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：在低维欧几里得空间中，贪婪支撑图是一种基础几何构造，过去三十年已被广泛研究，因为它具备优质支撑图的两项基本性质：恒定最大度与恒定轻量性。近期，埃普斯坦和科达班德证明二维欧几里得空间中的贪婪支撑图在强意义下存在次线性分隔子：$\mathbb{R}^2$中贪婪支撑图的任意$k$顶点子图均存在大小为$O(\sqrt{k})$的分隔子。然而该技术本质上是平面性的，无法推广至更高维度。他们提出疑问：对于任意常数$d\geq 3$，$\mathbb{R}^d$中的贪婪支撑图是否存在小规模分隔子？本文通过证明$\mathbb{R}^d$中贪婪支撑图的任意$k$顶点子图均存在大小为$O(k^{1-1/d})$的分隔子，解决了埃普斯坦与科达班德提出的这一问题。我们引入一项新技术，为任意几何图存在次线性分隔子提供简洁刻画——我们称之为$\tau$-瘦长性：若任意半径为$r$的球至多截断图中$r$条长度不小于$r$的边，则称该几何图是$\tau$-瘦长的。我们证明$\mathbb{R}^d$中任意$n$顶点$\tau$-瘦长几何图存在大小为$O(\tau n^{1-1/d})$的分隔子。进而通过证明贪婪支撑图是$O(1)$-瘦长的，推导出主要结论。实际上我们获得了更普适的结果，该结论适用于单位球图及$\mathbb{R}^d$中低分形维点集。我们的技术自然可推广至度量倍增空间。利用$\tau$-瘦长刻画，我们证明存在一个关于维度$d$的度量倍增空间的$(1+\epsilon)$-支撑图，具有恒定最大度与大小为$O(n^{1-\frac{1}{d}})$的分隔子；该结果解决了阿巴姆与哈-佩莱德十年前提出的开放问题。