Brief Announcement: A Tight Distributed Algorithm for All Pairs Shortest Paths and Applications

Given an unweighted and undirected graph, this paper aims to give a tight distributed algorithm for computing the all pairs shortest paths (APSP) under synchronous communications and the CONGEST(B) model, where each node can only transfer B bits of information along each incident edge in a round. The best previous results for distributively computing APSP need O(N+D) time where N is the number of nodes and D is the diameter [1,2]. However, there is still a B factor gap from the lower bound Ω(N/B+D) [1]. In order to close this gap, we propose a multiplexing technique to push the parallelization of distributed BFS tree constructions to the limit such that we can solve APSP in O(N/B+D) time which meets the lower bound. This result also implies a Θ(N/B+D) time distributed algorithm for diameter. In addition, we extend our distributed algorithm to compute girth which is the length of the shortest cycle and clustering coefficient (CC) which is related to counting the number of triangles incident to each node. The time complexities for computing these two graph properties are also O(N/B+D).