Computing farthest neighbors on a convex polytope

Otfried Cheong, Chan Su Shin, Antoine Vigneron

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Scopus citations


Let N be a set of n points in convex position in ℝ3. The farthest-point Voronoi diagram of N partitions ℝ3 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 log2 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 log2 n), and to perform farthest-neighbor queries on N in O(log2 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 log4 n) time.

Original languageEnglish (US)
Title of host publicationComputing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings
EditorsJie Wang
PublisherSpringer Verlag
Number of pages11
ISBN (Print)9783540424949
StatePublished - 2001
Externally publishedYes
Event7th Annual International Conference on Computing and Combinatorics, COCOON 2001 - Guilin, China
Duration: Aug 20 2001Aug 23 2001

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Other7th Annual International Conference on Computing and Combinatorics, COCOON 2001

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2001.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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