A combinatorial algorithm for all-pairs shortest paths in directed vertex-weighted graphs with applications to disc graphs

Abstract

We consider the problem of computing all-pairs shortest paths in a directed graph with non-negative real weights assigned to vertices. For an n x n 0-1 matrix C, let K C be the complete weighted graph on the rows of C where the weight of an edge between two rows is equal to their Hamming distance. Let MWT(C) be the weight of a minimum weight spanning tree of K C. We show that the all-pairs shortest path problem for a directed graph G on n vertices with non-negative real weights and adjacency matrix A G can be solved by a combinatorial randomized algorithm in time Õ(n 2√n+min{MWT(A G), MWT(A Gt)}) As a corollary, we conclude that the transitive closure of a directed graph G can be computed by a combinatorial randomized algorithm in the aforementioned time. We also conclude that the all-pairs shortest path problem for vertex-weighted uniform disk graphs induced by point sets of bounded density within a unit square can be solved in time Õ(n 2.75). © 2012 Springer-Verlag.