Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

2026-02-19

AI Summary

This work addresses the problem of constructing non-crossing Euclidean Steiner (1+ε)-spanners for point sets in the plane with the fewest possible edges, while guaranteeing that the length of the shortest path between any two points is at most (1+ε) times their Euclidean distance. By integrating techniques from computational geometry, planar graph construction, Steiner point strategies, and the disk-tube incidence theory from geometric measure theory, the authors improve the best-known upper bound on the number of edges from O(n/ε⁴) to O(n/ε^{3/2}). Furthermore, leveraging a generalization of the Szemerédi–Trotter theorem, they establish an almost matching lower bound of Ω(n/ε^{3/2−μ}) for any arbitrarily small μ > 0. This result achieves the current state-of-the-art sparsity and nearly reaches the theoretical limit of the problem.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Privacy-enhancing technologies

Abstract

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.

Problem

• Euclidean spanner
• noncrossing
• Steiner spanner
• sparsity
• geometric graph

Innovation

• noncrossing spanner
• Steiner spanner
• sparsity
• Euclidean geometry
• Szemerédi-Trotter theorem