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Annual Symposium on Theoretical Aspects of Computer Science

STACS 1994: STACS 94 pp 709–722Cite as

On Voronoi diagrams in the L p -metric in higher dimensions

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 775))

Abstract

We prove upper bounds on the number of L p -spheres passing through D+1 points in general position in D-space, and on the sum of the Betti numbers of the intersection of bisectors in the L p -metric, where p is an even positive integer. The bounds found, surprisingly, do not depend on p. The proofs for these bounds involve the techniques of Milnor [14] and Thorn [20] for finding bounds on the sum of the Betti numbers of algebraic varieties, but instead of the usual degree of polynomials we use their additive complexity, and apply results of Benedetti and Risler [2, 16]. Furthermore, using the theory of degree of mappings in D-space we prove that for even p the number of L p -spheres passing through D+1 points in general position is odd. Combined with results in [10, 11], our results clarify the structure of Voronoi diagrams based on the L p -metric (with even p) in 3-space.

This work was partially supported by Deutsche Forschungsgemeinschaft, grant Kl 655/2-1.

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Authors and Affiliations

  1. Praktische Informatik VI, Fern Universität Hagen, D-58084, Hagen, Germany

    Ngoc-Minh Lê

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  1. Ngoc-Minh Lê
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Patrice Enjalbert Ernst W. Mayr Klaus W. Wagner

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© 1994 Springer-Verlag Berlin Heidelberg

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Lê, NM. (1994). On Voronoi diagrams in the L p -metric in higher dimensions. In: Enjalbert, P., Mayr, E.W., Wagner, K.W. (eds) STACS 94. STACS 1994. Lecture Notes in Computer Science, vol 775. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-57785-8_184

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  • DOI: https://doi.org/10.1007/3-540-57785-8_184

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-57785-0

  • Online ISBN: 978-3-540-48332-8

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