On Euclidean Steiner $(1+\epsilon)$-Spanners

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 $\tilde{O}(\varepsilon^{-(d-1))/2})$ 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 $\Omega(\varepsilon^{-(d-1)/2})$ 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.