# Parallel Approximate Undirected Shortest Paths Via Low Hop Emulators

We present a (1+ε)-approximate parallel algorithm for computing shortest paths in undirected graphs, achieving poly(log n) depth and mpoly(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+ε)-approximation, all prior algorithms with poly(log n) depth perform at least Ω(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 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 poly(log n)-approximate single source shortest path (SSSP), Bourgain's embedding, metric tree embedding, and low diameter decomposition, all with poly(log n) depth and mpoly(log n) work. To boost the approximation ratio to (1+ε), 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+ε)-approximate uncapacitated minimum cost flow in poly(log n) depth using mpoly(log n) work. As a consequence, it also improves the state-of-the-art sequential running time from m· 2^O(√(log n)) to mpoly(log n).

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