# The aligned K-center problem

- Handle URI:
- http://hdl.handle.net/10754/561744
- Title:
- The aligned K-center problem
- Authors:
- Abstract:
- In this paper we study several instances of the aligned k-center problem where the goal is, given a set of points S in the plane and a parameter k ≥ 1, to find k disks with centers on a line ℓ such that their union covers S and the maximum radius of the disks is minimized. This problem is a constrained version of the well-known k-center problem in which the centers are constrained to lie in a particular region such as a segment, a line, or a polygon. We first consider the simplest version of the problem where the line ℓ is given in advance; we can solve this problem in time O(n log2 n). In the case where only the direction of ℓ is fixed, we give an O(n2 log 2 n)-time algorithm. When ℓ is an arbitrary line, we give a randomized algorithm with expected running time O(n4 log2 n). Then we present (1+ε)-approximation algorithms for these three problems. When we denote T(k, ε) = (k/ε2+(k/ε) log k) log(1/ε), these algorithms run in O(n log k + T(k, ε)) time, O(n log k + T(k, ε)/ε) time, and O(n log k + T(k, ε)/ε2) time, respectively. For k = O(n1/3/log n), we also give randomized algorithms with expected running times O(n + (k/ε2) log(1/ε)), O(n+(k/ε3) log(1/ε)), and O(n + (k/ε4) log(1/ε)), respectively. © 2011 World Scientific Publishing Company.
- KAUST Department:
- Publisher:
- Journal:
- Issue Date:
- Apr-2011
- DOI:
- 10.1142/S0218195911003597
- Type:
- Article
- ISSN:
- 02181959
- Sponsors:
- Author for correspondence; Supported by Korean Research Foundation Grant (KRF-2007-531-D00018).Supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010-0016416), and the HUFS Research Fund.

- Appears in Collections:
- Articles; Computer Science Program; Visual Computing Center (VCC); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.contributor.author | Braß, Peter | en |

dc.contributor.author | Knauer, Christian | en |

dc.contributor.author | Na, Hyeonsuk | en |

dc.contributor.author | Shin, Chansu | en |

dc.contributor.author | Vigneron, Antoine E. | en |

dc.date.accessioned | 2015-08-03T09:03:38Z | en |

dc.date.available | 2015-08-03T09:03:38Z | en |

dc.date.issued | 2011-04 | en |

dc.identifier.issn | 02181959 | en |

dc.identifier.doi | 10.1142/S0218195911003597 | en |

dc.identifier.uri | http://hdl.handle.net/10754/561744 | en |

dc.description.abstract | In this paper we study several instances of the aligned k-center problem where the goal is, given a set of points S in the plane and a parameter k ≥ 1, to find k disks with centers on a line ℓ such that their union covers S and the maximum radius of the disks is minimized. This problem is a constrained version of the well-known k-center problem in which the centers are constrained to lie in a particular region such as a segment, a line, or a polygon. We first consider the simplest version of the problem where the line ℓ is given in advance; we can solve this problem in time O(n log2 n). In the case where only the direction of ℓ is fixed, we give an O(n2 log 2 n)-time algorithm. When ℓ is an arbitrary line, we give a randomized algorithm with expected running time O(n4 log2 n). Then we present (1+ε)-approximation algorithms for these three problems. When we denote T(k, ε) = (k/ε2+(k/ε) log k) log(1/ε), these algorithms run in O(n log k + T(k, ε)) time, O(n log k + T(k, ε)/ε) time, and O(n log k + T(k, ε)/ε2) time, respectively. For k = O(n1/3/log n), we also give randomized algorithms with expected running times O(n + (k/ε2) log(1/ε)), O(n+(k/ε3) log(1/ε)), and O(n + (k/ε4) log(1/ε)), respectively. © 2011 World Scientific Publishing Company. | en |

dc.description.sponsorship | Author for correspondence; Supported by Korean Research Foundation Grant (KRF-2007-531-D00018).Supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2010-0016416), and the HUFS Research Fund. | en |

dc.publisher | World Scientific Pub Co Pte Lt | en |

dc.subject | approximate algorithm | en |

dc.subject | The k-center problem | en |

dc.title | The aligned K-center problem | en |

dc.type | Article | en |

dc.contributor.department | Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division | en |

dc.contributor.department | Computer Science Program | en |

dc.contributor.department | Visual Computing Center (VCC) | en |

dc.contributor.department | Geometric Algorithms Group | en |

dc.identifier.journal | International Journal of Computational Geometry & Applications | en |

dc.contributor.institution | Department of Computer Science, City College, NY, United States | en |

dc.contributor.institution | Institut für Informatik, Universität Bayreuth, Germany | en |

dc.contributor.institution | School of Computing, Soongsil University, Seoul, South Korea | en |

dc.contributor.institution | School of Electronics and Information Engineering, Hankuk University of Foreign Studies, South Korea | en |

kaust.author | Vigneron, Antoine E. | en |

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