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.

中文翻译：

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欧几里得空间中的贪婪扳手承认次线性分隔符

低维欧几里得空间中的贪婪扳手是一种基本的几何构造，已经被广泛研究了三十多年，因为它具有良好扳手的两个最基本的特性：恒定的最大度数和恒定的亮度。最近，Eppstein 和 Khodabandeh 表明 $\mathbb{R}^2$ 中的贪婪扳手在强意义上承认了一个次线性分隔符：$\mathbb{R}^2$ 中贪婪扳手的 $k$ 顶点的任何子图有一个大小为 $O(\sqrt{k})$ 的分隔符。他们的技术本质上是平面的，不能扩展到更高的维度。他们留下了在 $\mathbb{R}^d$ 中对于任何常量 $d\geq 3$ 的贪婪扳手存在一个小的分隔符作为一个开放问题。在本文中，我们通过证明 $\mathbb{R}^d$ 中贪婪扳手的 $k$ 顶点的任何子图都有一个大小为 $O(k^{1-1/d}) 的分隔符来解决 Eppstein 和 Khodabandeh 的问题$. 我们引入了一种新技术，它为任何几何图形提供了一个简单的表征，它具有我们称为 $\tau$-lanky 的次线性分隔符：如果任何半径为 $r$ 的球最多切割，则几何图形为 $\tau$-lanky $\tau$ 边在图中的长度至少为 $r$。我们证明 $\mathbb{R}^d$ 中 $n$ 个顶点的任何 $\tau$-lanky 几何图都有一个大小为 $O(\tau n^{1-1/d})$ 的分隔符。然后我们通过证明贪心扳手是 $O(1)$-lanky 来推导出我们的主要结果。我们确实获得了一个更一般的结果，它适用于 $\mathbb{R}^d$ 中的单位球图和低分形维数的点集。我们的技术自然延伸到加倍指标。我们使用 $\tau$-lanky 表征来表明存在一个 $(1+\epsilon)$-spanner 用于将维度 $d$ 的度量加倍，并且具有恒定的最大度数和大小为 $O(n^{ 1-\frac{1}{d}})$; 这一结果解决了十年前 Abam 和 Har-Peled 提出的一个悬而未决的问题。