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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2023
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| Accès en ligne: | https://arxiv.org/abs/2312.08803 |
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| _version_ | 1866929497502121984 |
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| author | Abrahamsen, Mikkel de Berg, Sarita Meijer, Lucas Nusser, André Theocharous, Leonidas |
| author_facet | Abrahamsen, Mikkel de Berg, Sarita Meijer, Lucas Nusser, André Theocharous, Leonidas |
| contents | Given a set of $n$ points in the Euclidean plane, the $k$-MinSumRadius problem asks to cover this point set using $k$ disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV~'12]; however, the running time of this algorithm is $O(n^{881})$, and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the $k$-MinSumRadius problem is that of small $k$. For the $2$-MinSumRadius problem, a near-quadratic time algorithm with expected running time $O(n^2 \log^2 n \log^2 \log n)$ was given over 30 years ago [Eppstein~'92].
We present the first improvement of this result, namely, a near-linear time algorithm to compute the $2$-MinSumRadius that runs in expected $O(n \log^2 n \log^2 \log n)$ time. We generalize this result to any constant dimension $d$, for which we give an $O(n^{2-1/(\lceil d/2\rceil + 1) + \varepsilon})$ time algorithm. Additionally, we give a near-quadratic time algorithm for $3$-MinSumRadius in the plane that runs in expected $O(n^2 \log^2 n \log^2 \log n)$ time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08803 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Clustering with Few Disks to Minimize the Sum of Radii Abrahamsen, Mikkel de Berg, Sarita Meijer, Lucas Nusser, André Theocharous, Leonidas Computational Geometry 68U05 F.2 Given a set of $n$ points in the Euclidean plane, the $k$-MinSumRadius problem asks to cover this point set using $k$ disks with the objective of minimizing the sum of the radii of the disks. After a long line of research on related problems, it was finally discovered that this problem admits a polynomial time algorithm [GKKPV~'12]; however, the running time of this algorithm is $O(n^{881})$, and its relevance is thereby mostly of theoretical nature. A practically and structurally interesting special case of the $k$-MinSumRadius problem is that of small $k$. For the $2$-MinSumRadius problem, a near-quadratic time algorithm with expected running time $O(n^2 \log^2 n \log^2 \log n)$ was given over 30 years ago [Eppstein~'92]. We present the first improvement of this result, namely, a near-linear time algorithm to compute the $2$-MinSumRadius that runs in expected $O(n \log^2 n \log^2 \log n)$ time. We generalize this result to any constant dimension $d$, for which we give an $O(n^{2-1/(\lceil d/2\rceil + 1) + \varepsilon})$ time algorithm. Additionally, we give a near-quadratic time algorithm for $3$-MinSumRadius in the plane that runs in expected $O(n^2 \log^2 n \log^2 \log n)$ time. All of these algorithms rely on insights that uncover a surprisingly simple structure of optimal solutions: we can specify a linear number of lines out of which one separates one of the clusters from the remaining clusters in an optimal solution. |
| title | Clustering with Few Disks to Minimize the Sum of Radii |
| topic | Computational Geometry 68U05 F.2 |
| url | https://arxiv.org/abs/2312.08803 |