Greedy Spanners in Euclidean Spaces Admit Sublinear Separators

The greedy spanner in a low dimensional Euclidean space is fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of good spanner: constant maximum degree and lightness. Recently, Eppstein Khodabandeh [28] showed \(\mathbb {R}^2 \) admits sublinear separator strong sense: any subgraph k vertices size \(O(\sqrt {k}) . Their technique inherently planar not extensible to higher dimensions. They left showing existence small for {R}^d d ≥ 3 an open problem. In this paper, we resolve problem by O ( 1 − 1/ ). We introduce new gives simple criterion graph have dub τ -lanky : if ball radius r cuts at edges length least graph. show n τn then derive our main result (1)-lanky. indeed obtain more general applies unit graphs point sets fractal dimensions Our naturally extends doubling metrics. use there exists (1 + ϵ)-spanner metrics dimension with \(O(n^{1-\frac{1}{d}}) ; resolves posed Abam Har-Peled [1] decade ago. another separator. -point metric \(O((n^{1-\frac{1}{d}}) \log \Delta) where Δ spread metric; factor log ) tightly connected fact that, unlike its counterpart, unbounded Finally, discuss algorithmic implications results.