## Abstract

We consider the p-piercing problem for axis-parallel rectangles. We are given a collection of axis-parallel rectangles in the plane and wish to determine whether there exists a set of p points whose union intersects all the given rectangles. We present efficient algorithms for finding a piercing set (i.e., a set of p points as above) for values of p = 1,2,3,4,5. The result for 4 and 5-piercing improves an existing result of O(n log^{3} n) and O(n log^{4} n) to O(n log n) time. The result for 5-piercing can be applied find an O(n log^{2} n) time algorithm for planar rectilinear 5-center problem, in which we are given a set S of n points in the plane, and wish to find 5 axis-parallel congruent squares of smallest possible size whose union covers S. We improve the existing algorithm for general (but fixed) p to O(n^{p-4} log n) running time, and we also extend our algorithms to higher dimensional space. We also consider the problem of piercing a set of rectangular rings.

Original language | English |
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Pages (from-to) | 219-233 |

Number of pages | 15 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 1999 |

## Keywords

- Algorithms
- Axis-parallel
- Computational geometry
- Piercing

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics