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
Hauptverfasser: De, Minati, Maheshwari, Anil, Mandal, Ratnadip
Format: Paper
Sprache:Englisch
Veröffentlicht: Ithaca Cornell University Library, arXiv.org 03.07.2024
Schlagworte:
Algorithms
Rectangles
ISSN:2331-8422
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Abstract 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.
AbstractList 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.
Author De, Minati
Mandal, Ratnadip
Maheshwari, Anil
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  givenname: Ratnadip
  surname: Mandal
  fullname: Mandal, Ratnadip
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