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 \(\mathcal{O}(m^* n + m \lg n + nT_\psi(m^*, n))\), where m * is the number of different edges contained in shortest paths and T ψ (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 \(\mathcal{O}(nT_\psi(m,n))\). 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 \(\mathcal{O}(m^*n + m \lg n)\) 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.