Light Euclidean Steiner Spanners in the Plane

Abstract

Lightness is a fundamental parameter for Euclidean spanners; it is the ratio of the spanner weight to the weight of the minimum spanning tree of a finite set of points in ${\mathbb{R}}^d$. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on $\epsilon >0$ and $d\in \mathbb{N}$ of the minimum lightness of $(1+\epsilon )$-spanners, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner $(1+\epsilon )$-spanners of lightness $O({\epsilon }^{-1}\log \Delta )$ in the plane, where $\Delta \ge \Omega (\sqrt{n})$ is the \emph{spread} of the point set, defined as the ratio between the maximum and minimum distance between a pair of points. They also constructed spanners of lightness $\tilde{O}({\epsilon }^{-(d+1)/2})$ in dimensions $d\ge 3$. Recently, Bhore and Tóth (2020) established a lower bound of $\Omega ({\epsilon }^{-d/2})$ for the lightness of Steiner $(1+\epsilon )$-spanners in ${\mathbb{R}}^d$, for $d\ge 2$. The central open problem in this area is to close the gap between the lower and upper bounds in all dimensions $d\ge 2$. In this work, we show that for every finite set of points in the plane and every $\epsilon >0$, there exists a Euclidean Steiner $(1+\epsilon )$-spanner of lightness $O({\epsilon }^{-1})$; this matches the lower bound for $d=2$. We generalize the notion of shallow light trees, which may be of independent interest, and use directional spanners and a modified window partitioning scheme to achieve a tight weight analysis.

29 pages, 14 figures. A 17-page extended abstract will appear in the Proceedings of the 37th International Symposium on Computational Geometry