## Abstract

We consider the p-piercing problem, in which we are given a collection of regions, and wish to determine whether there exists a set of p points that intersects each of the given regions. We give linear or near-linear algorithms for small values of p in cases where the given regions are either axis-parallel rectangles or convex c-oriented polygons in the plane (i.e., convex polygons with sides from a fixed finite set of directions). We also investigate the planar rectilinear (and polygonal) p-center problem, in which we are given a set S of n points in the plane, and wish to find p axis-parallel congruent squares (isothetic copies of some given convex polygon, respectively) of smallest possible size whose union covers S. We also study several generalizations of these problems. New results are a linear-time solution for the rectilinear 3-center problem (by showing that this problem can be formulated as an LP-type problem and by exhibiting a relation to Helly numbers). We give O(n log n)-time solutions for 4-piercing of translates of a square, as well as for the rectilinear 4-center problem; this is worst-case optimal. We give O(n polylog n)-time solutions for 4- and 5-piercing of axis-parallel rectangles, for more general rectilinear 4-center problems, and for rectilinear 5-center problems. 2-pierceability of a set of n convex c-oriented polygons can be decided in time O(c^{2}n log n), and the 2-center problem for a convex c-gon can be solved in O(c^{5}n log n) time. The first solution is worst-case optimal when c is fixed.

Original language | English |
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Pages | 122-132 |

Number of pages | 11 |

DOIs | |

State | Published - 1996 |

Event | Proceedings of the 1996 12th Annual Symposium on Computational Geometry - Philadelphia, PA, USA Duration: 24 May 1996 → 26 May 1996 |

### Conference

Conference | Proceedings of the 1996 12th Annual Symposium on Computational Geometry |
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City | Philadelphia, PA, USA |

Period | 24/05/96 → 26/05/96 |