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

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, for all $p,q\in S$, a $pq$-path of weight at most $(1+\varepsilon)|pq|$. 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 in the plane. It improves upon the baseline $O(\varepsilon^{{-2}\sqrt{n})$,} obtained by combining a tight bound for the weight of a Euclidean minimum spanning tree (MST) on $n$ points in $[0,1]^2$, 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. Our result generalizes to Euclidean $d$-space for every constant dimension $d\in \mathbb{N}$: The minimum weight of a Euclidean $(1+\varepsilon)$-spanner for $n$ points in the unit cube $[0,1]^d$ is $O_d(\varepsilon^{{(1-d2)/d}n{(d-1)/d})$,} and this bound is the best possible. For the $n\times n$ section of the integer lattice in the plane, we show that the minimum weight of a Euclidean $(1+\varepsilon)$-spanner is between $\Omega(\varepsilon^{-3/4}\cdot n^2)$ and $O(\varepsilon^{{-1}\log(\varepsilon{-1})\cdot} n^2)$. These bounds become $\Omega(\varepsilon^{-3/4}\cdot \sqrt{n})$ and $O(\varepsilon^{{-1}\log(\varepsilon{-1})\cdot} \sqrt{n})$ when scaled to a grid of $n$ points in the unit square. In particular, this shows that the integer grid is \emph{not} an extremal configuration for minimum weight Euclidean $(1+\varepsilon)$-spanners.