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 $O_d(\varepsilon^{(1-d^2)/d}n^{(d-1)/d})$, 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.

翻译：给定平面上的$n$个点集$S$和参数$\varepsilon>0$，欧几里得$(1+\varepsilon)$-生成子图是一个几何图$G=(S,E)$，使得对所有$p,q\in S$，存在一条$pq$-路径的权重至多为$(1+\varepsilon)\|pq\|$。我们证明了单位正方形$[0,1]^2$内$n$个点的欧几里得$(1+\varepsilon)$-生成子图的最小权重为$O(\varepsilon^{-3/2}\,\sqrt{n})$，且该界是最优的。上界基于一种新的平面生成子图算法，它改进了由$[0,1]^2$内$n$个点欧几里得最小生成树(MST)权重的紧界与欧几里得$(1+\varepsilon)$-生成子图轻量性（即生成子图权重与MST权重之比）的紧界结合得到的基线$O(\varepsilon^{-2}\sqrt{n})$。我们的结果推广到任意常数维度$d\in \mathbb{N}$的欧几里得$d$维空间：单位立方体$[0,1]^d$内$n$个点的欧几里得$(1+\varepsilon)$-生成子图的最小权重为$O_d(\varepsilon^{(1-d^2)/d}n^{(d-1)/d})$，且该界是最优的。对于平面整数格点的$n\times n$截面，我们证明欧几里得$(1+\varepsilon)$-生成子图的最小权重介于$\Omega(\varepsilon^{-3/4}\cdot n^2)$与$O(\varepsilon^{-1}\log(\varepsilon^{-1})\cdot n^2)$之间。当缩放为单位正方形内$n$个点的网格时，这些界变为$\Omega(\varepsilon^{-3/4}\cdot \sqrt{n})$和$O(\varepsilon^{-1}\log(\varepsilon^{-1})\cdot \sqrt{n})$。特别地，这表明整数网格并非最小权重欧几里得$(1+\varepsilon)$-生成子图的极端配置。