A unified approach for all pairs approximate shortest paths in weighted undirected graphs

Let G = (V, E) be a weighted undirected graph with n vertices and m edges, and let dG(u, v) be the length of the shortest path between u and v in G. In this paper we present a unified approach for obtaining algorithms for all pairs approximate shortest paths in weighted undirected graphs. For every integer k ≥ 2 we show that there is an Õ(n² + kn²−3^/km^2/k) expected running time algorithm that computes a matrix M such that for every u, v ∈ V : (equation presented) Previous algorithms obtained only specific approximation factors. Baswana and Kavitha [FOCS 2006, SICOMP 2010] presented a 2-approximation algorithm with expected running time of Õ(n² + m√n) and a 7/3-approximation algorithm with expected running time of Õ(n² + m²/³n). Their results improved upon the results of Cohen and Zwick [SODA 1997, JoA 2001] for graphs with m = o(n²). Kavitha [FSTTCS 2007, Algorithmica 2012] presented a 5/2-approximation algorithm with expected running time of Õ(n⁹/⁴). For k = 2 and k = 3 our result gives the algorithms of Baswana and Kavitha. For k = 4, we 1 get a 5/2-approximation algorithm with Õ(n⁵⁴ m2) expected running time. This improves upon the running time of Õ(n⁹/⁴) due to Kavitha, when m = o(n²). Our unified approach reveals that all previous algorithms are a part of a family of algorithms that exhibit a smooth tradeoff between approximation of 2 and 3, and are not sporadic unrelated results. Moreover, our new algorithm uses, among other ideas, the celebrated approximate distance oracles of Thorup and Zwick [STOC 2001, JACM 2005] in a non standard way, which we believe is of independent interest, due to their extensive usage in a variety of applications.