Towards Instance-Optimal Euclidean Spanners

Euclidean spanners are important geometric objects that have been extensively studied since the 1980s. The two most basic "compactness'' measures of a Euclidean spanner $E$ are the size (number of edges) $|E|$ and the weight (sum of edge weights) $\|E\|$. In this paper, we initiate the study of instance optimal Euclidean spanners. Our results are two-fold. We demonstrate that the greedy spanner is far from being instance optimal, even when allowing its stretch to grow. More concretely, we design two hard instances of point sets in the plane, where the greedy $(1+x \epsilon)$-spanner (for basically any parameter $x \geq 1$) has $\Omega_x(\epsilon^{-1/2}) \cdot |E_\mathrm{spa}|$ edges and weight $\Omega_x(\epsilon^{-1}) \cdot \|E_\mathrm{light}\|$, where $E_\mathrm{spa}$ and $E_\mathrm{light}$ denote the per-instance sparsest and lightest $(1+\epsilon)$-spanners, respectively, and the $\Omega_x$ notation suppresses a polynomial dependence on $1/x$. As our main contribution, we design a new construction of Euclidean spanners, which is inherently different from known constructions, achieving the following bounds: a stretch of $1+\epsilon\cdot 2^{O(\log^*(d/\epsilon))}$ with $O(1) \cdot |E_\mathrm{spa}|$ edges and weight $O(1) \cdot \|E_\mathrm{light}\|$. In other words, we show that a slight increase to the stretch suffices for obtaining instance optimality up to an absolute constant for both sparsity and lightness. Remarkably, there is only a log-star dependence on the dimension in the stretch, and there is no dependence on it whatsoever in the number of edges and weight.