On Euclidean Steiner (1+ε)-Spanners

Authors Sujoy Bhore , Csaba D. Tóth

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Author Details

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

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Sujoy Bhore and Csaba D. Tóth. On Euclidean Steiner (1+ε)-Spanners. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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}logΔ) 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Mathematics of computing → Paths and connectivity problems
  • Theory of computation → Computational geometry
  • Geometric spanner
  • (1+ε)-spanner
  • lightness
  • sparsity
  • minimum weight


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