## Abstract

We propose an implicit representation for the farthest Voronoi diagram of a set P of n points in the plane located outside a set R of m disjoint axes-parallel rectangular obstacles. The distances are measured according to the L1 shortest path (geodesic) metric. In particular, we design a data structure of size O(N^{1.5}) in O(N^{1.5} log^{2} N) time that supports O(N^{0.5} logN)- time farthest point queries (where N = m + n). We avoid computing the more complicated farthest neighbor Voronoi diagram, whose combinatorial complexity is Θ(mn). This allows one to compute the diameter (and all farthest pairs) of P in O(N^{1.5} log^{2}N) time. This improves the previous O(mnlogN) bound [1].

Original language | English |
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Pages | 243-246 |

Number of pages | 4 |

State | Published - 1 Jan 2005 |

Externally published | Yes |

Event | 17th Canadian Conference on Computational Geometry, CCCG 2005 - Windsor, Canada Duration: 10 Aug 2005 → 12 Aug 2005 |

### Conference

Conference | 17th Canadian Conference on Computational Geometry, CCCG 2005 |
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Country/Territory | Canada |

City | Windsor |

Period | 10/08/05 → 12/08/05 |

## ASJC Scopus subject areas

- Geometry and Topology
- Computational Mathematics