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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)

Abstract

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.

Keywords

Voronoi Diagram Voronoi Cell Implicit Representation Convex Polytope Connected Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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