## 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(nlog^{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(nlog^{2}n), and to perform farthest-neighbor queries on N in O(log^{2}n) time with high probability.

Original language | English (US) |
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Pages (from-to) | 47-58 |

Number of pages | 12 |

Journal | Theoretical Computer Science |

Volume | 296 |

Issue number | 1 |

DOIs | |

State | Published - 2003 |

Externally published | Yes |

Event | Computing and Combinatorics - Guilin, China Duration: Aug 20 2001 → Aug 23 2001 |

### Bibliographical note

Funding Information:This research was partially supported by the Hong Kong Research Grants Council and partially by grant No. R05-2002-000-00780-0 from the Korea Science & Engineering Foundation. Part of it was done when the 1rst two authors were at HKUST. ∗Corresponding author. E-mail addresses: otfried@cs.uu.nl (O. Cheong), cssin@control.hufs.ac.kr (C.-S. Shin), antoine@cs.ust.hk (A. Vigneron).

## Keywords

- 3D
- Computational geometry
- Farthest neighbors
- Farthest-point Voronoi Diagram
- Polytope

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science