On Line-Separable Weighted Unit-Disk Coverage and Related Problems
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929404101263360 |
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| author | Liu, Gang Wang, Haitao |
| author_facet | Liu, Gang Wang, Haitao |
| contents | Given a set $P$ of $n$ points and a set $S$ of $n$ weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of $P$. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of $P$ by a line $\ell$. We present an $O(n^{3/2}\log^2 n)$ time algorithm for the problem. This improves the previously best work of $O(n^2\log n)$ time. Our result leads to an algorithm of $O(n^{{7}/{2}}\log^2 n)$ time for the halfplane coverage problem (i.e., using $n$ weighted halfplanes to cover $n$ points), an improvement over the previous $O(n^4\log n)$ time solution. If all halfplanes are lower ones, our algorithm runs in $O(n^{{3}/{2}}\log^2 n)$ time, while the previous best algorithm takes $O(n^2\log n)$ time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_00329 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Line-Separable Weighted Unit-Disk Coverage and Related Problems Liu, Gang Wang, Haitao Computational Geometry Data Structures and Algorithms Given a set $P$ of $n$ points and a set $S$ of $n$ weighted disks in the plane, the disk coverage problem is to compute a subset of disks of smallest total weight such that the union of the disks in the subset covers all points of $P$. The problem is NP-hard. In this paper, we consider a line-separable unit-disk version of the problem where all disks have the same radius and their centers are separated from the points of $P$ by a line $\ell$. We present an $O(n^{3/2}\log^2 n)$ time algorithm for the problem. This improves the previously best work of $O(n^2\log n)$ time. Our result leads to an algorithm of $O(n^{{7}/{2}}\log^2 n)$ time for the halfplane coverage problem (i.e., using $n$ weighted halfplanes to cover $n$ points), an improvement over the previous $O(n^4\log n)$ time solution. If all halfplanes are lower ones, our algorithm runs in $O(n^{{3}/{2}}\log^2 n)$ time, while the previous best algorithm takes $O(n^2\log n)$ time. Using duality, the hitting set problems under the same settings can be solved with similar time complexities. |
| title | On Line-Separable Weighted Unit-Disk Coverage and Related Problems |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2407.00329 |