Online Class Cover Problem
In this paper, we study the online class cover problem where a (finite or infinite) family \(\cal F\) of geometric objects and a set \({\cal P}_r\) of red points in \(\mathbb{R}^d\) are given a prior, and blue points from \(\mathbb{R}^d\) arrives one after another. Upon the arrival of a blue point,...
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| Veröffentlicht in: | arXiv.org |
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| Hauptverfasser: | , , |
| Format: | Paper |
| Sprache: | Englisch |
| Veröffentlicht: |
Ithaca
Cornell University Library, arXiv.org
03.07.2024
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| Schlagworte: | |
| ISSN: | 2331-8422 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In this paper, we study the online class cover problem where a (finite or infinite) family \(\cal F\) of geometric objects and a set \({\cal P}_r\) of red points in \(\mathbb{R}^d\) are given a prior, and blue points from \(\mathbb{R}^d\) arrives one after another. Upon the arrival of a blue point, the online algorithm must make an irreversible decision to cover it with objects from \(\cal F\) that do not cover any points of \({\cal P}_r\). The objective of the problem is to place a minimum number of objects. When \(\cal F\) consists of axis-parallel unit squares in \(\mathbb{R}^2\), we prove that the competitive ratio of any deterministic online algorithm is \(\Omega(\log |{\cal P}_r|)\), and also propose an \(O(\log |{\cal P}_r|)\)-competitive deterministic algorithm for the problem. |
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| Bibliographie: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.2308.07020 |