2020/10/06 by Bhore, Sujoy, Tóth, Csaba D.
#Computational Geometry (cs.CG) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2010.02908
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 O(ε-(d+1)/2) for the minimum lightness in dimensions d≥ 3, and O(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-1logΔ) in the plane, where Δ∈ Ω(√(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-1log n) for n points in Euclidean plane.