Almost Shortest Paths with Near-Additive Error in Weighted Graphs

Michael Elkin, Yuval Gitlitz, Ofer Neiman

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

Abstract

Let G = (V,E,w) be a weighted undirected graph with n vertices and m edges, and fix a set of s sources S ⊆ V . We study the problem of computing almost shortest paths (ASP) for all pairs in S × V in both classical centralized and parallel (PRAM) models of computation. Consider the regime of multiplicative approximation of 1 + ϵ, for an arbitrarily small constant ϵ > 0 (henceforth (1 + ϵ)-ASP for S × V ). In this regime existing centralized algorithms require ω(min{|E|s, nω}) time, where ω < 2.372 is the matrix multiplication exponent. Existing PRAM algorithms with polylogarithmic depth (aka time) require work ω(min{|E|s, nω}). In a bold attempt to achieve centralized time close to the lower bound of m + ns, Cohen [10] devised an algorithm which, in addition to the multiplicative stretch of 1 + ϵ, allows also additive error of β Wmax, where Wmax is the maximum edge weight in G (assuming that the minimum edge weight is 1), and β = (log n)O( log 1/ρ ρ ) is polylogarithmic in n. It also depends on the (possibly) arbitrarily small parameter ρ > 0 that determines the running time O((m+ ns)nρ) of the algorithm. The tradeoff of [10] was improved in [15], whose algorithm has similar approximation guarantee and running time, but its β is (1/ρ)O( log 1/ρ ρ ). However, the latter algorithm produces distance estimates rather than actual approximate shortest paths. Also, the additive terms in [10, 15] depend linearly on a possibly quite large global maximum edge weight Wmax. In the current paper we significantly improve this state of affairs. Our centralized algorithm has running time O((m + ns)nρ), and its PRAM counterpart has polylogarithmic depth and work O((m+ ns)nρ), for an arbitrarily small constant ρ > 0. For a pair (s, v) ∈ S × V , it provides a path of length d(s, v) that satisfies d(s, v) ≤ (1 + ϵ)dG(s, v) + β W(s, v), where W(s, v) is the weight of the heaviest edge on some shortest s - v path. Hence our additive term depends linearly on a local maximum edge weight, as opposed to the global maximum edge weight in [10, 15]. Finally, our β = (1/ρ)O(1/ρ), i.e., it is significantly smaller than in [10, 15]. We also extend a centralized algorithm of Dor et al. [14]. For a parameter κ = 1, 2, . . ., this algorithm provides for unweighted graphs a purely additive approximation of 2(κ - 1) for all pairs shortest paths (APASP) in time O(n2+1/κ). Within the same running time, our algorithm for weighted graphs provides a purely additive error of 2(κ-1)W(u, v), for every vertex pair (u, v) ∈ (V 2 ) , with W(u, v) defined as above. On the way to these results we devise a suite of novel constructions of spanners, emulators and hopsets.

Original languageAmerican English
Title of host publication18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022
EditorsArtur Czumaj, Qin Xin
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772365
DOIs
StatePublished - 1 Jun 2022
Event18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022 - Torshavn, Faroe Islands
Duration: 27 Jun 202229 Jun 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume227

Conference

Conference18th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2022
Country/TerritoryFaroe Islands
CityTorshavn
Period27/06/2229/06/22

Keywords

  • PRAM
  • distance oracles
  • hopset
  • shortest paths

All Science Journal Classification (ASJC) codes

  • Software

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