## Abstract

In the p-piercing problem, a set S of n d-dimensional objects is given, and one has to compute a piercing set for S of size p, if such a set exists. We consider several instances of the 3-piercing problem that admit linear or almost linear solutions: (i) If S consists of axis-parallel boxes in R^{d}, then a piercing triplet for S can be found (if such a triplet exists) in O(n log n) time, for 3 ≤ d ≤ 5, and in O(n ^{[d/3]} log n) time, for d ≥ 6. Based on the solutions for 3 ≤ d ≤ 5, efficient solutions are obtained to the corresponding 3-center problem - Given a set P of n points in R^{d}, compute the smallest edge length λ such that P can be covered by the union of three axis-parallel cubes of edge length λ. (ii) If S consists of homothetic triangles in the plane, or of 4-oriented trapezoids in the plane, then a piercing triplet can be found in O(n log n) time.

Original language | English |
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Pages (from-to) | 249-259 |

Number of pages | 11 |

Journal | International Journal of Computational Geometry and Applications |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - 1 Jan 1999 |

## Keywords

- Geometric optimization
- Piercing set
- p-center

## ASJC Scopus subject areas

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