Faster Approximation of Distances in Graphs

Abstract

Let G = (V,E) be a weighted undirected graph on n vertices and m edges, and let d _G be its shortest path metric. We present two simple deterministic algorithms for approximating all-pairs shortest paths in G. Our first algorithm runs in $\tilde{O}(n^2)$ time, and for any u,v ∈ V reports distance no greater than 2d _G (u,v) + h(u,v). Here, h(u,v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u,v ∈ V reports distance no greater than (1 + ε)d _G (u,v) + 2h(u,v). The currently best known algorithm for Boolean matrix multiplication yields an O(n ^{2.24 + o(1)} ε ^− 3log(nε ^− 1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term.

We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in $\tilde{O}(m\sqrt{n}+n^2)$ time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for the diameter approximation problem. Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.