Minimum Weight Euclidean $(1+\varepsilon )$-Spanners

Abstract

Given a set S of n points in the plane and a parameter $\varepsilon >0$, a Euclidean $(1\,+\,\varepsilon )$-spanner is a geometric graph $G=(S,E)$ that contains a path of weight at most $(1+\varepsilon )\Vert pq\Vert _2$ for all $p,q\in S$. We show that the minimum weight of a Euclidean $(1+\varepsilon )$-spanner for n points in the unit square $[0,1]^2$ is $O(\varepsilon ^{-3/2}\,\sqrt{n})$, and this bound is the best possible. The upper bound is based on a new spanner algorithm that sparsifies Yao-graphs. It improves upon the baseline $O(\varepsilon ^{-2}\sqrt{n})$, obtained by combining a tight bound for the weight of an MST and a tight bound for the lightness of Euclidean $(1+\varepsilon )$-spanners, which is the ratio of the spanner weight to the weight of the MST. The result generalizes to d-space for all $d\in \mathbb {N}$: The minimum weight of a Euclidean $(1\,+\,\varepsilon )$-spanner for n points in the unit cube $[0,1]^d$ is $O_d(\varepsilon ^{(1-d^2)/d}n^{(d-1)/d})$, and this bound is the best possible. For the $n\times n$ section of the integer lattice, we show that the minimum weight of a Euclidean $(1+\varepsilon )$-spanner is between $\varOmega (\varepsilon ^{-3/4}n^2)$ and $O(\varepsilon ^{-1}\log (\varepsilon ^{-1})\, n^2)$. These bounds become $\varOmega (\varepsilon ^{-3/4}\sqrt{n})$ and $O(\varepsilon ^{-1}\log (\varepsilon ^{-1})\sqrt{n})$ when scaled to a grid of n points in $[0,1]^2$.

Research partially supported by NSF grant DMS-1800734.