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

Maor Akav, Liam Roditty

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations

Abstract

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/km2/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.

Original languageEnglish
Title of host publication29th Annual European Symposium on Algorithms, ESA 2021
EditorsPetra Mutzel, Rasmus Pagh, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772044
DOIs
StatePublished - 1 Sep 2021
Event29th Annual European Symposium on Algorithms, ESA 2021 - Vitual, Lisbon, Portugal
Duration: 6 Sep 20218 Sep 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume204
ISSN (Print)1868-8969

Conference

Conference29th Annual European Symposium on Algorithms, ESA 2021
Country/TerritoryPortugal
CityVitual, Lisbon
Period6/09/218/09/21

Bibliographical note

Publisher Copyright:
© Maor Akav and Liam Roditty; licensed under Creative Commons License CC-BY 4.0

Keywords

  • Approximate all pairs of shortest paths
  • Distance oracles
  • Graph algorithms

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