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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.03213 |
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| _version_ | 1866912057548341248 |
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| author | Acharyya, Ankush Keikha, Vahideh Saumell, Maria Silveira, Rodrigo I. |
| author_facet | Acharyya, Ankush Keikha, Vahideh Saumell, Maria Silveira, Rodrigo I. |
| contents | We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning interval} is as large as possible. A minimum color-spanning interval is an interval of minimum size that contains at least one point from a given interval of each color. We prove that if the input intervals are pairwise disjoint, the problem can be solved in $O(n)$ time, even for intervals of arbitrary length. For overlapping intervals, the problem becomes much more difficult. Nevertheless, we show that it can be solved in $O(n \log^2 n)$ time when $k=2$, by exploiting several structural properties of candidate solutions, combined with a number of advanced algorithmic techniques. Interestingly, this shows a sharp contrast with the 2-dimensional version of the problem, recently shown to be NP-hard. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_03213 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Computing largest minimum color-spanning intervals of imprecise points Acharyya, Ankush Keikha, Vahideh Saumell, Maria Silveira, Rodrigo I. Computational Geometry Data Structures and Algorithms We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning interval} is as large as possible. A minimum color-spanning interval is an interval of minimum size that contains at least one point from a given interval of each color. We prove that if the input intervals are pairwise disjoint, the problem can be solved in $O(n)$ time, even for intervals of arbitrary length. For overlapping intervals, the problem becomes much more difficult. Nevertheless, we show that it can be solved in $O(n \log^2 n)$ time when $k=2$, by exploiting several structural properties of candidate solutions, combined with a number of advanced algorithmic techniques. Interestingly, this shows a sharp contrast with the 2-dimensional version of the problem, recently shown to be NP-hard. |
| title | Computing largest minimum color-spanning intervals of imprecise points |
| topic | Computational Geometry Data Structures and Algorithms |
| url | https://arxiv.org/abs/2410.03213 |