## Abstract

Let N be a set of n points in convex position in ℝ^{3}. The farthest-point Voronoi diagram of N partitions ℝ^{3} into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in expected O(n log^{2} n) time. More precisely, we compute the combinatorial structure of G(N), the coordinates of its vertices, and the equation of the plane defining each edge of G(N). The algorithm allows us to solve the all-pairs farthest neighbor problem for N in expected time O(n log^{2} n), and to perform farthest-neighbor queries on N in O(log^{2} n) time with high probability. This can be applied to find a Euclidean maximum spanning tree and a diameter 2-clustering of N in expected O(n log^{4} n) time.

Original language | English (US) |
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Title of host publication | Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings |

Editors | Jie Wang |

Publisher | Springer Verlag |

Pages | 159-169 |

Number of pages | 11 |

ISBN (Print) | 9783540424949 |

DOIs | |

State | Published - 2001 |

Externally published | Yes |

Event | 7th Annual International Conference on Computing and Combinatorics, COCOON 2001 - Guilin, China Duration: Aug 20 2001 → Aug 23 2001 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 2108 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 7th Annual International Conference on Computing and Combinatorics, COCOON 2001 |
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Country/Territory | China |

City | Guilin |

Period | 08/20/01 → 08/23/01 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 2001.

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science