Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

A Euclidean noncrossing Steiner (1+ε) -spanner for a point set P ⊂ ℝ² is a planar straight-line graph that, for any two points a, b ∈ P, contains a path whose length is at most 1+ε times the Euclidean distance between a and b. We construct a Euclidean noncrossing Steiner (1+ε) -spanner with O (n/ε^3/2) edges for any set of n points in the plane. This result improves upon the previous best upper bound of O (n/ε⁴) 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+ε) -spanner has Ω_μ (n/ε^3/2-μ) edges for any μ > 0. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.