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 [28] 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 ≥ 3 as an open problem.

In this paper, we resolve the problem of Eppstein and Khodabandeh [28] 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 criterion for any geometric graph to have a sublinear separator that we dub τ-lanky: a geometric graph is τ-lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ-lanky geometric graph of n vertices in \(\mathbb {R}^d \) has a separator of size O(τn^{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 τ-lanky criterion to show that there exists a (1 + ϵ)-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 [1] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n-point metric space of doubling dimension d has a separator of size \(O((n^{1-\frac{1}{d}}) + \log \Delta) \) where Δ is the spread of the metric; the factor log (Δ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree. Finally, we discuss algorithmic implications of our results.

低维欧几里德空间中的贪婪扳手是一种基本的几何结构，已被广泛研究了 30 年，因为它具有优质扳手的两个最基本属性：常数最大度和常数亮度。最近，Eppstein 和 Khodabandeh [28] 表明 \(\mathbb {R}^2 \) 中的贪婪扳手在强意义上承认一个次线性分隔符： \(\mathbb {R} 中贪婪扳手的k个顶点的任何子图}^2 \) 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。他们将 \(\mathbb {R}^d \) 中任何常数d ≥ 3的贪婪扳手的小分隔符的存在作为一个未解决的问题。

在本文中，我们解决了 Eppstein 和 Khodabandeh [28] 的问题，方法是证明\(\mathbb {R}^d \) 中贪婪扳手的k个顶点的任何子图都有一个大小为O ( k ^{1 − 1 / d}）。我们引入了一种新技术，该技术为任何几何图形提供了一个简单的标准，使其具有我们称之为 τ -lanky 的次线性分隔符：如果任何半径为r的球最多切割长度至少为r的τ条边，则几何图形为τ -lanky图。我们证明了 \(\mathbb {R}^d \) 中任何τ -n个顶点的细长几何图都有一个大小为O ( τn ^{1 − 1/ d} )。然后，我们通过证明贪心扳手是O (1)-瘦长型来得出我们的主要结果。我们确实获得了适用于单位球图和 \(\mathbb {R}^d \) 中低分形维数点集的更一般结果。

我们的技术很自然地延伸到加倍指标。我们使用τ - 瘦长标准来证明存在一个 (1 + ϵ)-spanner 用于将维度d的度量加倍，其最大度数为常量，分隔符大小为 \(O(n^{1-\frac{1} {d}}) \); 这一结果解决了 Abam 和 Har-Peled [1] 十年前提出的一个开放性问题。然后，我们引入另一个简单的标准，用于将维度d的度量加倍以具有次线性分隔符的图。我们使用新的标准来证明一个n点度量空间的双维d的贪婪扳手有一个大小为 \(O((n^{1-\frac{1}{d}}) + \log \Delta) \) 其中Δ是度量的分布；因子对数 ( Δ) 与这样一个事实紧密相关，即与欧几里得对应物不同，加倍度量中的贪心扳手具有无界最大度数。最后，我们讨论了我们结果的算法含义。