Greedy Spanners in Euclidean Spaces Admit Sublinear Separators
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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 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 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+ϵ)-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 a decade ago.
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