Truly Optimal Euclidean Spanners
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Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any d-dimensional n-point Euclidean space, there exists a (1+ϵ)-spanner with nO(ϵ^-d+1) edges and lightness O(ϵ^-2d). Surprisingly, the fundamental question of whether or not these dependencies on ϵ and d for small d can be improved has remained elusive, even for d = 2. This question naturally arises in any application of Euclidean spanners where precision is a necessity. The state-of-the-art bounds nO(ϵ^-d+1) and O(ϵ^-2d) on the size and lightness of spanners are realized by the greedy spanner. In 2016, Filtser and Solomon proved that, in low dimensional spaces, the greedy spanner is near-optimal. The question of whether the greedy spanner is truly optimal remained open to date. The contribution of this paper is two-fold. We resolve these longstanding questions by nailing down the exact dependencies on ϵ and d and showing that the greedy spanner is truly optimal. Specifically, for any d= O(1), ϵ = Ω(n^-1/d-1): _ We show that any (1+ϵ)-spanner must have n Ω(ϵ^-d+1) edges, implying that the greedy (and other) spanners achieve the optimal size. _ We show that any (1+ϵ)-spanner must have lightness Ω(ϵ^-d), and then improve the upper bound on the lightness of the greedy spanner from O(ϵ^-2d) to O(ϵ^-d). We then complement our negative result for the size of spanners with a rather counterintuitive positive result: Steiner points lead to a quadratic improvement in the size of spanners! Our bound for the size of Steiner spanners is tight as well (up to lower-order terms).
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