Light euclidean steiner spanners in the plane

Sujoy Bhore, Csaba D. Tóth

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

4 Scopus citations

Abstract

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ℝd. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on ε > 0 and d ∈ N of the minimum lightness of a (1 + ε)-spanner, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner (1 + ε)-spanners of lightness O(ε1 log ∆) in the plane, where ∆ ≥ Ω(√n) is the spread of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness Õ(ε(d+1)/2) in dimensions d ≥ 3. Recently, Bhore and Tóth (2020) established a lower bound of Ω(εd/2) for the lightness of Steiner (1 + ε)-spanners in ℝd, for d ≥ 2. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions d ≥ 2. In this work, we show that for every finite set of points in the plane and every ε > 0, there exists a Euclidean Steiner (1 + ε)-spanner of lightness O(ε1); this matches the lower bound for d = 2. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.

Original languageEnglish
Title of host publication37th International Symposium on Computational Geometry, SoCG 2021
EditorsKevin Buchin, Eric Colin de Verdiere
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771849
DOIs
StatePublished - 1 Jun 2021
Externally publishedYes
Event37th International Symposium on Computational Geometry, SoCG 2021 - Virtual, Buffalo, United States
Duration: 7 Jun 202111 Jun 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume189
ISSN (Print)1868-8969

Conference

Conference37th International Symposium on Computational Geometry, SoCG 2021
Country/TerritoryUnited States
CityVirtual, Buffalo
Period7/06/2111/06/21

Keywords

  • Geometric spanner
  • Lightness
  • Minimum weight

ASJC Scopus subject areas

  • Software

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