Computing shortest paths with comparisons and additions

Abstract

We present an undirected all-pairs shortest paths (APSP) algorithm which runs on a pointer machine in time O(mnα(m,n)) while making O(mnlog α(m, n)) comparisons and additions, where m and n are the number of edges and vertices, respectively, and α(m, n) is Tarjan's inverse-Ackermann function. This improves upon all previous comparison & addition-based APSP algorithms when the graph is sparse, i.e., when m = o(n log n).At the heart of our APSP algorithm is a new single-source shortest paths algorithm which runs in time O(mα(m, n) + n log log r) on a pointer machine, where r is the ratio of the maximum-to-minimum edge length. So long as r