A Deterministic Distributed Algorithm for Exact Weighted All-Pairs Shortest Paths in O~(n3/2)\tilde{O}(n^{3/2}) Rounds

We present a deterministic distributed algorithm to compute all-pairs shortest paths(APSP) in an edge-weighted directed or undirected graph. Our algorithm runs in $\tilde{O}(n^{3/2})$ rounds in the Congest model, where $n$ is the number of nodes in the graph. This is the first $o(n^2)$ rounds deterministic distributed algorithm for the weighted APSP problem. Our algorithm is fairly simple and incorporates a deterministic distributed algorithm we develop for computing a `blocker set' \cite{King99}, which has been used earlier in sequential dynamic computation of APSP

New and Simplified Distributed Algorithms for Weighted All Pairs Shortest Paths

We consider the problem of computing all pairs shortest paths (APSP) and shortest paths for k sources in a weighted graph in the distributed CONGEST model. For graphs with non-negative integer edge weights (including zero weights) we build on a recent pipelined algorithm to obtain $\tilde{O}(\lambda^{1/4}\cdot n^{5/4})$ in graphs with non-negative integer edge-weight at most $\lambda$, and $\tilde{O}(n \cdot \bigtriangleup^{1/3})$ rounds for shortest path distances at most $\bigtriangleup$. Additionally, we simplify some of the procedures in the earlier APSP algorithms for non-negative edge weights in [HNS17,ARKP18]. We also present results for computing h-hop shortest paths and shortest paths from $k$ given sources. In other results, we present a randomized exact APSP algorithm for graphs with arbitrary edge weights that runs in $\tilde{O}(n^{4/3})$ rounds w.h.p. in n, which improves the previous best $\tilde{O}(n^{3/2})$ bound, which is deterministic. We also present an $\tilde{O}(n/\epsilon^2)$-round deterministic $(1+\epsilon)$ approximation algorithm for graphs with non-negative $poly(n)$ integer weights (including zero edge-weights), improving results in [Nanongkai14,LP15] that hold only for positive integer weights.Comment: arXiv admin note: text overlap with arXiv:1807.0882