On euclidean steiner (1 + ϵ)-spanners

Sujoy Bhore, Csaba D. Tóth

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

3 Scopus citations

Abstract

Lightness and sparsity are two natural parameters for Euclidean (1 + ϵ)-spanners. Classical results show that, when the dimension d ∈ N and ϵ > 0 are constant, every set S of n points in d-space admits an (1+ϵ)-spanners with O(n) edges and weight proportional to that of the Euclidean MST of S. Tight bounds on the dependence on ϵ > 0 for constant d ∈ N have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a (1 + ϵ)-spanner. They gave upper bounds of Õ(ϵ-(d+1)/2) for the minimum lightness in dimensions d ≥ 3, and Õ(ϵ-(d-1))/2) for the minimum sparsity in d-space for all d ≥ 1. They obtained lower bounds only in the plane (d = 2). Le and Solomon (ESA 2020) also constructed Steiner (1 + ϵ)-spanners of lightness O(ϵ-1 log Δ) in the plane, where Δ ∈ Ω(log n) is the spread of S, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner (1 + ϵ)-spanners. Using a new geometric analysis, we establish lower bounds of Ω(ϵ-d/2) for the lightness and Ω(ϵ-(d-1)/2) for the sparsity of such spanners in Euclidean d-space for all d ≥ 2. We use the geometric insight from our lower bound analysis to construct Steiner (1 + ϵ)-spanners of lightness O(ϵ-1 log n) for n points in Euclidean plane.

Original languageEnglish
Title of host publication38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021
EditorsMarkus Blaser, Benjamin Monmege
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771801
DOIs
StatePublished - 1 Mar 2021
Externally publishedYes
Event38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021 - Virtual, Saarbrucken, Germany
Duration: 16 Mar 202119 Mar 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume187
ISSN (Print)1868-8969

Conference

Conference38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021
Country/TerritoryGermany
CityVirtual, Saarbrucken
Period16/03/2119/03/21

Keywords

  • (1 + ϵ)-spanner
  • Geometric spanner
  • Lightness
  • Minimum weight
  • Sparsity

ASJC Scopus subject areas

  • Software

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