Fast constructions of lightweight spanners for general graphs

Michael Elkin, Shay Solomon

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

It is long known that for every weighted undirected n-vertex m-edge graph G = (V, E, ω), and every integer k ≥ 1, there exists a ((2k- 1) · (1 + ϵ))-spanner with O(n1+1/k) edges and weight O(k · n1/k · ω(MST(G)), for an arbitrarily small constant ϵ > 0. (Here ω(MST(G)) stands for the weight of the minimum spanning tree of G.) To our knowledge, the only algorithms for constructing sparse and lightweight spanners for general graphs admit high running times. Most notable in this context is the greedy algorithm of Althöfer et al. [1993], analyzed by Chandra et al. [1992], which requires O(m· (n1+1/k + n · log n)) time. In this article, we devise an efficient algorithm for constructing sparse and lightweight spanners. Specifically, our algorithm constructs ((2k - 1) · (1 + ϵ))-spanners with O(k · n1+1/k) edges and weight O(k · n1/k) · ω(MST(G)), where ϵ > 0 is an arbitrarily small constant. The running time of our algorithm is O(k ·m+ min{n · log n,m· α(n)}). Moreover, by slightly increasing the running time we can reduce the other parameters. These results address an open problem by Roditty and Zwick [2004].

Original languageEnglish
Article number29
JournalACM Transactions on Algorithms
Volume12
Issue number3
DOIs
StatePublished - 1 Apr 2016

Keywords

  • Graph spanners
  • Light spanners
  • Sparse graphs

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