Speeding up shortest path algorithms

TL;DR

This paper introduces an efficient all-pairs shortest path algorithm that leverages a black-box SSSP solver, significantly reducing complexity especially for graphs with limited edge diversity in shortest paths.

Contribution

The authors present a novel algorithm that improves shortest path computations by exploiting the structure of shortest paths and combining existing techniques like Johnson's reweighting.

Findings

Achieves faster all-pairs shortest path computation with complexity depending on $m^*$

Provides an optimal algorithm for directed acyclic graphs

Links sorting complexity on shortest paths to SSSP complexity

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 $T_\psi(m^*, n)$ 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 $\psi$ that runs in $\mathcal{O}(nT_\psi(m,n))$. In our algorithm we use $\psi$ as a black box and hence any improvement on $\psi$ 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.