Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

Abstract

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 [EK21] showed that the greedy spanner in R 2 admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in R 2 has a separator of size O( √ 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 R d for any constant d ≥ 3 as an open problem.In this paper, we resolve the problem of Eppstein and Khodabandeh [EK21] by showing that any subgraph of k vertices of the greedy spanner in 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 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 R d has a separator of size O(τ 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 R d .Our technique naturally extends to doubling metrics. We use the τ -lanky characterization to show that there exists a (1 + )-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size O(n 1− 1 d ); this result resolves an open problem posed by Abam and Har-Peled [AHP10] a decade ago. We then introduce another simple characterization for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new characterization to show that the greedy spanner of an n-point metric space of doubling dimension d has a separator of size O((n 1− 1 d ) + log ∆) where ∆ is the spread of the metric; the factor log(∆) 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.