On Euclidean Steiner $(1+\epsilon)$Spanners
Abstract
Lightness and sparsity are two natural parameters for Euclidean $(1+\varepsilon)$spanners. Classical results show that, when the dimension $d\in \mathbb{N}$ and $\varepsilon>0$ are constant, every set $S$ of $n$ points in $d$space admits an $(1+\varepsilon)$spanners with $O(n)$ edges and weight proportional to that of the Euclidean MST of $S$. Tight bounds on the dependence on $\varepsilon>0$ for constant $d\in \mathbb{N}$ have been established only recently. Le and Solomon (FOCS 2019) showed that Steiner points can substantially improve the lightness and sparsity of a $(1+\varepsilon)$spanner. They gave upper bounds of $\tilde{O}(\varepsilon^{(d+1)/2})$ for the minimum lightness in dimensions $d\geq 3$, and $\tilde{O}(\varepsilon^{(d1))/2})$ for the minimum sparsity in $d$space for all $d\geq 1$. They obtained lower bounds only in the plane ($d=2$). Le and Solomon (ESA 2020) also constructed Steiner $(1+\varepsilon)$spanners of lightness $O(\varepsilon^{1}\log\Delta)$ in the plane, where $\Delta\in \Omega(\sqrt{n})$ is the \emph{spread} of $S$, defined as the ratio between the maximum and minimum distance between a pair of points. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner $(1+\varepsilon)$spanners. Using a new geometric analysis, we establish lower bounds of $\Omega(\varepsilon^{d/2})$ for the lightness and $\Omega(\varepsilon^{(d1)/2})$ for the sparsity of such spanners in Euclidean $d$space for all $d\geq 2$. We use the geometric insight from our lower bound analysis to construct Steiner $(1+\varepsilon)$spanners of lightness $O(\varepsilon^{1}\log n)$ for $n$ points in Euclidean plane.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 arXiv:
 arXiv:2010.02908
 Bibcode:
 2020arXiv201002908B
 Keywords:

 Computer Science  Computational Geometry
 EPrint:
 16 pages, 5 figures