Speeding up shortest path algorithms

Abstract

Given an arbitrary, non-negatively weighted, directed graph G = (V, E) we present an algorithm that computes all pairs shortest paths in time O(m*n + mlgn + nTψ(m*, n)), where m* is the number of different edges contained in shortest paths and Tψ(m*, n) is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial n times application of ψ that runs in O(nTψ(m, n)). In our algorithm we use ψ as a black box and hence any improvement on ψ results also in improvement of our algorithm. Furthermore, a combination of our method, Johnson's reweighting technique and topological sorting results in an O(m*n+mlg n) all-pairs shortest path algorithm for arbitrarily-weighted directed acyclic graphs. In addition, we also point out a connection between the complexity of a certain sorting problem defined on shortest paths and SSSP. © Springer-Verlag 2012.