Linear-size hopsets with small hopbound, and constant-hopbound hopsets in RNC

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

20 Scopus citations

Abstract

For a positive parameter β, the β-bounded distance between a pair of vertices u,v in a weighted undirected graph G = (V,E,ω) is the length of the shortest u −v path in G with at most β edges, aka hops. For β as above and ϵ > 0, a (β,ϵ)-hopset of G = (V,E,ω) is a graph GH = (V,H,ωH ) on the same vertex set, such that all distances in G are (1 + ϵ)-approximated by β-bounded distances in G ∪ GH . Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with Ω(n log n) edges, or with a hopbound nΩ(1). In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on ϵ) (log log n)log log n+O(1). This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [11] for computing hopsets with a constant (i.e., independent of n) hopbound requires nΩ(1) time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [11]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.

Original languageEnglish
Title of host publicationSPAA 2019 - Proceedings of the 31st ACM Symposium on Parallelism in Algorithms and Architectures
PublisherAssociation for Computing Machinery
Pages333-341
Number of pages9
ISBN (Electronic)9781450361842
DOIs
StatePublished - 17 Jun 2019
Event31st ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2019 - Phoenix, United States
Duration: 22 Jun 201924 Jun 2019

Conference

Conference31st ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2019
Country/TerritoryUnited States
CityPhoenix
Period22/06/1924/06/19

Keywords

  • Hopsets
  • Shortest paths

ASJC Scopus subject areas

  • Software
  • Theoretical Computer Science
  • Hardware and Architecture

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