On Euclidean Steiner (1+ε)-Spanners
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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^-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.
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