Faster approximation of distances in graphs

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 Õ(n²) time, and for any u, v ε 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 ε 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 Õ(m√n + n²) 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.