Greedy Spanners in Euclidean Spaces Admit Sublinear Separators
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| Formato: | Preprint |
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2021
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| _version_ | 1866911890266914816 |
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| author | Le, Hung Than, Cuong |
| author_facet | Le, Hung Than, Cuong |
| contents | 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 $\mathbb{R}^2$ 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 $d\geq 3$ 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 $\mathbb{R}^d$ 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 $\mathbb{R}^d$ 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 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 $O(n^{1-\frac{1}{d}})$; this result resolves an open problem posed by Abam and Har-Peled a decade ago. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_06490 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Greedy Spanners in Euclidean Spaces Admit Sublinear Separators Le, Hung Than, Cuong Computational Geometry I.3.5; F.2.2 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 $\mathbb{R}^2$ 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 $d\geq 3$ 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 $\mathbb{R}^d$ 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 $\mathbb{R}^d$ 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 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 $O(n^{1-\frac{1}{d}})$; this result resolves an open problem posed by Abam and Har-Peled a decade ago. |
| title | Greedy Spanners in Euclidean Spaces Admit Sublinear Separators |
| topic | Computational Geometry I.3.5; F.2.2 |
| url | https://arxiv.org/abs/2107.06490 |