even where all the intervals are closed. Given an instance MIG, define a new instance P MIG (in which all intervals are closed), as follows: Let L(v) if v is closed Let P be an instance generated by the reduction in section 5.2. We shall prove that P has a realization if and only if P 0 has one. First, we observe that the construction introduced in the proof of theorem 5.16 has a special property: Let S be a realization in which the shortest non-zero length of an interval is C. S is called discrete if all the endpoints of its intervals are integer multiples of C. In that case, C is called the grid size of S. Remark 5.17. Fig. 12. The relative position of D 3and D 3forces the colors of the vertices to be different. By the proofs of lemma 5.15 and corollary 5.14, P has a realization if and only if it has a discrete realization, with grid size 1 . Lemma 5.18. If P has a realization, then P 0 has one. Proof. If P has a realization, then by remark 5.17, it has a discrete realization fI v g v2V (G) with grid size 1 . Construct the set of closed intervals fI 0 defined as follows: If I v is closed, let I 0 I v is open, let I 0 We lemma that this set is a realization of P 0 is discrete, the intervals I v and I u intersect if and only if I 0 v and I 0 u intersect, since if one (or both) of I v ; I u is open, then their overlap is at least 1 . Furthermore, clearly jI 0 Unfortunately, the converse of lemma 5.18 does not always hold for arbitrary MIG instances, as demonstrated in figure 13. We shall prove that the converse does hold for instances generated by the reduction in section 5.2. Fig. 13. In the MIG instance P on the left, the the numbers denote lengths, and the four intervals corresponding to the vertices marked 00 should be open. P has no realization, but P 0 has one, as shown on the right. Define the following order-oriented analog of MIG and MIG , respectively: Recognizing a Measured Interval Order with Specified Endpoints (MIO): INSTANCE: A partial order OE on a set V , a non-negative length L(v) for every and a function closedg. QUESTION: Is there an interval realization of (V; OE), in which the length of I v is exactly L(v), and I v is open if and only if MIO is the restriction of MIO to instances with all intervals closed. MIO can be solved in polynomial time [19, 24, 32], and that solution can be generalized to deal with open intervals and solve MIO as well. We need to generalize the notion of rigidness in the following manner: Definition 5.19. For a real p - 0, two realizations fI j g and fI 0 j g of the same interval graph are p-isometric if there exists a function that j for all j. We call U ' V (G) p-rigid in a MIO instance if in any two realizations of the instance, the sets of intervals realizing U are p-isometric. Note that in this case all endpoints of U are located at fixed distances from the leftmost endpoint, up to \Sigmap. Hence, every realization has the same length, up to \Sigmap. For an instance Q of MIO, define the following system of inequalities S(Q), on the variables fl v ffl If xOEy, and both x; y are closed: l x ffl If xOEy, and at least one of x; y is open: l x o, and both x; y are closed: l x o, and at least one of x; y is open: l x Q has a realization if and only if S(Q) has a feasible solution, since the left endpoints of the realization satisfy S(Q), and vice versa. Recall that D(S(Q)) is the distance graph of S(Q), as in the proof of lemma 3.2, and denote it D(Q), for short. Lemma 5.20. Let Q be an instance of MIO. If U is a strongly connected component in the union of all zero-weight cycles in D(Q), then U is rigid in Q. Proof. For vertices there is a zero-weight cycle c in D(Q) passing through both x and y. Let d (resp. \Gammad) be the weight of the path from x to y (resp. y to x) along c. Summing the inequalities in S(Q) along the two paths we get l y - l x respectively, implying l y \Gamma l every realization must satisfy S(Q), for every realization l y \Gamma l so U is rigid. The converse holds as well: Lemma 5.21. Let Q be a realizable instance of MIO, with U rigid in Q. Then for each x; y 2 U there is a zero-weight cycle in D(Q) containing both x and y. Proof. Suppose to the contrary x; y 2 U and there is no zero-weight cycle in D(Q) containing both x and y. Either there is a cycle in D(Q) through x and y, or there is no such cycle. If there is no cycle in D(Q) through x and y, then w.l.o.g. there is no path in D(Q) from x to y. fxg be the set of all vertices in V to which there is a path from y in D(Q) (including y itself). Then Q does not contain any inequalities v Let fI v g v2V be a realization for Q. Then fI realizes Q, with a different distance between the intervals corresponding to x and to y, contradicting the rigidness of U in Q. If there exists a cycle in D(Q) through x and y, then let c be such a cycle of minimum weight, and let l = w(c). By assumption l 6= 0, and since Q has a realization, by corollary 3.3 l ? 0. Let d be the weight of the path from x to y along c. For every \Delta, d d, consider a new directed graph D 0 (\Delta) obtained from D(Q) by adding two arcs xy and yx, both labeled -, with weights \Delta and \Gamma\Delta, respectively. Observe that adding the two arcs does not introduce any cycles of negative-weight or zero-weight cycles with strict arcs, into the graph. Hence, the augmented system corresponding to 0 (\Delta) has a realization. Moreover, in every realization of D 0 (\Delta), the distance between the left endpoints of x and y is \Delta. By choosing different values of \Delta, we contradict the rigidness of U . We can now generalize lemma 5.20: Lemma 5.22. Let Q be a realizable instance of MIO. For a non-negative p, let C be a union of cycles in D(Q), each of weight less than p. If U is a strongly connected component in C then U is jU jp-rigid in Q. Proof. For vertices by the definition of U , there is a simple path in C. Every edge x i x i+1 is in C, therefore there exists a path P i in C from x i+1 to x i s.t. x i x i+1 and a cycle in C. The concatenation of P path P from y to x in C. Moreover, the concatenation of P 0 and P is a cycle c in C (not necessarily simple) of weight at most (k \Gamma 1)p. Let we have established that w(c) = w(P d. Summing the inequalities in S(Q) along the two paths P 0 and P we get l y - l x realization of Q satisfies S(Q), so \Gammad - l x any two realizations are q-isometric. Lemma 5.23. Let OE) be a MIO instance, and let Q be the corresponding MIO instance (obtained by the transformation in (18)). Suppose U ' V (OE) is rigid in Q, and let then U is 2n 2 ffl-rigid in Q 0 . Proof. The weight of each arc in D(Q 0 ) changes by no more than 2ffl, compared to D(Q). Hence, the weight of every simple cycle changes by at most 2nffl. U is rigid in Q, hence, by lemma 5.21 it is contained in a strongly connected component W of a union of zero-weight cycles in D(Q). W is also a union of simple zero-weight cycles in D(Q). The weight of each such cycle in D(Q 0 ) is at most 2nffl. Hence, by lemma 5.22, W is 2jW jnffl-rigid in Q 0 , and so is U . We now return to the instance P generated by the reduction in the proof of theorem 5.16. Recall that P 0 is the instance obtained from P by the transformation (18). Suppose P 0 has a realization. Let OE be the corresponding interval order, and let Consider each of our gadgets: By lemma 5.4, every Switch is rigid in Q . A slight modification of lemma 5.7 shows that every F etters must be rigid in Q (since the directions of the Switches are set). By lemma 5.8 every F rame is rigid in Q, with the exception of its end fi-intervals. Hence, each of these gadgets is 2n 2 ffl-rigid in , by lemma 5.23. This imposes, up to small additive shifts, the relative positions of the intervals in each vertex (or edge) sub-instance. Define the function Col as in (2). We shall show that the choice of ffl makes these shifts sufficiently small so that the properties of the coloring are preserved. Lemma 5.24. For each exist C i , jC Proof. The proof is analogous to lemma 5.10: The relations (3)-(7) hold up to \Sigma2n 2 ffl. Hence, (8) holds up to \Sigma4n 2 ffl. Lemma 5.25. For every edge (i; j), Proof. The proof is analogous to lemma 5.13: The relations hold up to \Sigma2n 2 ffl. Relations hold up to \Sigma8n 2 ffl, as they involve up to four differences of endpoint distances. Let round(x) be the integer closest to x. Recall that ffl ! 1 0:2. By lemma 5.25, for every edge (i; j): jCol(i)\GammaCol(j)j - 0:6. Hence, round(Col(i)) 6= round(Col(j)). This proves that if there exist a realization to P 0 , by rounding the colors to the nearest integer we obtain a proper 3-coloring. By lemma 5.15 this implies the existence of a realization to P. Thus, P has a realization if and only if P 0 has one. Since the transformation described in (18) is polynomial, we conclude: Theorem 5.26. MIG is NP-complete. 5.4. Related Problems. In section 1 we have introduced the recognition problem of interval graph with individual lower and upper bounds on interval lengths (the BIG problem). Since MIG is a restriction of BIG and DCIG: Corollary 5.27. BIG and DCIG are NP-complete. When restricted to interval graphs with depth 0 decomposition trees (see [23] for a definition of the decomposition tree), i.e., to prime interval graphs, the MIG problem can be solved in polynomial time, using the algorithm devised in section 3 for UCO graphs. This depth bound is indeed tight, namely, when allowing deeper decomposition trees the problem is NP-complete: Proposition 5.28. MIG is NP-complete even when restricted to interval graphs with decomposition tree of depth 1. Proof. We shall see that besides the Switches ffi i (j) and D i (k), and the K 2 modules fc 3 (k); c 1 (k+1)g, there are no non-trivial modules in the interval graph constructed by the reduction in the proof of theorem 5.26: Let H be the graph obtained by contraction of the above modules. Suppose to the contrary that H contains a non-trivial module M , and suppose v; v; u are in the same vertex subgraph (or in the same edge subgraph) HU , then M " U is a non-trivial module in HU , contradicting the primality of the vertex subgraph (and the edge subgraph). Hence, are in different vertex/edge subgraphs. In this case, there are intervals in these subgraphs, which intersect only one out of u; v, in contradiction to M being a module. 6. Acknowledgments . We thank Phil Green for illuminating conversations on the physical mapping problems which motivated this work. We also thank Garth Isaak for helpful discussions. We further thank the referees for their useful remarks. --R Network Flows: Theory Maintaining knowledge about temporal intervals. Reasoning about plans. A linear time and space algorithm to recognize interval orders. On a routing problem. On the topology of the genetic fine structure. Testing for the consecutive ones property Establishing the order of human chromosome-specific DNA fragments On the detection of structures in attitudes and developmental processes. Linear time representation algorithms for proper circular arc graphs and proper interval graphs. Interval Orders and Interval Graphs. Classes of interval graphs under expanding length restrictions. Faster scaling algorithms for network problems. Khachiyan's algorithm for linear programming. Computers and Intractability: A Guide to the Theory of NP-Completeness A characterization of comparability graphs and of interval graphs. Algorithmic Graph Theory and Perfect Graphs. Graph sandwich problems. Complexity and algorithms for reasoning about time: A graph-theoretic approach Chromosomal region of the Cystic Fibrosis gene in yeast artificial chromosomes: a model for human genome mapping. Substitution decomposition on chordal graphs and applications. Discrete interval graphs with bounded representation. Mapping the genome: some combinatorial problems arising in molecular biology. Incidence matrices Transitive orientation of graphs with side constraints. An incremental linear time algorithm for recognizing interval graphs. Temporally Distributed Symptoms in Technical Diagnosis. New scaling algorithms for the assignment and minimum cycle mean problems. Scheduling interval ordered tasks. Satisfiability problems on intervals and unit intervals. Discrete Mathematical Models Partially ordered sets and their comparability graphs. Hyperplane arrangements Computation Structures. Proof of an interval satisfiability conjecture. --TR --CTR Peter Damaschke, Point placement on the line by distance data, Discrete Applied Mathematics, v.127 n.1, p.53-62, April

size constraints;distance constraints;interval graphs;NP-completeness;graph algorithms;computational biology

587957

Stack and Queue Layouts of Posets.

The stacknumber (queuenumber) of a poset is defined as the stacknumber (queuenumber) of its Hasse diagram viewed as a directed acyclic graph. Upper bounds on the queuenumber of a poset are derived in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. A lower bound of $\Omega(\sqrt n)$ is shown for the queuenumber of the class of n-element planar posets. The queuenumber of a planar poset is shown to be within a small constant factor of its width. The stacknumber of n-element posets with planar covering graphs is shown to be $\Theta(n)$. These results exhibit sharp differences between the stacknumber and queuenumber of posets as well as between the stacknumber (queuenumber) of a poset and the stacknumber (queuenumber) of its covering graph.

Introduction . Stack and queue layouts of undirected graphs appear in a variety of contexts such as VLSI, fault-tolerant processing, parallel processing, and sorting networks (Pemmaraju [13]). In a new context, Heath, Pemmaraju, and Ribbens [8, 13] use queue layouts as the basis of an efficient scheme to perform matrix computations on a data driven network. Bernhart and Kainen [1] introduce the concept of a stack layout, which they call book embedding. Chung, Leighton, and Rosenberg [3] study stack layouts of undirected graphs and provide optimal stack layouts for a variety of classes of graphs. Heath and Rosenberg [10] develop the notion of queue layouts and provide optimal queue layouts for many classes of undirected graphs. Heath, Leighton, and Rosenberg [7] study relationships between queue and stack layouts of undirected graphs. In some applications of stack and queue layouts, it is more realistic to model the application domain with directed acyclic graphs (dags) or with posets, rather than with undirected graphs. Various questions that have been asked about stack and queue Department of Computer Science, Virginia Polytechnic Institute and State University, Blacks- burg, VA 24061-0106 y Department of Computer Science, University of Iowa, Iowa City, IA 52242 layouts of undirected graphs acquire a new flavor when there are directed edges (arcs). This is because the direction of the arcs imposes restrictions on the node orders that can be considered. Heath, Pemmaraju, and Trenk [9, 13] initiate the study of stack and queue layouts of dags and provide optimal stack and queue layouts for several classes of dags. In this paper, we focus on stack and queue layouts of posets. Posets are ubiquitous mathematical objects and various measures of their structure have been defined. Some of these measures are bumpnumber, jumpnumber, length, width, dimension, and thickness [2, 6]. Nowakowski and Parker [12] define the stacknumber of a poset as the stacknumber of its Hasse diagram viewed as a dag. They derive a general lower bound on the stacknumber of a planar poset and an upper bound on the stacknumber of a lattice. Nowakowski and Parker conclude by asking whether the stacknumber of the class of planar posets is unbounded. Hung [11] shows that there exists a planar poset with stacknumber 4; moreover, no planar poset with stacknumber 5 is known. Sys/lo [15] provides a lower bound on the stacknumber of a poset in terms of its bump- number. He also shows that, while posets with jumpnumber 1 have stacknumber at most 2, posets with jumpnumber 2 can have an arbitrarily large stacknumber. The organization of this paper is as follows. Section 2 contains definitions. In Section 3, we derive upper bounds on the queuenumber of a poset in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. In Section 4, we show that the queuenumber of the class of planar posets is unbounded. In a complementary upper bound result, we show that the queuenumber of a planar poset is within a small constant factor of its width. In Section 5, we show that the stacknumber of the class of n-element posets with planar covering graphs is \Theta(n). In Section 6, the decision problem of recognizing a 4-queue poset is defined; Heath, Pemmaraju, and Trenk [9, 13] show that the problem is NP-complete. In Section 7, we present several open questions and conjectures concerning stack and queue layouts of posets. 2. Definitions. This section contains the definitions of stack and queue layouts of undirected graphs, dags, and posets. Other measures of the structure of posets are also defined. E) be an undirected graph without multiple edges or loops. A k-stack layout of G consists of a total order oe on V along with an assignment of each edge in E to one of k stacks, s Each stack s j operates as follows. The vertices of V are scanned in left-to-right (ascending) order according to oe. When a vertex v is encountered, any edges assigned to s j that have v as their right endpoint must be at the top of the stack and are popped. Any edges that are assigned to s j and have left endpoint v are pushed onto s j in descending order (according to oe) of their right endpoints. The stacknumber SN(G) of G is the smallest k such that G has a k-stack layout. G is said to be a k-stack graph if k. The stacknumber of a class of graphs C, denoted by SN C (n), is the function of the natural numbers that equals the least upper bound of the stacknumber of all graphs in C with at most n vertices. We are interested in the asymptotic behavior of SN C (n) or in whether SN C (n) is bounded above by a constant. A k-queue layout of G consists of a total order oe on V along with an assignment of each edge in E to one of k queues, Each queue q j operates as follows. The vertices of V are scanned in left-to-right (ascending) order according to oe. When a vertex v is encountered, any edges assigned to q j that have v as their right endpoint must be at the front of the queue and are dequeued. Any edges that are assigned to q j and have left endpoint v are enqueued into q j in ascending order (according to oe) of their right endpoints. The queuenumber QN(G) of G is the smallest k such that G has a k-queue layout. The queuenumber of a class of graphs C, denoted by QN C (n), is the function of the natural numbers that equals the least upper bound of the queuenumber of all graphs in C with at most n vertices. We are interested in the asymptotic behavior of QN C (n) or in whether QN C (n) is bounded above by a constant. For a fixed order oe on V , we identify sets of edges that are obstacles to minimizing the number of stacks or queues. A k-rainbow is a set of k edges such that i.e., a rainbow is a nested matching. Any two edges in a rainbow are said to nest. A k-twist is a set of k edges such that i.e., a twist is a fully crossing matching. Any two edges in a twist are said to cross. A rainbow is an obstacle for a queue layout because no two edges that nest can be assigned to the same queue, while a twist is an obstacle for a stack layout because no two edges that cross can be assigned to the same stack. Intuitively, we can think of a stack layout or a queue layout of a graph as a drawing of the graph in which the vertices are laid out on a horizontal line and the edges appear as arcs above the line. In a stack layout no two edges that intersect can be assigned to the same stack, while in a queue layout no two edges that nest can be assigned to the same queue. Clearly, the size of the largest twist (rainbow) in a layout is a lower bound on the number of stacks (queues) required for that layout. Heath and Rosenberg [10] show that the size of the largest rainbow in a layout equals the minimum queue requirement of the layout. Proposition 2.1. (Heath and Rosenberg, Theorem 2.3 [10]) Suppose E) is an undirected graph, and oe is a fixed total order on V . If G has no rainbow of more than k edges with respect to oe, then G has a k-queue layout with respect to oe. In contrast, the size of the largest twist in a layout may be strictly less than the minimum stack requirement of the layout (see [10], Proposition 2.4). The definitions of stack and queue layouts are now extended to dags by requiring that the layout order be a topological order. Following a common distinction, we use vertices and edges for undirected graphs, but nodes and arcs for directed graphs. Suppose that E) is an undirected graph and that ~ E) is a dag whose arc set ~ E is obtained by directing the edges in E. A topological order of ~ G is a total order oe on V such that (u; v) 2 ~ layout of the 6 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU dag ~ E) is a k-stack (k-queue) layout of the graph G such that the total order is a topological order of ~ G. As before, SN( ~ G) is the smallest k such that ~ G has a k-stack layout and QN( ~ G) is the smallest k such that ~ G has a k-queue layout. A partial order is a reflexive, transitive, anti-symmetric binary relation. A poset is a set V with a partial order - (see Birkhoff [2] or Stanton and White [14]). The cardinality jP j of a poset P equals jV j. We only consider posets with finite cardinality in this paper. We write v. The Hasse diagram ~ E) of a poset is a dag with arc set ~ there is no w such that u (see Stanton and White [14]). A Hasse diagram is a minimal representation of a poset because it contains none of the arcs implied by transitivity of -. The stacknumber SN(P ) of a poset P is SN( ~ H(P )), the stacknumber of its Hasse diagram. Similarly, the queuenumber QN(P ) of a poset P is QN( ~ H(P )), the queuenumber of its Hasse diagram. Fig. 1 gives an example of a 2-stack poset, while Fig. 2 gives an example of a 2-queue poset. The underlying undirected graph, H(P ), of ~ H(P ) is called the covering graph of P . Clearly, for any poset P , we have and The stacknumber and the queuenumber of the covering graphs of the posets in both Fig. 1 and Fig. 2 are 1. A poset P is planar if its Hasse diagram ~ H(P ) has a planar ae- ae- @@@I @ @ @ @I Fig. 1. A 2-stack poset. ae- -ae- oe- ae- QQQQQk ae ae ae ae ae? Fig. 2. A 2-queue poset. embedding in which all arcs are drawn as straight line segments with the tail of each arc strictly below its head with respect to a Cartesian coordinate system; call such an embedding of any dag an upwards embedding. Without loss of generality, we may always assume that no two nodes of ~ H(P ) are on the same horizontal line. (If two nodes are on the same horizontal line, a slight vertical perturbation of either of them yields another upwards embedding with the nodes on different horizontal lines). Given an upwards embedding of a dag, the y coordinates of the nodes give a topological order on the nodes from lowest to highest called the vertical order. Note that the covering graph H(P ) may be planar even though the poset P is not. Fig. 3 shows an example 8 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU ae- @@@I ae- ae- @ @ @ @I @ @ @ @ @ @ @ @ @I Fig. 3. A non-planar poset whose covering graph is planar. of a non-planar poset whose covering graph is planar. Let fl be a fixed topological order on ~ are adjacent in fl if there is no w such that spine arc in ~ with respect to fl is an arc (u; v) in ~ H(P ) such that u and v are adjacent in fl. A break in ~ H(P ) with respect to fl is a pair (u; v) of adjacent elements such that u ! fl v and (u; v) is not an arc in ~ connection C in ~ H(P ) with respect to fl is a maximal sequence of elements other words a connection is a maximal path of spine arcs without a break. Since ~ contains no transitive arcs, there can be no non-spine arcs between nodes in a connection. The breaknumber BN(fl; P ) of a topological order fl of ~ is the number of breaks in ~ H(P ) with respect to fl. The jumpnumber of P , denoted by JN(P ), is the minimum of BN(fl; P ) over all topological orders fl on ~ A chain in a poset P is a set of elements such that . The length L(P ) of a poset P is the maximum cardinality of any chain in P . An antichain in a poset P is a subset of elements of S that does not contain a chain of size 2. The width W (P ) of a poset P is the maximum cardinality of any antichain in P . 3. Upper Bounds on Queuenumber. In this section we derive upper bounds on the queuenumber of a poset in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph. 3.1. Jumpnumber and Queuenumber. Sys/lo [15] proves the following relationship between the jumpnumber and the stacknumber of posets. Proposition 3.1. (Sys/lo [15]) For any poset P with JN(P 2. If J 2 is the infinite class of posets having jumpnumber 2, then SNJ 2 n). In contrast, we show that, for any poset P , the queuenumber of P is at most the jumpnumber of P plus 1. Moreover, we show that this bound is tight within a small constant factor. Theorem 3.2. For any poset P , QN(P 1. For every n - 2, there exists a poset P such that jP Proof. For the upper bound on queuenumber, suppose that P is any poset and that JN(P be a topological order on ~ H(P ) that has exactly k breaks connections. Lay out ~ according to fl and label these connections from left to right. Let any two nonspine arcs such that u 1 and u 2 are in C i and v 1 and v 2 are in C j , where 1 and nest, then one of that nests over the other arc) is a transitive arc. Since ~ contains no transitive arcs, do not nest. This suggests the following assignment of arcs to queues. Assign all non-spine arcs between pairs of connections C i and C j , where to queue q ' . Assign all the spine arcs to a queue q 0 . Hence, we use k queues for non-spine arcs and one queue for spine arcs, for a total of k queues. For the lower bound on queuenumber, construct the Hasse diagram of a poset P from the complete bipartite graph K E) by directing all the edges from vertices in V 1 to vertices in V 2 . All topological orders on ~ layouts. Hence, JN(P The lower bound follows. Proposition 3.1 and Theorem 3.2 lead to the following corollary. Corollary 3.3. There exists a class of posets P for which the ratio QNP (n) is unbounded. Looking ahead, Theorem 4.2 shows the existence of a class of posets P for which the reciprocal ratio QNP (n)=SNP (n) is unbounded. 3.2. Length and Queuenumber. To prove the next theorem, we need the following lemma that gives a bound on the queuenumber of a layout of a graph whose vertices have been rearranged in a limited fashion. Lemma 3.4. (Pemmaraju [13]) Suppose that oe is an order on the vertices of an m-partite graph E) that yields a k-queue layout of G. Let oe 0 be an order on the vertices of G in which the vertices in each set V i consecutively and in the same order as in oe. Then oe 0 yields a layout of G in 2(m \Gamma 1)k queues. Theorem 3.5, the main result of this section, gives an upper bound on the queue- number of a poset in terms of its length and the queuenumber of its covering graph. Theorem 3.5. For any poset P , There exists an infinite class of posets P such that LP Proof. Suppose P is any poset, ~ E), and QN(H(P oe be a total order on V that yields a k-queue layout of H(P ). The nodes of ~ H(P ) can be labeled by a function l follows. Let ~ all the nodes with indegree 0 in ~ with the label 1. Delete all the labeled nodes in ~ H 0 to obtain ~ In general, label the nodes with indegree 0 in ~ with the label i + 1. Delete the labeled nodes in ~ H i to obtain ~ By an inductive proof, it can be checked that the labeling so obtained satisfies the required conditions. Let V ig. For any arc (u; v) 2 ~ is an L(P )-partite dag. Define the total order fl on the nodes of ~ 1. The elements in each set V i contiguously and in the order prescribed by oe. 2. The elements in V i occur before the elements in V i+1 for all Since every arc in ~ H(P ) is from a node in V i to a node in V j , 1 a topological order on ~ yields a layout that requires no more queues. We now prove the second part of the theorem. For each n - 2, let e. Let the complete bipartite graph K E) be such that jV 1 and We get the Hasse diagram of a poset P of size n by directing the edges in K p;q from V 1 to V 2 . Clearly, L(P and Pemmaraju [13] present different proofs of the following formula that gives the precise queuenumber of an arbitrary complete bipartite Therefore, Let P be the class of all posets constructed in the manner described above. The second part of the theorem follows. Note that Theorem 3.5 holds for dags as well as for posets as its proof does not rely on the absence of transitive arcs. Theorem 3.5 leads to the following corollary. Corollary 3.6. For any poset P , Suppose P is a class of posets such that there exists a constant K with all We conjecture, but have been unable to show, that the upper bound in Theorem 3.5 is tight, within constant factors, for larger values of L(P ) also. 3.3. Width and Queuenumber. In this section, we establish an upper bound on the queuenumber of a poset in terms of its width. We need the following result of Dilworth. Lemma 3.7. (Dilworth [4]) Let be a poset. Then V can be partitioned into For a poset be a partition of V into W (P ) chains. Define an i-chain arc as an arc in ~ H(P ), both of whose end points belong to is an arc whose tail belongs to chain Z i and whose head belongs to chain Z j . Theorem 3.8. The largest rainbow in any layout of a poset P is of size no greater than Hence, the queuenumber of any layout of P is at most W (P Proof. Fix an arbitrary topological order of ~ partition of V into W (P ) chains. For any i, no two i-chain arcs nest, since ~ 14 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU contains no transitive arcs. Therefore, the largest rainbow of chain arcs has size no greater than W (P ). If i 6= j then no two (i; j)-cross arcs can nest without one of them being a transitive arc. Therefore, the largest rainbow of cross arcs has size no greater than W (P 1). The size of the largest rainbow is at most W (P By Proposition 2.1, the theorem follows. The bound established in the above theorem is not known to be tight. In fact, we believe that the queuenumber of a poset is bounded above by its width (see Conjecture 1 in Section 7). 4. The Queuenumber of Planar Posets. In this section, we first show that the queuenumber of the class of planar posets is unbounded. We then establish an upper bound on the queuenumber of a planar poset in terms of its width. 4.1. A Lower Bound on the Queuenumber of Planar Posets. We construct a sequence of planar posets P n with jP n). In fact, we determine the queuenumber of P n almost exactly. To prove the theorem, we need the following result of Erd-os and Szekeres. Lemma 4.1. (Erd-os and Szekeres [5]) Let be a sequence of distinct elements from a set X. Let ffi be a total order on X. Then either contains a monotonically increasing subsequence of size d e or a monotonically decreasing subsequence of size d e with respect to ffi . The proof of Theorem 4.2 constructs the desired sequence of posets. Theorem 4.2. For each n - 1, there exists a planar poset P n with 3n+3 elements such that l p \Upsilon Proof. Suppose n - 1. Define three disjoint sets U; V , and W as follows: ng ng . The planar poset P is given by Fig. 4 shows the Hasse diagram of P 4 . Let oe be an arbitrary order on the elements of S. The elements of U appear in the order in oe, and all elements of W appear between u n and v n . Define a total order ffi on the elements of W by w 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i 'j 'i @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ Fig. 4. The planar poset P4 . is an increasing sequence of nodes in W with respect to ffi . Since w appears after u in any topological order of ~ the following sequence of nodes is a subsequence of oe: Therefore, the set f(u i j kg is a k-rainbow in oe. Similarly, if is a decreasing sequence of nodes in W with respect to ffi, then the set f(w kg is a k-rainbow in oe. By Lemma 4.1, in oe, there is an increasing subsequence of size l p or a decreasing subsequence of size l p with respect to ffi . Thus there is a rainbow of size l p in any topological order on ~ l p . This is the desired lower bound. To prove the upper bound, we give a layout of P n in d queues. Let e, and let e. Partition W \Gamma fw 0 g into s nearly equal-sized subsets as follows: ng Construct an order oe on the elements of S by first placing the elements in U [ in the order Now place the elements of W \Gamma fw 0 g between u 0 and v 0 such that the elements belonging to each set W i appear contiguously and the sets themselves appear in the order Within each set W i place the elements in increasing order with respect to Fig. 5 schematically represents the constructed order. The arcs from U to W form Fig. 5. Schematic layout of planar poset Pn . s mutually intersecting rainbows each of size at most t. Therefore t queues suffice for these arcs. The arcs from W to V form s nested twists each of size at most t. Therefore s queues suffice for these arcs. Since no two arcs, one from U to W and the other from W to V nest, they can all be assigned to the same set of s queues. An additional queue is required for the remaining arcs. This is a layout of P n in d queues. Therefore, We believe that the upper bound in the above proof can be tightened to exactly match the lower bound. In fact, we have been able to show that for l p The situation for stacknumber of planar posets is somewhat different in that there is no known example of a sequence of planar posets with unbounded stacknumber. Two observations about the sequence P n in Theorem 4.2 are in order. The first observation is that SN(P n 2. A 2-stack layout of ~ shown in Fig. 6. The second observation is that the stacknumber and the queuenumber of H(P n ) is 2. A 2-queue layout of H(P 4 ) is shown in Fig. 7. Theorem 4.2 and the above observations imply the following corollaries. ff \Phi ff \Phi ff \Phi ff \Omega \Psi\Omega \Psi\Omega \Psi\Omega \Psi\Omega \Psi\Omega \Psi l l l l l l l l l l l l l l l Fig. 6. A 2-stack layout of the planar poset P4 . ff \Phi ff \Phi ff \Phi ff \Omega \Psi\Omega \Psi\Omega \Psi\Omega \Psi ff l l l l l l l l l l l l l l l Fig. 7. A 2-queue layout of the covering graph of P4 . Corollary 4.3. There exists a class P of planar posets such that QNP (n) =\Omega \Gammap Corollary 4.4. There exists a class P of planar posets such that QNP (n) QN H(P) (n) While Theorem 4.2 establishes a lower bound of \Omega\Gamma n) on the queuenumber of the class of n-element planar posets, a matching upper bound is not known (see Conjecture 2 in Section 7). 4.2. An Upper Bound on the Queuenumber of Planar Posets. In this section, we show that the queuenumber of a planar poset is bounded above by a small constant multiple of its width. The bound is a consequence of the following theorem, the proof of which occupies the remainder of the section. Theorem 4.5. For any planar poset P where ~ contains at least one arc and for any upward embedding of ~ H(P ), the layout of ~ by the vertical order oe has queuenumber less than 4W (P ). Before the proof of Theorem 4.5, we present some definitions, some observations, and a series of three lemmas. First, we fix notation and terminology to use through-out the section. Suppose that poset with a given upwards embedding of ~ oe be the vertical order on V . Now suppose that the size of a largest rainbow in the vertical order of ~ H(P ) is k - 1. By Proposition 2.1, the queuenumber of this layout is k. Focus on a particular k-rainbow whose arcs are these arcs the rainbow arcs ; in particular, the arc is the rainbow arc of a i and of b i . The nodes in the set are bottom nodes, and the nodes in the set are top nodes. Let y(v) denote the y-coordinate of a node v in the upwards embedding. Suppose (a and (a are distinct rainbow arcs. Since these arcs nest in the vertical order oe, we know that maxfy(a i )g. More generally, The horizontal line defined by the equation In moving along this line from left to right, we encounter these intersections in a definite \Omega \Omega OE \Theta \Theta \Theta \Theta \Theta \Theta \Theta \Theta \Theta \ThetaffiP Z Z Z ZZ- ae ae? A A A A A A A A A A A A A A A A A A @ @ @I a 5 a 6 a 4 a 3 a 1 a 2 Fig. 8. An example of rainbow arcs. order. By re-indexing the rainbow arcs, we may assume that these intersections are encountered in the order (a call this the left-to-right order of the rainbow arcs. Fig. 8 illustrates an upwards embedding of a Hasse diagram with 6. The arcs are indexed in left-to-right order. Define the left-to-right total order -LR on A (respectively, B) by a i -LR a j (respectively, j. If a i -LR a j , we say that a i is to the left of a j and that a j is to the right of a i . These notions of left and right do not always correspond to our normal understanding of these notions when looking at an upwards embedding. For example, in Figure 8, the x-coordinate of a 1 is greater than that of a 2 , though a 1 !LR a 2 and hence a 1 is to the left of a 2 . We consistently use left and right with 22 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU respect to the order -LR . A bottom chain is any chain of bottom nodes, and a top chain is any chain of top nodes. In Figure 8, the set fa 1 ; a 3 ; a 4 g is a bottom chain, while the set fa 2 ; a 3 ; a 5 g is not. If C is a chain of P and u; v 2 V , then the closed interval from u to v is the subchain C[u; and the open interval from u to v is the subchain C(u; vg. Subchains C(u; v] and C[u; v), the corresponding half-open intervals, are defined analogously. For any bottom chain C, the extent of C is min a j 2C that is, the extent is the distance from the leftmost node in C to the rightmost node in C, measured in rainbow arcs. The extent of a top chain is defined analogously. Suppose C is any chain. We say that C covers the nodes it contains. If D is a path in ~ H(P ) that contains every node of C, then D covers C. Note that there must be at least one path in ~ H(P ) that covers C. In what follows, we show that more than k=4 chains are required to cover the set A[B. Since W (P ) is the minimum number of chains required to cover all the nodes in the poset, it follows that k=4 ! W (P ) and therefore QN(P As the proof is long and tedious, we give here an informal overview. Start with a partition CA of A into bottom chains and a partition CB of B into top chains. Because each element of is a chain, there is a path in ~ covering it. Thinking of each such path as a vertex, we construct a graph G that contains an edge connecting a pair of vertices if the corresponding paths in ~ H(P are connected by a rainbow arc. It is easy to see that G is planar if the paths in ~ covering the chains in CA [A B are pairwise non- intersecting. The construction of a collection of pairwise non-intersecting paths that cover the chains of CA [ CB is not always possible. This leads us to the weaker notion of a crossing of two chains and to the construction of G from chains rather than paths. Since the final step of the proof requires G to be planar, we first show (Lemmas 4.7 and 4.8) that all crossings between pairs of chains can be eliminated. Applying Euler's formula to the resulting planar G finally yields the bound in Theorem 4.5. At this point, we restrict our argument to bottom nodes, as the corresponding argument for top nodes is similar. If C is any bottom chain, the order in which its elements appear with respect to - P is constrained by the rainbow arcs. In particular, we make the following observation. Observation 1. Suppose C is a bottom chain whose nodes occur in the following order with respect to - For any i with Similarly, for any i with Intuitively, if the chain starts going to the right after c i , then the remainder of the chain must be to the right of the rainbow arc of c i . The rainbow arc of c i is a barrier to the chain reaching a bottom node to the left of c i . For example, in Fig. 8, the rainbow arc (a 5 ; b 5 ) is a barrier to any path originating at a 6 . Since a 5 ! P a 6 and a 5 !LR a 6 , no bottom chain containing both a 5 and a 6 has a node a i ? P a 6 to the left of a 5 . By Lemma 3.7, there is a partition of A into at most W (P ) chains. Let CA be such a partition. Let C 1 2 CA have the order and let C have the order These two bottom chains cross if there exist c c p !LR d r !LR c q !LR d s or c p ?LR d r ?LR c q ?LR d s ; in such a case, the 4-tuple crossing of C 1 and C 2 . Since c p and c q are related by - P , there is a directed path D 1 in ~ Similarly, there is a directed path D 2 in ~ Lemma 4.6. D 1 and D 2 have at least one node in common. Proof. Without loss of generality, assume that c p !LR d r !LR c q !LR d s . Consider the polygonal path consisting of the horizontal ray from c p to \Gamma1, followed by the line segments completed by the horizontal ray from d s to 1. Let R be the region of the plane consisting of this polygonal path and all points below it. (Fig. 9 illustrates the region R derived from Fig. 8 with crossing Topologically, R is a 2-dimensional ball with a single boundary point removed. Topologically, D 1 and D 2 are paths in the plane with endpoints on the boundary of R. By Observation 1, neither path can cross either of the two infinite oe a 2 a 4 a 5 a 1 Fig. 9. The region R. rays. Also, neither path can pass above the rainbow arc of c q or d r , because every top node is higher than any bottom nodes in the upwards embedding of ~ either path crosses one of the three line segments of the polygonal path and proceeds outside of R, then that path must return to the polygonal path at a higher point on the same line segment. In essence, we can disregard any excursions outside of R and assume, from a topological viewpoint, that both paths remain within R. The nodes of D 1 and D 2 alternate along the polygonal path. Hence, these paths must intersect topologically, and D 1 and D 2 must have at least one node in common. A node that D 1 and D 2 have in common is an intersection of C 1 and C 2 . Note that an intersection need not be a bottom node. In Fig. 8, the chains fa 1 ; a 3 ; a 4 g and cross and have the intersection v, which is not a bottom node. Observation 2. Since, with respect to - P , an intersection associated with the crossing and c q and between d r and d s , we have these 26 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU relations: The following observation is helpful in constructing pairs of non-crossing chains. Observation 3. Suppose C do not cross. If no d c with respect to -LR or if no d with respect to -LR , then C 1 and C 2 do not cross. We wish to be able to assume that CA does not contain a pair of crossing chains. The first of two steps in justifying that assumption is to show that we can replace two crossing chains with two non-crossing chains according to the following lemma. The replacing pair is further constrained to satisfy the 5 properties in the lemma. The need for Properties 1, 2, and 3 is clear. Property 4 states that, if the original pair crosses, then the replacing pair is smaller, in a precise technical sense, than the original hence the process of replacement of a crossing pair by a noncrossing pair cannot be repeated forever. Property 5 allows us to identify the minima in the replacing this property is a technical condition useful only within the inductive proof of the lemma. Lemma 4.7. Suppose C 1 and C 2 are disjoint bottom chains. Then there exists a function NC that yields a pair of bottom chains (C 0 properties: 1. C 0 2. C 0 are disjoint ; 3. C 0 2 do not cross ; 4. The sum of extents does not increase: if equality holds and if C 1 and C 2 cross, then the minimum extent decreases: and 5. Chain minima are preserved: Proof. In addition to our previous notation for C 1 and C 2 , we define By Observation 1, either c choose a path D 1 from c 1 to fi that covers the subchain C 1 choose a path D 1 from c 1 to ff that covers the subchain C 1 Similarly, if d choose a path D 2 from d 1 to ffi that covers the subchain C 2 [d choose a path 28 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU D 2 from d 1 to fl that covers the subchain C 2 [d 1 ; fl]. By Observation 1, both paths are monotonic with respect to -LR . We proceed to show the lemma by induction on the pair (m; n). Recall that m is the cardinality of C 1 and n is the cardinality of C 2 . The base cases are all pairs (m; n) with either In these cases, C 1 and C 2 do not cross, and setting yields the desired pair of bottom chains. For the inductive case, we assume that m - 2, that n - 2, and that the lemma holds for (m show that the lemma then holds for C 1 and C 2 . Without loss of generality, we assume ff !LR fl. There are now three main cases depending on the relative order of ff, fi, fl, and ffi with respect to !LR . Case 1: ff !LR fi !LR fl !LR ffi . In this case, C 1 and C 2 do not cross and the lemma trivially holds. Case 2: ff !LR fl !LR fi !LR ffi . In this case, C 1 and C 2 necessarily cross. There are four subcases. Case 2.1: necessarily contain at least one intersection. Let v be the intersection that occurs first in going from ff to fi on D 1 . The subpath D 0 1 of D 1 from c 1 to v does not meet the subpath D 0 2 of D 2 from d 1 to v until v. Hence, unless D 0 consists only of d 1 (that is, d one of D 0 1 and D 0is above the other in the upwards embedding. D 0 1 cannot be above D 0 because the rainbow arc of d 1 is a barrier to D 0 going above d 1 . Hence, either D 0 consists only of d 1 or D 0 2 is above D 0 1 . There are two subcases, depending on the relative order of c 2 and v according to P . Case 2.1.1: 2 is on D 0 1 and the rainbow arc of c 2 must not be a barrier for D 0 2 , we have c 2 !LR d 1 . Let (C 0 we set 0). For this case only, we provide a full proof that the lemma holds for leaving the details for the remaining cases to the reader. We employ the properties that hold for (C 0 by the inductive hypothesis. By Property 5 of the inductive hypothesis, C 0 are bottom chains with c 2 . We must show that C 0 is a bottom chain. If d have since any path in ~ between c 1 and d j must cross D 0 and v. In any case, for any d j 2 C 0 2 ) is a pair of bottom chains, as required. We now establish that properties. 1. By Property 1 of the inductive hypothesis, C 0 2. By Property 2 of the inductive hypothesis, C 0and C 0are disjoint. Since are disjoint. 3. By Property 3 of the inductive hypothesis, C 0 2 do not cross. Since there is no node of C 2 between c 1 and c 2 . Also, by Observation 1 there is no node in C 1 that is between c 1 and c 2 . Therefore there is no node in C 0 hence by Observation 3, C 0 2 do not cross. Since there is no node of C 0 with respect to -LR , 2 do not cross by Observation 3. 4. To be definite, let of the induction hypothesis and the fact that c 1 !LR c 2 !LR fi, we have and, if equality holds and if C If hC 0 then we are done. So assume that hC 0 1 , a contradiction to C 0 not crossing. Hence Then we have Hence Property 4 holds for (C 0 5. By Property 5 of the inductive hypothesis, c and This completes the full proof for the case c 2 ! P v. Case 2.1.2: . For this case, let c a y . We have y. Consider the relative left-to-right positions of c 2 and d 2 . First suppose that d 2 !LR c 2 . Since d 2 !LR c 2 !LR ffi , no node in C 2 (d 2 ; d n ] is between d 1 and d 2 . Since the subpath of D 2 from d 2 to ffi must go below or through must be above d 2 in the vertical order. Hence no node of C 1 is between d 1 and d 2 . Let (C 0 pair of noncrossing chains. Set We need to show that 4. By Property 4 of the inductive hypothesis, Calculate If hC 0 holds. So assume hC 0 (that is C 0 2. Since d 2 2 C 0and C 0 2 do not cross, 1 . Hence hC 0 We have Hence Property 4 holds. Now suppose that c 2 !LR d 2 . There are finally three subcases to consider. Case 2.1.2.1: d 2 !LR ffi and c 2 !LR fi. Let (C 0 ]). There are no nodes of C 1 [ C 2 between d 1 and c 2 . So set 2 is a chain that does not cross 1 . By Property 4 of the inductive hypothesis, and, if equality holds, then either fc 1 do not cross, or We proceed to show that Property 4 holds for C 0and fd 1 and hence If this inequality is strict, then Property 4 holds. If equality holds, then one of two possibilities holds. First suppose that fc 1 do not cross. In that case, we have fi !LR d 2 !LR ffi and Second suppose that Then 34 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU For both possibilities, Property 4 holds. We conclude that the desired pair of chains. Case 2.1.2.2: d 2 !LR ffi and c are chains, and they do not cross. Setting give the desired pair of chains. Since and Property 4 holds. Case 2.1.2.3: d is leftmost and d 2 rightmost in C 1 [ C 2 , the pair (C 0 2 ) is also noncrossing. Let a of the inductive hypothesis, we have We proceed to show Property 4 for (C 0 If this inequality is strict, then we are done. Otherwise, hC 0 We have Hence Property 4 holds for (C 0 Case 2.2: . In this case, C 1 and C 2 always cross. If we succeed in replacing these with two non-crossing chains C 0 2 having the same nodes, then maxLR C 0 2 . Hence, Property 4 follows easily for every (C 0 constructed for this case. Again, let v be the first intersection of D 1 and D 2 . If v 2 A, then all of C 1 (v; c m ] is to the right of v, and all of C 2 (v; d n ] is to the left of v. If v 62 36 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU setting gives the desired pair of chains. If v 2 setting gives the desired pair of chains. In either case, C 0 2 do not cross. If v 62 A, then the argument is a bit more involved. Otherwise, if c 2 !LR fl, then let (C 0 the desired pair of chains. If fi !LR d 2 , then let (C 0 Setting gives the desired pair of chains. Hence suppose fl !LR c 2 and d 2 !LR fi. Since the rainbow arcs of c 2 and d 2 are barriers, we have . By Observation 1, there are four possibilities. Case 2.2.1: C 1 is to the left of c 2 and C 2 (d 2 ; d n ] is to the left of d 2 . If remains to the right of d 2 , then set for all j - 2. Hence is a chain, and there are no nodes of c 2 and d 1 . Let (C 0 gives the desired pair of chains. Case 2.2.2: C 1 is to the left of c 2 and C 2 (d 2 ; d n ] is to the right of d 2 . Let gives the desired pair of chains. Case 2.2.3: C 1 is to the right of c 2 and C 2 (d 2 ; d n ] is to the left of d 2 . Here do not cross. Setting gives the desired pair of chains. Case 2.2.4: C 1 is to the right of c 2 and C 2 (d 2 ; d n ] is to the right of d 2 . This is the left-to-right mirror image of 2.2.1. The same argument applies, mutatis mutandis. Case 2.3: This case cannot occur because the rainbow arcs of c 1 and d 1 are barriers to the paths D 1 and D 2 . It would require both D 1 to go below d 1 and D 2 to go below c 1 , which is impossible. Case 2.4: This case is the left-to-right mirror image of Case 2.1. The same argument applies, mutatis mutandis. Case 3: ff !LR fl !LR ffi !LR fi. In this case, C 1 and C 2 may cross. There are again four subcases. Case 3.1: First suppose c 2 !LR d 1 . Let (C 0 desired pair of chains is Suppose d 1 !LR c 2 !LR ffi . Then D 1 and D 2 necessarily have an intersection before c 2 and before ffi . This is handled as in Case 2.1. Suppose ffi !LR c 2 and c 2 6= fi. is to the right of c 2 , C 2 is between c 1 and c 2 , and C 1 and C 2 do not cross. Finally, suppose ffi !LR c 2 and c Since all of C 1 with respect to -LR , and since 2 do not cross. The desired pair of chains is 2 ). It is necessary to justify Property 4. Let where min is taken with respect to -LR . There are three subcases. 38 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU Case 3.1.1: - !LR fl. Note that all of C is to the right of d and fl are unrelated with respect to - P or if 2 , since . being to the left of fl, is unrelated to every node in C 2 [d 2 ; d n ]; again 1 , we have hC 0 Applying Property 4, we must have cross. It follows that hC 0 if C 1 and C 2 cross. Case 3.1.2: case is the same as Case 2.2. For all the possibilities in that case, we get that hC 0 as desired. Case 3.1.3: . In this case, C 0 do no cross. Hence, neither do C 0 and C 0Case 3.2: c . In this case, C 1 and C 2 do not cross, as the rainbow arc of d 1 is a barrier to D 1 crossing D 2 . Case 3.3: This case is the left-to-right mirror image of Case 3.2. Case 3.4: This case is the left-to-right mirror image of Case 3. The second and last step in justifying the assumption converts any CA into a C 0 A that has no pair of crossing chains. Lemma 4.8. Suppose CA is a set of disjoint bottom chains of minimum cardinality that covers A. Then there exists a set C 0 A of disjoint bottom chains that covers A such that jC 0 no pair of chains in C 0 A cross. Proof. If CA contains no pair of crossing chains, then C 0 is the set required for the lemma. Otherwise, let C be a pair of chains that cross. By Lemma 4.7, there exist chains C 0 2 such that by substituting these chains for C 1 and C 2 , we get the set C 00 which is also a set of bottom chains of minimum cardinality that covers A. By Property 4, either (i) the sum of the extents of chains in C 00 A is strictly less than the sum of the extents of chains in CA or ig. Since every chain has extent at least 0, repeated substitution of a pair of crossing chains by a pair of non-crossing chains must eventually reduce the sum of the extents of the chains. Again, since every chain has extent at least 0, the sum of the extents of the chains cannot reduce infinitely, and hence we must eventually arrive at a set C 0 A that contains no pair of non-crossing chains. This set C 0 A is the set required for the lemma. We are finally prepared to prove our main result. Proof of Theorem 4.5. By Lemma 4.8, we may assume that CA contains no pair of crossing chains. Now let CB be a partition of B into at most W (P ) chains. Similarly, we may assume that CB contains no pair of crossing chains. Consider an arbitrary bottom chain C and an arbitrary top chain C 0 . It is possible that a rainbow arc connects a node in C to a node in C 0 . However, it is not possible for more than one rainbow arc to connect C and C 0 , for then one of the rainbow arcs (the "longest" one) would be a transitive arc in ~ H(P ). For example, in Figure 8, we cannot have a bottom chain and a top chain C for then there is a path from b 2 to b 1 and (a We now construct a bipartite graph contains an edge between there is a rainbow arc connecting C to C 0 . Since every rainbow arc connects exactly one bottom chain to exactly one top chain, there is exactly one edge in G for every rainbow arc; that is, k. Since there is no pair of crossing bottom chains and no pair of crossing top chains, G is planar. As an example, Figure 10 illustrates a graph E) obtained from the poset of Figure 8. In particular, and 'j 'i `j 'i 'j 'i 'j 'i OEAE OEAE OEAE OEAE \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega \Omega @ @ @ @ @ @ @ @ @ @ @ a 3 ; a 4 Fig. 10. A bipartite planar graph E) corresponding to the poset in Figure 8 According to Euler's formula for planar graphs, we have (1) where f is the number of faces in a planar embedding of G and t is its number of connected components. If G consists of a single edge, then Otherwise, since G is bipartite, we have the following inequality: (2) Combining Equations 1 and 2, we obtain We know that and that both jC A j and jC B j are at most W (P ). Substituting 42 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU into Equation 3, we obtain Hence, the queuenumber of ~ H(P ) with respect to oe is less than 4W (P ). Corollary 4.9. For any planar poset P where ~ contains at least one arc, We believe that this result can be improved to show that, for any poset P , there exists a W (P)-queue layout of ~ H(P ). See Conjecture 1 in Section 7. 5. Stacknumber of Posets with Planar Covering Graphs. In this section, we construct, for each n - 1, a 3n-element poset R n such that H(R n ) is planar and hence has stacknumber at most 4 (see Yannakakis [19]), but such that the stacknumber of the class 1g is not bounded. Theorem 5.1. For each n - 1, there exists a poset R n such that jR n Proof. Suppose n - 1. Define three disjoint sets U , V , and W as follows ng ng STACK AND QUEUE LAYOUTS OF POSETS 43 The poset R is given by Fig. 11 shows H(R 4 ). Aside. While the covering graph H(R n ) is clearly planar, the poset R n is not planar. This can be seen as follows. In any upward embedding of ~ in the plane, the nodes have increasing y-coordinates. Thus, any point in the plane whose y-coordinate is between the y-coordinates of u 1 and v 2 lies either on the left or on the right of the path Now add the nodes w 1 and w 2 to the embedding. Their y-coordinates are between the y-coordinates of u 1 and v 2 because of u 44 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU oe oe oe oe oe oe 'j 'i 'j 'i `j 'i 'j 'i 'j 'i `j 'i 'j 'i 'j 'i `j 'i 'j 'i 'j 'i `j 'i Fig. 11. The covering graph of R4 . If both w 1 and w 2 are embedded on the same side of D, then the paths must cross somewhere. If w 1 and w 2 are embedded on different sides of D, then the line segment (w line segment in D. End Aside. To prove the lower bound on SN(R n ), let oe be any topological order on ~ The order oe contains the elements of U [ V in the order and the elements of W in the order w . The elements of W are mingled among the elements of U [ V . Suppose occur before u n in oe, while occur after u n . Then the arcs form a k-twist, while the arcs form an (n \Gamma k)-twist. Hence, Therefore, dn=2e, as desired. The proof of the upper bound is constructive. An n-stack layout of R n is obtained by laying out the elements of U [ V in the only possible order, and then placing each immediately after u i for all n. The assignment of arcs to stacks is as follows. Assign each arc in the set f(u assign each arc in the set f(u stack s n . Note that no two arcs assigned to the same stack intersect. The only arcs remaining to be assigned are the arcs in the set The arcs (v do not intersect any other arc and can be assigned to any stack. Each arc assigned to stack s i+1 and arc assigned to stack s 1 . An n-stack layout of R n is obtained. The upper bound follows. Two observations about the poset R n constructed in the above proof are in order. The first observation is that QN(R n 2. A 2-queue layout of R 4 is shown in Fig. 12. In general, the total order used in the n-stack layout of R n described in the above ff \Phi ff \Phi ff \Phi ff \Omega \Psi\Omega \Psi\Omega \Psi l l l l l l l l l l l l Fig. 12. A 2-queue layout of R4 . ff \Phi ff \Phi ff \Phi ff ff \Phi ff \Phi ff \Phi ff \Phi ff \Phi ff \Phi ff l l l l l l l l l l l l Fig. 13. A 2-stack layout of the covering graph of R4 . proof yields a 2-queue layout of R n . The second observation is that the stacknumber and the queuenumber of the covering graph H(R n ) is 2. A 2-stack layout of H(R 4 ) is shown in Fig. 13. In general, a 2-stack layout of H(R n ) can be obtained because hamiltonian planar graph [1]. Theorem 5.1 and the above observations lead to the following corollaries. Corollary 5.2. There exists a class of posets such that QNR (n) Corollary 5.3. There exists a class of posets R n such that 6. NP-completeness Results. Heath and Rosenberg [10] show that the problem of recognizing a 1-queue graph is NP-complete. Since a 1-stack graph is an outerplanar graph, it can be recognized in linear time (Sys/lo and Iri [16]). But Wigderson [17] shows that the problem of recognizing a 2-stack graph is NP-complete. Heath, Pemmaraju, and Trenk [9, 13] show that the problem of recognizing a 4-queue poset is NP-complete. Formally, the decision problem for queue layouts of posets is POSETQN. POSETQN INSTANCE: A poset P . Does P have a 4-queue layout? Theorem 6.1. (Heath, Pemmaraju, and Trenk [9, 13]) The decision problem POSETQN is NP-complete. Since the Hasse diagram of a poset is a dag, this result hold for dags in gen- eral. This result is in the spirit of the result of Yannakakis [18] that recognizing a 3-dimensional poset is NP-complete. 48 LENWOOD S. HEATH AND SRIRAM V. PEMMARAJU 7. Conclusions and Open Questions. In this paper, we have initiated the study of queue layouts of posets and have proved a lower bound result for stack layouts of posets with planar covering graph. The upper bounds on the queuenumber of a poset in terms of its jumpnumber, its length, its width, and the queuenumber of its covering graph, proved in Section 3, may be useful in proving specific upper bounds on the queuenumber of various classes of posets. We believe that the upper bound of W (P ) 2 on the queuenumber of an arbitrary poset P , proved in Section 3, and the upper bound of 3W (P on the queuenumber of any planar poset P , proved in Section 4 are not tight. We conjecture that: Conjecture 1. For any poset P , QN(P We have established a lower bound of \Omega\Gamma p n) on the queuenumber of the class of planar posets. We believe that this bound is tight and conjecture that: Conjecture 2. For any n-element planar poset P , QN(P n). We conjecture that another upper bound on the queuenumber of a planar poset P is given by its length L(P ). We believe that it is possible to embed a planar poset in an "almost" leveled-planar fashion with L(P ) levels. (See Heath and Rosenberg [10] for a definition of leveled-planar embeddings.) From such an embedding, a queue layout of P in L(P ) queues should be obtainable. Therefore we conjecture that: Conjecture 3. For any planar poset P , QN(P In Section 5, we show that the stacknumber of the class of n-element posets having planar covering graphs is \Theta(n). However the stacknumber of the more restrictive class of planar posets is still unresolved. Acknowledgments . This research was partially supported by National Science Foundation Grant CCR-9009953. We thank Praveen Paripati for his helpful com- ments. We are also grateful for the helpful comments of the referees, especially the elucidation of an error in the original statement of Theorem 4.5. --R The book thickness of a graph Lattice Theory Embedding graphs in books: a layout problem with applications to VLSI design A decomposition theorem for partially ordered sets Thickness of ordered sets Comparing queues and stacks as mechanisms for laying out graphs Sparse matrix-vector multiplication on a small linear array Stack and queue layouts of directed acyclic graphs Laying out graphs using queues A planar poset which requires 4 pages. Ordered sets Exploring the Powers of Stacks and Queues via Graph Layouts Springer Verlag The complexity of the Hamiltonian circuit problem for maximal planar graphs The complexity of the partial order dimension problem --TR --CTR Josep Daz , Jordi Petit , Maria Serna, A survey of graph layout problems, ACM Computing Surveys (CSUR), v.34 n.3, p.313-356, September 2002

jumpnumber;poset;hasse diagram;stack layout;book embedding;queue layout

587958

Task Scheduling in Networks.

Scheduling a set of tasks on a set of machines so as to yield an efficient schedule is a basic problem in computer science and operations research. Most of the research on this problem incorporates the potentially unrealistic assumption that communication between the different machines is instantaneous. In this paper we remove this assumption and study the problem of network scheduling, where each job originates at some node of a network, and in order to be processed at another node must take the time to travel through the network to that node.Our main contribution is to give approximation algorithms and hardness proofs for fully general forms of the fundamental problems in network scheduling. We consider two basic scheduling objectives: minimizing the makespan and minimizing the average completion time. For the makespan, we prove small constant factor hardness-to-approximate and approximation results. For the average completion time, we give a log-squared approximation algorithm for the most general form of the problem. The techniques used in this approximation are fairly general and have several other applications. For example, we give the first nontrivial approximation algorithm to minimize the average weighted completion time of a set of jobs on related or unrelated machines, with or without a network.

Introduction Scheduling a set of tasks on a set of machines so as to yield an efficient schedule is a basic problem in computer science and operations research. It is also a difficult problem and hence, much of the research in this area has incorporated a number of potentially unrealistic assumptions. One such assumption is that communication between the different machines is instantaneous. In many application domains, however, such as a network of computers or a set of geographically-scattered repair shops, decisions about when and where to move the tasks are a critical part of achieving efficient resource allocation. In this paper we remove the assumption of instantaneous communication from the traditional parallel machine models and study the problem of network scheduling, in which each job originates at some node of a network, and in order to be processed at another node must take the time to travel through the network to that node. Until this work, network scheduling problems had either loose [2, 4] or no approximation algo- rithms. Our main contribution is to give approximation algorithms and hardness proofs for fully general forms of the fundamental problems in network scheduling. Our upper bounds are robust, as they depend on general characteristics of the jobs and the underlying network. In particular, our algorithmic techniques to optimize average completion time yield other results, such as the first nontrivial approximation algorithms for a combinatorial scheduling question: minimization of average weighted completion time on unrelated machines, and the first approximation algorithm for a problem motivated by satellite communication systems. (To differentiate our network scheduling models from the traditional parallel machine models, we will refer to the latter as combinatorial scheduling models.) Our results not only yield insight into the network scheduling problem, but also demonstrate contrasts between the complexity of certain combinatorial scheduling problems and their network variants, shedding light on their relative difficulty. An instance of the network scheduling problem consists of a network non-negative edge lengths '; we define ' max to be the maximum edge length. At each vertex v i in the network is a machine M i . We are also given a set of n jobs, J Each job J j originates, at time 0, on a particular origin machine M and has a processing requirement define p max to be max 1-j-n p j . Each job must be processed on one machine without interruption. Job J j is not available to be processed on a machine M 0 until time d(M the length of the shortest path in G between M i and M k . We assume that the M i are either identical on every machine) or that they are unrelated (J j takes time p ij on M i , and the may all be different). In the unrelated machines setting we define p . The identical and unrelated machine models are fundamental in traditional parallel machine scheduling and are relatively well understood [3, 10, 11, 12, 15, 17, 25]. Unless otherwise specified, in this paper the machines in the network are assumed to be identical. An alternative view of the network scheduling model is that each job J j has a release date, a time before which it is unavailable for processing. In previous work on traditional scheduling models a job's release date was defined to be the same on all machines. The network model can be characterized by allowing a job J j 's release date to be different on different machines; J j 's release date on M k is d(M One can generalize further and consider problems in which a job's release date can be chosen arbitrarily for all m machines, and need not reflect any network structure. Almost all of our upper bounds apply in this more general setting, whereas our lower bounds all apply when the release dates have network structure. We study algorithms to minimize the two most basic objective functions. One is the makespan or maximum completion time of the schedule; that is, we would like all jobs to finish by the earliest time possible. The second is the average completion time. We define an ff-approximation algorithm to be a polynomial-time algorithm that gives a solution of cost no more than ff times optimal. 1.1 Previous Work The problem of network scheduling has received some attention, mostly in the distributed setting. Deng et. al. [4] considered a number of variants of the problem. In the special case in which each edge in the network is of unit length, all job processing times are the same, and the machines are identical, they showed that the off-line problem is in P . It is not hard to see that the problem is NP-Complete when jobs are allowed to be of different sizes; they give an off-line O(log(m' max ))- approximation algorithm for this. They also give a number of results for the distributed version of the problem when the network topology is completely connected, a ring or a tree. Awerbuch, Kutten and Peleg [2] considered the distributed version of the problem under a novel notion of on-line performance, which subsumes the minimization of both average and maximum completion time. They give distributed algorithms with polylogarithmic performance guarantees in general networks. They also characterize the performance of feedback-based approaches. In addition they derived off-line approximation results similar to those of Deng et. al [2, 20]. Alon et. al. [1] proved an m) lower bound on the performance of any distributed scheduler that is trying to minimize schedule length. Fizzano et. al. [5] give a distributed 4:3-approximation algorithm for schedule length in the special case in which the network is a ring. Our work differs from these papers by focusing on the centralized off-line problem and by giving approximations of higher quality. In addition, our approximation algorithms work in a more general setting, that of unrelated machines. 1.2 Summary of Results We first focus on the objective of minimizing the makespan, and give a 2-approximation algorithm for scheduling jobs on networks of unrelated machines; the algorithm gives the same performance guarantee for identical machines as a special case. The 2-approximation algorithm matches the best known approximation algorithm for scheduling unrelated machines with no underlying network [17]. Thus it is natural to ask whether the addition of a network to a combinatorial scheduling problem actually makes the problem any harder. We resolve this question by proving that the introduction of the network to the problem of scheduling identical machines yields a qualitatively harder problem. We show that for the network scheduling problem, no polynomial-time algorithm can do better than a factor of 4 3 times optimal unless even in a network in which all edges have length one. Comparing this with the polynomial approximation scheme of Hochbaum and Shmoys [10] for parallel machine scheduling, we see that the addition of a network does indeed make the problem harder. Although the 2-approximation algorithm runs in polynomial time, it may be rather slow [21]. We thus explore whether a simpler strategy might also yield good approximations. A natural approach to minimizing the makespan is to construct schedules with no unforced idle time. Such strategies provide schedules of length a small constant factor times optimal, at minimal computational cost, for a variety of scheduling problems [6, 7, 15, 24]. We call such schedules busy schedules, and show that for the network scheduling problem their quality degrades significantly; they can be as much as an\Omega log log m factor longer than the optimal schedule. This is in striking contrast to the combinatorial model (for which Graham showed that a busy strategy yields a 2-approximation algorithm [6]). In fact, even when release dates are introduced into the identical machine scheduling problem, if each job's release date is the same on all machines, busy strategies still give a )-approximation guarantee [8, 9]. Our result shows that when the Combinatorial Network min. makespan, identical machines ff min. makespan, identical machines, log log m log log m Busy schedules min. makespan, unrelated machines 3=2 min. avg. completion time unrelated machines n) min. avg. wtd. completion time unrelated machines, release dates ff - O(log 2 n) Figure 1: Summary of main algorithms and hardness results. The notation x ! ff - y means that we can approximate the problem within a factor of y, but unless the problem within a factor of x. Unreferenced results are new results found in this paper. release dates of the jobs are allowed to be different on different machines busy scheduling degrades significantly as a scheduling strategy. This provides further evidence that the introduction of a network makes scheduling problems qualitatively harder. However, busy schedules are of some quality; we show that they are of length a factor of O log log m longer than optimal. This analysis gives a better bound than the (O(log m' max )) bound of previously known approximation algorithms for identical machines in a network [2, 4, 20]. We then turn to the NP-hard problem of the minimization of average completion time. Our major result for this optimality criterion is a O(log 2 n)-approximation algorithm in the general setting of unrelated machines. It formulates the problem as a hypergraph matching integer program and then approximately solves a relaxed version of the integer program. We can then find an integral solution to this relaxation, employing as a subroutine the techniques of Plotkin, Shmoys and Tardos [21]. In combinatorial scheduling, a schedule with minimum average completion time can be found in polynomial time, even if the machines are unrelated. The techniques for the average completion time algorithm are fairly general, and yield an O(log 2 n)-approximation for minimizing the average weighted completion time. A special case of this result is an O(log 2 n)-approximation algorithm for the NP-hard problem of minimizing average weighted completion time for unrelated machines with no network; no previous approximation algorithms were known, even in the special case for which the machines are just of different speeds [3, 15]. Another special case is the first O(log 2 n)-approximation algorithm for minimizing the average completion time of jobs with release dates on unrelated machines. No previous approximation algorithms were known, even for the special case of just one machine [15]. The technique can also be used to give an approximation algorithm for a problem motivated by satellite communication systems [18, 26]. We also give a number of other results, including polynomial-time algorithms for several special cases of the above-mentioned problems and a 5 -approximation for a variant of network scheduling in which each job has not only an origin, but also a destination. A summary of some of these upper bounds and hardness results appears in Figure 1. A line of research which is quite different from ours, yet still has some similarity in spirit, was started by Papadimitriou and Yannakakis [19]. They modeled communication issues in parallel machine scheduling by abstracting away from particular networks and rather describing the communication time between any two processors by one network-dependent constant. They considered the scheduling of precedence-constrained jobs on an infinite number of identical machines in this model; the issues involved and the sorts of theorems proved are quite different from our results. Although all of our algorithms are polynomial-time algorithms, they tend to be rather inefficient. Most rely on the work of [21] as a subroutine. As a result we will not discuss running times explicitly for the rest of the paper. Makespan In this section we study the problem of minimizing the makespan for the network scheduling problem. We first give an algorithm that comes within a factor of 2 of optimal. We then show that this is nearly the best we can hope for, as it is NP-hard to approximate the minimum makespan within a factor of better than 4 3 for identical machines in a network. This hardness result contrasts sharply with the combinatorial scenario, in which there is a polynomial approximation scheme [10]. The 2-approximation algorithm is computationally intensive, so we consider simple strategies that typically work well in parallel machine scheduling. In another sharp contrast to parallel machine scheduling, we show that the performance of such strategies degrades significantly in the network setting; we prove an\Omega log log m lower bound on the performance of any such algorithm. We also show that greedy algorithms do have some performance guarantee, namely O( log m log log m ). Finally we consider a variant of the problem in which each job has not only an origin, but also a destination, and give a 5 -approximation algorithm. 2.1 A 2-Approximation Algorithm For Makespan In this section we describe a 2-approximation algorithm to minimize the makespan of a set of jobs scheduled on a network of unrelated machines; the same bound for identical machines follows as a special case. Let U be an instance of the unrelated network scheduling problem with optimal schedule length D. Assuming that we know D, we will show how to construct a schedule of length at most 2D. This can be converted, via binary search, into a 2-approximation algorithm for the problem in which we are not given D [10]. In the optimal schedule of length D, we know that the sum of the time each job spends travelling and being processed is bounded above by D. Thus, job J j may run on machine M i in the optimal schedule only if: In other words, the length of an optimal schedule is not altered if we allow job J j to run only on the machines for which (1) is satisfied. Formally, for a given job J j , we will denote by Q(J j ) the set of machines that satisfy (1). If we restrict each J j to only run on the machines in Q(J j ), the length of the optimal schedule remains unchanged. combinatorial unrelated machines scheduling problem (Z) as follows: If the optimal schedule for the unrelated network scheduling problem has length D, then the optimal solution to the unrelated parallel machine scheduling problem (2) is at most D. We will use the 2-approximation algorithm of Lenstra, Shmoys and Tardos [17] to assign jobs to machines. The following theorem is easily inferred from [17]. Theorem 2.1 (Lenstra, Shmoys, Tardos [17]) Let Z be an unrelated parallel machine scheduling problem with optimal schedule of length D. Then there exists a polynomial-time algorithm that finds a schedule S of length 2D. Further, S has the property that no job starts after time D. Theorem 2.2 There exists a polynomial-time 2-approximation algorithm to minimize makespan in the unrelated network scheduling problem. Proof: Given an instance of the unrelated network scheduling problem, with shortest schedule of length D, form the unrelated parallel machine scheduling problem Z defined by (2) and use the algorithm of [17] to produce a schedule S of length 2D. This schedule does not immediately correspond to a network schedule because some jobs may have been scheduled to run before their release dates. However, if we allocate D units of time for sending all jobs to the machines on which they run, and then allocate 2D units of time to run schedule S, we immediately get a schedule of length 3D for the network problem. By being more careful, we can create a schedule of length 2D for the network problem. In schedule S, each machine M i is assigned a set of jobs S i . Let jS i j be the sum of the processing times of the jobs in S i and let S max i be the job in S i with largest processing time on machine call its processing time p max . By Theorem 2.1 and the fact that the last job run on machine i is no longer than the longest job run, we know that jS i denote the set of jobs i . We form the schedule for each machine i by running job S max i at time by the jobs in S 0 In this schedule the jobs assigned to any machine clearly finish by time 2D; it remains to be shown that all jobs can be routed to the proper machines by the time they need to run there. Job must start at time conditions (1) and (2) guarantee that it arrives in time. The remaining jobs need only arrive by time D; conditions (1) and (2) guarantee this as well. Thus we have produced a valid schedule of length 2D. Observe that this approach is fairly general and can be applied to any problem that can be characterized by a condition such as (2). Consider, for example the following very general problem, which we call generalized network scheduling with costs. In addition to the usual unrelated network scheduling problem, the time that it takes for job J j to travel over an edge is dependent not only on the endpoints of the edge, but also on the job. Further, there is a cost c ij associated with processing job J j on machine M i . Given a schedule in which job J j runs on machine M -(j) , the cost of a schedule is Given any target cost C, we define s(C) to be the minimum length schedule of cost at most C. Theorem 2.3 Given a target cost C, we can, in polynomial time, find a schedule for the generalized network scheduling problem with makespan at most 2s(C) and of cost C if a schedule of cost C exists. Proof: We use similar techniques to those used for Theorem 2.2. We first modify Condition (1) so that d(\Delta; \Delta) depends on the job as well. We then use a generalization of the algorithm of Lenstra, Shmoys and Tardos for unrelated machine scheduling, due to Shmoys and Tardos [25] which, given a target cost C finds a schedule of cost C and length at most twice that of the shortest schedule of cost C. The schedule returned also has the property that no job starts after time D, so the proof of Theorem 2.2 goes through if we use this algorithm in place of the algorithm of [17]. 2.2 Nonapproximability Theorem 2.4 It is NP-complete to determine if an instance of the identical network scheduling problem has a schedule of length 3, even in a network with ' Proof: See Appendix. Corollary 2.5 There does not exist an ff-approximation algorithm for the network scheduling problem with even in a network with ' Proof: Any algorithm with ff ! 4=3 would have to give an exact answer for a problem with a schedule of length 3 since an approximation of 4 would have too high a relative error. It is not hard to see, via matching techniques, that it is polynomial-time decidable whether there is a schedule of length 2. We can show that this is not the case when the machines in the network can be unrelated. Lenstra, Shmoys and Tardos proved that it is NP-Complete to determine if there is a schedule of length 2 in the traditional combinatorial unrelated machine model [17]. If we allow multiple machines at one node, their proof proves Theorem 2.6. If no zero length edges are allowed, i.e. each machine is forced to be at a different network node, this proof does not work, but we can give a different proof of hardness, which we do not include in this paper. Theorem 2.6 There does not exist an ff-approximation algorithm for the unrelated network scheduling problem with ff ! 3=2 unless even in a network with ' 2.3 Naive Strategies The algorithms in Section 2.1 give reasonably tight bounds on the approximation of the schedule length. Although these algorithms run in polynomial time, they may be rather slow [21]. We thus explore whether a simpler strategy might also yield good approximations. A natural candidate is a busy strategy: construct a busy schedule, in which, at any time t there is no idle machine M i and idle job J j so that job J j can be started on M i at time t. Busy strategies and their variants have been analyzed in a large number of scheduling problems (see [15]) and have been quite effective in many of them. For combinatorial identical machine scheduling, Graham showed that such strategies yield a In this section we analyze the effectiveness of busy schedules for identical machine network scheduling. Part of the interest of this analysis lies in what it reveals about the relative hardness of scheduling with and without an underlying network; namely, the introduction of an underlying network can make simple strategies much less effective for the problem. 2.3.1 A Lower Bound We construct a family of instances of the network scheduling problem, and demonstrate, for each instance, a busy schedule which is \Omega\Gamma log log m ) longer than the shortest schedule for that instance. The network E) consists of ' levels of nodes, with level nodes. Each node in level is connected to every node in level by an edge of length 1. Each machine in levels ae jobs of size 1 at time 0. The machines in level ' initially receive no jobs. The optimal schedule length for this instance is 2 and is achieved by each machine in level taking exactly one job from level i \Gamma 1. We call this instance I. See Figure 2. The main idea of the lower bound is to construct a busy schedule in which machine M always processes a job which originated on M , if such a job is available. This greediness "prevents" the scheduler from making the much larger assignment of jobs to machines at time 2 in which each job is assigned to a machine one level away. To construct a busy schedule S, we use algorithm B, which in Step t constructs the subschedule of S at time t. Step t: Phase 1: Each machine M processes one job that originated at M , if any such jobs remain. We call such jobs local to machine M . r r r r r Level 1 Level 2 . Level L Figure 2: Lower Bound Instance for Theorem 2.8. Circles represent processors, and the numbers inside the circles are the number of jobs which originate at that processor at time 0. Levels i and are completely connected to each other. The optimal schedule is of length 2 and is achieved by shifting each job to a unique processor one level to its right. Phase 2: Consider the bipartite graph G has one vertex representing each job that is unprocessed after Phase 1 of time t, Y contains one vertex representing each machine which has not had a job assigned to it in Phase 1 of Step t, and (x; y) 2 A if and only if job x originated a distance no more than t \Gamma 1 from machine y. Complete the construction of S at time t by processing jobs on machines based on any maximum matching in G . It is clear that S is busy. When we apply algorithm B to instance I, the behavior follows a well-defined pattern. In Phase 2 of Step 2, all unprocessed jobs that originated in level are processed by distinct processors in level '. During Phase 2 of Step 3, all unprocessed jobs that originated in levels are processed by machines in levels This continues, so that at Step i an additional (i \Gamma 1) levels pass their jobs to higher levels and all these jobs are processed. This continues until either level 1 passes its jobs, or processes its own jobs. We characterize the behavior of the algorithm more formally in the following lemma. Lemma 2.7 Let j(i; t) be the number of local jobs of processor i still unprocessed after Phase 2 of Step t and let lev(i) be the level number of processor i. Then for all times t - 2, if ae - t, then Proof: We prove the lemma by induction on t. During Phase 2 of Step 2, the only edges in the graph G connect levels ' and ' \Gamma 1. There are ae '\Gamma1 nodes in level ' and ae '\Gamma2 (ae \Gamma 1) remaining jobs local to machines in level ' \Gamma 1, so the matching assigns all the unprocessed jobs in level to level '. Machines in level 1 to process local jobs during Phase 1. As a result, all the neighbors of machines in levels 1 to are busy in Phase 1 and cannot process jobs local to these machines during Phase 2. The number of local jobs on these machines, therefore, decreases only by 1. Thus the base case holds. Assume the lemma holds for all greater than b as well. We now show that j(i; t 0 level b+x has ae b+x\Gamma1 processors. Level has at most ae \Delta ae b+x\Gamma(t 0 local jobs remaining. If t 0 - 2 then there are enough machines on level b + x to process all the remaining jobs local to level b another of the highest-numbered levels have their local jobs completed during time t 0 . Thus at time t 0 we have Since we assumed sufficiently large initial workloads on all processors on levels by the induction hypothesis, for all machines in levels less than distance them have local jobs remaining after time will be assigned a local job during Phase 1 of Step t 0 . Therefore all machines i such that lev(i) any jobs to higher levels and j(i; t 0 Depending on the relative values of ae and ', either the machine in level 1 processes all of the jobs which originated on it, or some of those jobs are processed by machines in higher-numbered levels. Balancing these two cases we get the following theorem: Theorem 2.8 For the family of instances of the identical machine network scheduling problem defined above, there exist busy schedules of length a log log m ) longer than optimal. Proof: The first case in (3) will apply to level 1 when 1 This inequality does not hold when 2', but it does hold when 2' then the schedule length is 2', while if ae ! 2' then the jobs in level 1 will be totally processed in their level, which takes ae time. Therefore the makespan of S is at most min( ae). Given that the total number of machines is calculation reveals that min(c '; ae) is maximized at log log m ). Thus S is a busy schedule of length '( log log m ) longer than optimal. Note that this example shows that several natural variants of busy strategies, such as scheduling a job on the machine on which it will finish first, or scheduling a job on the closest available processor, also perform poorly. 2.3.2 An Upper Bound In contrast to the lower bound of the previous subsection, we can prove that busy schedules are of some quality. Given an instance I of the network scheduling problem, we define C (I) to be the length of a shortest schedule for I and C A (I) to be the length of the schedule produced by algorithm A; when it causes no confusion we will drop the I and use the notation C Definition 2.9 Consider a busy schedule S for an instance I of the identical machines network scheduling problem. Let p j (t) be the number of units of job J j remaining to be processed in schedule S at time t, and W be the total work remaining to be processed in schedule S at time t. Lemma 2.10 W iC Proof: We partition schedule S into consecutive blocks what happens in each block of schedule S to an optimal schedule S of length C for instance I. Consider a job J j that was not started by time C in schedule S, and let M j be the machine on which job J j is processed in schedule S . This means that in block B 1 machine M j is busy for units of time during job J j 's slot in schedule S - the period of time during which job J j was processed on machine M j in schedule S . Hence for every job J j that is not started in block there is an equal amount of unique work which we can identify that is processed in block B 1 , implying that WC max Successive applications of this argument yields W iC which proves the lemma for 2. To obtain the stronger bound W iC we increase the amount of processed work which we identify with each unstarted job. Choose i - 3 and consider a job J j which is unstarted in schedule S at the start of block B i+1 , namely at time iC . Assume for the sake of simplicity that in every block B k of schedule S, only one job is processed in job J j 's slot (the time during which job J j would be processed if block B k was schedule S ). Assume also that this job is exactly of the same size as job J multiple jobs are processed the argument is essentially the same. Let J r be the job that took job J j 's slot in block B r , for r - 2. We will show that J j could have been processed in J r 's slot in block B i for all 2. Figure 2.3.2 illustrates the network structure used in this argument. Assume that job J j originated on machine M , and that job J r originated on machine M or , and that job J j was processed on machine M j in schedule S . Then d(M since job J j was processed on machine M j in schedule S , and d(M or ; M j ) - rC since job J r was processed in job J j 's slot in block B r . Thus d(M consequently J j could have run in job J r 's slot in any of blocks B We focus on block B i . Since J j was not processed in and schedule S is busy, some job must have been processed during job J r 's slot in block We identify this work with job J note that no work is ever identified with more than one job. When we consider the (i\Gamma2) different jobs which were processed in J j 's slot in blocks and consider the jobs that were processed in their slots in B i , we see that with each job J j unstarted at time iC , we can uniquely identify units of work that was processed in block O <_ <_ O r r <_ Figure 3: If J r takes J j 's slot in B r , then the machine on which J j originates, M , is at most a distance of (r r , the machine on which J r runs in S . Thus J j could have been run in J r 's slot in block i, . If all these slots were not full in block B i , then job J j would have been started in one of them. Including the work processed during job J j 's slot in block B i , we obtain Corollary 2.11 During time iC max at most m=(2i!) machines are completely busy. Proof: We have W 0 - mC . Therefore, by Lemma 2.10, we have W iC machine that is completely busy from time iC does C work during that time and therefore at most m=(2i!) machines can be completely busy. To get a stopping point for the recurrence, we require the following lemma: Lemma 2.12 In any busy schedule, if at time t all remaining unprocessed jobs originated on the same machine, the schedule is no longer than t Let M be the one machine with remaining local jobs. Let W be the amount of work from machine M that is done by machine M i in the optimal schedule. Clearly equals the amount of work that originated on machine M . Because there is no work left that originated on machines other than M , each machine M i can process at least W work from machine M in the next C steps. If after C steps, all the work originating on machine M is done, then we have finished. Otherwise, some machine M i processed less than W work during this time, which means there was no more work for it to take. Therefore after C steps all the jobs that originated on machine M have started. Because no job is longer than C suffices to finish all the jobs that have started. We are now ready to prove the upper bound: Theorem 2.13 Let A be any busy scheduling algorithm and I an instance of the identical machine network scheduling problem. Then C A log log m C Proof: If a machine ever falls idle, all of its local work must be started. Otherwise it would process remaining local work. Thus by Corollary 2.11, in O( lg m time, the number of processors with local work remaining is reduced to 1. By Lemma 2.12, when the number of processors with remaining local work is down to one, a constant number of extra blocks suffice to finish. 2.4 Scheduling with Origins and Destinations In this subsection we consider a variant of the (unrelated machine) network scheduling problem in which each job, after being processed, has a destination machine to which it must travel. Specif- ically, in addition to having an origin machine M , job J j also has a terminating machine M t j begins at machine M , travels distance d(M ) to machine M d j , the machine it gets processed on, and then proceeds to travel for d(M d j units of time to machine M t j . We call this problem the point-to-point scheduling problem. Theorem 2.14 There exists a polynomial-time 5 -approximation algorithm to minimize makespan in the point-to-point scheduling problem. Proof: We construct an unrelated machines scheduling problem as in the proof of Theorem 2.2. In this setting the condition on when a job J j can run on machine M i depends on the time for J j to get to M i , the time to be processed there, and the time to proceed to the destination machine. Thus a characterization of when job J j is able to run on machine M i in the optimal schedule is Now, for a given job J j , we define Q(J j ) to be the set of machines that satisfy (4). We can then form a combinatorial unrelated machines scheduling problem as follows: We then approximately solve this problem using [17] to obtain an assignment of jobs to machines. Pick any machine M i and let J i be the set of jobs assigned to machine M i . By Theorem 2.1 we know that the sum of the processing times of all of the jobs in J i except the longest is at most D. We partition the set of jobs J i into three groups, and place each job into the lowest numbered group which is appropriate: 1. J 0 i contains the job in J i with the longest processing time, 2. J 1 contains jobs for which d(M 3. J 2 contains jobs for which d(M i ) be the sum of the processing times of the jobs in group J k 2. As noted above, We will always schedule J 1 i in a block of D consecutive time steps, which we call B. The first p(J 1 steps will be taken up by jobs in J 1 i while the last p(J 2 steps will be taken up by jobs in J 2 . Note that there may be idle time in the interior of the block. We consider two possible scheduling strategies based on the relative sizes of p(J 1 Case 1:(p(J 1 In this case we first run the long job in J 0 by condition (4) it finishes by time D. We then run block B from time D to 2D. Since p(J 1 the jobs in J 1 all finish by time 3D=2 and by condition (4) reach their destinations by time 5D=2. By the definition of J 2 for any job J i is scheduled to complete processing by time 2D, it will arrive at its destination by time 5D=2. Case 2: (p(J 1 first run block B from time D=2 to 3D=2. We then start the long job in J 0 i at time 3D=2; by condition (4) it arrives at its destination by time 5D=2. Since p(J 2 machine M i need not start processing any job in J 2 hence we are guaranteed that they have arrived at machine M i by that time. By definition of J 1 i all of its jobs are available by time D=2; it is straightforward from condition (4) that all jobs arrive at their destinations by time 5D=2. We can also show that the analysis of this algorithm is tight, for algorithms in which we assign jobs to processors using the linear program defined in [17] using the processing times specified by Equation 5. Let D be the length of the optimal schedule. Then we can construct instances for which any such schedule S has length at least 5=2D \Gamma 1. Consider a set of k+1 jobs and a particular machine M i . We specify the largest of these jobs to have size D and to have M i as both its origin and destination machine. We specify that each of the other k jobs are of size D=k and have distance to both their origin and destination machines. The combinatorial unrelated machines algorithm may certainly assign all of these jobs to M i , but it is clear that any schedule adopted for this machine will have competion time at least ( 5 2k )D. 3 Average Completion Time 3.1 Background We turn now to the network scheduling problem in which the objective is to minimize the average completion time. Given a schedule S, let C S j be the time that job J j finishes running in S. The average completion time of S is 1 whose minimization is equivalent to the minimization of . Throughout this section we assume without loss of generality that n - m. We have noted in Section 1 that our network scheduling model can be characterized by a set of and a set of release dates r ij , where J j is not available on m i until time r ij . We noted that this is a generalization of the traditional notion of release dates, in which r will refer to the latter as traditional release dates; the unmodified phrase release date will refer to the general r ij . The minimization of average completion time when the jobs have no release dates is polynomial-time solvable [3, 12], even on unrelated machines. The solution is based on a bipartite matching formulation, in which one side of the bipartition has jobs and the other side (machine, position) pairs. Matching J j to (m corresponds to scheduling J j in the kth-from-last position on m i ; this edge is weighted by kp ij , which is J j 's contribution to the average completion time if J j is kth from last. When release dates are incorporated into the scheduling model, it seems difficult to generalize this formulation. Clearly it can not be generalized precisely for arbitrary release dates, since even the one machine version of the problem of minimizing average completion time of jobs with release dates is strongly NP-hard [3]. Intuitively, even the approximate generalization of the formulation seems difficult, since if all jobs are not available at time 0, the ability of J j to occupy position on m i is dependent on which jobs precede it on m i and when. Release dates associated with a network structure do not contain traditional release dates as a subclass even for one machine, so the NP-completeness of the network scheduling problem does not follow immediately from the combinatorial hardness results; however, not surprisingly, minimizing average completion time for a network scheduling problem is NP-complete. Theorem 3.1 The network scheduling problem with the objective of minimum average completion time is NP-complete even if all the machines are identical and all edge lengths are 1. Proof: See Appendix. In what follows we will develop an approximation algorithm for the most general form of this problem. We will follow the basic idea of utilizing a bipartite matching formulation; however we will need to explicitly incorporate time into the formulation. In addition, for the rest of the section we will consider a more general optimality criterion: average weighted completion time. With each J j we associate a weight w j , and the goal is to minimize . All of our algorithms handle this more general case; in addition they allow the nm release dates r ij to be arbitrary and not necessarily derived from the network structure. 3.2 Unit-Size Jobs We consider first the special case of unit-size jobs. Theorem 3.2 There exists a polynomial-time algorithm to schedule unit-size jobs on a network of identical machines with the objective of minimizing the average weighted completion time. Proof: We reduce the problem to minimum-weight bipartite matching. One side of the bipartition will have a node for each job J j , 1 - j - n, and the other side will have a node [m to be described below. An edge included if J j is available on m i at time t, and the inclusion of that edge in the matching will represent the scheduling of J j on m i from time t to t + 1. Release dates are included in the model by excluding an edge will not be available on m i by time t. To determine the necessary sets T i , we observe that there is no advantage in unforced idle time. Since each job is only one unit long, there is no reason to make it wait for a job of higher weight that is about to be released. It is clear, therefore, that setting T would suffice, since no job would need to be scheduled more than n time later than its release date. This this can be reduced to O(n), but we omit the details for the sake of brevity. By excluding edges which do not give job J j enough time to travel between the machine on which runs and the destination machine M d j , we can prove a similar theorem for the point-to-point scheduling problem, defined in Section 2.4. Theorem 3.3 There exists a polynomial-time algorithm to solve the point-to-point scheduling problem with the objective of minimizing the average weighted completion time of unit-size jobs. 3.3 Polynomial-Size Jobs We now turn to the more difficult setting of jobs of different sizes and unrelated machines. The minimization of average weighted completion time in this setting is strongly NP-hard, as are many special cases. For example, the minimization of average completion time of jobs with release dates on one machine is strongly NP-hard [16]; no approximation algorithms were known for this special case, to say nothing of parallel identical or unrelated machines, or weighted completion times. If there are no release dates, namely all jobs are available at time 0, then minimization of average weighted completion time is NP-hard for parallel identical machines. A small constant factor approximation algorithm was known for this problem [14], but no approximation algorithms were known for the more general cases of machines of different speeds or unrelated machines. We introduce techniques which yield the first approximation algorithms for several other problems as well, which we discuss in Section 3.5. Our approximation algorithm for minimum average completion time begins by formulating the scheduling problem as a hypergraph matching problem. The set of vertices will be the union of two sets, J and M , and the set of hyperedges will be denoted by F . J will contain n vertices J j , one for each job, and M will contain mT vertices, where T is an upper bound on the number of time units that will be needed to schedule this instance. The time units will range over g. M will have a node for each (machine, time) pair; we will denote the node that corresponds to machine M i at time t as [m i ; t]. A hyperedge e 2 F represents scheduling a job J j on machine M i from time t 1 to t 2 by including nodes J The cost of an edge e, denoted by c e , will be the weighted completion time of job J j if it is scheduled in the manner represented by e. There will be one edge in the hypergraph for each feasible scheduling of a job on a machine; we exclude edges that would violate the release date constraints. The problem of finding the minimum cost matching in the hypergraph can be phrased as the following integer program I. We use decision variable x e 2 f0; 1g to denote whether hyperedge e is in the matching. minimize e subject to X (i;t)2e Two considerations suggest that this formulation might not be useful. The formulation is not of polynomial size in the input size, and in addition the following theorem suggests that calculating approximate solutions for this integer program may be difficult. Theorem 3.4 Consider an integer program in the form I which is derived from an instance of the network scheduling problem with identical machines, with the c e allowed to be arbitrary. Then there exists no polynomial-time algorithm A to approximate I within any factor unless Proof: For an arbitrary instance of the network scheduling problem construct the hypergraph matching problem in which an edge has weight W ?? n if it corresponds to a job being completed later than time 3 and give all other edges weight 1. If there is a schedule of length 3 then the minimum weight hypergraph matching is of weight n; otherwise the weight is at least W ; therefore an ff-approximation algorithm with ff ! W would give a polynomial-time algorithm to decide if there was a schedule of length 3 for the network scheduling problem, which by Theorem 2.4 would imply In order to overcome this obstacle, we need to seek a different kind of approximation to the hypergraph matching problem. Typically, an approximate solution is a feasible solution, i.e. one that satisfies all the constraints, but whose objective value is not the best possible. We will look for a different type of solution, one that satisfies a relaxed set of constraints. We will then show how to turn a solution that satisfies the relaxed set of constraints into a schedule for the network scheduling problem, while only introducing a bounded amount of error into the quality of the approximation. We will assume for now that p max - n 3 . This implies that the size of program I is polynomial in the input size. We will later show how to dispense with the assumption on the size of p max via a number of rounding and scaling techniques. We begin by turning the objective function of I into a constraint. We will then use the standard technique of applying bisection search to the value of the objective function. Hence for the remainder of this section we will assume that C, the optimal value to integer program I, is given. We can now construct approximate solutions to the following integer linear program (J (i;t)2e e This integer program is a packing integer program, and as has been shown by Raghavan [22], Raghavan and Thompson [23] and Plotkin, Shmoys and Tardos [21], it is possible to find provably good approximate solutions in polynomial time. We briefly review the approach of [21], which yields the best running times. Plotkin, Shmoys and Tardos [21] consider the following general problem. The Packing Problem: 9?x 2 P such that Ax - b, where A is an m \Theta n nonnegative matrix, b ? 0, and P is a convex set in the positive orthant of R n . They demonstrate fast algorithms that yield approximately optimal integral solutions to this linear program. All of their algorithms require a fast subroutine to solve the following problem. The Separation Problem: Given an m-dimensional vector y - 0, find ~ x 2 P such that A. The subroutine to solve this problem will be called the separating subroutine. An approximate solution to the packing problem is found by considering the relaxed problem and approximating the minimum - such that this is true. Here the value - characterizes the "slack" in the inequality constraints, and the goal is to minimize this slack. Our integer program can be easily put in the form of a packing problem; the equality constraints (7) define the polytope P and the inequality constraints (8,9) make up Ax - b. The quality of the integral solutions obtained depends on the width of P relative to Ax - b, which is defined by a It also depends on d, where d is the smallest integer such that any solution returned by the separating routine is guaranteed to be an integral multiple of 1 d . Applying equation (10) to compute ae for polytope P (defined by (7)) yields a value that is at least n, as we can create matchings (feasible schedules) whose cost (average completion time) is much greater than C, the optimal average completion time. In fact, many other packing integer programs considered in [21] also, when first formulated, have large width. In order to overcome this obstacle, [21] gave several techniques to reduce the width of integer linear programs. We discuss and then use one such technique here, namely that of decomposing a polytope into n lower-dimensional polytopes, each of which has smaller width. The intuition is that all the non-zero variables in each equation of the form (7) are associated with only one particular job. Thus we will be able to decompose the polytope into n polytopes, one for each job. We will then be able to optimize individually over each polytope and use only the inequality constraints (8) and (9) to describe the relationships between different jobs. We now procede in more detail. We say that a polytope P can be decomposed into a product of n polytopes the coordinates of each vector x can be partitioned into our polytope can be decomposed in this way, and we can solve the separation problem for each polytope P l , then we can apply a theorem of [21] to give an approximately optimal solution in polynomial time. In particular, let - be the optimum value of J . The following theorem is a specialization of Theorem 2.11 in [21] to our problem, and describes the quality of integral solutions that can be obtained for such integer programs. Theorem 3.5 [21] Let ae l be the width of P l and - . Let fl be the number of constraints in Ax - b, and let - log fl). Given a polynomial-time separating subroutine for each of the P l , there exists a polynomial-time algorithm for J which gives an integral solution with We will now show how to reformulate J so that we will be able to apply this theorem. Polytope (from can indeed be decomposed into n different polytopes: to those equality constraints which include only J j . In order to keep the width of the P j small, we also include into the definition of P j the constraint x for each edge e which includes J j and has c e ? C; this does not increase the optimal value of the integer program. We integrate each of these new constraints into the appropriate polytope P j , and decompose consists of those components of x which represent edges that include J j . In other words, P l is defined by J l 2e e: This yields the following relaxation L: subject to (i;t)2e e To apply Theorem 3.5 we must (1) demonstrate a polynomial-time separating subroutine and ae, d and fl. The decomposition of P into n separate polytopes makes this task much easier. The separating subroutine must find x l 2 P l that minimizes cx l ; however, since the vector that is 1 in the eth component and 0 in all other components is in P l for all e such that J l 2 e and c e - C, the separating routine reduces merely to finding the minimum component c e 0 of c and returning the vector with a 1 in position e 0 and 0 everywhere else. An immediate consequence of this is that d = 1. Recall as well that the assumption that p max - n 3 implies that fl is upper bounded by a polynomial in n. It is not hard to see that - ae is 1; HERE IT IS. therefore (-ae=d) log fl(-ae=d) log(flnd)) By employing binary search over C and the knowledge that the optimal solution has can obtain an invalid "schedule" in which as many as O(-) jobs are scheduled at one time. If p max is polynomial in n and m then we have a polynomial-time algorithm; therefore we have proven the following lemma. Lemma 3.6 Let C be the solution to the integer program I and assume that jM j is bounded by mn 4 . There exists a polynomial-time algorithm that produces a solution x such that j2e x (i;t)2e x e x x This relaxed solution is not a valid schedule, since O(log n) jobs are scheduled at one time; however, it can be converted to a valid schedule by use of the following lemma. Lemma 3.7 Consider an invalid schedule S for a set of jobs with release dates on m unrelated parallel machines, in which at most - jobs are assigned to each machine at any time. If W is the average weighted completion time of S, then there exists a schedule of average weighted completion time at most -W , in which at most one job is assigned to each machine at any time, Proof: Consider a job J j scheduled in S; let its completion time be C S . If we schedule the jobs on each machine in the order of their completion times in S, never starting one before it's release date, then in the resulting schedule 1. J j is started no earlier than its release date, 2. J j finishes by time at most -C S . Statement 1 is true by design of the algorithm. Statement 2 is true since at most -C S work from other jobs can complete no later than C S in schedule S, and jobs run simultaneously in schedule S can run back-to-back with no intermediate idle time in our expanded schedule. Therefore, job J j is started by time -C S completed by time -C S . Combining the last two lemmas with the observation that p max - n 3 implies jM j - mn 4 yields the following theorem. Theorem 3.8 There is a polynomial-time O(log 2 n)-approximation algorithm for the minimization of average weighted completion time of a set of jobs with machine-varying release dates on unrelated machines, under the assumption that the maximum job sizes are bounded by p 3.4 Large Jobs Since the p ij are input in binary and in general need not be polynomial in n and m, the technique of the last section can not be applied directly to all instances, since it would yield superpolynomial- size formulations. Therefore we must find a way to handle very large jobs without impacting significantly on the quality of solution. It is a standard technique in combinatorial scheduling to partition the jobs into a set of large jobs and a set of small jobs, schedule the large jobs, which are scaled to be in a polynomially- bounded range, and then schedule the small jobs arbitrarily and show that their net contribution is not significant, (see e.g. [24]). In the minimization of average weighted completion time, however, we must be more careful, since the small jobs may have large weights and can not be scheduled arbitrarily. We employ several steps, each of which increases the average weighted completion time by a small constant factor. With more care we could reduce the constants introduced by each step; however since our overall bound is O(log 2 n) we dispense with this precision for the sake of clarity of exposition. The basic idea is to characterize each job by the minimum value, taken over all machines, of its (release date processing time) on that machine. We then group the jobs together based on the size of their minimum . The jobs in each group can be scaled down to be of polynomial size and thus we can construct a schedule for the scaled down versions of each group. We then scale the schedules back up, correct for the rounding error, and show that this does not affect the quality of approximation by more than a constant factor. We then apply Lemma 3.9 (see below) to show that the makespan can be kept short simultaneously. The resulting schedules will be scheduled consecutively. However, since we have kept the makespan from growing too much, we have an upper bound on the start time of each subsequent schedule and thus we can show that the the net disturbance of the initial schedules to the latter schedules will be minimal. We now proceed in greater detail. Let m(J g. Note that there are at most n nonempty J i , one for each of the n jobs. We will employ the following lemma in order to keep the makespan from growing too large. Lemma 3.9 A schedule S for J k can be converted, in polynomial time, to a schedule T of makespan at most 2n k+1 such that C T Proof: Remove all jobs from S that complete later than time n k+1 , and, starting at time n k+1 , schedule them arbitrarily on the machine on which they run most quickly. This will take at most n k+1 time, so therefore any rescheduled job J j satisfies C T . We now turn to the problem of scheduling each J l with a bounded guarantee on the average completion time. Lemma 3.10 There exists an O(log 2 n)-approximation algorithm to schedule each J l . In addition the schedule for J l has makespan at most 2n l+1 . Proof: Let A be the algorithm referred to in Theorem 3.8. We will use A to find an approximately optimal solution S l for each J l . A can not be applied directly to J l since the sizes of the jobs involved may exceed n 3 , so we apply A to a scaled version of J l . For all j such that J j 2 J l and for all i, set p 0 c and r 0 c. Note that on at least one machine i , for each job J j , p 0 We use A to obtain an approximate solution to the scaled version of J l of average weighted completion time W . Although some of the p 0 may still be large, Lemma 3.9 indicates that restricting the hypergraph formulation constructed by A to allow completion times no later than time only affect the quality of approximation by at most a factor of 2. Therefore jM j, the number of (machine, time) pairs, is O(mn 3 ). Note that some of the p 0 ij may be 0, but it is still important to include an edge in the hypergraph formulation for each job of size 0. Now we argue that interpreting the solution of the scaled instance as a solution to the original instance J l does not degrade the quality of approximation by more than a constant factor. The conversion from the scaled instance to the original instance is carried out by multiplying p ij (which has no impact on quality of approximation) and then adding to each r ij and p ij the residual amount that was lost due to the floor operation. The additional residual amounts of the release dates contribute at most a total of n l\Gamma1 time to the makespan of the schedule, since jr therefore the entire contribution to the makespan is bounded above by n \Theta n . By a similar argument, the entire contribution of the residual amounts of the processing times to the makespan is bounded above by n l\Gamma1 . So in the conversion from p ij to we add at most 2n l\Gamma1 to the makespan of the schedule for J l . However, n l\Gamma1 is a lower bound on the completion time of any job in J l . Therefore, even if this additional time were added to the completion time of every job, the restoration of the residual amounts of the r ij and p ij degrades the quality of the approximation to average completion time by at most a constant factor. Finally, to satisfy the makespan constraint, we apply Lemma 3.9. We now construct two schedules S o and S e . In S o we consecutively schedule in S e we consecutively schedule :. For the sake of clarity our schedule will have time of length 2n i+1 dedicated to each S i even if S i has no jobs. Lemma 3.11 Let J o be the set of jobs scheduled in S o and J e the set of jobs scheduled in S e . The average weighted completion time of S o is within a factor of O(log 2 n) of the best possible for similarly for S e and J e . Proof: The subschedule for any set J i scheduled in S o or S e begins by time since J i is scheduled after J i\Gamma2 ; J and the makespan of J l is at most 2n l+1 . Since n i\Gamma1 is a lower bound on the completion time of any job in J i , in the combined schedule S o or S e , each job completes within a small constant factor of its completion time in S i . We now combine S o and S e by superimposing them over the same time slots. This creates an infeasible schedule in which the sum of completion times is just the sum of the completions times in but in which there may be two jobs scheduled simultaneously. We then use Lemma 3.7 to combine S o and S e to obtain a schedule S ff for all the jobs, whose average weighted completion time is within a factor of O(log 2 n) of optimal. Theorem 3.12 There is a polynomial-time O(log 2 n)-approximation algorithm for the minimization of average weighted completion time of a set of jobs with machine-varying release dates on unrelated machines. 3.5 Scheduling with Periodic Connectivity The hypergraph formulation of the scheduling problem can model time-varying connectivity between jobs and machines; e.g. a job can only be processed during certain times on each machine. In this section we show how to apply our techniques to scheduling problems of periodic connectivity under some modest assumptions on the length of the period and job sizes. Definition 3.13 The periodic scheduling problem is defined by n jobs, m unrelated machines, a period P , and for each time unit of P a specification of which jobs are allowed to run on which machines at that time. Theorem 3.14 Let I be an instance of the periodic scheduling problem in which p max is polynomial in n and m, and let the optimum makespan of I be L. There exists a polynomial-time algorithm which delivers a schedule of makespan O(log n)(L Proof: As above, we assume that L is known in advance, and then use binary search to complete the algorithm. We construct the integer program (i;t)2e Lg. We include an edge in the formulation if and only if it is valid with respect to the connectivity conditions. We then use Theorem 3.8 to produce a relaxed solution that satisfies j2e x (i;t)2e x x Let the length of this relaxed schedule be L; L - L. We construct a valid schedule of length concatenating O(log n) blocks of length L. At the end of each block we will have to wait until the start of the next period to begin the next block; hence we obtain an overall bound of O(log n)(L Note that we are assuming that the entire connectivity pattern of P is input explicitly; if it is input in some compressed form then we must assume that P is polynomial in n and m. One motivation for such problems is the domain of satellite communication systems [18, 26]. One is given a set of sites on Earth and a set of satellites(in Earth orbit). Each site generates a sequence of communication requests; each request is potentially of a different duration and may require communication with any one of the satellites. A site can only transmit to certain satellites at certain times, based on where the satellite is in its orbit. The connectivity pattern of communication opportunities is periodic, due to the orbiting nature of the satellites. The goal is to satisfy all communication requests as quickly as possible. We can use our hypergraph formulation technique to give an O(log n)-approximation algorithm for the problem under the assumption that the p j are bounded by a polynomial, since the rounding techniques do not generalize to this setting. Acknowledgments We are grateful to Phil Klein for several helpful discussions early in this research, to David Shmoys for several helpful discussions, especially about the upper bound for average completion time, to David Peleg and Baruch Awerbuch for explaining their off-line approximation algorithm to us, and to Perry Fizzano for reading an earlier draft of this paper. --R Lower bounds on the competitive ratio for mobile user tracking and distributed job scheduling. Competitive distributed job scheduling. Deterministic load balancing in computer networks. Job scheduling in rings. Bounds for certain multiprocessor anomalies. Bounds on multiprocessing anomalies. Bounds for naive multiple machine scheduling with release times and deadlines. Approximation schemes for constrained scheduling problems. Using dual approximation algorithms for scheduling problems: theoretical and practical results. A polynomial approximation scheme for machine scheduling on uniform processors: using the dual approximation approach. Minimizing average flow time with parallel machines. Reducibility among combinatorial problems. Worst case bound of an lrf schedule for the mean weighted flow-time problem Rinnooy Kan Rinnooy Kan Mobile satellite communication systems: Toward global personal communications. Towards an architecture-independent analysis of parallel algo- rithms Private communication Fast approximation algorithms for fractional packing and covering problems. Probabilistic construction of deterministic algorithms: approximating packing integer programs. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Improved approximation algorithms for shop scheduling problems. Scheduling parallel machines with costs. Mobile satellite services for travelers. --TR --CTR Dekel Tsur, Improved scheduling in rings, Journal of Parallel and Distributed Computing, v.67 n.5, p.531-535, May, 2007 Cynthia A. Phillips , R. N. Uma , Joel Wein, Off-line admission control for general scheduling problems, Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms, p.879-888, January 09-11, 2000, San Francisco, California, United States S. Muthukrishnan , Rajmohan Rajaraman, An adversarial model for distributed dynamic load balancing, Proceedings of the tenth annual ACM symposium on Parallel algorithms and architectures, p.47-54, June 28-July 02, 1998, Puerto Vallarta, Mexico 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 algorithm;networks;NP-completeness;scheduling

587963

Determining When the Absolute State Complexity of a Hermitian Code Achieves Its DLP Bound.

Let g be the genus of the Hermitian function field $H/{\mathbb F}_{q^2}$ and let $C_{\cal L}(D,mQ_{\infty})$ be a typical Hermitian code of length n. In [Des. Codes Cryptogr., to appear], we determined the dimension/length profile (DLP) lower bound on the state complexity of $C_{\cal L}(D,mQ_{\infty})$. Here we determine when this lower bound is tight and when it is not. For $m\leq \frac{n-2}{2}$ or $m\geq \frac{n-2}{2}+2g$, the DLP lower bounds reach Wolf's upper bound on state complexity and thus are trivially tight. We begin by showing that for about half of the remaining values of m the DLP bounds cannot be tight. In these cases, we give a lower bound on the absolute state complexity of $C_{\cal L}(D,mQ_{\infty})$, which improves the DLP lower bound.Next we give a "good" coordinate order for $C_{\cal L}(D,mQ_{\infty})$. With this good order, the state complexity of $C_{\cal L}(D,mQ_{\infty})$ achieves its DLP bound (whenever this is possible). This coordinate order also provides an upper bound on the absolute state complexity of $C_{\cal L}(D,mQ_{\infty})$ (for those values of $m$ for which the DLP bounds cannot be tight). Our bounds on absolute state complexity do not meet for some of these values of m, and this leaves open the question whether our coordinate order is best possible in these cases.A straightforward application of these results is that if $C_{\cal L}(D,mQ_{\infty})$ is self-dual, then its state complexity (with respect to the lexicographic coordinate order) achieves its DLP bound of $\frac {n}{2}-\frac{q^2}{4}$, and, in particular, so does its absolute state complexity.

Introduction Let C be a linear code of length n. Many soft-decision decoding algorithms for C (such as the Viterbi algorithm and lower complexity derivatives of it) take place along a minimal trellis for C. The complexity of trellis decoding algorithms can be measured by various trellis complexities. The most common one is the state complexity s(C) of C, which varies with the coordinate order of C. Since the number of operations required for Viterbi decoding of C is proportional to s(C), it is desirable that s(C) be small. A classical upper bound for s(C) is the Wolf bound dim(C)g, [9]. It is well-known that if C is a Reed-Solomon code, then W(C). Let [C] denote the set of codes equivalent to C by a change of coordinate order. We write s[C] for the minimum of s(C) over all coordinate orders of C and call it the absolute state complexity of Research supported by the U. K. Engineering and Physical Sciences Research Council under Grant L88764 at the Algebraic Coding Research Group, Centre for Communications Research, University of Bristol. Copyright 2000, Society for Industrial and Applied Mathematics. C. (We note that state-complexity notation and terminology varies in the literature. For example, state complexity is called minimal trellis size in [2]; absolute state complexity is called absolute minimal trellis size in [2] and minimal state complexity in [13].) Finding a coordinate order of C that achieves s[C] is called the 'art of trellis decoding' in [10] since exhaustive computation of s(C) over all possible coordinate orders of C is infeasible, even for quite short codes. An important step towards attaining this goal is determining good lower bounds on s[C]. The dimension/length prole (DLP) of C is a deep property which is equivalent to the generalised weight hierarchy (GWH) of C. (For a survey of GWH, see [15].) The DLP of C is independent of the coordinate order of C and provides a natural lower bound r(C) for s[C]. For example, if C is a Reed-Solomon code, then so that s[C] is as bad as possible and uninteresting. However, determining when important. An obvious and useful way of doing this is to nd a coordinate order of C for which In particular this provides one route to the art of trellis decoding. It is also important to develop methods for determining when r(C) < s(C), and in these cases to improve on r(C). Geometric Goppa codes generalise Reed-Solomon codes. Hermitian codes are widely studied geometric Goppa codes which are longer than Reed-Solomon codes and have very good parameters for their lengths. Let q be a xed prime power, we write CL (D; mQ1 ) for a typical Hermitian code of length n dened over F q 2 . In [5], we determined r(CL (D; mQ1 using some of the GWH of Hermitian codes obtained in [11, 16]. (The complete GWH of Hermitian codes has subsequently appeared in [1].) From [5], we have s(CL (D; mQ1 so we restrict ourselves to the interesting Hermitian codes i.e. to CL (D; mQ1 Here we determine precisely when r(CL (D; mQ1 In the process, we exhibit a good coordinate order which often gives s(CL (D; mQ1 )) < W(CL (D; mQ1 )). We also improve on the DLP bound (when it is strictly less than the state complexity). 'Points of gain and fall' were introduced in [3, 4, 6, 7] to help determine the state complexity of certain generalisations of Reed-Muller codes. For these codes, the points of gain and fall had particularly nice characterisations. For Hermitian codes however, their characterisation is not quite as nice and so our approach is slightly dierent. We describe a coordinate order giving and characterise the points of gain and fall of Cm . We also characterise these points of gain and fall in terms of runs. This has the advantage of greatly reducing (from n to the number of trellis depths needed to nd s(Cm ). The paper is arranged as follows. Section 2 contains terminology, notation and some previous results that will be used throughout the paper. The paper proper begins with Section 3. Here we show that for m 2 I(n; g), just under half of the Hermitian codes cannot attain their DLP bound. In these cases we give an improvement of the DLP bound, written r { (CL (D; mQ1 )). The main goal of Section 4 is to characterise the points of gain and fall of Cm in runs. In Section 5 we determine s(Cm ) using Section 4. We show that s(Cm just over half the Thus we have determined precisely when the DLP bound for Hermitian codes is tight. Furthermore { (Cm ) for around a further quarter (respectively 1=q) of m 2 I(n; g) when q is odd (respectively even). In conclusion, we have found s[Cm ] for three quarters (respectively one half) of the m 2 I(n; g) when q is odd (respectively even). For the remaining m 2 I(n; g), we do not know a better coordinate order (than that described in Section 4) nor a better bound (than that given in Section 3). Thus, although we have reduced the possible range of s[Cm ], some of its actual values remain open. Finally, our method of characterising points of gain and fall is essentially the same as the one used to determine r(CL (D; mQ1 )) in [5] and may be able to be used quite generally in determining DLP bounds and state complexity. We would like to thank Paddy Farrell for his continued interest and support of our work. An initial account of some of these results was given in [8]. The state complexity of Hermitian codes has also been studied in [13]. For a stronger version of [13, Proposition 1] (an application of Cliord's theorem), see [5, Proposition 3.4]. Also, Example 5.11 below generalizes the main result of [13] to arbitrary self-dual Hermitian codes. Notation and Background State complexity. Let C be a linear code of length n and 0 i n. The state space dimension of C at depth i is where C 0g: The state complexity of C is ng. It is well known that s(C ? simple upper bound on s(C) (and hence on s[C]) is the Wolf bound dim(C)g. We write [C] for the set of codes equivalent to C by a change of coordinate order i.e. C if and only if there exists a permutation (l n) such that C Cg. Then we dene the absolute state complexity of C to be The dimension/length prole (DLP) of C is (k ig. Clearly dim(C (C). The DLP bound on s i (C) is and the DLP bound on s(C) is ng. We will use DLP bound to mean for some C. It is well known that r(C ? Since r(C) is independent of coordinate order of C, r(C) s[C]. If we say that C is DLP-tight; e.g. if then C is DLP-tight. Hermitian codes. Our terminology and notation for Hermitian codes for the most part follow [14]. We write H=F q 2 for the Hermitian function eld. Thus over F q 2 and y q is the minimal polynomial of y over F q 2 [x]. The genus of H=F q 2 is We write PH for the set of places of H=F q 2 and DH for the divisor group of H=F q 2 . For for the valuation of z at Q. Thus v Q (z) < 0 if and only if Q is a pole of z and v Q (z) > 0 if and only if Q is a zero of z. Also (z) 2 DH is given by Q2PH vQ (z)Q and for A 2 DF , There are q 3 places of degree one in PH . One of these is the place at innity, which we denote Q1 . We denote the others as . For the rest of the paper, unless otherwise stated We put . For an integer m, L(mQ1 The Hermitian codes over F q 2 are CL (D; mQ1 permutation (l Strictly speaking the code C(D;mQ1 ) depends on the permutation (l n) and may be better denoted CL (Q l 1 this notation is cumbersome and CL (D; mQ1 ) is standard. Unless otherwise stated, when we write CL (D; mQ1 ) we have some xed but arbitrary coordinate order in mind. From the usual expression for the dimension of geometric Goppa codes, dim(CL (D; mQ1 When m is understood, stated otherwise. The abundance of CL (D; mQ1 ) is dim(mQ1 D). For m < n, the abundance is 0 and the code is non-abundant. For , so we restrict our attention to m 2 m, the dual of CL (D; mQ1 ) is be the pole number sequence of Q1 . Also, for ig. Thus [1; 1) is the set of pole numbers of Q1 , (r) is the rth pole number and 1 [R 1 g. We note that From [14, Proposition VI.4.1] we deduce that We note that, for m < n, State complexity of Hermitian codes. For 0 i n we put D (where (l xed but arbitrary permutation of We deduce that In particular These identities yield s(CL (D; mQ1 Thus we will almost exclusively be interested in m 2 I(n; In fact, since restrict our attention to deducing results for It is convenient to put J(n; Using results of [11, 16], [5, Proposition 5.1] shows that for m 2 I(n; g), which is used to prove Theorem 2.1 ([5, Theorem 5.5]) For is attained at m 2g 2 cq and equals min l um If CL (D; mQ1 ) is DLP-tight then we just say m is DLP-tight. 3 When the DLP bound is not tight by [5, Proposition 4.3, Example 4.9], we have r(CL (D; mQ1 r(CL (D; mQ1 where CL (D; mQ1 ) can have any coordinate order. Such m are therefore DLP-tight and we are reduced to determining which m 2 I(n; g) are DLP-tight. We note that n 3+ 2g < n, so that the codes that we are interested in are non-abundant. In this section we determine the m 2 I(n; g) which are not DLP-tight, i.e with s[CL (D; mQ1 )] > r(CL (D; mQ1 )): The coordinate order of CL (D; mQ1 ) is arbitrary, so it suces to show that Table 1: Table of New Notation xed prime power r { (CL (D; mQ1 Improved DLP bound for m 2 I(n; g) Denition 3.11 { (CL (D; mQ1 (Theorem 3.9 and Corollary 3.10) Finite places of degree one in PH ab Elements of F q 2 such that q+1 ac Elements of F q 2 such that q Element of P 1 H such that x(Q a;b;c ab and y(Q a;b;c Cm Element of [CL (D; mQ1 )] with coordinate order given in Section 4 of points of gain and fall of Cm fall (m) jP gain (m) \ [1; i]j and jP fall (m) \ [1; i]j by (j; gain 0; q q2; q depending on M : dened before Proposition 4.8 fall 0; q+q2 depending on M : dened before Proposition 4.8 3: Our approach has three steps. (i) we prove the key lemma, Lemma 3.2, and indicate how this can be used to show that m is not DLP-tight (Example 3.3) (ii) a generalisation of the key lemma (Lemma 3.4) and an application 3.5. We indicate how this can be used to improve on the DLP bound by more than one (Example 3.6). (iii) an application of Proposition 3.5 to improve the DLP bound for m 2 I(n; g), Theorem 3.9 and Corollary 3.10. We conclude Section 3 with a table of the improved DLP bound for small values of q and an analysis of the proportion of those m 2 I(n; g) for which our bound is strictly better than the DLP bound (Proposition 3.12). The key lemma. We begin with a clarication of Equations (3) and (4). Lemma 3.1 For dim(mQ1 D and s i (CL (D; mQ1 only if there is equality in both. Proof. The rst part follows from [5, Lemma 4.1] and the fact that the gonality sequence of the pole number sequence of Q1 by [12, Corollary 2.4]). The second part then follows from (3) and (4). 2 We note that Lemma 3.1 implies that a coordinate order is inecient, in the sense of [9], if and only if there exists an i, 0 dim(mQ1 D i;+ ). To show the stronger result that s(CL (D; mQ1 require a stronger condition on i, namely that it satises so that s i (CL (D; mQ1 This stronger condition is clearly more likely to hold attains or is close to attaining r(CL (D; mQ1 )). For now, we concentrate on determining when the equalities in (5) cannot hold. For these equalities to hold, dim(mQ1 D must change with respectively. We shall see that it is possible for both (i.e. it is possible that both are pole numbers of Q1 ). Lemma 3.2 For m n 2+g, it is not possible that dim(mQ1 D and dim(mQ1 D i;+ Proof. We assume that dim(mQ1 D derive a contradiction. Suppose we have z is a principal divisor of H=F q 2 (e.g. as in the proof of [14, Proposition VII.4.2]), say nQ1 and vQ l i so that by [14, Lemma I.4.8]) dim((2m n)Q1 +Q l i Now (2g 2)Q1 is a canonical divisor of H=F q 2 (e.g. by [14, Lemma VI.4.4] or because 2g 2 is the gth pole number of Q1 and [14, Proposition I.6.2]). Thus dim((2m n)Q1 by the Riemann-Roch theorem, so from (6), Again by the Riemann-Roch theorem dim((2g so that and hence L((2g 2 2m+n)Q1 Q l i giving the required contradiction. 2 Example 3.3 Let 3. We show is not DLP-tight. From (2), we have [1; 11. From Theorem 2.1, r(CL (D; mQ1 and similarly, r 14 (CL (D; mQ1 implies that s i (CL (D; mQ1 3.1 then implies that dim(mQ1 D 13; dim(mQ1 D 14;+ and since the coordinate order of CL (D; mQ1 ) is arbitrary, m is not DLP-tight. We will see in Section 5 that 14 and 15 are DLP-tight. Generalisation of the key lemma. Since dim(mQ1 D i dim(mQ1 Lemma 3.2 can be restated as: for m n 2+ g, either dim(mQ1 D i dim(mQ1 D i 1;+ ). This generalises as Lemma 3.4 For or (ii) dim(mQ1 D i;+ ) dim(mQ1 D i t;+ Proof. Suppose that dim(mQ1 D tc. So there are such that dim(mQ1 D dim(mQ1 so that, since jfi t contradicting Lemma 3.2. 2 The following application of Lemmas 3.1, 3.4 is a straightforward consequence of (3),(4). Proposition 3.5 For Example 3.6 Let We have (e.g. by the Riemann-Roch Theorem). From (2), the rst few pole numbers of Q1 are [1; 16g. From Theorem 2.1, we have r(CL (D; mQ1 so that, from (4), 8g. Thus Proposition 3.5 gives We shall see in Section 5 that s[CL (D; mQ1 Improvement on the DLP bound. We show how Proposition 3.5 can be used to improve on the DLP bound generally. First, we introduce some useful notation: q is even and q if q is odd. For a xed m 2 J(n; g), we put We easily deduce: Lemma 3.7 (i) for q odd, 0 M q 3 We begin by reinterpreting Theorem 2.1 in terms of M and M . Lemma 3.8 For m 2 J(n; g), the DLP bound is attained at Proof. If u; v are dened as in Theorem 2.1, then The result now follows from the fact that the DLP bound is attained at m 2g Next we give our improvement on the DLP bounds for m 2 J(n; g). The size of the improvement is given by We note that (m) > 0 if and only if q q2 2 or q M M q q 2 . Theorem 3.9 For Proof. First assume that q q2 is attained at We have two subcases. (a) For q q2 2 we have 0 < t M +q 2 . Now, from (2), M 1), so that j 1 [m t, and Proposition 3.5 gives 2: (b) For q M M q 2+q2we have From (2), 2 (N) since M q 2, so that j 1 [m+i n t+1; m+i Suppose now that q M M q q 2 . From Lemma 3.8, r(CL (D; mQ1 )) is attained at 3.5. Now again we have two subcases. (a) For q M M q M +1 2 we have 0 < t M +1 q 2 . From (2) (M +1 q t, and Proposition 3.5 gives s[CL (D; mQ1 )] r(CL (D; mQ1 (b) For q M M q q 2 we have M From (2) so that j 1 [m so that from Proposition 3.5, Corollary 3.10 For Proof. Easy consequence of Theorem 3.9, the denition of (m). 2 Definition 3.11 For m 2 I(n; g), we put r { (CL (D; mQ1 We note that for m 2 I(n; g), r { (CL (D; mQ1 In Table 2 we have written r { (m) for r { (CL (D; mQ1 ) and the DLP bound is calculated using Theorem 2.1. The bold face entries are those for which r { (CL (D; mQ1 (The values of r { (CL (D; mQ1 can of course be deduced from (7).) Table 2: r { (CL (D; mQ1 r { (m) 3 r { (m) 11 11 11 r { (m) 26 27 27 28 28 28 r { (m) 53 53 54 54 55 56 56 56 56 56 r { (m) 151 151 152 153 153 154 155 156 156 156 r { (m) 157 157 157 158 159 159 159 159 159 159 159 r { (m) 228 229 230 231 231 232 233 234 234 234 235 236 236 236 r { (m) 237 238 238 238 238 239 239 239 239 240 240 240 240 240 We conclude this section by calculating the proportion of m 2 I(n; g) for which (m) > 0. Proposition 3.12 if q is odd2 Proof. We note rst that jI(n; 1. Recall from the denition of (m) that Next we note that j 1 (0; This follows from the denition of (m) for when q is odd and from n 2+ when q is even. Now, xing We note that the restriction M q 1 does not aect this. We also note that for q even and M 2 , the restriction of M Thus the result follows from is even:Thus, for large q at least, r { (CL (D; mQ1 )) improves on r(CL (D; mQ1 )) for just under half the We shall see in Section 5 that m is DLP-tight when r { (CL (D; mQ1 )) fails to improve on r(CL (D; mQ1 )). 4 A Good Coordinate Order We describe a 'good' coordinate order for Hermitian codes, denoting the code in [CL (D; mQ1 )] with this coordinate order by Cm . After recalling the notions of points of gain and fall for a linear code, we give the most natural description of the points of gain and fall of Cm in Propositions 4.2 and 4.4. We conclude by characterising the points of gain and fall of Cm as 'runs' in Theorem 4.10 (which we will use in Section 5 to derive a formula for s(Cm ) .) The good coordinate order. As noted at the beginning of Section 3, for m n 2 coordinate orders of CL (D; mQ1 ) are equally bad with regard to state complexity. Thus we are interested in m 2 I(n; g). Recall that H=F q 2 has places of degree one viz. Q1 , and the nite places of degree one, for some xed but arbitrary ordering (Q l 1 H . Thus the order of P 1 H determines the coordinate order of CL (D; mQ1 ). As in [14], for each (; there exists a unique H , such that x(Q We now describe an order of P 1 H giving Cm 2 [CL (D; mQ1 )]. First we relabel the elements of P 1 as Q a;b;c for certain integers a; b; c. We write f0; Now for each a 2 F q nf0g there exist such that q for q. Thus for each a 2 F q n f0g, 0 c q 1 and 0 b q, there exists H , such that x(Q a;b;c H . For exist Thus the remaining q elements of P 1 H , which we write as Q 0;0;c for 0 c q 1, are such that We note that Q When a, b or c takes any of its possible values we write Q ;b;c , Q a;;c or Q a;b; . Note that for a we have q. Thus there are q places of the form Q 0;; and for 1 a q 1 there are q 2 1 places of the form Q a;; . We rst describe the ordering of P 1 H giving Cm 2 [CL (D; mQ1 )] for m 2 J(n; g). This uses lexicographic order of t-tuples of integers: (i only if there exists u such that is dened by simply using the order of H . For q M 1 , Cm is dened by the'Order O2' of P 1 H into three sets Then Order O2 of P 1 H is given by putting P 1 3 by Q 1;b;c < Q 2 by Q a;b;c < Q a 0 ;b 0 ;c 0 if (a; b; c) < (a For the coordinate order of Cm is dened to be that of Cm ? . From now on Q i denotes the ith element of P 1 H ordered as above. Thus The points of gain and fall of Cm . Points of gain and fall were introduced in [3, 6]. For this paragraph, C is a length n linear code with dimension k. We note that dim(C i; ) (as dened in Section 2) increases in unit steps from 0 to k and dim(C i;+ ) decreases in unit steps from k to 0 as i increases from 0 to n. If 0 i n then i is a point of gain of C if dim(C i;+ i is a point if fall of C if dim(C These denitions are motivated by (1). We note that there are k points of gain and k points of fall. Points of gain and fall describe the local behaviour of a minimal trellis, [6], and being able to give a succinct characterisation of them for particular families of codes has been useful in calculating for their state complexity, e.g. [3, 6]. The same proves to be the case here. We note that, as in [6], i is a point of gain of Cm if and only if i is the 'initial point' of a codeword of Cm i.e. if and only if there exists z 2 L(mQ1 ) such that Similarly i is a point of fall of Cm if and only if i is the 'nal point' of a codeword of Cm i.e. if and only if there exists z 2 L(mQ1 ) such that We write P gain (C) and P fall (C) for the sets of points of gain and fall of C. With P fall (C): We also write P gain (m) := P gain (Cm ) and P fall (m) := P fall (Cm ). We will need a function closely related to . Dene We have [1; Im() from [14]. We note that and for m < Proposition VII.4.3]. For 0 a q 1, we put and Also we put a=0 A(a) and We will determine the initial and nal points of certain z 2 H=F q 2 of the form that (x ab )(Q a 0 ;b only if a = a only if a = a course, we are interested in when Lemma 4.1 If (j; l) 2 1 [0; m], Proof. We put z Using the facts that (i) vQ1 fQ1g. Hence (j; l) 2 1 [0; m] implies that z jl 2 L(mQ1 ). 2 Proposition 4.2 (O1 ordering of P 1 1. P gain 2. P fall Proof. We order the set A by ab < a 0 b 0 if and only if (a; b) < (a and only if Q a;b; < Q a 0 ;b 0 ; . For 0 d q 2 1, we write d for the (d 1)st element of A. Thus a(d) by Thus and (y ac We begin with P gain (m). For (j; l) 2 1 [0; m] we put We note that jq (j; l) m n 2 which implies that j < q 2 +q 1 that u gain jl and z gain jl are well-dened for all (j; l) 2 1 [0; m]. Now u gain only Hence the initial point of z gain jl is jq Also, by Lemma 4.1, z gain Finally, each (j; l) 2 1 [0; m] gives a dierent point of gain of Cm and, since j 1 [0; are all the points of gain. Similarly for points of fall. 2 We use Proposition 4.2 to determine s(Cm ) for To do this we use and so we put gain fall fall (Cm Example 4.3 If C 4 is our rst example of a geometric Goppa code with where the latter is given by Theorem 2.1.) Proof. The coordinate order of C 4 is Q In the notation of Proposition 4.2, we have Now P gain (4) is the set of initial points of z gain jl , where (j; l) 2 1 [0; 4]. These are given in the table below. The third column in the table gives the 'initial place' i.e. the Q a;b;c such that Q where i is the initial point. (j; l) z gain jl Initial Place Initial Point Thus P gain given by the nal points of z fall jl such that (j; l) 2 (j; l) z fall jl Final Place Final Point Thus P fall using (9) we have gain giving For 2: Proposition 4.4 (O2 ordering of P 1 gain (m) and P fall fall (m) where Proof. We recall that P 1 3 were dened in (8). We note that 1 , so that writing and for gain (q 2 and for gain (q 3 . We begin by showing that P 1 gain (m) P gain (m). For (j; l) 2 1 [0; m] such that 0 j q and we exhibit an element of L(mQ1 ) with initial point Thus v gain l (Q a;;c only if a = 1, and 0 c l 1 and u gain (Q a;b; only if l (taking q). Hence the initial point of z gain jl is 1. Also, from Lemma 4.1, z gain gain (m) P gain (m). Next we show that P 2 gain (m) P gain (m). We order A n A(1) by ab < a 0 b 0 if and only if (a; b) < for the (d+1)st element of AnA(1), where 0 d q 2 q 2. (This is dierent from the labelling in the proof of Proposition 4.2 since we do not include A(1) in the relabelling.) We dene a(d) by writing set z gain We note that jq (j; l) m gain (q which implies that j q 2 q 2. Thus jl and z gain are well-dened for all (j; l) 2 jl (Qd a(d)c l 1. Thus Therefore the initial point of z gain is gain (q and (z gain Hence z gain gain P gain (m) and gain (m) P gain (m). it remains to show that jP 1 To do this we exhibit a bijection 1 [0; m] ! gain (m). First, for (j; l) 2 1 [0; m] we map (j; l) to l(q gain (m) if Now we are left with dening a bijection F : gain (m) by F (j; (j It is easy to check that F maps into P 2 gain (m) and F is one-to-one since for gain l q 1 and Finally we prove F is onto. For gain (m), such that (j (j It is straightforward to see that (i) (j This completes the proof for P gain (m). Similarly for the points of fall. 2 Example 4.5 If { (CL (D; 13Q1 using Theorem 3.9, but s(C 13 Proof. The coordinate order of C 13 is We use the notation of the proof of Proposition 4.4. We note that gain = 1. Thus for 0 j q and 0 l gain 1, jq gain (13) is the set of initial points of z gain which are as follows. (j; l) z gain jl Initial Place Initial Point (3; Thus gain (13). Now we have so that 2. Then P 2 gain (13) is the set of initial points of z gain such that (j; l) 2 1 [0; 13 gain (q giving the following. Initial Place Initial Point (0; (0; Thus for P fall (13). We have P gain From Propositions 4.2 and 4.4 we have, if (i) 0 M q M 2or (ii) q M M q or (iii) In these cases the following useful property holds. Remark 4.6 For a length n code C, if P fall In particular, for m 2 J(n; g), if (i) q is odd and 0 M q M 2or q M M q or (ii) q is even and 0 M q, then s i (Cm n. The same holds for Proof. The proof is similar to that of [6, Proposition 2.5], and in fact can be modied to hold for branch complexity as in [6, Proposition 2.5]. We put P i;+ Of course, with gain (C) and P i;+ fall (C) for any linear code C. The condition P fall implies that also fall (C) and P gain (C): Thus, from (9), we have gain (C) is ordered by O1 then as in the proof of Proposition 4.2 so that by O2, P gain Thus O2 is strictly better than O1 for m=14. If C 2 [CL (D; 15Q1 )] is ordered by O1, then then as in the proof of Proposition 4.2, P gain But if C is ordered by O2, we get P gain Thus O1 is strictly better than O2 for m=15. To summarise, for { (CL (D; mQ1 )). Thus, in these cases s(Cm and the coordinate order for Cm is optimal with regard to s(Cm ). In fact, except for Another characterisation of the points of gain and fall of Cm . We now characterise gain (m) and P fall (m) as runs i.e. as sequences of non-contiguous intervals of integers. This is useful since s(Cm ) must be attained at the end of a run of points of gain. Thus to determine s(Cm ), we only need to nd the maximum of s i (Cm ) over those i that end a run of points of gain, i.e. over those i such that We begin by combining Propositions 4.2 and 4.4 for a common development of the cases (i) 0 2 or q M . First we extend the denitions of gain and fall as follows: and Proposition 4.8 For gain (m) and P fall fall (m) where Proof. From the examples above and Remark 4.7, we can assume that q 4. For q M 2 M q M +1, the result is just a restatement of Proposition 4.4. Also, for the result states that P gain in agreement with Proposition 4.2. So we are reduced to m such that q M Rewriting see that P 1 qg. We claim that P 2 1g. Firstly, if 0 j q and 4. Thus we need to show that If k is in the left-hand side, (l In either case, 0 j q, 0 l q 1 and l(q that k is in the right-hand side. The reverse inclusion is similar. The result now follows from Proposition 4.2 since for q M gain (m)+1 and Lemma 4.9 If 2. Proof. Straightforward using Lemma 3.7. 2 Theorem 4.10 For 1. P gain (m) is the union of (b) fm 2g gain (c) fm 2g gain and 2. P fall (m) is the union of (a) [n m+ 2g (b) fn m+ 2g (c) fn m+ 2g Proof. As in the proof of 4.8, we assume that q 4. We will use the fact that For convenience we put R 1 e q We show that R 1 gain (m) P gain (m) in two steps. First we note that P 1 since for q 4, 0 j q and 0 l gain 1 q 1, (j; l) Next we show that [ gain (q gain (m). Now from (10) we have for and 0 l q 1: Also, if 0 j and 0 l q 1 then, again using (10), (j; l) so that (j; l) 2 gain (m). Next we show that R 2 gain gain 1. Then, from (10), and using (10), gain (m) if (e that R 2 gain gain (m). If q 1 gain e q 1 and 0 f q 1 e then and f gain , so that R 3 gain gain (m). Thus gain (m) P gain (m) and it suces to show that j that jP gain [ gain (m) The proof for P fall (m) is similar and we omit the details. 2 5 When the DLP bound is tight Here we use Theorem 4.10 to determine s(Cm ). We know (from Corollary 3.10 and Proposition 3.12) that s[CL (D; mQ1 )] > r(Cm ) for just under half of the m in the range I(n; g). We show that for the remaining m in this range, s(Cm As a consequence, we have determined s[CL (D; mQ1 )] and a coordinate order that achieves s[CL (D; mQ1 )] for such m. For those m with s(Cm ) > r(CL (D; mQ1 )) we compare the upper bound, s(Cm ), on s[CL (D; m1 )] with the lower bound r { [(C L (D; mQ1 )] given in Corollary 3.10. When q is odd, these bounds meet for over three-quarters of those m in I(n; g), but when q is even, the bounds meet for only a little over one half of those m in I(n; g). Determining s(Cm ). As discussed in Section 4, it suces to nd the maximum of s i (Cm ) over those i such that (m). From Theorem 4.10, there are only q i. Thus concentrating on these i is signicantly simpler. So we calculate s values of i (in Proposition 5.5) by determining P gain (m) and P fall (m) (in Lemmas 5.1 and 5.4). We determine which of these i gives the largest s i (Cm ) (in Lemma 5.6). This enables us to write Theorem 5.7). Early on we introduce a variable which plays a crucial role in the proofs and statements of many of the results and we end with a table of s(Cm 8g. We begin by determining s(Cm We note rst that were dened just before Proposition 4.8. As noted above, s i (Cm From Theorem 4.10 such i are either (i) of the form m 2g gain or (ii) of the form m 2g Thus putting we have From gain (m); so we wish to determine P gain (m) and P fall (m) for 1. The rst of these is straightforward. Lemma 5.1 For Proof. Since 1. For 1 e q 2 gain , Theorem 4.10 gives e The rst case follows since In the second case, q e fall (m) it is convenient to introduce some more notation. For xed m we put Thus norm is 0, 1 or 2 depending on whether Also we put 3: In Lemma 5.4 and Propositions 5.5 we will see a symmetry between the roles of e in P gain (m) and e in P fall (m). We will see in Lemma 5.6 that s i e (Cm ) is maximised near and hence appears naturally in our formula for s(Cm ). Lemma 5.2 q 1 2q 3. Proof. First, it follows from Lemma 3.7 that is even and M > 0 is even and M Next, clearly 2q 2, with equality only if M 1. However, from Lemma 2 so that norm 1. 2 Now, in order to use Theorem 4.10 to calculate P fall (m), we need to write i e as n m+2g+ fall preferably non-negative, integer e 0 and 0 f q 1. We could then determine an expression for fall (m) in terms of e 0 and f in a similar way to the proof of Lemma 5.1, except that f would add complications. This would give us an expression for s i e (Cm ) in terms of e, e 0 and f . To maximise this over 1 e q 1 we would need to relate e 0 and f to e. Fortunately these relationships are reasonably simple. Lemma 5.3 Let m 2 I(n; g) and 1 e q 1. If we write then e e for q 1 gain e q 1. In particular e 0 0. Also if e q Proof. For Now giving norm )q which implies that e. Similarly, for q 1 gain e q 1 we get For the second part we have q 1 (from Lemma 5.2) and f e (from the rst part). Thus We show that, for e q it is not possible that Firstly implies that e q 1 gain . Also imply that e fall . Thus q 1 gain e fall so that, adding gain to both sides, Now, as in (12), implies that either (i) 2 and Each of these clearly contradicts (13). 2 Lemma 5.4 For Proof. We as in Lemma 5.3 and work from Theorem 4.10. First, if e 0 q, i.e. if e q, then P We note also that, for e q, q Next, if q 1 fall e 0 q 1, i.e. if q the last equality following from the second part of Lemma 5.3. Finally (since e 0 0 by Lemma and q since q 1 and f e, by Lemma 5.3. 2 We use the convention that, for b 0, a In particular a a for a 0 0 for a 0, a 1 for a 0 0 for a < 0, a a 1 a 1 where b 1. Lemmas 5.1 and 5.4, together with (9), give Proposition 5.5 For Now we determine for which e, 1 e q 1, s i e (Cm ) is maximised. Lemma 5.6 For 1. at 2. at Proof. From Proposition 5.5, with q e we have s i e (Cm maximising s i e (Cm ) is equivalent to minimising (e) over 1 e q 1. Now, for 0 e q 1, q e Thus, since 0 q 2 gain e 1, we have First, for 0 e q implies that (e) (e 1) 2q so that (e) is minimised over 1 e q + 1 at 1. Thus it is sucient to determine where (e) is minimised over q We note that, since 2q 3 (Lemma 5.2), l m Similarly, for q that if e b 1. if d e q 2. if b e. This leaves the case b . In this case, the above analysis implies that (e) is minimised at either b d Also we have fall gain so that (e) is minimised at d e. Finally we note that if 2q 3 2 fall then 2q so that adding sides and dividing by 2 gives l m and we are in case 1 above. Also if we have d we are in case 1. Similarly for 2q 6 2 fall we are in case 2 above. 2 Proposition 5.5 and Lemma 5.6 give us Theorem 5.7 For Proof. The result follows since 1. for 2q 6 2 fall , q 2 gain b 2. for 3. for 2q 4 2 gain , q 2 gain d For example, 2q 6 2 fall implies that b so that The other equalities and inequalities follow similarly. 2 Of course, Theorem 5.7 essentially gives the values of s(Cm ) for I(n; g) since Table Comparing these values of s(Cm ) with the values of r { (CL (D; mQ1 )) given in Table 2 (where r { (CL (D; mQ1 )) is as dened in Denition 3.11), we have s(Cm { (CL (D; mQ1 )) except for 281g. In particular, s(Cm ) achieves the DLP bound for Cm for q 2 f2; 3; 4; 5; 7; 8g and m 2 I(n; g) when this is not excluded by Corollary 3.10 i.e. whenever the entry for m or m ? in Table 2 is not in boldface. Table 3: s(Cm ) for q 2 f2; 3; 4; 5; 7; 8g and m 2 J(n; g) 28 28 28 Comparing s(Cm ) with r { (CL (D; mQ1 )) We start by reinterpreting r(CL (D; mQ1 )) in terms of in Theorem 5.8. We use this to calculate (in Proposition 5.9) and hence to show (in Corollary 5.10) that s(Cm this is not excluded by Corollary 3.10 . This means that s(Cm ) achieves the DLP bound for Cm for just over half of those m in the range [ We then compare s(Cm ) with r { (CL (D; mQ1 )) in Table 4 and see that s(Cm ) achieves the bound r { (CL (D; mQ1 )) for approximately a further quarter of those m in [ n 1; n 3+2g] if q is odd but only for about a further 1=q of those m in [ n 1; n 3+ 2g] if q is even. Previously we partitioned J(n; g) into three subintervals, according to whether 0 M q M 2 Now we consider a ner partition and say that according to whether (A) 0 M q 2 2 or We compare s(Cm ) with r { (CL (D; mQ1 )), by reinterpreting Theorems 3.9 and 5.7 using (A){(E). Theorem 5.8 If m 2 J(n; g), then r(CL (D; mQ1 Proof. Take u and v as in the statement of Theorem 2.1. It is straightforward to show, using the characterisation of (u; v) given in the proof of Lemma 3.8, that if m satises (A), (C) or 2. Thus Theorem 2.1 implies that, for m satisfying (A), (C) or (E), r(CL (D; mQ1 min and for m satisfying (B) or (D), r(CL (D; mQ1 min l m First, for m satisfying (A), (C) or (E) we have (i) d +1e q M 1 if norm 2 f0; 1g or (ii) 2. Also gain or (iii) for . Thus, for m satisfying (A), (C) or (E), r(CL (D; mQ1 )) is equal to as required. Similarly, for m satisfying (B) or (D) (so that norm 1) it is easy to see that (by considering the cases that 1. Thus, for m satisfying (B) or (D), r(CL (D; mQ1 as required. 2 Before comparing s(Cm ) with r { (CL (D; mQ1 )), we compare it with r(CL (D; mQ1 )). To do this we rene (A){(E) as follows: if m satises (C) then we say that m satises (C1), (C2) or (C3) if Proposition 5.9 For Proof. Using it is straightforward to see that if 1. if m satises (A), (B), (D) or (C3), then 2q 2 fall 4, 2. if m satises (C1) or (E), then 2q 6 fall or 3. if m satises (C2), then Also, if The result then follows from Theorems 5.7 and 5.8 noting that, for cases (B) and (D), gain cases (C1) and (E) with M q 1, It follows from Proposition 5.9 that s(Cm ) achieves the DLP bound for Cm as often as this is possible. We state this as Corollary 5.10 For only if Proof. Since for and s(Cm suces to show the result for m 2 J(n; g). It follows from the denition of (m) for such m that only if (i) m satises (A) or (ii) m satises (C3) or (iii) q. These are exactly the values of M for which Proposition 5.9 implies that Example 5.11 If Cm is self-dual, then r(Cm where Cm has the lexicographic coordinate order. In particular, s[Cm We know that q is a power of 2, g). From the denitions, 4 by Theorem 5.8. The result now follows since We remark that the main result of [13] is Example 5.11 with q 4. Corollary 5.10 and Proposition 3.12 imply that r(Cm ) is attained for just over half the m 2 I(n; g). Explicitly, the proportion of these m for which the DLP bound is attained is 1+ 1 for q odd and 1+ 3q 5 for q even. Of course Corollary 5.10 implies that if m satises (A), (C3) or M = q is odd, then s[CL (D; mQ1 The bounds on s[CL (D; mQ1 )] given by Theorem 3.9 and Proposition 5.9 for all m in J(n; g) (and hence implicitly also for are given in Table 4. The lower bound is Table 4: Table of Bounds on s[CL (D; mQ1 Lower Bound Upper Bound satises r(CL (D; mQ1))+ r(CL (D; mQ1))+ Range (D) M +M r { (CL (D; mQ1 )) and the upper bound is s(Cm ). The entries for both bounds are the amount by which they exceed r(CL (D; mQ1 )). The range is the upper bound minus the lower bound. As well as those m for which s(Cm implies that { (CL (D; mQ1 for those m 2 J(n; g) such that if q is odd Hence (15) also holds for those In all these cases except M 2 and M s[CL (D; mQ1 For 3 we have s[CL (D; mQ1 2: For q odd, this gives q 2 1values of m 2 I(n; g) for which s[CL (D; mQ1 )] is determined but is strictly greater than r(CL (D; mQ1 )). Thus, for q odd, the total proportion of those m in I(n; g) for which we have determined s[CL (D; mQ1 )] is2 For q even, it gives q 2 values of m 2 I(n; g) for which s[CL (D; mQ1 )] is determined but is strictly greater than r(CL (D; mQ1 )). Thus, for q even, the total proportion of those m 2 I(n; g) for which we have determined s[CL (D; mQ1 Thus we have determined s[CL (D; mQ1 )] for over three quarters of those m in I(n; g) when q is odd but only for something over one half of those m in I(n; g) when q is even. For q odd, the rst m for which s[CL (D; mQ1 )] is not determined is (when it is either 56 or 57), and for q even the rst m for which s[CL (D; mQ1 )] is not determined is it is either 236 or 237). --R The Weight Hierarchy of Hermitian codes. On the state complexity of some long codes. On the trellis structure of GRM codes. Lower bounds on the state complexity of geometric Goppa codes. On trellis structures for Reed-Muller codes On a family of abelian codes and their state complexities. Bounds on the state complexity of geometric Goppa codes. Foundation and methods of channel encoding. On the generalized Hamming weights of geometric Goppa codes. On special divisors and the two variable zeta function of algebraic curves over Bounds on the State Complexity of Codes from the Hermitian Function Field and its Sub Algebraic Function Fields and Codes. Geometric approach to higher weights. On the weight hierarchy of geometric Goppa codes. --TR --CTR T. Blackmore , G. H. Norton, Lower Bounds on the State Complexity of Geometric Goppa Codes, Designs, Codes and Cryptography, v.25 n.1, p.95-115, January 2002

hermitian code;dimension/length profile bound;state complexity

587964

Efficiency of Local Search with Multiple Local Optima.

The first contribution of this paper is a theoretical investigation of combinatorial optimization problems. Their landscapes are specified by the set of neighborhoods of all points of the search space. The aim of the paper consists of the estimation of the number N of local optima and the distributions of the sizes $(\alpha_j)$ of their attraction basins. For different types of landscapes we give precise estimates of the size of the random sample that ensures that at least one point lies in each attraction basin. A practical methodology is then proposed for identifying these quantities ($N$ and $(\alpha_j)$ distributions) for an unknown landscape, given a random sample of starting points and a local steepest ascent search. This methodology can be applied to any landscape specified with a modification operator and provides bounds on search complexity to detect all local optima. Experiments demonstrate the efficiency of this methodology for guiding the choice of modification operators, eventually leading to the design of problem-dependent optimization heuristics.

Introduction . In the eld of stochastic optimization, two search techniques have been widely investigated during the last decade: Simulated Annealing [25] and Evolutionary Algorithms (EAs) [6, 7]. These algorithms are now widely recognized as methods of order zero for function optimization as they impose no condition on function regularity. However, the e-ciency of these search algorithms, in terms of the time they require to reach the solution, is strongly dependent on the choice of the modication operators used to explore the landscape. These operators in turn determine the neighborhood relation of the landscape under optimization. This paper provides a new methodology allowing to estimate the number and the sizes of the attraction basins of a landscape specied in relation to some modication operator. This allows one to derive bounds on the probability that one samples a point in the basin of the global optimum for example. Further, this method could be used for guiding the choice of e-cient problem-dependent modication operators or representations. Formally, a landscape can be denoted by E) where f is the function to optimize and the modication operator that is applied to elements of the search space E. The structure of the landscape, heavily depends on the choice of the modication operators, which in turn may depend on the choice of the representation (the coding of the candidate solutions into binary or gray strings for example). Hence, before the optimization process can be started, there is a number of practical choices (representation and operators) that determine the landscape structure. Consequently, these choices are often crucial for the success of stochastic search algorithms. Some research has studied how the tness landscape structure impacts the potential search di-culties [13, 21, 22, 26]. It is shown that every complex tness landscape can be represented as an expansion of elementary landscapes {one term in the Fourier expansion{ which are easier to search in most cases. This result has been applied to Centre de Mathematiques Appliquees, Ecole Polytechnique, 91128 Palaiseau Cedex, France y Corresponding author, Tel: (33).1.69.33.46.30 Fax: (33).1.69.33.30.11, E-mail: Josselin.Garnier@polytechnique.fr J. Garnier and L. Kallel solve a di-cult NP-complete problem [20] (the identication of minimal nite k-state automaton for a given input-output behavior), using evolutionary algorithms. Other theoretical studies of search feasibility consider the whole landscape as a tree of local optima, with a label describing the depth of the attraction basin at each node [16, 19]. Such a construction naturally describes the inclusion of the local attraction basins present in the landscape. These studies investigate tree structures that ensure a minimal correlation between the strength of the local optima and their proximity to the global optimum, with respect to an ultra-metric distance on the tree. However, from a practical point of view, the tree describing the repartition of local optima is unknown and too expensive in terms of computational cost to determine for a given landscape. The lack of an e-cient method at reasonable cost that allows one to characterize a given landscape, motivates the construction of heuristics for extracting a priori statistical information about landscape di-culty, for example based on random sampling of the search space. We cite from the eld of evolutionary algorithms: Fitness Distance relations, rst proposed in [8] and successfully used to choose problem dependent random initialization procedures [11, 14]; Fitness Improvement of evolution operators, rst proposed in [5], then extended and successfully used to choose binary crossover operators [12] and representations [9]. However, even if such heuristics can guide the a priori choice of some EA parameters, they do not give signicant information about landscape structure, for instance, recent work suggests that very dierent landscapes (leading to dierent EA behaviors) can share the same tness distance relation [18, 10]. Further, the e-ciency of such summary statistics is limited to the sampled regions of the space, and therefore does not necessarily help the long term convergence results as implicitly illustrated in [12] for example. This gives strong motivation for developing tools that allow one to derive a more global (beyond the sampled regions) information on the landscape at hand, relying on an implicit assumption of stationarity of the landscape. Along that line, this paper proposes a new method to identify the number and the repartition of local optima with respect to a given neighborhood relation of a given landscape. The proposed method applies to any neighborhood relation specied with a modication operator, and hence provides a practical tool to compare landscapes obtained with dierent operators and representations. The framework is the following. We assume that the search space E can be split into the partition E 1 ,.,E N of subspaces which are attraction basins of local maxima of the tness function. We also assume that there exists a local search algorithm (for example a steepest ascent) which is able to nd from any point of the search space the corresponding local maximum: The basic problem consists in detecting all local maxima m j . This is equivalent to nding a way to put a point in all attraction basins because the local search algorithm will complete the job. We shall develop the following strategy. First we shall study the direct problem, which consists in studying the covering of the search space by a collection of points randomly distributed when the partition Second we shall deal with the inverse problem which consists in estimating the number of local maxima from information deduced from the covering. Direct problem (Section 4): One puts M points randomly in the search space. The question is the following: Given the statistical distribution of the relative sizes of E-ciency of local search with multiple local optima 3 Fig. 1. Schematic representations of the search space E with points have been randomly placed on both pictures. As a result there is at least one point in each attraction basin in the left picture, but not in the right picture, where E 4 is empty. the attraction basins and their number N , what is the probability pN;M that at least one point lies in every attraction basin ? This probability is very important. Indeed, using the local search algorithm, it is exactly equal to the probability to detect all local maxima of the function. Inverse problem (Section 5): The statistical distribution of the relative sizes of the attraction basins and their number are assumed to be known for computing pN;M in Section 4. Unfortunately, this is rarely the case in practical situations, and one wants to estimate both. The strategy is to put randomly M initial points in the search space and to detect the corresponding local maxima by the local search algorithm. The data we collect is the set ( j ) j1 of the number of maxima detected with j initial points. Of course 0 is unknown (number of local maxima of the landscape that have not been detected). The question is the following: How can the total number of local e-ciently estimated from the set lower bound is but we aim at constructing a better estimator. The paper is divided into three parts. First, Section 4 addresses the direct problem of sample sizing in the case of basins of random sizes then in the case of basins of equal sizes. Second Section 5 is devoted to the estimation of the distribution of the relative basins sizes for an unknown landscape, using a random sample from the search space. This is achieved by a two step methodology: Section 5.2 starts by considering a parametrized family of laws for the relative sizes of basins, for which it derives the corresponding covering of the search space (law of ( j )). Then Section 5.3 comments on how these results can be practically used for characterizing the sizes of basins of an unknown landscape. For instance, it proposes to compare the covering of an unknown landscape (given by the empirically observed ( j ) values) to the coverings studied in Section 5.2. Finally, the last part of the paper (Section devoted to some experiments that validate (Section 6.1) and illustrate (Section 6.2) the methodology: First, a landscape is purposely designed to test the reliability of the method according to the size of the random sample, and to the number of local optima (recall the theoretical results are asymptotic with respect to N and M ). Second, the method is used to investigate some problems, known to be di-cult to optimize for EAs. For each problem, we also compare the landscapes related to dierent mutation operators. 2. Notations and Denitions. Consider a tness f R, and a neighborhood relation induced by a modication operator , such that the number of dierent -neighbors (neighbors that can be obtained by one application of to x) of x is 'bounded'. In the following, we denote by N the number of local optima of L, 4 J. Garnier and L. Kallel and by ( j ) the random variables describing the sizes of the attraction basins of L (normalized to the average size). As shown in [23, 24], a local improvement algorithm is e-cient to nd quickly a local optimum starting from some given point. Among the possible algorithms we present the Steepest Ascent (SA) also called optimal adjacency algorithm in [23]: Steepest Ascent Algorithm (SA). Input: A tness R, an operator and a point X 2 E. Algorithm: Modify X by repeatedly performing the following steps: - Record, for all -neighbors of X denoted by i (X): (i; f( i (X))) chosen such that f( i (X)) reaches the highest possible value (this is the steepest ascent). - Stop when no strictly positive improvement in -neighbors tnesses has been found. Output: The point X, denoted by SA (X). The SA algorithm thus consists in selecting the best neighbors after the entire neighborhood is examined. An alternative algorithm, the so-called First Improvement consists in accepting the rst favorable neighbor as soon as it is found, without further searching. Note that in the FI case there are extra free parameters which are the order in which the neighborhood is searched. As pointed out in [15, p. 470], the steepest ascent is often not worth the extra computation time, although it is sometimes much quicker. Nevertheless our focus in this paper is not a complete optimization of the computational time, so we let this problem as an open question. Definition 2.1. Attraction basin: The attraction basin of a local optimum m j is the set of points of the search space such that a steepest ascent algorithm starting ends at the local optimum m j . The normalized size of the attraction basin of the local optimum m j is then equal to k=jEj. Remarks. 1. This denition of the attraction basins yields a partition of the search space into dierent attraction basins, as illustrated in Figure 1. The approach proposed in this paper is based on this representation of the search space into a partition of attraction basins, and could be generalized to partitions dened with alternative denitions of attraction basins. 2. In the presence of local constancy in the landscape, the above denition of the steepest ascent (and hence also the related denition of the attraction basins) is not rigorous. For instance, if the ttest neighbors of point p have the same tness value, then the steepest ascent algorithm at point p have to make a -random or user dened- choice. Nevertheless, even in the presence of local constancy, the comparison of the results (distribution of ( j obtained with dierent steepest ascent choices, may give useful information about the landscape and guide the best elitism strategy: 'move' to tter points, or 'move' to strictly tter points only. 3. Summary of the results. Given a distribution of ( j ), we determine Mmin , the minimal size of a random sample of the search space, in order to sample at least one point in each attraction basin of the landscape. Two particular cases are investigated. 1. Deterministic conguration: all the attraction basins have the same size (( j ) are deterministic). 2. Random conguration: the sizes of the attraction basins are completely random are uniformly distributed). In both congurations, we give the value of Mmin as a function of the number of local optima N . For instance, a random sample of size E-ciency of local search with multiple local optima 5 *m *m *m *m Fig. 2. Schematic representation of the search space E with its attraction basins and the 4 corresponding local maxima m 1 ,.,m 4 . In the left picture we have put randomly chosen. We apply the search algorithm and detect 3 maxima according to the right picture, so that we have 5. the deterministic conguration (resp. for the random conguration), ensures that a point is sampled in each attraction basin with probability exp( 1=a). We then address the inverse problem of identifying the distribution of the normalized of the attraction basins, for an unknown landscape. Some direct analysis is rst required as discussed below. Direct analysis. Consider a random sample (X uniformly chosen in the search space. For each steepest ascent starting from X i (with the modication operator(s) at hand ) ends at the local optimum SA (X i ). Dene j as the number of local optima (m : ) that are reached by exactly j points from (X i ) (see an example in Figure 2): Proposition 5.1 gives the distribution of ( j ) for a family of parametrized distributions for asymptotically with respect to N and M . More precisely, if (Z j ) j=1;:::;N denotes a family of positive real-valued independent random variables with Gamma distributions whose densities are: z and , then the expected number j; (j a j a=M=N Moreover, the ratio M=N is the unique solution of: r (1) The latter equation is then used to nd a good estimator of N , with observed values of the variables j , as explained below. Inverse problem. Given an unknown landscape, we then propose to characterize the distribution of ( j ) through the empirical estimation of the distribution of the random family ( j ). In fact, by construction, the distribution of ( j ) and that of 6 J. Garnier and L. Kallel are tightly related: We experimentally determine observed values taken by (random sampling and steepest ascent search). Then, for each value, we use a 2 test to compare the observed law for to the law should (theoretically) obey if the law of ( j ) were Law . Naturally, we nd a (possible) law for only if one of the latter tests is positive. Otherwise, we only gain the knowledge that does not obey the law Law . Note also that the method can be used to determine sub-parts of the search space with a given distribution for ( j ). In case the law of , Eq. (1) is used to nd a good estimator of N . Last, Section 6 validates the methodology of Section 5, by considering known landscapes with random and deterministic sizes of basins, showing that the estimations of the number of local optima N are accurate, even if M is much smaller than N . Fur- ther, we apply the methodology on unknown landscapes, and show that the Hamming binary and gray F1 landscapes contain much more local optima than the 3-bits- ip landscapes. 4. Direct problem. We assume that the search space E can be split into the partition of subspaces which are attraction basins of local maxima m 1 ,.,m N of the tness function. Let us put a sample of M points randomly in the search space. We aim at computing the probabilities pN;M that at least one point of the random sample lies in each attraction basin. Proposition 4.1. If we denote by j := jE j j=jEj the normalized size of the j-th attraction basin, then: (2) Proof. Let us denote by A j the event: there is no point in The probability of the intersection of a collection of events A j is easy to compute. For any 1 there is one initial point chosen uniformly in E then we have If there are M initial points chosen uniformly and independently in E, then: On the other hand, 1 pN;M is the probability that at least one of the attraction basin contains no point, which reads as: The result thus follows from the inclusion-exclusion formula [28, Formula 1.4.1a]. Proposition 4.1 gives an exact expression for pN;M which holds true whatever but is quite complicated. The following corollaries show that the expression of pN;M is much simpler in some particular congurations. Corollary 4.2. 1. If the attraction basins all have the same size j 1=N (the so-called D-conguration), then: E-ciency of local search with multiple local optima 7 2. If moreover the numbers of attractors and initial points are large N 1 and 3. Let us denote by MD the number of points which are necessary to detect all local maxima. Then in the asymptotic framework N 1, MD obeys the distribution of where Z is an exponential variable with mean 1. An exponential variable with mean 1 is a random variable whose density with respect to the Lebesgue measure over R + is Proof. The rst point is a straightforward application of Proposition 4.1. It is actually referenced in the literature as the \coupon-collector" problem. The fact that MD =(N ln N) converges in probability to 1 is also well-known. The corollary is going one step further by exhibiting the statistical distribution of M d N ln N . Let us assume that We begin by establishing an estimate of First note C k Second ln(1 x) x for any 0 x 1 so that As a consequence, uniformly with respect to e: We thus have pN;M;k ea k =k! for all k uniformly with respect to N . Choosing some K 1, we can write from Eq. (3): pN;M eX a k It is easy to check that, for any xed k: N k C k as N !1, so that: lim sup pN;M eX a k This holds true for any K, so that we take the limit K ! 1 which gives the result of the second point. The third point then follows readily from the identity P(M d stands for the integral part of the real number x. 8 J. Garnier and L. Kallel Corollary 4.3. 1. If the sizes of the attraction basins are random (the so-called R-conguration), in the sense that their joint distribution is uniform over the simplex of and the numbers of attractors and initial points are large: N 1 and a > 0, then: 2. Let us denote by MR the number of points which are necessary to detect all local maxima. Then in the asymptotic framework N 1, MR obeys the distribution of where Z is an exponential variable with mean 1. A construction of the R-conguration is the following. Assume that the search space E is the interval [0; 1). Choose N 1 points (a i uniformly over [0; 1] and independently. Consider the order statistics (a (i) ) i=;1;:::;N 1 of this sample, that is to say permute the indices of these points so that a (0) := 0 a (1) ::: a (N 1) a (N) := 1. Denote the spacings by a (j) a (j 1) for Note that j is also called the j-th coverage. If the j-th attraction basin E j is the interval [a (j 1) ; a (j) ), then the sizes of the attraction basins obey a uniform distribution over the simplex SN . Proof. From Eq. (2) and the relation stands for the expectation with respect to ( j ) j=1;:::;N whose distribution is uniform over SN . As pointed out in Ref. [28, Section 9.6a], the probability distribution of the sum of any k of the N coverages j is described by the repartition function given by Formula 9.6.1-[28], which shows that it admits a density q N;k () with respect to the Lebesgue measure over [0; 1]: We can thus write a closed form expression for E-ciency of local search with multiple local optima 9 We shall rst prove an estimate of pN;M;k . Step 1. pN;M;k k! We have N !=(N k)! N k and (N 1)!=(N k 1)! (N 1) k N k . For any k = 0; :::; N we also have (M +N 1 k)!=(M +N 1)! M k . Substituting these inequalities into Eq. (9) establishes the desired estimate. Step 2. For any xed k, if On the one hand, C On the other hand: is bounded by 1 and converges to exp( as) as N ! 1, the dominated convergence theorem implies that: which yields the result by Eq. (8). Step 3. Convergence of pN;M when We rst choose some K 1. We have from the result of Step 1: pN;M X Substituting the result of Step 2 into Eq. (10) shows that: lim sup pN;M X a k This inequality holds true for any K, so letting K ! 1 completes the proof of the corollary. It follows from the corollaries that about N ln N points are needed in the D- conguration to detect all maxima, while about N 2 points are needed to expect the same result in the R-conguration. This is due to the fact that there exists very small attraction basins in the R-conguration. Actually it can be proved that the smallest attraction basin in the R-conguration has a relative size which obeys an exponential distribution with mean N 2 (for more detail about the asymptotic distribution concerning order statistics we refer to [28]). That is why a number of points of the order of N 2 is required to detect this very small basin. Mean values. The expected value of MD is: where C is the Euler's constant whose value is C ' 0:58. The expected value of MR =N 2 is equal to innity. This is due to the fact that the tail corresponding to exceptional large values of MR is very important: P(MR N 2 a) J. Garnier and L. Kallel Standard deviations. The normalized standard deviation, which is equal to the standard deviation divided by the mean, of the number of points necessary to detect all local maxima in the D-conguration is equal to: which goes to 0 as which proves in particular that MD =(N ln N) converges to 1 in probability. This is of course not surprising. The D-conguration has a deterministic environment, since all basins have a xed size, so that we can expect an asymptotic deterministic behavior. The situation is very dierent in the R-conguration which has a random environment, and it may happen that the smallest attraction basin be much smaller than its expected size N 2 . That is why the uctuations of MD , and especially the tail corresponding to exceptional large values, are very important. 5. Inverse problem. 5.1. Formulation of the problem. We now focus on the inverse problem. We look for the number N of local maxima of the tness function and also some pieces of information on the distribution of the sizes of the corresponding attraction basins. We assume that we can use an algorithm that is able to associate to any point of the search space the corresponding local maximum. In order to detect all local maxima, we should apply the algorithm to every point of the search space. Nevertheless this procedure is far too long since the search space has a large cardinality. Practically we shall apply the algorithm to M points that will be chosen randomly in the search space E. The result of the search process can consequently be summed up by the following set of observed values (j 1): number of maxima detected with j points: Our arguments are based upon the following observations. First note that is the number of detected maxima. It is consequently a lower bound of the total number of local maxima N , but a very rough estimate in the sense that it may happen that many maxima are not detected, especially those whose attraction basins are small. Besides N represents less information than the complete set ( j ) j1 . By a clever treatment of this information, we should be able to nd a better estimate of than N . 5.2. Analysis. The key point is that the distribution of the set j is closely related to the distribution of the sizes of attraction basins. Let us assume that the relative of the attraction basins can be described by a distribution parametrized by some positive number as follows. Let (Z j ) j=1;:::;N be a sequence of independent random variables whose common distribution has density p with respect to the Lebesgue measure over (0; 1): 2 where is the Euler's Gamma function dt. The density p is the so-called Gamma density with parameters ( is a positive integer then p is a negative-binomial distribution. E-ciency of local search with multiple local optima 11 z Fig. 3. Probability density of the sizes of the attraction basins under H for dierent the expected value of Z 1 is 1 and its standard deviation is 1= p . In the following we shall say that we are under H if the relative sizes of the attraction basins can be described as (Z 1 and the distribution of Z j has density p . Note that the large deviations principle (Cramer's theorem [1, Chapter 1]) applied to the sequence (Z j ) yields that for any x > 0 there exists c TN exp( Nc which shows that, in the asymptotic framework N 1, the ratio Z j =N stands for the relative size j up to a negligible correction. The so-called D and R congurations described in Section 4 are particular cases of this general framework: - For so that we get back the deterministic D- - For 1, the Z j 's obey independent exponential distributions with mean 1, and the family obeys the uniform distribution over SN [17]. The important statement is the following one. Proposition 5.1. Under H the expected values j; of the j 's can be computed for any N , M , and In the asymptotic framework N 1, if can be expanded as: (j a j Proof. Under H , the probability that j of the M points lie in the k-th attraction basin can be computed explicitly: (j points in E k 3 Applying the procedure described in [1] establishes that c J. Garnier and L. Kallel stands for the expectation of Z j with distribution p . Accordingly, in terms of the Z i 's this expression reads: (j points in E k where . The random variables Z k and Z k are independent. The probability density of Z k is p given by Eq. (12). The random variable Z k is the sum of independent random variables with densities p , so that its probability density is [4, p. 47, Formula 2.3]: z (N 1) Accordingly: (j points in E k Z 1dz Z 1dzp (z)p z By the change of variables z) and z we get: (j points in E k Z 1du Z 1dvv N The integral with respect to v is straightforward by denition of the Gamma function. The integral with respect to u can be obtained via tabulated formulae [4, p. 47, formula 2.5]. This gives the explicit formula (14) for j; since (j points in E If N 1 and then we have N (N a) j+ , and N j M !=(M j)! ! a j as N !1. This proves the asymptotic formula (15). In particular, the distribution of the j 's under the D-conguration is Poisson in the asymptotic framework N 1: while it is geometric under the R-conguration: From Eq. (15) we can deduce that the following relation is satised by the ratio r E-ciency of local search with multiple local optima 13 5.3. Estimator of the number of local maxima. We have now su-cient tools to exhibit a good estimator of the number of local maxima. We remind the reader of the problem at hand. We assume that some algorithm is available to determine from any given point the corresponding local maximum. We choose randomly M points in the search space and detect the corresponding local maxima. We thus obtain a set of values ( j ) j1 as dened by (11). We can then determine from the set of values 0 is the most probable, or at least which H 0 is the closest conguration of the real underlying distribution of the relative sizes of the attraction basins. The statistics used to compare observed and expected results is the so-called 2 goodness of t test [27, Section 8.10], which consists rst in calculating for each where is the set of the indices j for which j 1. Obviously a large value for T indicates that the corresponding j; are far from the observed ones, that is to say H is unlikely to hold. Conversely, the smaller T , the more likely H holds true. In order to determine the signicance of various values of T : , we need the distribution of the statistics. A general result states that if the hypothesis H does hold true, then the distribution of T 0 is approximatively the so-called 2 -distribution with degrees of freedom equal to the cardinality of the set minus 1. Consequently we can say that the closest conguration of the real underlying distribution of the relative sizes of the attraction basins is H 0 is given by: Furthermore, we can estimate the accuracy of the conguration H 0 by referring T to tables of the 2 -distribution 1 degrees of freedom. A value of T much larger than the one indicated in the tables mean that none of the congurations hold true. Nevertheless, H 0 is the closest distribution of the real one. Remark. The distribution theory of 2 goodness of t statistic can be found in [3, Chapter 30]. The result is in any case approximate, and all the poorer as they are many expected j; less than ve. These cases must be avoided by combining cells. But power in the tail regions is then lost, where dierences are more likely to show up. Dening 0 as (17), we denote by the quantity: From Eq. (16), under H 0 the ratio M=N is the unique solution of: Consequently, once we have determined is a good estimator of the 14 J. Garnier and L. Kallel gc Simul. 3 Nb. of optima 9100Estimated Visited Exact gc Simul. 3 Nb. of optima 91000Estimated Visited Exact Fig. 4. Basins with random uniform sizes: The left gures plot the 2 test results (i.e. the values of T comparing the empirically observed distribution to the family of -parametrized distributions. The right gures plot for dierent values the estimation of the number of local optima computed by Eq. (18). These estimations are very robust (only one estimation is plotted) and are accurate for 1. The same gures also show the visited numbers of optima actually visited by the steepest ascent ( The numerical simulations exhibit unstable results for the 2 test for small N values and 6. Experiments. Given a landscape L, the following steps are performed in order to identify a possible law for the number and sizes of the attraction basins of L, among the family of laws Law studied above. 1. Choose a random sample (X uniformly in E. 2. Perform a steepest ascent starting from each X i up to SA (X i ). 3. Compute dened as the number of local optima reached by exactly j initial points X i . 4. Compare the observed law of to the laws of ( dierent values, using the 2 test. To visualize the comparison of the last item, we propose to plot the obtained 2 value for dierent values. We also plot the corresponding 2 value below which the test is positive with a condence of 95 %. 6.1. Experimental validation. The results obtained in Section 5 are asymptotic with respect to the number of local optima N and the size of the random sample M . Hence before the methodology can be applied, some experimental validation is required in order to determine practical values for M and N for which the method is reliable. This is achieved by applying the methodology to determine the distribution of (normalized sizes of the attraction basins) in two known purposely constructed landscapes: The rst contains basins with random sizes, the second contains basins with equal sizes. Results are plotted in Figures 4-5 and 6. Samples with smaller sizes than those shown in these gures yield j values which are not rich enough to allow a signicant E-ciency of local search with multiple local optima 15 gc Simul. 3 Nb. of optima 9100Estimated Visited Exact gc Simul. 3 Nb. of optima Estimated Visited Exact gc Simul. 3 Nb. of optima Estimated Visited Exact Fig. 5. The same as in Figure 4 with dierent values for N and M . Stable results are obtained when N increases and M is bounded (M min(2000; 3N) here). The estimation of N corresponding to the smallest 2 value ( very accurate. test comparison. For instance, the 2 test requires that observed j are non-null for some j > 1 at least (some initial points are sampled in the same attraction basin). In case all initial points are sampled in dierent attraction basins the 2 test comparison is not signicant. These experiments give practical bounds on the sample sizes (in relation to the number of local optima) for which the methodology is reliable: The numerical simulations exhibit unstable results for the 2 test for 4). When N increases and M is bounded (M min(2000; 3N) in the experiments), results become stable and accurate (Figures 5). Further, we demonstrate that the estimation of number of local optima is accurate, even when initial points visit a small number of attraction basins of the landscape (Figure 6). This situation is even more striking in the experiments of the following section on Baluja F1 problems. 6.2. The methodology at work. Having seen that the methodology is a powerful tool, provided that the information obtained for is rich enough, we apply it to investigate the landscape structure of the di-cult gray and binary coded F1 Baluja problems [2], for a 1-bit- ip and 3-bit- ips neighborhood relations. J. Garnier and L. Kallel gc Simul. 3 Nb. of optima Estimated Visited Exact gc Nb. of optima Estimated Visited Exact 2: log scale c N=10^5, M=500 Nb. of optima 3. Estimation 1 Estimation 2 Visited Exact N=10^5, M=500 Fig. 6. Basins with Deterministic equal sizes: The 2 results are stable for smaller sample sizes than those of the random conguration. The bottom gures correspond to the case and 500, where the 2 test is not signicant, yet the predicted number of local optima is very accurate! With 500 initial points, 497 local optima have been visited, while there are actually 10 5 optima. Yet, formula (18) is able to estimate the true number with an error of less than 30% when the adequate value is used. Gray-coded Baluja F1 functions. Consider the function of k variables It reaches its maximum value among 10 7 at point 0). The Gray-encoded F1g and binary F1b versions, with respectively 2, 3 and 4 variables encoded on 9 bits are considered. This encoding consequently corresponds to the binary search space with Considering the 1-bit- ip mutation (Hamming landscape), Figure 7 shows that the distribution of the sizes of the basins is closer to the random conguration than to the deterministic one, and that the estimated number of local optima is similar for the binary and gray codings. On the other hand, considering the 3-bit- ip mutation Figure 8), the estimated number of local optima drops signicantly for both problems: less than 250 for both binary and gray landscapes, whereas the Hamming landscape E-ciency of local search with multiple local optima 17 gc bits F1g, M=500 Nb. of optima Estimated Visited bits F1g, M=500 gc bits F1b, M=500 Nb. of optima Estimated Visited bits F1b, M=500 Fig. 7. The di-cult Baluja 27-bits F1 gray (F1g) and binary (F1b) landscapes with a 1-bit- ips mutation. Experiments with samples of sizes show the same results for the 2 test, and the corresponding estimations of the number of local maxima converge to a stable value around 4000. contains thousands of local optima (Figure 7). Experiments at problem sizes l = carried out in addition to the plotted ones (l = 27), leading to similar results for both F1g and F1b problems: The number of local optima of the 3-bit- ips landscape is signicantly smaller than that of the Hamming landscape. For example, when there are less than 10 local optima in the 3-bit- ips landscape versus hundreds in the Hamming landscape. the estimations for the Hamming landscape show about two times more local optima for the gray than for the binary encoding (resp. 45 000 and 25 000). Still but for the 3-bit- ips landscape, the estimated number of local optima drops respectively to 1400 and 1000. A new optimization heuristic. A simple search strategy for solving di-cult problems naturally follows from the methodology presented in this paper: Once the number N and distribution of the attraction basins is estimated following the guidelines summarized in the beginning of Section 6, generate a random sample whose size is set to if the sizes of the basins are close to the deterministic conguration if the sizes of the basins are close to random). Then a simple steepest ascent starting from each point of the sample, ensures that the global optimum is found with probability exp( 1=a). In the 27-bits F1 problem, this heuristic demonstrates to be very robust and ecient in solving the problem with the 3-bits- ip operator. Using a 3-bits- ip mutation steepest ascent, an initial random sample of 5 points (versus 100 with 1-bit- ip mu- tation) is enough to ensure that one point at least lies in the global attraction basin (experiments based on 50 runs)! This is due to the fact that the basin of the global optimum is larger than the basins of the other local optima. In order to detect all attraction basins, we can estimate the required sample size to 62500 (250 250 using Corollary 4.3 and the estimation of in the experiments of Figure 8). J. Garnier and L. Kallel gc Nb. of optima Visited bits F1g, M=500 gc bits F1b, M=500 Nb. of optima Exact bits F1b, M=500 Fig. 8. The di-cult Baluja 27-bits F1 gray (F1g) and binary (F1b) landscapes with a 3- bit- ips mutation: the number of local optima drops signicantly compared to the Hamming 1-bit- ip landscape. These results are conrmed by experiments using samples of sizes 2000 and which give the same estimation for the number of local optima. 6.3. Discussion. This paper provides a new methodology allowing to estimate the number and the sizes of the attraction basins of a landscape specied in relation to some modication operator. This allows one to derive bounds on the probability that one samples a point in the basin of the global optimum for example. Further, it allows to compare landscapes related to dierent modication operators or representations, as illustrated with the Baluja problem. The e-ciency of the proposed method is certainly dependent on the class of laws of of the attraction basins) for which the distribution of is known. We have chosen a very particular family of distributions p for representing all possible distributions for the relative sizes of attraction basins. The constraints for this choice are twofold and contradictory. On the one hand, a large family of distributions is required to be sure that at least one of them is su-ciently close of the observed repartition On the other hand, if we choose an over-large family, then we need a lot of parameters to characterize the distributions. It is then very di-cult to estimate all parameters and consequently to decide which distribution is the closest to the observed one. That is why the choice of the family is very delicate and crucial. We feel that the particular family p that has been chosen (12) fullls determinant conditions. First it contains two very natural distributions, the so-called D and R congurations that we have studied with great detail. Second it is characterized by a single parameter easy to estimate. Third it contains distributions with a complete range of variances, from 0 (the D-conguration) to innity, by going through 1 (the R-conguration). However, the experiments with the Baluja problem, appeal for rening the class of laws of ( j ) around basins with random sizes. We may propose where Z j are independent and identically distributed with one of the distributions of E-ciency of local search with multiple local optima 19 the bidimensional family p ;- (:), The parameter - characterizes the distribution of the sizes of the small basins, since ;- (z) z - 1 as z ! 0, while characterizes the distribution of the sizes of the large basins, since the decay of p ;- (z) as z !1 is essentially governed by e z . The density p ;- is the so-called Gamma density with parameters ( 2.2]. This family presents more diversity than the family p (:) we have considered in Section 5.2. The expected value of j is under p (j a j a=M=N The method of estimating the number of local minima described in Section 5.3 could then be applied with this family. To apply our method we have also made a crucial choice which consists in executing the local search algorithm from randomly distributed points. We do so because we have no a priori information on the landscape at hand. However, assume for a while that we have some a priori information about the tness function, for instance its average value. Consequently we could hope that starting with points whose tnesses are better than average would improve the detection of the local maxima. Nevertheless extensive computer investigations of some particular problems have shown that this is not the case [15, p. 456], possibly because a completely random sampling of starting points allows one to get a wider sample of local optima. A rst application of the methodology presented in this paper is to compare landscapes obtained when dierent operators are used (k-bit ip binary mutations for dierent k values for example). However, the complexity of this method is directly related to size of the neighborhood of a given point. Hence, its practical usefulness to study k-bit- ip landscapes is limited when k value increases. Hence, it seems most suited to investigate dierent representations. Its extension to non-binary representations is straightforward, provided that a search algorithm that leads to the corresponding local optimum can be provided for each representation. Further, this methodology can be used to determine sub-parts of the search space, such that obey a particular law, hence guiding a hierarchical search in dierent subparts of the space. Note nally that the distributions of the sizes of basins do not fully characterize landscape di-culty. Depending on the relative position of the attraction basins, the search still may range from easy to di-cult. Additional information is necessary to compare landscapes di-culty. Further work may address such issues to extract additional signicant information in order to guide the choice or the design of problem dependent operators and representations. --R Grandes d An empirical comparison of seven iterative and evolutionary function optimization heuristics Mathematical methods of statistics An introduction to probability theory and its applications Genetic algorithms in search Adaptation in natural and arti Fitness distance correlation as a measure of problem di-culty for genetic algorithms Convergence des algorithmes g On functions with a Alternative random initialization in genetic algorithms A priori predictions of operator e-ciency The genetic algorithm and the structure of the Memetic algorithms and the Algorithms and com- plexity Spin glass theory and beyond Ultrametricity for Physicists Genetic algorithms Complex systems and binary networks Landscapes and their correlation functions Local improvement on discrete structures Hill climbing with multiple local optima Theory and Applications Correlated and uncorrelated Chapman and Hall Mathematical statistics --TR --CTR Sheldon H. Jacobson , Enver Ycesan, Analyzing the Performance of Generalized Hill Climbing Algorithms, Journal of Heuristics, v.10 n.4, p.387-405, July 2004

combinatorial complexity;neighborhood graph;local search;randomized starting solution

587975

Reserving Resilient Capacity in a Network.

We examine various problems concerning the reservation of capacity in a given network, where each arc has a per-unit cost, so as to be "resilient" against one or more arc failures. For a given pair (s,t) of nodes and demand T, we require that, on the failure of any k arcs of the network, there is sufficient reserved capacity in the remainder of the network to support an (s,t) flow of value T. This problem can be solved in polynomial time for any fixed k, but we show that it is NP-hard if we are required to reserve an integer capacity on each arc. We concentrate on the case where the reservation has to consist of a collection of arc-disjoint paths: here we give a very simple algorithm to find a minimum cost fractional solution, based on finding successive shortest paths in the network. Unlike traditional network flow problems, the integral version is NP-hard: we do, however, give a polynomial time $\frac{15}{14}$-approximation algorithm in the case k=1 and show that this bound is best possible unless

Introduction 1.1 Overview A commonly encountered network design problem is that of reserving capacities in a network so as to support some given set of pairwise traffic demands. Algorithms for this network capacity allocation problem have been developed by a number of groups, see for example [6, 8, 9, 21, 22, 23]. These are primarily based on polyhedral methods. One significant drawback to the capacity reservation problem discussed, and especially to the successive shortest path approach, is that if we simply reserve capacity along a single path, we make ourselves totally vulnerable to the failure of any arc (or node) along our chosen path. In many practical settings, this is not acceptable, and we wish to reserve our capacities so as to allow for the failure of any one arc of the network. Several groups have recently addressed this issue of "resilience" or "survivability" in network design problems, e.g., [2, 3, 5, 10, 14, 15, 20, 24, 25, 26], to name a few. As with the preceding batch of references, these are also primarily based on polyhedral or branch and cut methods (although computationally, these problems prove to be dramatically more difficult to solve in practice (cf [10])) and and hence if they terminate, they usally produce an optimality certificate. This aspect of this approach is very desirable in situations where (i) costs and data are 'certain' and (ii) there is time available to solve the optimization problem off-line. These conditions are often not met, however, and consequently many of the network planning tools in the telecommunication industry solve the 'vanilla' form of the capacity allocation problem by successively solving a shortest path problem for each demand pair incrementing loads on the arcs of the shortest path by the demand for that pair. 1 In its favour, this approach is fast and allows for trivial implementation in software. Unfortunately, one easily concocts examples where this approach produces solutions arbitrarily far from the optimum. On the other hand, one may also produce examples where the exact methods do not terminate or exhibit poor running times; in addition, they may require substantial mathematical sophistication on the part of its implementors. Another situation where single source-destination pair algorithms are sought is in on-line settings. This area is becoming increasingly important as network management becomes a key concern of network operators. This is largely driven by the changing nature of demands from their customers. In particular, bursty or short-term requests for connectivity are becoming increasingly common. The present paper is dedicated adapting the successive shortest path heuristic to be able to find resilient capacity reservations. Thus we restrict ourselves throughout to the special case of the problem with a single source-destination pair of nodes; we show that even this case presents some surprising difficulties. This case may be equally viewed as an extension of finding a minimum cost flow for a single source-destination pair. (Overviews of previous computational and theoretical work on such survivability and augmentation problems can be found in [18] and [17].) We adopt the viewpoint that the network, with specified nodes s and t, is given to us, along with a per-unit cost c a for each arc a, and that we are free to reserve, once and for all, as much capacity as we like on whatever arcs we choose. Our objective is to find a "reservation vector" x minimizing the total cost a c a x a , subject to supporting a given target amount T of traffic from s to t, even if any one (or, more generally, any k) of the arcs in the network fails. This rough description of the problem admits many different versions, depending on the type of network we are dealing with, the way we are required to recover from arc failures, and especially on any structure imposed on the vector of reserved capacities itself. In this paper, we consider two types of constraints on the capacity reservation vector. 1 Costs are often altered so that they exponentially increase as the load on an edge approaches its capacity. 1. Integrality: We may be forced to reserve capacity in discrete amounts, so that our reservation vector must be integral. 2. Structural: It may be required that our reservation vector be formed by selecting a collection of arc-disjoint (s; t)-paths (i.e., directed paths from s to t in the network), and assigning a capacity to each path - we call such a reservation a diverse-path reservation. We begin Section 2 by studying resilient diverse-path reservations. This version arose out of a problem encountered by British Telecom which was solved by two of the authors in a consultancy report [13] - this work forms the basis of Section 2.1. The research described in this paper is partly an attempt to determine conditions under which such diverse-path solutions will be optimal in general. Diverse-path solutions have several features which are attractive to network planners. For a start, a diverse-path routing may be "hardwired" at the terminating nodes, thus decreasing routing complexity. If a traffic flow control is centralized, then this allows load balancing of traffic over the collection of diverse paths. Even if this is not the case, as in a noncooperative network, the restoration phase is much simpler since an arc failure may be treated as a path failure, and traffic routed along the path may be shifted to the remaining non-failed paths. Finally, like shortest paths, they are conceptually simple to visualize. In Section 1.3 we give an overview of different types of resilience and restoration and discuss some of the issues involved. In Sections 2.1 and 2.2 we describe an extremely simple algorithm to find a minimum cost resilient diverse-path reservation. If the paths are pre-specified, this may be used to find the optimal integral reservation. If the paths are not specified, the integer version of this problem turns out to be NP-hard. Here instead we give a polytime 1 14 -approximate algorithm in the case of resilience against the failure of one arc, and show this bound is the best possible (if P 6= NP). Similar results hold if k takes other values, or is unrestricted. In Section 3, we discuss the general problem of resilience with no side-constraints. We give several examples of basic optimal solutions, showing that we do not always obtain anything resembling a diverse-path reservation in the general case. We show that the integral resilience problem is NP-hard. However, we do show that the naive algorithm that allocates capacity T to the arcs of a cheapest pair of arc-disjoint (s; t)-paths gives a solution with at most k times the cost of an optimal solution. 1.2 Notation and Definitions We start with basic definitions and facts concerning flows in networks. Throughout, we suppose (sometimes implicitly) that we are given a directed graph (network) with node set V and arc set A. We shall always assume that D comes with two nodes permanently fixed as the source (or origin) s and the destination (or sink) t. We also suppose that we are given a rational number T (usually an integer) representing the required traffic flow from s to t through the network D in the case of failure. Finally, we are also given a vector (c a ) of non-negative rational (again, usually integer) costs on the arcs a of D. Even though there may be parallel arcs, we often use the notation (u; v) when no confusion arises. For any S ' V , let D (S) (or simply if the context is clear) denote the set of arcs with tail in S and head in denote the set of arcs S). For a node v 2 V , we we denote by [S; S 0 ] the set of arcs with tail in S and head in S 0 . We call an (s; t)-set if s 2 S and t called s-minimal if the graph induced by S contains a directed path from s to each node of S. Let Q+ denote the set of non-negative rational numbers, so that Q A is the set of all assignments of a non-negative rational to each member of the arc-set A, which we will frequently view as a vector. For any vector x 2 Q A and A 0 ' A, we denote by a2A 0 x a . We let I(x) denote the support of x, that is A vector x 2 Q A is an (s; t) flow vector (or simply a flow) if x(ffi and (therefore) x(ffi (t)). In this case, the value of the flow is For any rational M and (s; t)-set S, the M-cut constraint for S is: i.e., the total capacity of arcs leaving S is at least M . We define the M-cut polyhedron as for each (s; t)-set S Network flow theory asserts that for a given vector x 2 Q A , there is a flow vector w - x of value M if and only if x 2 C(M;D). It is an exercise to show that in (1) we need only include the cut constraints corresponding to s-minimal sets. A vertex of a polyhedron P ' R n is an extreme point of P, i.e., a vector in P that is not a convex combination of two other vectors in P. For a polyhedron P in R n , a vector y 2 P is a vertex if and only if there is some linearly independent set of n inequalities defining P that are satisfied by y with equality. Since we consider only polyhedra in R n linear function is bounded below on P, then it attains its minimum at a vertex; such a vertex is called a basic optimal solution to the minimization problem. 1.3 Types of Resilience We consider vectors of reserved capacities on the arcs of D such that if any k arcs of D are deleted, then the remaining arcs have sufficient capacity. A vector x 2 Q A is (T; k)-resilient if, for each set K of k arcs, the capacities on arcs in A \Gamma K are sufficient to admit an (s; t) flow of value T . As in the previous section, this is equivalent to requiring that x satisfies the constraint x(ffi for each (s; t)-set S and each subset K ae size k. In fact, it is easily shown that we need only require these constraints for s-minimal sets S. We also call a (T; 1)-resilient reservation T -resilient, and for the remainder of this section we do indeed restrict ourselves to the case Everything here goes through in much the same way for the case As with the standard network flow problem, the problem of finding a minimum cost T -resilient reservation vector can be expressed as an optimization problem over a certain polyhedron, which we now describe. For any rational T , (s; t)-set S, and arc e (S), the partial T -cut constraint associated with the is the constraint x(ffi T: The resilience polyhedron is defined by the system of all partial cut constraints. for each (s; t)-set S and e 2 Note that R(T; D) is empty if there is an (s; t)-set S with of size at most 1, and otherwise R(T; D) is full-dimensional. Also note that if x 2 R(T; D), then setting any x results in a vector in C(T; D). Thus if we reserve the capacities x, then even if an arc fails, there is still enough capacity to build an (s; t) flow of value T . Conversely, any vector x not in R(T; D) fails the partial T -cut constraint for some pair (S; e) and hence if e fails, there is not sufficient capacity in the network to support a flow of value T . Therefore R(T; D) consists exactly of the T -resilient vectors, and the problem of finding a minimum cost T -resilient reservation is that of minimizing the linear function a2A c a x a over R(T; D). A consequence of this formulation is that there is a polynomial-time algorithm to find a minimum cost T -resilient vector. Indeed, it is easily seen that the separation problem for R(T; D) amounts to solving jAj maximum flow problems. Still better, as noted previously by several researchers, we can rephrase the problem as that of finding one (s; t) flow vector y a of size T for each failing arc (i.e., with y a along with a common upper bound x - y a whose cost is to be minimized: this formulation constitutes a linear program with a polynomially bounded number of variables and constraints. This is not offered as a practical approach however, and the remainder of the paper addresses the task of finding more direct combinatorial algorithms. Let us give an example, which also illustrates a possible limitation of our notion of resilience. Consider the network D in Figure 1. For appropriate costs, the arc values in Figure 1(a) form a (unique) minimum cost 6-resilient reservation vector: in other words, the vector shown is a vertex of the polytope R(6; D). A "working flow" of value 6 for this network is displayed in Figure 1(b), where the numbers in brackets denote the "spare capacity" for this flow. Note however that if the arc (u; t) were to fail, and we are asked to redirect the one unit of traffic currently flowing along the path s ! then this cannot be done without altering flow values on paths which were unaffected by the arc's failure. This might not be acceptable in some practical situations. (a)213 s (b) s Figure 1: A resilient reservation vector with It is easily seen that this situation does not arise if we demand that our reservation vector x consist of a collection of arc-disjoint paths, which is one motivation for considering that restriction. Another way to refine the notion of resilience so as to address this issue is to replace the partial T -cut constraints by stronger inequalities. The net partial T -cut constraint associated with a pair (S; e) is: is net T -resilient if it satisfies all the net partial T -cut constraints; we denote by N (T; D) the set of all such vectors. For instance, the reservation vector in Figure 1(a) fails the net partial 6-cut constraint for (S; e) where It is not hard to show that, given a net T -resilient reservation vector x, and any flow of value T , thought of as a combination of traffic flowing down (s; t)-paths, on the failure of any arc, flow of value T can be restored without altering the routing of any unaffected traffic. Net resilience is of course a much stronger requirement than resilience, but it does share most of the good computational properties. For more details, see the technical report [11], which is a longer version of the present paper. We close this section with a brief discussion of other types of resilience. We have concentrated on the case where we require resilience against one or more arc failures. One may also wish to guard against node failures. We remark only that this problem can be formulated using our previous models by applying standard splitting operations on nodes other than s and t. Another possibility is that we may wish to guard against losing some fraction ff 2 [0; 1] of the flow on each arc, giving, for each (S; e), the constraint: A vector x satisfying these constraints will be called )-resilient. Note that if ff = 0, this leads to a formulation for traditional network flows. If it is the partial T -cut constraint again. The effect of intermediate values of ff can essentially be analyzed by duplicating arcs. To be precise, suppose that ff = p=q is rational, and that we require an (ff; T )-resilient reservation x in a network D. To model this, we form D 0 by taking q copies of each arc of D, each with the same cost as in D. It is easy to check that a minimum cost reservation vector for D 0 that is qT -resilient against the loss of any p arcs can be constructed from a minimum cost (ff; T )-resilient reservation in D by duplicating the reservation on each set of multiple arcs. One final possibility is that the structure of the reservation vector x when all arcs are functional is of paramount importance. A likely scenario is that there is some value M such that the vector x is required to admit an M -flow, that is, there is an (s; t) flow f of value M with f - x, or maybe even that x itself is required to be an M -flow. We don't discuss such constraints in this paper, but we plan to discuss this situation in a future paper [12], where we also consider the effect of imposing upper bounds on the capacities that can be reserved on each arc. Resilient Diverse-Path Reservations As we have mentioned, our aim with resilience problems is often to reduce to the case where the reservation has to form a collection of arc-disjoint paths, and we term such a reservation a diverse- path reservation. We thus need to consider how we solve the problem once it has been reduced to this special case. We start with the case where the collection of diverse paths is fixed ahead of time. This forms the basis for much of what follows and so we present it in detail. The material in the first two parts of this section, written for a non-technical audience, appears as [13]. A more thorough handling of the polyhedron considered herein (including a complete linear description of the integer hull) has been given by Bienstock and Muratore - see [10]. The case can also be regarded as a special case of a problem treated by Bartholdi, Orlin and Ratliff [7]; our methods give a somewhat simpler solution in this special case. 2.1 Reservations on a Fixed Set of Paths Suppose that we are given a network and two integers T and k, together with a source s and a destination t, along with a fixed set of diverse paths from s to t on which capacity may be reserved for (s; t) traffic. We want to ensure that, in the absence of any k of the paths P i , there is sufficient capacity on the remaining paths to carry a given amount T of traffic from s to t. To accomplish this requires us to fix a capacity x i for each path P i , and give each arc on path P i capacity . The cost of reservations can be calculated from the per-unit total costs c i of the paths P i . Thus in practice we can think of the network as consisting of just the two nodes s and t, with the P i being single arcs of cost c i from s to t. A diverse-path reservation as above is (T; k)-resilient if the total amount of reserved capacity, excluding any k arcs, is at least T . We may assume that the paths P i are numbered in increasing order of per-unit cost c i , so we can state the problem formally as follows. k-Failure Allocation Problem. Given a demand T , and a sequence of costs c find non-negative real numbers x subject to the conditions For any k-set S ' We shall also consider the integral version of this problem. A result that is fundamental in much of the rest of the paper is that the optimal allocation of capacities (for any costs satisfying (4)) is always achieved by some vector z j;k of the form ae for some j between k m. Thus we need only find the appropriate j for which the cost is minimized. Of course, the single-failure problem is just the special case of the k-failure problem with this case we denote by z j the vector z j;1 . Theorem 1 An optimal solution to the k-Failure Allocation Problem is obtained at one of the solutions z j;k . Proof. Because of the symmetry of the situation, we know that there is an optimal solution such that x 1 - x 2 - xm . Thus we lose nothing by including these inequalities as constraints. Once we do this, we see that, if the constraint x all the other resilience constraints given by the removal of k of the paths are automatically satisfied. Thus we may reformulate the problem as follows. Given a demand T , and a sequence of costs c 1 - c 2 - c m , find non-negative real numbers to minimize subject to the constraints x 1 We note for future reference that the same reformulation goes through if the x i are all constrained to be integers. Consider a basic optimal solution for the resulting linear program which necessarily satisfies m linearly independent inequalities with equality. If there are j non-zero variables at the optimum, then the only possibility is that all of the inequalities x 1 are satisfied with equality, i.e., that x This is just the solution z j;k and the result follows. Choosing amongst the various solutions z j;k is evidently not hard, and in fact the structure of the problem allows a particularly simple procedure for doing this. The cost of solution z j;k is which is best thought of as the average of c 1 . , and c j . We are trying to minimize A j;k , over the range of possible values Note that the A j;k are decreasing in j up to the minimum, which is attained for the last j where increasing thereafter: this unimodality property will be a recurring theme. Thus for each . If this is the case, then we terminate and z j;k is optimal. If we reach z m;k is the optimal solution. allocation problems with integer capacity allocations To find the optimal solution in the case where the allocated capacities x i are required to be integers (and the demand T is an integer), we will show that the following procedure suffices. First find the optimal solution z j;k to the original (non-integral) k-failure Allocation Problem. Then (if the z j;k i are not already integers) consider the two integer solutions "nearest" to z j;k , as follows. (a) Set r equal to either bT=(j \Gamma k)c or dT=(j \Gamma k)e. (Here dae denotes the next integer above the real number a, and bac the next integer below.) Note that r may be zero if T (b) If r is one of the two chosen values and r is nonzero, we attempt to construct a solution x with all the non-zero x i , except possibly one, equal to r. To do this, we set l (The choice of l ensures that could have l ? m, but this is not possible with Note that this is a feasible solution, since removing any k of the x i leaves capacity at least (l which is constructed to be at least T . (c) We now have either one or two candidate integral solutions, corresponding to the two choices of r in (a). We denote by z j;k;+ the solution with it is feasible), and by z j;k;\Gamma the solution with (which is always feasible). To finish, just calculate the costs of the two solutions, and choose the lower. Theorem 2 Suppose we have an instance of the k-Failure Allocation Problem in which the optimal solution is z j;k . Then the optimal solution x with all the x i integer is either Proof. We work with the reformulation of the problem as in the beginning of the proof of Theorem 1, which, as we noted, is also valid for the integer case. Suppose that x is an optimal integer solution. Clearly, since the only constraints on the first k variables are that x 1 - x 2 - x k+1 , we will have at the optimum. Now suppose that some x j is non-zero, but that not all of x are equal. Then let i be the minimum index with x the previous observation. Also let x l be the last non-zero variable, so increasing x i by one, and decreasing x l by one. This keeps the solution feasible, since all the variables remain in the right order, and x unaltered. Also, this operation does not increase the cost. We have thus shown that one may restrict attention to integral solutions with the following form: for some j with k we have all of x 1 , . , all of x equal to 0. If the value of x j is q, then the common value of the earlier x i is which will be an integer, at least q. At this point, there are still potentially a large number of candidate solutions; in fact, there is one for each integer value of x 1 at least T=m, namely to set e, and We next observe that the integer solution given above is a convex combination of the two "fractional" solutions z j;k and z j \Gamma1;k . To be precise, our solution is z Thus each of our candidate integral solutions is a convex combination of two consecutive z i;k s. Let A be the set of all such convex combinations; we think of A as a "path" with vertices corresponding to z We note that any vector on this path gives a feasible solution. Next observe that the first coordinate values are decreasing along A, and the values for any vector in A are thus determined by the first coordinate x 1 . Moreover, this solution will be integral if and only if x 1 is an integer. Also, since the costs of the vertices z i;k are unimodal, and cost is a linear function between each pair of vertices, the solution cost is unimodal along A, with the minimum obtained at the vertex corresponding to some z j;k , with first coordinate equal to, say, r. These arguments show that the lowest cost integral solution among our candidates, and so the overall optimal integer solution, is the solution on A corresponding to taking x 1 to be either dre or brc, i.e., taking either z j;k;+ or z j;k;\Gamma . This completes the proof. We close this section with a bound relating the costs of the optimal integral and fractional solutions. Proposition 3 For any j ? k - 1, let O be the cost of a solution z j;k and O 0 be the cost of the integral solution z j;k;\Gamma . Then O T . Proof. Recall that, in the solution z j;k , the j cheapest paths are chosen, each with capacity k). The "rounded" solution z j;k;\Gamma is obtained from this by taking the l \Gamma 1 cheapest paths with capacity k)e, and the next cheapest path with capacity The first observation is that the average cost per unit of reservation in z j;k;\Gamma is no greater than that in z j;k . Thus O 0 =O is at most the ratio of the total numbers of units of capacity reserved in the two allocations. In z j;k , a total of jT=(j \Gamma units of capacity are allocated, while in z j;k;\Gamma , the total is (l \Gamma 1)x Hence we have O 0 O Now (j by definition of x, so we have the estimate as claimed. Corollary 4 If O I is the cost of the optimal integral solution to an instance of the k-failure allocation problem with target flow T , and OF the cost of the optimal fractional solution to the same problem, then O I Consideration of the proof of Proposition 3 shows that the ratio (1 in general be improved. Indeed, if our network consists of a large number M of paths of cost 1, then it is easy to see that O I 2.2 Diverse-Path Reservations without Specified Paths We now consider what happens when we are still required to find a resilient reservation consisting of a set of diverse paths in a given network D, but we are not restricted as to what paths we may use. We give a fast algorithm for the fractional case, but show that the integral case is NP-hard. We start with the fractional case. The results of Section 2.1 imply that the optimal solution will have as support the arcs of some j ? k diverse paths, with each arc in the support given capacity T=(j \Gamma k). Of course, j can only take integer values up to the (s; t)-connectivity -(D) of D. We may take advantage of this structure and apply the successive shortest path method - c.f. [1] - for minimum cost flow problems, thus only needing to solve -(D) shortest path problems. For an arc a = (u; v) 2 A, we let a \Gamma denote an "artificial" arc (v; u), not present in D. In the course of the following algorithm, we construct a series of auxiliary digraphs D j , each of which will contain exactly one of each pair a; a \Gamma . We assume that we are given a digraph D with -(D) ? k. f While (D j contains a directed (s; t)-path) Let Q j be a minimum c j -cost directed (s; t)-path in D j z j+1;k be the vector obtained by assigning to each arc in P j+1 then Output(z j;k ) and Quit are the same as D if a 2 R remove a \Gamma , and include a with cost c j+1 if a 2 F remove a, and include a \Gamma with cost c j+1 EndWhile We will also refer to the version of the algorithm which does not terminate early and thus generates a reservation vector z j;k for every Proposition 5 (c.f. [1]) Let c be a nonnegative vector of arc costs in a network A). The algorithm Paths finds a minimum cost (T; k)-resilient diverse-path reservation. To establish correctness, we need two facts. First, for each j, the collection P j induces a minimum cost collection of j diverse (s; t)-paths; this follows from the correctness of the successive shortest path method. This implies that each solution z j;k is the minimum cost solution using j paths, and hence the minimum cost (T; k)-resilient vector is among these vectors z j;k . Moreover, traditional flow theory implies that for each is a minimum cost flow of value subject to the capacities on each arc. Second, as we now show, the sequence cost(z j;k ) is unimodal for termination is justified. Proposition 6 Let h, i and j be such that k ! h Proof. Suppose the contrary: there exists h, i, j, with and cost(z i;k i\Gammak and choose - 2 (0; 1) such that i is a flow of value does not exceed T=(i \Gamma 1) on any arc. Thus by the remarks preceding the proposition, cost(z 0 maxfcost(z h;k ); cost(z j;k )g, a contradiction. 2 integer diverse-path reservations We now consider the problem of finding a minimum cost T -resilient integral reservation whose support is a collection of diverse paths. Thus we denote by idp the problem each instance of which consists of a network with two specified nodes s; t together with nonnegative integer costs on the arcs and an integer T . An optimal solution for an instance will be a minimum cost T -resilient reservation obtained by reserving integer capacities on a collection of diverse (s; t)-paths. The results of the previous section once again show that the support of an optimal such solution will consist of a collection of diverse (s; t)-paths the arcs of the first reserve a commonamount, r, of capacity, and the last path's arcs will reserve capacity r. 2 We now show that the subproblem of idp with denoted by 3-idp, is NP-hard. Let 2div-paths denote the problem of determining whether a given digraph D, with four distinct nodes contains a pair of arc-disjoint paths Fortune, Hopcroft and Wyllie [16] show that this problem is NP-complete. Suppose that we are given an instance of 2div-paths as above. Construct a digraph obtained from D by adding new nodes s; t as well as the arcs (s; 2, 1 and 2 respectively. All remaining arcs will have cost zero. This is our instance of 3-idp. From the preceding section, we deduce that an optimal 3-resilient reservation on diverse paths will either have support on (i) 2 diverse paths, in which case capacity 3 is reserved on each of the arcs of these paths, or (ii) 3 diverse paths in which case two of the paths will have reserved capacity 2 and the third capacity 1. Note that the cheapest collection of 2 diverse paths has cost 5 and hence any solution of the form (i) will have cost at least 15. Next note that if there exists a positive solution to the instance of 2div- paths, with P i a path between s then by assigning capacity 2 to the arc (s; t) and the arcs of P 1 , and capacity 1 to the arcs of P 2 we obtain a solution to 3-idp of cost 14. Conversely, if the instance of 2div-paths has no solution, then any "3-path" solution to 3-idp will use only paths of cost 3, from which we deduce that the reservation will cost at least 15. Thus the optimal solution to the instance of 3-idp is 14 if and only if the instance of the 2-disjoint path problem has a positive solution. Note that the above result shows that approximating 3-idp to within a factor of (1 + 1) is NP- hard. On the other hand, Proposition 3 implies that applying the rounding procedure to an optimal fractional solution will yield a (1+ 1)-approximation to the optimal integral solution. We now improve this latter bound. We continue to restrict attention to the case allow T to take arbitrary integer values. Consider the polynomial time algorithm A (based on Paths) that finds, for each value of j, the fractional solution z j;1 based on some cheapest set of j diverse paths, and the two "rounded" integer solutions z j;1;\Gamma and z j;1;+ , and chooses the best among all of the integer solutions. The algorithm A In essence we thus need to solve an integer 2-multicommodity flow problem where both commodities have the same origin and destination. can fail to find the optimal integer solution because it may use a minimum cost set of l paths in which the costs are distributed "more evenly" between the paths (in particular, the most expensive path is cheaper) than in some other (not necessarily even minimum cost) set of l paths. All we know is that an optimal solution for an instance of idp will have the same form as either z l;1;\Gamma or z l;1;+ for some l, since it will arise from a similar rounding process applied to some collection of diverse paths. Let OF be the cost of a fractional optimum solution, O I the cost of an integer optimum solution, and OA the best solution among those considered by the algorithm, i.e., the value returned by A. Clearly we have OF - O I - OA . We prove the following result. Theorem 7 The algorithm A is a 1 14 -approximate algorithm for idp with that is, for each instance: OA - 15O I . Moreover, there is no ffl-approximate algorithm for idp with latter statement, of course follows from the previous example showing that the decision version of idp with We note that the quality of approximation by the algorithm depends greatly on the input T . If we view A as an infinite collection of algorithms fA T g 1 T=1 , each restricted to instances with a fixed value of T , then many of these are ffl-approximate algorithms with 14 . Indeed, Proposition 3 tells us that, for each T , OA I . Furthermore, we note in the course of the proof that, for exactly solves the subproblem T -idp. Proof. Suppose an optimum integer solution z uses paths each with reserved capacity r, and path P l+1 , with reserved capacity y. We may assume that 0 - y - r, l - 2, and that r is either bT=(l \Gamma 1)c or dT =le, with y, since otherwise there is a solution of one of the given forms with no greater cost using the same set of paths, by Theorem 2. We also have that Clearly we can assume that O I 6= OA . In particular, this means that otherwise one of z l+1;1 or z l;1 would be an integral solution found by A whose cost was at most that of z . Thus The values of T and r determine y and l and, as we noted before the proof, we may assume that 7. There are thus only a finite number of possible forms of z (in just 23 pairs (T; r) satisfy all the restrictions mentioned so far), and we shall rule all these out using the same basic method. At this point, let us observe that there are no cases with or 12; for these values of T , any r not dividing T exactly is not of the form bT=(l \Gamma 1)c or dT =le for any integer l. In particular, we may assume that we have T - 3. We require a lower bound on the cost of z . Notice that z can be written as y times the characteristic vector of some set of l times the characteristic vector of some set of l diverse paths. Let C i be the cost of reserving one unit of capacity on the arcs in a cheapest collection of i diverse paths. So we have O I = C(z Our algorithm A considers some integer solution of the same form as z (i.e., the same values of using some set of l +1 paths of cost C l+1 . The cost of this solution is at most what it would be if all the l +1 paths had the same cost, which is (lr +y)C l+1 =(l +1). So OA is at most this quantity, i.e., We aim for a similar bound on C l , and to get this we need to look at a solution produced by A on at most l paths. Accordingly, let r there is some integer solution with reserved capacity r 0 on the first m ! l paths from C l , and v - r 0 on one further path, with total of reserved capacities on all the paths equal to T Our algorithm will have looked at an integer solution with a cost at least as good as some solution of this form, and the average cost of a path in any solution of this form is at most C l =l, as in the proof of Proposition 3. So we have OA - We conclude that O I - After a little manipulation, this becomes O I G: We could now run through all the 23 cases separately and show that G - 1=15 in each case. Let us proceed slightly more systematically. First, we consider all cases with l = 2. In this case, we have r which implies that r Next, if This gives l - 2. Assume from now on that l; r - 3. This implies that T - 7 and that r ! T=2. If r we have G 1), and we are done if 4. On the other hand, if thus the only two cases with are From here on in, there seems to be no great saving on dealing with all the cases individually. Here are all the cases not so far ruled out. All the values for G above are less than 1=15, so the theorem is proved. It is clear that this technique can be used to prove similar results for other values of k. If k is unrestricted, it turns out that the algorithm Paths is a 1 5 -approximation algorithm for idp, and that this is best possible, i.e., there is no ffl-approximation algorithm for idp with omit the details. Resilient Reservations with no Restrictions 3.1 Examples of Vertices of R(T;D) Although the problem of finding minimum cost resilient reservations can be solved in polynomial time, we have been unable to find a truly practical algorithm for the problem. It is natural to believe that there might be an algorithm which uses some generalization of the pivot operation (i.e., cycle augmentations) for standard minimum cost flows. In order to explain why it is likely to be difficult to find any such algorithm, we consider various examples of vertices of polyhedra R(T; D) - these are basic optimal solutions to the resilience problem, and we also term these basic resilient reservations - which are far from the pleasant diverse-path reservations we have been working with so far. We have already seen one such example in Figure 1: two more are given in Figures 2 and 3 below. Here and throughout this section, we restrict attention to the single-failure case. Figure 2: A basic resilient reservation with a cycle Figure 3: A vertex of both R(T; D) and N (T; D) By definition, x is a vertex of R(T; D) if (i) x 2 R(T; D) and (ii) there is some subset of jAj linearly independent constraints, from the system of non-negativity and partial T -cut constraints, which are satisfied with equality by x. If the partial T -cut inequality for (S; e), where S is an (s; t)-set and e an arc in is called a critical arc, and S is called a tight set, for the reservation vector x. One hope of understanding the structure of the vertices of R(T; D) is to understand the role of the critical arcs. One may apply uncrossing techniques to show that if a is a critical arc, then there is no tight set S for which a 2 For suppose this were the case, let e be a critical arc for S and let S 0 be a second (s; t)-set for which x(ffi the left hand side is also equal to x(ffi In particular, a Consider the case where e we have that contradiction. The other cases follow similarly. This immediately implies the following. Proposition 8 If x 2 R(T; D) and all arcs in I(x) are critical, then I(x) is acyclic. Figure above shows a vertex of R(T; D) for the given digraph D in which the reservation vector contains a cycle. The two arcs forming the cycle are not critical. Figure 4 shows another vertex x of a polyhedron R(T; D) for which there are non-critical arcs a (the two arcs not incident with s and with x a ? 0. (This example is also a basic optimal solution for the node-deletion version of the problem; this reservation is resilient against the failure of any internal node.) s Figure 4: A basic resilient reservation for both standard and node-deletion versions Further understanding of the structure of the vertices has eluded us; we offer two conjectures. Conjecture 3.1 Let x be a vertex of R(T; D). ffl There exists a directed path from s to t consisting only of critical arcs. ffl There exists an (s; t)-set S such that the reserved capacities on all arcs in I(x) " have the same value, and each arc in this set is critical. Finally, we observe that basic resilient reservations may have components of the form pT q for any rational Indeed for any such p; q, Figure 5 displays such a vertex. 3.2 NP-completeness of Integer Resilience Problems, and an Approximation Algorithm We next show that the problem of finding a minimum cost integral T -resilient vector is NP-hard, even in the single-failure case. We couch the problem as a decision problem. Integer Resilience Instance: a digraph D, with integer costs c ij on the arcs, with a single source s and destination t, a target resilience T (integer), and a target cost C (integer). q-p arcs s (pT)/q Figure 5: A vertex of R(T; D) and N (T; D) with a component a given rational multiple of T Question: is there an integer reservation vector x on the arcs of D such that c \Delta x - C, and such that x is T -resilient? Theorem 9 Integer Resilience is strongly NP-complete. Proof. Certainly the problem is in NP, since checking T -resilience simply involves finding flows of value T in the networks obtained by removing individual edges. To prove the problem is NP-complete, we give a reduction from 3d-matching. Recall that an instance of 3d-matching consists of three sets A, B, C of size n, and a collection T of m "triangles" each containing exactly one element from each of A, B and C; the question is whether A can be written as the disjoint union of n triangles from T . Suppose that we are given an instance of 3d-matching as above. We show how to construct an instance of Integer Resilience with m+ 3n arcs each of cost 1, n, or 2n, such that there is a (4m 1)-resilient integer reservation of cost at most (2n only if the original instance did possess a 3D-matching. We take one node of D for each triangle abc 2 T , one node for each element of A nodes s and t. We take four parallel arcs, each of cost 1, from s to each node corresponding to an element of T . Each node abc has seven arcs leaving it: four, of cost 2n, go directly to t and one, of cost n, to each of the constituent elements a, b and c. Finally, there is a single arc of cost n from each element of A t. Consider any T -resilient integer reservation vector x of cost at most (2n First note that the reservation x must dominate a flow of value otherwise there is a cut of capacity at most T . Look first at the set A 0 of "expensive" arcs not incident with s. A flow of value necessarily costs 2n(T + 1) on A 0 . Thus the cost of x on A 0 is at least 2n(T + 1), and equality is only possible if every arc in A 0 has reserved capacity just 1. Thus on A 0 , the reservation looks like the characteristic vector of a set of T diverse paths. Such a set of paths must include every arc arriving at t, and uses between 4 and 7 arcs from each node of T . For be the number of reserved arcs leaving v. The cost of any integer T -resilient reservation not of this form must be at least 2n(T arcs. Now consider the reservation vector x on the set A 1 of arcs leaving s. Certainly the total cost of the reservation on A 1 is greater than T must have reserved capacity at least 2. Thus we see that, since the total cost of x is at most (2n the reservation on A 0 does indeed form a set of T diverse paths, with d(v) 2 f4; 5; 6; 7g starting from each v 2 T . For the reservations on the four arcs between s and v must total at least d(v), and the sum of any three of them must be at least after deleting an arc there still exists a T -flow). The minimum cost of such a reservation between s and v is just 4 if since then only 1 need be reserved on each arc. For d(v) ? 4, one routinely checks that such a reservation costs at least since arcs of capacity at least 2 are needed. So the total cost of x on A 1 consistent with the values d(v) is at least plus the number N of elements v for which We know that v. Therefore N - n, with equality if and only if just n elements of T , and for the remainder. In this case, the cost of x on A 1 is otherwise is at least T Hence, if there is such a reservation vector x of total cost just (2n are just while the remainder have 4. A collection of T paths on A 0 subject to these conditions must involve n elements of T for which the 3n arcs from these elements to A [ B [C go to distinct nodes, i.e., a 3D-matching in the original instance. Conversely, given a 3D-matching U , we can find a T -resilient reservation by reversing the argument. We reserve capacity 1 on all arcs entering t, and all arcs leaving elements of U ; we further reserve capacity 1 on arcs from s to elements of T n U , and capacity 2 on arcs from s to elements of U . It is easy to check that this reservation is T -resilient, and has the required cost. Of course, the requirement that all reservations be integers is crucial. However, notice that the reservation we emerge with is actually net T -resilient as well, so the corresponding problem for net resilience is also NP-hard. On the positive side, there is a simple k-approximate algorithm for (T; k)-resilience in a network D - in the fractional or integral case - namely to find and output the minimum cost diverse paths reservation using which of course reserves capacity T on each arc of a cheapest set of k paths. The following result states that this is indeed a k-approximate algorithm. Proposition 10 If x is a (T; k)-resilient vector, then cx - 1 Proof. Let x be a minimum cost (T; k)-resilient vector. Define x 0 by setting, for each arc a, Let S be any (s; t)-set. We claim that x Suppose first that there is a set K of k+1 , and so If instead x a ! T k+1 for all arcs a those in a set K of k arcs. Then x(ffi and so x This proves our claim. Now, since x for each (s; t)-set S, there exists an (s; t) flow x 00 of value (k+1)T such that x 00 - x 0 , so in particular no arc has reserved capacity more than T ; thus x 00 is a (T; k)-resilient vector. As we remarked earlier, the diverse path reservation z k+1;k is a minimum cost (s; t) flow of value subject to an upper bound T on the flow through any given arc. It follows that cz k+1;k - Thus we have that the minimum cost diverse-path reservation has at most k times the cost of an optimal (T; k)-resilient reservation. Of course, this is of greatest interest in the case 1. The best possible: consider the following example. The network D has three nodes s, u and t. There are k cost 1 from s to u, and r arcs of large cost c from u to t. The minimum cost (1; k)-resilient reservation involves giving capacity 1 to the arcs between s and u, and capacity 1 r\Gammak to the arcs between u and t, at a cost of k However, the minimum cost diverse-path reservation, z k+1;k , uses only k+1 of the arcs from u to t, at a total cost of (k +1)(1+c). For large c and r, the ratio between these two costs can be made arbitrarily close to k+ 1: the optimal diverse-path reservation is almost twice as expensive as the optimal (1; k)-resilient reservation. As usual, one would expect that the normal situation is even better than suggested by Proposition 10, and in many practical settings the optimal diverse-path reservation (which can be found easily, as shown in Section 2.2) will not be very far from the overall optimal solution in cost. Given the simplicity of a diverse-path solution, it is likely that this is a good solution to adopt. 4 Applications to More than one Source-Destination Pair We now consider the problem where we are given a collection of node pairs as a collection of failure rates T i , a collection of integers k Each commodity i must reserve capacity in a network D which is (T )-resilient for the source-destination One approach to tackling this problem is to insist on diverse-path reservation vectors. We have seen that such reservations may cost more, even in the 1-commodity case, but also have several practical advantages. This approach also allows us to formulate the problem in a similar manner to the well-known multicommodity capacity allocation problem where there is a flow demand between each commodity pair and one must find routings of the flow such that capacity cost in the network is minimized. Suppose that, for each commodity i, we look for a reservation using j i diverse paths, each with capacity . If each arc a is equipped with an integer upper bound M a on the total capacity that can be reserved on a, and a cost c a , then one may pose this as a mixed integer program as follows. min a c a y a a for each arc a a 2 f0; 1g for each a and i Of course, there are still many choices of for each commodity, and we must branch on these in order to find the global optimum. Acknowledgements : The research of the first and second authors is supported by the EU-HCM grant TMRX-CT98 0202 DONET. They would also like to acknowledge support from DIMACS during extended visits to Bell Labs. Some of the first author's research was also carried out while visiting the University of Memphis. The authors are grateful for insightful remarks and encouragement from Gautam Appa, Dan Bienstock, Fan Chung, Michele Conforti, Bharat Doshi, Susan Powell, Paul Seymour and Mihalis Yannakakis. A major inspiration for our work on this paper was Dr. Ewart Lowe, of British Telecom, who tragically died in a diving accident on May 22nd, 1998, off the coast of Normandy. Ewart introduced the authors to many mathematical problems in telecommunications. He also acted as mentor to the final author during his projects for British Telecom. We dedicate this paper to the memory of his inspiration, generosity, and his unbounded enthusiasm which will be greatly missed by all who knew him. --R Network Flows - Theory Capacity and survivability models for telecommunications networks Combinatorial online optimization in practice Modeling and solving the single facility line restoration problem Network design using cut inequalities Cyclic scheduling via integer programs with circular ones Minimum cost capacity installation for multicommodity network flows Capacitated network design - polyhedral structure and computation Strong inequalities for capacitated survivable network design prob- lems Some strategies for reserving resilient capacity Reserving resilient capacity with upper bound constraints Resilience strategy for a single source-destination pair An optimal spare-capacity assignment model for survivable networks with hop limits The directed subgraph homeomorphism problem Connectivity and network flows Design of survivable networks Optimal capacity placement for path restoration in mesh survivable networks Modelling and solving the capacitated network loading problem The convex hull of two core capacitated network design problems Modelling and solving the two facility capacitated network loading problem Polyhedral properties of the network restoration problem - with the convex hull of a special case Working Paper OR 323-97 Two strategies for spare capacity placement in mesh restorable networks --TR --CTR Friedrich Eisenbrand , Fabrizio Grandoni, An improved approximation algorithm for virtual private network design, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia

network flows;resilience;capacity reservation

587996

Graphs with Connected Medians.

The median set of a graph G with weighted vertices comprises the vertices minimizing the average weighted distance to the vertices of G. We characterize the graphs in which, with respect to any nonnegative vertex weights, median sets always induce connected subgraphs. The characteristic conditions can be tested in polynomial time (by employing linear programming) and are immediately verified for a number of specific graph classes.

Introduction . Given a (finite, connected) graph G one is sometimes interested in finding the vertices minimizing the total distance to the vertices u of G; where the distance d(u; x) between u and x is the length of a shortest path connecting u and x: The subgraph induced by all minima of F need not be connected: actually every (possibly disconnected) graph can be realized as such a "median" subgraph of another graph (Slater [23]); see Hendry [18] for further information and pertinent references. Here we will focus on the weighted version of the median problem, which arises with one of the basic models in discrete facility location [25] and with majority consensus in classification and data analysis [3, 9, 10, 11, 22]. A weight function is any mapping - from the vertex set to the non-negative real numbers, which is not the zero function (in order to avoid trivialities). The total weighted distance of a vertex x in G is given by A vertex x minimizing this expression is a median (vertex) of G with respect to -; and the set of all medians is the median set Med(-): By a local median one means a vertex x such that F - (x) does not exceed F - (y) for any neighbour y of x: Denote by Med loc (-) the set of all local medians. We will consider the following questions: Research supported in part by the Alexander von Humboldt Stiftung ffl When are all median sets Med(-) of a graph connected? ffl When does Med loc (-) hold for all weight functions - ? If we allow only 0-1 weight functions -; then recognition of graphs with Med loc is an NP -complete problem [2]. We will show that connectivity of median sets with respect to arbitrary weight functions turns out to be equivalent to the condition that for each function F - all local minima are global. This property can also be formulated as a certain convexity condition. In the next section we investigate weakly convex functions on graphs, and in Section 3 we then obtain the basic characterizations of graphs with connected medians. One of those equivalent conditions, weak convexity of F - for each weight function -, can be formulated as a linear programming problem, thus allowing to recognize graphs with connected medians in polynomial time. This LP condition, however, entails a lot of redundancy and cannot be read off from other graph properties, which are known to imply median connectedness. Striving for less redundancy in the requirements, we establish the main result in Section 4 by employing LP duality to a particular instance of the original LP formulation. This theorem can conveniently be used to derive median properties in several specific classes of graphs, as is demonstrated in the final section. 2. Weakly convex functions. A real-valued function f defined on the vertex set V of a graph G is said to be weakly convex if for any two vertices u; v; and a real number - between 0 and 1 such that -d(u; v) and (1 \Gamma -)d(u; v) are integers, there exists a vertex x such that Weakly convex functions were introduced by Arkhipova and Sergienko [1] under the name "r-convex functions"; see also Lebedeva, Sergienko, and Soltan [19]. The interval I(u; v) between two vertices u and v consists of all vertices on shortest u; v-paths, that is: For convenience we will use the short-hand I to denote the "interior" of the interval between u and v: Lemma 1. For a real-valued function f defined on the vertex set of a graph G the following conditions are equivalent: (i) f is weakly convex; (ii) for any two non-adjacent vertices u and v there exists w 2 I ffi (u; v) such that (iii) any two vertices u and v at distance 2 have a common neighbour w with Proof. (i) Consider two vertices u and v at distance d(u; shortest paths connecting u and v select a path is as small as possible. Condition (iii) implies that 2f(w i as points in the plane R Connecting the consecutive points by segments, we will get a graph of a piecewise-linear function. This function is necessarily convex (in the usual sense), because it coincides on P with the function f: From this we conclude that Recall that a real-valued function f defined on a path holds only say that a function f defined on the vertex set of a graph G is pseudopeakless if any two vertices of G can be joined by a shortest path along which f is peakless [15]. Equivalently, f is pseudopeakless if for any two non-adjacent vertices u; v there is a vertex w 2 I ffi (u; v) such that f(w) - maxff(u); f(v)g and equality holds only if The key property of pseudopeakless functions is their unimodality, that is, every local minimum of f is global. The proof is simple: let u be a global minimum of f and v a local minimum of G: Consider a shortest path P between u and v along which f is peakless. Then for any neighbour w of v we have f(w) - maxff(u); f(v)g; whence as required. Note that every weakly convex function is pseudopeakless. The following result constitutes a partial converse. Remark 1. A real-valued function f defined on the vertex set of a graph G is pseudopeakless if and only if the composition ffffif is weakly convex for some strictly isotone transformation ff of the reals. Proof. The property of being pseudopeakless is clearly invariant under strictly isotone transformations of the range. Conversely, let f be a pseudopeakless function taking n distinct values a 1 ! a ff be a strictly isotone map which assigns to each a i the We assert that the composition ffffif is weakly convex. For given vertices u and v at distance 2, let w be their common neighbour such that f is peakless along the path consequently as required. 2 In what follows we will apply Lemma 1 to the functions F - of G: The simplest instance is given by the weight function - assigning 1 to a distinguished vertex w and 0 otherwise: then just measures the distance in G to that vertex. We say that G is meshed [6] if the function d(\Delta; w) is weakly convex for every choice of w: A somewhat weaker property is used in subsequent results. Three vertices u; v; w of a G are said to form a metric triangle uvw if the intervals I(u; v); I(v; w); and I(w; u) pairwise intersect only in the common end vertices. If d(u; triangle is called equilateral of size k: Remark 2. Every metric triangle in a meshed graph G is equilateral. To see this, suppose the contrary: let uvw be a metric triangle in G with d(u; w) ! d(v; w): The above definition of weak convexity applied to the pair u; v; and the number provides us with a neighbour x of v which necessarily belongs to 3. Basic characterizations. We commence by giving first answers to the questions raised in the introduction. Proposition 1. For a graph G the following conditions are equivalent: (i) Med loc weakly convex for all -; is pseudopeakless for all -; (iv) all level sets fx : F - (x) -g induce isometric subgraphs; (v) all median sets Med(-) induce isometric subgraphs; (vi) all median sets Med(-) are connected. Any of the conditions (i) to (vi) is equivalent to the analogous condition with the additional requirement that - be positive. The following observation is basic for the proof of Proposition 1. Lemma 2. If the function F - is not weakly convex on the vertex set V of a graph G for some weight function -; then there exist a positive weight function - + and vertices u; v at distance 2 such that Med(- Proof. If F - is not weakly convex, then by Lemma 1 there exist two vertices u and v at distance 2 such that 2F - (w) ? F - (u) +F - (v) for all (common neighbours) w 2 I ffi (u; v): This inequality can be maintained under sufficiently small positive perturbations of - : viz., add any ffi satisfying to all weights, yielding the new weights - 0 for all w 2 I ffi (u; v); that is, the initial inequality remains valid with respect to the thus perturbed weight function. We may therefore assume that - is actually positive. We stipulate that denote the maximum value of Define a new positive weight function - + by for every vertex x outside I(u; v): For w 2 I ffi (u; v) one obtains Therefore consists only of u and v: This concludes the proof. 2 Proof of Proposition 1. The implications (ii) are trivial, while (vi) ) (ii) is covered by Lemma 2. It remains to verify that (i) implies (ii). Suppose by way of contradiction that some function F - is not weakly convex. By Lemma 2 there exist a positive weight function - + and vertices u; v at distance 2 such that Pick any ffl satisfying and define a new weight function - 0 by for all x Therefore both u and v are local minima of F - 0 ; but establishes the implication (i) ) (ii). The same arguments can be applied to prove that the analogous conditions (i + ) to (vi additionally requiring the weight functions to be positive are all equivalent. Since (i)) (i is trivial and (vi covered by Lemma 2, the proof is complete. 2 In view of Lemma 1(iii) and Proposition 1, all median sets are connected if and only if for each pair u; v of vertices at distance 2 the following system of linear inequalities is unsolvable in -: Since LP problems can be solved in polynomial time, we thus obtain the following result. Corollary 1. The problem to decide whether all median sets Med(-) of a graph G are connected is solvable in polynomial time. 4. The main result. In order to obtain a convenient characterization of median con- nectedness, we will restrict the supports of the weight functions - to be tested and then take advantage of LP duality in the following form: Remark 3. Let u; v be any vertices of a graph G with d(u; W be a nonempty subset of I ffi (u; v) and X be a nonempty subset of the vertex set of G: Then for every weight function - with support included in X there exists some w 2 W such that if and only if there exists a weight function j with support included in W such that for every Proof. Let D denote the submatrix of D uv (as just defined) with rows and columns corresponding to W and X; respectively. The asserted equivalence is then a particular instance of LP duality, as formulated by J.A.Ville (cf. [16, p.248]) in terms of systems of linear inequalities: For vertices x and y define the following superset of I(x; Proposition 2. The median sets Med(-) in a graph G are connected for all weight functions - if and only if (1) all metric triangles are equilateral and (2) for any vertices u; v with every weight function - whose support fx included in the set J(u; v) there exists a vertex w 2 I ffi (u; v) such that 2F - (w) - F - (u) Proof. Necessity immediately follows from Remark 2 and Proposition 1. To prove sufficiency assume that (1) and (2) hold but there exists a weight function - for which F - is not weakly convex. Then by Lemma 1 there are vertices u and v at distance 2 such that fixing u and v; we may assume that - was chosen so that this inequality still holds and the number of vertices from the support of - outside J(u; v) is as small as possible. By (2) we can find a vertex x with -(x) ? 0 and To arrive at a contradiction, we will slightly modify - by setting the weight of x to zero (and possibly transferring its old weight to some vertex in J(u; v)): We distinguish three cases. Case 1: jd(u; 2: Then 2d(w; Hence the weight function - 0 defined by contrary to the minimality assumption on -: Case 2: jd(u; Without loss of generality assume that d(v; be a vertex in I(u; I(v; x) at maximal distance to x: Then x From the latter equality and d(u; t be a vertex in I(u; v) " I(v; x 0 ) at maximal distance to v: Since the vertices u; t; x 0 must constitute an equilateral metric triangle. Therefore d(u; x 2: Define a new weight function - For every w 2 I ffi (u; v) we have either or 2: Hence and further again a contradiction to the choice of -: Case 3: Let x 0 be a vertex in I(u; x) " I(v; x) at maximal distance to x: Since all metric triangles of G are equilateral and d(u; we obtain that 2: Define the new weight function - 0 as in Case 2. Then and for every w 2 I ffi (u; v) a final contradiction. 2 In view of Remark 2 we may require in Proposition 2 (as well as the Theorem below) that G be meshed instead that all metric triangles in G be equilateral. Let M(u; v) denote the set of those vertices of J(u; v) which are equidistant to u and v : The neighbourhood N(x) of a vertex x consists of all vertices adjacent to x: Note that if Lemma 3. If all median sets of a graph G are connected, then for any vertices u and v at distance 2 there exist (not necessarily distinct) vertices s; t 2 I ffi (u; v) such that for all vertices y 2 M(u; v): Proof. We define a weight function - for which the hypothesis that F - be pseudopeakless implies the asserted inequality: let and Since weights are zero outside M(u; v); we have F - v) such that F - is peakless along the (shortest) path (u; s; v); that is, F - We claim that 2: Indeed, otherwise holds and either there are two distinct vertices there exists some vertex z 2 M(u; I 3: In the first case we would get and in the second case both giving a contradiction. Now, if all vertices in M(u; v) \Gamma fsg are equidistant to s and u (as well as v), then and thus is the required solution to the asserted inequality. Else, there exists some whence 2: Consequently, and therefore s; t constitutes the required vertex pair in this case. 2 Lemma 4. Let u and v be vertices at distance 2 in a graph G; for which all median sets are connected. Select a maximal subset S of I ffi (u; v) with the property that for each vertex there exists a vertex t 2 S (not necessarily distinct from s) such that d(s; y) weight function - with support included in J(u; v) there exists some vertex w Proof. By Lemma 3, the set S is nonempty. Now, assume the contrary and among all weight functions violating our assertion, choose a function - for which the set is as small as possible. This set contains some vertex x because F - is weakly convex. Pick First suppose that d(x; distinct vertices new weight function - 0 by -(w) otherwise. Then Hence and for all w 2 I ffi (u; v) distinct from x; contrary to the choice of -: Therefore, x is adjacent to all vertices possibly one vertex. Now, suppose that d(x; for the modified weight function - 0 defined by -(w) otherwise we have Then we obtain the same inequalities as in the preceding case and thus again arrive at a contradiction. We conclude that d(x; If x is adjacent to all other vertices of I ffi (u; v); then 2d(x; z 2 M(u; v); z 6= x; and hence we could adjoin x to S; contrary to the maximality of S: Therefore, by what has been shown, I ffi (u; v) contains exactly one vertex y distinct from x which is not adjacent to x: Since 2 there exists a vertex z 2 M(u; v) such that necessarily new weight function - 0 by -(w) otherwise. Then giving the same contradiction as before. This concludes the proof. 2 Now we are in a position to formulate the principal result. Theorem. For a graph G all median sets are connected if and only if the following conditions are satisfied: (i) all metric triangles of G are equilateral; (ii) for any vertices u and v at distance 2 there exist a nonempty subset S of I ffi (u; v) and a weight function j with support included in S having the two properties: (ff) every vertex s 2 S has a companion t 2 S (not necessarily distinct from s) such that and (fi) the joint weight of the neighbours of any x 2 J(u; v) \Gamma M(u; v) from S is always at least half the total weight of S : Proof. First assume that G is a graph with connected medians. By Proposition 2 all metric triangles of G are equilateral. Let u; v be a pair of vertices at distance 2. The nonempty subset S of I ffi (u; v) provided by Lemma 4 satisfies the inequality in (ff) for each s 2 S and companion t: Moreover, Lemma 4 guarantees that for every weight function - with support included in J(u; v) there exists a vertex w 2 S with F - (u) duality as formulated in Remark 3 for yields a weight function j with support included in S such that for every x 2 J(u; v) the weighted sum of all and therefore (fi) holds. If x 2 M(u; v); then d(s; x) - 2 for all s 2 S and In the latter case is the companion of This yields the inequality 0; and by interchanging the role of s and t; we infer that 2: Conversely, if conditions (i) and (ii) are satisfied, then by virtue of LP duality (Remark conditions (1) and (2) of Proposition 2 are fulfilled, whence G has connected medians. 2 Corollary 2. All median sets in a meshed graph G are connected whenever the following condition is satisfied: any two vertices u and v with #I ffi (u; v) - there exist (not necessarily vertices s; t 2 I ffi (u; v) such that If, moreover, G satisfies the stronger condition requiring in addition that d(s; t) - 1; then Med(-) induces a complete subgraph for every positive weight function -: Proof. First observe that the inequality of ( ) also holds in the case that u and v are at distance 2 with a unique common neighbour s: Indeed, if x 2 J(u; v); then necessarily since G is meshed. To see that ( ) implies condition (ii) of the Theorem, put trivially satisfied. For x 2 J(u; we have d(u; x) +d(v; is meshed, and therefore x is adjacent to at least one of s; t; thus establishing (fi): To prove the second statement, observe that the inequality in ( ) actually holds for all vertices x of G: Indeed, for each vertex x select a vertex x 0 from I(u; x) " I(v; x) at maximal distance to x: Then x 0 belongs to J(u; v) and Adding up all these inequalities each multiplied by the corresponding weight -(x) then yields where strict inequality holds exactly when one of the former inequalities is strict for x with positive, u and v cannot both belong to Med(-) under these circumstances. 2 In general, we can neither dispense with weight functions nor impose any fixed upper bound on the cardinalities of the sets S occuring in the Theorem, as the following example shows. Example. Let G be the chordal graph with 2 vertices comprising a set of mutually adjacent vertices, two vertices u and v adjacent to all of and vertices wZ associated with certain subsets Z of S; namely the sets fx 1 and fx k-subsets Y of fy such that each vertex wZ is adjacent to u and all vertices from Z: Then a weight function j with support included in I for the pair u; v if and only if for each of the above sets Z (which come in complementary pairs) one has z2Z which is equivalent to Thus, in order to satisfy (fi); we are forced to take a weight function j having a large support on which it is not constant. Note that (ff) for u; v is trivially fulfilled with any choice of from S: Finally, every other pair of vertices at distance 2 meets condition ( ) of Corollary when either replacing u by wZ and selecting from Z or replacing u; v by wZ ; w Z 0 setting This shows that G has connected medians. 5. Particular cases. A number of graph classes [12] consist of particular meshed graphs, for instance, the classes of chordal graphs (and more generally, bridged graphs), Helly graphs, and weakly median graphs (see below), respectively. In the case of chordal graphs the first statement of Corollary 2 is due to Wittenberg [26, Theorem 1.1]. Whenever a chordal graph satisfies condition ( ), then any pair s; t selected in ( ) necessarily satisfies comes into full action for the class of Helly graphs. A Helly graph G (alias "absolute retract of reflexive graphs") has the characteristic "Helly property" that for any vertices non-negative integers r the system of inequalities admits a solution x whenever holds [8]. Since this Helly property for (as formulated in Lemma 1(iii)) of the function d(\Delta; z); all Helly graphs are meshed. To verify condition ( ) (with d(s; t) - 1), first observe that the Helly property for N(v) and thus M(u; 2: Now, all vertices in are pairwise at distance - 2; whence the Helly property guarantees a common neighbour s: Similarly, the vertices from (N(v) " J(u; v)) [ fug have a common neighbour t: Necessarily s; t 2 I ffi (u; v) and d(s; t) - 1 hold, and we have therefore established the following observation, which implies the result of Lee and Chang [20] on strongly chordal graphs (being special Helly graphs): Corollary 3. All median sets Med(-) of a Helly graph G are connected, and moreover, if - is positive, then Med(-) induces a complete graph in G: In some particular classes of meshed graphs the pair s; t meeting ( ) can always be selected by the following trivial rule: given vertices u; v at distance 2, choose any pair s; t from I ffi (u; v) for which d(s; t) is as large as possible. Evidently, a meshed graph G satisfies condition ( ) with this selection rule provided that the following two requirements are met: is a complete subgraph for vertices u and v at distance 2, then d(s; induce a 4-cycle where d(s; Notice that in (xx) we could replace the last equality by an inequality because the inequality would imply J(s; in a meshed graph and thus the reverse inequality would follow by symmetry. We will now show that the so-called weakly median graphs satisfy (x) and (xx). A graph is weakly median [14] if (i) any three distinct common neighbours of any two distinct vertices u and v always induce a connected subgraph and (ii) all functions d(\Delta; z) satisfy the following condition, stronger than weak convexity: if u; v are at distance 2 and Condition (ii) is also referred to weak modularity; see [4]. Trivially, weakly modular graphs are meshed. Remark 4. A weakly modular graph G satisfies conditions (x) and (xx) if and only if G does not contain any of the graphs of Fig.1 as an induced subgraph. In particular, weakly median graphs have connected medians. @ @ @ @ @ @ @ @ @ @ \Theta \Theta \Theta \Theta \Theta @ @ @ @ @ @ @ @ @ @ \Theta \Theta \Theta \Theta \Theta @ @ @ @ @ @ @ @ s x s x s (a) (b) (c) Fig. 1 Proof. If G includes an induced subgraph from Fig. 1, then s; t; u; v; x violate either (xx) or (x). Conversely, assume that G is weakly modular but violates (x) or (xx) for some vertices s; t; u; v; x: Then, as x 2 J(u; v); we have d(u; x)+d(v; x) - 3 by weak modularity. If necessarily s; t; u; v; x induce one of the first two graphs in Fig. 1. Otherwise, say, As G is meshed, u; v; x have some common neighbour w: If w is not adjacent to both s and t; then we are back in the preceding case with w playing the role of x: Therefore we may assume that w is a common neighbour of s and t: If s and t are not adjacent, then s; t; u; v; w induce a subgraph isomorphic to Fig. 1b. Otherwise, s and t are adjacent, and we obtain the graph of Fig. 1c as an induced subgraph. Finally, note that the latter graph minus v as well as Fig. 1a,b are all forbidden in a weakly median graph. 2 Inasmuch as pseudo-median graphs and quasi-median graphs are weakly median, Remark 4 (in conjunction with Proposition 1) generalizes some results from [3, 7, 17, 24]. Another class of meshed graphs fulfilling (x) and (xx) is associated with matroids. A matroid M can be defined as a finite set E together with a collection B of subsets (referred to as the bases of M) such that for there exists some e The basis graph of M is the (necessarily connected) graph whose vertices are the bases of M and edges are the pairs B; B 0 of bases differing by a single exchange. We immediately infer from the characterization established by Maurer [21] that the basis graph of every matroid is meshed (but not weakly modular in general) and satisfies (x) and (xx). Corollary 4. The basis graph of every matroid has connected medians. In view of Corollary 4 and Proposition 1 one can solve the median problem in the basis graph of a matroid with the greedy algorithm. Given a weight function - on assign a weight to each element e 2 E by Applying the greedy algorithm one finds a base B maximizing the function B 7! We assert that B is a minimum for the function F - in the basis graph. Indeed, if is any neighbour of B ; then Hence B 2 Med loc (-) =Med(-); as required. --R On conditions of coincidence of local and global extrema in optimization problems (in Russian) A Helly theorem in weakly modular space decomposition via amalgamation and Cartesian multiplication Weak Cartesian factorization with icosahedra Graph Theory Dismantling absolute retracts of reflexive graphs The Geometry of Geodesics Separation of two convex sets in convexity structures Discrete Math. Medians of pseudo-median graphs (in Russian) On graphs with prescribed median I On conditions of coincidence of local and global minima in problems of discrete optimization Matroid basis graphs I The median procedure in a formal theory of consensus Medians of arbitrary graphs The solution of the Weber problem for discrete median metric spaces (in Russian) a survey I Local medians in chordal graphs --TR --CTR Victor Chepoi , Clmentine Fanciullini , Yann Vaxs, Median problem in some plane triangulations and quadrangulations, Computational Geometry: Theory and Applications, v.27 n.3, p.193-210, March 2004

majority rule;local medians;LP duality;medians;graphs

587999

Single Machine Scheduling with Release Dates.

We consider the scheduling problem of minimizing the average weighted completion time of n jobs with release dates on a single machine. We first study two linear programming relaxations of the problem, one based on a time-indexed formulation, the other on a completion-time formulation. We show their equivalence by proving that a O(n log n) greedy algorithm leads to optimal solutions to both relaxations. The proof relies on the notion of mean busy times of jobs, a concept which enhances our understanding of these LP relaxations. Based on the greedy solution, we describe two simple randomized approximation algorithms, which are guaranteed to deliver feasible schedules with expected objective function value within factors of 1.7451 and 1.6853, respectively, of the optimum. They are based on the concept of common and independent $\alpha$-points, respectively. The analysis implies in particular that the worst-case relative error of the LP relaxations is at most 1.6853, and we provide instances showing that it is at least $e/(e-1) \approx 1.5819$. Both algorithms may be derandomized; their deterministic versions run in O(n2) time. The randomized algorithms also apply to the on-line setting, in which jobs arrive dynamically over time and one must decide which job to process without knowledge of jobs that will be released afterwards.

Introduction We study the single-machine scheduling problem with release dates in which the objective is to minimize a weighted sum of completion times. It is dened as follows. A set ng of n jobs has to be scheduled on a single disjunctive machine. Job j has a processing time and is released at time r j 0. We assume that release dates and processing times are integral. The completion time of job j in a schedule is denoted by C j . The goal is to nd a non-preemptive schedule that minimizes where the w j 's are given positive weights. M.I.T., Department of Mathematics, Room 2-351, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. y University of British Columbia, Faculty of Commerce and Business Administration, Vancouver, B.C., Canada V6T 1Z2. Email: maurice.queyranne@commerce.ubc.ca z M.I.T., Sloan School of Management, Room E53-361, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. x Technische Universitat Berlin, Fakultat II { Mathematik und Naturwissenschaften, Institut fur Mathematik, MA 6-1, Strae des 17. Juni 136, 10623 Berlin, Germany. Email: skutella@math.tu-berlin.de { PeopleSoft Inc., San Mateo, CA, USA 94404. Email: Yaoguang Wang@peoplesoft.com In the classical scheduling notation [12], this problem is denoted by 1j r . It is strongly NP-hard, even if w One of the key ingredients in the design and analysis of approximation algorithms as well as in the design of implicit enumeration methods is the choice of a bound on the optimal value. Several linear programming based as well as combinatorial lower bounds have been proposed for this well studied scheduling problem, see for example, Dyer and Wolsey [9], Queyranne [22], and Queyranne and Schulz [23], as well as Belouadah, Posner and Potts [4]. The LP relaxations involve a variety of dierent types of variables which, e. g., either express whether job j is completed at time t (non-preemptive time-indexed relaxation), or whether it is being processed at time t (preemptive time-indexed relaxation), or when job j is completed (completion time relax- ation). Dyer and Wolsey show that the non-preemptive time-indexed relaxation is stronger than the preemptive time-indexed relaxation. We will show that the latter relaxation is equivalent to the completion time relaxation that makes use of the so-called shifted parallel inequalities. In fact, it turns out that the polyhedron dened by these inequalities is supermodular and hence one can optimize over it by using the greedy algorithm. A very similar situation arises in [24]. The greedy solution may actually be interpreted in terms of the following preemptive schedule, which we call the LP schedule: at any point in time it schedules among the available jobs one with the largest ratio of weight to processing time. Uma and Wein [38] point out that the value of this LP solution coincides with one of the combinatorial bounds of Belouadah, Posner and Potts based on the idea of allowing jobs to be split into smaller pieces that can be scheduled individually. We show that the optimal value of 1j r j j is at most 1:6853 times the lower bound given by any of these three equivalent relaxations | the preemptive time-indexed relaxation, the completion time relaxation or the combinatorial relaxation in [4]. We prove this result on the quality of these relaxations by converting the (preemptive) LP schedule into a non-preemptive schedule. This technique leads to approximation algorithms for 1j r j j Recall that a { approximation algorithm is a polynomial-time algorithm guaranteed to deliver a solution of cost at most times the optimal value. A randomized {approximation algorithm is a polynomial-time algorithm that produces a feasible solution whose expected objective function value is within a factor of of the optimal value. The technique of converting preemptive schedules to non-preemptive schedules in the design of approximation algorithms was introduced by Phillips, Stein and Wein [21]. More specically, showed that list scheduling in order of the completion times of a given preemptive schedule produces a non-preemptive schedule while increasing the total weighted completion time by at most a factor of 2. In the same paper they also introduced a concept of -points. This notion was also used by Hall, Shmoys and Wein [13], in connection with the non-preemptive time-indexed relaxation of Dyer and Wolsey to design approximation algorithms in various scheduling environments. For our purposes, the -point of job j in a given preemptive schedule is the rst point in time at which an -fraction of j has been completed. When one chooses dierent values of , sequencing in order of non-decreasing -points in the same preemptive schedule may lead to dierent non-preemptive schedules. This increased exibility led to improved approximation algorithms: Chekuri, Motwani, Natarajan and Stein [6] for Goemans [11] for 1j r chose at random and analyzed the expected performance of the resulting randomized algorithms. We will show that, using a common value of for all jobs and an appropriate probability distribution, sequencing in order of -points of the LP schedule has expected performance no worse than 1:7451 times the optimal preemptive time-indexed LP value. We also prove that by selecting a separate value j for each job j, one can improve this bound to a factor of 1:6853. Our algorithms are inspired by and partly resemble the algorithms of Hall et al. [13] and Chekuri et al. [6]. In contrast to Hall et al. Reference and/or O-line On-line type of schedule deterministic randomized deterministic Phillips et al. [21] Hall et al. [13] 4 4 Schulz [26] 3 Hall et al. [14] 3 3 Chakrabarti et al. [5] 2:8854 Combining [5] and [14] 2:4427 -schedule for Table 1: Summary of approximation bounds for 1j r j . An -schedule is obtained by sequencing the jobs in order of non-decreasing -points of the LP schedule. The use of job-dependent j 's yields an )-schedule. The results discussed in this paper are below the second double line. Subsequently, Anderson and Potts [2] gave a deterministic 2{competitive algorithm. For the unit-weight problem 1j r j the rst constant-factor approximation algorithm is due to Phillips, Stein and Wein [21]. It has performance ratio 2, and it also works on-line. Further deterministic 2{competitive algorithms were given by Stougie [36] and Hoogeveen and Vestjens [15]. All these algorithms are optimal for deterministic on-line algorithms [15]. Chekuri, Motwani, Natarajan and Stein [6] gave a randomized e=(e 1){approximation algorithm, which is optimal for randomized on-line algorithms [37, 39]. we exploit the preemptive time-indexed LP relaxation, which, on the one hand, provides us with highly structured optimal solutions and, on the other hand, enables us to work with mean busy times. We also use random -points. The algorithm of Chekuri et al. starts from an arbitrary preemptive schedule and makes use of random -points. They relate the value of the resulting -schedule to that of the given preemptive schedule, and not to that of an underlying LP relaxation. While their approach gives better approximations for 1j r insights on limits of the power of preemption, the link of the LP schedule to the preemptive time-indexed LP relaxation helps us to obtain good approximations for the total weighted completion time. Variants of our algorithms also work on-line when jobs arrive dynamically over time and, at each point in time, one has to decide which job to process without knowledge of jobs that will be released afterwards. Even in this on-line setting, we compare the value of the computed schedule to the optimal (o-line) schedule and derive the same bounds (competitive ratios) as in the o- line setting. See Table 1 for an account of the evolution of o-line and on-line approximation results for the single machine problem under consideration. The main ingredient to obtain the results presented in this paper is the exploitation of the structure of the LP schedule. Not surprisingly, the LP schedule does not solve the strongly NP-hard [16] preemptive version of the problem, 1j r However, we show that the LP schedule solves optimally the preemptive problem with the related objective function where M j is the mean busy time of job j, i. e., the average point in time at which the machine is busy processing j. Observe that, although 1j r are equivalent optimization problems in the non-preemptive case (since C are not when considering preemptive schedules. The approximation techniques presented in this paper have also proved useful for more general scheduling problems. For the problem with precedence constraints 1j r sequencing jobs in order of random -points based on an optimal solution to a time-indexed relaxation leads to a 2:7183{approximation algorithm [27]. A 2{approximation algorithm for identical parallel machine scheduling is given in [28]; the result is based on a time-indexed LP relaxation an optimal solution of which can be interpreted as a preemptive schedule on a fast single machine; jobs are then assigned randomly to the machines and sequenced in order of random j -points of this preemptive schedule. For the corresponding scheduling problem on unrelated parallel machines R j r j j performance guarantee of 2 can be obtained by randomized rounding based on a convex quadratic programming relaxation [33], which is inspired by time-indexed LP relaxations like the one discussed herein [28]. We refer to [32] for a detailed discussion of the use of -points for machine scheduling problems. Signicant progress has recently been made in understanding the approximability of scheduling problems with the average weighted completion time objective. Skutella and Woeginger [34] developed a polynomial-time approximation scheme for scheduling identical parallel machines in the absence of release dates, Subsequently, several research groups have found polynomial-time approximation schemes for problems with release dates such as and see the resulting joint conference proceedings publication [1] for details. We now brie y discuss some practical consequences of our work. Savelsbergh, Uma and Wein [25] and Uma and Wein [38] performed experimental studies to evaluate, in part, the quality of the LP relaxation and approximation algorithms studied herein, for 1j r and related scheduling problems. The rst authors report that, except for instances that were deliberately constructed to be hard for this approach, the present formulation and algorithms \deliver surprisingly strong experimental performance." They also note that \the ideas that led to improved approximation algorithms also lead to heuristics that are quite eective in empirical experiments; furthermore they can be extended to give improved heuristics for more complex problems that arise in practice." While the authors of the follow-up study [38] report that when coupled with local improvement the LP-based heuristics generally produce the best solutions, they also nd that a simple heuristic often outperforms the LP-based heuristics. Whenever the machine becomes idle, this heuristic starts non-preemptively processing an available job of largest w j =p j ratio. By analyzing the dierences between the LP schedule and this heuristic schedule, Chou, Queyranne and Simchi-Levi [7] have subsequently shown the asymptotic optimality of this on-line heuristic for classes of instances with bounded job weights and bounded processing times. The contents of this paper are as follows. Section 2 is concerned with the LP relaxations and their relationship. We begin with a presentation and discussion of the LP schedule. In Section 2.1 we then review a time-indexed formulation introduced by Dyer and Wolsey [9] and show that it is solved to optimality by the LP schedule. In Section 2.2 we present the mean busy time relaxation (or completion time relaxation) and prove, among other properties, its equivalence to the time-indexed formulation. Section 2.3 explores some polyhedral consequences, in particular the fact that the mean busy time relaxation is (up to scaling by the job processing times) a supermodular linear program and that the \job-based" method for constructing the LP schedule is equivalent to the corresponding greedy algorithm. Section 3 then deals with approximation algorithms derived from these LP relaxations. In Section 3.1 we present a method for constructing ( j )-schedules, which allows us to analyze and bound the job completion times in the resulting schedules. In Section 3.2 we derive simple bounds for -schedules and ( j )-schedules, using a deterministic common or uniformly distributed random j 's. Using appropriate probability distributions, we improve the approximation bound to the value of 1.7451 for -schedules in Section 3.3 and to the value of 1.6853 for ( j )-schedules in Section 3.4. We also indicate how these algorithms can be derandomized in O(n 2 ) time for constructing deterministic schedules with these performance guarantees. In Section 3.5 we show that our randomized approximations also apply in an on-line setting and, in Section 3.6 we present a class of \bad" instances for which the ratio of the optimal objective function value and our LP bound is arbitrarily close to e 1:5819. This constant denes a lower bound on the approximation results that can be obtained by the present approach. We conclude in Section 4 by discussing some related problems and open questions. Relaxations In this section, we present two linear programming relaxations for 1j r j j We show their equivalence and discuss some polyhedral consequences. For both relaxations, the following preemptive schedule plays a crucial role: at any point in time, schedule (preemptively) the available job with highest w j =p j ratio. We assume (throughout the paper) that the jobs are indexed in order of non-increasing ratios w 1 wn pn and ties are broken according to this order. Therefore, whenever a job is released, the job being processed (if any) is preempted if the released job has a smaller index. We refer to this preemptive schedule as the LP schedule. See Figure 1 for an example of an LP schedule.011011011 Figure 1: An LP schedule for a 4-job instance given by r 1 r 4 5. Higher rectangles represent jobs with larger weight to processing time ratio. Time is shown on the horizontal axis. Notice that this LP schedule does not in general minimize preemptive schedules. This should not be surprising since the preemptive problem 1j r (strongly) NP-hard [16]. It can be shown, however, that the total weighted completion time of the LP schedule is always within a factor of 2 of the optimal value for 1j r this bound is tight; see [29]. The LP schedule can be constructed in O(n log n) time. To see this, we now describe an implementation, which may be seen as \dynamic" (event-oriented) or, using the terminology of [19], \machine-based" and can even be executed on-line while the jobs dynamically arrive over time. The algorithm keeps a priority queue [8] of the currently available jobs that have not yet been completely processed, with the ratio w j =p j as the key and with another eld indicating the remaining processing time. A scheduling decision is made at only two types of events: when a job is released, and when a job completes its processing. In the former case, the released job is added to the priority queue. In the latter case, the completed job is removed from the priority queue. Then, in either case, the top element of the priority queue (the one with highest w ratio) is processed; if the queue is empty, then move on to the next job release; if there is none, then all jobs have been processed and the LP schedule is complete. This implementation results in a total of O(n) priority queue operations. Since each such operation can be implemented in O(log n) time [8], the algorithm runs in O(n log n) time. The LP schedule can also be dened in a somewhat dierent manner, which may be seen as \static" or \job-based" [19]. Consider the jobs one at a time in order of non-increasing w j =p j . Schedule each job j as early as possible starting at r j and preempting it whenever the machine is busy processing another job (that thus came earlier in the w j =p j ordering). This point-of-view leads to an alternate O(n log n) construction of the LP schedule, see [10]. 2.1 Time-Indexed Relaxation Dyer and Wolsey [9] investigate several types of relaxations of 1j r j j the strongest ones being time-indexed. We consider the weaker of their two time-indexed formulations, which they call formulation (D). It uses two types of variables: y being processed during time interval [; + 1), and zero otherwise; and t j represents the start time of job j. For simplicity, we add p j to t j and replace the resulting expression by C j ; this gives an equivalent relaxation. subject to (D) where T is an upper bound on the makespan of an optimal schedule (for example, We refer to this relaxation as the preemptive time-indexed relaxation. The expression for C j given in (1) corresponds to the correct value of the completion time if job j is not preempted; an interpretation in terms of mean busy times is given in the next section for the case of preemptions. Observe that the number of variables of this formulation is pseudo- polynomial. If we eliminate C j from the relaxation by using (1), we obtain a transportation problem [9] and, as a result, y j can be assumed to be integral: Lemma 2.1. There exists an optimal solution to (D) for which y j 2 f0; 1g for all j and . As indicated in [9], (D) can be solved in O(n log n) time. Actually, one can derive a feasible solution to (D) from the LP schedule by letting y LP j be equal to 1 if job j is being processed in Theorem 2.2. The solution y LP derived from the LP schedule is an optimal solution to (D). Proof. The proof is based on an interchange argument. Consider any optimal 0=1-solution y to (D). If there exist j < k and > r j such that y replacing y j and y by 0, and y j and y k by 1, we obtain another feasible solution with an increase in the objective function value of ( ) 0. The resulting solution must therefore also be optimal. By repeating this interchange argument, we derive that there exists an optimal solution y such that there do not exist j < k and > r j such that y This implies that the solution y must correspond to the LP schedule. In particular, despite the pseudo-polynomial number of variables in the LP relaxation (D) an optimal solution can be obtained e-ciently. We will make use of this fact as well as of the special structure of the LP schedule in the design and analysis of the approximation algorithms, see Section 3. We note again that in spite of its nice properties the preemptive time-indexed LP relaxation (D) solves neither 1j r . In the former case, the processing of a job in the LP solution may fail to be consecutive; in the latter case equation (1) does not necessarily dene the completion time of a job in the preemptive LP schedule, as will be shown in the next lemma. 2.2 Mean Busy Time Relaxation Given any preemptive schedule, let I j be the indicator function of the processing of job j at time t, i. e., I j (t) is 1 if the machine is processing j at time t and 0 otherwise. To avoid pathological situations, we require that, in any preemptive schedule, when the machine starts processing a job, it does so for a positive amount of time. Given any preemptive schedule, we dene the mean busy time M j of job j to be the average time at which the machine is processing j, i. e., I For instance, in the example given in Figure 1, which will be used throughout this paper, the mean busy time of job 4 is 5:5. We rst establish some important properties of M j in the next two lemmas. Lemma 2.3. For any preemptive schedule, let C j and M j denote the completion and mean busy time, respectively, of job j. Then for any job j, we have only if job j is not preempted. Proof. If job j is processed without preemption, then I j is not processed during some interval(s) of total length L > 0 between times C R T I j (t) must be processed during some time interval(s) of the same total length L before C j p j . Therefore, I and the proof is complete. Let S N denote a set of jobs and dene Let I S (t) := j2S I j (t). Since, by the machine capacity constraint, I S (t) 2 f0; 1g for all t, we may view I S as the indicator function for job set S. We can thus dene the mean busy time of set S as M S := 1 R T dt. Note that we have I j (t) So, unlike its start and completion time, the mean busy time of a job set is a simple weighted average of the mean busy times of its elements. One consequence of this observation is the validity of the shifted parallel inequalities (3) (see, e.g., [10, 23, 24]) below. Lemma 2.4. For any set S of jobs and any preemptive schedule with mean busy time vector M , we have and equality holds if and only if all the jobs in S are scheduled without interruption from r min (S) to r min (S) Proof. Note that R T r min (S) I S (t) t dt ; that I S and I S (t) 1 for t r min (S) ; and that R T r min (S) I S (t) is minimized when I S M S is uniquely minimized among all feasible preemptive schedules when all the jobs in S are continuously processed from r min (S) to r min (S) + p(S). This minimum value is p(S)(r min (S) +2 p(S)) and is a lower bound for in any feasible preemptive schedule. As a result of Lemma 2.4, the following linear program provides a lower bound on the optimal value of 1j r and hence on that of 1j r j j subject to (R) The proof of the following theorem and later developments use the notion of canonical decompositions [10]. For a set S of jobs, consider the schedule which processes jobs in S as early as possible, say, in order of their release dates. This schedule induces a partition of S into such that the machine is busy processing jobs in S exactly in the disjoint We refer to this partition as the canonical decomposition of S. A set is canonical if it is identical to its canonical decomposition, i. e., if its canonical decomposition is fSg. Thus a set S is canonical if and only if it is feasible to schedule all its jobs in the time interval [r min (S); r min (S)+p(S)). Note that the set our example is canonical whereas the subset f1; 2; 3g is not; it has the decomposition ff3g; f1; 2gg. Let is the canonical decomposition of S N . Then Lemma 2.4 implies that is a valid inequality for the mean busy time vector of any preemptive schedule. In other words, relaxation (R) may be written as: min Theorem 2.5. Let M LP j be the mean busy time of job j in the LP schedule. Then M LP is an optimal solution to (R). Proof. By Lemma 2.4, M LP is a feasible solution for (R). To prove optimality of M LP , we construct a lower bound on the optimal value of (R) and show that it is equal to the objective function value of M LP . Recall that the jobs are indexed in non-increasing order of the w k(i) denote the canonical decomposition of [i]. Observe that for any vector where we let w n+1 =p n+1 := 0. We have therefore expressed as a nonnegative combination of expressions sets. By construction of the LP schedule, the jobs in any of these canonical sets S i are continuously processed from r min (S i ' ) to r min (S i ) in the LP schedule. Thus, for any feasible solution M to (R) and any such canonical set ' we have r min (S i where the last equation follows from Lemma 2.4. Combining this with (5), we derive a lower bound on feasible solution M to (R), and this lower bound is attained by the LP schedule. >From Theorems 2.2 and 2.5, we derive that the values of the two relaxations (D) and (R) are equal. Corollary 2.6. The LP relaxations (D) and (R) of yield the same optimal objective function value, i. e., weights w 0. This value can be computed in O(n log n) time. Proof. For the equivalence of the lower bounds, note that the mean busy time M LP j of any job j in the LP schedule can be expressed as y LP where y LP is the solution to (D) derived from the LP schedule. The result then follows directly from Theorems 2.2 and 2.5. We have shown earlier that the LP schedule can be constructed in O(n log n) time. Although the LP schedule does not necessarily minimize over the preemptive sched- ules, Theorem 2.5 implies that it minimizes over the preemptive schedules. In addition, by Lemma 2.3, the LP schedule is also optimal for both preemptive and non-preemptive problems it does not preempt any job. For example, this is the case if all processing times are equal to 1 or if all jobs are released at the same date. Thus, the LP schedule provides an optimal solution to problems 1j r and to This was already known. In the latter case it coincides with Smith's ratio rule [35]; see Queyranne and Schulz [24] for the former case. 2.3 Polyhedral Consequences We now consider some polyhedral consequences of the preceding results. Let P 1 D be the feasible region dened by the constraints of relaxation (D) when In addition, we denote by PR := fM 2 R the polyhedron dened by the constraints of relaxation (R). Theorem 2.7. (i) Polyhedron PR is the convex hull of the mean busy time vectors M of all preemptive sched- ules. Moreover, every vertex of PR is the mean busy time vector of an LP schedule. (ii) Polyhedron PR is also the image of P 1 D in the space of the M-variables under the linear mapping dened by for all j 2 N: Proof. (i) Lemma 2.4 implies that the convex hull of the mean busy time vectors M of all feasible preemptive schedules is contained in PR . To show the reverse inclusion, it su-ces to show that (a) every extreme point of PR corresponds to a preemptive schedule; and (b) every extreme ray of PR is a direction of recession for the convex hull of mean busy time vectors. Property (a) and the second part of statement (i) follow from Theorem 2.5 and the fact that every extreme point of PR is the unique minimizer of note that the extreme rays of PR are the n unit vectors of R N . An immediate extension to preemptive schedules and mean busy times of results in Balas [3] implies that these unit vectors of R N are directions of recession for the convex hull of mean busy time vectors. This completes the proof of (i). (ii) We rst show that the image M(P 1 D is contained in PR . For this, let y be a vector in P 1 D and S N with canonical decomposition g. By denition of M(y) j , we have r min (S ' The inequality follows from the constraints dening P 1 and the interchange argument which we already used in the proof of Theorem 2.2. This shows M(y) 2 PR and thus M(P 1 To show the reverse inclusion, we use the observation from the proof of part (i) that PR can be represented as the sum of the convex hull of the mean busy time vectors of all LP schedules and the nonnegative orthant. Since, by equation (6), the mean busy time vector M LP of any LP schedule is the projection of the corresponding 0=1-vector y LP , it remains to show that every unit vector e j is a direction of recession for M(P 1 D ). For this, x an LP schedule and let y LP and denote the associated 0=1 y-vector and mean busy time vector, respectively. For any job j 2 N and any real > 0, we need to show that M LP Let max := argmaxfy LP Ng. Choose such that y LP choose an integer k otherwise. In the associated preemptive schedule, the processing of job j that was done in interval [; +1) is now postponed, by time units, until interval [ +; ++1). Therefore, its mean busy time vector k for all k 6= j. Let 0 := =p j , so . Then the vector M LP + e j is a convex combination of M and y be the corresponding convex combination of y LP and y 0 . Since P 1 D is convex then y D and, since the mapping M is linear, M LP In view of earlier results for single machine scheduling with identical release dates [22], as well as for parallel machine scheduling with unit processing times and integer release dates [24], it is interesting to note that the feasible set PR of the mean busy time relaxation is, up to scaling by the job processing times, a supermodular polyhedron: Proposition 2.8. The set function h dened in (4) is supermodular. Proof. Consider any two elements any subset S N n fj; kg. We may construct an LP schedule minimizing using the job-based method by considering rst the jobs in S and then job k. (Note that considering the jobs in any sequence leads to a schedule minimizing because jobs are weighted by their processing times in this objective function). By denition (4) the resulting mean busy times M LP satisfy and Note that job k is scheduled, no earlier than its release date, in the rst p k units of idle time left after the insertion of all jobs in S. Thus M LP k is the mean of all these p k time units. Similarly, we may construct an LP schedule, whose mean busy time vector will be denoted by f considering rst the jobs in S, so f f job k, so f has been inserted after subset S was scheduled, job k cannot use any idle time interval that is earlier than those it used in the former schedule M LP |and some of the previously available idle time may now be occupied by job j, causing a delay in the mean busy time of job k; thus we have f k and therefore f This su-ces to establish that h is supermodular. An alternate proof of the supermodularity of h can be derived, as in [10], from the fact, observed by Dyer and Wolsey and already mentioned above, that relaxation (D) becomes a transportation problem after elimination of the C j 's. Indeed, from an interpretation of Nemhauser, Wolsey and Fisher [20] of a result of Shapley [31], it then follows that the value of this transportation problem as a function of S is supermodular. One of the consequences of Proposition 2.8 is that the job-based method to construct an LP schedule is just a manifestation of the greedy algorithm for minimizing over the supermodular polyhedron PR . We nally note that the separation problem for the polyhedron PR can be solved combina- torially. One can separate over the family of inequalities by trying all possible values for r min (S) (of which there are at most n) and then applying a O(n log n) separation routine of Queyranne [22] for the problem without release dates. The overall separation routine can be implemented in O(n 2 ) time by observing that the bottleneck step in Queyranne's algorithm | sorting the mean busy times of the jobs | needs to be done only once for the whole job set. Provably Good Schedules and LP Relaxations In this section, we derive approximation algorithms for 1j r j j that are based on converting the preemptive LP schedule into a feasible non-preemptive schedule whose value can be bounded in terms of the optimal LP value This yields results on the quality of both the computed schedule and the LP relaxations under consideration since the value of the computed schedule is an upper bound and the optimal LP value is a lower bound on the value of an optimal schedule. In Section 3.6 below, we describe a family of instances for which the ratio between the optimal value of the 1j r problem and the lower bounds ZR and ZD is arbitrarily close to e 1:5819. This lower bound of e e 1 sets a target for the design of approximation algorithms based on these LP relaxations. In order to convert the preemptive LP schedule into a non-preemptive schedule we make use of so-called -points of jobs. For 0 < 1 the -point t j () of job j is the rst point in time when an -fraction of job j has been completed in the LP schedule, i. e., when j has been processed for p j time units. In particular, t j (1) is equal to the completion time and we dene to be the start time of job j. Notice that, by denition, the mean busy time M LP j of job j in the LP schedule is the average over all its -points We will also use the following notation: For a xed job j and 0 < 1 we denote the fraction of job k that is completed in the LP schedule by time t j () by k (); in particular, j The amount of idle time that occurs between time 0 and the start of job j in the LP schedule is denoted by idle . Note that k and idle implicitly depend on the xed job j. By construction, there is no idle time between the start and completion of job j in the LP schedule; therefore we can express j's -point as For a given 0 < 1, we dene the -schedule as the schedule in which jobs are processed non-preemptively as early as possible and in the order of non-decreasing -points. We denote the completion time of job j in this schedule by C . The idea of scheduling non-preemptively in the order of -points in a preemptive schedule was introduced by Phillips, Stein and Wein [21], and used in many of the subsequent results in the area. This idea can be further extended to individual, i. e., job-dependent j -points t 1. We denote the vector consisting of all j 's by := Then, the ( j )-schedule is constructed by processing the jobs as early as possible and in non-decreasing order of their j -points; the completion time of job j in the ( j )-schedule is denoted by C . Figure compares an -schedule to an ( j )-schedule both derived from the LP schedule in Figure 1. In the sequel we present several results on the quality of -schedules and ( j )-schedules. These results also imply bounds on the quality of the LP relaxations of the previous section. The main result is the construction of a random ( j )-schedule whose expected value is at most a factor 1:6853 of the optimal LP value Therefore the LP relaxations (D) and (R) deliver a lower bound which is at least 0:5933 ( 1:6853 1 ) times the optimal value. The corresponding randomized algorithm can be implemented on-line; it has competitive ratio 1:6853 and running time O(n log n); it can also be derandomized to run o-line in O(n 2 ) time. We also investigate the case of a single common and show that the best -schedule is always within a factor of 1:7451 of the optimum. Figure 2: A non-preemptive -schedule (for and an ( j )-schedule shown above and below the LP schedule, respectively. Notice that there is no common value that would lead to the latter schedule. 3.1 Bounding the completion times in ( j )-schedules To analyze the completion times of jobs in ( j )-schedules, we consider non-preemptive schedules of similar structure that are, however, constructed by a slightly dierent conversion routine which we call ( j )-Conversion: Consider the jobs j 2 N in order of non-increasing j -points t iteratively change the preemptive LP schedule to a non-preemptive schedule by applying the following steps: i) remove the j p j units of job j that are processed before t j leave the machine idle during the corresponding time intervals; we say that this idle time is caused by job j; ii) delay the whole processing that is done later than t iii) remove the remaining (1 j )-fraction of job j from the machine and shrink the corresponding time intervals; shrinking a time interval means to discard the interval and move earlier, by the corresponding amount, any processing that occurs later; iv) process job j in the released time interval Figure 3 contains an example illustrating the action of ( j )-Conversion starting from the LP schedule of Figure 1. Observe that in the resulting schedule jobs are processed in non-decreasing order of j -points and no job j is started before time t . The latter property will be useful in the analysis of on-line ( j )-schedules. Figure 3: Illustration of the individual iterations of ( j )-Conversion. Lemma 3.1. The completion time of job j in the schedule constructed by equal to Proof. Consider the schedule constructed by )-Conversion. The completion time of job j is equal to the idle time before its start plus the sum of processing times of jobs that start no later than j. Since the jobs are processed in non-decreasing order of their j -points, the amount of processing before the completion of job j is The idle time before the start of job j can be written as the sum of the idle time idle that already existed in the LP schedule before j's start plus the idle time before the start of job j that is caused in steps i) of ( j )-Conversion; notice that steps iii) do not create any additional idle time since we shrink the aected time intervals. Each job k that is started no later than j, i. e., such that k units of idle time, all other jobs k only contribute units of idle time. As a result, the total idle time before the start of job j can be written as The completion time of job j in the schedule constructed by is equal to the sum of the expressions in (9) and (10); the result then follows from equation (8). It follows from Lemma 3.1 that the completion time C j of each job j in the non-preemptive schedule constructed by hence is a feasible schedule. Since the ( j )-schedule processes the jobs as early as possible and in the same order as the ( j )-Conversion schedule, we obtain the following corollary. Corollary 3.2. The completion time of job j in an ( j )-schedule can be bounded by 3.2 Bounds for -schedules and ( j )-schedules We start with a result on the quality of the -schedule for a xed common value of . Theorem 3.3. For xed , (i) the value of the -schedule is within a factor max of the optimal LP value; in particular, for 2 the bound is 1 2. Simultaneously, (ii) the length of the -schedule is within a factor of 1 + of the optimal makespan. Proof. While the proof of (ii) is an immediate consequence of (8) and Corollary 3.2, it follows from the proof of Theorem 2.5 that for (i) it is su-cient to prove that, for any canonical set S, we have Indeed, using (5) and Lemma 2.4 it would then follow that proving the result. Consider now any canonical set S and let us assume that, after renumbering the jobs, the ordering is not necessarily anymore in non-increasing order of w j =p j ). Fix now any job j 2 S. From Corollary 3.2, we derive that where k := k () represents the fraction of job k processed in the LP schedule before t j (). Let R denote the set of jobs k such that t k () < r min (S) (and thus k ). Since S is a canonical set, the jobs in S are processed continuously in the LP schedule between r min (S) and therefore every job k with k is either in S or in R. Observe that implies that p(R) 1 r min (S). We can thus simplify (12) Since the jobs in S are scheduled with no gaps in [r min (S); r min (S) + p(S)], we have that Combining (13) and (14), we derive that Multiplying by p j and summing over S, we get: which implies (11). In the sequel we will compare the completion time C j of every job j with its \completion time" in the LP schedule. However, for any xed common value of , there exist instances which show that this type of job-by-job analysis can give a bound no better than 1+ One can also show that, for any given value of , there exist instances for which the objective function value of the -schedule can be as bad as twice the LP lower bound. In view of these results, it is advantageous to use several values of as it appears that no instance can be simultaneously bad for all choices of . In fact, the -points develop their full power in combination with randomization, i. e., when a common or even job-dependent j 's are chosen randomly from (0; 1] according to an appropriate density function. This is also motivated by equation (7) which relates the expected -point of a job under a uniform distribution of to the LP variable M LP . For random values j , we analyze the expected value of the resulting )-schedule and compare it to the optimal LP value. Notice that a bound on the expected value proves the existence of a vector ( j ) such that the corresponding ( j )-schedule meets this bound. Moreover, for our results we can always compute such an ( j ) in polynomial time by derandomizing our algorithms with standard methods; see Propositions 3.8 and 3.13. Although the currently best known bounds can only be achieved for ( j )-schedules with job-dependent j 's, we investigate -schedules with a single common as well. On the one hand, this helps to better understand the potential advantages of ( j )-schedules; on the other hand, the randomized algorithm that relies on a single admits a natural derandomization. In fact, we can easily compute an -schedule of least objective function value over all between 0 and 1; we refer to this schedule as the best-schedule. In Proposition 3.8 below, we will show that there are at most n dierent -schedules. The best-schedule can be constructed in O(n 2 ) time by evaluating all these dierent schedules. As a warm-up exercise for the kind of analysis we use, we start by proving a bound of 2 on the expected worst-case performance ratio of uniformly generated ( j )-schedules in the following theorem. This result will then be improved by using more intricate probability distributions and by taking advantage of additional insights into the structure of the LP schedule. Theorem 3.4. Let the random variables j be pairwise independently and uniformly drawn from (0; 1]. Then, the expected value of the resulting ( j )-schedule is within a factor 2 of the optimal LP value Proof. Remember that the optimal LP value is given by To get the claimed result, we prove that EU [C denotes the expectation of a function F of the random variable when the latter is uniformly distributed. The overall performance follows from this job-by-job bound by linearity of expectations. Consider an arbitrary, but xed job j 2 N . To analyze the expected completion time of j, we rst keep j xed, and consider the conditional expectation EU [C the random variables j and k are independent for each k 6= j, Corollary 3.2 and equation (8) yield EU [C To obtain the unconditional expectation EU [C we integrate over all possible choices of EU [C Z 1EU [C the last equation follows from (7). We turn now to deriving improved results. We start with an analysis of the structure of the LP schedule. Consider any job j, and assume that, in the LP schedule, j is preempted at time s and its processing resumes at time t > s. Then all the jobs which are processed between s and t have a smaller index; as a result, these jobs will be completely processed between times s and t. Thus, in the LP schedule, between the start time and the completion time of any job j, the machine is constantly busy, alternating between the processing of portions of j and the complete processing of groups of jobs with smaller index. Conversely, any job preempted at the will have to wait at least until job j is complete before its processing can be resumed. We capture this structure by partitioning, for a xed job j, the set of jobs N n fjg into two subsets N 1 and denote the set of all jobs that are processed between the start and completion of job j. All remaining jobs are put into subset N 1 . Notice that the function k is constant for jobs k 2 N 1 ; to simplify notation we write k := k ( j ) for those jobs. For k 2 N 2 , the fraction of job j that is processed before the start of job k; the function k is then given by We can now rewrite equation (8) as Plugging (15) into equation (7) yields and Corollary 3.2 can be rewritten as where, for k 2 N 2 , we have used the fact that k k ( j ) is equivalent to j > k . The expressions (15), (16), and (17) re ect the structural insights that we need for proving stronger bounds for ( j )-schedules and -schedules in the sequel. As mentioned above, the second ingredient for an improvement on the bound of 2 is a more sophisticated probability distribution of the random variables j . In view of the bound on C given in (17), we have to cope with two contrary phenomena: On the one hand, small values of k keep the terms of the form (1 on the right-hand side of (17) small; on the other hand, choosing larger values decreases the number of terms in the rst sum on the right-hand side of (17). The balancing of these two eects contributes to reducing the bound on the expected value of C . 3.3 Improved bounds for -schedules In this subsection we prove the following theorem. Theorem 3.5. Let 0:4675 be the unique solution to the equation satisfying 0 < < 1. Dene c := 1+ e < 1:7451 and - := 1 1+ 0:8511. If is chosen according to the density function then the expected value of the resulting random -schedule is bounded by c times the optimal LP value Before we prove Theorem 3.5 we state two properties of the density function f that are crucial for the analysis of the corresponding random -schedule. Lemma 3.6. The function f given in Theorem 3.5 is a density function with the following properties: Property (i) is used to bound the delay to job j caused by jobs in N 1 which corresponds to the rst summation on the right-hand side of (17). The second summation re ects the delay to caused by jobs in N 2 and will be bounded by property (ii). Proof of Lemma 3.6. A short computation shows that . The function f is a density function since Z -e In order to prove property (i), observe that for 2 [0; -] For 2 (-; 1] we therefore get Property (ii) holds for 2 (-; 1] since the left-hand side is 0 in this case. For 2 [0; -] we have e This completes the proof of the lemma. Proof of Theorem 3.5. In Lemma 3.6, both (i) for denotes the expected value of a random variable that is distributed according to the density f given in Theorem 3.5. Thus, using inequality (17) and Lemma 3.6 we derive that C the last inequality follows from the denition of N 1 and k and the last equality follows from (16). Notice that any density function satisfying properties (i) and (ii) of Lemma 3.6 for some value c 0 directly leads to the job-by-job bound E f [C for the corresponding random -schedule. It is easy to see that the unit function satises Lemma 3.6 with c which establishes the following variant of Theorem 3.4. Corollary 3.7. Let the random variable be uniformly drawn from (0; 1]. Then, the expected value of the resulting -schedule is within a factor 2 of the optimal LP value The use of an exponential density function is motivated by the rst property in Lemma 3.6; notice that the function 7! (c 1)e satises it with equality. On the other hand, the exponential function is truncated in order to reduce the term in the second property. In fact, the truncated exponential function f in Theorem 3.5 can be shown to minimize c 0 ; it is therefore optimal for our analysis. In addition, there exists a class of instances for which the ratio of the expected cost of an -schedule, determined using this density function, to the cost of the optimal LP value is arbitrarily close to 1:745; this shows that the preceding analysis is essentially tight in conjunction with truncated exponential functions. Theorem 3.5 implies that the best-schedule has a value of at most 1:7451 ZR . The following proposition shows that the randomized algorithm that yields the -schedule can be easily derandomized because the sample space is small. Proposition 3.8. There are at most n dierent -schedules; they can be computed in O(n 2 ) time. Proof. As goes from 0 + to 1, the -schedule changes only whenever an -point, say for reaches a time at which job j is preempted. Thus, the total number of changes in the -schedule is bounded from above by the total number of preemptions. Since a preemption can occur in the LP schedule only whenever a job is released, the total number of preemptions is at most n 1, and the number of -schedules is at most n. Since each of these -schedules can be computed in O(n) time, the result on the running time follows. 3.4 Improved bounds for ( j )-schedules In this subsection, we prove the following theorem. Theorem 3.9. Let 0:4835 be the unique solution to the equation )e satisfying 0 < < 1. Dene - := =- < 1:6853. Let the 's be chosen pairwise independently from a probability distribution over (0; 1] with the density function Then, the expected value of the resulting random ( j )-schedule is bounded by c times the optimal LP value The bound in Theorem 3.9 yields also a bound on the quality of the LP relaxations: Corollary 3.10. The LP relaxations (D) and (R) deliver in O(n log n) time a lower bound which is at least 0:5933 ( 1:6853 1 ) times the objective function value of an optimal schedule. Following the lines of the last subsection, we state two properties of the density function g that are crucial for the analysis of the corresponding random ( j )-schedule. Lemma 3.11. The function g given in Theorem 3.9 is a density function with the following properties: denotes the expected value of a random variable that is distributed according to g. Notice the similarity of Lemma 3.11 and Lemma 3.6 of the last subsection. Again, properties (i) and (ii) are used to bound the delay to job j caused by jobs in N 1 and N 2 , respectively, in the right-hand side of inequality (17). Property (i) for yield E g [] c 1. Proof of Lemma 3.11. A short computation shows that . It thus follows from the same arguments as in the proof of Lemma 3.6 that g is a density function and that property (i) holds. In order to prove property (ii), we rst compute Z -e Property (ii) certainly holds for 2 (-; 1]. For 2 [0; -] we get e )e e e This completes the proof of the lemma. Proof of Theorem 3.9. Our analysis of the expected completion time of job j in the random )-schedule follows the line of argument developed in the proof of Theorem 3.4. First we consider a xed choice of j and bound the corresponding conditional expectation E g [C In a second step we bound the unconditional expectation E g [C by integrating the product over the interval (0; 1]. For a xed job j and a xed value j , the bound in (17) and Lemma 3.11 (i) yield The last inequality follows from (15) and . Using property (ii) and equation (16) yields The result follows from linearity of expectations. While the total number of possible orderings of jobs is log n) , we show in the following lemma that the maximum number of ( j )-schedules is at most 2 n 1 . We will use the following observation. Let q j denote the number of dierent pieces of job j in the LP schedule; thus q j represents the number of times job j is preempted plus 1. Since there are at most n 1 preemptions, we have that Lemma 3.12. The maximum number of ( j )-schedules is at most 2 n 1 and this bound can be attained. Proof. The number of ( j )-schedules is given by . Note that q is not preempted in the LP schedule. Thus, 1). By the mean inequality, we have that Y Furthermore, this bound is attained if q and this is achieved for example for the instance with Therefore, and in contrast to the case of random -schedules, we cannot aord to derandomize the randomized 1:6853{approximation algorithm by enumerating all ( j )-schedules. We instead use the method of conditional probabilities [18]. >From inequality (17) we obtain for every vector an upper bound on the objective function value of the corresponding ( j )-schedule, denotes the right-hand side of inequality (17). Taking expectations and using Theorem 3.9, we have already shown that where c < 1:6853. For each job j 2 N let g denote the set of the intervals for j corresponding to the q j pieces of job j in the LP schedule. We consider the jobs one by one in arbitrary order, say, Assume that, at step j of the derandomized algorithm, we have identied intervals Q d Using conditional expectations, the left-hand side of this inequality is Since there exists at least one interval Q j' 2 Q j such that Therefore, it su-ces to identify such an interval Q d satisfying (18) and we may conclude that Having determined in this way an interval Q d j for every job note that the )-schedule is the same for all 2 Q d Q d Q d n . The (now deterministic) objective function value of this ( j )-schedule is as desired. For every checking whether an interval Q d amounts to evaluating O(n) terms, each of which may be computed in constant time. Since, as observed just before Lemma 3.12, we have a total of follows that the derandomized algorithm runs in O(n 2 ) time. Proposition 3.13. The randomized 1:6853{approximation algorithm can be derandomized; the resulting deterministic algorithm runs in O(n 2 ) time and has performance guarantee 1:6853 as well. Constructing provably good schedules on-line In this subsection we show that our randomized approximation results also apply in an on-line setting. There are several dierent on-line paradigms that have been studied in the area of scheduling; we refer to [30] for a survey. We consider the setting where jobs continually arrive over time and, for each time t, we must construct the schedule until time t without any knowledge of the jobs that will arrive afterwards. In particular, the characteristics of a job, i. e., its processing time and its weight become only known at its release date. It has already been shown in Section 2 that the LP schedule can be constructed on-line. Unfortunately, for a given vector ( j ), the corresponding ( j )-schedule cannot be constructed on-line. We only learn about the position of a job k in the sequence dened by non-decreasing -points at time t k ( k ); therefore we cannot start job k at an earlier point in time in the on-line setting. On the other hand, however, the start time of k in the ( j )-schedule can be earlier than its k -point t k ( k ). Although an ( j )-schedule cannot be constructed on-line, the above discussion reveals that the following variant, which we call on-line-( j )-schedule, can be constructed on-line: For a given vector ( j ), process the jobs as early as possible in the order of their j -points, with the additional constraint that no job k may start before time t k ( k ). See Figure 4 for an example. We note that this idea of delaying the start of jobs until su-cient information for a good decision is available was in this setting introduced by Phillips, Stein and Wein [21]. Notice that the non-preemptive schedule constructed by straints; its value is therefore an upper bound on the value of the on-line-( j )-schedule. Our analysis in the last subsections relies on the bound given in Corollary 3.2 which also holds for the schedule constructed by 3.1. This yields the following results. Theorem 3.14. For any instance of the scheduling problem 1j r j j a) choosing 2 and constructing the on-line-schedule yields a deterministic on-line algorithm with competitive ratio 1 2:4143 and running time O(n log n); b) choosing the j 's randomly and pairwise independently from (0; 1] according to the density function g of Theorem 3.9 and constructing the on-line-( j )-schedule yields a randomized on-line algorithm with competitive ratio 1:6853 and running time O(n log n). The competitive ratio 1:6853 in Theorem 3.14 beats the deterministic on-line lower bound 2 for the unit-weight problem 1j r 36]. For the same problem, Stougie and Vestjens proved the lower bound e randomized on-line algorithms [37, 39]. Figure 4: The on-line schedule for the previously considered instance and j -points. The LP schedule is shown above for comparison. 3.6 Bad instances for the LP relaxations In this subsection, we describe a family of instances for which the ratio between the optimal value of the 1j r j j problem and the lower bounds ZR and ZD is arbitrarily close to e These instances I n have n 2 jobs as follows: one large job, denoted job n, and small jobs denoted 1. The large job has processing time and release date r Each of the n 1 small jobs j has zero processing time, weight release date r Throughout the paper, we have assumed that processing times are non-zero. In order to satisfy this assumption, we could impose a processing time of 1=k for all small jobs, multiply all processing times and release dates by k to make the data integral, and then let k tend to innity. For simplicity, however, we just let the processing time of all small jobs be 0. The LP solution has job n start at time 0, preempted by each of the small jobs; hence its mean busy times are: M LP Its objective function value is Notice that the completion time of each job j is in fact equal to M LP such that the actual value of the preemptive schedule is equal to ZR . Now consider an optimal non-preemptive schedule C and let c n 0, so k is the number of small jobs that can be processed before job n. It is then optimal to process all these small jobs at their release dates, and to start processing job n at date r just after job k. It is also optimal to process all remaining jobs k just after job n. Let C k denote the resulting schedule, that is, C k otherwise. Its objective function value is n(n 1) . Therefore the optimal schedule is C n 1 with objective function value . As n grows large, the LP objective function value approaches e 1 while the optimal non-preemptive cost approaches e. Even though polynomial-time approximation schemes have now been discovered for the problem [1], the algorithms we have developed, or variants of them, are likely to be superior in practice. The experimental studies of Savelsbergh et al. [25] and Uma and Wein [38] indicate that LP-based relaxations and scheduling in order of j -points are powerful tools for a variety of scheduling problems. Several intriguing questions remain open. Regarding the quality of linear programming relaxations, it would be interesting to close the gap between the upper (1:6853) and lower (1:5819) bound on the quality of the relaxations considered in this paper. We should point out that the situation for the strongly NP-hard [16] problem 1j r is similar. It is shown in [29] that the completion time relaxation is in the worst case at least a factor of 8=7 and at most a factor of 4=3 o the optimum; the latter bound is achieved by scheduling preemptively by LP-based random -points. Chekuri et al. [6] prove that the optimal non-preemptive value is at most e=(e 1) times the optimal preemptive value; our example in Section 3.6 shows that this bound is tight. Dyer and Wolsey [9] propose also a (non-preemptive) time-indexed relaxation which is stronger than the preemptive version studied here. This relaxation involves variables for each job and each time representing whether this job is being completed (rather than simply processed) at that time. This relaxation is at least as strong as the preemptive version, but its worst-case ratio is not known to be strictly better. For randomized on-line algorithms, there is also a gap between the known upper and lower bound on the competitive ratios that are given at the end of Section 3.5. For deterministic on-line algorithms, the 2{competitve algorithm of Anderson and Potts [2] is optimal. Acknowledgements The research of the rst author was performed partly when the author was at C.O.R.E., Louvain- la-Neuve, Belgium and supported in part by NSF contract 9623859-CCR. The research of the second author was supported in part by a research grant from NSERC (the Natural Sciences and Research Council of Canada) and by the UNI.TU.RIM. S.p.a. (Societa per l'Universita nel riminese), whose support is gratefully acknowledged. The research of the third author was performed partly when he was with the Department of Mathematics, Technische Universitat Berlin, Germany. The fourth author was supported in part by DONET within the frame of the TMR Programme (contract number ERB FMRX-CT98-0202) while staying at C.O.R.E., Louvain-la-Neuve, Belgium, for the academic year 1998/99. The fth author was supported by a research fellowship from Max-Planck Institute for Computer Science, Saarbrucken, Germany. We are also grateful to an anonymous referee whose comments helped to improve the presentation of this paper. --R Introduction to Algorithms Rinnooy Kan Rinnooy Kan Rinnooy Kan and P. Randomized Algorithms ", Mathematical Programming, 82, 199-223 (1998). An extended abstract appeared under the title \Scheduling jobs that arrive over time" " in R. Burkard and cited as personal communication in --TR --CTR Jairo R. Montoya-Torres, Competitive Analysis of a Better On-line Algorithm to Minimize Total Completion Time on a Single-machine, Journal of Global Optimization, v.27 n.1, p.97-103, September Leah Epstein , Rob van Stee, Lower bounds for on-line single-machine scheduling, Theoretical Computer Science, v.299 n.1-3, p.439-450, R. N. Uma , Joel Wein , David P. Williamson, On the relationship between combinatorial and LP-based lower bounds for NP-hard scheduling problems, Theoretical Computer Science, v.361 n.2, p.241-256, 1 September 2006 Martin W. P. Savelsbergh , R. N. Uma , Joel Wein, An Experimental Study of LP-Based Approximation Algorithms for Scheduling Problems, INFORMS Journal on Computing, v.17 n.1, p.123-136, Winter 2005 Martin Skutella, Convex quadratic and semidefinite programming relaxations in scheduling, Journal of the ACM (JACM), v.48 n.2, p.206-242, March 2001 F. Afrati , I. Milis, Designing PTASs for MIN-SUM scheduling problems, Discrete Applied Mathematics, v.154 n.4, p.622-639, 15 March 2006 Rolf H. Mhring , Andreas S. Schulz , Frederik Stork , Marc Uetz, Solving Project Scheduling Problems by Minimum Cut Computations, Management Science, v.49 n.3, p.330-350, March

on-line algorithm;LP relaxation;approximation algorithm;scheduling

588006

Binary Clutters, Connectivity, and a Conjecture of Seymour.

A binary clutter is the family of odd circuits of a binary matroid, that is, the family of circuits that intersect with odd cardinality a fixed given subset of elements. Let A denote the 0,1 matrix whose rows are the characteristic vectors of the odd circuits. A binary clutter is ideal if the polyhedron $\{ x \geq {\bf 0}: \; Ax \geq {\bf 1} \}$ is integral. Examples of ideal binary clutters are st-paths, st-cuts, T-joins or T-cuts in graphs, and odd circuits in weakly bipartite graphs. In 1977, Seymour [J. Combin. Theory Ser. B, 22 (1977), pp. 289--295] conjectured that a binary clutter is ideal if and only if it does not contain ${\cal{L}}_{F_7}$, ${\cal{O}}_{K_5}$, or $b({\cal{O}}_{K_5})$ as a minor. In this paper, we show that a binary clutter is ideal if it does not contain five specified minors, namely the three above minors plus two others. This generalizes Guenin's characterization of weakly bipartite graphs [J. Combin. Theory Ser., 83 (2001), pp. 112--168], as well as the theorem of Edmonds and Johnson [ Math. Programming, 5 (1973), pp. 88--124] on T-joins and T-cuts.

INTRODUCTION A clutter H is a finite family of sets, over some finite ground set E(H), with the property that no set of H contains, or is equal to, another set of H. A clutter is said to be ideal if the polyhedron fx 2 R jE(H)j is an integral polyhedron, that is, all its extreme points have 0; 1 coordinates. A clutter H is trivial if f;g. Given a nontrivial clutter H, we write A(H) for a 0,1 matrix whose columns are indexed by E(H) and whose rows are the characteristic vectors of the sets S 2 H. With this notation, a nontrivial clutter H is ideal if and only if fx is an integral polyhedron. Given a clutter H, a set T E(H) is a transversal of H if T intersects all the members of H. The clutter b(H), called the blocker of H, is defined as follows: E b(H) E(H) and b(H) is the set of inclusion-wise minimal transversals of H. It is well known that b b(H) Hence we say that H; b(H) form a blocking pair of clutters. Lehman [14] showed that, if a clutter is ideal, then so is its blocker. A clutter is said to be binary if, for any S 1 contains, or is equal to, a set of H. Given a clutter H and i 2 E(H), the contraction H=i and deletion H n i are clutters defined as follows: fig, the family H=i is the set of inclusion-wise minimal members of Hg. Contractions and deletions can be performed sequentially, and the result does not depend on the order. A clutter obtained from H by a set of deletions J d and a set of contractions J c , (where J c \J called a minor of H and is denoted by HnJ d =J c . It is a proper minor if J c [ J d 6= ;. A clutter is said to be minimally nonideal (mni) if it is not ideal but all its proper minors are ideal. Date: March 2000, revised December 2001. Key words and phrases. Ideal clutter, signed matroid, multicommodity flow, weakly bipartite graph, T -cut, Seymour's conjecture. Classification: 90C10, 90C27, 52B40. This work supported in part by NSF grants DMI-0098427, DMI-9802773, DMS-9509581, ONR grant N00014-9710196, and DMS 96-32032. EJOLS AND BERTRAND GUENIN The clutter OK5 is defined as follows: E(OK5 ) is the set of 10 edges of the complete graph K 5 and OK5 is the set of odd circuits of K 5 (the triangles and the circuits of length 5). The 10 constraints corresponding to the triangles define a fractional extreme point 3 ) of the associated polyhedron fx 1g. Thus OK5 is not ideal and neither is its blocker. The clutter LF7 is the family of circuits of length three of the Fano matroid (or, equivalently, the family of lines of the Fano plane), i.e. E(LF7 and The fractional point 3 ) is an extreme point of the associated polyhedron, hence LF7 is not ideal. The blocker of LF7 is LF7 itself. The following excluded minor characterization is predicted. Seymour's Conjecture [Seymour [23] p. 200, [26] (9.2), (11.2)] A binary clutter is ideal if and only if it has no LF7 , no OK5 , and no b(OK5 ) minor. Consider a clutter H and an arbitrary element t 62 E(H). We for the clutter with E(H Hg. The clutter Q 6 is defined as follows: E(Q 6 ) is the set of edges of the complete graph K 4 and Q 6 is the set of triangles of K 4 . The clutter Q 7 is defined as follows: Note that the first six columns of A(Q 7 ) form the matrix A b(Q 6 ) The main result of this paper is that Seymour's Conjecture holds for the class of clutters that do not have 6 and Q 7 minors. Theorem 1.1. A binary clutter is ideal if it does not have LF7 , OK5 , b(OK5 6 or Q 7 as a minor. Since the blocker of an ideal binary clutter is also a ideal, we can restate Theorem 1.1 as follows. Corollary 1.2. A binary clutter is ideal if it does not have LF7 , OK5 , b(OK 5 6 ) as a minor. We say that H is the clutter of odd circuits of a graph G if E(H) is the set of edges of G and H the set of odd circuits of G. A graph is said to be weakly bipartite if the clutter of its odd circuits is ideal. This class of graphs has a nice excluded minor characterization. Theorem 1.3 (Guenin [10]). A graph is weakly bipartite if and only if its clutter of odd circuits has no OK5 minor. The class of clutters of odd circuits is closed under minor taking (Remark 8.2). Moreover, one can easily check that OK5 is the only clutter of odd circuits among the five excluded minors of Theorem 1.1 (see Remark 8.3 and [20]). It follows that Theorem 1.1 implies Theorem 1.3. It does not provide a new proof of Theorem 1.3 however, as we shall use Theorem 1.3 to prove Theorem 1.1. Consider a graph G and a subset T of its vertices of even cardinality. A T -join is an inclusion-wise minimal set of edges J such that T is the set of vertices of odd degree of the edge-induced subgraph G[J ]. IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 3 A T -cut is an inclusion-wise minimal set of edges -(U ) := U is a set of vertices of G that satisfies jU \ T j odd. T -joins and T -cuts generalize many interesting special cases. If then the T -joins (resp. T -cuts) are the st-paths (resp. inclusion-wise minimal st-cuts) of G. If then the T -joins of size jV j=2 are the perfect matchings of G. The case where T is identical to the set of odd-degree vertices of G is known as the Chinese postman problem [6, 12]. The families of T -joins and T -cuts form a blocking pair of clutters. Theorem 1.4 (Edmonds and Johnson [6]). The clutters of T -cuts and T -joins are ideal. The class of clutters of T -cuts is closed under minor taking (Remark 8.2). Moreover, it is not hard to check that none of the five excluded minors of Theorem 1.1 are clutters of T -cuts (see Remark 8.3 and [20]). Thus Theorem 1.1 implies that the clutter of T -cuts is ideal, and thus that its blocker, the clutter of T -joins, is ideal. Hence Theorem 1.1 implies Theorem 1.4. However, we shall also rely on this result to prove Theorem 1.1. The paper is organized as follows. Section 2 considers representations of binary clutters in terms of signed matroids and matroid ports. Section 3 reviews the notions of lifts and sources, which are families of binary clutters associated to a given binary matroid [20, 29]. Connections between multicommodity flows and ideal clutters are discussed in Section 4. The material presented in Sections 2, 3 and 4 is not all new. We present it here for the sake of completeness and in order to have a unified framework for the remainder of the paper. In Sections 5, 6, 7 we show that minimally nonideal clutters do not have small separations. The proof of Theorem 1.1 is given in Section 8. Finally, Section 9 presents an intriguing example of an ideal binary clutter. 2. BINARY MATROIDS AND BINARY CLUTTERS We assume that the reader is familiar with the basics of matroid theory. For an introduction and all undefined terms, see for instance Oxley [21]. Given a matroid M , the set of its elements is denoted by E(M ) and the set of its circuits by ). The dual of M is written M . The deletion minor M n e of M is the matroid defined as follows: E(M n and )g. The contraction minor M=e of M is defined as (M n e) . Contractions and deletions can be performed sequentially, and the result does not depend on the order. A matroid obtained from M by a set of deletions J d and a set of contractions J c is a minor of M and is denoted by M n J d =J c . matroid M is binary if there exists a 0; 1 matrix A with column set E(M ) such that the independent sets of M correspond to independent sets of columns of A over the two element field. We say that A is a representation of M . Equivalently, a 0; 1 matrix A is a representation of a binary matroid M if the rows of A span the circuit space of M . If C 1 and C 2 are two cycles of a binary matroid then C 1 4 C 2 is also a cycle of M . In particular this implies that every cycle of M can be partitioned into circuits. Let M be a binary matroid and E(M ). The pair (M; ) is called a signed matroid, and is called the signature of M . We say that a circuit C of M is odd (resp. even) if jC \ j is odd (resp. even). The results in this section are fairly straightforward and have appeared explicitly or implicitly in the literature [8, 13, 20, 23]. We include some of the proofs for the sake of completeness. EJOLS AND BERTRAND GUENIN Proposition 2.1 (Lehman [13]). The followingstatements are equivalent for a clutter: (i) H is binary; (ii) for every contains, or is equal, to an element of H. Proposition 2.2. The odd circuits of a signed matroid (M; ) form a binary clutter. Proof. Let circuits of (M; ). Then L := C 1 4 C 2 4 C 3 is a cycle of M . Since each of C parity, so does L. Since M is binary, L can be partitioned into a family of circuits. One of these circuits must be odd since jL \ j is odd. The result now follows from the definition of binary clutters (see Section 1). Proposition 2.3. Let F be a clutter such that ; 62 F . Consider the following properties: (i) for all C F and e 2 C 1 \ C 2 there exists C 3 2 F such that e 62 C 3 . (ii) for all C there exists C 3 2 F such that C 3 C 1 4 C 2 . If property (i) holds then F is the set of circuits of a matroid. If property (ii) holds then F is the set of circuits of a binary matroid. Property (i) is known as the circuit elimination axiom. Circuits of matroids satisfy this property. Note that property (ii) implies property (i). Both results are standard, see Oxley [21]. Proposition 2.4. Let H be a binary clutter such that ; 62 H. Let F be the clutter consisting of all inclusion- wise minimal, non-empty sets obtained by taking the symmetric difference of an arbitrary number of sets of H. Then H F and F is the set of circuits of a binary matroid. Proof. By definition, F satisfies property (ii) in Proposition 2.3. Thus F is the set of circuits of a binary matroid M . Suppose for a contradiction there is S 2 H F . Then there exists S 0 2 F such that S 0 S. Thus S 0 is the symmetric difference of a family of, say t, sets of H. If t is odd then, Proposition 2.1 implies that S 0 contains a set of H. If t is even then, Proposition 2.1 implies that S 0 4 S contains a set of H. Thus S is not inclusion-wise minimal, a contradiction. Consider a binary clutter H such that ; 62 H. The matroid defined in Proposition 2.4 is called the up matroid and is denoted by u(H). Proposition 2.1 implies that every circuit of u(H) is either an element of H or the symmetric difference of an even number of sets of H. Since H is a binary clutter, sets of b(H) intersect with odd parity exactly the circuits of u(H) that are elements of H. Hence, Remark 2.5. A binary clutter H such that ; 62 H is the clutter of odd circuits of (u(H); ) where 2 b(H). Moreover, this representation is essentially unique. Proposition 2.6. Suppose that the clutters of odd circuits of the signed matroid (N; ) and (N are the same and are not trivial. If N and N 0 are connected then To prove this, we use the following result (see Oxley [21] Theorem 4.3.2). Theorem 2.7 (Lehman [13]). Let e be an element of a connected binary matroid M . The circuits of M not containing e are of the form C are circuits of M containing e. We shall also need the following observation which follows directly from Proposition 2.3. IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 5 Proposition 2.8. Let (M; ) be a signed matroid and e an element not in E(M ). Let F := fC [ eveng. Then F is the set of circuit of a binary matroid. Proof of Proposition 2.6. Let M (resp. M 0 ) be the matroid constructed from Proposition 2.8. By construction the circuits of M and M 0 using e are the same. Since N is connected and H is not trivial, M and M 0 are connected. It follows from Theorem 2.7 that and in particular . By the same argument and Remark 2.5, In a binary matroid, any circuit C and cocircuit D have an even intersection. So, if D is a cocircuit, the clutter of odd circuits of (M; ) and (M; 4 D) are the same (see Zaslavsky [28]). Let e 2 E(M ). The deletion (M; )ne of (M; ) is defined as (M n e; feg). The contraction (M; )=e of (M; ) is defined as follows: if e 62 then (M; )=e := (M=e; ); if e 2 and e is not a loop then there exists a cocircuit D of M with e 2 D and (M; )=e := (M=e; 4D). Note if e 2 is a loop of M , then H=e is a trivial clutter. A minor of (M; ) is any signed matroid which can be obtained by a sequence of deletions and contractions. A minor of (M; ) obtained by a sequence of J c contraction and J d deletions is denoted (M; )=J c n J d . Remark 2.9. Let H be a the clutter of odd circuits of a signed matroid (M; ). If J c does not contain an odd circuit, then H=J c n J d is the clutter of odd circuits of the signed matroid (M; )=J c n J d . Let M be a binary matroid and e an element of M . The clutter P ort(M; e), called a port of M , is defined as )g. Proposition 2.10. Let M be a binary matroid, then P ort(M; e) is a binary clutter. Proof. By definition S 2 P ort(M; e) if and only if S [ feg is an odd circuit of the signed matroid (M; feg). We may assume P ort(M; e) is nontrivial, hence in particular, e is not a loop of M . Therefore, there exists a cocircuit D that contains e. Thus P ort(M; e) is the clutter of odd circuits of the signed matroid (M=e; D 4 feg). Proposition 2.2 states that these odd circuits form a binary clutter. Proposition 2.11. Let H be a binary clutter. Then there exists a binary matroid M with element e 2 E(M ) E(H) such that P ort(M; Proof. If define M to have element e as a loop. If ; 62 H, we can represent H as the set of odd circuits of a signed matroid (N; ) (see Remark 2.5). Construct a binary matroid M from (N; ) as in Proposition 2.8. Then P Proposition 2.12 (Seymour [23]). P ort(M; e) and P ort(M ; e) form a blocking pair. Proof. Proposition 2.10 implies that P ort(M; e) and P ort(M ; e) are both binary clutters. Consider T 2 is a circuit of M . For all is a circuit of M . Since T [feg and S[feg have an even intersection jS \T j is odd. Thus we proved: for all is To complete the proof it suffices to show: for all T there is odd (Proposition 2.1). Thus T 0 [ feg intersects every circuit of M using e with even parity. It follows from Theorem 2.7 that T 0 [ feg is orthogonal to the space spanned by the circuits of M , i.e. T 0 [ feg is a cycle of M . It follows that there is a circuit of M of the form T [ feg where T T 0 . Hence, required. EJOLS AND BERTRAND GUENIN 3. LIFTS AND SOURCES Let N be a binary matroid. For any binary matroid M with element e such that M=e, the binary clutter P ort(M; e) is called a source of N . Note that H is a source of its up matroid u(H). For any binary matroid M with element e such that e, the binary clutter P ort(M; e) is called a lift of N . Note that a source or a lift can be a trivial clutter. Proposition 3.1. Let N be a binary matroid. H is a lift of N if and only if b(H) is a source of N . Proof. Let H be a lift of N , i.e. there is a binary matroid M with M n Proposition 2.12, we have that b(H) is a source of N . Moreover, the implications can be reversed. It is useful to relate a description of H in terms of excluded clutter minors to a description of u(H) in terms of excluded matroid minors. Theorem 3.2. Let H be a binary clutter such that its up matroid u(H) is connected, and let N be a connected binary matroid. Then u(H) does not have N as a minor if and only if H does not have H 1 or H 2 as a minor, a source of N and H 2 is a lift of N . To prove this we will need the following result (see Oxley [21] Proposition 4.3.6). Theorem 3.3 (Brylawski [3], Seymour [25]). Let M be a connected matroid and N a connected minor of M . For any i 2 E(M ) E(N ), at least one of M n i or M=i is connected and has N as a minor. Proof of Theorem 3.2. Let M := u(H) and let 2 b(H). Remark 2.5 states that H is the clutter of odd circuits of (M; ). Suppose first that H has a minor H 1 that is a source of N . Remark 2.9 implies that H 1 is the clutter of odd circuits of a signed minor (N 0 ; 0 ) of (M; ). Since N is connected, H 1 is nontrivial and therefore Proposition 2.6 implies In particular N is a minor of M . Suppose now that H has a is a lift of N . Let e be the element of E(H implies that H +is the clutter of odd circuits of a signed minor of (M; ). Since H 2 is a lift of N there is a connected M 0 with element e such that ^ 2 is the clutter of odd circuits of M 0 is a minor of M and so is Now we prove the converse. Suppose that M has N as a minor and does not satisfy the theorem. Let H be such a counterexample minimizing the cardinality of E(H). Clearly, N is a proper minor of M as otherwise is a source of N . By Theorem 3.3, for every i 2 E(M ) E(N ), one of M n i and M=i is connected and has N as a minor. Suppose M=i is connected and has N as a minor. Since i is not a loop of M , it follows from Remark 2.9 that H=i is nontrivial and is a signed minor (M=i; 0 ) of (M; ). Proposition 2.6 implies contradicts the choice of H minimizing the cardinality of E(H). Thus, for every connected and has an N minor. Suppose for some because of Remark 2.9 and Proposition 2.6 u(H n a contradiction to the choice of H. Thus for every or equivalently, all odd circuits of (M; ) use i. As even circuits of M do not use i. We claim that E(M ) E(N not and let j 6= i be an element of E(M ) E(N ). The set of circuits of (M; ) using j is exactly the set of odd circuits. It follows that the elements must be in series in M . But then M n i is not connected, IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 7 a contradiction. Therefore E(M . As the circuits of (M; ) using i are exactly the odd circuits of (M; ), it follows that column i of A(H) consists of all 1's. Thus is a lift of N . Next we define the binary matroids F 7 ; F 7 and R 10 . For any binary matroid N , let BN be a 0,1 matrix whose rows span the circuit space of N (equivalently BN is a representation of the dual matroid N ). Square identity matrices are denoted I . Observe that R I Given a binary matroid N , let M be a binary matroid with element e such that M=e. The circuit space of M is spanned by the rows of a matrix of the form [BN jx], where x is a 0,1 column vector indexed by e. Assuming M is connected, we have (up to isomorphism), the following possible columns x for each of the three aforementioned matroids x x x Note that (1),(2) are easy and (3) can by found in [24] (p. 357). The rows of the matrix [BF7 jx b ] (resp. span the circuit space of a matroid known as AG(3; 2) (resp. S 8 ). If [BN jx] is a matrix whose rows span the circuits of M , then by definition of sources, P ort(M; e) is a source of N . Thus, Remark 3.4. 7 has a unique source, namely Q 6 . F 7 has three sources: sources including b(OK5 ) (when Luetolf and Margot [16] have enumerated all minimally nonideal clutters with at most 10 elements (and many more). Using Remark 3.4, we can then readily check the following. Proposition 3.5. Let H be the clutter of odd circuits of a signed matroid (M; ). or H is ideal. 7 , then H is ideal. 4. MULTICOMMODITY FLOWS In this section, we show that a binary clutter H is ideal exactly when certain multicommodity flows exist in the matroid u(H). This equivalence will be used in Sections 6 and 7 to show that minimally nonideal EJOLS AND BERTRAND GUENIN binary clutters do not have small separations. Given a set S, a function i2T p(i). Consider a signed matroid (M; F ). The set of circuits of M that have exactly one element in common with F , is denoted F . Let be a cost function on the elements of M . Seymour [26] considers the following two statements about the triple (M; F; p). For any cocircuit D of M : There exists a function F We say that the cut condition holds if inequality (4.1) holds for all cocircuits D. We say that M is F -flowing with costs p if statement (4.2) holds; the corresponding solution is an F -flow satisfying costs p. M is F - flowing [26] if, for every p for which the cut condition holds, M is F -flowing with costs p. Elements in F are called demand (resp. capacity) elements. It is helpful to illustrate the aforementioned definitions in the case where M is a graphic matroid [9]. For a demand edge f , p(f) is the amount of flow required between its endpoints. For a capacity edge e, p(e) is the maximum amount of flow that can be carried by e. Then M is F -flowing with costs p when a multicommodity flow meeting all demands and satisfying all capacity constraints exists. The cut condition requires that for every cut the demand across the cut does not exceed its capacity. When F consists of a single edge f and when M is graphic then M is f-flowing [7]. The cut condition states p(D \F ) p(D F sides, we obtain p(F Remark 4.1. The cut condition holds if and only if p(F ) p(D 4 F ) for all cocircuits D. Let H be the clutter of odd circuits of (M; F ). We define: (a) (H; (b) By linear programming duality we have: (H; p) (H; p). When write (H) for (H; p) and (H) for (H; p). Proposition 4.2. Let H be the clutter of odd circuits of a signed matroid (M;F ) and let only if the cut condition holds. -flowing with costs p. (iii) If (C) > 0 for a solution to (4.2), then CF for all F 2 b(H) with p(F Proof. We say that a set X E(M ) is a (feasible) solution for (a) if its characteristic vector is. Consider (i). Suppose (H; We can assume that F is an inclusion-wise minimal solution of (a) and thus D be any cocircuit of M and consider any S 2 H. Since S is a circuit of M , jD \ Sj is even and since H is binary, jF \Sj is odd. Thus j(D 4F ) \Sj is odd. It follows that D4F is a transversal of H. Therefore, D 4 F is a feasible solution to (a) and we have p(F ) p(D 4 F ). Hence, by Remark 4.1, the IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 9 cut condition holds. Conversely, assume the cut condition holds and consider any set X that is feasible for (a). We need to show p(F ) p(X). We can assume that X is inclusion-wise minimal, i.e. that X 2 b(H). Observe that F and X intersect circuits of M with the same parity. Thus D := F 4 X is a cocycle of M . Since the cut condition holds, by Remark 4.1, p(F ) p(D 4 F Consider (ii). Suppose (H; it follows from linear programming duality that F is an optimal solution to (a). Let y be an optimal solution to (b). Complementary slackness states: if jF \ Cj > 1, then the corresponding dual variable y F y C , for all e 2 E(M ). Complementary slackness states: if e 2 F , then F Hence, choosing every CF satisfies (4.2). Conversely, suppose is a solution to (4.2). For each e 2 F such that C: F reduce the values C on the left hand side until equality holds. Since C contains no element of F other than e, we can get equality for every e 2 F . So we may assume F Now y is a feasible solution to (b) and F; y satisfy all complementary slackness conditions. Thus F and y must be a pair of optimal solutions to (a) and (b) respectively. Finally, consider (iii). From (ii) we know there is an optimal solution y to (b) with y C > 0. By complementary slackness, it follows that jF \ F that are optimal solutions to (a). The last proposition implies in particular that, if M is F -flowing with costs p, then the cut condition is satisfied. We say that a cocircuit D is tight if the cut condition (4.1) holds with equality, or equivalently (Remark 4.1) if p(F Proposition 4.3. Suppose M is F -flowing with costs p and let D be a tight cocircuit. If C is a circuit with Proof. We may assume C\D 6= ;. As CF , it follows that ffg. Moreover, C\D 6= ffg, since M is binary. To complete the proof, it suffices to show that there is no pair of elements Suppose for a contradiction that we have such a pair and let F As D is tight, p(F It follows from Proposition 4.2(iii) that CF 0 . But Corollary 4.4. Let H be the clutter of odd circuits of a signed matroid (M; F ). (i) If H is ideal then M is F -flowing with costs p, for all satisfies the cut condition. (ii) If H is nonideal then M is not F 0 -flowing with costs p, for some minimizes p(F 0 ). Proof. Consider (i). Proposition 4.2 states (H; Because H is ideal, (H; This implies by Proposition 4.2(ii) that M is F -flowing with costs p. Consider (ii). If H is nonideal then for some be an optimal solution to (a). states M is not F 0 -flowing with costs p. We leave the next result as an easy exercise. Corollary 4.5. A binary clutter H is ideal if and only if u(H) is F -flowing for every F 2 b(H). Consider the case where H = OK5 . Let F be a set of edges of K 5 such that E(K 5 ) F induces a K 2;3 . (the graphic matroid of K 5 ) is not F -flowing. EJOLS AND BERTRAND GUENIN 5. CONNECTIVITY, PRELIMINARIES be a partition of the elements E of a matroid M and let r be the rank function. M is said to have a k-separation then the separation is said to be strict. A matroid M has a k-separation only if its dual M does (Oxley [21], 4.2.7). A matroid is k-connected if it has no (k 1)-separation and is internally k-connected if it has no strict (k 1)-separation. A 2-connected matroid is simply said to be connected. We now follow Seymour [24] when presenting k-sums. Let M be binary matroids whose element sets E(M 1 may intersect. We define M 1 4M 2 to be the binary matroid on E(M 1 the cycles are all the subsets of E(M 1 of the form C 1 4 C 2 where C i is a cycle of M i , 2. The following special cases will be of interest to us: Definition 5.1. is the 1-sum of M f is not a loop of M 1 or M 2 . Then M 1 4M 2 is the 2-sum of M a circuit of both M 1 and M 2 . Then M 1 4M 2 is the 3-sum of M We denote the k-sum of M 1 and M 2 as Mk M 2 . The elements in E(M i are called the markers of M i . As an example, for 3, the k-sum of two graphic matroids corresponds to taking two graphs, choosing a k-clique from each, identifying the vertices in the clique pairwise and deleting the edges in the clique. The markers are the edges in the clique. We have the following connection between k-separations and k-sums. Theorem 5.2 (Seymour [24]). Let M be a k-connected binary matroid and k 2 f1; 2; 3g. Then M has a k-separation if and only if it can be expressed as Mk M 2 . Moreover, M 1 (resp. M 2 ) is a minor of M obtained by contracting and deleting elements in E(M 2 We say that a binary clutter H has a (strict) k-separation if u(H) does. Remark 5.3. H has a 1-separation if and if A(H) is a block diagonal matrix. Moreover, H is ideal if and only if the minors corresponding to each of the blocks are ideal. Recall (Proposition 2.11) that every binary clutter H can be expressed as P ort(M; e) for some binary matroid M with element e. So we could define the connectivity of H to be the connectivity of the associated matroid M . The two notions of connectivity are not equivalent as the clutter LF7 illustrates. The matroid AG(3; 2) has a strict 3-separation while F 7 does not, but P ort(AG(3; 2); and LF7 is the clutter of odd circuits of the signed matroid F 7 Chopra [4] gives composition operations for matroid ports and sufficient conditions for maintaining ide- alness. This generalizes earlier results of Bixby [1]. Other compositions for ideal (but not necessarily binary clutters) can be found in [19, 17, 18]. Novick-Seb-o [20] give an outline on how to show that mni binary clutters do not have 2-separations, the argument is similar to that used by Seymour [26](7.1) to show that k-cycling matroids are closed under 2-sums. We will follow the same strategy (see Section 6). Proving that mni binary clutters do not have 3-separations is more complicated and requires a different approach (see IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 11 Section 7). In closing observe that none of LF7 ; OK5 and b(OK5 ) have strict 4-separations. So if Seymour's Conjecture holds, then mni binary clutters are internally 5-connected. 6. 2-SEPARATIONS Let (M;F ) be a signed matroid with a 2-separation We say that is a part of (M;F ) if it is a signed minor of (M; F ). It is not hard to see that at most two choices of F i can give distinct signed matroids Therefore (M;F ) can have at most four distinct parts. In light of Remark 2.5 we can identify binary clutters with signed matroids. The main result of this section is the following. Proposition 6.1. A binary clutter with a 2-separation is ideal if and only if all its parts are ideal. To prove this, we shall need the following results. Proposition 6.2 (Seymour [24]). If connected if and only if M 1 and M 2 are connected. Proposition 6.3 (Seymour [24]). Let M be a binary matroid with a 2-separation two circuits of M . If C 1 Proposition 6.4 (Seymour [24]). Let choose any circuit C of M such that C\E 1 and C \E 2 6= ;. Let j. For any f 2 C Proof of Proposition 6.1. Let H be a binary clutter with a 2-separation, without loss of generality that M is connected. Remark 2.5 states that H is the clutter of odd circuits of (M; F ). If H is ideal, then so are all its parts by Remark 2.9. Conversely, suppose all parts of (M; F ) are ideal. Consider any p : E(M Because of Corollary 4.4(ii), it suffices to show that M is F -flowing with costs p. Observe that the cut condition is satisfied because of Proposition 4.2(i). Since M has a 2-separation, it can be expressed as M2 M 2 . Throughout this proof, denote arbitrary distinct elements of f1; 2g. Define F be the marker of M i . Since f i is not a loop, there is a cocircuit D i of M i using f i . Let i denote the smallest value of where D i is any cocircuit of M i using f i . In what follows, we let D i denote some cocircuit where the minimum is attained. Expression (*) gives the difference between the sum of the capacity elements and the sum of the demand elements in D i , excluding the marker f i . Thus is a cocycle of M and the cut condition is satisfied, we must have: Claim 1. If i > 0, then there is an even circuit of (M uses marker f i . Proof of Claim: Suppose for a contradiction that all circuits C of M i that use f i , satisfy jC \ F i j odd. Then intersects all these circuits with even parity. By hypothesis M is connected and, because of Proposition 6.2, so is M i . We know from Theorem 2.7 that all circuits that do not use the marker f i are the EJOLS AND BERTRAND GUENIN symmetric difference of two circuits that do use f i . It follows that D intersects all circuits of M i with even parity. Thus D is a cocycle of M i . But expression (*) is nonpositive for cocycle D. D can be partitioned into cocircuits. Because the cut condition holds, expression (*) is nonpositive for the cocircuit that uses f i , a contradiction as i > 0. 3 2. If i < 0, then there is an odd circuit of (M uses marker f i . Proof of Claim: Suppose, for a contradiction, that all circuits C of M i that use f i , satisfy jC \ F i j even. By the same argument as in Claim 1, we know that in fact so do all circuits of M i . This implies that F and F j intersect each circuit of M with the same parity. As F is inclusion-wise minimal must have ;. But this implies that expression (*) is non negative, a contradiction. 3 a part of (M; F ). Proof of Claim: From Claim 2 (resp. Claim 1), there is an odd (resp. even) circuit C using f j of (M Proposition 6.3 implies that elements are in series in M n (E j C). Proposition 6.4 implies that M i is obtained from M n (E j C) by replacing series elements of C \E j by a unique element f j . The required signed minor is (M; F any element of C Because suffices to consider the following cases. Case 1: 1 0; 2 0. We know from Proposition 6.4 that M i is a minor of M (where no loop is contracted) say M n J d =J c . For be the signed minor (M; F ) n J d =J c . Since (M a part of (M; F ), it is ideal. So in particular (M be defined as follows: Let D be a cocircuit of M i n f i . The inequality p(D \ F i ) p(D F i ) follows from i 0 when D [ f i is a cocircuit of M i and it follows from the fact that the cut condition holds for when D is a cocircuit of M i . Therefore the cut condition holds for It follows from Corollary 4.4(i) that each of these signed matroids has an F i -flow satisfying costs p i . Let i be the corresponding function satisfying (4.2). By scaling p, we may assume i for each circuit in . Let L i be the multiset where each circuit C in appears i (C) times. Define L j similarly. The union (with repetition) of all circuits in L i and L j correspond to an F -flow of M satisfying costs p. Case 2: i < 0; j > 0. Because of Claim 3, there are parts (M defined as follows: p i (f be defined as follows: Since we can scale p, we can assume that the F i -flow of M i satisfying costs p i is a multiset L i of circuits and that the F j [ff j g-flow of M j satisfying costs p j is a multiset l can be partitioned into L l is a demand element for the flow L j , jL j is a capacity element for the flow j. Let us define a collection of circuits of M as follows: include all circuits of L i . Pair each circuit C 1 with a different circuit C 1 , and add to the collection the circuit included in C i 4C j that contains the element of F . The resulting collection corresponds to a F -flow of M satisfying costs p. IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 13 7. 3-SEPARATIONS The main result of this section is the following, Proposition 7.1. A minimally nonideal binary clutter H has no strict 3-separation. The proof follows from two lemmas, stated next and proved in sections 7.1 and 7.2 respectively. Lemma 7.2. Let H be a minimally nonideal binary clutter with a strict 3-separation There exists a set F 2 b(H) of minimum cardinality such that F E 1 or F E 2 . Let (M; F ) be a signed matroid with a strict 3-separation be the triangle common to both M 1 and M 2 . Let be obtained by deleting from M i a (possibly empty) set of elements of C 0 . We call of (M; F ) if it is a signed minor of (M; F ). Lemma 7.3. Let (M; F ) be a connected signed matroid with a strict 3-separation suppose -Flowing with costs p if the cut condition is satisfied and all parts of (M; F ) are ideal. Proof of Proposition 7.1. Suppose H is a mni binary clutter that is connected with a strict 3-separation. Re-mark 2.5 states that H is the clutter of odd circuits of a signed matroid (M;F ). Consider defined by We know (see Remark 7.5) that (H; p) > (H; p). From Lemma 7.2 and Remark 2.5, we may assume F E 1 and p(F It follows from Proposition 4.2(i) that the cut condition holds. Since the separation of (M; F ) is strict, all parts of (M; F ) are proper minors, and hence ideal. It follows therefore from Lemma 7.3 that M is F -flowing with costs p. Hence, because of Proposition 4.2(ii), (H; 7.1. Separations and blocks. In this section, we shall prove Lemma 7.2. But first let us review some results on minimally nonideal clutters. For every clutter H, we can associate a 0; 1 matrix A(H). Hence we shall talk about mni 0; 1 matrices, blocker of 0; 1 matrices, and binary 0; 1 matrices (when the associated clutter is binary). The next result on mni 0,1 matrices is due to Lehman [15] (see also Padberg [22], Seymour [27]). We state it here in the binary case. Theorem 7.4. Let A be a minimally nonideal binary 0,1 matrix with n columns. Then nonideal binary as well, the matrix A (resp. B) has a square, nonsingular row submatrix A (resp. B) with entries in every row and columns, rs > n. Rows of A (resp. B) not in A (resp. B) have at least r entries. Moreover, A (rs n)I , where J denotes an n n matrix filled with ones. It follows that ( 1 r ) is a fractional extreme point of the polyhedron fx 2 R n 1g. Hence, Remark 7.5. If H is a minimally nonideal binary clutter, then (H) > (H). The submatrix A is called the core of A. Given a mni clutter H with we define the core of H to be the clutter H for which A( A. Let H and be binary and mni. Since H;G are binary, for all H and T 2 G, we have jS \T j odd. As A (rs n)I , for every S 2 H, there is exactly one set G called the mate of S such that jS \ T (rs n). Note that if A is binary then rs EJOLS AND BERTRAND GUENIN Proposition 7.6. Let A be a mni binary matrix. Then no column of A is in the union of two other columns. Proof. Bridges and Ryser [2] proved that square 0; 1 matrices B that satisfy A commute, i.e. A T (rs n)I. Thus col( rs ng. Hence there is no ng fig such that col( A; i), for otherwise contradicting the equation A T (rs n)I. Proposition 7.7 (Guenin [10]). Let H be a mni binary clutter and e 2 E(H). There exists such that S 1 Proposition 7.8 (Guenin [10]). Let H be a mni binary clutter and S 1 H. If S then either Proposition 7.9. Let H be a mni binary clutter and let 2. Proof. Let T be the mate of S. Then jT \ Sj 3 and jT \ S Proposition 7.10 (Luetolf and Margot [16]). Let H be a mni binary clutter. Then H). Further- more, if T is a transversal of H and jT then T is a transversal of H. We shall also need, Proposition 7.11 (Seymour [24]). Let M be a binary matroid with 3-separation . Then there exist circuits such that every circuit of M can be expressed as the symmetric difference of a subset of circuits in fC 2 g. Throughout this section, we shall consider a signed matroid (M; F ) with a 3-separation will denote the corresponding circuits of Proposition 7.11. Let H be the clutter of odd circuits of (M; F ). We shall partition b(H) into sets Proposition 7.12. If S 1 contains a set of b(H). Proof. Let Note that since S 1 circuits C of M , jS 1 \ Cj and jS 2 \ Cj have the same parity. This implies that if C is a circuit where C intersects S 0 and S 1 with the same parity. It also implies, together with the definition of B i , that S 0 intersects with the same parity as S 1 . It follows from Proposition 7.11 that S 0 and S 1 intersect all circuits of M with the same parity. Proof of Lemma 7.2. Let G denote the blocker of H and let be the sets partitioning G. We will denote by G the core of G. It follows that G can be partitioned into sets . Assume for a contradiction that for all S 2 We will say that a set with forms an E 1 -block if, for all pairs of sets Similarly we define E 2 -blocks. IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 15 Claim 1. For each nonempty is either an or an E 2 -block. Proof of Claim: Consider S 1 Proposition 7.12 states that (S 1 contains a set G. Proposition 7.8 implies that S Moreover, by hypothesis neither S 0 \E 1 nor S 0 \E 2 is empty. Since chosen arbitrarily, the result follows. 3 For any nonempty . We define E( to be equal to S \ is an and to S \ is an E 2 -block. Let r (resp. s) be the cardinality of the members of H)j. As H is binary r 3 and s 3. 2. Let U E( G) be a set that intersects E( . Then U is a transversal of G and jU j Proof of Claim: Clearly U is a transversal of G, thus jU j ( G). Proposition 7.10 states ( 3. Let U; U 0 be distinct transversals of G. If ( 2. Proof of Claim: Proposition 7.10 imply that U and U 0 are minimum transversals of G. Hence, U; U The result now follows from Corollary 7.9. 3 4. None of the Proof of Claim: Let U be a minimum cardinality set that intersects E( r 3, it follows from Claim 2 that at most one of the can be empty. Assume for a contradiction that one of the is empty. It follows from Claim 2 and the choice of U that each of E( E( are pairwise disjoint (otherwise U contains an element common to at least 2 of E( E( 1 be distinct elements of E( E( E( contradict Claim 3. Thus jE( similarly, jE( are not all E 1 -blocks and not -blocks. Thus w.l.o.g. we may assume -blocks and 3 is an E 2 -block. Let t 1 be any element in E 1 E( E( be the unique element in E( the column of A( indexed by t 1 is included in the column of A( indexed by t 2 , a contradiction to Proposition 7.6. 3 Consider first the case where every is an E 1 -block. Suppose that no two E( has four columns that add up to the vector of all ones. By Theorem 7.4, each of these columns has s ones and therefore 4s. Furthermore the four elements that index these columns form a transversal of G and therefore r 4 (see Claim 2). This contradicts Theorem 7.4 stating that rs > n. Thus two E( intersect, say . For otherwise a contradiction to rs > n. Let t be any element of E( any element of E( E( 4 g. It follows from Claim 2 that r = 3. It follows from Claim 3 that each of E( E( cardinality one, and E( E( contains a unique element e. Since there are no dominated columns in A( we have that E( E( a contradiction to the hypothesis that the 3-separation is strict. Consider now the case where -blocks and -blocks. Suppose there exists that is not in any of E( 4g. Assume without loss of generality that e EJOLS AND BERTRAND GUENIN Then column e of A( is included in the union of any two columns f 1 2 E( E( contradiction to Proposition 7.6. Thus every element of E(H) is in E( there is e 2 E( E( g. Then implies that partitioned into E( E( partitioned into E( E( 4, then we can use Claim 3 to show that for each i 2 contradiction as then jE 2. Thus be a minimum transversal of G. Suppose both u; v 2 E( E( is a transversal. It then follows that T intersects all sets of parity, a contradiction as H is binary. Thus we may assume w 2 E( all sets in It follows that, for any x 2 E( E( 3 ), the set fw; x; yg is a transversal of G, a contradiction to 3. Hence for any transversal each element of T is in a different E( may assume E( E( E( It follows that for any x 2 E( wg is a transversal and thus by Claim 3 E( contains a unique element t. Since jE 1 j > 2, we cannot have a transversal E( E( E( would imply jE( Hence every minimum transversal contains t, a contradiction to Theorem 7.4. Finally, consider the case where -blocks and B 4 is an E 2 -block. Note that every t is in some E( 3g. Otherwise the corresponding column t of A( is dominated by any E( Suppose there is t 2 E( E( are distinct elements in 3g. Proposition 7.7 states there exist three sets of G that intersect exactly in t. This implies jE( Now since E( E( there is a column in say E( E( Similarly, a contradiction to jE 1 j > 3. Thus E( E( and therefore either (1) for some distinct E( E( is a partition of E( E( E( for each distinct 3g. By considering sets U containing one element of E( intersecting each of E( E( E( use Claim 3 to show that jE 1 j 2 in Case (1) and jE 1 j 3 in Case (2), a contradiction. 7.2. Parts and minors. In this section, we prove Lemma 7.2 Consider the matroid with exactly three elements which form a circuit C 0 . Let I 0 ; I 1 be disjoint subsets of C 0 . We say that a signed matroid (N; ) is a fat triangle is obtained from C 0 by adding a parallel element for every signed binary matroid with a circuit C of M is a simple circuit of type We say that a cocircuit D has a small intersection with a simple circuit C if either: D \ and the unique element in C \ is in D. Lemma 7.13. Let (M; ) be a signed binary matroid with a circuit C (1) Let I C 0 be such that for all i 2 I there is a simple circuit C i of type i. Suppose for all distinct small intersection with the simple circuits in fC Ig. Then the fat triangle (;; I) is a minor of (M; ). (2) Let C 1 be a simple circuit of type 1. Suppose we have a cocircuit D 12 where D 12 \C D 12 has a small intersection with C 1 . If C 1 f1g [ f2g is dependent then C 1 f1g [ f2g contains an odd circuit using 2 and the fat triangle (f3g; f2g) is a minor of (M; ). IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 17 (3) Suppose for each we have a simple circuit C i of type i. Suppose we have a cocircuit D 12 where D 12 \C small intersection with C 1 and C 2 . If both C 1 f1g [ f2g and C 2 f2g [ f1g are independent, then the fat triangle (;; f1; 2g) is a minor of (M; ). Proof. Throughout the proof distinct elements of C 0 . Let us prove (1). For each i 2 I let f i be the unique element in C i \. For each D jk either: D jk \C or D jk \ C is an element not in . Let E 0 be the set of elements in C 0 or in any of C i (a) If g i exists then f i is in each of D 12 ; D 13 ; D 23 and g i is in D jk but not D ij ; D ik . (b) If g i does not exists but f i does then f i is in D ij ; D ik but not in D jk . . Observe that (a) and (b) imply respectively (a') and (b'). (a') If g i exists then f i 62 and g i 2 . (b') If g i does not exist then f i 2 . Let (N; ) be the minor of (M; ) obtained by deleting the elements not in E 0 and then contracting the elements not in C 0 [ . It follows from (a') and (b') that if C 0 is a circuit then (N; ) is the fat triangle (;; I). Otherwise some element i 2 C 0 is a loop of N , say 1. Then there is a circuit C of M such that does not intersect D 12 and D 23 with the same parity. Consider any e 2 C C 0 such that e is in some cocircuit D ij . Since e 62 , it follows from (a') and (b') that I and that g i exists. But then (a) implies that e 2 D 12 \ D 13 \ D 23 . It follows that C cannot intersect D 12 and D 23 with the same parity, a contradiction. Let us prove (2). Let f be the unique element in C 1 \. By hypothesis there is a circuit C in C 1 f1g[ f2g. Since C 1 is a circuit 2 2 C. Since D 12 has a small intersection C 1 \ D fg. It follows that be the circuit using 3 in C 1 4C4C 0 . Since C 0 is a circuit, 3 is not a loop, hence contains at least one element say g. Observe that f2; fg is an odd cycle of (N; ) and that f3; gg and C 0 are even cycles of (N; ). Hence, if C 0 is a circuit of N then (N; ) is the fat triangle (f3g; f2g). Because D 12 is a cocircuit of M , f1; 2; fg is a cocycle of N , in particular 1; 2; f are not loops. If 3 is a loop of N then there is a circuit S C 1 f1; 2; f; gg[ f3g of (M; ). But C 0 4 S is a cycle and C 0 4 S C 1 , a contradiction as C 1 is a circuit. Let us prove (3). Let M 0 be obtained from M by deleting all elements not in C 0 [ small intersection with C 1 and C 2 we have is a signed minor of (M; ). Choose a minor N of M 0 which is minimal and satisfies the following properties: (i) C 0 is a circuit of N , there exist circuits C i of N such that C i \C Note that by hypothesis M satisfies properties (i)-(iv) and thus so does M 0 . Hence N is well defined. We will show that jC 1 in N . Then (N; f1; 2g) is a minor of (M; ) and after resigning on the cocircuit containing we obtain the fat triangle (;; f1; 2g). There is no circuit S C 1 f1g [ f3g of N , for otherwise there exists a cycle C 1 4 S 4C 0 C 1 f1g [ f2g, a contradiction with (iii). Hence, EJOLS AND BERTRAND GUENIN 2. C 1 \ C Proof of Claim: Otherwise define N 0 := N=(C 1 \ C 2 ). Note that N 0 satisfies (ii)-(iv). Suppose (i) does not hold for N 0 , i.e. C 0 is a cycle but not a circuit of N 0 . Then 3 is a loop of N 0 . Thus there is S C 1 \C 2 such that S [ f3g is a circuit of N , contradicting Claim 1. 3 Assume for a contradiction jC 3. There exists a circuit S of N . Proof of Claim: Let e 2 C 1 f1g and consider (N is not a circuit of N 0 . Then 2 or 3 is a loop of N . But then either f2; eg or f3; eg is a circuit of N . In the former case it contradicts (iii), in the latter it contradicts Claim 1. Hence (i) holds for N 0 . Trivially (iii) holds for N 0 as well. Suppose (ii) does not hold, then C 2 is not a circuit of N 0 . It implies there exists a circuit S C 2 [ feg f2g of N . Then S is the required circuit. Suppose (iv) does not hold. Then there is a circuit S N , and S 4 C 1 contains the required circuit. 3 Let S be the circuit in the previous claim. Since C are circuits, S \C 1 are non-empty. Let C 0 2 be the circuit in C 2 4 S which uses 2. Note that N n (E(N properties (i)-(iii) using 2 instead of C 2 . Thus, by minimality, (iv) is not satisfied for C 0 contains a circuit C 0 1 . 2 is a circuit, 1 2 C 0 . By the same argument as above, (iii) is not satisfied for C 0 contains a circuit C 00 using 2. Since C 00 it follows that C 00 2 is a circuit). Therefore, f1g. But the cycle C 0 contradicts the fact that C 0 is a circuit. Lemma 7.14. Let (M; ) be an ideal signed binary matroid with a circuit C Suppose we have such that the cut condition is satisfied. Then there exists Q+ which satisfies the following properties: (i) p 0 satisfies the cut condition; (ii) p 0 and 0g. There is a -flow (1) The fat triangle (;; I) is a signed minor of (M; ) and circuits C such that (C) > 0. Or after possibly relabeling elements of C 0 we have p 0 (3) The fat triangle (f3g; f2g) is a signed minor of (M; ) and (4) for all odd circuits C with (C) > 0 and C \ C contains an odd circuit using 2. Proof. Claim 1. We can assume that there exists Q+ such that properties (i)-(iii) hold. For distinct be the minimum of p 0 (D jg. We then have (after possibly relabeling the elements of C 0 ) the following cases, either: (a) IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 19 Proof of Claim: Choose which satisfies the following prop- erties: the cut condition holds for (i)-(iii) holds for p 0 . Suppose (a) does not hold. Then we may assume (after relabeling) that 23 > 0 and that Consider first the case where 12 > 0. Then 2 is in no tight cocircuit, it follows from the choice of p 0 that then for all circuits C such that holds. Moreover, (1) is satisfied since (M; )n(E(M ) C 0 ) is the Thus we may assume relabeling 2 and 3 we satisfy (b). Hence we can assume is in no tight cocircuit, thus p 0 holds. Thus we may assume defined as hold for ^ p. Suppose p(3). Thus we may assume for each distinct there is a cocircuit D where D \ C which is tight for p. It follows that (a) holds. 3 Throughout the proof distinct elements of C 0 . Let p 0 be the costs given in Claim 1. implies that there is a -flow, ij be the cocircuits of M for which D ij \ C Consider first case (a) of Claim 1, i.e. D ij is tight for all distinct We will show that (1) and (2) hold. Let C be any circuit with (C) > 0. Then jC 1. Suppose there is an element i in C 0 \ C . Proposition 4.3 states C is the unique element in C \ . Thus C \ C is in a tight cocircuit, thus if p 0 (i) > 0 then there is a circuit C i with Moreover, (2) implies that C i is a simple circuit of type i. Proposition 4.3 implies that D 12 ; D 13 ; D 23 all have small intersections with each of the simple circuits. Then (1) follows from Proposition 7.13(1). Consider case (b) of Claim 1, i.e. 2. Let C be a circuit with (C) > 0. If i 2 C \ f1; 2g, then C i is a simple circuit of type i. This follows from the fact that 3 62 C (as p 0 and that jC \ f1; (because of Proposition 4.3 and the fact that D 12 is tight). The case where p 0 has already been considered (see proof of Claim 1). Suppose for some f be the unique element in . The minor (M; )n(E(M is the fat triangle (;; fig) and both (1) and (2) hold. Thus p 0 (1) > 0; p 0 (2) > 0. Suppose now for all i 2 f1; 2g there exists a circuit C i with states that these circuits are simple circuits of type i. Then (2) holds and Proposition 7.13(3) implies that (M; ) contains the fat triangle (;; f1; 2g), i.e. (1) holds. Thus we may assume, for some i 2 f1; 2g that for all circuits C i such that (C i dependent. If interchange the labels 2 and 1. Since we had we get in that case p 0 (2) Proposition 7.13(2) implies that for all circuits C 1 with contains an odd circuit using 1 and that (M; ) contains the fat triangle (f3g; f2g) as a minor. Together with this implies (3) and (4) hold. EJOLS AND BERTRAND GUENIN We are now ready for the proof of the main lemma. Proof of Lemma 7.3. Since M has a strict 3-separation, a triangle. Throughout this proof i; j; k will denote distinct elements of C 0 . Recall that F E 1 . Let 1 denote the smallest value of some cocircuit of M 1 with D ij \ C jg. Expression (*) gives the difference between the sum of the capacity elements and the sum of the demand elements in D ij , excluding the marker C 0 . Denote by D 1 ij the cocircuit for which the minimum is attained in (*). Let 2 ij denote the smallest value of some cocircuit of M 2 with jg. In what follows, we let D 2 the cocircuit for which p(D 2 ij . For each Proof of Claim: We have 2 jk . Thus 2. 1 Proof of Claim: 1 But the last expression is non negative since the cut condition holds for (M; F; p). 3 signed minor of (M; F ). Proof of Claim: Theorem 5.2 implies that M 1 is a minor of M obtained by contracting and deleting elements in 4. The cut condition is satisfied for Proof of Claim: Since the cut condition holds for (M; F; p) the cut condition is satisfied for all cocircuits of 1 disjoint from C 0 . Let D be a cocircuit of M 1 such that D\C ij . It follows from Claim 2 that the previous expression is non-negative. 3 implies that (M 1 of (M; F ) and hence its clutter of odd circuits is ideal. Together with implies that (M 1 the hypothesis of Lemma 7.14. It follows that M 1 is F -flowing with costs p 0 1 is as described in the lemma) and either case 1 or case 2 occurs. Case 1: Statements (1) and (2) hold. We define I := 5. (M 0 is a signed minor of (M; F ). Proof of Claim: Statement (1) says that the fat triangle (;; I) is a signed minor of (M is equal to showed that (M3 M 2 ) nJ d =J IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 21 6. The cut condition is satisfied for (M 0 Proof of Claim: It suffices to show the cut condition holds for cocircuits D that intersect C 0 . Suppose D \ states that p 0 ij . Thus implies that (M 0 is a part of (M; F ) and hence its clutter of odd circuits is ideal. It follows from Claim 6 and Corollary 4.4(i) that M 0 2 is I-flowing with costs p 2 . Since we can scale p (and hence p 0 1 and may assume that the F -flow of M 1 satisfying costs p 0 1 is a multiset L 1 of circuits and that the I-flow of M 0 satisfying costs p 2 is a multiset L 2 of circuits. Because of Statement (2), L 1 can be partitioned into L 1and L 1 . Because can be partitioned into L 2 1 (i) for each i 2 I, jL 1 us define a collection of circuits of M as follows: include all circuits of L 1 , and for every i 2 I pair each circuit C i with a different circuit and add to the collection the circuit included in C 1 4C 2 that contain the element of F . The resulting collection corresponds to a F -flow of M satisfying costs p. Case 2: statements (3) and (4) hold (after possibly relabeling C 0 ). that the fat triangle (f3g; f2g) is a signed minor of (M 1 ; F ). Proceeding as in the proof of Claim 5 we obtain the following. 7. (M 0 is a signed minor of (M; F ). 1 (1) and 8. The cut condition is satisfied for (M 0 Proof of Claim: Consider first a cocircuit D of M 0 2 such that D \ C 3g. Let us check D does not violate the cut condition. The following expression should be non negative: p 2 (D f2g) p 2 (D \ states Consider a cocircuit D of M 0 2 such that 2 2 D but 3 62 D. Let us check D does not violate the cut condition. The following expression should be non negative: p 2 (D f2g) p 2 (D \ is a cocircuit of M 2 , It follows that p(D C 0 imply that M 0 2 is f2g-flowing with costs p 2 . We may assume that the F -flow of M 1 satisfying costs p 0 1 is a multiset L 1 of circuits. Because of Statement (4), L 1 can be partitioned into L 1 . We may assume that the f2g-flow of M 0 satisfying costs p 2 is a multiset L 2 of circuits. Since C 2 L 2 implies Cf2g and since 1 62 E(M 0 can be partitioned into L 2 , and L 2 22 G - EJOLS AND BERTRAND GUENIN 9. (i) jL 1 j. Proof of Claim: Let us prove (i). 2 is a demand element for the flow L 2 , thus jL 2 (2). 3 is a capacity element for the flow L 2 , thus jL 2 where the last inequality follows from the fact that 2 is a capacity element for the flow L 1 . Let us prove (ii). Let us define a collection of circuits of M as follows: (a) include all circuits of L 1 every circuit 2 with a different circuit C - such a pairing exists because of Claim 9(i) - and add to the collection pair as many circuits C 1 of L 1 1 to as many different circuits C 2 of L 2 1 as possible, and add to the collection C 1 remaining circuits C 1 of L 1 1 to circuits of L 2 not already used in (b). Such a pairing exists because of Claim 9(ii). Statement (4) says that C 1 f1g [ f2g contains an odd circuit 1 . For every pair C add to the collection the cycle C 0 for each cycle C in the collection only keep the circuit included in C that contains the element of F . The resulting collection corresponds to an F -flow of M satisfying costs p. 8. SUFFICIENT CONDITIONS FOR IDEALNESS We will prove Theorem 1.1 in this section, i.e. that a binary clutter is ideal if it has none of the following minors: LF7 , OK5 , b(OK5 6 and Q 7 . The next result is fairly straightforward. Proposition 8.1 (Novick and Seb-o [20]). H is a clutter of odd circuits of a graph if and only if u(H) is graphic. H is a clutter of T -cuts if and only if u(H) is cographic. Remark 8.2. The class of clutters of odd circuits and the class of clutters of T -cuts is closed under minor taking. This follows from the previous proposition, Remark 2.9 and the fact that the classes of graphic and cographic matroid are closed under taking (matroid) minors. We know from Remark 3.4 that b(Q 6 is a source of F 7 , and Q 6 is a source of F 7 . Thus Proposition 8.1 implies, Remark 8.3. Q 7 and Q 6 are not clutters of odd circuits or clutters of T -cuts. We use the following two decomposition theorems. Theorem 8.4 (Seymour [24]). Let M be a 3-connected and internally 4-connected regular matroid. Then is graphic or M is cographic. Theorem 8.5 (Seymour [24, 26]). Let M be a 3-connected binary matroid with no F Then M is regular or Corollary 8.6. Let H be a binary clutter such that u(H) has no F 7 minor. If H is 3-connected and internally 4-connected, then H is one of b(Q 7 or one of the 6 lifts of R 10 , or a clutter of odd circuits or a clutter of T-cuts. IDEAL BINARY CLUTTERS, CONNECTIVITY, AND A CONJECTURE OF SEYMOUR 23 Proof. Since H is 3-connected, u(H) is 3-connected. So, by Theorem 8.5, u(H) is regular or In the latter case, Remark 3.4 implies that H is one of b(Q 7 . Thus we can assume that u(H) is regular. By hypothesis, u(H) is internally 4-connected and therefore, by Theorem 8.4, is graphic or u(H) is cographic. Now the corollary follows from Proposition 8.1 and Remark 3.4. We are now ready for the proof of the main result of this paper. Proof of Theorem 1.1. We need to prove that, if H is nonideal, then it contains LF7 , OK5 , b(OK5 6 or 7 as a minor. Without loss of generality we may assume that H is minimally nonideal. It follows from Remark 5.3 and propositions 6.1 and 7.1 that H is 3-connected and internally 4-connected. Consider first the case where u(H) has no F 7 minor. Then, by Corollary 8.6 either: (i) H is one of b(Q 7 (ii) H is one of the 6 lifts of R 10 , or (iii) H is a clutter of odd circuits, or (iv) H is a clutter of T-cuts. Since H is minimally nonideal, it follows from Proposition 3.5 that if (i) occurs then occurs then occurs then, by Theorem 1.3, occur because of Theorem 1.4. Now consider the case where u(H) has an F 7 minor. It follows by Theorem 3.2 that H has a minor H 1 a source of F 7 and H 2 is a lift of F 7 . Proposition 3.1 states that the lifts of F 7 are the blockers of the sources of F 7 . Remark 3.4 states that the sources of F 7 are b(Q 7 ), LF7 or b(Q 6 that F 7 has only one source, namely Q 6 . This implies that H 6 and H 7 or b(LF7 ) has an LF7 minor and b b(Q has a Q 6 minor, the proof of the theorem is complete. One can obtain a variation of Theorem 1.1 by modifying Corollary 8.6 as follows: Let H be a binary clutter such that u(H) has no F 7 minor. If H is 3-connected and internally 4-connected, then H is b(Q 7 ), 6 ) or one of the 6 lifts of R 10 or a clutter of odd circuits or a clutter of T-cuts. Following the proof of Theorem 1.1, this yields: A binary clutter is ideal if it does not have an LF7 , OK5 , b(OK5 minor. But this result is weaker than Corollary 1.2. Other variations of Theorem 1.1 can be obtained by using Seymour's Splitter Theorem [24] which implies, since u(H) is 3-connected and u(H) 6= F 7 , that u(H) has either S 8 or AG(3; 2) as a minor. Again, by using Proposition 3.2, we can obtain a list of excluded minors that are sufficient to guarantee that H is ideal. 9. SOME ADDITIONAL COMMENTS Corollary 8.6 implies the following result, using the argument used in the proof of Theorem 1.1. Theorem 9.1. Let H be an ideal binary clutter such that u(H) has no F 7 minor. If H is 3-connected and internally 4-connected, then H is one of b(Q 7 or one of the 5 ideal lifts of R 10 , or a clutter of odd circuits of a weakly bipartite graph, or a clutter of T-cuts. A possible strategy for resolving Seymour's Conjecture would be to generalize this theorem by removing the assumption that u(H) has no F 7 minor, while allowing in the conclusion the possibility for H to also be a clutter of T-joins or the blocker of a clutter of odd circuits in a weakly bipartite graph. However, this is not possible as illustrated by the following example. EJOLS AND BERTRAND GUENIN Let T 12 be the binary matroid with the following partial representation.6 6 4 This matroid first appeared in [11]. It is self dual and satisfies the following properties: (i) For every element t of T 12 , T 12 =t is 3-connected and internally 4-connected. (ii) For every element t of T 12 , T 12 =t is not regular. We are indebted to James Oxley (personal communication) for bringing to our attention the existence of the matroid T 12 and pointing out that it satisfies properties (i) and (ii). Let t be any element of T 12 and let Because of (i), T 12 3-connected and internally 4-connected and thus so is H. Because of (ii), T 12 not graphic or cographic thus Proposition 8.1 implies that H is not a clutter of T -cuts and not a clutter of odd circuits. We know from Proposition 2.12 that Thus, b(H) is also 3-connected, internally 4-connected, and H is not the clutter of T -joins or the blocker of the clutter of odd circuits. However, it follows from the results of Luetolf and Margot [16] that the clutter H is ideal. --R On the length-width inequality for compound clutters Combinatorial designs and related systems. A decomposition for combinatorial geometries. Composition for matroids with the Fulkerson property. Euler tours and the chinese postman. Blocking and anti-blocking pairs of polyhedra Graphs and polyhedra A characterization of weakly bipartite graphs. A generalization of a graph result of D. Graphic programming using odd or even points (in chinese). A solution of the Shannon switching game. On the width-length inequality On the width-length inequality and degenerate projective planes A catalog of minimally nonideal matrices. The anti-join composition and polyhedra Polyhedral properties of clutter amalgam. Composition operations for clutters and related polyhedra. On combinatorial properties of binary spaces. Matroid Theory. Lehman's forbidden minor characterization of ideal 0 1 matrices. The Matroids with the Max-Flow Min-Cut property Decomposition of regular matroids. A note on the production of matroid minors. European J. On Lehman's width-length characterization Signed graphs. Biased graphs. --TR --CTR Bertrand Guenin, Integral Polyhedra Related to Even-Cycle and Even-Cut Matroids, Mathematics of Operations Research, v.27 n.4, p.693-710, November 2002 Grard Cornujols , Bertrand Guenin, Ideal clutters, Discrete Applied Mathematics, v.123 n.1-3, p.303-338, 15 November 2002

ideal clutter;signed matroid;connectivity;multicommodity flow;t-cut;seymour's conjecture;weakly bipartite graph;separation

588327

Geometric Computation of Curvature Driven Plane Curve Evolutions.

We present a new numerical scheme for planar curve evolution with a normal velocity equal to $F(\kappa)$, where $\kappa$ is the curvature and F is a nondecreasing function such that F(0)=0 and either $x\mapsto F(x^3)$ is Lipschitz with Lipschitz constant less than or equal to 1 or $F(x)=x^\gamma$ for $\gamma\geq 1/3$. The scheme is completely geometrical and avoids some drawbacks of finite difference schemes. In particular, no special parameterization is needed and the scheme is monotone (that is, if a curve initially surrounds another one, then this remains true during their evolution), which guarantees numerical stability. We prove consistency and convergence of this scheme in a weak sense. Finally, we display some numerical experiments on synthetic and real data.

Introduction In this paper, we investigate the evolution of a closed smooth plane curve, when each point of the curve moves with a normal velocity depending on the curvature of the curve at this point. More precisely, we study evolution of a curve C obeying the equation @t where N(s; t) is the inner normal vector to the curve at the point with parameter s at evolution time t. Equations of type (1) model phenomena in physics or material science. They also play an important role in digital image analysis. Indeed, it was proved in [1] that any image analysis process satisfying some reasonable properties and invariance (essentially causality, stability, invariance with respect to isometries and contrast changes) is described by an equation of type (1), or more precisely by the corresponding grey level evolution described by the scalar equation @t which has to be considered in the viscosity sense [6]. In this equation x z e-mail: moisan@cmla.ens-cachan.fr x Mail adress: CMLA, ENS-CACHAN, 61 Avenue du Pr'esident Wilson, 94 235 Cachan Cedex FRANCE. Phone: is the curvature of the level line passing through x. Equation (2) means that the level lines of u move with respect to Equation (1). The case F (with the convention F for ff 2 R and x 2 R) is particularly interesting since it is the only contrast invariant equation that also commutes with affine transformations preserving the area (this set is named the special affine group, and is composed of the mappings of the type x 7! Ax is a linear transformation such that det A = 1). The case F has extensively been studied ([8, 9]), and existence, regularity, and vanishing in finite time have been proved. The affine invariant case F x 1=3 has also been studied in [2, 22, 23]. For results have been exposed in [25]. The equivalence between curve motion and contrast invariant smoothing was proved in [7]. In this paper, we focus on a possible numerical algorithm solving (1). According to [7], if u is a continuous real-valued function in R 2 , we can solve Equation (2) by applying this algorithm to all level lines of u. Of course, this implies that these level lines must not intersect during their evolution. We thus require that, like Equation (1) the algorithm satisfies an inclusion principle, meaning that the order (with respect to inclusion) is respected. Among the first attempts to solve Equation (1), Osher and Sethian ([19, 24]) solve (2) by introducing the signed distance function to the curve at time Unfortunately, this algorithm satisfies neither inclusion principle nor rotation invariance. In addition, the evolution a priori depends on the chosen distance function, since the scalar algorithm is not contrast invariant. Moreover, the data (and thus the CPU time) becomes rapidly huge when a high precision is requested. Some attempts on curves by finite difference schemes have also been made ([15, 16]) with interesting results, but the nongeometric nature of the scheme still prevents inclusion principle from being satisfied. Moreover, for evolution driven by a power a the curvature larger than 1, the discretization of the curves are enclined to becoming sparser around points with high curvature, thus preventing a good accuracy. On the contrary, a completely different scheme has been implemented in [17, 18] for the affine invariant case. It is fully geometrical and also satisfies inclusion principle (implying numerical stability). In [11], a theoretical algorithm for moving hypersurface by a power of Gauss Curvature has also been studied. We generalize and implement this algorithm in the plane for nonconvex curves and for more general functions of the curvature. We just mention that a numerical approach on curves has the advantage that the resolution is not limited by the pixel size, which allows a very high precision. Moreover, the computation time is far shorter than in a scalar approach. On the other hand, topology changes (e.g. a single curve breaking into two connected components) are automatically handled in a scalar approach. This should not be a real problem for plane curves evolution since it is likely that no topological changes occur. This has been proved at least in [9] for the Mean Curvature Flow and in [2] for Affine Curve evolution. On the contrary, it is known that topology changes do occur in higher dimension. In addition, it seems difficult (at least not trivial) to generalize our algorithm in higher dimensions even for hypersurfaces in a three dimensional space. The paper is as follows. We first give in section x2 some preliminary definitions and introduce some operators on sets. This provides operators on curves which are boundaries of sets. These operators are consistent with curvature dependent differential operators. They satisfy some monotonicity and continuity properties, allowing to extend them to real valued functions. We prove some consistency results and give the proof of convergence without entering too much into details since we prefer to focus on numerical applications. In x3, we adapt the operator previously defined to the special case of evolution driven by a power of the curvature and in x4, we give an algorithm with the same scaling covariance properties as the curve Scale Space. We then show in x5 some numerical experiments in the convex case as well as in the nonconvex case. 2 Definition and Properties We first give some notations (previously used in [18]). Let C a semi-closed curve in R 2 (that is an oriented simple curve dividing the plane in exactly two connected components) and K the interior of C (which is the bounded component of R 2 nC when C is closed). We suppose that C is oriented such that K lies on "the left" when C is positively described. More rigorously, if we assume that the plane is counterclockwise oriented, the inner normal is such that the tangent vector and the inner normal form an orthonormal direct basis. We assume that a smooth parameterization is defined on C (for example piecewise C 1 ). A chord is a segment of the form ]C(s); C(t)[ that does not intersect any point of C with parameter between s and t. A chord set C s;t is the connected set enclosed by a chord ]C(s); C(t)[ and the curve C(]s; t[). We say that C s;t is a oe-chord set and (s; t) is a oe-chord if L 2 (C s;t and if the area of any chord set strictly included in C s;t is strictly less than oe. We denote by K oe (C) the set of oe-chord sets. a oe-chord of K and C s;t the associated oe-chord set. We call chord arc distance of C (or of C s;t ) the number ffi(C([s; t]); [C(s); C(t)]) where ffi is the Hausdorff semi- distance (in particular it is not commutative) defined by For x in R 2 , we also denote by ffi s;t (x) the signed distance from x to the oriented line ]C(s); C(t)[, that is are in R 2 , we denote by [x; y] the determinant of the 2 \Theta 2 matrix with columns x and y). We also denote by K oe (C) the sets of positive oe-chords i.e the chord-sets C s;t satisfying In the same way, we can define oe the set of negative chord sets. Remark that for positive chord sets, the chord-arc distance is nothing but sup ffi s;t (x) for x 2 C([s; t]) and for negative chord sets, the chord-arc distance is To finish with notations, we set We now define a mapping on the set of plane sets that we shall assume piecewise regular for sake of simplicity. Let G be a nondecreasing 1-Lipschitz function defined in R+ and such that K be a set in R 2 whose oriented boundary assume that L 2 (K) ? oe. For C s;t a chord set of K with chord-arc distance h, we write oe (C s;t ae !oe 2=3 'oe In the sequel, we shall briefly say that oe (C s;t ) is a modified chord set. We remark that the right-hand term of the inequality above is nonnegative because of the Lipschitz assumption on G. Hence, a modified oe-chord set is always included in its associated oe-chord set. On Figure 1, we represent a oe-chord and its modified chord. The modified oe-chord set is filled. PSfrag replacements oe !oe 2=3 G( h s Figure 1: A oe-chord set before and after transform. K be the interior of a piecewise C 1 semi-closed curve. We define oe (A): (4) We will refer to E oe as an erosion operator and to E oe (K) as the eroded of K at scale oe. Remark 2. The algorithm in [18] corresponds to Remark 3. We can generalize Definition 1 to sets with several connected components by applying the erosion to each component. Lemma 4 Let K be a convex set of R 2 . Then E oe (K) is also convex. Proof. We can write Kn oe (A); which proves that E oe K is convex as an intersection of convex sets (each of them is the intersection between K and a half plane). Lemma 5 If K is a smooth compact set, then E oe (K) is a compact set. Proof. This is obvious since K is an intersection of compact sets. The proposition below is crucial in a theoretical point of view as well in a numerical one since it is necessary to obtain a stable numerical algorithm. Proposition 6 (Inclusion Principle) Let K 1 ae K 2 . Assume that G is nondecreasing and 1-Lipschitz. Then Proof. Assume that x 2 K 2 and x 62 E oe (K 2 ). We prove that x 62 E oe (K 1 ). If x 62 K 1 then Assume now that x 2 K 1 . By assumption, there exists a oe 0 -chord of K 2 (that we denote by C) with oe 0 oe such that x belongs to the modified chord-set. The Lipschitz condition on G implies that x also belongs to the oe 0 -chord set. The same oe 0 -chord delimits in K 1 a unique chord-set containing x. Let oe 00 be its area : we have oe 00 oe 0 and thus oe 00 oe. It then suffices to prove that this chord excludes x from E oe (K 1 ). Consider the situation illustrated in Figure 2. are the chord-arc distances of C in K 1 and K 2 . are the chord-arc distances of the associated modified chords. (iii) l is the difference of length between K 1 and K 2 in the direction that is orthogonal to the chords, i.e. It is enough to prove that l + l 1 l 2 . But, we know that l oe 2=3 !G( h 2 is 1-Lipschitz, we conclude. PSfrag replacements l l 1 l 2 x oe 00 Figure 2: Inclusion Principle. Proposition 7 Assume that is of class C 2 . Let M 2 C such that the curvature of C at M is not equal to 0. Then lim !oe 2=3 is the positive part of , defined by Proof. Assume that C is concave at M (that is the curvature at M is strictly negative in a neighborhood of M ). Then for oe small enough, any oe-chord (s; t) such that M 2 C([s; t]) is a (strictly) negative oe-chord. Hence M 2 E oe (K) and the proposition follows from Assume now that C is strictly convex at M (thus in a neighborhood of M ). Then for any chord parallel to the tangent at M enclosing a oe-chord set containing M , the chord-arc distance h satisfies which can be easily established for a parabola, then for any regular curve by approximation (see [18] for example). Then the result follows from 3 and 4, which imply !oe 2=3 and the fact that G is continuous. The following property is a continuity property allowing to extend E oe to any plane set, and then to define an operator acting on functions with real values. Proposition 8 (Continuity) Let K n a sequence of compact smooth sets. Let Assume that K is also a smooth set. Then Proof. Since K ae K n for any n, by monotonicity we have also the first part of the equality. In order to prove the reverse inclusion, we can assume that the family K n is nonincreasing. Without loss of generality, we also suppose that (K n ) converges to K for the Hausdorff distance between compact sets. Assume that x 62 E oe (K). By definition, there is a chord (s; t) with area not more than oe such that the modified chord excludes x. Since oe (K) is closed, its complementary is an open set; thus we can assume that the area of C s;t is strictly less than oe. In K n it also defines a chord and for n large enough, the area of this chord is also less than oe (by using convergence of measures). Moreover, as K n tends to K for the Hausdorff distance, the chord-arc distance also converges. Since G is continuous, this implies that the chord excludes x in K n for n large enough. Hence x 62 "E oe (K n ) and this ends the proof. Remark 9. The compactness assumption in the proposition above is far from necessary. It suffices for example that the boundary of the sets is locally convex or concave. This ensures that the erosion is local when oe is small. We can then conclude by the same kind of arguments. Note also that the continuity property allows to define the erosion on any closed set by approximating closed sets by smooth closed sets. We can now extend E oe to real-valued functions in R 2 . First, if u R, we define as usual the level set with value , the set Applying a theorem by Matheron (see [14]) yields the following F a set of real valued functions in R 2 such that the level sets of the elements of F are compact and smooth. Then, we can extend E oe to elements of F by setting It is equivalent to define E oe (u) by its level sets, This uniquely defines a monotone, translation invariant operator commuting with nondecreasing continuous functions. We also define a dual operator D oe (called dilation operator) by the subscript designing the complementary set in R 2 . This operator satisfies the same properties as E oe except the consistency result where the positive part of the curvature has to be replaced by the negative part. By standard arguments ([4], [10], [20]), we can derive the following consistency result on the operator acting on functions. Proposition 11 Let u be a C 3 function and x a point such that Du(x) 6= 0 and (u)(x) 6= 0. Then Proof. The whole proof is not very difficult but a bit technical and long. Thus, we do not enter into all details and we shall skip some points. The aim is to prove that E oe (u) only depends on local features of u. Choose ff such that oe oe tends to 0 (note that there is no incompatibility; choose r = oe 1=4 for instance). By using translation invariance and contrast invariance, we assume that of the proof is the following. If r is small enough, the curvature of the level lines of u has a strict sign in D(0; r). As a consequence, for any oe-chord, we can estimate the chord-arc distance by Equation (5). Moreover the same kind of approximation (made on a circle or a parabola) shows that the length of the chord is of order ( oe ) 1=3 . Hence, any oe-chord intersecting D(0; r asymptotically included in D(0; r(1 (because of the choice of r). Assume first that (u)(0) ? 0. Define u+ and u \Gamma by and u+ elsewhere (we can replace the infinite value by very large numbers). The global inequalities enough, the level lines of u are uniformly strictly concave in D(0; r). Thus, there is no positive oe-chord of level sets of u \Gamma and u+ intersecting D(0; r Hence r Assume now that ! 0. Define in D(0; r) and in D(0; r) and elsewhere. The constant k is chosen such that for oe small enough, we have PSfrag replacements Figure 3: Case ? 0. On the left, a level set of u \Gamma . The oriented boundary is the bold line. The level set is the bounded connected component delimited by the curve. The dashed line is the boundary of the eroded set. On the right, a level set of u+ : the boundary is the bold oriented line and the level set is the unbounded component. In both cases, there is no positive oe-chord set intersecting D(0; r Hence, the erosion has no effect. This is possible since we assumed that u is C 3 . Monotonicity yields As v and w have trivial level sets out of D(0; r), it is quite easy to estimate their image by E oe . We now use the consistency result (Proposition 7). The only trick is that the level lines of u and v are not parabola in the canonical form. Nevertheless, with some few arguments we can be led back to this situation ([4, 10, 20]). The computation of the eroded level sets of v and w is drawn on Figure 4. From this, it is no longer difficult to prove that and are constants depending on D 2 u(0). We can apply the same result to the dilation operator D oe to obtain the second part of the proposition. For the next proposition, we first extend G to make it odd, that is if x ! 0, we set \GammaG(\Gammax). Proposition 12 Let u be a C 3 function. Suppose that Du(x) 6= 0 and (u)(x) 6= 0. Then Proof. This follows from the fact that : 1) E oe and D oe are monotone and commute with addition of constants, 2) near a point with gradient and curvature different from zero, the arguments developed in the previous proposition are uniform. In order to be complete, we just give without proof (it is simple when introducing the affine erosion operator ([18])) a lemma controlling the behavior of E oe at critical points. PSfrag replacements Figure 4: Case ! 0. On the left, a level line of v (oriented bold line). The eroded line is the dashed line. On the right, the same thing with w. lim oe 2=3 the limit being taken when y and oe tend to 0 independently. This and the consistency result above allow to deduce the convergence result we now enounce (see [21]). We denote T Theorem 14 Let u 0 bounded and uniformly continuous in R 2 . For oe ? 0, let define R 2 \Theta R! R by Then, when oe tends to 0, u oe converges locally uniformly to the unique viscosity solution of the equation @t The usual definition of viscosity solution ([6]) is no longer valid here because of the possible singularity of the operator at critical points. An extended notion of solution is defined in [3] and the solution still exists and is unique. 3 Curvature Power 3.1 Approximation of power functions In this section, we show that the previous study can be adapted to the particular case of power functions F not 1-Lipschitz on the whole real line, we define where is the largest positive number at which the power function x 3fl has a derivative less than 1. As G is 1-Lipschitz, we can then apply E oe to this function. We could think that this scheme is not consistent with motion by curvature power. Indeed, if the curvature is too large for fixed oe then it may happen that the chord arc distance is also very large and the erosion is then given by the linear part of G. Nevertheless, we shall see that by an adequate scaling, G is not evaluated in its linear part. From now on, we slightly change the notations in the definition of E oe . This will simplify the statements in the case of power functions. For a oe-set C s;t , we now set oe (C s;t and we still define the erosion operator by oe (A): (23) When the scale tends to 0, h also tends to 0 and G is not taken in its linear part. The fact that we get an operator consistent with a power function is due to the homogeneity properties of power functions (this implies that except for power functions, this new definition of the erosion operator makes no sense). We can adapt the proof of the inclusion principle 6 to prove that the modified erosion operator E oe still satisfies this inclusion principle. Consistency on curves (circles is enough !) is easy to establish. Continuity is not a problem as well. Thus, by using Matheron's Theorem, we can extend this erosion operator to an operator acting on functions. The following proposition asserts that this operator is consistent with a power of the curvature. Proposition 15 Let u be a C 3 function. Let x be such that Du(x) 6= 0 and (u)(0) 6= 0. Then Moreover, consistency is locally uniform. Proof. As usual, we assume that In a first time, assume that (u)(0) ? 0. We use the same locality argument as in the first proof of consistency in this chapter. Since the curvature of u in a small ball with radius r ? 0 is bounded from below by a positive constant, say the level sets of u have no positive oe-chord in D(0; r). We set We use the same locality argument as in the first proof of consistency as above: For oe small enough, the level sets of u \Gamma have no positive oe-chord in D(0; r). Thus the erosion has no effect upon u \Gamma in D(0; r). By using monotonicity, we have and the result is proved in the case ? 0. Let us now come to the most difficult case: Assume that cxy be the Taylor expansion of u at the origin with parameter and let If r is chosen small enough, we have v(x) u(x) in D(0; r). By extending v by \Gamma1 out of D(0; r) this remains true everywhere. Moreover, we can assume that the curvature of the level lines of v is still strictly negative. Indeed, its value is is the value at the origin. We want to approximate E oe (v)(0). Let now j ? 0 be also small, such that the curvatures of the level lines of v is larger than that in this part of the proof the curvatures are all negative, hence a circle with a small curvature will also have a small radius). We can again invoke the same locality property of the erosion in the case of a curve with a strictly negative curvature: We know that the chord-arc distance is equal to O(oe 2=3 the length of the chord is a We deduce from this that the oe-chord sets containing 0 must be included in a ball with radius O(oe 1=3 ). A small computation shows that the modified corresponding oe-chord sets are included in a ball with radius O(oe fl ). The constant in these terms are clearly uniform because the curvature is bounded from above by a negative constant. In particular, they do not depend on " and j. Let now x be in a D(0; r). We call C \Gamma"\Gammaj (x) the disk of curvature that is tangent to the level line of v at x and that is in the same side as v(x) (x). Because of the comparison of the curvatures, we can still assume that r is small enough such that we have the inclusion Now, since the erosion operator is local, we also have Assume that E oe (v)(0) . By definition, this means that 0 62 E oe ( (u)). By using inclusion principle, we deduce that 0 62 E oe (C \Gamma"\Gammaj (x)) for any point such that . On the level line of v with the same value , we can find a unique point x such that x and the normal at x are colinear (this is due to the strict convexity of the level lines and the theorem of intermediary values). This point is also characterized by the fact that the distance between the origin and the tangent to the level line is minimal. Hence, the modified oe-chord set of C \Gamma"\Gammaj also contains the origin. Let be the coordinates of x . A simple calculation gives Since x and Du(x ) are colinear, we have By using consistency on disks, we have jx From this, we finally deduce where we have approximated jDv(x )j by p up to a O(oe 2fl ) term. This analysis can be performed since eventhough the constant were not explicited, we already stress that they do not depend upon " and j. We then deduce that Let now search an upper bound to E oe (u)(0). We do not repeat all the arguments since there will be some similarity with the research of a lower bound. We approximate u by its Taylor expansion and define 2cxy such that u w is a small ball of radius r with enough, the curvature of the level lines of w is smaller than which can also be chosen negative if r is small enough. The locality of the oe-chords still holds. We now define C +"+j (x) as above; its radius is equal r is small enough, for any x the level set w(x) (w) is included in C +"+j (x) inside D(0; r). The rest of the proof is still an application of comparison principle and the asymptotic behavior on erosion on disks. Assume that E oe (w)(0) . This means that In particular is a above. This implies that x ! 3fl We use the characterization (24) of x and deduce that we must have Since, this is true for any " ? 0 and j ? 0, we also obtain Again, we can pass to the limit since the o(1) term is uniform in " and j, since all the curvatures may be taken bounded by above by a strictly negative constant. Thus, if ! 0, we have With the case 0, this gives the result. The case of the dilation can be deduced by the relation D oe (\Gammau). The uniform consistency follows from the fact that the oe-chords are uniformly bounded in some ball with radius O(oe 2=3 ) and the constants of these terms are bounded as soon as the curvature have an absolute value strictly more that a positive constant. Uniform consistency yields consistency for the alternate operator. Corollary At critical points, we need to describe the behavior of the erosion in the following manner. Let R be a C 2 function such that tends to 0. Lemma any sequence of points x n tending to 0, we have lim oe 2fl where the limit is taken as n and oe tends to their respective limit independently and T h designs either E oe or D oe or D oe This lemma can be easily established by finding some estimates on the radius of the circle after erosion. We can then prove the following convergence theorem. Theorem 3 . Let T the alternate dilation-erosion for the curvature power function x fl . Let u 0 in BUC(R 2 ) and define u oe by Then, when oe tends to 0, u oe tends locally uniformly to the unique viscosity solution of the equation @t with initial value u 0 . As soon as the power fl is more than 1, then the usual notion of viscosity solution is not appropriate since the elliptic operator jDuj(curv u) fl is singular at critical points. Ishii and Souganidis proved in [12] that existence and uniqueness where still true if test functions were restricted to a class of functions with flat critical point. This flatness is given by the same conditions in the previous lemma. This point apart, the scheme of the proof is standard so we do not explicit it. The only new point is the previous lemma in the case where test functions are stationnary. In this case, the lemma directly gives the solution and we leave the rest of the proof to the reader. 3.2 Scale Covariance We denote by H the dilation with ratio , that is H t be the evolution semi-group of @t that is, is the curve evolving according to Equation (25) above (the solution exists and is unique at least for short times for smooth initial data (see [25]). The semi-group S t satisfies the relation Indeed, let C 1 (t) the evolving curve defined by Then, it is simple to check that C 1 satisfies Equation (25) with initial condition H C(0). This is exactly what Equation (26) asserts. The erosion operator E oe does not satisfy the same covariance property. We thus define a modified operator O a 2 where a and oe are positive parameters depending upon t (and possibly on C). We want O t to satisfy covariance Property (26), like S t . This will be true as soon as and oe(H C; We also want O t to be consistent with S t , that is, for any convex set K with C 3 boundary, O t the term o(t) being measured with the Hausdorff distance (we also use an abusive notation by denoting S t (K) the set whose boundary is S t (@K)). By using consistency result above, we see that a, oe and t must be linked by the relation Assume that oe ? 0 is fixed. If a is chosen large enough such that, for any oe chord of K with chord-arc distance h the inequality a ff fl (30) holds, then the modified chord-arc distance is then given by G near the origin, that is Equation (21) is not involved. Precisely, for M 2 @K (with smooth boundary) consistency writes a a a Notice that it is interesting to take the smallest possible value of a (given by the case of equality in (30)) in order to get the largest possible scale step t from (29). We can summarize these results in the following Proposition 19 Let h(A) denote the chord-arc distance of a chord A. Then, the operator O t defined by (27) with sup t! \Gamma3fl a 3fl \Gamma1 \Delta 1=2fl (32) is consistent with (25) and satisfy the same scaling property as S t in (26). Remark 20. (Error Analysis) In order to obtain consistency, it is not necessary that h a ff fl for all (oe; a); for instance, if we fix a and define O t by (27) and (29), we still have consistency since inequality (30) holds for oe small enough. However if a and oe do not satisfy (30), the difference between O t C and S t C is of the same magnitude as t (since Equation(21) is involved). On the contrary, this difference is O(t 2 ) if a and oe satisfy (30) (because of Consistency 5). 4 Algorithm 4.1 General method Each iteration of the operator defined above involves three parameters: the scale step t (that can be viewed as a time step), the erosion area oe and the saturation length a. These three quantities have to satisfy (32), which leaves only one degree of freedom. A usual numerical scheme would consider t as the free parameter (the time step, related to the required precision), and then define a and oe from t. In the present case, this would not be a good choice for two reasons. First, a is defined as an explicit function of oe but as an implicit function of t, which suggests that oe may be a better (or at least simpler) free parameter than t. Second, t has no geometrical interpretation in the scheme we defined, contrary to oe which corresponds in some way to "the scale at which we look at the curve". In particular, oe is constrained by the numerical precision at which the curve is known: roughly speaking, if C is approximated by a polygon with a precision " (corresponding, for example, to the Hausdorff distance between the both of them), then we must have oe AE " 2 in order that the effect of the erosion at each iteration overcomes the effect of the spatial quantization. For all these reasons, we choose to fix oe as the free parameter, and then compute t and a using (32). If the scale step t obtained this way is too large, we can simply adjust it by reducing oe while keeping the same value of a. We propose the following algorithm for the evolution of a convex set K at final scale T with area precision oe. 1. Let 2. While - For each oe-set of K t , compute the chord-arc distance. - Set a to the maximal value of these distances. decrease oe in order to keep the previous equality. Apply operator O t to K t , yielding K t+ffit . Increment t by ffi t. In practice, it is of course impossible to deal with all the oe-chords. In fact, the curve is a polygonal line and we take the chords with an end point equal to a vertex of the polygon. 4.2 Computation of the erosion The boundary of O t (K) is included in the envelop of all the modified chords. To obtain an approximation of this set, we explicitly determine the position of the unique point of each chord belonging to the envelop. This result is a generalization of the middle point property exposed in [18], Lemma 21 (Middle point Property) Let K be a strictly convex set. Let A 1 be a oe-set of K with oe-chord C 1 . Let A 2 be another oe-set, and let C 2 be its oe-chord. Then, when dH tends to 0, the intersection point of C 1 and C 2 tends to the middle point of C 1 . Proof. (quoted from [18]). Let ' be the geometrical angle between C 1 and C 2 . If A 1 and A 2 are close enough, C 1 and C 2 intersect at a point that we call I('). We also call r 1 (') and r 2 (') the length of the part of the chord C 2 on each side of I('). Since C 1 and C 2 are oe-chords, we PSfrag replacements Figure 5: The middle point property. This implies that lim(r 1 tends to 0. The boundary of O t (K) is included in the envelop of the "modified" oe-chords. In the case these chords are the oe-chords themselves. For all other values of fl, this is no longer true and we have to compute the position of the intersection of closer and closer chords. This is the purpose of Proposition 22 Let K be a strictly convex set, and a oe-chord of K. Consider P the farthest point of C([s; t]) from C. If (L; h) are the coordinates of P in the direct orthonormal referential whose origin is the middle point of C and whose first axis is directed by C(t)-C(s), then the contribution of the modified chord arising from C is either void or the unique point with coordinates a a in the same referential. Proof. Examine the situation on Figure 6. Let C ' the oe-chord making an angle ' with C. We search the coordinates of the intersection point of the modified chord of C and C ' when ' tends to 0. We set x(') the abscissa of the point . By the middle point property, we know that ' the modified chord of C ' . The distance between these two chords is a is the chord-arc distance of C ' . Let (L('); H(0)) be the coordinates of the common point of C 0 and C 0 ' . Elementary but a bit fastidious geometry proves PSfrag replacements' Figure The modified middle point property. that a implying that the limit point we are looking for has coordinates given by (33). Remark 23. In the case of the general function of the curvature and without introducing the scaling preventing saturation phenomenon, the coordinates of the point are !oe 2=3 !oe 2=3 As precised in the proposition, the limit point may not belong to the boundary of O t (K). Indeed @O t (K) is in general strictly included in the envelop of the modified oe-chords. In general, this envelop is not even the boundary of a convex set! Nevertheless, if we know that C is a convex curve then it is simple to decide whether a point has to be kept or not by comparing its position with adjacent modified chord. Hence we can remove the bad points and obtain a convex set. For example, on Figure 7, we display the envelop of the modified oe-chords of a square. If the set is a "corner", the explicit computation can be made and shows the same behavior in the corner as in Figure 7. The eroded set is obtained by removing the parts with cusps in the corners of the square. 5 Numerical Experiments To finish, we display numerical experiments, first in the case of convex sets. By a change of scale variable, we implemented an approximation of the equation @t For this rescaling, scale and space are homogeneous (precisely, if T t maps C to C(t) by this equation, we check that T t Figure 7: Envelop of the modified oe-chords for a square 2). The result is not convex. The bold line is what is to be kept. 5.1 Closed convex curves The first example is the the case of circles. The radius is explicitly computable for the scale space since Remark that the extinction scale for a circle with initial radius R(0) is R(0) for any fl. On Figure 8, we display the evolution of a circle with radius 10, for 9. Figure 8: Evolution of circles for displayed at scale 0, 1,., 9 (fast computation). In Table 1, we give the theoretical and computed radius for several values of fl and for a circle with initial radius equal to 10. We performed two sets of experiments with different precisions, and for each of them we give the CPU time on a Pentium II 366MHz, the number of performed iterations and the final obtained radius. In Figures 9 and 10, we display the evolution of convex closed polygons. On each figure, the display scales are the same for all the different values of fl. 5.2 Unclosed curves evolution Until now, we have only studied the evolution of compact convex sets. The boundary of such sets is a closed convex curve. It is possible to make nonclosed convex curves evolve by fixing their end points as in [5]. This is equivalent to symmetrize and periodize the curve. The steady state to this evolution is a segment if the two ends are disjoint. If they are equal, then a singularity fast computation slow computation R theo R comp # iter CPU 2.0 6.47 6.39 112 1.22 6.45 421 27 3.0 7.65 7.56 108 1.38 7.64 400 28 10.0 9.66 9.62 100 1.19 9.66 214 15 Table 1: The shortening at scale 9 of an initial circle with radius 10 is considered. For different values of fl, we give the theoretical value of the final radius (R theo ), the corresponding values obtained with different precisions (fast or slow computation), the number of performed iterations and the used CPU time. Figure 9: Evolution of a triangle. Up-left: occurs in finite time. Such an evolution is displayed on Figure 11 for a nonclosed convex curve for several values of fl. As can be derived from [8, 9, 2], if C moves by Equation 25 and is locally the graph of a function, y then y satisfies the equation We use this equation to determine whether a convex curve with distinct fixed end points becomes a straight line in finite time. Proposition 24 Let u 0 be a strictly convex function on [\Gamma1; 1] with u Figure 10: Evolution of a pentagon. Up-left: be the solution of Then, if identically zero in finite time. If fl 1, then the steady state is attained for infinite time. Proof. From Equation (37), we deduce that u is subsolution of the following equation On the other hand, we can derive from Equation (37) an equation for u 0 . This equation is also parabolic. By maximum principle, the supremum of u 0 is attained at time that u is supersolution of For both Equations (38), (39), we can compute separable solutions of the type g(t)f(x). Both functions f and g then satisfy an ordinary differential equation that is explicitely solvable for g and can be expressed in terms of elliptic functions for f . We can then check that the time components becomes null in finite time if and only if fl ! 1. In order to conclude, it suffices to bound u 0 fromabove if 1 and from below if fl 1 by a adequate f (the spatial part of the separable solution) and to apply maximum principle. Figure 11: Evolution of an angle for different powers of the curvature 2:0). The displayed curves correspond to scales that are integer multiples of a same fixed value. Let us interpret this experiment in terms of image processing. Imagine that we process the filtering of a grey level image by applying (25) to its level lines, as done in [13] for This will have smoothing effects, and in particular one can expect to remove pixellization effects. The periodic structure corresponding to the initial state of Figure 5.2 is a "staircase" line that corresponding to the discretization of a perfect straight line (oriented at 45 ffi ). As the evolution scale increases, this infinite staircase is smoothed and eventually becomes a perfect straight line in finite time if fl ! 1. A natural question now arises: how choose fl in order to smooth these staircase effects with the smallest possible damaging effects on the image? In other terms, what power of the curvature regularizes discrete lines in the shortest time? The previous proposition asserts that only powers smaller than 1 can straigthen a staircase in finite time. Experiments of Figure 5.2 corroborate this result and indicate that the straightening time increases with fl. But this result has to be counterbalanced in the following. By smoothing an image, we would like to remove small undesirable details while keeping the rest of the image unchanged. The results and experiments on circles (Figure 8) tend to prove that large powers can do this in a better way than small powers (i.e. when a circle with initial radius 9 disappears at scale with radius 10 is less changed for large values of fl). 5.3 Generalization to nonconvex sets An algorithm for nonconvex curves has been proposed in [18] for the affine erosion apply the same method in the general case. It consists in splitting a curve into its convex and concave components. This decomposition is unique and well defined. By this method, we do not find inflexion points but inflexion segments and we define an inflexion point as the middle point of an inflexion segment. We then apply the erosion operator to all the convex components by fixing the end points (which are the inflexion points defined above). Once this is done we gather all the parts to form a new curve. We reapply the decomposition and the erosion to this new curve. Notice that near an inflexion point, the ending segments of the joining convex components do not stay parallel in general; thus, inflexion points have have no reason to stay still. Practically, this is what we observe. Moreover, as fl increases, they seem to move more and more slowly, which also seems logical. On Figures 12 and 13, we display the result of the algorithm on nonconvex curves. Nevertheless, we do not have any strict justification for the convex component decomposition and the displacement of inflexion points should be studied more carefully. Acknowledgements . The authors would like to thanks Jean-Michel Morel for all valuable conversations and advice. Figure 12: Scale space of a "T" shape: the curves are displayed at the same evolution scale. From left to right and top to bottom: fl =0.4, 1, 2, 3. CPU times are respectively 5, 9, 23 and Figure 13: Scale space of a hand curve: the curves are displayed at the same evolution scale. From left to right and top to bottom: initial curve,fl =0.4, 1, 2. CPU times are respectively 6, --R Axioms and fundamental equations of image processing. On the affine heat equation for nonconvex curves. Convergence of approximation schemes for fully nonlinear second order equations. Partial differential equations and mathematical morphology. Is scale space possible User's guide to viscosity solution of second order partial differential equations. Motion of level sets by mean curvature. The heat equation shrinking convex plane curves. The heat equation shrinks embedded plane curves to round points. Partial Differential Equations and Image Iterative Filtering. An approximation scheme for gauss curvature flow and its convergence. Generalized motion of noncompact hypersurfaces with velocity having arbitrary growth on the curvature tensor. Geometric multiscale representation of numerical images. Random Sets and Integral Geometry. Solution of nonlinear curvature driven evolution of plane convex curves. Traitement num'erique d'images et de films Affine plane curve evolution: A fully consistent scheme. Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulation Approximation de propagation de fronts avec ou sans termes locaux. Approximation of viscosity solution by morphological filters. Affine invariant scale space. On affine plane curve evolution. Curvature and the evolution of fronts. The generalized curve shortening problem. --TR --CTR L. Alvarez , A.-P. Blanc , L. Mazorra , F. Santana, Geometric Invariant Shape Representations Using Morphological Multiscale Analysis, Journal of Mathematical Imaging and Vision, v.18 n.2, p.145-168, March

curve evolution;image processing;level set methods;viscosity solutions;numerical approximation

588362

On the Accuracy of the Finite Volume Element Method Based on Piecewise Linear Polynomials.

We present a general error estimation framework for a finite volume element (FVE) method based on linear polynomials for solving second-order elliptic boundary value problems. This framework treats the FVE method as a perturbation of the Galerkin finite element method and reveals that regularities in both the exact solution and the source term can affect the accuracy of FVE methods. In particular, the error estimates and counterexamples in this paper will confirm that the FVE method cannot have the standard O(h2) convergence rate in the L2 norm when the source term has the minimum regularity, only being in L2, even if the exact solution is in H2.

Introduction . In this paper, we consider the accuracy of finite volume element methods for the following elliptic boundary value problem: Find such that where# is a bounded convex polygon in R 2 with boundary # uniformly positive definite matrix in # and the source term has enough regularity so that this boundary value problem has a unique solution in a certain Sobolev space. Finite volume (FV) methods have a long history as a class of important numerical tools for solving di#erential equations. In the early literature [26, 27] they were investigated as the so-called integral finite di#erence methods, and most of the results were given in one-dimensional cases. FV methods also have been termed as box schemes, generalized finite di#erence schemes, and integral-type schemes [20]. Generally speaking, FV methods are numerical techniques that lie somewhere between finite di#erence and finite element methods; they have a flexibility similar to that of finite element methods for handling complicated solution domain geometries and boundary conditions; and they have a simplicity for implementation comparable to finite di#er- ence methods with triangulations of a simple structure. More important, numerical solutions generated by FV methods usually have certain conservation features that # Received by the editors March 10, 2000; accepted for publication (in revised form) May 25, 2001; published electronically January 30, 2002. This work was partially supported by NSF grant DMS-9704621, U.S. Army grant ARO-39676-MA, and by the NSERC of Canada. http://www.siam.org/journals/sinum/39-6/36887.html Institute for Scientific Computation, Texas A&M University, College Station, (ewing@isc.tamu.edu). This author was supported by NSF grants DMS-9626179, DMS-9706985, DMS-9707930, NCR9710337, DMS-9972147, INT-9901498; EPA grant 825207; two generous awards from Mobil Research and Development; and Texas Higher Education Coordinating Board Advanced Research and Technology Program grants 010366-168 and 010366-0336. # Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 (tlin@math.vt.edu). - Department of Mathematics, University of Alberta, Edmonton, AL T6G 2G1 Canada (ylin@hilbert.math.ualberta.ca). are desirable in many applications. However, the analysis of FV methods lags far behind that of finite element and finite di#erence methods. Readers are referred to [3, 6, 17, 21, 22, 25] for some recent developments. The FVE method considered in this paper is a variation of the FV method, which also can be considered as a Petrov-Galerkin finite element method. Much has been published on the accuracy of FVE methods using conforming linear finite elements. Some early work published in the 1950s and 1960s can be found in [26, 27]. Later, the authors of [20] and their colleagues obtained optimal-order H 1 error estimates and superconvergence in a discrete H 1 norm. They also obtained L 2 error estimates of the following form: where u and u h comprise the solution of (1.1) and its FVE solution, respectively. Note that the order in this estimate is optimal, but its regularity requirement on the exact solution seems to be too high compared with that for finite element methods having an optimal-order convergence rate when the exact solution is in W 2,p or H Optimal-order H 1 estimates and superconvergence in a discrete H 1 norm also have been given in [3, 17, 21, 22, 25] under various assumptions on the above form for equations or triangulations. More recently, the authors of [7, 8] presented a framework based on functional analysis to analyze the FVE approximations. The authors in [11] obtained some new error estimates by extending the techniques of [20]. The authors of [14, 15] considered FVE approximations for parabolic integrodi#erential equations, covering the above boundary value problems as a special case, in both one and two dimensions. All the authors obtained optimal-order H 1 and W 1,# error estimates and superconvergence in H 1 and W 1,# norms. In addition, they found an optimal-order L # error estimate in the following form: which is in fact an error estimate without any logarithmic factor. However, all the estimates obtained by these authors require that the exact solution have H 3 regularity. To the best of our knowledge, there have been no results indicating whether the above W 3,p regularity is necessary for the FVE solution with conforming linear finite elements to have the optimal-order convergence rate. On the other hand, it is well known that in many applications the exact solution of the boundary value problem cannot have W 3,p or H 3 regularity. In fact, the regularities of the source f , the coe#cient, and the solution domain all can abate the regularity of the exact solution. A typical case is the regularity of the solution domain that may force the exact solution not to be in W 3,p or H 3 even for the best possible coe#cient A and source term f , such as constant functions. It has been noticed that the regularity of the source term may a#ect the convergence rate of an FVE solution. The counterexample in [18] showed that the FVE solution with the conforming linear elements cannot have the optimal L 2 convergence rate if the exact solution is in H 2 but the source term f is only in L 2 . On the other hand, the author of [6] found an optimal error estimate for the FVE solution with the nonconforming Crouzeix-Raviart linear element under the assumption that the exact solution is in H 2 and the source term f is in H 1 , but did not state whether this H 1 regularity of f is necessary for the FVE method presented there. The central aim of this paper is to show how, by both error estimates and coun- terexamples, the regularity of the source term f can a#ect the convergence rate of the FVE solution with conforming linear elements. The results indicate that, unlike the finite element method, the H 2 regularity of the exact solution indeed cannot guarantee the optimal convergence rate of the conforming linear FVE method if the source term has a regularity worse than H 1 , assuming that the coe#cient is smooth enough. Namely, we will present the following error estimate: which leads to the optimal convergence rate of the FVE method only if f # H # with # 1. Note first that, except for special cases such as when the dimension of# is one or the solution domain has a boundary smooth enough, the H 1 regularity of the source term does not automatically imply the H 3 regularity of the exact solution. On the other hand, the H 3 regularity of the exact solution will lead to the H 1 regularity of the source term when the coe#cient is smooth enough, and this error estimate reduces to one similar to estimates obtained in [11, 20]. Also, this error estimate is optimal from the point of view of the best possible convergence rate and the regularity of the exact solution. Moreover, counterexamples given in this paper indicate that the regularity of the source term cannot be reduced. Hence, we believe this is a more general error estimate than those in the literature. In fact, the FVE method is a Petrov-Galerkin finite element method in which the test functions are piecewise constant. As we will see later, the nonsmoothness in the test function demands a stronger regularity of the source term than the Galerkin finite element method. Also, our view of the FVE method as a Petrov-Galerkin finite element method suggests that we treat the FVE method as a perturbation of the Galerkin finite element method [6, 20] so that we can derive optimal-order L 2 , H 1 , and L # error estimates with a minimal regularity requirement just like finite element methods except for the additional smoothness assumption on the source term f . This error estimation framework also enables us to investigate superconvergence of the FVE method in both H 1 and W 1,# norms using the regularized Green's functions [23, 29] and to obtain the uniform convergence of the FVE method similar to that in [24] for the finite element method. To summarize, we observe that the FVE method not only preserves the local conservation of certain quantities of the solution (problem but also has optimal-order convergence rates in all usual norms. The additional smoothness requirement on the source term f is necessary due to the formulation of the method. The results of this paper can easily be extended to cover more complicated models. For example, most of the results and analysis framework are still valid if the di#erential equation contains a convection term # - (b u) (see [21] and [22]) and the symmetry of the tensor coe#cient A(x) is not critical. Also, one may consider Neumann and Robin boundary conditions on the whole or a part of the boundary # In fact, the FVE method was introduced in [2] as a consistent and systematic way to handle the flux boundary conditions for finite di#erence methods. We also refer readers to [1, 19] for FVE approximations of nonlinear problems, to [12] for an immersed FVE method to treat boundary value problems with discontinuous coe#cients, and to [13] for the mortar FVE methods with domain decomposition. This paper is organized as follows. In section 2, we introduce some notation, formulate our FVE approximations in piecewise linear finite element spaces defined on a triangulation, and recall some basic estimates from the literature. All error estimates are presented in the pertinent subsections of section 3. Section 4 is devoted to counterexamples demonstrating that smoothness of the source term is necessary in order for the FVE method to have the optimal-order convergence rate. 2. Preliminaries. 2.1. Basic notation. We will use the standard notation for Sobolev spaces consisting of functions that have generalized derivatives of order s in the spaces L p(#7 The norm of W s,p is defined by |#s with the standard modification for In order to simplify the notation, we denote s(# and skip the index and# whenever possible; i.e., we will use We denote by H 1 0(# the subspace of H 1(# of functions vanishing on the boundary # in the sense of traces. Finally, H -1(# denotes the space of all bounded linear functionals on H 1 0(#7 For a functional f # H its action on a function u 0(# is denoted by (f, u), which represents the duality pairing between H 0(#1 To avoid confusion, we use (-) to denote both the L 2(#89#401 product and the duality pairing between H For the polygonal domain# we now consider a quasi-uniform triangulation T h consisting of closed triangle elements K such K. We will use N h to denote the set of all nodes or vertices of T h , is a vertex of element K # T h and p # -# and we let N 0 For a vertex x i # N h , we denote by #(i) the index set of those vertices that, along with x i , are in some element of T h . We then introduce a dual mesh T # h based on T h ; the elements of T # h are called control volumes. There are various ways to introduce the dual mesh. Almost all approaches can be described by the following general scheme: In each element K # T h consisting of vertices x i , x j , and x k , select a point q in K, and select a point x ij on each of the three edges x i x j of K. Then connect q to the points x ij by straight lines # ij,K . Then for a vertex x i we let V i be the polygon whose edges are # ij,K in which x i is a vertex of the element K. We call V i a control volume centered at x i . Obviously, we have and the dual mesh T # h is then defined as the collection of these control volumes. Figure 1 gives a sketch of a control volume centered at a vertex x i . We call the control volume mesh T # h regular or quasi-uniform if there exists a positive constant C > 0 such that here h is the maximum diameter of all elements There are various ways to introduce a regular dual mesh T # h depending on the choices of the point q in an element K # T h and the points x ij on its edges. In Fig. 1. Control volumes with barycenter as internal point and interface # ij of V i and V j . this paper, we use a popular configuration in which q is chosen to be the barycenter of an element K # T h , and the points x ij are chosen to be the midpoints of the edges of K. This type of control volume can be introduced for any triangulation T h and leads to relatively simple calculations for both two- and three-dimensional problems. In addition, if T h is locally regular, i.e., there is a constant C such that dual mesh h is also locally regular. Other dual meshes also may be used. For example, the analysis and results of this paper for all the error estimates in the H 1 norm are still valid if the dual mesh is of the so-called Voronoi type [21]. 2.2. The FVE method. We now let S h be the standard linear finite element space defined on the triangulation T h , is linear for all K # T h and v| and its dual volume element space S # h , h and v| Obviously, h } and S # # i are the standard nodal basis functions associated with the node x i , and # i are the characteristic functions of the volume V i . Let I h : be the usual interpolation operators, i.e., I h where Then, the FVE approximation u h of (1.1) is defined as a solution to the following problem: Find u h # S h such that or h . Here the bilinear form a(u, v) is defined as follows: where n is the outer-normal vector of the involved integration domain. Note that the bilinear form a(u, v) has di#erent definition formulas according to the function spaces involved. We hope that this will not lead to serious confusion but rather will simplify tremendously the notation and the overall exposition of the material. To describe features of the bilinear forms defined in (2.3), we first define some discrete norms on S h and S # h , is the distance between x i and x j . In the lemmas below, we assume that the lines of discontinuity (if any) of the matrix A(x) are aligned with edges of the elements in the triangulation T h and that the entries of the matrix A(x) are C 1 -functions over each element of T h . Lemma 2.1 (see, e.g., [7, 21]). There exist two positive constants C 0 , C 1 > 0, independent of h, such that Lemma 2.2 (see, e.g., [7, 21]). There exist two positive constants C 0 , C 1 > 0, independent of h and h 0 > 0, such that for all 0 < h # h 0 , 3. Error estimates for the FVE method. 3.1. Optimal-order H 1 error estimates. We first consider the error of the FVE solution u h in the H 1 norm. We start with the following two lemmas. Lemma 3.1. For any u h , v h # S h , we have with j#Nh and Moreover, if A is in W 1,# , then there is a positive constant C > 0, independent of h, such that Proof. For the proof, see [12, 13]. Lemma 3.2. Assume that u h is the FVE solution defined by (2.1). Then we have Proof. The proof follows directly from Lemma 3.1. Theorem 3.3. Assume that u and u h are the solutions of (1.1) and (2.1), re- spectively, -1+# with 0 < # 1, and A # W 1,# . Then we have ||f ||u|| 1+# . Proof. By (3.1) and (1.1), we see that for # Notice that from Lemma 2.2 and the approximation theory we have the proof is then completed by combining these inequalities. Remark. The main idea in the proof above is motivated by [6], which is somewhat di#erent from those ideas in [3, 17, 20, 21, 25]. The approach is also more direct and simpler because the key identity (3.2) allows us to employ the standard error estimation procedures developed for finite element methods. In particular, the estimate for ||I h u - u h || is not needed in this proof. Moreover, the estimate here describes how the regularities of the exact solution and the source term can independently a#ect the accuracy of the FVE solution. 3.2. Optimal-order L 2 error estimates. In this section, we derive an optimal- order L 2 error estimate for the FVE method with the minimal regularity assumption for the exact solution u. This error estimate also will show how the error in the L 2 norm depends on the regularity of the source term. The following lemma gives another key feature of the bilinear form in the FVE method. Lemma 3.4. Assume that u h , v h # S h . Then we have Proof. It follows from Green's formula that and j#Nh Then the proof is completed by taking the di#erence of these two identities. Theorem 3.5. Assume that u and u h are the solutions of (1.1) and (2.1), respec- tively, and u # H 2 2,# . Then there exists a positive constant C > 0 such that Proof. Let 0 be the solution of # , and # . Then we have ||w|| 2 # ||u - u h || 0 . By Theorem 3.3 we have Then by Lemma 3.4, where the J i 's are defined for u h , w h # S h by and the continuity of #u - n on each #K is used. Since the dual mesh is formed by the barycenters, we have (w h - I # h w h so that where f K is the average value of f on K. Similarly, using the fact that A # W 2,# , we have For J 3 , according to the continuity of #u - n and the shape of the control volume, we have AK is a function designed in a piecewise manner such that for any edge E of a triangle and x c is the middle point of E. Since |A(x)- AK | # h||A|| 1,# , we have from Theorem 3.3 that Thus, it follows by taking w therefore, we have and the proof is completed. Corollary 3.6. Assume that u # H # with 0 < # 1, and A # W 2,# . Then we have Proof. Let f h be the L 2 projection of f into S h and consider S(u, linear operator from H s s > 0. For any (u, f) # H s -H -1+s , we let -1+s . Then, by Theorem 3.5, we have Hence, according to the theory of interpolation spaces [4, 5], we have which in fact is (3.7). Remark. When the source term f is in H 1 , the order of convergence in Theorem 3.5 is optimal with respect to the approximation capability of finite element space. Note that, in many applications, the H 1 regularity of f does not imply the W 3,# or regularity of the exact solution required by the L 2 norm error estimates in the literature. Moreover, counterexamples presented in the next subsection indicate that the regularity assumption on f cannot be reduced. The result in Theorem 3.5 reveals how the regularities of the exact solution and the source term can a#ect the error of the FVE solution in the L 2 norm, and this is a more general result than those in the literature. 3.3. Superconvergence in the H 1 norm. In a way similar to the finite element solution with linear elements, we can show that the FVE solution has a certain superconvergence in the H 1 norm when the exact solution has a stronger regularity and the partition used has a better quality. Specifically, throughout this subsection we assume that the involved partition for the FVE solution is uniform or piecewise uniform without any interior meeting points. This requirement might be relaxed (see, for example, [29]), but we would rather use this simpler assumption to present our basic idea. We first recall the following superconvergence estimates for the Lagrange interpolation [9, 28, 29, 30] from finite element theory. Lemma 3.7. Assume that u # W 3,p 0(# . We have Theorem 3.8. Assume that f # H 2,# . Then we have Proof. It follows from Lemma 3.7 that Following a similar argument used in the proof of Theorem 3.5, we see that because I h u - u h is in S h . The result of this theorem follows by combining these two inequalities. We can use one of the applications of the above superconvergence property of the FVE solution to obtain a maximum norm error estimate. Corollary 3.9. Under the assumptions of Theorem 3.8 and u # W 2,# 3(# , we have logh Proof. The proof follows from Theorem 3.8 and from the approximation theory stating that logh logh We remark that this result is not optimal with respect to the regularity required on the exact solution u. This excessive regularity can be removed according to the result in the following subsection. 3.4. Error estimates in maximum norm. Now we turn to the L # norm and W 1,# norm error estimates for the FVE solution. First, we recall from [10, 16, 23, 29] the definition and estimates on the regularized Green's functions. For a point 2(# to be the solution of the equation in# , is a smoothed #-function associated with the point z, which has the following properties: Let G z h be the finite element approximation of the regularized Green's function, i.e., a(G z -G z Following [29], for a given point z # we define # z G z by G z+#z -G z |#z| for any fixed direction L in R 2 , where #z//L means that #z is parallel to L. Clearly, # z G z satisfies The finite element approximation # z G z h of # z G z is then defined by It is well known that the functions G z and # z G z have the following properties [29]: For any w # H 1 where P h is an L 2 -projection operator on S h , i.e., Moreover, the following estimates have been established in the literature [10, 16, 23, 29]: #G z -G z # z G z ||G z ||# z G z #u# 2,# with constant C > 0 independent of h and z. First, let us consider the W 1,# norm error estimate. Theorem 3.10. Assume that u # W # , and A # W 1,# . Then there exist positive constants C > 0 and h 0 > 0 independent of u such that for Proof. It follows from (3.8) that For the second term on the right-hand side, we have (f, # z G z logh For the third term, by the definition of E h given in Lemma 3.1 and the fact that # z G z is a piecewise linear polynomial, we have logh ||u|| 1,# . Thus, we obtain logh so that we have for some h 0 > 0, such that 0 < h # h 0 , logh Applying this inequality and (3.13) in leads to the result of this theorem. The following theorem gives a maximum norm error estimate for the FVE solution. Theorem 3.11. Assume that u # W 1,# , and A # W 2,# . Then there exist constants C > 0 and h 0 > 0, independent of u, such that for all Proof. We follow an idea similar to the proof of the previous theorem, but we now use the regularized Green's function G z and its finite element approximation G z as follows: -G z -G z -G z The functionals J 1 , J 2 , and J 3 above are defined in the same way as given in the proof of Theorem 3.5. For J 1 (u h , G z h ), from (3.6) we have ||G z logh ||f || 1,# . Similarly, we have ||G z We know by Theorem 3.10 that # Ch logh Therefore, there exists a small h 0 > 0 such that for 0 < h # h 0 , logh As for J 3 (u h , G z ), we note that G z is a piecewise linear polynomial and Thus, it is easy to see from Theorem 3.10 and (3.11) that ||G z logh Combining the estimates obtained above for the J i 's, we have logh This together with (3.13) completes the proof. The following theorem gives a superconvergence property in the maximum norm for the FVE solution. Theorem 3.12. Under the same conditions as in Theorem 3.11, we have Proof. It follows from the properties of # z G z h and # z G z and from Lemma 3.7 that ||u|| 3,# z G z We see from (3.6) and (3.11) that logh ||f || 1,# . When h > 0 is small, we also have logh For h ), we have |# z G z because # z G z h is piecewise linear in each element K # T h . Finally, the proof is completed by combining the above estimates. 3.5. Uniform convergence for u in H 1 In many applications, the exact solution u of (1.1) may be in the space H 1(#5 but not in H 1+# for any # > 0. In this situation, the authors of [24] showed that for any # > 0, there exists h such that for all 0 < h # h 0 , we have for the Galerkin finite element solution u h # S h (or the Ritz projection of u into S h of the exact solution of (1.1)). This implies that u h converges to u uniformly even though there is no order of convergence for u h . The following theorem shows that the FVE solution also has this uniform convergence feature. Theorem 3.13. Assume that A is uniformly continuous and f # L . Let 0(# and u h # S h be the solutions of (1.1) and (2.1), respectively. Then for any # > 0, there exists h such that for all 0 < h # h # , the following holds: Proof. As in the proof in Theorem 3.3, we have uniformly continuous in # for any # 0 > 0, there exists h . Thus, by Lemma 3.7 we can take h # (0, h 0 ) to obtain is defined in Lemma 3.1. By Lemma 3.2, we have Thus it follows from the triangle inequality that Lemma 2 of [24] indicates that for any # 1 > 0, there exists h Notice that the constant C > 0 above is independent of u, f , and A; therefore, the theorem follows from the last two inequalities. 4. Counterexamples. In this section, we will present two examples to show that, when the source term f(x, y) is only in L 2(#7 the FVE solution generally cannot have the optimal second-order convergence rate even if the exact solution u(x, y) has the usual H 2 regularity. The first example is based on theoretical error estimates, while the second is presented through numerical computations. We also provide an example to corroborate the optimal error estimate obtained in this paper under the condition that the exact solution u is in H 2 and the source term f is in H 1 . 4.1. A one-dimensional example. First, we consider an example in one dimension, but is not in H 1 (0, 1) if 0 # < 1/2. Clearly this problem has an exact solution, x 2-# which is in the space H 2 (0, 1). Let T h be the uniform partition of the interval [0, 1] such that x 0, 1, . , N and x 1. Let S h be the piecewise linear finite element space. Let u f # S h be the finite element solution of (4.1) defined by and let u h be the FVE solution. Then we have with e #h Our main task is to show that there exists a constant C > 0 such that This inequality and together imply that the FVE solution cannot have the optimal L 2 norm convergence rate for 0 < # < 1/2. We start with the estimates of the error function e(x) at the nodes. Let G(x, y) be the Green's function defined by Then, we have x k-1/2 xk Now we will estimate the J l 's one by one under the assumptions that # . For J 1 and J 6 it easily follows from a simple calculation that For J 5 , using the definition of x # h and integration by parts, we have x -# dx # x 1-# x k+1/2 x 1-# dx # . Note that Thus there is a positive constant C 5 independent of h such that because of the error estimate for the trapezoidal quadrature formula. Now consider J 3 and J 4 . First rewrite J 3 x k-1/2 xk x k-1/2 x k-1/2 xk x k-1/2 x k-1/2 )xdx Clearly, we have and Hence For calculation similar to that for J 5 we have x 1-# x 1/2 x 1-# dx # . Letting applying the error formula for the trapezoidal quadrature rule, we have x 1/2 x 1/2 x x 1/2 x 1/2 Hence For J 7 we have x k-1/2 xk - 1. It is obvious that Hence Finally, it follows from the above estimates for the J i 's that there is a positive constant independent of h, such that for all x k # [1/3, 2/3], which in turn implies that for all small h > 0 due to the equivalence of the discrete and continuous norms on S h given in Lemma 2.1. This clearly indicates that the convergence rate of the FVE solution for this example cannot be O(h 2 On the other hand, our discussion in subsection 3.2 shows that the FVE solution can have the optimal convergence rate when the exact solution u is in H 2 and the source term f is in H 1 . This is supported by the following example. We consider the following boundary value problem: where a a Table errors of the FVE solutions for various partition sizes h. The boundary conditions are chosen so that dt is the exact solution to this boundary value problem. Note that u is piecewise smooth, u # is continuous, but u # is discontinuous at Hence, in this example, the right-hand side function f is H 1 (0, 1), but the exact solution to the boundary value problem is only in H 2 (0, 1). The L 2 errors of the FVE solutions with linear finite elements corresponding to various mesh sizes h are listed in Table 1. The involved calculations were carried out such that is one of the mesh points in the partitions used. Linear regression indicates that the data in this table satisfy which suggests the optimal convergence rate, and the data are in agreement with the error estimate obtain in subsection 3.2. 4.2. A two-dimensional example. We consider the following boundary value problem: where# is the unit square (0, 1) - (0, 1). It is easy to see that the exact solution to this boundary value problem is which is in H 2 but not in H 3(#2 On the other hand, the source term 5 is just in L We have applied the FVE method (2.1) to generate the FVE solution u h (x, y) to this boundary value problem by the usual uniform partition T h of the unit square with the partition size h. Due to the lack of regularity in the source term, an exact Table Errors of the FVE solutions for various partition sizes h. integration formula is used to carry out all the quadratures in (2.1) that involve the source term f(x, y). In fact, we can show that for each triangle #A 1 A 2 A 2 with vertices z 3 we have f(x)dx with Note that this formula is valid only if the vertices of the triangle #A 1 A 2 A 2 have distinct coordinate values. This is true when #A 1 A 2 A 2 is a triangle used in the integration over a control volume. Table 2 contains the errors of the FVE solutions for this boundary problem with various typical partition sizes h. In this table, is the usual L 2 error of an FVE solution u h (x, y). Obviously, the FVE solutions in these computations do not seem to have the standard second-order convergence because the error is not reduced by a factor of 4 when the partition size is reduced by a factor of 2. Also see the counterexample in [18]. 5. Conclusion. In this paper, we have considered the accuracy of FVE methods for solving second-order elliptic boundary value problems. The approach presented herein combines traditional finite element and finite di#erence methods as a variation of the Galerkin finite element method, revealing regularities in the exact solution and establishing that the source term can a#ect the accuracy of FVE methods. Optimal- estimates and superconvergence also have been discussed. The examples presented above show that the FVE method cannot have the standard O(h 2 ) convergence rate in the L 2 norm when the source term has the minimum regularity in L 2 , even if the exact solution is in H 2 . --R A box scheme for coupled systems resulting from microsensor thermistor problems A new finite-element formulation for convection-di#usion problems Some error estimates for the box method Notes Math. The Analysis of A finite Volume On the finite Volume On the accuracy of the finite Volume Element analysis method and superconvergence High Accuracy Theory of Finite Element Methods estimates in L 2 The immersed finite Volume The Mortar Finite Volume Finite Volume Finite Volume Eine On first and second order box schemes On the finite Volume Piecewise linear Petrov-Galerkin error estimates for the box method Generalized Di Finite Volume Finite Volume Some optimal error estimates for piecewise linear finite element approximations Some new estimates for Ritz-Galerkin methods with minimal regularity assumptions Homogeneous di Homogeneous di Superconvergence in Galerkin Finite Element Methods Superconvergence Theory for Finite Element Methods A survey of superconvergence techniques in finite element methods --TR --CTR Zhoufeng Wang , Zhiyue Zhang, The characteristic finite volume element methods for the two-dimensional generalized nerve conduction equation, Neural, Parallel & Scientific Computations, v.15 n.1, p.27-44, March 2007 Rajen K. Sinha , Jrgen Geiser, Error estimates for finite volume element methods for convection-diffusion-reaction equations, Applied Numerical Mathematics, v.57 n.1, p.59-72, January 2007 Chunjia Bi , Hongxing Rui, Uniform convergence of finite volume element method with Crouzeix-Raviart element for non-self-adjoint and indefinite elliptic problems, Journal of Computational and Applied Mathematics, v.200 n.2, p.555-565, March, 2007 Guoliang He , Yinnian He, The finite volume method based on stabilized finite element for the stationary Navier-Stokes problem, Journal of Computational and Applied Mathematics, v.205 n.1, p.651-665, August, 2007

finite volume;error estimates;elliptic;counterexamples

588405

A Mortar Finite Element Method Using Dual Spaces for the Lagrange Multiplier.

The mortar finite element method allows the coupling of different discretization schemes and triangulations across subregion boundaries. In the original mortar approach the matching at the interface is realized by enforcing an orthogonality relation between the jump and a modified trace space which serves as a space of Lagrange multipliers. In this paper, this Lagrange multiplier space is replaced by a dual space without losing the optimality of the method. The advantage of this new approach is that the matching condition is much easier to realize. In particular, all the basis functions of the new method are supported in a few elements. The mortar map can be represented by a diagonal matrix; in the standard mortar method a linear system of equations must be solved. The problem is considered in a positive definite nonconforming variational as well as an equivalent saddle-point formulation.

Introduction . Discretization methods based on domain decomposition techniques are powerful tools for the numerical approximation of partial differential equa- tions. The coupling of different discretization schemes or of nonmatching triangulations along interior interfaces can be analyzed within the framework of the mortar methods [6, 7]. In particular, for time dependent problems, diffusion coefficients with jumps, problems with local anisotropies as well as corner singularities, these domain decomposition techniques provide a more flexible approach than standard conforming formulations. One main characteristic of such methods is that the condition of pointwise continuity across the interfaces is replaced by a weaker one. In a standard primal approach, an adequate weak continuity condition can be expressed by appropriate orthogonality relations of the jumps of the traces across the interfaces of the decomposition of the domain [6, 7]. If a saddle point formulation arising from a mixed finite element discretization is used, the jumps of the normal components of the fluxes are relevant [29]. To obtain optimal results, the consistency error should be at least of the same order as the best approximation error. Most importantly, the quality of the a priori error bounds depends strongly on the choice of weak continuity conditions at the interfaces. Section 2 contains a short overview of the mortar finite element method restricted to the coupling of P 1 -Lagrangian finite elements and a geometrically conforming sub-division of the given region. We briefly review the definition of the discrete Lagrange multiplier space and the weak continuity condition imposed on the product space as it is given in the literature. In Section 3, we introduce local dual basis functions, which span the modified Lagrange multiplier space. We also give an explicit formula of projection-like operators and establish stability estimates as well as approximation properties. Section 4 is devoted to the proof of the optimality of the modified nonconforming variational problem. It is shown that we can define a nodal basis function satisfying the constraints at the interface and which at the same time has local support. This is a great advantage of this modified method compared with the Math. Institut, Universitat Augsburg, Universitatsstr. 14, D-86 159 Augsburg, Germany. Email: wohlmuth@math.uni-augsburg.de, http://wwwhoppe.math.uni-augsburg.de/~wohlmuth standard mortar methods. Central results such as uniform ellipticity, approximation properties and consistency error are given in separate lemmas. A saddle point formu- lation, which is equivalent to these nonconforming variational problems is considered in Section 5. Here, the weak continuity condition at the interface enters explicitly in the variational formulation. As in the standard mortar case, we obtain a priori estimates for the discretization error for the Lagrange multiplier. Here, we analyze the error in the dual norm of H 1=2 00 , as well as in a mesh dependent L 2 -norm. Finally, in Section 6, numerical results indicate that the discretization errors are comparable with the errors obtained when using the original mortar method. 2. Problem setting. We consider the following model problem where\Omega is a bounded, polygonal domain in IR 2 , and that f 2 L Furthermore, we assume a 2 L 1 (\Omega\Gamma to be an uniformly positive function and 0 b 2 L We will consider a non-overlapping decomposition of\Omega into polyhedral subdo- Each subdomain\Omega k is associated with a family of shape regular simplicial triangulations is the maximum of the diameters of the elements in T hk . The sets of vertices and edges of the subdomains\Omega k and of\Omega are denoted by P hk , respectively. We use P 1 -conforming finite elements individual subdomains and enforce the homogeneous Dirichlet boundary conditions on @\Omega k . We restrict ourselves to the geometrical conforming situation where the intersection between the boundary of any two different subdomains l, is either empty, a vertex, or a common edge. We only call it an interface in the latter case. The mortar method is characterized by introducing Lagrange multiplier spaces given on the interfaces. A suitable triangulation on the interface is necessary for the definition of a discrete Lagrange multiplier space. Each interface @\Omega k is associated with a one dimensional triangulation, inherited either from T hk or from T h l . In general, these triangulations do not coincide. The interface in question will be denoted by its triangulation is given by that and\Omega l , respectively. We call the inherited one dimensional triangulation on \Gamma kl and \Gamma lk , \Sigma kl and \Sigma lk , respectively with the elements of \Sigma kl and \Sigma lk being edges of T hk and T h l , respectively. We remark that geometrically \Gamma lk and \Gamma kl are the same. Thus, each @\Omega k can be decomposed, without overlap, into where denotes the subset of f1; Kg such that @\Omega k is an interface for l 2 M(k): The union of all interfaces S can be decomposed uniquely in A Mortar Method Using Dual Spaces 3 Here, M(k) such that for each set fk; lg, 1 k K, l 2 l 2 M(k) or k 2 M(l) but not both. The elements of f\Gamma kl j 1 k K; l 2 M(k)g are called the mortars and those of f\Gamma lk j 1 k K; l 2 M(k)g the non-mortars. The choice of mortars and non-mortars is arbitrary but fixed. We note that the discrete Lagrange multiplier space will be associated with the non-mortars. To simplify the analysis, we will assume that the coefficients a and b are constant in each subdomain, with a k := a It is well known that the unconstrained product space Y is not suitable as a discretization of (2.1). We also note that in case of non-matching meshes at the interfaces, it is in general not possible to construct a global continuous space with optimal approximation properties. It is shown [6, 7] that weak constraints across the interface are sufficient to guarantee an approximation and consistency error of order h if the weak solution u is smooth enough. The nonconforming formulation of the mortar method is given by: Find ~ u h 2 e h such that where a(v; w) := R\Omega arv 1(\Omega Here, the global space e V h is defined by e where the bilinear form b(\Delta; \Delta) is given by the duality pairing on S Y 1(\Omega Y Y and [v] := v j\Omega l . Here, (H 1 denotes the dual space of H 1 Of crucial importance is the suitable choice of f M h in (2.2) f Y Y f where in general the local space f chosen as a modified trace space of finite element functions in S can be found satisfying is the trace space of S f subspace of W h (\Gamma lk ) of codimension 2 and given by f contains an endpoint of \Gamma lk g and N lk := dim f denotes the number of elements in \Sigma lk . Here, we assume that N e 2. The nodal basis functions fOE i g N lk of f are associated with the interior vertices given by The space f its nodal basis functions fOE i g N lk are illustrated in Figure 2.1; for a detailed analysis of f Fig. 2.1. Lagrange multiplier space Let us remark that continuity was imposed at the vertices of the decomposition in the first papers about mortar methods. However, this condition can be removed without loss of stability. Both these settings guarantee uniform ellipticity of the bilinear form a(\Delta; \Delta) on e as well as a best approximation error and a consistency error of O(h) [6, 7]. Combining the Lemmas of Lax Milgram and Strang, it can be shown that a unique solution of (2.2) exists and that the discretization error is of order h if the solution of (2.1) is smooth; see [6, 7]. In a second, equivalent approach the space f explicitly plays the role of a Lagrange multiplier space. This approach is studied in [4] and used further in [11, 25, 26]. The resulting variational formulation gives rise to a saddle-point problem: Find M h such that In particular, it can be easily seen that the first component of the solution of (2.3) is the unique solution of (2.2). Observing that ~ h is an approximation of the normal derivative of u on the interface, it makes sense to consider a priori estimates for suitable norms. Here n lk is the outer unit normal restricted to lk . This issue was first addressed in [4] where a priori estimates in the (H 1=2 were established. Similar bounds are given in [26] for a weighted L 2 -norm. As in the general saddle-point approach [13], the essential point is to establish adequate inf-sup conditions; such bounds have been established with constants independent of h for both these norms; see [4, 26]. In the following, all constants are generic depending on the local ratio between the coefficients b and a, the aspect ratio of the elements and subdomains but not on the mesh size and not on a. We use standard Sobolev notations and Y stand for the broken H 1 -norm and semi-norm. The dual space of a Hilbert space X is denoted by X 0 and the associated dual norm is defined by (2. A Mortar Method Using Dual Spaces 5 3. Dual basis functions. The crucial point for the unique solvability of (2.2) and (2.3) is the definition of the discrete space f M h . As we have seen, the discrete space of Lagrange multipliers is closely related to the trace space in the earlier work on mortar methods; these spaces are only modified in the neighborhood of the interface boundaries where the degree of the elements of the test space is lower. We note that it has been shown only recently, see [23], that for Pn -conforming finite elements the finite dimensional space of piecewise polynomials of only degree can be used instead of degree n in the definition of the Lagrange multiplier space without losing the optimality of the discretization error . However, in none of these studies has duality been used to construct an adequate finite element space for the approximation of the Lagrange multiplier. We recall that the Lagrange multiplier in the continuous setting represents the flux on the interfaces. Even if the weak solution of (2.1) is does not have to be continuous on the interfaces. This observation has motivated us to introduce a new type of discrete Lagrange multiplier space. We note that local dual basis functions have been used in [22] to define global projection-like operators which satisfy stability and approximation properties; in this paper we use the same dual basis functions to define the discrete Lagrange multiplier space. Let oe be an edge and e be a polynomial space satisfying P 0 (oe) ae e 2g, be a basis satisfying R oe OE oe;i ds 6= 0. We can then define a dual basis f/ oe;i g N by the following relation Z oe Z oe The definition (3.5) guarantees that f/ oe;i g N is well defined. Each / oe;i can be written as a linear combination of the OE oe;i , 1 i N and the coefficients are obtained by solving a N \Theta N mass matrix system. Furthermore (3.5) yields Z oe ds and thus 1. The f/ oe;i g N also form a linearly independent set. To see this, let us assume that R oe ds R oe 0: As a consequence, we obtain e Ng. Let us consider the case that f oe;i g 2 are the nodal basis functions of Then, the dual basis is given by 2: Based on these observations, we introduce a global space M h (\Gamma lk ) on each non- 6 BARBARA I. WOHLMUTH be the nodal basis function of f introduced in Section 2. Then, each OE i can be written as the sum of its local contributions =: OE oe;i where the local contributions are linearly independent. We set e g. In particular by construction, it is guaranteed that P 0 (oe) ae e Using the local dual basis functions on each oe, the global basis functions of M h (\Gamma lk ) are defined as The support of / i is the same as that of OE i and the f/ i g N lk form a linear independent system. Figure 3.2 depicts the two different types of dual basis functions. G lk G lk Fig. 3.2. Dual basis functions in the neighborhood of the boundary of \Gamma lk (left) and in the interior (right) Remark 3.1. The following global orthogonality relation holds Z Z oe Z oe Z We note the similarity with (3.5). The central point in the analysis of the consistency and approximation error will be the construction of adequate projection-like operators. We refer to [6, 7] for the standard mortar approach. Here, we use different operators onto M h (\Gamma lk ), f and W h (\Gamma lk defined by R ds R ds A dual operator M h (\Gamma lk ), is now given by R ds R ds A detailed discussion of this type of operator can be found in [22]. It is easy to see that P lk and Q lk restricted to M h (\Gamma lk ) and f respectively, is the identity A Mortar Method Using Dual Spaces 7 In addition, using (3.6), (3.7) and (3.8), we find that for v; w 2 L 2 (\Gamma lk ), and it therefore makes sense to call Q lk a dual operator to P lk . Furthermore, the operators are L 2 -stable. We have 0;oe R ds R ds R where the domain D oe is defined by Here, we have used the fact that D oe contains at most three elements and that independently of the length of the edges. The same type of estimate holds true for Q lk 0;oe R ds R ds R 0;D oe Thus, P lk and Q lk are L 2 -projection-like operators which preserve the constants. Using (3.9) and (3.10), it is easy to establish an approximation result. Lemma 3.2. There exist constants such that for C Proof. The proof of (3.12) follows by applying the Bramble-Hilbert Lemma and using the stability (3.10) and the identity (3.9); it is important that the constants are contained in the space M h (\Gamma lk ). For each v we define a constant c v in the following way Z where jD oe j is the length of D oe . We remark that the constant c v depends only on the values of v restricted on D oe . Now, by means of P lk c oe jvj s;D oe : The global estimate (3.12) is obtained by summing over all local contributions and observing that each oe 0 is only contained in a fixed number of D oe . Although dim f we get the same type of estimate as (3.14) for Q lk instead of P lk by using (3.11). For the estimate (3.13) in the dual norm, we use the definition (2.4) R ds R ds In a next step, we consider the last integral in more detail. Using (3.14) for Q lk instead of P lk and setting 0;oe CjOEj 2 (D oe ) Summing over all oe 2 \Sigma lk and using that the sum over jOEj 2 is bounded by Combining this upper bound with (3.14) gives (3.13). 4. Nonconforming formulation. Replacing the space f M h in the definition of e we get a new nonconforming space The original nonconforming variational problem (2.2) is then replaced by: Find h such that In what follows, we analyze the structure of an element v h 2 V h . Each v 2 X h restricted to a non-mortar side \Gamma lk can be written as are defined in Section 2 and OE 0 and OE N lk +1 are the nodal basis functions of W h (\Gamma lk ) associated with the two endpoints of \Gamma lk . The following lemma characterizes the elements of V h . Lemma 4.1. Let v 2 X h restricted on \Gamma lk be given as in (4.17). Then, and only if for each non-mortar \Gamma lk R (v ds R ds The proof follows easily from (4.17) and the global orthogonality relation (3.6). As in case of e h the values of a function v 2 V h at the nodal Lagrange interpolation points in the interior, p i , 1 i N lk , of any non-mortar \Gamma lk are uniquely determined by its values on the corresponding mortar side \Gamma kl and the values at the endpoints of A Mortar Method Using Dual Spaces 9 . The nodal values in the interior of the non-mortars \Gamma lk are obtained by combining (4.18) with a basis transformation. In particular, these values can be directly obtained by the simple formula j\Omega l R ds R ds For the two interior nodal points p 1 and pN lk next to the endpoints of \Gamma lk , we get j\Omega l R (v ds R ds j\Omega l (p N lk R (v j\Omega l (pN lk +1 )OE N lk +1 ) ds R ds Here, we have used that v j\Omega l j\Omega l (p N lk +1 and that are identically 1 on the edges next to the endpoints of \Gamma lk . We note that by definition of the basis functions there exist Z Z Z Z OE N lk ds: If we have a closer look at the nodal basis functions of e we realize that there is a main difference in the structure of the basis functions. Figures 4.3 and 4.4 illustrate this difference for the special situation that we have a uniform but nonmatching triangulation on the mortar and the non-mortar side. nodal basis functions mortar side nodal basis functions non-mortar side nodal basis functions non-mortar side Fig. 4.3. Nodal basis function on a mortar side (left) and on the non-mortar side in e and in V h (right) In Figure 4.3, the mortar side is associated with the finer triangulation whereas in Figure 4.4 it is associated with the coarser one. As in the standard finite element context, nodal basis functions can be defined for contained in a circle of diameter Ch: This is in general not possible for e h . In the latter case, the support of a nodal basis function associated with a nodal point on the mortar side is a strip of length j\Gamma lk j and width h, see Figure 4.5, and the locality of the basis functions is lost. We conclude this section, by establishing a priori bounds for the discretization error. As in [6, 7] a mortar projection will be a basic tool in the analysis of the best approximation error. We now use the new Lagrange multiplier space M h in nodal basis functions mortar side0.20.61 nodal basis functions non-mortar side0.20.61 nodal basis functions non-mortar side Fig. 4.4. Nodal basis function on a mortar side (left) and on the non-mortar side in e and in V h (right) lk G lk Fig. 4.5. Support of a nodal basis function in e the definition of suitable projection-like operators. For each non-mortar side the new mortar projection is given by \Pi Z By using (4.19) and (4.20), it can be easily seen that the operator \Pi lk is well defined. To analyze the approximation error, it is sufficient to show that the mortar projection is uniformly stable in suitable norms. The stability in the L 2 - and H 1 -norms is given in the following lemma. Lemma 4.2. The mortar projection \Pi lk is L 2 - and H 1 -stable Proof. Using the explicit representation (4.19) and (4.20) where v has to be replaced by v and v j\Omega l j\Omega l (p N lk +1 ) have to be set to zero, (4.22) is obtained. It can be easily seen that even the local estimate holds true. By means of an inverse inequality, we find for each p 2 W h (\Gamma lk satisfying const. and R ds R ds We remark that if @D oe " @ \Gamma lk 6= ;, then p was set to zero. However, due to the boundary conditions of v we obtain kvk 0;D oe Ch oe jvj 1;D oe in this case. A Mortar Method Using Dual Spaces 11 The mortar projection can be extended to the space H 1=2 in the following way: Z Then, an interpolation argument together with Lemma 4.2 gives the H 1=2 -stability Ckvk It is of interest to compute the stability constant in (4.22) in the special case of a uniform triangulation of \Gamma lk with h := joej. Then, are the two endpoints of oe. Using the mortar definition and summing over all elements in \Sigma lk , we get In general, the constants in a priori estimates depend on the coefficients. Here, we will give a priori estimates which depends explicitly on the coefficient a. For each ff k in the following way sup min min a j ~ sup a j We note that ff k and ~ ff k are bounded by 2 if the non-mortar side is chosen as that with the smaller value of a. The uniform ellipticity of the bilinear form a(\Delta; \Delta) on V h \Theta V h is important for the a priori estimates. For the standard mortar space, it is well known, see [6, 7, 8]. Moreover in [8], it is shown that the bilinear form a(\Delta; \Delta) is uniform elliptic on Y \Theta Y , where Y 1(\Omega Z The starting point of the proof is a suitable Poincar'e-Friedrichs type inequality. For general considerations on Poincar'e-Friedrichs type inequalities in the mortar situation, we refer to [24]. In [17, Theorem IV.1], it is shown that the ellipticity constant does not depend on the number of subdomains. A similar estimate is given for the three field formulation in [14]. We refer to [17] for a detailed analysis of the constants in the a priori estimates in terms of the number of subdomains and their diameter. Observing that V h is a subspace of Y , it is obvious that that the bilinear form a(\Delta; \Delta) on V h \Theta V h is uniform elliptic. 4.1. Approximation property. To establish an approximation property for h , we follow [6, 7]. One central point in the analysis is an extension theorem. In [9], a discrete extension is used such that the H 1 -norm of the extension on\Omega k is bounded by a constant times the H 1=2 -norm on the boundary @\Omega k . The support of such an extension is in general\Omega k and it is assumed that the triangulation is quasi- uniform. However, it can be generalized to the locally quasi-uniform case. Combining the approximation property of using the mortar projection \Pi lk , we obtain the following lemma. Lemma 4.3. Under the assumption that u 2 the best approximation error is of order h s , vh 2Vh k a k kuk 2 where the ff k are defined in (4.25). Proof. The proof follows exactly the same lines as for e h and the Laplace operator; we therefore omit the details and refer to [6, 7]. For each subdomain\Omega k , we use the Lagrange interpolation operator I k . Then, we define w I k v. We note that w h is not in general, contained in V h . To obtain an element in V h , we have to add appropriate corrections. For each interface \Gamma lk , we consider the jump apply the mortar projection. The result is extended as a discrete harmonic function into the interior of\Omega l . Finally, we define H l (\Pi lk [w h ]) where H l denotes the discrete harmonic extension operator kH l vk1;\Omega l Ckvk H 1(@\Omega l ) see [9, Lemma 5.1]. Here, \Pi lk [w h ] is extended by zero onto vanishes outside\Omega l . By construction, we have Z Z and thus v h 2 V h . A coloring argument yields, a l k\Pi lk [w h ]k 2 C a k h 2s a l k\Pi lk [w h ]k 2 C a k h 2s a l C a k h 2s a l l kuk 2 C ff k a k h 2s A Mortar Method Using Dual Spaces 13 Here, we have used the stability of the harmonic extension; see [9], the stability of the mortar projection (4.24) and the approximation property of the Lagrange interpolant. 4.2. Consistency error. The space V h is in general not a subspace of H 1 Therefore, we are in a nonconforming setting and in addition to uniform ellipticity and the approximation property we need to consider the consistency error [10] to obtain a stable and convergent finite element discretization. In Strang's second Lemma, the discretization error is bounded by the best approximation error and the consistency error [10]. Lemma 4.4. The consistency error for [arun lk is of order h s sup R a @u @n lk ds k a k kuk 2 1where ff k is defined in (4.25). Proof. The proof generalizes that given for e V h in [6, 7]. Here, the Lagrange multiplier space M h is used and we also consider the effect of discontinuous coefficients. By the definition of V h , we have Z and thus Z a @n lk Z (a @n lk @n lk ds: where P lk is defined in (3.7). Using a duality argument and the continuity of the trace, we get Z a @n lk ds where := a @u @n lk . To replace, in the last inequality, the H 1 -norm by the H 1 -semi- norm, we take into account that j\Omega l where \Pi lk denotes the L 2 -projection onto piecewise constant functions on \Gamma lk . In the duality argument, the H 1=2 -norm can therefore be replaced by the H 1=2 -semi-norm kw hj\Omega l j\Omega l C j\Omega l j\Omega l Finally, Lemma 3.2, which states the approximation property of P lk in the (H 1=2 norm, yields l jj H l min(a l juj 14 BARBARA I. WOHLMUTH Remark 4.5. In case that the coefficient a is smaller on the non-mortar side then ff k is bounded by 2 independently of the jumps in a. Otherwise the upper bound for ff k depends on the jumps in a. A possibly better bound might depend on the ratio of the mesh size across the interface; see (4.25). However, numerical results have shown that in case of adaptive mesh refinement controlled by an a posteriori error estimator, remains bounded independently of the jump in the coefficients; see [26, 27]. Using Lemmas 4.3 and 4.4, we obtain a standard a priori estimate for the modified mortar approach (4.16). Under the assumptions that [arun lk ff k a k h 2(s\Gamma1) Remark 4.6. The a priori estimates in the literature [6, 7] are often given in the following form This is weaker than the estimate (4.26), since for generally we only have K 4.3. A priori estimates in the L 2 -norm. Finite element discretizations pro- vide, in general, better a priori estimates in the L 2 -norm than in the energy norm. In particular, if we assume H 2 -regularity, we have the following a priori estimate for in the L 2 -norm The proof can be found in [11] and is based on the Aubin-Nitsche trick. In addition, the nonconformity of the discrete space has to be taken into account. An essential role in the proof of the a priori bound is the following lemma. It shows a relation between the jumps of an element across the interfaces and its nonconformity. The same type of result for v 2 e V h can be found in [26]. Lemma 4.7. The weighted L 2 -norm of the jumps of an element v 2 V h is bounded by its nonconformity a l 0;oe inf ~ Proof. The proof follows the same ideas as in case for v h 2 e use the orthogonality of the jump and the definition (3.8) to obtain j\Omega l Now, it is sufficient to consider an interface \Gamma lk at a time, and we find a l a l j\Omega l j\Omega l (v A Mortar Method Using Dual Spaces 15 Using (3.15) and the continuity of the trace operator, we get for each w a l a l jv j\Omega l a l jv Summing over the subdomains and using the definition for ~ ff k give the assertion. Using the dual problems: Find h such that gives a @w @n lk ds Z a @u @n lk ds Then, the H 2 -regularity, Lemma 3.2, Lemma 4.7, and observing that the jump of an element in V h is orthogonal on M h yield a l 0;oe a l 0;oe a l 0;oe a l 0;oe := a @w @n lk . Using the a priori estimate for the energy norm (4.26), we obtain an a priori estimate for the L 2 -norm. The following lemma gives the a priori estimate for the modified mortar approach. Lemma 4.8. Assuming H 2 -regularity, the discretization error in the L 2 - norm is of order h 2 . 5. Saddle point formulation. A saddle point formulation for mortar methods was introduced in [4]. In particular, a priori estimates involving the (H 1=2 for the Lagrange multiplier were established in that paper whereas estimates in a weighted L 2 -norm were given in [26]. Here, we analyze the error in the Lagrange multiplier for both norms and obtain a priori estimates of the same quality as for the standard mortar approach. The norm for the Lagrange multiplier is defined by l2M(k)a l Y Y The weight a \Gamma1 l is related to the fact that we use the energy norm for in the a priori estimates. Working within the saddle point framework, the approximation property on V h , which is given in Lemma 4.3, is a consequence of the approximation property on X h , the continuity of the bilinear form b(\Delta; \Delta), and an inf-sup condition [13]. A discrete inf-sup condition is necessary to obtain a priori estimates for the Lagrange multiplier. The saddle point problem associated with the new nonconforming formulation (4.16) involves the space (X We get a new saddle point problem, with exactly the same structure as (2.3): Find The inf-sup condition, established in [4] for the pairing (X Lemma 5.1. There exists a constant independent of h such that sup c: Proof. Using the definition of the dual norm (2.4), we get OE2H2 OE2H2 C sup OE2H2 The maximizing element in W h (\Gamma lk called and a v lk 2 X h is defined in the following way j\Omega n\Omega l \Omega l where OE h is extended by zero on We then find and a(v lk 21;\Omega l . Finally, we set a l and observe that a(v h ; v h only if coloring argument gives Summing over all interfaces yields C l2M(k)a l By construction, we have found for each C A Mortar Method Using Dual Spaces 17 The proof of the inf-sup condition (5.28) together with the approximation Lemma 3.2 and the first equation of the saddle point problem gives an a priori estimate similar to (4.26) for the Lagrange multiplier. Lemma 5.2. Under the assumptions u 2 Q K [arun lk the following a priori estimate for the Lagrange multiplier holds true C ff k a k h 2(s\Gamma1) Proof. Following [4] and using the first equation of the saddle point problem, we get Taking (5.29) into account, we find that the inf-sup condition even holds if the supremum over X h is replaced by the supremum over a suitable subspace of X h . For the proof of (5.30), we start with (5.29) and not with the inf-sup condition (5.28) constructed as in the proof of Lemma 5.1. We recall that w is defined as a linear combination of discrete harmonic functions a l where w 1. A coloring argument shows that the energy norm of w is bounded by the H 1=2 00 -dual norm of h \Gamma h , moreover we find a l a l K l2M(k)a l combining (5.31) and (5.32), we obtain kwk Applying the triangle inequality, choosing h using we find that (3.13) yields, for C a l C ff k a k h 2 Here, we have used that restricted on \Gamma lk is arun lk and a trace theorem. We note that in spite of Lemma 3.2 we cannot obtain a priori estimates of order h for the norm of the dual of H 1=2 (S). This is due to the fact that the inf-sup condition (5.28) cannot be established for that norm. Remark 5.3. The a priori estimate (5.30) also holds if we replace the (H 1=2 norm by the weighted L 2 -norm a l Using (3.12) and the techniques of the proof of Lemma 5.2, it is sufficient to have a discrete inf-sup condition similar to (5.28) for the weighted L 2 -norm, i.e. sup c: The only difference in the proof is the definition of v lk . Instead of using a discrete harmonic extension onto\Omega l , we use a trivial extension by zero, i.e. we set all nodal values on @\Omega l n\Gamma lk and on\Omega l to zero. Then, v lk is non zero only on a strip of length j\Gamma lk j and width h l and a(v lk ; v lk ) is bounded form below and above by a l 0;oe . 6. Numerical results. We get a priori estimates of the same quality for the error in the weak solution and the Lagrange multiplier as in the standard mortar case [4, 6, 7]. In contrast to e , we can define nodal basis functions for V h which have local supports. Efficient iterative solvers for linear equation systems arising from mortar finite element discretization are very often based on the saddle point formulation or work with the product space X h instead of the nonconforming mortar space. Different types of efficient iterative solvers are developed in [1, 2, 3, 11, 15, 16, 19, 20, 18, 25]. However, most of these techniques require that each iterate satisfies the constraints exactly. In most studies of multigrid methods, these constraints have to be satisfied even in each smoothing step [11, 12, 18, 25]. If we replace e the constraints are much easier to satisfy, since instead of solving a mass matrix system, the nodal values on the non-mortar side can be given explicitly. Fig. 6.6. Decomposition and initial triangulation (left) and solution (right) (Example 1) Here, we will present some numerical results illustrating the discretization errors for the standard and the new mortar methods in the case of P 1 Lagrangian finite elements. We recall that in the standard mortar approach the Lagrange multipliers belong to f M h whereas we use M h in the new method. We have used a multigrid method which satisfies the constraints in each smoothing step; see [11, 25] for a A Mortar Method Using Dual Spaces 19 discussion of the standard mortar case. This multigrid method can be also applied without any modifications to our modified mortar setting. It does not take advantage of the diagonal mass matrix on the non-mortar side of the new formulation. To obtain a speedup in the numerical computations, special iterative solvers for the new mortar setting have to be designed. We will address this issue in a forthcoming paper [28]. We start with an initial triangulation T 0 , and obtain the triangulation T l on level l by uniform refinement of T l\Gamma1 . Both discretization techniques have been applied to the following test example: where the right hand side f and the Dirichlet boundary conditions are chosen so that the exact solution is yy. The solution and the initial triangulation are given in Figure 6.6. The domain is decomposed into nine subdomains defined by and the triangulations do not match at the interfaces. We observe two different situations at the interface, e.g. the isolines of the solution are almost parallel at @\Omega 12 whereas at @\Omega 21 the angle between the isolines and the interface is bounded away from zero. In case that the isolines are orthogonal on the interface the exact Lagrange multiplier will be zero. Table Discretization errors (Example 1) standard approach modified approach Lagrange multiplier f level # elem. L 2 -err. energy err. L 2 -err. energy err. In Table 6.1, the discretization errors are given in the energy norm as well as in the for the two different mortar methods. We observe that the energy error is of order h whereas the error in the L 2 -norm is of order h 2 . There is no significant difference in the accuracy between the two mortar algorithm. The discretization errors in the energy norm as well as in the L 2 -norm are almost the same. Fig. 6.7. Decomposition and initial triangulation (left) and solution (right) (Example 2) In our second example, we consider the union square with a slit decomposed into four subdomains, see Figure 6.7. Here, the right hand side f and the Dirichlet boundary conditions of are chosen so that the exact solution is given by sin OE. The solution has a singularity in the center of the domain. We do not have H 2 -regularity, and we therefore cannot expect an O(h) behavior for the discretization error in the energy norm. Table Discretization errors (Example 2), Energy error in 1e \Gamma 01 standard approach modified approach Lagrange multiplier f level # elem. L 2 -err. energy err. L 2 -err. energy err. 2:069586 The discretization errors are compared in Table 6.2. In this case, we observe a difference in the performance of the different mortar methods. The L 2 -error of the modified mortar method is asymptotically better than that of the standard method. The situation is different for the energy error; the standard mortar approach gives slightly better results. A non-trivial difference can only be observed in this example where there is no H 2 -regularity. In that case, the modified mortar method gives better results in the L 2 -norm. Our last example illustrates the influence of discontinuous coefficients. We consider the diffusion equation \Gammadiv the coefficient a is dis- continuous. The unit square\Omega is decomposed into four as in Figure 6.8. Fig. 6.8. Decomposition and initial triangulation (left) and solution (right) (Example The coefficients on the subdomains are given by a The right hand side f and the Dirichlet boundary conditions are chosen to match a given exact solution, solution is continuous with vanishing [arun] on the interfaces. Because of the discontinuity of the coefficients, we use a highly non-matching triangulation at the interface, see Figure 6.8. A Mortar Method Using Dual Spaces 21 The discretization errors in the energy norm as well as in the L 2 -norm are given for the two different mortar algorithms in Table 6.3. We observe that the energy error is of order h. As in Example 1, there is only a minimal difference in the performance of the two mortar approaches. Table Discretization errors (Example 3), Energy error in 1e \Gamma 01 standard approach modified approach Lagrange multiplier f level # elem. L 2 -err. energy err. L 2 -err. energy err. The following two figures illustrate the numbers given in Tables 6.1 - 6.3. In Figure 6.9, the errors in the energy norm are visualized whereas in Figure 6.10 the errors in the L 2 -norm are shown. In each figure a straight dashed line is drawn below the obtained curves to indicate the asymptotic behavior of the discretization errors.1100 1000 10000 100000 in the energy norm Number of elements Example 1 standard modified in the energy norm Number of elements Example 2 standard modified in the energy norm Number of elements Example 3 standard modified Fig. 6.9. Discretization errors in the energy norm versus number of elements0.0010.1100 1000 10000 100000 in the norm Number of elements Example 1 standard modified in the norm Number of elements Example 2 standard modified 100 1000 10000 100000 in the norm Number of elements Example 3 standard modified Fig. 6.10. Discretization errors in the L 2 -norm versus number of elements In Examples 1 and 2, almost from the beginning on the predicted order h for the energy norm and the order h 2 for the L 2 -norm can be observed. In these two examples only one plotted curve for the standard and the new mortar approach can be seen. The numerical results are too close to see a difference in the pictures. In Example 2, where we have no full H 2 -regularity, the asymptotic starts late. We observe for both 22 BARBARA I. WOHLMUTH mortar methods an O(h 1=2 ) behavior for the discretization error in the energy norm. During the first refinement steps the error decreases more rapidly. For the L 2 -norm the asymptotic rate is given by O(h 3=2 ). Moreover, it seems to be the case that the new mortar method performs asymptotically better than the standard one. However, this cannot be observed for other examples without full regularity. Acknowledgment The author would like to thank Professor Olof B. Widlund for his continuous help and for fruitful discussions as well as the referees for their valuable comments. --R Substructuring preconditioners for finite element methods on nonmatching grids. Substructuring preconditioners for the Q 1 mortar element method. Iterative substructuring preconditioners for mortar element methods in two dimensions. The mortar finite element method with Lagrange multipliers. The mortar element method for three dimensional finite elements. Domain decomposition by the mortar element method. A new nonconforming approach to domain de- composition: The mortar element method Raffinement de maillage en elements finis par la methode des joints. Iterative methods for the solution of elliptic problems on regions partitioned into substructures. A multigrid algorithm for the mortar finite element method. Stability estimates of the mortar finite element method for 3- dimensional problems Mixed and hybrid finite element methods. estimates for the three-field formulation with bubble stabilization submitted to Math A hierarchical preconditioner for the mortar finite element method. Adaptive macro-hybrid finite element methods On the mortar finite element method. Multigrid for the mortar finite element method. Analysis and parallel implementation of adaptive mortar finite element methods. Domain decomposition with nonmatching grids: Augmented Lagrangian approach. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Convergence results for non-conforming hp methods: The mortar finite element method Poincar'e and Friedrichs inequalities for mortar finite element methods. The coupling of mixed and conforming finite element dis- cretizations Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers. Mortar finite element methods for discontinuous coefficients. Multigrid methods based on the unconstrained product space arising from mortar finite element discretizations. A mixed finite element discretization on non-matching multiblock grids for a degenerate parabolic equation arising in porous media flow --TR --CTR Q. Hu, Numerical integrations and unit resolution multipliers for domain decomposition methods with nonmatching grids, Computing, v.74 n.2, p.101-129, March 2005 Dan Stefanica, Parallel FETI algorithms for mortars, Applied Numerical Mathematics, v.54 n.2, p.266-279, July 2005 Ralf Unger , Matthias C. Haupt , Peter Horst, Application of Lagrange multipliers for coupled problems in fluid and structural interactions, Computers and Structures, v.85 n.11-14, p.796-809, June, 2007 S. Heber , M. Mair , B. I. Wohlmuth, A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems, Applied Numerical Mathematics, v.54 n.3-4, p.555-576, August 2005 Barbara I. Wohlmuth, An a Posteriori Error Estimator for Two-Body Contact Problems on Non-Matching Meshes, Journal of Scientific Computing, v.33 n.1, p.25-45, October 2007

a priori estimates;lagrange multiplier;nonmatching triangulations;dual norms;mortar finite elements

588410

Modified Adaptive Algorithms.

It is well known that the adaptive algorithm is simple and easy to program but the results are not fully competitive with other nonlinear methods such as free knot spline approximation. We modify the algorithm to take full advantages of nonlinear approximation. The new algorithms have the same approximation order as other nonlinear methods, which is proved by characterizing their approximation spaces. One of our algorithms is implemented on the computer, with numerical results illustrated by figures and tables.

Introduction . It is common knowledge that nonlinear approximation methods are better, in general, than their linear counterparts. In the case of splines, nonlinear approximation puts more knots where the function to be approximated changes rapidly, which results in dramatic improvements in approximating functions with singularities. There are various satisfactory results on free knot spline approxi- mation, in which knots are chosen at one's will. Most related theorems are proved by showing the existence of certain balanced partitions (a more accurate description will be given later). This may cause di#culties in practice, since it is often numerically expensive to find such balanced partitions. Then, there is so-called adaptive approximation by piecewise polynomial (PP) functions, in which only dyadic intervals are used in the partition. Adaptive approximation draws great attention because of its simplicity in nature. As a price to pay for the simplicity, its approximation power is slightly lower than that of its free knot counterpart. Moreover, it is not known exactly what kind of functions can be approximated to a prescribed order; that is, there is no characterization of adaptive approximation spaces. We point out here that when we say adaptive algorithms in this paper, we mean those that approximate a given (univariate) function by PP functions/splines. There are other kinds of adaptive algorithms; some are characterized in the literature (see [10] for an example). In this paper, we shall modify the existing adaptive algorithms in two ways. The resulting algorithms have the same approximation power as free knot spline approximation while largely keeping the simplicity of adaptive approximation. In the next section, we shall state some known results on free knot spline approximation. After describing our algorithms in section 3, in section 4 we shall give our main results, which are parallel to those on free knot spline approximation given in the next section. Numerical implementation and examples will be the contents of the last section. # Received by the editors March 17, 1999; accepted for publication (in revised form) February 9, 2000; published electronically August 29, 2000. http://www.siam.org/journals/sinum/38-3/35356.html Department of Mathematics and Computer Science, Georgia Southern University, Statesboro, GA 30460 (yhu@gasou.edu). # Department of Mathematics, University of Manitoba, Winnipeg, MB, Canada R3T 2N2 (kkopotun@math.vanderbilt.edu). This author was supported by NSF grant DMS 9705638. - Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804 (xmy944f@mail.smsu.edu). 1014 Y.-K. HU, K. A. KOPOTUN, AND X. M. YU We emphasize that we consider only the univariate case in this paper. The idea of merging cubes was initially introduced and used by Cohen et al. in their recent paper [11] on multivariate adaptive approximation. The resulting partition consists of rings, which are cubes with a (possibly empty) subcube removed. Their algorithm produces near minimizers in extremal problems related to the space BV (R 2 ). The authors further explored this algorithm in [21]. In particular, we were able to obtain results on extremal problems related to the spaces V #,p (R d ) of functions of "bounded variation" and Besov spaces B # (R d ). This algorithm is ready to implement for some settings, depending on the value of p (if L p norm is chosen) and order of local polynomials (it is more di#cult for r > 1), though the bookkeeping may be messy. On the other hand, this algorithm is designed for the multivariate case. Its univariate version would not only be much more complex than necessary, but would also produce one-dimensional rings, that is, unions of the subintervals not necessarily neighboring, which are unpleasant and, as it turned out, unnecessary. Our modified algorithms take advantage of the simplicity of the real line topology, simply merging neighboring intervals, thus resulting in partitions consisting of only intervals. These algorithms cannot be easily generalized to multivariate setting, since a procedure of emerging neighboring cubes may generate very complicated and undesirable sets in a partition. This also makes it much more di#cult to establish Jackson inequalities for local ap- proximants. We refer the interested reader to [21], where one can find that the proof of Jackson inequality on a ring is already di#cult enough. For these reasons, we strongly believe that simpler and more e#cient univariate algorithms are necessary. 2. Preliminaries. Throughout this paper, when we say that f, the function to be approximated, belongs to L p (I), we mean f # L p (I) if 0 < p < #, and f # C(I) r is an integer, 0 < # < r and 0 < p, q #, then the Besov space is the set of all functions f # L p (I) such that the semi-(quasi)norm sup is finite, where # r is the usual rth modulus of smoothness. The (quasi)norm for defined by We also define a short notation for a special case that is used frequently in the theory: If there is no potential confusion, especially in the case I = [0, 1], the interval I will be omitted in the notation for the sake of simplicity. For example, L p stands for L p [0, 1] are quasi-normed, complete, linear spaces continuously embedded in a Hausdor# space X, then the K-functional for all f # defined as K(f, t, X 0 , This can be generalized if we replace # X1 by a quasi-seminorm | - | X1 on K(f, t, X 0 , MODIFIED ADAPTIVE ALGORITHMS 1015 The interpolation space (X 0 , consists of all functions < #, where |f | (X0,X1 ) #,q sup When studying an approximation method, it is very revealing to know its approximation spaces, which we now define. Let functions in a quasi-normed linear space X be approximated by elements of its subsets # n , . , which are not necessarily linear but are required to satisfy the assumptions any a #= 0; does not depend on n; (v) # n=0 # n is dense in X; (vi) Any f # X has a best approximation from each # n . All approximant sets in this paper satisfy these assumptions. Denoting we define the approximation space A # to be the set of all f # X for which E n (f) is of order n -# in the sense that the following seminorm is finite: |f | A # sup The general theorem below enables one to characterize an approximation space by merely proving the corresponding Jackson and Bernstein inequalities (see [13, sections 7.5 and 7.9], [9], and [15]). Theorem A. Let Y := Y # > 0, be a linear space with a semi-(quasi)norm | - | Y that is continuously embedded in X. If {# n } satisfies the six assumptions above, and Y satisfies the Jackson inequality and the Bernstein inequality then for all 0 < #, 0 < q #, the approximation space A # By a partition of the interval [0, 1] we mean a finite set of subintervals whose union # n I . The (nonlinear) Y.-K. HU, K. A. KOPOTUN, AND X. M. YU spaces # n,r of all PP functions of order r on [0, 1] with no more than n > 0 pieces are defined by I#P I (x)# I (x), |P| # n}, where P I are in P r-1 , the space of polynomials of degree < r, and # I are the characteristic functions on I. # 0,r is defined as {0}. These # all assumptions (i)-(vi) on {# n } (see p. 3). The degree of best approximation of a function f by the elements of # n,r is denoted by # n,r (f) p := E(f, # n,r ) p . Remark . Some authors use the notation # (n-1)r,r in place of # n,r , since PP functions can be viewed as special kinds of splines with each interior break point x i , a knot of multiplicity r. Also in use is PP n,r . Following general notation in nonlinear approximation, we use the first subscript for the number of coe#cients in the approximant. See [13], [14], [17], [26]. Strictly speaking, all n- piece PP function of order r only form a proper subset of the free knot spline space (n-1)r,r , but this subset has the same approximation power in L p as the whole space (see Theorem 12.4.2 of [13]). In his 1988 paper [23] (also see [24] and [13, section 12.8]), Petrushev characterized the approximation space A # using the Besov spaces; see the following theorem. Theorem B. Let 0 < p < #, n > 0, and 0 < # < r. Then we have and Therefore for 0 < q # and 0 < # < r A # In particular, if # A # The inequality (2.4) can be proved by finding a balanced partition according to the function s (f, x)| # dsdt in the sense that (see [13] for details of the proof). In fact, many Jackson-type inequalities can be proved by showing the existence of a balanced partition (see, e.g., Theorems 12.4.3, 5, and 6 in [13], Theorem 1.1 in [19], and parts of Theorems 2.1 and 4.1 in [17]). We state here Theorem 12.4.6 of [13], given by Burchard [8] in 1974 for the case (see also de Boor [3]). MODIFIED ADAPTIVE ALGORITHMS 1017 Theorem C. Let r and n be positive integers, and let # := (r monotone function, then #,p be the space of functions f # L p [0, 1] for which the variation |f | V #,p I#P is finite, where the sup is taken over all finite partitions P of [0, 1]. Following [17] (see also Brudnyi [7] and Bergh and Peetre [1]), we define a modulus of smoothness f, 0<h#t sup The following theorem, which is due to DeVore and Yu [17], provides characterization of A # using interpolation spaces involving V #,p . Theorem D. Let 0 < p #, 0 < # < r, and # approximation by elements from {# n,r } # 0 , we have the Jackson inequality and the Bernstein inequality Therefore A # In particular, if p < #, A # The Jackson inequality (2.12) follows from the definition of # f, t) #,p and the existence for any f # V #,p of an S # n,r with n := which can be proved (see [17]) by showing the existence of a balanced partition such that and then defining S by where P i are best L p approximations to f on I i from the space P r-1 . Y.-K. HU, K. A. KOPOTUN, AND X. M. YU 3. Adaptive algorithms. 3.1. The original adaptive algorithm. More than likely it will be hard to find an exactly balanced partition numerically. An algorithm of this sort by Hu [20], for instance, uses two nested loops (there is another level of loop that increases the number of knots). This is probably one of the reasons why much attention is paid to adaptive approximation, which selects break points by repeatedly cutting the intervals into two equal halves, and produces PP functions with dyadic break points, which can be represented by finite binary numbers of the form m - 2 -k , . Denote the spaces of such PP functions by # d n,r and their approximation errors E(f, # d n,r (f) p . We now describe the original adaptive algorithms in the univariate setting. Let E be a nonnegative set function defined on all subintervals of [0, 1] which satisfies E(I) # E(J) if I # J ; uniformly as |I| # 0. Given a prescribed tolerance # > 0, we say that an interval I is good if E(I) #; otherwise it is called bad. We want to generate a partition G := G(#, E) of [0, 1] into good intervals. If [0, 1] is good, then is the desired partition; otherwise we put [0, 1] in B, which is a temporary pool of bad intervals. We then proceed with this B and divide every interval in it into two equal pieces and test whether they are good, in which case they are moved into G, or bad, in which case they are kept in B. The procedure terminates when resulting intervals are good and are in G), which is guaranteed to happen by (3.2). The set function E(I) usually depends on the function f that is being approximated and measures the error of approximation of f on I, such as # I G(x) dx in (2.9), thus will be called the (error) measure of I throughout this paper. In the simplest case, E(I) is taken as the local approximation error of f on I # [0, 1] by polynomials of degree < r: E(I) and the corresponding approximant on G is defined by (2.15). This gives an error where |G| is the number of intervals in G. One can estimate in di#erent ways a n (f) p := a n (f, E) p := inf |G| 1/p #, where the infimum is taken over all # > 0 such that and Solomjak [2] and DeVore [12] for estimates for functions f in Sobolev spaces. Other estimates can be found in Rice [25], de Boor and Rice [6], and DeVore and Yu [18] and the references therein. We only mention the following two results. Theorem E (see [18, Theorem 5.1]). Let If f # C r (0, 1) with |f (r) (x)| #(x), where # L # is a monotone function such that I MODIFIED ADAPTIVE ALGORITHMS 1019 where C 1 is an absolute constant, then we have a n (f) # Cn -r # . Note that compared with Theorem C with theorem has an extra requirement (3.5) on #. Theorem F (see [18, Corollary 3.3]). Let 0 < p < # > 0, and q > # := a # (L q ) , we see (3.7) is weaker than (2.4), which is for free knot spline approximation. The reason for this is not hard to see: adaptive algorithms not only select break points from a smaller set of numbers (that is, the set of all finite binary numbers), but they also do it in a special order. Consider as an example, a good free knot approximant will have most knots very close to 0 (see examples in [20] and Table 5.2 later in this paper). However, an adaptive algorithm needs at least n - 1 knots, 2 before it can put one at 2 -n and thus needs more knots than a free knot spline algorithm. Although one classifies adaptive approximation as a special kind of free knot spline approximation (since the knots sequence depends on the function to be approximated), one is far from free when choosing knots. It is considered "more restrictive" (DeVore and Popov [14]) than free knot spline approximation. We should point out that all theorems mentioned in this subsection are of a Jackson-type, that is, so-called direct theorems. Bernstein inequalities (closely related to inverse theorems, sometimes referred to also as inverse theorems themselves) for free knot splines, such as (2.5) and (2.13), are valid for all splines, including PP functions produced by adaptive algorithms. The problem is that all Jackson inequalities for the original adaptive algorithms are not strong enough to match those Bernstein inequalities in the sense of Theorem A. From this point of view, Theorems E and F are weaker than they look. We do not know exactly what kind of functions can be approximated by the original adaptive algorithms to a prescribed order, that is, we can not characterize their approximation spaces A # q . They do not fully exploit the power of nonlinear approximation, and sometimes they generate too many intervals, many of which may have an error measure much smaller than #. As mentioned above, there are two major aspects in which adaptive approximation is di#erent from free knot spline approximation: (a) a smaller set of numbers to choose knots from and (b) a special, and restrictive, way to select knots from the set. It turns out that (b) is the reason for its drawback. Although it is also the reason why adaptive approximation is simple (and we want to keep it that way), it does not mean we have to keep all the knots it produces. In this paper, we modify the usual adaptive algorithm in two ways. The idea is that of splitting AND merging intervals/cubes used in a recent paper by Cohen et al. [11]. The two new algorithms generate partitions of [0, 1] with fewer dyadic knots which are nearly balanced in some sense. In section 4, we prove that they have the same approximation order as that of free knot splines. 3.2. Algorithm I. We start with the original adaptive procedure with some # > 0, which generates a partition of [0, 1] into good intervals. The number N # may be much larger than it has to be. To decrease it, we merge some of the intervals I # i . We begin with I # 1 and check the union of I # 1 and I # 2 . If it is still a good interval, that is, if its measure E(I # #, we add I # 3 to the union and 1020 Y.-K. HU, K. A. KOPOTUN, AND X. M. YU check whether E(I # 3 ) #, and we proceed until we find the largest good union I # k in the sense that but E(I # We name I # 1 # I # k as I 1 . If k < N # , we continue with I # k+1 and find the next largest good union as I 2 . At the end of this procedure, we obtain a modified partition consisting of N # N # good intervals for which each union J i := I i # I i+1 is bad, This partition is considered nearly balanced. For the size of N we have # . 3.3. Algorithm II. Our second algorithm generates a nearly balanced partition in another way. It does not make heavy use of prescribed tolerance #; rather, it merges intervals with relatively small measures while dividing those with large ones. As in the ordinary adaptive algorithms, we start with dividing [0, 1] into two intervals I 1 and I 2 of equal length. However, this is where the similarity ends. We then compare measures E(I 1 ) and E(I 2 ) and divide the interval with larger measure into two equal pieces. In the case of equal measure, we divide, rather randomly, the one on the left. Now we have three intervals and are ready for the three-step loop below. Step 1. Assume there is currently a partition {I i } k I j has the largest measure among all I i . If E(I j+1 is a fixed parameter, we check the union of I j+1 # I j+2 to see whether its measure E(I j+1 # I j+2 ) < M . If so, add the next interval I j+3 into the union and check its measure again. We continue until we get a largest union # j+m1 I i whose measure is less than M, and replace this union by the intervals it contains. Then, if j +m 1 < k, we find the next largest union # j+m1+m2 I i in the same manner and replace these intervals by their union. Furthermore, we do the same to the intervals to the left of I j (but keep I j intact). In this way we obtain a new partition with (the old) I j still having the largest measure. This partition is nearly balanced in the sense that the measure of the union of any two consecutive new intervals is no less than # (because these new intervals were largest unions of old intervals). At the end of this step we renumber the new intervals and update the value of k. Step 2. Check whether the new partition produced in Step 1 is satisfactory using an application-specific criterion, for instance, whether k has reached a prescribed value n or the error is reduced to a certain level. If not, continue with Step 3; otherwise define the final spline by (2.15) and terminate the algorithm. Step 3. Divide the interval with the largest measure into two equal pieces, renumber the intervals, update the values of k and M, and then go back to Step 1. Remark. In Step 1, if I l and I l+1 are the two newest intervals (two "brothers" with equal length), one needs only to check I l-1 # I l if l - 1, l #= j, and/or I l+1 # I l+2 since other unions of two consecutive intervals have measures no MODIFIED ADAPTIVE ALGORITHMS 1021 less than the value of M in the previous iteration, which is, in turn, no less than the current M . We stated it in the way above only because it shows the purpose of the step more clearly. It should be pointed out that one needs to be careful about the stopping criterion in Algorithm II. For example, if it is applied to the characteristic function after two iterations we will always have The break point # 2/2 in this example can be replaced by any number in (0, 1) which does not have a finite binary representation such as 0.4. If k # used as the sole stopping criterion, the algorithm will fall into infinite loop. Fortunately, the error in this example still tends to 0; therefore, infinite loop can be avoided by adding error checking in the criterion. The next lemma shows this is the case in general. Lemma 3.1. Let E be an interval function satisfying (3.1) and (3.2), and let prescribed. Then the criterion will terminate Algorithm II. Proof. We show that if k never exceeds n, then as the number of iterations goes to #. Let 0 < # < 1 be fixed. Let with the max taken at one moment. Fix this M and denote the group of all subintervals in the partition with "large" errors by G be as in Step 1, changing from iteration to iteration. We have # M from now on. We first make a few observations. Since the interval currently having the largest measure is always in G M , each iteration cuts a member of G M . However, the algorithm will not merge any member I i # G M with another interval because E(I i any union of I i with another interval would have even larger measure by (3.1). By (3.2), there exists # > 0 such that |I i | > # for any I i # G M . Note all intervals in a partition are disjoint, thus the total length of the intervals in G M is no larger than 1, and its cardinal number |G # M | # 1/#. From these observations, we conclude the following. When an iteration cuts a member I i of G M into two "children" of equal length, one of the three cases will happen: (a) neither child of I i belongs to G M , thus |I i | > # is removed from the total length of G exactly one of the children belongs to it (hence having a length > #) and the other child, with the same length |I i |/2 > #, is removed from G or (c) both children belong to it. The case (a) decreases |G # M | by 1, (b) keeps it unchanged, and (c) increases it by 1. Now one can see that at most # 3/# +1 iterations will empty G M , since at least one third of them will be cases (a) or (b) to keep |G # M | # 1/#, which will remove all the total length of G M , thus emptying it. This reduces the maximum error by a factor # < 1. Repeat this enough times and the maximum error will eventually tend to 0. Although (3.2) does not say anything about the convergence rate of E(I) as |I| # 0, and the proof of the above lemma may make it sound extremely slow, one can expect a fairly fast convergence in most cases. For example, in the case if f is in the generalized Lipschitz space Lip #, p) := Lip #, L p [a, b]), 0 < # < r, that is, if |f | Lip # := |f | Lip #,p) := sup 1022 Y.-K. HU, K. A. KOPOTUN, AND X. M. YU then for any I # [a, b] |f | Lip # . We feel it is safe to say that most functions in applications belong to Lip #, p) with an # reasonably away from 0, at least on subintervals not containing singularities, thus halving an interval often reduces its error by a factor of 2 # . A natural question that may arise here is: How complex are the new algorithms? We give brief comparisons below to answer this question. Algorithm I is straight for- ward. It is the original adaptive algorithm with a second (merging) phase added. This phase consists of no more than merging attempts, where N # is the number of subintervals the original algorithm generates, and N that of the final subintervals. As for Algorithm II, there are two major di#erences from the original version. The first one, as mentioned in the remark after the algorithm description, is: up to two merging attempts are made after cutting each interval. The other one is in the book-keeping. In the original version, a vector is needed to record errors on all intervals (or to indicate which intervals are bad), while Algorithm II keeps the index of the interval that has the largest error E(I) in a scalar variable, in addition to the vector containing all errors. This requires a search for the largest element in the vector after each cutting or merging operation. One can see from above that the new algorithms are not much more complex in terms of programming steps. The added CPU time, in terms of the number of results mainly from the evaluations of the error measure E(I) required by merging operations. Our estimate is that either algorithm uses two or three times as much CPU time as the original algorithm. More information on CPU time will be given in section 5 together with numerical details. 4. Approximation power of the algorithms. We now show that our modified adaptive algorithms have the full power of nonlinear approximation. More precisely, we prove that they produce piecewise polynomials satisfying the very same Jackson inequalities for free knot spline approximation (with possibly larger constants on the right-hand side since the partitions are not exactly balanced). As we mentioned earlier, the corresponding Bernstein inequalities hold true for all splines; therefore we are really proving that the approximation spaces for the modified adaptive algorithms are the same as those for free knot spline approximation. We state below our results as three main theorems, parallel to Theorems B, C, and D, respectively. In fact, we can prove most results of this kind for our algorithms, such as Kahane's theorems and its generalization [13, Theorems 12.4.3 and 5], but the proofs would be too similar to the ones below. We recall that throughout this paper, I j denotes the interval with largest measure among all I i in the partition, the union of any two consecutive intervals J has a measure E(J i ) > E(I j ), and J i is called bad in Algorithm I. All PP functions on the resulting partitions are defined by (2.15). Theorem 4.1. Let n and r be positive integers, and let 0 < p < #, 0 < # < r, then the two modified adaptive algorithms (with defined in (2.8) or (ii) functions S of (2.15) that satisfy the Jackson inequality (4. MODIFIED ADAPTIVE ALGORITHMS 1023 From Theorem A we obtain the approximation space A # product. It turns out to be the same as A # which is not surprising since # d n,r is dense in # n,r . The surprising part is that one can get such an approximant using a simple adaptive algorithm. Corollary 4.2. Let 0 < p < #, 0 < q #, 0 < # < r, and . For approximation by PP functions in # d n,r , we have A # In particular, A # Proof of Theorem 4.1. The proofs of the theorem in the cases (i) and (ii) are very similar. We only consider (i) and remark that, in the case (ii), the inequality plays the major role. PP approximants produced by Algorithm I. Let E(I) := # I G(x) dx, where G is as in (2.8), and # := We claim that the number N of intervals it produces is no greater than 2n+1. Indeed, by (3.8) The rest of the proof of (4.1) is similar to that of (2.4) (cf. section 12.8, p. 386 of [13]); we sketch it here for completeness. It is proved in [13] that for any f # B # [0, 1], M is equivalent to |f | B # [0, 1] with constants of equivalence depending only on r and #, and that for such an f Define the approximant S by (2.15) and we have here in the fifth step we have used the equality # and in the last step we have used the equivalence of M and |f | B # . PP approximants produced by Algorithm II. Let E(I), M, and # be the same as above, and use (3.9) as stopping criterion in Step 2. If the algorithm terminates due 1024 Y.-K. HU, K. A. KOPOTUN, AND X. M. YU to (thus giving less than n pieces), it is the same situation as with Algorithm I. Otherwise we have n pieces when it terminates, and (4.1) follows: Cn Theorem 4.3. Under the conditions of Theorem C, the modified adaptive algorithms (with E(I) := # I #(x) # dx and # := n produce PP approximants S in # d n,r that satisfy the Jackson inequality: Proof of Theorem 4.3. PP approximants produced by Algorithm I. Let E(I) := # I as the Taylor polynomial for f of degree r - 1 at the point x i+1 (not best we have (see equation (4.15) in Chapter 12 of [13]) Using (4.6) in place of (4.4), then (4.5) for p < # can be proved by arguments very similar to those in the proof of Theorem 4.1 by Algorithm I. We also refer the reader to the proof of Theorem C in [13]. For #, the estimate of N is the same and we need only to replace # N by PP approximants produced by Algorithm II. Let E(I), #, and M be the same as above. Use (3.9) again as the stopping criterion in Step 2. If the algorithm terminates because it is the same situation as in Algorithm I. Otherwise, for p/# MODIFIED ADAPTIVE ALGORITHMS 1025 where we have used the inequality (4.6) in the second step, and the last one. For #, we make similar changes to those in Algorithm I: Theorem 4.4. Let n and r be positive integers, and let 0 < p, #, 0 < # < r, then the two modified adaptive algorithms (with E(I) #,p ) produce PP functions S of (2.15) that satisfy the Jackson inequality Using Theorems 4.4 and A we have the following characterization of A # Corollary 4.5. For approximation by PP functions in # d n,r we have A # In particular, if p < #, A # Proof of Theorem 4.4. It su#ces to show (2.14) since (4.7) immediately follows from it with any t > 0 and n := (see the end of section 2). We only prove it for p < #. The case of can be verified by making changes similar to those in the proof of the L# case in the previous theorem. PP approximants produced by Algorithm I. Let E(I) p and #,p . From (3.8), the number N of intervals the algorithm produces can be estimated as Indeed, if N > 2n (otherwise, it's done) we have #,p Cn #/p #,p where we have used the definition (2.11) of # f, t) #,p . Since 1 - #/p, this gives Cn. Now (2.14) follows, since #,p . PP approximants produced by Algorithm II. We set E(I) := E r (f, I) # , and use #,p in the stopping criterion (3.9). If it stops because have exactly the same situation as with Algorithm I, with 1026 Y.-K. HU, K. A. KOPOTUN, AND X. M. YU the same partition; otherwise there are n intervals when it terminates. In the latter case, we have 5. Numerical implementation and examples. Theoretically, the two algorithms have the same approximation power. However, when it comes to numerical implementation, we prefer Algorithm II since it directly controls the number of polynomial pieces n, while # in Algorithm I is neither a power of n nor a tolerance for (though it is closely related to both). We implemented Algorithm II on the computer, using Fortran 90 and mainly for 2. The error measure used in the code is 2 unless we have a better one to use, such as # I # I |f (r) | # for the square root function in the first example in this section. The L 2 norm of f on the interval I estimated by the composite Simpson rule for integral and its L# norm is estimated by are equally distributed nodes, n p is a program parameter roughly set as 6 times r, and . The best L 2 polynomial approximant on I i , discretized by (5.1) as an overdetermined n p - r system of linear equations for the least squares method, is calculated by either QR decomposition or singular value decomposition by calling LINPACK subroutines Sqrdc and Sqrsl, or Ssvdc (or their double precision counterparts). The latter takes longer but we did not see any di#erence in the first four or five digits of the local approximation errors they computed; thus we did not test it extensively. The L# version of algorithm is basically the same, except that we use estimated by (5.2). The local polynomials P I (and the global smooth splines) are still obtained by the least squares method, that is, still best L 2 approximants. This is common in the literature, and it is justified by the fact that the best L 2 polynomial approximant is also a near-best L# polynomial on the same interval; see Lemma 3.2 of DeVore and Popov [16]. The number of polynomial pieces is used as the main termination criterion, while # in (3.9) is set to a small value mainly to protect the program from falling into infinite loops, rather than the sophisticated ones as in proofs in the previous section. It turned out that infinite loop is not a problem. A nonfull rank matrix in the least squares method is a problem, which happens far before it falls into an infinite loop. This is because if I i is too small, the machine will have di#culties distinguishing the n p MODIFIED ADAPTIVE ALGORITHMS 1027 points needed in (5.1). Therefore, we added a third condition to protect the program from failing: stop the program when We also added a second part in the code, namely, finding an L 2 smooth spline approximation to the function with the knot sequence {t i } n+r i=2-r , where the interior knots a < t 2 < t 3 < - < t n < b are the break points of the PP function obtained by Algorithm II, used as single knots, and the auxiliary knots are set as t b. Despite the fact that the partitions are guaranteed to be good only for PP functions, they usually work well for smooth splines, too. De Boor gave some theoretical justification in the discussion of his subroutine Newnot [4, Chapter XII]. The least square objective function for finding this smooth spline - S is set as 5r+1, is the number of equal pieces into which we cut each subinterval I are the points resulted from such cutting, and the weights w j are chosen so that (5.4) becomes a composite trapezoidal rule for the integral # b a # f(x) - dx: The actual calculation of the B-spline coe#cients of are the B-splines with the knot sequence {t i } scaled so that done by de Boor's subroutine L2Appr in [4, Chapter XIV]. We used the source code of all the subroutines in the book from the package PPPACK on the Internet. We tested our code on a SUN UltraSparc, with a clock frequency 167MHz, 128MB of RAM, and running Solaris 2.5.1. The speed is so fast that it is not an issue here: for finding break points, it is somewhere from 0.015 second for to 0.1 second for printing minimum amount of messages on the screen, and it is less than 10% of these for computing smooth splines. We also tested the code on a 300 MHz Pentium II machine with 64 MB of RAM running Windows NT 4.0. The speed is at least three times as fast. None of the problems we tested used more than 0.1 second. (The reason for the great di#erence in speed may be that the SUN we used is a file server, not ideal for numerical computation.) There is still room for improvement in e#ciency. For example, one can use a value of n p , larger than what we use, at the beginning and decrease it as n increases (and the error on each subinterval decreases). The value of n s should be related to n, too, for the same reason. The main cost of CPU time is the evaluation of the error measure E(I) for each subinterval I. We use estimated by QR decomposition, as an exam- ple. Each such problem involves n p function evaluations, and (n p - r operations required in QR decomposition, plus some more for estimating the error from the resulting matrices. Each cutting of intervals requires two E(I) evaluations, 1028 Y.-K. HU, K. A. KOPOTUN, AND X. M. YU Table Approximation order of and each merging attempt requires one. Our numerical experiments show that a typical run resulting in n subintervals cuts intervals about 2n time. Each cutting results in up to two attempts of merging subintervals. That gives about 8n least squares problems, each of which involves n p function evaluations plus about n p r 2 arithmetic operations. In view of the approximation order we proved in the previous section, and the fact that n p is roughly a multiple of r, we think it pays to use a relatively large r, at least 4 or 5. For 5, the error will reach the machine epsilon (single precision) when n is somewhere between 30 and 70 in most cases. We use the square root function to test the PP function approximation order. This function is only in the Lipschitz space Lip( 1 thus the approximation order is only 1/2 for splines with equally spaced knots in the L# norm, no matter what their order r is. By Theorem 4.3, we should have e n := #f -S n # p # Cn -r , where S n is the function consisting of n polynomial pieces computed by Algorithm II using I |f (r) (x)| # dx, and we have combined # in the theorem into the constant C. After the knot sequence has been found, QR decomposition is used at the end of the program on each subinterval to estimate e n . Since the error decreases fast for double precision had to be used in QR decomposition for large values of n. Assume that what we actually obtain from the code is e where # is the approximation order. Since log e plot the points in the plane, they should form a line. Since such a plot zigzags very much, we calculated the least squares line fitting to find the order. Table 5.1 gives values of # for di#erent r using both L 2 and L# norms. We should mention that the points values of n are too low and ruin the obvious line pattern formed by those for larger n, thus we give two values of #, one from the points for and the other from As can be seen from the table, the latter values are right around or even exceed r. Remark. We tried some power of E r (f, I) p for E(I) and felt, in view of (4.6), it would yield a better balance of subintervals, thus a higher order. But the orders so obtained were well below r (4.46 for e.g. The reason might be that I # is additive, but (power of) E r (f, I) p is not. To illustrate the advantage of interval merging, we compare the original adaptive algorithm and our modified ones with the function log 2 -m This function is in C # , and is decreasing and convex on [0, 1] with . Note that since f is decreasing on [0, 1]. Table 5.2 shows comparison in numbers of knots produced for the same approximation error by the original adaptive algorithm and our Algorithm II. Both programs try to put first knots near where the graph is very steep. The original algorithm has to, as pointed out early, lay down knots 2 one by one before reaching an error of MODIFIED ADAPTIVE ALGORITHMS 1029 Table Comparison in numbers of interior knots produced by the original and modified adaptive algorithms for the same error in approximating Original Alg. while Algorithm II, after trying all these knots one at a time and merging all but the last interval, puts the very first knot at 2 -23 . It is interesting to watch how Algorithm II moves a knot toward a better position in successive iterations without increasing the total number of pieces. The following screen output shows that in iterations 1 and 2 the program moves the break point 0.5 to 0.25 and then to 0.125, while the error decreases form 0.5 to 0.47; in iterations 3-22 it moves the break point all the way to 2 -23 # with the error decreased to 0.27. What happened internally is, in iteration 1, e.g., it cuts the interval [0, 0.5] into [0, 0.25] and [0.25, 0.5]. Since the error on the union of [0.25, 0.5] and [0.5, 1] is smaller than that on [0, 0.25], it then merges the two intervals into [0.25, 1]. The net e#ect of these steps is moving the break point 2 -1 to 2 -2 . Iteration 0: # of errors= L_\infty error on [a, Iteration 1: # of errors= L_\infty error on [a, Iteration 2: # of errors= L_\infty error on [a, (Many lines deleted.) Iteration 22: # of errors= 2.70000E-01 2.30000E-01 L_\infty error on [a, Y.-K. HU, K. A. KOPOTUN, AND X. M. YU Table Approximation errors to the Runge function on [-5, 5]. 9 We now consider the infamous Runge function, which is also in C # but, on the other hand, is hard to interpolate or approximate. Lyche and M-rken [22] approximated it by the knot removal algorithm, and Hu [20] approximated it by balancing the rth derivative of the function on subintervals in two nested loops. Here and in the rest of the paper, we use 4. In Table 5.3, we compare our results with those of Lyche and M-rken (LM) [22] and Hu [20]. For the same number of knots (that is, list our errors measured in # 2 / # b - a for the PP function S n and the smooth spline - also that of - measured in L# norm. We divide the L 2 norm by since it is more comparable to the L# norm, which is what LM and Hu used. The errors by LM are estimated from figures in [22]. Because of the simple nature of our algorithm, we only expected to compete with their results by splines with two or three times as many knots. It turns out that our approximation errors are almost as good as theirs, which were produced by more sophisticated methods. By now, the reader may begin to wonder: what is the e#ect of the parameter of Algorithm II, used in Lemma 3.9 to guarantee the termination of Algorithm II. We tried functions we tested, it worked excellently except that the number of polynomial pieces went up and down a few times with the square root function using dx, in which case used instead. It is true that in theory it might get into an infinite loop, but since our goal is to find a nearly balanced partition, better in this aspect, provided infinite loop does not happen. It did not. As a matter of fact, sometimes we feel the need for a value slightly larger than 1, e.g., with symmetric functions such as the Runge function. What happens with # 1 is that if there are two subintervals having the same largest measure at the moment, symmetric about the center of the interval, then the outcome of the next iteration, which processes the subinterval on the left, will very often interfere with the processing of the subinterval on the right later. It may not make the approximation error worse, at least not by much, it is just that the knot sequence becomes unsymmetrical, thus unnatural and unpleasant. Furthermore, most algorithms in the literature produce symmetric knots for symmetric functions; it would be hard to compare our results with theirs. For these minor reasons, we used in preparation of Table 5.3. In the next example, we consider the PP function which has a jump at # 2/2. As we mentioned in the discussion before Lemma 3.1, since # 2/2 has no finite binary representation, this function can never be approximated exactly by a PP function with dyadic break points. The program (with cutting and merging around the jump (since the number of pieces is always 3 after two iterations), until it is stopped by the criterion (5.3), resulting in t MODIFIED ADAPTIVE ALGORITHMS 1031 500 600 700 800 900 1000 11000.611.41.82.2-.04 Temperature Fig. 5.1. Titanium Heat Data (circles). The final spline (solid line) has 15 interior knots. The errors for preapproximation (dotted) and for the final spline (dashed) use scales on the right. and 0.70710754. The PP function matches f exactly on the computer screen since the two points are indistinguishable. One can very well combine them into a single break point, thus virtually reproducing f . The original adaptive algorithm, in contrast, would put many many knots around the jump while trying to narrow the subinterval containing the jump: 0.5, 0.75, 0.625, 0.6875, 0.71875, . All these knots are useless except the newest two. In practice, one often wants to approximate discrete data points other than known functions as in the previous examples. In this case, we preapproximate the points by a spline with as many parameters as we wish to use, then apply our algorithm to this spline. For smooth-looking data, we interpolate the data by a C 1 cubic spline with knots at the data points, using de Boor's subroutine Cubspl in [4]. This worked very well. We produced some sample data points from the Runge function and square root function and applied this approach to them. It resulted in virtually the same knot sequences as those generated by directly approximating the original functions. In the real world, however, it is likely that the data will contain errors. If the data points are interpolated, one can see small wiggles in the graph, which tricks the program laying knots in areas where the curve is otherwise flat. One such example is the Titanium Heat Data (experimentally determined), see [4, Chapter XIII], and also LM [22] and Hu [20]. In Figure 5.1 the reader can see wiggles on both the left and right. De Boor [4, Chapter XIV] suggests that the data be approximated by a less smooth spline. We absolutely agree. For the same reason, we used fewer knots for preapproximating spline in the flat parts at both ends, than we did near the high 1032 Y.-K. HU, K. A. KOPOTUN, AND X. M. YU peak around 900 # , trying to ignore the wiggles. In fact, we used almost the same knot sequence for preapproximating spline as in Figure 4 of [20]. Table Approximation errors to the Titanium Heat Data. Obtained by Order # of knots Error Alg. II 4 11 0.070 Alg. II 4 15 0.031 Since de Boor, LM, and Hu all used L# norm for approximating these data, we also used the L# version of our program. Figure 5.1 shows a cubic spline approximation to the Titanium Data obtained by this method. It has 15 interior knots with an error of 0.031. Table 5.4 gives a comparison of our results with those by others on the same data. Acknowledgments . We are deeply indebted to Professor Ron DeVore, who inspired us by discussing the excellent ideas in [11] during our visit to the University of South Carolina. We want to thank him and Professors Pencho Petrushev and Albert Cohen for providing us with drafts of their manuscript [11]. Credit is also due to Professor Dietrich Braess, the editor of this paper, and the referees, whose opinions and suggestions helped very much in improving the manuscript. As a matter of fact, we reshaped the last section during the communication with them. --R On the space Vp (0 Piecewise polynomial approximation of functions of classes W Good approximation by splines with variable knots A Practical Guide to Splines Least squares cubic spline approximation II-Variable knots An adaptive algorithm for multivariate approximation giving optimal convergence rates Spline approximation and functions of bounded variation Splines with optimal knots are better Jackson and Bernstein-type inequalities for families of commutative operators in Banach spaces Adaptive wavelet methods for elliptic operator equations-Convergence rates Nonlinear approximation and the space BV (R 2 A note on adaptive approximation in Function Spaces and Applications Interpolation of Besov spaces Degree of adaptive approximation Convexity preserving approximation by free knot splines An algorithm for data reduction using splines with free knots On multivariate adaptive approximation A data reduction strategy for splines with applications to the approximation of functions and data Direct and converse theorems for spline and rational approximation and Besov spaces Rational Approximation of Real Functions Basic Theory --TR

adaptive algorithms;nonlinear approximation;data reduction;besov spaces;degree of approximation;piecewise polynomials;splines;modulus of smoothness;approximation spaces

588414

The Best Circulant Preconditioners for Hermitian Toeplitz Systems.

In this paper, we propose a new family of circulant preconditioners for ill-conditioned Hermitian Toeplitz systems A x= b. The preconditioners are constructed by convolving the generating function f of A with the generalized Jackson kernels. For an n-by-n Toeplitz matrix A, the construction of the preconditioners requires only the entries of A and does not require the explicit knowledge of f. When f is a nonnegative continuous function with a zero of order 2p, the condition number of A is known to grow as O(n2p). We show, however, that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p+1 outliers. Moreover, the smallest eigenvalue is uniformly bounded away from zero. Hence the conjugate gradient method, when applied to solving the preconditioned system, converges linearly. The total complexity of solving the system is therefore of O(n log n) operations. In the case when f is positive, we show that the convergence is superlinear. Numerical results are included to illustrate the effectiveness of our new circulant preconditioners.

Introduction An n-by-n matrix A n with entries a ij is said to be Toeplitz if a a i\Gammaj . Toeplitz systems of the form A n occur in a variety of applications in mathematics and engineering E-mail: rchan@math.cuhk.edu.hk. Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong. Research supported in part by Hong Kong Research Grants Council Grant No. CUHK 4207/97P and CUHK DAG Grant No. 2060143. y E-mail: mhyipa@hkusua.hku.hk. Department of Mathematics, The University of Hong Kong, Pokfu- lam Road, Hong Kong. z E-mail: mng@maths.hku.hk. Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong. Research supported in part by HKU CRCG grant no. 10201939. [7]. In this paper, we consider the solution of Hermitian positive definite Toeplitz systems. There are a number of specialized fast direct methods for solving such systems in O(n 2 ) operations, see for instance [22]. Faster methods requiring O(n log 2 n) operations have also been developed, see [1]. Strang in [21] proposed using the preconditioned conjugate gradient method with circulant matrices as preconditioners for solving Toeplitz systems. The number of operations per iteration is of order O(n log n) as circulant systems can be solved efficiently by fast Fourier transforms. Several successful circulant preconditioners have been introduced and analyzed; see for instance [11, 5]. In these papers, the given Toeplitz matrix A n is assumed to be generated by a generating function f , i.e., the diagonals a j of A n are given by the Fourier coefficients of f . It was shown that if f is a positive function in the Wiener class (i.e., the Fourier coefficients of f are absolutely summable), then these circulant preconditioned systems converge superlinearly [5]. However, if f has zeros, the corresponding Toeplitz systems will be ill-conditioned. In fact, for the Toeplitz matrices generated by a function with a zero of order 2p, their condition numbers grow like O(n 2p ), see [19]. Hence the number of iterations required for convergence will increase like O(n p ), see [2, p.24]. Tyrtyshnikov [23] has proved that the Strang [21] and the T. Chan [11] preconditioners both fail in this case. To tackle this problem, non-circulant type preconditioners have been proposed, see [6, 4, 18, 16]. The basic idea behind these preconditioners is to find a function g that matches the zeros of f . Then the preconditioners are constructed based on the function g. These approaches work when the generating function f is given explicitly, i.e., all Fourier coefficients of f are available. However, when we are given only a finite n-by-n Toeplitz system, i.e., only fa j g jjj!n are given and the underlying f is unknown, then these preconditioners cannot be constructed. In contrast, most well-known circulant preconditioners, such as the Strang and the T. Chan preconditioners, are defined using only the entries of the given Toeplitz matrix. Di Benedetto in [3] has proved that the condition numbers of the preconditioned matrices by sine transform preconditioners are uniformly bounded. However, the preconditioners themselves may be singular or indefinite in general. Our aim in this paper is to develop a family of positive definite circulant preconditioners that work for ill-conditioned Toeplitz systems and do not require the explicit knowledge of f , i.e., they require only fa j g jjj!n for an n-by-n Toeplitz system. Our idea is based on the unified approach proposed in Chan and Yeung [9], where they showed that circulant preconditioners can be derived in general by convolving the generating function f with some kernels. For instance, convolving f with the Dirichlet kernel D bn=2c gives the Strang preconditioner. They proved that for any positive 2- periodic continuous function f , if C n is a kernel such that the convolution product C n f tends to f uniformly on [\Gamma-], then the corresponding circulant preconditioned matrix n A n will have clustered spectrum. In particular, the conjugate gradient method will converge superlinearly when solving the preconditioned system. This result turns the problem of finding a good preconditioner to the problem of approximating f with C n f . Notice that D bn=2c f , being the partial sum of f , depends solely on the first bn=2c Fourier coefficients fa j g jjj!bn=2c of f . Thus the Strang preconditioner, and similarly for other circulant preconditioners constructed through kernels, does not require the explicitly knowledge of f . In this paper, we construct our preconditioners by approximating f with the convolution product K m;2r f that matches the zeros of f and depends only on fa j g jjj!n . Here K m;2r is chosen to be the generalized Jackson kernels, see [15]. Since K m;2r are positive kernels, our preconditioners are positive definite for all n. In comparison, the Dirichlet kernel D n is not positive and hence the Strang preconditioner is indefinite in general. We will prove that if f has a zero of order 2p, then K m;2r f matches the zero of f when r ? p. Using this result, we can show that the spectra of the circulant preconditioned matrices are uniformly bounded except for at most 2p outliers, and that their smallest eigenvalues are bounded uniformly away from zero. It follows that the conjugate gradient method, when applied to solving these circulant preconditioned systems, will converge linearly. Since the cost per iteration is O(n log n) operations, see [7], the total complexity of solving these ill-conditioned Toeplitz systems is of O(n log n) operations. In the case when f is positive, we show that the spectra of the preconditioned matrices are clustered around 1 and thus the method converges superlinearly. The case where f has multiple zeros is more involved and will be considered in a future paper. This paper is an expanded version of the proceedings paper [10] where some of the preliminary results were reported. Recently Potts and Steidl [17] have proposed skew- circulant preconditioners for ill-conditioned Toeplitz systems. Their idea is also to use convolution products that match the zeros of f to construct the preconditioners. In par- ticular, they have used the generalized Jackson kernels and the B-spline kernels proposed in [8] in their construction. However, in order to guarantee that the preconditioners are positive definite, the position of the zeros of f is required which in general may not be readily available. In contrast, our circulant preconditioners can be constructed without any explicit knowledge of the zeros of f . The outline of the paper is as follows. In x2, we give an efficient method for computing the eigenvalues of the preconditioners. In x3 we show that K m;2r f matches the zeros of f . We then analyze the spectrum of the preconditioned matrices in x4. Numerical results are given in x5 to illustrate the effectiveness of our preconditioners in solving ill-conditioned Toeplitz systems. Concluding remarks are given in x6. 2 Construction of Circulant Preconditioners Let C 2- be the space of all 2-periodic continuous real-valued functions. The Fourier coefficients of a function f in C 2- are given by a \Gamma- f(')e \Gammaik' d'; a \Gammak for all k. Let A n [f ] be the n-by-n Hermitian Toeplitz matrix with the j)th entry given by a i\Gammaj , We will use C 2- to denote the space of all nonnegative functions in C 2- which are not identically zero. We remark that the Toeplitz matrices A n [f ] generated by f are positive definite for all n, see [6, Lemma 1]. Conversely, if f 2 C 2- takes both positive and negative values, then A n [f ] will be non- definite. In this paper, we only consider f being positive definite Hermitian Toeplitz matrices. We say that ' 0 is a zero of f of order p if f(' 0 is the smallest positive integer such that f (p) (' 0 ) 6= 0 and f (p+1) (') is continuous in a neighborhood of ' 0 . By Taylor's theorem, p! for all ' in that neighborhood. Since f is nonnegative, c ? 0 and p must be even. We remark that the condition number of A n [f ] generated by such an f grows like O(n p ), see [19]. In this paper, we will consider f having a single zero. The general case where f has multiple zeros is more complicated and will be considered in a future paper. The systems A n [f will be solved by the preconditioned conjugate gradient method with circulant preconditioners. It is well known that all n-by-n circulant matrices can be diagonalized by the n-by-n Fourier matrix F n , see [7]. Therefore, a circulant matrix is uniquely determined by its set of eigenvalues. For a given function f , we define the circulant preconditioner C n [f ] to be the n-by-n circulant matrix with its j-th eigenvalue given by We note that C n [f Hence the matrix-vector multiplication [f ]v, which is required in each iteration of the preconditioned conjugate gradient method, can be done in O(n log n) operations by fast Fourier transforms. Clearly if f is a positive function, then C n [f ] is positive definite. In the following, we will use the generalized Jackson kernel functions sin( m' to construct our circulant preconditioners. Here k m;2r is a normalization constant such that \Gamma- K m;2r 1. It is known that k m;2r is bounded above and below by constants independent of m, see [15, p.57] or (11) below. We note that K m;2 (') is the Fej'er kernel and K m;4 (') is the Jackson kernel [15, p.57]. For any m, the Fej'er kernel K m;2 (') can be expressed as where see for instance [9]. Note that \Gamma- K m;2 1. By (2), we see that K m;2r (') is the r-th power of K m;2 (') up to a scaling. Hence we have where the coefficients b (m;2r) k can be obtained by convolving the vector (b (m;2) times and this can be done by fast Fourier transforms, see [20, pp.294-296]. Thus the cost of computing the coefficients fb (m;2r) k g for all jkj - is of order O(rm log m) operations. In order to guarantee that \Gamma- K m;2r can normalize b (m;2r) 0 to 1=(2-) by dividing all coefficients b (m;2r) k by 2-b (m;2r) The convolution product of two arbitrary functions defined as (g h)(') j \Gamma- When we are given an n-by-n Toeplitz matrix A n [f ], our proposed circulant preconditioner is By (3) and (4), since a k e ik' , the convolution product of K m;2r f is given by (K m;2r a k b (m;2r) where 0; otherwise: Clearly, K m;2r f depends only on a k for jkj ! n, i.e., only on the entries of the given n-by-n Toeplitz matrix A n [f ]. Notice that by (1), to construct our preconditioner we only need the values of K m;2r f at 2-j=n for n. By (6), these values can be obtained by taking one fast Fourier transform of length n. Thus the cost of constructing C n [K m;2r f ] is of O(n log n) operations. We remark that the Strang [21] and the T. Chan circulant preconditioners [11] for are just equal to C n [D bn=2c f ] and C n [K n;2 f ] respectively where D bn=2c is the Dirichlet kernel and K n;2 (') is the Fej'er kernel, see [9]. 3 Properties of the Kernel K m;2r In this section, we study some properties of K m;2r in order to see how good the approximation of f by K m;2r f will be. These properties are useful in the analysis of our circulant preconditioners in x4. First we claim that our preconditioners are positive definite. Lemma 3.1 2- . The preconditioner C n [K m;2r f ] is positive definite for all positive integers m, n and r. Proof: By (2), K m;2r (') is positive except at 2- is nonnegative and not identically zero, the function (K m;2r f)(') j \Gamma- is clearly positive for all ' 2 [\Gamma-]. Hence by (1), the preconditioners C n [K m;2r f ] are positive definite. In the following, we will use ' to denote the function ' defined on the whole real line R. For clarity, we will use ' 2- to denote the periodic extension of ' on [\Gamma-], i.e. ' 2- 1 below). It is clear that ' 2- 2 C 2- . We first show that K m;2r ' 2p 2- matches the order of the zero of ' 2p 2- at Lemma 3.2 Let p and r be positive integers with r ? p. Then \Gamma- where Proof: The first two equalities in (7) are trivial by the definition of ' 2- . For the last equality, since '=- sin('=2) - '=2 on [0; -], we have by (2) \Gamma- Z -sin 2r Z m- 0sin 2r u aeZ 1sin 2r u Z 1sin 2r u oe aeZ 1u 2p du Z 11 oe On the other hand, we also have \Gamma- Z -sin 2r Z 1sin 2r u By setting \Gamma- Putting (11) back into (9) and (10), we then have (8). We remark that using the same arguments as in (10), we can show that i.e., the T. Chan preconditioner does not match the order of the zeros of ' 2p at when p - 1. We will see in x5 that the T. Chan preconditioner does not work for Toeplitz matrices generated by functions with zeros of order greater than or equal to 2. Next we estimate (K m;2r ' 2p 2- )(OE) for OE 6= 0. In order to do so, we first have to replace the function ' 2p 2- in the convolution product by ' 2p defined on R. Lemma 3.3 Let p be a positive integer. Then \Theta and Proof: To prove (13), we first claim that By the definition of 2- , we have (see Figure 1) For \Gamma-=2], we have For \Gamma-=2], we have Thus we have (16). By (16), we see that \Gamma- \Gamma- \Theta K m;2r ' 2p (' Similarly, we also have 5- Thus, we have (13). To prove (14), we just note that As for (15), we have With Lemmas 3.2 and 3.3, we show that K m;2r ' 2p 2- and ' 2p are essentially the same away from the zero of ' 2p 2- . Figure 1: The functions Theorem 3.4 Let p and r be positive integers with r ? p and dn=re. Then there exist positive numbers ff and fi independent of n such that for all sufficiently large n, Proof: We see from Lemma 3.3 that for different values of OE, (K m;2r ' 2p 2- )(OE) can be replaced by different functions. Hence, we proceed the proof for different ranges of values of OE. We first consider OE 2 [-=n; -=2]. By the binomial expansion, \Gamma- \Gamma- For odd k, \Gamma- K m;2r (t)t k OE \Gamma2k \Gamma- K m;2r (t)t 2k dt: By (7), we then have \Gamma K m;2r ' 2p where by (8), c k;2r are bounded above and below by positive constants independent of m by (5), -=r -m=n - OEm, we have Thus by (18), Hence by (14), (17) follows for OE 2 [-=n; -=2]. The case with OE 2 [\Gamma-=2; \Gamma-=n] is similar to the case where OE 2 [-=n; -=2]. Next we consider the case OE 2 [-=2; -]. Note that \Theta K m;2r ' 2p \Gamma- \Gamma- dt where is a degree 4p polynomial without the constant term. By (7), we have \Gamma- c 2j;2r Thus \Theta c 2j;2r 2- for OE 2 [-=2; -], we have \Theta which is clearly bounded independent of n. For the lower bound, we use the fact that 2- for OE 2 [-=2; -] in (19), then we have \Theta K m;2r ' 2p c 2j;2r c 2j;2r for sufficiently large n (and hence large m), the last expression is bounded uniformly from below say by - 2p =2. Combining (20), (21) and (15), we see that (17) holds for OE 2 [-=2; -] and for n sufficiently large. The case where OE 2 [\Gamma-; \Gamma-=2] can be proved in a similar way as above. Using the fact that \Gamma- we obtain the following corollary which deals with functions having a zero at fl 6= 0. Corollary 3.5 Let fl 2 [\Gamma-], p and r be positive integers with r ? p and Then there exist positive numbers ff and fi, independent of n, such that for all sufficiently large n, Now we can extend the results in Theorem 3.4 to any functions in C 2- with a single zero of order 2p. Theorem 3.6 Let f 2 C 2- and have a zero of order 2p at fl 2 [\Gamma-]. Let r ? p be any integer and dn=re. Then there exist positive numbers ff and fi, independent of n, such that for all sufficiently large n, Proof: By the definition of zeros (see x2), 2- g(') for some positive continuous function g(') on [\Gamma-]. Write (K m;2r f) (OE) \Deltag(OE) Clearly the last factor is uniformly bounded above and below by positive constants. By Corollary 3.5, the same holds for the second factor when -=n As for the first factor, by the Mean Value Theorem for integrals, there exists a i 2 [\Gamma-] such that Hence where g min and g max are the minimum and maximum of g respectively. Thus the theorem follows. So far we have considered only the interval -=n - now show that the convolution product K m;2r f matches the order of the zero of f at the zero of f . Theorem 3.7 Let f 2 C 2- and have a zero of order 2p at fl 2 [\Gamma-]. Let r ? p be any integer and dn=re. Then for any jOE \Gamma flj -=n, we have (K m;2r f) Proof: We first prove the theorem for the function . By the binomial theorem, \Gamma- \Gamma- Since \Gamma- K m;2r (t)t j we have for jOEj -=n, \Gamma- \Gamma- By (7), (8) and (5), we then have Hence by (14), On the other hand, from (22), (8) and (5), we have (K m;2r ' 2p )(OE) - \Gamma- O Hence by (14) again, Thus the theorem holds for 2- . In the general case where 2- g(') for some positive function g 2 C 2- , by the Mean Value Theorem for integrals, there exists a i 2 [\Gamma-] such that (K m;2r Hence min \Delta (K m;2r ' 2p for all OE 2 [\Gamma-]. Here g min and g max are the minimum and maximum of g respectively. From the first part of the proof, we already see that (K m;2r ' 2p is of O 1=n 2p for all jOE \Gamma flj -=n, hence the theorem follows. 4 Spectral Properties of the Preconditioned Matrices 4.1 Functions with a Zero In this subsection, we analyze the spectra of the preconditioned matrices when the generating function has a zero. We will need the following lemma. Lemma 4.1 [4, 16] Let f 2 C 2- . Then A n [f ] is positive definite for all n. Moreover if 2- is such that 0 ! ff - f=g - fi for some constants ff and fi, then for all n, x A n [g]x Next, we have our first main theorem which states that the spectra of the preconditioned matrices are essentially bounded. Theorem 4.2 Let f 2 C 2- and have a zero of order 2p at fl. Let r ? p and Then there exist positive numbers ff ! fi, independent of n, such that for all sufficiently large n, at most 2p are outside the interval [ff; fi]. Proof: For any function g 2 C 2- , we let ~ [g] to be the n-by-n circulant matrix with the j-th eigenvalue given by there is at most one j such that j2-j=n \Gamma flj ! -=n, by (1), ~ is a matrix of rank at most 1. By assumption, positive function g in C 2- . We use the following decomposition of the Rayleigh quotient to prove the theorem: x A n [f ]x x A n sin 2p x x A n sin 2p x x ~ sin 2p x x ~ sin 2p x x ~ x ~ x ~ x ~ We remark that by Lemma 4.1 and the definitions (1) and (23), all matrices in the factors in the right hand side of (24) are positive definite. As g is a positive function in C 2- , by Lemma 4.1, the first factor in the right hand side of (24) is uniformly bounded above and below. Similarly, by (23), the third factor is also uniformly bounded. The eigenvalues of the two circulant matrices in the fourth factor differ only when j2-j=n \Gamma flj -=n. But by Theorem 3.6, the ratios of these eigenvalues are all uniformly bounded when n is large. The eigenvalues of the two circulant matrices in the last factor differ only when j2-j=n -=n. But by Theorem 3.7, their ratios are also uniformly bounded. It remains to handle the second factor. Define s 2p (') j sin 2p ( '\Gammafl i.e., s 2p (') is a p-th degree trigonometric polynomial in '. Recall that for any function the convolution product of the Dirichlet kernel D n with h is just equal to the nth partial sum of h, i.e., (D n j=\Gamman b j e ij' . Hence for n - 2p, (D bn=2c s 2p Since C n [D bn=2c s 2p (')] is the Strang preconditioner for A n [s 2p (')], see [9], C n [s 2p (')] will be the Strang preconditioner for A n [s 2p (')] when n - 2p. As s 2p (') is a p-th degree trigonometric polynomial, A n [s 2p (')] is a band Toeplitz matrix with half bandwidth p+ 1. Therefore when n - 2p, by the definition of the Strang preconditioner, R where R p is a p-by-p matrix, see [21]. Thus A n [s 2p where the n-by-n matrix R n is of rank at most 2p + 1. Putting this back into the numerator of the second factor in (24), we have x A n [f ]x x ~ x ~ x ~ x ~ x ~ x A n [f ]x x R n x Notice that for all sufficiently large n, except for the last factor, all factors above are uniformly bounded below and above by positive constants. We thus have x A n [f ]x when n large, where Hence for large n, x If R n has q positive eigenvalues, then by Weyl's theorem [13, p.184], at most q eigenvalues of C n [K m;2r f are larger than ff max . By using a similar argument, we can prove that at most 2p are less than ff min . Hence the theorem follows. Finally we prove that all the eigenvalues of the preconditioned matrices are bounded from below by a constant independent of n. Hence the computational cost for solving this class of n-by-n Toeplitz systems will be of O(n log n) operations. Theorem 4.3 Let f 2 C 2- and have a zero of order 2p at fl. Let r ? p and Then there exists a constant c independent of n, such that for all n sufficiently large, all eigenvalues of the preconditioned matrix C are larger than c. Proof: In view of the proof of Theorem 4.2, it suffices to get a lower bound of the second Rayleigh quotient in the right hand side of (24). Equivalently, we have to get an upper bound of ae(A \Gamma1 denotes the spectral radius and We note that by the definition (23), ~ the zero matrix or is given by F diag for some j such that j2-j=n \Gamma flj ! -=n. Thus By Lemma 4.1, A \Gamma1 positive definite. Thus the matrix A is similar to the symmetric matrix A \Gamma1=2 Hence we have ae A A \Gamma1=2 A \Gamma1=2 A \Gamma1=2 A By [6, Theorem 1], we have Hence the last term in (26) is of O(1). It remains to estimate the first term in (26). According to (25), we partition A \Gamma1 as A are p-by-p matrices. Then by (25), ae A '- where the last equality follows because the 3-by-3 block matrix in the equation has vanishing central column blocks. In [3, Theorem 4.3], it has been shown that R p , B 11 , B 13 and all have bounded ' 1 -norms and ' 1 -norms. Hence using the fact that ae(\Delta) - k we see that (27) is bounded and the theorem follows. By combining Theorems 4.2 and 4.3, the number of preconditioned conjugate gradient iterations required for convergence is of O(1), see [3]. Since each PCG iteration requires O(n log n) operations (see [7]) and so is the construction of the preconditioner (see x2), the total complexity of the PCG method for solving Toeplitz systems generated by 2- is of O(n log n) operations. 4.2 Positive Functions In this subsection, we consider the case where the generating function is strictly positive. We note that the spectrum of A n [f ] is contained in [f min are the minimum and maximum values of f , see [6, Lemma 1]. Since f min ? 0, A n [f is well-conditioned. In [9], it was shown that for such f , the spectrum of C f ]A n [f ] is clustered around 1 and the PCG method converges superlinearly. Recall that is just the T. Chan circulant preconditioner. In the following, we generalize this result to other generalized Jackson kernels. First, it is easy to show that (K m;2r f) (OE) - f max . Thus the whole spectrum of C is contained in [f min =f i.e. the preconditioned system is also well-conditioned. We now show that its spectrum is clustered around 1. Theorem 4.4 Let f 2 C 2- be positive. Then the spectrum of C clustered around 1 for sufficiently large n. Here Proof: We first prove that K m;2r f converges to f uniformly on [\Gamma-]. For - ? 0, be the modulus of continuity of f . It has the property that see [15, p.43]. By the uniform continuity of f , for each " ? 0, there exists a ffi ? 0 such that !(f; ffi) ! ". \Gamma- \Gamma- \Gamma- \Gamma- \Gamma- bounded by a constant independent of n (cf. the proof of Lemma 3.2 for Therefore, K m;2r f converges uniformly to f . By [9, Theorem 1], the spectrum of C clustered around 1 for sufficiently large n. As an immediate consequence, we can conclude that when f is positive and C n [K m;2r \Lambdaf is used as the preconditioner, the PCG method converges superlinearly, see for instance [5]. 5 Numerical Experiments In this section, we illustrate by numerical examples the effectiveness of the preconditioner solving Toeplitz systems. For comparisons, we also test the Strang [21] and the T. Chan [11] circulant preconditioners. In the following, m is set to dn=re. Example 1: The first set of examples is on mildly ill-conditioned Toeplitz systems where the condition numbers of the systems grow like O(n ' ) for some ' ? 0. They correspond to Toeplitz matrices generated by functions having zeros of order ', see [19]. Because of the ill-conditioning, the conjugate gradient method will converge slowly and the number of iterations required for convergence grows like O(n '=2 ) [2, p.24]. However, we will see that using our preconditioner C n [K m;2r f ] with 2r ? ', the preconditioned system will converge linearly, i.e., the number of iterations required for convergence is independent of n. We solve Toeplitz systems A n [f by the preconditioned conjugate gradient method for twelve nonnegative test functions. Since the functions are nonnegative, the so generated are all positive definite. We remark that if f takes negative values, then A n [f ] will be non-definite for large n. As mentioned in x2, the construction of our preconditioners for an n-by-n Toeplitz matrix requires only the n diagonal entries fa j g jjj!n of the given Toeplitz matrix. knowledge of f is required. In the tests, the right-hand side vectors b are formed by multiplying random vectors to A n [f ]. The initial guess is the zero vector and the stopping criteria is jjr q jj 2 =jjr 0 jj is the residual vector after q iterations. Tables 1-4 show the numbers of iterations required for convergence for different choices of preconditioners. In the table, I denotes no preconditioner, S is the Strang preconditioner [21], K m;2r are the preconditioners from the generalized Jackson kernel K m;2r defined in (2) and is the T. Chan preconditioner [11]. Iteration numbers more than 3,000 are denoted by "y". We note that S in general is not positive definite as the Dirichlet kernel D n is not positive, see [9]. When some of its eigenvalues are negative, we denote the iteration number by "-" as the PCG method does not apply to non-definite systems and the solution thus obtained may be inaccurate. The first two test functions in Table 1 are positive functions and therefore correspond to well-conditioned systems. Notice that the iteration number for the non-preconditioned systems tends to a constant when n is large, indicating that the convergence is linear. In this case, we see that all preconditioners work well and the convergence is fast, see Theorem 4.4 and [9]. I Table 1: Numbers of iterations for well-conditioned systems. The four test functions in Table 2 are nonnegative functions with single or multiple zeros of order 2 on [\Gamma-]. Thus the condition numbers of the Toeplitz matrices are growing like O(n 2 ) and hence the numbers of iterations required for convergence without using any preconditioners is increasing like O(n). We see that for these functions, the T. Chan preconditioner does not work. This is to be expected from the fact that the order of does not match that of ' 2 at see (12). However, we see that K m;4 , K m;6 and K m;8 all work very well as predicted from our convergence analysis in x4. When the order of the zero is 4, like the two test functions in Table 3, the condition number of the Toeplitz matrices will increase like O(n 4 ) and the matrices will be very ill-conditioned even for moderate n. We see from the table that both the Strang and the T. Chan preconditioners fail. For the T. Chan preconditioner, the failure is also to be expected from the fact that the order of K m;2 ' 4 does not match that of ' 4 at (12). As predicted by our theory, K m;6 and K m;8 still work very well. The numbers of iterations required for convergence are roughly constant independent of n. In Table 4, we test functions that our theory does not cover. The first two functions are not differentiable at their zeros. The last two functions are functions with slowly decaying Fourier coefficients. We found numerically that the minimum values of I 36 79 170 362 753 1544 53 141 293 547 1113 2213 I Table 2: Numbers of iterations for functions with order 2 zeros. I 26 42 71 161 167 247 24 35 58 106 144 196 KN;4 15 17 20 24 26 26 15 KN;8 Table 3: Numbers of iterations for functions with order 4 zeros. I I Table 4: Numbers of iterations for other functions. and jkj!1024jkj 0:5 +1 e ik' are approximately equal to 0.3862 and 0.4325 respectively. Hence the last two test functions are approximately zero at some points in [\Gamma-]. Table 4 shows that the K m;2r preconditioners still perform better than the Strang and the T. Chan preconditioners. To further illustrate Theorems 4.2 and 4.3, we give in Figures 2 and 3 the spectra of the preconditioned matrices for all five preconditioners for We see that the spectra of the preconditioned matrices for K m;6 and K m;8 are in a small interval around 1 except for one to two large outliers and that all the eigenvalues are well separated away from 0. We note that the Strang preconditioned matrices in both cases have negative eigenvalues and they are not depicted in the figures. Example 2: In image restoration, because the blurring is an averaging processing, the resulting matrix is usually strongly ill-conditioned in the sense that its condition number grows exponentially with respect to its size n. In contrast, the condition numbers of the mildly ill-conditioned matrices considered in Example 1 are increasing like polynomials of n only. Regularization techniques have been used for some time in mathematics and engineering to treat these strongly ill-conditioned systems. The idea is to restrict the solution in some smooth function spaces [14]. This approach has been adopted in the circulant preconditioned conjugate gradient method and is very successful when applied to ground-based astronomy [7]. To illustrate the idea, we use a "prototype" image restoration problem given in [12]. Strang Preconditioner (has negative eigenvalues) T. Chan Preconditioner K m,4 Jackson Preconditioner K m,6 Jackson Preconditioner K m,8 Jackson Preconditioner Figure 2: Spectra of preconditioned matrices for Strang Preconditioner (has negative eigenvalues) T. Chan Preconditioner K m,4 Jackson Preconditioner K m,6 Jackson Preconditioner K m,8 Jackson Preconditioner Figure 3: Spectra of preconditioned matrices for Consider a 100-by-100 Toeplitz matrix A with (i; entries given by ae 0; if where -oe Blurring matrices of this form (called the truncated Gaussian blur) occur in many image restoration contexts and are used to model certain degradations in the recorded image. The condition number of A is approximately 2.3\Theta10 6 . Thus if no regularization is used, the result obtained will be very inaccurate. In our experiment, we solve the regularized least squares problem min x as suggested in [12]. The problem is equivalent to the normal equations (ffI which we solve by the preconditioned conjugate gradient method. We choose the solution vector x with its entries given by see [12], and then we compute noise vector is added to b where each component of the noise vector is taken from a normal distribution with mean zero and standard deviation . The stopping criteria is jjr q jj 2 =jjr 0 jj is the residual vector after q iterations. We choose the optimal regularization parameter ff such that it minimizes the relative error between the computed solution x(ff) of the normal equations and the original solution x given in (28), i.e. ff minimizes . By trial and error, it is found to be \Gamma6 up to one digit of accuracy. The preconditioner we used for the normal equations is of the form ff I is chosen to be S, T , K m;4 , K m;6 , and K m;8 . The corresponding numbers of iterations required for convergence are equal to 21; 33; 22; 22, and 23 respectively. The number of iterations without preconditioning is 171. The relative error of the regularized solution is about 3:1 \Theta 10 \Gamma1 . In contrast, it is about 6:9 \Theta 10 +2 if no regularization is used. Thus we see that our preconditioners also work for strongly ill-conditioned systems after it is regularized. 6 Concluding Remarks We remark that even for mildly ill-conditioned matrices with condition number of order then the matrix A n will be very ill-conditioned already for moderate regularization is also needed in this case. Once the system is regularized, our preconditioner C n [K m;8 f ] will work even if p ? 6, cf. Example 2 in x5 for instance. Hence in general, the circulant preconditioner C n [K m;8 f ] should be able to handle all cases, whether the matrix A n is well-conditioned, mildly ill-conditioned, or very ill-conditioned but regularized. --R Superfast solution of real positive definite Toeplitz systems Finite Element Solution of Boundary Value Problems Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices Circulant Preconditioners for Hermitian Toeplitz Systems Toeplitz Preconditioners for Toeplitz Systems with Nonnegative Generating Functions Conjugate Gradient Methods for Toeplitz Systems Circulant Preconditioners from B-Splines Circulant Preconditioners Constructed from Kernels Circulant Preconditioners for Ill-Conditioned Hermitian Toeplitz Matrices An Optimal Circulant Preconditioner for Toeplitz Systems An algorithm for the regularization of ill-conditioned Matrix Analysis The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind Approximation of Functions Preconditioners for Ill-Conditioned Toeplitz Matrices Preconditioners for Ill-Conditioned Toeplitz Systems Constructed from Positive Kernels Preconditioning Strategies for Hermitian Toeplitz Systems with Nondefinite Generating Functions On the extreme eigenvalues of Hermitian (block) Toeplitz matrices Introduction to Applied Mathematics A Proposal for Toeplitz Matrix Calculations An Algorithm for the Inversion of Finite Toeplitz Matrices Circulant Preconditioners with Unbounded Inverses --TR --CTR Weiming Cao , Ronald D. Haynes , Manfred R. Trummer, Preconditioning for a Class of Spectral Differentiation Matrices, Journal of Scientific Computing, v.24 n.3, p.343-371, September 2005

preconditioned conjugate gradient method;kernel functions;toeplitz systems;circulant preconditioner

588420

Existence Verification for Singular Zeros of Complex Nonlinear Systems.

Computational fixed point theorems can be used to automatically verify existence and uniqueness of a solution to a nonlinear system of n equations in n variables ranging within a given region of n-space. Such computations succeed, however, only when the Jacobi matrix is nonsingular everywhere in this region. However, in problems such as bifurcation problems or surface intersection problems, the Jacobi matrix can be singular, or nearly so, at the solution. For n real variables, when the Jacobi matrix is singular, tiny perturbations of the problem can result in problems either with no solution in the region, or with more than one; thus no general computational technique can prove existence and uniqueness. However, for systems of n complex variables, the multiplicity of such a solution can be verified. That is the subject of this paper.Such verification is possible by computing the topological degree, but such computations heretofore have required a global search on the (n-1)-dimensional boundary of an n-dimensional region. Here it is observed that preconditioning leads to a system of equations whose topological degree can be computed with a much lower-dimensional search. Formulas are given for this computation, and the special case of rank-defect one is studied, both theoretically and empirically.Verification is possible for certain subcases of the real case. That will be the subject of a companion paper.

Introduction . Given an approximate solution - x to a nonlinear system of equations F is useful in various contexts to construct bounds around - x in which it is proven that there exists a unique solution x # , F continuously di#erentiable F for which the Jacobian det(F # (x #= 0 and for which that Jacobian is well conditioned, interval computations have no trouble proving that there is a unique solution within small boxes with x # reasonably near the center; see [8], [16], [23]. However, if F # conditioned or singular, such computations necessarily must fail. In the singular case, for some classes of systems F arbitrarily small perturbations of the problem can lead to no solutions or an even number of solutions, so multiplicity verification is not logical. In contrast, verification is always possible if F maps C n into C n . Here, algorithms are developed for the multiplicity of such solutions for F The algorithms are presented in the context of solutions that lie near the real line of complex extensions of real systems. (Such solutions arise, for example, in bifurcation problems.) However, the algorithms can be generalized to arbitrary solutions z # C n with z not necessarily near the real line. Also, verification is possible for singular solutions of particular general classes of We will cover this in a separate paper. # Received by the editors September 10, 1999; accepted for publication (in revised form) February 21, 2000; published electronically July 19, 2000. This work was supported by National Science Foundation grant DMS-9701540. http://www.siam.org/journals/sinum/38-2/36107.html Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504 (rbk@ louisiana.edu, dian@louisiana.edu). # Institut f?r Mathematik, Universit?t Wien, Strudhofgasse 4, A-1050 Wien, Austria (neum@ cma.univie.ac.at). SINGULAR COMPLEX ZEROS VERIFICATION 361 1.1. Previous work, related material, and references. The emphasis in this paper is on rigorous verification of existence of a zero of a system of nonlinear equations in a small region containing an approximate, numerically computed solu- tion. Verification for F : R n with the Jacobi matrix of F nonsingular at points x with F done with computational fixed point theorems based on interval Newton methods. Such methods are introduced, for example, in the books [2], [8], [11], [16], [21], and [23]. The techniques in this paper for handling singularities are based on the topological degree. Introductions to degree theory include parts of [3] (in German) or [20]. A basic computational procedure for the degree over large regions appears in Stenger [27]. Stynes [28], [29] and Kearfott [12], [13], [14] derived additional formulas and algorithms based on Stenger's results. These degree computation procedures, however, involved heuristics, and the result was not guaranteed to be correct. Aberth [1] based a verified degree computation method on interval Newton methods and a recursive degree-computation formula such as Theorem 2.2 below. The work here di#ers from this previous work in two important aspects: . The algorithms here execute in polynomial time with respect to the number of variables and equations, 1 and . the algorithms here assume at least second-order smoothness, and are meant to compute the degree over small regions containing the solution, over which certain asymptotic approximations are valid. The treatment of verified existence represented in this paper involves computation of the topological degree in n-dimensional complex space. In loosely related work, Vrahatis et al. develop an algorithm for computing complex zeros of a function of a complex variable in [31]. Finally, most of the literature we know on specialized methods for finding complex zeros, verified or otherwise, of equations and systems of equations deals with polynomial systems. Along these lines, continuation methods, as introduced in [6] and [22], figure prominently. The article [4] contains methods for determining the complex zeros of a single polynomial, while [7] and [9] contain verified methods for determining the complex zeros of a single polynomial. 1.2. Notation. We assume familiarity with the fundamentals of interval arith- metic; see [16, 23] for an introduction in the present context. (The works [2], [8], [24] also contain introductory material.) Throughout, scalars and vectors will be denoted by lower case, while matrices will be denoted by upper case. Intervals, interval vectors (also called "boxes"), and interval matrices will be denoted by boldface. For instance, an interval vector, denotes an interval matrix. Real n-space will be denoted by R n , while the set of n-dimensional interval matrices will be denoted by IR n-n . Similarly, complex n-space will be denoted by C n . The midpoint of an interval or interval vector x will be denoted by m(x). The nonoriented boundary of a box x will be denoted by #x while its oriented boundary will be denoted by b(x) (see section 2). 1.3. Traditional computational existence and uniqueness. Computational existence and uniqueness verification rests on interval versions of Newton's method. Typically, such computations can be described as evaluation of a related interval 1 The general degree computation problem is NP-complete; see [26]. 362 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER operator implies existence and uniqueness of the solution of To describe these, we review the following definition. Definition 1.1 (see [23, p. 174], etc. Let F : R n . The matrix A is said to be a Lipschitz matrix for F over x provided for every x # x and y # x, F (x) - A. Most interval Newton methods for F : R n abstractly, are of the general where v is computed to contain the solution set to the interval linear system and where, for initial uniqueness verification, A is generally a Lipschitz matrix 2 for F over the box (interval vector) x and - x # x is a guess point. We sometimes write F # (x) in place of A, since the matrix can be an interval extension of the Jacobi matrix of F . Uniqueness verification traditionally depends on regularity of the matrix A. We have the following lemma. Lemma 1.2. (see [16], [23]). Suppose - is the image under the interval Newton method (formula (1.1)), where v is computed by any method that bounds the solution set to the interval linear system (1.2), and - x # x. Then A is regular. The method of bounding the solution set of (1.2) to be considered here is the interval Gauss-Seidel method, defined by the following definition. Definition 1.3. The preconditioned interval Gauss-Seidel image GS(F ; x, - x) of a box x is defined as GS(F ; x, - x i is defined sequentially for to n by where and where - is an initial guess point, Y A # IR n-n and Y F (-x) are the matrix and right-hand-side vector for the preconditioned interval system Y A(x- -Y F (-x), Y # R n-n is a point preconditioning matrix, Y i denotes the ith row of Y , and A j denotes the jth column of A. Lemma 1.2 applies when N(F x), provided we specify that x) be in the interior 3 int(x) of x. In particular, we have the following theorem. Theorem 1.4 (see [16], [23]). Suppose F : x # R n A is a Lipschitz matrix such as an interval extension F # (x) of the Jacobi matrix. If - x is the image under an interval Newton method as in formula (1.1) and - x # int(x), then there is a Various authors have proven Theorem 1.4; see [16], [23]. In particular, Miranda's theorem can be used to easily prove Theorem 1.4 for see [19], [30], or [16, p. 60]. For worked-out examples, see [18, p. 3] or [17]. However, see [16, 25] for techniques for using slope matrices. 3 We must specify the interior because of the intersection step in Definition 1.3. SINGULAR COMPLEX ZEROS VERIFICATION 363 Inclusion in the interval Gauss-Seidel method is possible because the inverse midpoint preconditioner reduces the interval Jacobi matrix to approximately a diagonal matrix. In the singular case, an incomplete factorization for the preconditioner leads to an approximate diagonal matrix in the upper (n-1)- (n-1) submatrix, but with approximate zeros in the last row. We discovered the methods in this paper by viewing the interval Gauss-Seidel method on this submatrix, then applying special techniques to the preconditioned nth function. 1.4. A simple singular example. Consider the following example. Example 1. Take and Even though there is a unique root x F is as in Example 1, the interval Gauss-Seidel method cannot prove this, since the In fact, the interval Jacobi matrix is computed to be , and the midpoint matrix is m(F # ). The midpoint matrix, often used as the preconditioner Y , is singular. 4 Symbolic methods can be used to show that Example 1 has a unique solution at arbitrarily small perturbations of the problem result in either no solutions or two solutions. Consider the following example. Example 2. Take and . Here, |#| is very small. The system in Example 2 has two solutions for # < 0 and no solutions for # > 0. Roundout in computer arithmetic and, perhaps, uncertainties in the system itself due to modelling or measurement uncertainties, however, make it impossible to distinguish systems such as in Example 2 for di#erent #, especially when computer arithmetic is used as part of the verification process. In such instances, no verification is possi- ble. However, if F is viewed as a complex function of two variables, then, for all # su#ciently small, F has two solutions in a small box in C 2 containing the real point (0, 0). More generally, we can extend an n-dimensional box in R n to an n-dimensional box in C n by adding a small imaginary part to each variable. If the system can be extended to an analytic function in complex n-space (or if it can be extended to a function that can be approximated by an analytic function), then the topological degree gives the number of solutions, counting multiplicities, within the small region in complex space. (See section 2 for an explanation of multiplicity.) For example, Alternate preconditioners can nonetheless be computed; see [16]. However, it can be shown that uniqueness cannot be proven in this case; see [16], [23]. 364 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER the degree of the system in Example 2 within an extended box in complex space can be computed to be 2, regardless of whether # is negative, positive, or zero. (See the numerical results in section 8.) The topological degree corresponds roughly to algebraic degree in one dimension; for example, the degree of z n in a small region in containing 0 is n. 1.5. Organization of this paper. A review of properties of the topological degree, to be used later, appears in section 2. The issue of preconditioning appears in section 3. Construction of the box in the complex space appears in section 4. Several algorithms have previously been proposed for computing the topological degree [1], [12], [28], but these require computational e#ort equivalent to finding all solutions to 4n (2n-1)-dimensional nonlinear systems within a given box, or worse. In section 5, a reduction is proposed that allows computation of the topological degree with a search in a space of dimension equal to the rank defect of the Jacobian matrix. A theorem is proven that further simplifies the search. In section 6, the actual algorithm is presented and its computational complexity is given. Test problems and the test environment are described in section 7. Numerical results appear in section 8. Future directions appear in section 9. 2. Review of some elements of degree theory. The topological degree or Brouwer degree, well known within algebraic topology and nonlinear functional anal- ysis, is both a generalization of the concept of a sign change of a one-dimensional continuous function and of the winding number for analytic functions. It can be used to generalize the concept of multiplicity of a root. The fundamentals will not be reviewed here, but we refer to [3], [5], [12]. We present only the material we need. Here we explain what we mean by "multiplicity." Actually, there is a more general concept index (see [5, Chapter I]) for an isolated zero. The topological degree is equal to the sum of the indices of zeros in the domain. The index is always positive in our context. For this reason, we use the more suggestive term multiplicity as an alternative term for index. Suppose that F : D # C n # C n is analytic. Then the real and imaginary components of F and its argument z # C n may be viewed as real components in R 2n . by - R 2n by - we have the following property of topological degree D, 0), and relationships between D, 0) and the solutions of the system Theorem 2.1 (see [5], [20], etc. Suppose F : D # C n # C n is analytic, with F (z) #= 0 for any z #D, and suppose - D and - D # R 2n are defined as above. Then D, D, only if there is a solution z # D, F (z # D, is equal to the number of solutions z # D, F (z # counting multiplicities. (4) If the Jacobi matrix F # (z # ) is nonsingular at every z # D with F (z # then D, is equal to the number of solutions z # D, F (z # The following three theorems lead to the degree computation formula in Theorem 5.1 in section 5, the formula used in our computational scheme. Theorem 2.2. (see [27, section 4.2]). Let D be an n-dimensional connected, oriented region in R n and are continuous SINGULAR COMPLEX ZEROS VERIFICATION 365 functions defined in D. Assume F #= 0 on the oriented boundary b(D) of D, b(D) can be subdivided into a finite number of closed, connected (n - 1)-dimensional oriented subsets # k and there is a on the oriented boundary b(# k n-1 ) of has the same sign at all solutions of there are any, on # k Choose s # {-1, +1} and let K 0 denote the subset of the integers k # {1, . , r} such that has solutions on # k n-1 and sgn(f p at each of those solutions. Then The formula in Theorem 2.2 is a combination of formulas (4.15) and (4.16) in [27]. The orientation of D is positive and the orientations of # k positive or negative, are induced by the orientation of D. If we assume that the Jacobi matrices of F-p are nonsingular at all solutions of depending on whether # k n-1 has positive orientation or negative orientation, and JF-p (x) is the determinant of the Jacobi matrix of F-p at x. (See Theorem 5.2 and Theorem 7.2 in Chapter I of [5].) Thus we can simplify the formula in Theorem 2.2 as follows. Theorem 2.3. Suppose the conditions of Theorem 2.2 are satisfied and, addi- tionally, the Jacobi matrix of F-p is nonsingular at each solution of for each k # K 0 (s). Then depending on whether # k n-1 has positive orientation or negative orientation, and JF-p (x) is the determinant of the Jacobi matrix of F-p at x. In our context, the region D is an n-dimensional box The boundary #x of x consists of 2n (n - 1)-dimensional boxes The following theorem, necessary for the main characterization used in our algo- rithm, is a basic property of oriented domains in n-space and follows from definitions such as in [3]. See [18, pp. 7-8] for a detailed derivation in terms of oriented simplices. Theorem 2.4. If x is positively oriented, then the induced orientation of x k is and the induced orientation of x k is The oriented boundary b(x) can be divided into x k and x k the associated orientations. Also, F #= 0 on b(x) is the same as F #= 0 on #x. 366 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER z }| { Fig. 3.1. A singular system of rank n - p preconditioned with an incomplete LU factorization, where "#" represents a nonzero element. Now fix a p between 1 and n. Then F-p ) is the same as F-p . For this fixed p, let K 0 denote the subset of the integers k # {1, . , n} such that has solutions on x k and sgn(f p at these solutions, and let K 0 denote the subset of the integers k # {1, . , n} such that has solutions on x k and sgn(f p at these solutions, where s # {-1, +1}. Then, by Theorem 2.3, we have the following theorem. Theorem 2.5. Suppose F #= 0 on #x, and suppose there is that (1) F-p #= 0 on #x k or #x k , has the same sign at all solutions of there are any, on x k or (3) the Jacobi matrices of F-p are nonsingular at all solutions of Then sgn #F-p x#x sgn #F-p 3. On preconditioning. The inverse midpoint preconditioner approximately diagonalizes the interval Jacobi matrix when F # (and well enough conditioned). This preconditioner can be computed with Gaussian elimination with partial pivoting. We can compute (to within a series of row permutations) an LU factorization of the midpoint matrix m F # (x) . The factors L and U may then be applied to actually precondition the interval linear system. When the rank of F # Gaussian elimination with full pivoting can be used to reduce F # (x) to approximately the pattern shown in Figure 3.1. Actually, an incomplete factorization based on full pivoting will put the system into a pattern that resembles a permutation of the columns of the pattern in Figure 3.1. However, for notational simplicity, there is no loss here in assuming exactly the form in Figure 3.1. SINGULAR COMPLEX ZEROS VERIFICATION 367 In the analysis to follow, we assume that the system has already been precondi- tioned, so that it is, to within second-order terms with respect to w(x), of the form in Figure 3.1. Here we concentrate on the case p=1, although the idea can be applied to the general case. 4. The complex setting and system form. Below, we assume R n can be extended to an analytic function in C n . small box that will be constructed centered at the approximate solution - x is near a point x # with F such that #-x - x # is much smaller than the width of the box x, and width of the box x is small enough so that mean value interval extensions lead, after preconditioning, to a system like Figure 3.1, with small intervals replacing the zeros. (4) F has been preconditioned as in Figure 3.1, and F # space of dimension 1. The following representation is appropriate under these assumptions: . to complex space: x iy, with y in a small box is centered at (0, . , 0). Define z # iy)), and v k (x, y) #(f k Then, if preconditioning based on complete factorization of the midpoint matrix for F # (x) is used, the first-order terms are eliminated in the pattern of Figure 3.1, and, , #xn (-x)y n +O #(x - , and (-x)y k y l +O #(x - , . 5. Simplification of a degree computation procedure. To use Theorem 2.5 to compute the topological degree directly in a verification algorithm would require a global search of the 4n (2n-1)-dimensional faces of the 2n-dimensional box z for zeros of - F-p . This is an inordinate amount of work for a verification process 368 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER that would normally require only a single step of an interval Newton method in the nonsingular case. However, if the system is preconditioned and in the form described in section 3 and section 4, the verification can be reduced to 4n-4 interval evaluations and four one-dimensional searches. To describe the simplification, define Similarly define y k and y k . Also define F-un To compute the degree F , z, 0), we will consider - F-un on the boundary of z. The boundary of z consists of the 4n faces x 1 , x 1 Observe that, for F-un #xn |#fk /#xn (-x)| # w(x n ). Similarly, - F-un F-un F-un on y k implies w(y k )/|#f k /#x n (-x)| # w(y n ). Thus if x n is chosen so that min , then it is unlikely that u k (x, . Similarly, if y n is chosen so that min , then it is unlikely that v k (x, on either y k or y k . Here, the coe#cient 1 2 is to take into consideration the fact that u k (x, y) # #xn #xn (-x)y n are only approximate equalities. (When #f k /#x n there is no restriction on w(x n ) or w(y n ) due to w(x k ) or w(y k ).) By constructing the box z in this way, we can eliminate search of 4n - 4 of the 4n faces of the boundary of z, since we have arranged to verify - F-un (x, y) #= 0 on each of these faces. Elimination of these 4n - 4 faces needs only 4n - 4 interval eval- uations. Then, we need only to search the four faces x n , x n , y n , and y n for solutions of - F-un regardless of how large n is. This reduces total computational cost dramatically, since searching a face is expensive. Based on this, the following theorem underlies our algorithm in section 6.1. Theorem 5.1. Suppose , and v k #= 0 on y k and y k has a unique solution on x n and x n with y n in the interior of y n , and - has a unique solution on y n and y n with x n in the interior of (3) at the four solutions of - condition 2; and (4) the Jacobi matrices of - F-un are nonsingular at the four solutions of - in condition 2. SINGULAR COMPLEX ZEROS VERIFICATION 369 Then F , z, F-un (x,y)=0 un (x,y)>0 sgn F-un xn=xn F-un (x,y)=0 un (x,y)>0 sgn F-un F-un (x,y)=0 un (x,y)>0 sgn F-un F-un (x,y)=0 un (x,y)>0 sgn F-un Proof. Condition 1 implies - conditions 2 and 3 imply - F #= 0 on #z. Condition 1 implies - F-un #= 0 on #x k , #x k consists of 2(n - 1) (2n - 2)-dimensional boxes, each of which is either embedded in some x k , x k or is embedded in y n or y n . Thus, by conditions 2 and 3, - F-un #= 0 on #x n . Similarly, - F-un #= 0 on #x n , #y n and #y n . Thus condition 1 in Theorem 2.5 is satisfied. Condition 2 in Theorem 2.5 is automatically satisfied since either has no solutions or a unique solution on x k , x k Then, with condition 4, the conditions of Theorem 2.5 are satisfied. The formula is thus obtained with The conditions of Theorem 5.1 will be satisfied when the system is that as described in section 3 and section 4, the box z is constructed as in (5.1) and (5.2), and the quadratic model is accurate. (See Theorem 5.2 and its proof of the results when all the approximations are exact.) In Theorem 5.1, the degree consists of contributions of the four faces we search. We can compute the degree contribution of each of the four faces, then add them to get the degree. In Theorem 5.1 we choose We can also choose s = -1. That doesn't make any di#erence in our context if we ignore higher order terms in the values of u n at the solutions of - on the four faces x n , x n , y n , and y n . To be specific, the four values of u n are , , , , 370 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER respectively, where # is defined in (5.3). When we choose w(y k ) the same (or roughly the same) as w(x k ), the values of u n as a function of y n (or y n ) will be the same (or roughly the same) as the values of u n as a function of x n - we ignore higher order terms, the cost of verifying u n < 0 and searching for solutions of - is approximately the same as the cost of verifying u n > 0 and searching for solutions of - Next we will give a theorem that will further reduce the search cost by telling us how we should search. Define Theorem 5.2. If the approximations of (4.1) and (4.2) are exact, if we construct the box z as in (5.1) and (5.2), and if #= 0, then F , z, 2. Proof. Under the assumptions, Due to the construction of the box z, u 1. Next we locate the solutions of - (1) On x n , Plugging (5.8) and (5.9) into (5.6) and (5.7), we get Then SINGULAR COMPLEX ZEROS VERIFICATION 371 since #= 0. Thus by (5.9) Therefore - has a unique solution (-x, - on x n . Plugging (5.12) into (5.10), we get the u n value at this solution, which is Next we compute the determinant of the Jacobi matrix of - F-un at this solu- tion. Noting (5.4), (5.5), and (5.7), we have F-un (2) Similarly, on x n , - has a unique solution (-x, - y) on x n . The u n value at this solution is The determinant of the Jacobi matrix of - F-un at this solution is F-un (3) On y n , Plugging (5.18) and (5.19) into (5.6) and (5.7), we get 372 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER Then since #= 0. Thus by (5.18), Therefore - has a unique solution (-x, on y n . Plugging (5.22) into (5.20), we get the u n value at this solution, which is Next, as in (5.15), we compute the determinant of the Jacobi matrix of - F-un at this solution. Noting (5.4), (5.5), and (5.7), we have F-un #. Similarly, - has a unique solution (-x, - y) on y n . The u n value at this solution is n . The determinant of the Jacobi matrix of - F-un at this solution is F-un Finally, we can use the formula in Theorem 5.1 to compute the topological degree F , z, 0). If # > 0, then we know from (5.14), (5.16), (5.24), and (5.26) that at the solutions of - We also know the signs of the determinants of the Jacobi matrices at the two solutions from (5.15) and (5.17). Therefore, F , z, 2. If # < 0, then we know from (5.14), (5.16), (5.24), and (5.26) that u n > 0 at the solutions of - y n . We also know the signs of the determinants of the Jacobi matrices at the two solutions from (5.25) and (5.27). Therefore F , z, also in this case. The proof of Theorem 5.2 tells us approximately where we can expect to find the solutions of - on the four faces we search and the value of the degree we can expect when the approximations (4.1) and (4.2) are accurate. From (4.1), we know that if x n is known precisely, formally solving u k (x, for x k gives sharper bounds - larly, if y n is known precisely, formally solving v k (x, sharper bounds SINGULAR COMPLEX ZEROS VERIFICATION 373 n- 1. Thus when we search x n (or x n ) for solutions of - 0, we can first get sharper bounds for x k , 1 # k # n-1, since x n is known precisely. Then, for a small subinterval y 0 n of y n , we can solve v k (x, y k to get sharper bounds - Thus we get a small subface of x n (or x n ) over which we can either use an interval Newton method to verify the existence and uniqueness of the zero of - F-un or use mean-value extensions to verify that - F-un has no zeros, depending on whether y 0 n is in the middle of y n or not. Thus the process reduces to searching over a one-dimensional interval y n . This further reduces the search cost. We can similarly search y n or y n . 6. The algorithm and its computational complexity. 6.1. Algorithm. The algorithm consists of three phases. In the box-setting phase, we set the box z. In the elimination phase, we verify that u k #= 0 on x k and , and v k #= 0 on y k and y k 1. In the search phase, we verify the unique solution of - in the interior of y n , and on y n and y n with x n in the interior of x n , compute the signs of u n and the signs of the Jacobi matrices of - F-un at the four solutions of - compute the degree contributions of the 4 faces x n , x n , y n , and y n according to the formula in Theorem 5.1, and finally add the contributions to get the degree. Algorithm Box-setting phase 1. Compute the preconditioner of the original system, using Gaussian elimination with full pivoting. 2. Set the widths of x k and y k (see explanation below), for 1 # k # n - 1. 3. Set the widths of x n and y n as in (5.1) and (5.2). Elimination phase 1. Do for (a) Do for x k and x k i. Compute the mean-value extension of u k over that face. ii. If 0 # u k , then stop and signal failure. (b) Do for y k and y k i. Compute the mean-value extension of v k over that face. ii. If 0 # v k , then stop and signal failure. Search phase 1. Do for x n and x n (a) i. Use mean-value extensions for u k (x, to solve for x k to get sharper bounds - x k with width O #(x - x, y)# 2 ii. If - return the degree contribution of that face as iii. Update x k . (b) i. Compute the mean-value extension u n over that face. ii. If u n < 0, then return the degree contribution of that face as 0. (c) Construct a small subinterval y 0 n of y n which is centered at 0. (d) i. Use mean-value extensions for v k (x, to solve for y k to get sharper bounds - y k with width O max(#(x - thus getting a subface x 0 n ) of x n (or x n .) ii. If - #, then stop and signal failure. 374 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER up an interval Newton method for - F-un to verify existence and uniqueness of a zero in the subface x 0 ii. If the zero cannot be verified, then stop and signal failure. (f) Inflate y 0 n as much as possible subject to verification of existence and uniqueness of the zero of - F-un over the corresponding subface, and thus get a subinterval y 1 n of y n . (g) In this step, we verify - solutions when y n . n has two separate parts; we denote the lower part by y l n and the upper part by y u n . We present only the processing of the lower part. The upper part can be processed similarly. A. Use mean-value extensions for v k (x, to solve for y k to get sharper bounds for y k , 1 # k # n - 1, and thus to get a subface of x n (or x n ). B. Compute the mean-value extensions - F -un over the subface obtained in the last step. F -un , then bisect y l update the lower part as a new y l and cycle. F -un , then exit the loop. ii. Do A. If y 1 exit the loop. B. y l C. Use mean-value extensions for v k (x, to solve for y k to get sharper bounds for y k , 1 # k # n - 1, and thus to get a subface of x n (or x n ). D. Compute the mean-value extensions - F -un over the subface obtained in the last step. F -un , then cycle. F -un , then y l , mid(y l Compute the mean-value extension of u n over x 0 .) ii. If u n < 0, then return the degree contribution of that face as 0. F-un F-un ii. If 0 # - F-un F-un stop and signal failure. (j) Use the formula in Theorem 5.1 to compute the degree contribution of that face. 2. Do for y n and y n (a) Same as step 1(a) except change x k to y k , - x k to - to v k . (b) Same as step 1(b). (c) Same as step 1(c) except change y 0 n to x 0 n , y n to x n , and 0 to - x n . (d) Same as step 1(d) except change y k to x k , - y k to - n to y 0 n to y 0 to y n , and x n to y n . Same as step 1(e) except change x 0 n to y 0 n and x 0 n to y 0 n . (f) Same as step 1(f) except change y 0 n to x 0 n to x 1 n , and y n to x n . (g) Same as step 1(g) except change y n \ y 1 n to x n \ x 1 n . Same as step 1(h) except change x 0 n to y 0 n and x 0 n to y 0 SINGULAR COMPLEX ZEROS VERIFICATION 375 (i) Same as step 1(i) except change F-un F-un F-un F-un (j) Same as step 1(j). 3. Add the degree contributions of the four faces obtained in steps 1 and 2 to get the degree. End of algorithm An explanation of the algorithm 1. In the box-setting phase, in step 2, the width w(x k ) of x k depends on the accuracy of the approximate solution - x of the system F should be much larger than |-x k - x # k |. At the same time, w(x k ) should not be too large, since the quadratic model needs to be accurate over the box. 2. In the search phase, in step 1(b) (or 2(b)), we check the sign of u n on that face and discard that face at the earliest possible time if u n < 0 on that face, since we know the degree contribution of that face is 0 according to the formula in Theorem 5.1. This will save time significantly if it happens that on that face. It did happen for all the test problems. (See section 8 for the test results.) 3. In the search phase, in step 1(e) (or 2(e)), we precondition the system - F-un before we use an interval Newton method, so that the method will succeed (see section 1.3 and section 3). The system - F-un is nonsingular over the subfaces under consideration. 4. In the search phase, in step 1(f) (or 2(f)), we first expand the subinterval n at both ends. If existence and uniqueness of the zero of - F-un can be verified over the corresponding subface, then we expand the subinterval by 2# at both ends, then 4# and so on until existence and uniqueness verification fails. 5. In the search phase, in step 1(g) (or 2(g)), the underlying idea is that the farther away the interval y l n is from the interval y 0 whose corresponding subface of x n (or x n ) contains a unique solution of - or the narrower the interval y l is, the more probable it is that we can verify that - F-un over the subface of x n (or x n ) corresponding to y l n . 6.2. Computational complexity. Derivation of the computational complexity Box-setting phase: Step 1 is of order O n 3 . Step 2 is of order O (n). Step 3 is of order O n 2 . Thus, the order of this phase is O n 3 . Elimination phase: Step 1(a)i and 1(b)i are of order O n 2 . Step 1(a)ii and 1(b)ii are of order O (1). Thus, the order of this phase is O n 3 . Search phase: Step 1(a) and 2(a) are of order O n 3 . Step 1(b) and 2(b) are of order O n 2 . Step 1(c) and 2(c) are of order O (1). Step 1(d) and 2(d) are of order O n 3 . Step 1(e) and 2(e) are of order O n 3 . Step 1(f) and 2(f) are of order N inf l *O n 3 . (See explanation below.) Step 1(g) and 2(g) are of order N proc *O n 3 . (See explanation below.) Step 1(h) and 2(h) are of order O n 2 . Step 1(i) and 2(i) are of order O n 3 . Step 1(j) and 2(j) are of order O (1) . The last step of this phase is of order O (1) too. Thus, the order of this phase is O n 3 376 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER The order of the overall algorithm is thus O n 3 . Remark. The order of the algorithm cannot be improved, since computing preconditioners of the original system and the system - F-un is necessary and computing each preconditioner is of order O n 3 . 7. Test problems and test environment. 7.1. Test problems. Before describing the test set, we introduce one more problem. Motivated by [10, Lemma 2.4], we considered systems of the following form. Example 3. Set the matrix corresponding to central di#erence discretization of the boundary value problem . The parameter t was chosen to be equal to t is the largest eigenvalue of A. The homotopy h in Example 3 has a simple bifurcation point at where the two paths cross obliquely. That is, there are two solutions to all t near t 1 and on either side of t 1 . Furthermore, the quadratic terms in the Taylor expansion for f do not vanish at The test set consists of Example 1, Example 2 with and Example 3 with For all the test problems, we used (0, 0, . , 0) as a good approximate solution to the problem F Actually, it's the exact solution in Example 1 and Example 3. w(x k ) and w(y k ) were set to 10 -3 for 1 # computed automatically by the algorithm. In fact, can also be computed automatically by the algorithm, depending on the accuracy of the approximate solution. At present, we used the known true solutions to Example 1 and Example 3 and the known approximate solution to Example 2 to test the algorithm and set the widths apparently small but otherwise arbitrary. For all the problems, the algorithm succeeded and returned a degree of 2. 7.2. Test environment. The algorithm in section 6.1 was programmed in the Fortran 90 environment developed and described in [15], [16]. Similarly, all the functions in the test problems were programmed using the same Fortran 90 system, and internal symbolic representations of the functions were generated prior to execution of the numerical tests. In the actual tests, generic routines then interpreted the internal representations to obtain both floating point and internal values. The LINPACK routines DGECO and DGESL were used in step 1 of the box-setting phase, and in step 1(e), 2(e), 1(f), and 2(f) of the search phase to compute the preconditioners. (See the algorithm and its explanation in section 6.1.) The Sun Fortran 90 compiler version 1.2 was used on a Sparc Ultra model 140 with optimization level 0. Execution times were measured using the routine DSECND. All times are given in CPU seconds. 8. Numerical results. We present the numerical results in Table 8.1 and some statistical data in Table 8.2. The column labels of Table 8.1 are as follows. Problem: names of the problems identified in section 7.1. n: number of independent variables. Success: whether the algorithm was successful. Degree: topological degree returned by the algorithm. CPU time: CPU time in seconds of the algorithm. Time ratio: This applies only to Example 3. It's the ratio of two successive CPU times. SINGULAR COMPLEX ZEROS VERIFICATION 377 Table Numerical results. Problem n Success Degree CPU time Time ratio Example 2 Example 2 Example 3 5 Example 3 20 Example 3 Example 3 160 Example 3 320 Table Statistical data. Problem Example Example 2 Example 2 Example Example Example 3 Example 3 160 Example 3 The column labels of Table 8.2 are as follows. Problem: names of the problems identified in section 7.1. n: number of independent variables. number of inflations the algorithm did in step 1(f) or 2(f) for the indicated face x n , x n , y n , or x y . number of subintervals of y n \ y 1 n the algorithm processed in step 1(g) or subintervals of x n \ x 1 n the algorithm processed in step 2(g), i.e., the number of y l 's plus number of y u n 's in step 1(g) or number of x l 's plus number of x u n 's in step 2(g) for the indicated face x n , x n , y n , or x y . We can see from Table 8.1 that the algorithm was successful on each problem in the test set. The overall algorithm is O n 3 , but the are many O n 3 and O n 2 steps. Some steps have many O n 3 and O n 2 substeps, and some of the substeps still have many O n 2 structures. Thus, when n was small, those lower order structures had significant influence on the CPU time. However, for the larger n in the examples tried, the O n 3 terms dominated. We can see this from the time ratios of Example 3 in Table 8.1. In Table 8.2, in each problem there were two faces of x n , x n , y n , and y n for which N inf l = 0. This is because the algorithm verified that u n < 0 on each of those two faces in step 1(b) or 2(b), and returned a degree contribution of each of those 378 R. BAKER KEARFOTT, JIANWEI DIAN, AND A. NEUMAIER two faces as 0. Thus, the algorithm didn't proceed to step 1(f) or 2(f). For the same reason, those two faces. For the remaining two faces for which the algorithm did proceed to step 1(f) or 2(f), N inf l is small. In step 1(g) or 2(g), which immediately follows the inflations, N Example 1 and Example 2. This is because the inflations had covered the whole interval y n . More significant is that N proc = 2 in Example 3 regardless of small n or large n. This is because only one interval was processed to verify that - has no solutions when x n # x l n and only one interval was processed to verify that solutions when x n . This means that the algorithm was quite e#cient. 9. Conclusions and future work. When we tested the algorithm, we took advantage of knowing the true solutions (see section 7.1. For this reason, we set arbitrarily. But we plan to have the algorithm eventually compute these, based on the accuracy of the approximate solution obtained by a floating point algorithm and the accuracy of the quadratic model. We presented an algorithm which was designed to work for the case that the rank deficiency of the Jacobian matrix at the singular solution is one. But the analysis in section 5 and the algorithm in section 6.1 can be generalized to general rank deficiency. Also, at present, it is assumed that the second derivatives # 2 fn #xk#x l don't vanish simultaneously at the singular solution. In fact, the analysis in section 5 and the algorithm in section 6.1 can be generalized to the general case that the derivatives of f n of order 1 through r (r # 2) vanish simultaneously at the singular solution. Computing higher order derivatives, however, may be expensive. Those two generalizations can also be combined, i.e., any rank deficiency and any order of derivatives of f n that vanish. We will pursue these generalizations in the future. Modification of the algorithm to verify complex roots that are not lying near the real axis is possible. Another future direction of this study is to apply the algorithms to bifurcation problems and other physical models. Finally, verification is possible in a multidimensional analogue of odd-multiplicity roots. We are presently writing up theoretical and experimental results for this situation. --R Computation of topological degree using interval arithmetic Introduction to Interval Computations New York Direkte Verfahren zur Berechnung der Nullstellen von Polynomen Fixed Points and Topological Degree in Nonlinear Analysis Fixed Points Circular arithmetic and the determination of polynomial zeros Global Optimization Using Interval Analysis Applied and Computational Complex Analysis. Computing the Degree of Maps and a Generalized Method of Bisection A summary of recent experiments to compute the topological degree A Fortran 90 environment for research and prototyping of enclosure algorithms for nonlinear equations and global optimization Continuous Problems Rigorous global optimization and the GlobSol package. Existence Verification for Singular Zeros of Nonlinear Systems The Poincar-e-Miranda theorem Degree Theory Methods and Applications of Interval Analysis Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems Interval Methods for Systems of Equations New Computer Methods for Global Optimization Verification methods for dense and sparse systems of equations Optimal Solution of Nonlinear Equations An algorithm for the topological degree of a mapping in R n An Algorithm for the Numerical Calculation of the Degree of a Mapping An algorithm for numerical calculation of the topological degree A short proof and a generalization of Miranda's existence theorem The topological degree theory for the location and computation of complex zeros of Bessel functions --TR --CTR Jianwei Dian , R. Baker Kearfott, Existence verification for singular and nonsmooth zeros of real nonlinear systems, Mathematics of Computation, v.72 n.242, p.757-766, 1 April R. Baker Kearfott , Jianwei Dian, Verifying topological indices for higher-order rank deficiencies, Journal of Complexity, v.18 n.2, p.589-611, June 2002 B. Mourrain , M. N. Vrahatis , J. C. Yakoubsohn, On the complexity of isolating real roots and computing with certainty the topological degree, Journal of Complexity, v.18 n.2, p.612-640, June 2002

interval computations;complex nonlinear systems;topological degree;verified computations;singularities