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• W2989439999 abstract "We present a $(1+varepsilon)$-approximate parallel algorithm for computing shortest paths in undirected graphs, achieving $mathrm{poly}(log n)$ depth and $mmathrm{poly}(log n)$ work for $n$-nodes $m$-edges graphs. Although sequential algorithms with (nearly) optimal running time have been known for several decades, near-optimal parallel algorithms have turned out to be a much tougher challenge. For $(1+varepsilon)$-approximation, all prior algorithms with $mathrm{poly}(log n)$ depth perform at least $Omega(mn^{c})$ work for some constant $c>0$. Improving this long-standing upper bound obtained by Cohen (STOC'94) has been open for $25$ years. We develop several new tools of independent interest. One of them is a new notion beyond hopsets --- low hop emulator --- a $mathrm{poly}(log n)$-approximate emulator graph in which every shortest path has at most $O(loglog n)$ hops (edges). Direct applications of the low hop emulators are parallel algorithms for $mathrm{poly}(log n)$-approximate single source shortest path (SSSP), Bourgain's embedding, metric tree embedding, and low diameter decomposition, all with $mathrm{poly}(log n)$ depth and $mmathrm{poly}(log n)$ work. To boost the approximation ratio to $(1+varepsilon)$, we introduce compressible preconditioners and apply it inside Sherman's framework (SODA'17) to solve the more general problem of uncapacitated minimum cost flow (a.k.a., transshipment problem). Our algorithm computes a $(1+varepsilon)$-approximate uncapacitated minimum cost flow in $mathrm{poly}(log n)$ depth using $mmathrm{poly}(log n)$ work. As a consequence, it also improves the state-of-the-art sequential running time from $mcdot 2^{O(sqrt{log n})}$ to $mmathrm{poly}(log n)$." @default.
• W2989439999 created "2019-11-22" @default.
• W2989439999 creator A5054641291 @default.
• W2989439999 creator A5062439923 @default.
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• W2989439999 date "2019-11-05" @default.
• W2989439999 modified "2023-10-15" @default.
• W2989439999 title "Parallel Approximate Undirected Shortest Paths Via Low Hop Emulators" @default.
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• W2989439999 doi "https://doi.org/10.48550/arxiv.1911.01956" @default.
• W2989439999 hasPublicationYear "2019" @default.
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