An almost 2-approximation for all-pairs of shortest paths in subquadratic time

Maor Akav, Liam Roditty

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

6 Scopus citations

Abstract

Let G = (V, E) be an unweighted undirected graph with n vertices and m edges. Dor, Halperin, and Zwick [FOCS 1996, SICOMP 2000] presented an Õ(n2)-time algorithm that computes estimated distances with a multiplicative approximation of 3. Berman and Kasiviswanathan [WADS 2007] improved the approximation of Dor et al. and presented an Õ(n2)time algorithm that produces for every u, v ∈ V an estimate (Equation presented)(u, v) such that: dG(u, v) ≤ (Equation presented)(u, v) ≤ 2dG(u, v) + 1. We refer to such an approximation as an (α, β)-approximation, where α is the multiplicative approximation and β is the additive approximation. A prerequisite for an O(n2ε)-time algorithm, where ε ∈ (0, 1), is a data structure that uses O(n2−δ) space, for some δ ≥ ε, and answers queries in constant time. An O(n2ε)-time (3, 0)-approximation algorithm became plausible after Thorup and Zwick [STOC 2001, JACM 2005] presented their approximate distance oracles, and in particular an O(n1.5)-space data structure that reports a (3, 0)-approximate distance in O(1) time. Indeed, using Thorup and Zwick distance oracles together with more ideas, Baswana, Gaur, Sen, and Upadhyay [ICALP 2008] improved the running time of Dor et al., and obtained an O(n2ε) time algorithm, at the cost of introducing also an additive approximation. They presented an algorithm that in Õ(m+ n23/12) expected running time constructs an O(n1.5)-space data structure, that in O(1) time reports a (3, 14)-approximate distance. An O(n2ε)-time (2, 1)-approximation algorithm became plausible only after Pǎtraşcu and Roditty [FOCS 2010, SICOMP 2014] presented an O(n5/3)-space data structure that reports (2, 1)-approximate distances in O(1) time. However, only few years ago, Sommer [ICALP 2016] obtained an Õ(n2) time algorithm that computes a (2, 1)-distance oracle with Õ(n5/3) space. This leads to the following natural question of whether Ω(n2) time is a lower bound for any (3−α, β)-approximation, where α ∈ (0, 1), and β is constant. In this paper we show that this is not the case by presenting an algorithm that for every ε ∈ (0, 1/2) computes in Õ(m) + n2−Ω time an Õ(n1 56 )-space data structure that in O(1/ε) time reports, for every u, v ∈ V , an estimate (Equation presented)(u, v) such that: (Equation presented). Our result improves, simultaneously, the running time and the multiplicative approximation of the Õ(n2)-time (3, 0)-approximation algorithm of Dor et al. at the cost of introducing also an additive approximation.

Original languageEnglish
Title of host publication31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
EditorsShuchi Chawla
PublisherAssociation for Computing Machinery
Pages1-11
Number of pages11
ISBN (Electronic)9781611975994
StatePublished - 2020
Event31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States
Duration: 5 Jan 20208 Jan 2020

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume2020-January

Conference

Conference31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
Country/TerritoryUnited States
CitySalt Lake City
Period5/01/208/01/20

Bibliographical note

Publisher Copyright:
Copyright © 2020 by SIAM

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