$0.01$

$\underset{\pm 0.10}{2.15}$

$(\operatorname{\bm{\theta}}_{\text{client}}^{(t)}=\operatorname{\bm{\theta}}_{% \text{client}}^{(0)}$

$r=64$

$297.78$

$\mathbf{0.43}$

$\operatorname{\mathbf{v}}_{i}\in\mathcal{V}$

$0.76$

$\underset{\pm 0.66}{45.89}$

$0$

$\rho$

$0.26$

$0.95$

$p\approx 8.69\times 10^{-8}$

$0.69$

$47.32$

$\mathbf{0.69}$

$2.30$

$\mathbf{A}\in\mathbb{R}^{d\times r}$

$\operatorname{\bm{\theta}}_{\text{client}}^{(t)}$

$500$

$\operatorname{\mathbf{v}}_{i}$

$0.78$

$308$

$\mathbf{0.36}$

$-\frac{1}{|\mathcal{D}^{(t)}_{\bigtriangledown}|}\sum_{\operatorname{\mathbf{d% }}^{(t)}\in\mathcal{D}^{(t)}_{\bigtriangledown}}\log p_{(\cdot)}\big{(}% \operatorname{\mathbf{d}}^{(t)}\big{|}\operatorname{\bm{\theta}}_{(\cdot)}^{(t% )}),$

$\operatorname{\mathbf{pr}}_{\text{client}}$

$p$

$0.66$

$2.65$

$\operatorname{\bm{\theta}}_{\text{client}}^{(0)}$

$25$

$277.25$

$\%$

$57.16$

$0.74$

$0.77$

$\tau=0.5$

$256$

$4.3$

$\operatorname{\bm{\theta}}_{\text{agent}}^{(0)}$

$0.90$

$0.80$

$4$

$0.8$

$9000$

$\operatorname{\bm{\theta}}_{\text{client}}$

$F_{1}$

$\mathbf{55.25}$

$0.60$

$\operatorname{\mathbf{v}}_{\text{next}}\in\text{Children}(\operatorname{% \mathbf{v}}_{i})$

$\displaystyle\geq bx_{j}^{\prime}+\sum_{i>j}x_{i}^{\prime}+(b-1)\sum_{i>j}x_{i}$

$x_{i}^{\prime}\geq x_{i}$

$A_{1}$

$\displaystyle(b^{k}-b^{k-1}-(b-1)^{k}+b^{k-1})x_{1}$

$(u,w)$

$C_{1},C_{2}$

$P:(u,v)\cup P^{\prime}$

$e_{>i}$

$(u_{1},u_{2},\dots,u_{k},v)$

$\displaystyle=\frac{(b-1)^{k-1}}{b^{k}-(b-1)^{k}}x.$

$(v,z)\in N(u)\times N(u)$

$1\leq i\leq k$

$x_{\tau+1},\dots,x_{k}\mapsto\textsf{Chunk-Shortest-Edge}(k-\tau,x-\delta)$

$\sum_{i}x_{i}

$\displaystyle=bx+c(v\to t)-c(u,w)-c(w\to t)=\delta+(b-1)x$

$p(e_{k}^{O})<\beta$

$i\geq\tau$

$\displaystyle=\begin{cases*}c(u,y)+cost[y,y,i]&if $(y,z)\in\mathcal{P}^{\prime% }(u,y)$\\ \infty&otherwise\end{cases*}$

$y^{*}\leq\frac{x}{k-1}$

$x-\delta$

$p(e;b_{i})$

$\displaystyle=\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}\left(\frac{b^{k}-(b-1)^{k-1}% (b-1+1)}{b^{k}-(b-1)^{k}}\right)x+c(v\to t)$

$(s,s_{1})$

$i+j\leq k$

$\displaystyle=\left(\frac{b^{k-1}}{b^{k-1}-(b-1)^{k-1}}-1\right)x$

$\begin{array}[]{ll}p(e_{\tau})&=bx_{\tau}+c(u_{\tau+1}\to t)\\ &=bx_{\tau}+\sum_{i=\tau+1}^{k}x_{i}+c(v\to t)\hfill\mbox{(shortest path from $u_{\tau+1}$ follows the chunking)}\\ &=bx_{\tau}+x-\sum_{i=1}^{\tau}x_{i}+c(v\to t)\\ &=b\cdot\frac{\delta}{\tau}-\tau\cdot\frac{\delta}{\tau}+x+c(v\to t)\hfill% \mbox{(substituing $x_{i}=\delta/\tau$ for $i\leq\tau$)}\\ &=\frac{b\delta}{\tau}+c(u,w)+c(w\to t)\hfill\mbox{(since $\delta=x+c(v\to t)-% c(u,w)-c(w\to t)$).}\end{array}$

$(u_{3},z)$

$\displaystyle\frac{(b-1)(b^{k-1}-(b-1)^{k-1})+b^{k-1}}{b^{k-1}-(b-1)^{k-1}}x_{1}$

$\sum_{l>i}x_{l}\leq x$

$\displaystyle(b-1)x_{1}+x$

$c^{n}$

$\delta/k$

$p(e_{i})=bx_{i}+c(u,w)+c(w\to t)$

$\alpha=\beta$

$p(e_{i};b_{1})\leq\alpha_{u}^{(j)}$

$p(e_{i}^{C})=bx_{i}^{C}+c(u_{i}^{C}\to t)$

$x_{i}-x_{i}^{\prime}\geq 0$

$p(e_{i})=\beta^{\prime}$

$\displaystyle\min_{(v,z)\in\mathcal{P}(u,y)}\min(C_{1}(u,v,y),C_{2}(u,y,z),C_{% 3}(u,v,y,z)).$

$c(u,v)$

$\beta^{*}=p(e_{i})$

$p(e_{j};b_{1})\leq\alpha_{u}^{(1)}$

$\displaystyle=\frac{x}{1-\left(\frac{b-1}{b}\right)^{k-\tau+1}}+c(v\to t)-c(w% \to t)-c(u,w).$

$x_{1}$

$O(|E|^{2}k^{3}\log k+|V|)$

$\beta\leftarrow\frac{x-\delta}{z_{\tau}}+c(v\to t)$

$min\_bottleneck\leftarrow\min(\alpha,\beta)$

$\delta/\tau$

$(y,z)$