Euclidean Steiner Spanners: Light and Sparse

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 $S$. In a recent breakthrough, Le and Solomon (2019) established the precise dependencies on $\varepsilon>0$, for constant $d\in \mathbb{N}$, of the minimum lightness and sparsity of $(1+\varepsilon)$-spanners, and observed 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 $d\geq 1$. In this work, we improve several bounds on the lightness and sparsity of Euclidean Steiner $(1+\varepsilon)$-spanners. 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 constant $d\geq 2$. Our lower bound constructions generalize previous constructions by Le and Solomon, but the analysis substantially simplifies previous work, using new geometric insight, focusing on the directions of edges. Next, we show that for every finite set of points in the plane and every $\varepsilon\in (0,1]$, 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.