Light Euclidean Spanners with Steiner Points

Abstract

The FOCS'19 paper of Le and Solomon, culminating a long line of research on Euclidean spanners, proves that the lightness (normalized weight) of the greedy $(1+ε)$-spanner in $\mathbb{R}^d$ is $\tilde{O}(ε^{-d})$ for any $d = O(1)$ and any $ε= Ω(n^{-\frac{1}{d-1}})$ (where $\tilde{O}$ hides polylogarithmic factors of $\frac{1}ε$), and also shows the existence of point sets in $\mathbb{R}^d$ for which any $(1+ε)$-spanner must have lightness $Ω(ε^{-d})$. Given this tight bound on the lightness, a natural arising question is whether a better lightness bound can be achieved using Steiner points. Our first result is a construction of Steiner spanners in $\mathbb{R}^2$ with lightness $O(ε^{-1} \log Δ)$, where $Δ$ is the spread of the point set. In the regime of $Δ\ll 2^{1/ε}$, this provides an improvement over the lightness bound of Le and Solomon [FOCS 2019]; this regime of parameters is of practical interest, as point sets arising in real-life applications (e.g., for various random distributions) have polynomially bounded spread, while in spanner applications $ε$ often controls the precision, and it sometimes needs to be much smaller than $O(1/\log n)$. Moreover, for spread polynomially bounded in $1/ε$, this upper bound provides a quadratic improvement over the non-Steiner bound of Le and Solomon [FOCS 2019], We then demonstrate that such a light spanner can be constructed in $O_ε(n)$ time for polynomially bounded spread, where $O_ε$ hides a factor of $\mathrm{poly}(\frac{1}ε)$. Finally, we extend the construction to higher dimensions, proving a lightness upper bound of $\tilde{O}(ε^{-(d+1)/2} + ε^{-2}\log Δ)$ for any $3\leq d = O(1)$ and any $ε= Ω(n^{-\frac{1}{d-1}})$.