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 [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.