Constructing light spanners deterministically in near-linear time

Stephen Alstrup, Søren Dahlgaard, Arnold Filtser, Morten Stöckel, Christian Wulff-Nilsen

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

7 Scopus citations

Abstract

Graph spanners are well-studied and widely used both in theory and practice. In a recent breakthrough, Chechik and Wulff-Nilsen [11] improved the state-of-the-art for light spanners by constructing a (2k − 1)(1 + ε)-spanner with O(n1+1/k) edges and Oε(n1/k) lightness. Soon after, Filtser and Solomon [19] showed that the classic greedy spanner construction achieves the same bounds. The major drawback of the greedy spanner is its running time of O(mn1+1/k) (which is faster than [11]). This makes the construction impractical even for graphs of moderate size. Much faster spanner constructions do exist but they only achieve lightness Ωε(kn1/k), even when randomization is used. The contribution of this paper is deterministic spanner constructions that are fast, and achieve similar bounds as the state-of-the-art slower constructions. Our first result is an Oε(n2+1/k+ε') time spanner construction which achieves the state-of-the-art bounds. Our second result is an Oε(m + n log n) time construction of a spanner with (2k − 1)(1 + ε) stretch, O(log k · n1+1/k) edges and Oε(log k · n1/k) lightness. This is an exponential improvement in the dependence on k compared to the previous result with such running time. Finally, for the important special case where k = log n, for every constant ε > 0, we provide an O(m + n1+ε) time construction that produces an O(log n)-spanner with O(n) edges and O(1) lightness which is asymptotically optimal. This is the first known sub-quadratic construction of such a spanner for any k = ω(1). To achieve our constructions, we show a novel deterministic incremental approximate distance oracle. Our new oracle is crucial in our construction, as known randomized dynamic oracles require the assumption of a non-adaptive adversary. This is a strong assumption, which has seen recent attention in prolific venues. Our new oracle allows the order of the edge insertions to not be fixed in advance, which is critical as our spanner algorithm chooses which edges to insert based on the answers to distance queries. We believe our new oracle is of independent interest.

Original languageEnglish
Title of host publication27th Annual European Symposium on Algorithms, ESA 2019
EditorsMichael A. Bender, Ola Svensson, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771245
DOIs
StatePublished - 1 Sep 2019
Event27th Annual European Symposium on Algorithms, ESA 2019 - Munich/Garching, Germany
Duration: 9 Sep 201911 Sep 2019

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume144
ISSN (Print)1868-8969

Conference

Conference27th Annual European Symposium on Algorithms, ESA 2019
Country/TerritoryGermany
CityMunich/Garching
Period9/09/1911/09/19

Keywords

  • Deterministic dynamic distance oracle
  • Efficient construction
  • Light spanners
  • Spanners

ASJC Scopus subject areas

  • Software

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