Computing farthest neighbors on a convex polytope

Otfried Cheong*, Chan Su Shin, Antoine Vigneron

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

15 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(nlog2n) 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(nlog2n), and to perform farthest-neighbor queries on N in O(log2n) time with high probability.

Original languageEnglish (US)
Pages (from-to)47-58
Number of pages12
JournalTheoretical Computer Science
Issue number1
StatePublished - 2003
Externally publishedYes
EventComputing and Combinatorics - Guilin, China
Duration: Aug 20 2001Aug 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: (O. Cheong), (C.-S. Shin), (A. Vigneron).


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

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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