eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-08-31
4:1
4:18
10.4230/LIPIcs.ESA.2021.4
article
A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs
Akav, Maor
1
Roditty, Liam
1
Bar-Ilan University, Ramat Gan, Israel
Let G = (V,E) be a weighted undirected graph with n vertices and m edges, and let d_G(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+kn^{2-3/k}m^{2/k}) expected running time algorithm that computes a matrix M such that for every u,v ∈ V:
d_G(u,v) ≤ M[u,v] ≤ (2+(k-2)/k)d_G(u,v).
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+m^{2/3}n). 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(n^{9/4}).
For k = 2 and k = 3 our result gives the algorithms of Baswana and Kavitha. For k = 4, we get a 5/2-approximation algorithm with Õ(n^{5/4}m^{1/2}) expected running time. This improves upon the running time of Õ(n^{9/4}) 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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol204-esa2021/LIPIcs.ESA.2021.4/LIPIcs.ESA.2021.4.pdf
Graph algorithms
Approximate All Pairs of Shortest Paths
Distance Oracles