Light Euclidean Steiner Spanners in the Plane
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.
Geometric spanner
lightness
minimum weight
Mathematics of computing~Approximation algorithms
Mathematics of computing~Paths and connectivity problems
Theory of computation~Computational geometry
15:1-15:17
Regular Paper
https://arxiv.org/abs/2012.02216
Sujoy
Bhore
Sujoy Bhore
Université Libre de Bruxelles, Belgium
https://orcid.org/0000-0003-0104-1659
Research on this paper was supported by the Fonds de la Recherche Scientifique-FNRS under Grant no MISU F 6001.
Csaba D.
Tóth
Csaba D. Tóth
California State University Northridge, Los Angeles, CA, USA
Tufts University, Medford, MA, USA
https://orcid.org/0000-0002-8769-3190
Research on this paper was partially supported by the NSF award DMS-1800734.
10.4230/LIPIcs.SoCG.2021.15
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Sujoy Bhore and Csaba D. Tóth
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