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On the construction of abstract voronoi diagrams, II

  • R. Klein
  • K. Mehlhorn
  • S. Meiser
Session I-3: Invited Talk
Part of the Lecture Notes in Computer Science book series (LNCS, volume 450)

Abstract

Abstract Voronoi Diagrams are defined by a system of bisecting curves in the plane, rather than by the concept of distance [K88a,b]. Mehlhorn, Meiser, Ó' Dúnlaing [MMO] showed how to construct such diagrams in time O(n log n) by a randomized algorithm if the bisecting curves are in general position. In this paper we drop the general position assumption. Moreover, we show that the only geometric operation in the algoithm is the construction of a Voronoi diagram for five sites. Using this operation, abstract Voronoi diagrams can be constructed in a purely combinatorial manner. This has the following advantages: On the one hand, the construction of a five-site-diagram is the only operation depending on the particular type of bisecting curves and we can therefore apply the algorithm to all concrete diagrams by simply replacing this operation. On the other hand, this is the only operation computing intersection points; thus, problems arising from instable numerical computations can occur only there.

Key words

Voronoi diagrams randomized algorithms 

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References

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    F. Aurenhammer (1988): Voronoi Diagrams — A survey, tech. report 263, Institutes for Information Processing, Graz Technical University, AustriaGoogle Scholar
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    R. Klein (1988): Abstract Voronoi Diagrams, Habilitationsschrift, Mathematics Faculty of University of Freiburg i. Br., LNCS 400Google Scholar
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  10. [MMO]
    K. Mehlhorn, S. Meiser, C. Ó' Dúnlaing (1989): On the Construction of Abstract Voronoi Diagrams, Journal of Discrete & Computational Geometry, to appearGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • R. Klein
    • 1
  • K. Mehlhorn
    • 2
  • S. Meiser
    • 2
  1. 1.FB Mathematik/Praktische softwareorientierte InformatikUniversität-GHS-EssenEssen 1West Germany
  2. 2.FB InformatikUniversität des SaarlandesSaarbrückenWest Germany

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