The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh showed that the greedy spanner in $\mathbb{R}^2$ admits a sublinear separator in a strong sense: any subgraph of $k$ vertices of the greedy spanner in $\mathbb{R}^2$ has a separator of size $O(\sqrt{k})$. Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in $\mathbb{R}^d$ for any constant $d\geq 3$ as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh by showing that any subgraph of $k$ vertices of the greedy spanner in $\mathbb{R}^d$ has a separator of size $O(k^{1-1/d})$. We introduce a new technique that gives a simple characterization for any geometric graph to have a sublinear separator that we dub $\tau$-lanky: a geometric graph is $\tau$-lanky if any ball of radius $r$ cuts at most $\tau$ edges of length at least $r$ in the graph. We show that any $\tau$-lanky geometric graph of $n$ vertices in $\mathbb{R}^d$ has a separator of size $O(\tau n^{1-1/d})$. We then derive our main result by showing that the greedy spanner is $O(1)$-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in $\mathbb{R}^d$. Our technique naturally extends to doubling metrics. We use the $\tau$-lanky characterization to show that there exists a $(1+\epsilon)$-spanner for doubling metrics of dimension $d$ with a constant maximum degree and a separator of size $O(n^{1-\frac{1}{d}})$; this result resolves an open problem posed by Abam and Har-Peled a decade ago.
翻译:低维的 Euclidean 空间的贪婪范围是一个基本的几何构造。 30年来, 一直对它进行了广泛研究, 因为它拥有一个高维的两种最基本的特性: 恒定最大度和恒定光度。 最近, Eppstein 和 Khodabandeh 显示, 贪婪范围在$\ mathbb{ R\\\\ 2 $ 中有一个亚线性分隔符: 任何以美元为单位的贪婪范围在 $ mathb{ R\ 2 中, 任何以美元为单位的垂直值在 美元中, 以美元为单位的平价计算。 任何以美元为单位的平面值在美元中, 以美元为单位的平面值在美元中, 以美元为单位的平面值在美元中, 以美元显示任何美元为单位的平面值。