On euclidean steiner (1 + ϵ)-spanners

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 log Δ) 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.