Truly Optimal Euclidean Spanners

Abstract: 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 $(1+\epsilon)$-spanner with $nO(\epsilon<sup>{-d+1})$ edges and lightness $O(\epsilon<sup>{-2d})$. Surprisingly, the fundamental question of whether or not these dependencies on $\epsilon$ 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 $nO(\epsilon<sup>{-d+1})$ and $O(\epsilon<sup>{-2d})$ on the size and lightness of spanners are realized by the {\em 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 $\epsilon$ and $d$ and showing that the greedy spanner is truly optimal. Specifically, for any $$d= O(1), \epsilon = \Omega({n}<sup>{-\frac{1}{d-1}})$$: - We show that any $(1+\epsilon)$-spanner must have $n \Omega(\epsilon<sup>{-d+1})$ edges, implying that the greedy (and other) spanners achieve the optimal size. - We show that any $(1+\epsilon)$-spanner must have lightness $\Omega(\epsilon<sup>{-d})$, and then improve the upper bound on the lightness of the greedy spanner from $O(\epsilon<sup>{-2d})$ to $O(\epsilon<sup>{-d})$. 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).