Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

arXiv:2602.17801v1 Announce Type: new
Abstract: A Euclidean noncrossing Steiner $(1+epsilon)$-spanner for a point set $Psubsetmathbb{R}^2$ 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.