Faster Replacement Paths

TL;DR

This paper presents improved algorithms for the replacement paths problem in directed graphs, achieving faster runtimes using fast matrix multiplication techniques, especially for small integer weights.

Contribution

The authors develop new algorithms that improve the runtime for replacement paths, leveraging fast matrix multiplication and achieving near-optimal complexity for small weights.

Findings

New algorithms run in O(M n^{} polylog(n)) time for >2

Achieve O(n^{2+\u03b5}) runtime for =2 and any >0

Show that replacement paths can be easier than all pairs shortest paths for small weights

Abstract

The replacement paths problem for directed graphs is to find for given nodes s and t and every edge e on the shortest path between them, the shortest path between s and t which avoids e. For unweighted directed graphs on n vertices, the best known algorithm runtime was \tilde{O}(n^{2.5}) by Roditty and Zwick. For graphs with integer weights in {-M,...,M}, Weimann and Yuster recently showed that one can use fast matrix multiplication and solve the problem in O(Mn^{2.584}) time, a runtime which would be O(Mn^{2.33}) if the exponent \omega of matrix multiplication is 2. We improve both of these algorithms. Our new algorithm also relies on fast matrix multiplication and runs in O(M n^{\omega} polylog(n)) time if \omega>2 and O(n^{2+\eps}) for any \eps>0 if \omega=2. Our result shows that, at least for small integer weights, the replacement paths problem in directed graphs may be easier than the related all pairs shortest paths problem in directed graphs, as the current best runtime for the latter is \Omega(n^{2.5}) time even if \omega=2.