Faster Algorithms for All-Pairs Small Stretch Distances in Weighted Graphs

Abstract

Let G = (V,E) be a weighted undirected graph, with non-negative edge weights. We consider the problem of efficiently computing approximate distances between all pairs of vertices in G. While many efficient algorithms are known for this problem in unweighted graphs, not many results are known for this problem in weighted graphs. Zwick [15] showed that for any fixed ε> 0, stretch (1 + ε) distances between all pairs of vertices in a weighted directed graph on n vertices can be computed in \(\tilde{O}(n^{\omega})\) time assuming that edge weights in G are not too large, where ω< 2.376 is the exponent of matrix multiplication and n is the number of vertices in G. It is known that finding distances of stretch less than 2 between all pairs of vertices in G is at least as hard as Boolean matrix multiplication of two n ×n matrices. It is also known that all-pairs stretch 3 distances can be computed in \(\tilde{O}(n^2)\) time and all-pairs stretch 7/3 distances can be computed in \(\tilde{O}(n^{7/3})\) time. Here we consider efficient algorithms for the problem of computing all-pairs stretch (2 + ε) distances in G, for any 0 < ε< 1.We show that all pairs stretch (2 + ε) distances for any fixed ε> 0 in G can be computed in expected time O(n 9/4) assuming that edge weights in G are not too large. This algorithm uses a fast rectangular matrix multiplication subroutine. We also present a combinatorial algorithm (that is, it does not use fast matrix multiplication) with expected running time O(n 9/4) for computing all-pairs stretch 5/2 distances in G.