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.
Categories: cs.DS cs.DC