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 $\varepsilon>0$ and $d\in \mathbb{N}$ of the minimum lightness of $(1+\varepsilon)$spanners, and observed that additional Steiner points can substantially improve the lightness. Le and Solomon (2020) constructed Steiner $(1+\varepsilon)$spanners of lightness $O(\varepsilon^{1}\log\Delta)$ in the plane, where $\Delta\geq \Omega(\sqrt{n})$ 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 $\tilde{O}(\varepsilon^{(d+1)/2})$ in dimensions $d\geq 3$. Recently, Bhore and Tóth (2020) established a lower bound of $\Omega(\varepsilon^{d/2})$ for the lightness of Steiner $(1+\varepsilon)$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 $d\geq 2$. In this work, we show that for every finite set of points in the plane and every $\varepsilon>0$, there exists a Euclidean Steiner $(1+\varepsilon)$spanner of lightness $O(\varepsilon^{1})$; 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.02216
 Bibcode:
 2020arXiv201202216B
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Discrete Mathematics
 EPrint:
 29 pages, 14 figures. A 17page extended abstract will appear in the Proceedings of the 37th International Symposium on Computational Geometry