Computing Farthest Neighbors on a Convex Polytope

  • Otfried Cheong
  • Chan-Su Shin
  • Antoine Vigneron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2108)


Let N be a set of n points in convex position in ∝3. The farthest-point Voronoi diagram of N partitions ∝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 log2 n) time. More precisely, we compute the combinatorial structure of G(N), the coordinates of its vertices, and the equation of the plane de?ning 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.


Voronoi Diagram Voronoi Cell Implicit Representation Convex Polytope Connected Subgraph 
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  1. 1.
    Pankaj K. Agarwal, J. Matouísek, and Subhash Suri Farthest neighbors, maximum spanning trees and related problems in higher dimensions. Comput. Geom. Theory Appl., 1(4):189–201, 1992.zbMATHMathSciNetGoogle Scholar
  2. 2.
    A. Aggarwal and D. Kravets A linear time algorithm for finding all farthest neighbors in a convex polygon. Information Processing letters, 31(1):17–20, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Sergei N. Bespamyatnikh An efficient algorithm for the three-dimensional diameter problem. In Proc. 9th ACM-SIAM Sympos. Discrete Algorithms, pages 137–146, 1998.Google Scholar
  4. 4.
    K.L. Clarkson and P.W. Shor Applications of random sampling in computational geometry, II. Discrete Comput. Geom., 4:387–421, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    K. Mulmuley Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, Englewood Cliffs, NJ, 1994.Google Scholar
  6. 6.
    F.P. Preparata and M.I. Shamos Computational Geometry: An Introduction. Springer-Verlag, New York, NY, 1985.Google Scholar
  7. 7.
    E. Ramos Construction of 1-d lower envelopes and applications. In Proc. 13th Annu. ACM Sympos. Comput. Geom., pp. 57–66, 1997.Google Scholar
  8. 8.
    E. Ramos An optimal deterministic algorithm for computing the diameter of a 3-d point set. In Proc. 16th Annu. ACM Sympos. Comput. Geom., page to appear, 2000.Google Scholar
  9. 9.
    P.M. Vaidya An O(n log n) algorithm for the all-nearest-neighbors problem. Discrete Comput. Geom., 4:101–115, 1989.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Otfried Cheong
    • 1
  • Chan-Su Shin
    • 2
  • Antoine Vigneron
    • 3
  1. 1.Institute of Information and Computing SciencesUtrecht UniversityNetherlands
  2. 2.Department of Computer ScienceKAISTKorea
  3. 3.Department of Computer ScienceThe Hong Kong University of Science and TechnologyChina

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