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 (J. Assoc. Comput. Mach. 49:289–317, 2002) showed that for any fixed e>0, stretch 1+e distances (a path in G between u,v∈V is said to be of stretch t if its length is at most t times the distance between u and v in G) between all pairs of vertices in a weighted directed graph on n vertices can be computed in $\tilde{O}(n^{\omega})$ time, where ω 0 in G can be computed in expected time O(n 9/4). 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. This combinatorial algorithm will serve as a key step in our all-pairs stretch 2+e distances algorithm.