## Abstract

We consider the problem of reporting the pairwise enclosures in a set of n axes-parallel rectangles in IR^{2}, which is equivalent to reporting dominance pairs in a set of n points in IR^{4}. Over a decade ago, Lee and Preparata [LP82] gave an O(n log^{2} n + k)-time and O(n)-space algorithm for these problems, where k is the number of reported pairs. Since that time, the question of whether there is a faster algorithm has remained an intriguing open problem. In this paper, we give an algorithm which runs in O (n log n log log n + k log log n) time and uses O(n) space. Thus, although our result is not a strict improvement over the Lee-Preparata algorithm for the full range of k, it is, nevertheless, the first result since [LP82] to make any progress on this long-standing open problem. Our algorithm is based on the divide-and-conquer paradigm. The heart of the algorithm is the solution to a red-blue dominance reporting problem (the "merge" step). We give a novel solution for this problem which is based on the iterative application of a sequence of non-trivial sweep routines. This solution technique should be of independent interest. We also present another algorithm whose bounds match the bounds given in [LP82], but which is simpler. Finally, we consider the special case where the rectangles have at most a constant number, a, of different aspect ratios, which is often the case in practice. For this problem, we give an algorithm which runs in O(αnlog n + k) time and uses O(n) space.

Original language | English (US) |
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Title of host publication | Proceedings of the 11th Annual Symposium on Computational Geometry, SCG 1995 |

Publisher | Association for Computing Machinery |

Pages | 162-171 |

Number of pages | 10 |

ISBN (Electronic) | 0897917243 |

DOIs | |

State | Published - Sep 1 1995 |

Event | 11th Annual Symposium on Computational Geometry, SCG 1995 - Vancouver, Canada Duration: Jun 5 1995 → Jun 7 1995 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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Volume | Part F129372 |

### Other

Other | 11th Annual Symposium on Computational Geometry, SCG 1995 |
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Country/Territory | Canada |

City | Vancouver |

Period | 6/5/95 → 6/7/95 |

### Bibliographical note

Funding Information:*Department of Computer Minnesota, Minneapolis, MN {PWPta. j anardan}@cs. umn. edu. t The research of these authors was supported in part by NSF grant CCR-92-00270. Part of this work was done while PG was visiting the Max-Planck-Institut fiir Informatik. PG thanks the MPI and the International Computer Science Institute for partial support. ‘Max-Planck-Institut fiir Informatik, D-66123 Saarbriic-ken, Germany. Email: michiel~mpi-sb mpg. de. This author was supported by the ESPRIT Basic Research Actions Program, under contract No. 7141 (project ALCOM II). $DIMACS, Rutgers University, Piscataway, NJ 08855, U.S.A. E-mail: bhaskar~dimacs. rutgers. edu.

Publisher Copyright:

© 1995 ACM.