Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

A Euclidean noncrossing Steiner $(1+\epsilon)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+\epsilon$ times the Euclidean distance between $a$ and $b$. We construct a Euclidean noncrossing Steiner $(1+\epsilon)$-spanner with $O(n/\epsilon^{3/2})$ edges for any set of $n$ points in the plane. This result improves upon the previous best upper bound of $O(n/\epsilon^{4})$ obtained nearly three decades ago. We also establish an almost matching lower bound: There exist $n$ points in the plane for which any Euclidean noncrossing Steiner $(1+\epsilon)$-spanner has $\Omega_\mu(n/\epsilon^{3/2-\mu})$ edges for any $\mu>0$. Our lower bound uses recent generalizations of the Szemer\'edi-Trotter theorem to disk-tube incidences in geometric measure theory.