TY - GEN

T1 - Light euclidean steiner spanners in the plane

AU - Bhore, Sujoy

AU - Tóth, Csaba D.

N1 - Funding Information:
Funding Sujoy Bhore: Research on this paper was supported by the Fonds de la Recherche Scientifique-FNRS under Grant no MISU F 6001. Csaba D. Tóth: Research on this paper was partially supported by the NSF award DMS-1800734.
Publisher Copyright:
© Sujoy Bhore and Csaba D. Tóth; licensed under Creative Commons License CC-BY 4.0 37th International Symposium on Computational Geometry (SoCG 2021).

PY - 2021/6/1

Y1 - 2021/6/1

N2 - 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(ε−1 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(ε−d/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(ε−1); 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.

AB - 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(ε−1 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(ε−d/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(ε−1); 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.

KW - Geometric spanner

KW - Lightness

KW - Minimum weight

UR - http://www.scopus.com/inward/record.url?scp=85108218931&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SoCG.2021.15

DO - 10.4230/LIPIcs.SoCG.2021.15

M3 - Conference contribution

AN - SCOPUS:85108218931

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 37th International Symposium on Computational Geometry, SoCG 2021

A2 - Buchin, Kevin

A2 - de Verdiere, Eric Colin

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 37th International Symposium on Computational Geometry, SoCG 2021

Y2 - 7 June 2021 through 11 June 2021

ER -