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Light Euclidean Steiner Spanners in the Plane

Authors Sujoy Bhore , Csaba D. Tóth

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Author Details

Sujoy Bhore
  • Université Libre de Bruxelles, Belgium
Csaba D. Tóth
  • California State University Northridge, Los Angeles, CA, USA
  • Tufts University, Medford, MA, USA

Funding

  • Bhore, Sujoy: Research on this paper was supported by the Fonds de la Recherche Scientifique-FNRS under Grant no MISU F 6001.
  • Tóth, Csaba D.: Research on this paper was partially supported by the NSF award DMS-1800734.

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Sujoy Bhore and Csaba D. Tóth. Light Euclidean Steiner Spanners in the Plane. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.SoCG.2021.15

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 ℝ^d. 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}logΔ) in the plane, where Δ ≥ 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 ℝ^d, 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Paths and connectivity problems
  • Theory of computation → Computational geometry
Keywords
  • Geometric spanner
  • lightness
  • minimum weight

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