A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929267699351552 |
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| author | de Berg, Mark Theocharous, Leonidas |
| author_facet | de Berg, Mark Theocharous, Leonidas |
| contents | Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside $\mathcal{P}$. We prove that there exists, for any $\varepsilon>0$, a set $\mathcal{C} \subset P$ of size $O(1/\varepsilon^2)$ such that the following holds: for any query point $q$ inside the polygon $\mathcal{P}$, the geodesic distance from $q$ to its furthest neighbor in $\mathcal{C}$ is at least $1-\varepsilon$ times the geodesic distance to its further neighbor in $P$. Thus the set $\mathcal{C}$ can be used for answering $\varepsilon$-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of the size of $P$. The coreset can be constructed in $O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$ time. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_04513 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon de Berg, Mark Theocharous, Leonidas Computational Geometry Let $\mathcal{P}$ be a simple polygon with $m$ vertices and let $P$ be a set of $n$ points inside $\mathcal{P}$. We prove that there exists, for any $\varepsilon>0$, a set $\mathcal{C} \subset P$ of size $O(1/\varepsilon^2)$ such that the following holds: for any query point $q$ inside the polygon $\mathcal{P}$, the geodesic distance from $q$ to its furthest neighbor in $\mathcal{C}$ is at least $1-\varepsilon$ times the geodesic distance to its further neighbor in $P$. Thus the set $\mathcal{C}$ can be used for answering $\varepsilon$-approximate furthest-neighbor queries with a data structure whose storage requirement is independent of the size of $P$. The coreset can be constructed in $O\left(\frac{1}{\varepsilon} \left( n\log(1/\varepsilon) + (n+m)\log(n+m)\right) \right)$ time. |
| title | A Coreset for Approximate Furthest-Neighbor Queries in a Simple Polygon |
| topic | Computational Geometry |
| url | https://arxiv.org/abs/2403.04513 |