Algorithmica

, Volume 9, Issue 4, pp 329–356 | Cite as

A semidynamic construction of higher-order voronoi diagrams and its randomized analysis

  • Jean -Daniel Boissonnat
  • Olivier Devillers
  • Monique Teillaud
Article

Abstract

Thek-Delaunay tree extends the Delaunay tree introduced in [1], and [2]. It is a hierarchical data structure that allows the semidynamic construction of the higher-order Voronoi diagrams of a finite set ofn points in any dimension. In this paper we prove that a randomized construction of thek-Delaunay tree, and thus of all the order≤k Voronoi diagrams, can be done inO(n logn+k3n) expected time and O(k2n) expected storage in the plane, which is asymptotically optimal for fixedk. Our algorithm extends tod-dimensional space with expected time complexityO(k⌈(d+1)/2⌉+1n⌊(d+1)/2⌋) and space complexityO(k⌈(d+1)/2⌉n⌊(d+1)/2⌋). The algorithm is simple and experimental results are given.

Key words

Computational geometry Dynamic algorithm Randomized complexity analysis Orderk Voronoi diagram 

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Copyright information

© Springer-Verlag New York Inc. 1993

Authors and Affiliations

  • Jean -Daniel Boissonnat
    • 1
  • Olivier Devillers
    • 1
  • Monique Teillaud
    • 1
  1. 1.Institut National de Recherche en Informatique et AutomatiqueSophia-Antipolis cedexFrance

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