A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an n × n 0 − 1 matrix C, let KC 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 KC . We show that the all-pairs shortest path problem for a directed graph G on n vertices with nonnegative real weights and adjacency matrix AG can be solved by a combinatorial randomized algorithm in time Õ(n2 √ n+min{MWT (AG),MWT (AtG)}) 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 uniform disk graphs, with nonnegative real vertex weights, induced by point sets of bounded density within a unit square can be solved in time Õ(n).