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Voronoi diagrams of moving points in the plane

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Graph-Theoretic Concepts in Computer Science (WG 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 570))

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Abstract

Consider a set of n points in the Euclidean plane each of which is continuously moving along a given trajectory. At each instant in time, the points define a Voronoi diagram. As the points move, the Voronoi diagram changes continuously, but at certain critical instants in time, topological events occur that cause a change in the Delaunay diagram. In this paper, we present a method of maintaining the Voronoi diagram over time, while showing that the number of topological events has a nearly cubic upper bound of O(n2λs(n)), where λs,(n) is the maximum length of an (n, s)-Davenport-Schinzel sequence and s is a constant depending on the motions of the point sites. In the special case of points moving at constant speed along straight lines, we get s = 4, implying an upper bound of O(n 32α(n)), where α(n) is the extremely slowly-growing inverse of Ackermann 's function. Our results are a linear-factor improvement over the naive quartic bound on the number of topological events.

In addition, we show that if only k points are moving (while leaving the other n− k points fixed), there is an upper bound of O(k n λs(n) + (n−k)2 λs(k)) on the number of topological events, which is nearly quadratic if k is constant.

We give a numerically stable algorithm for the update of the topological structure of the Voronoi diagram, using only O(log n) time per event (which is worst-case optimal per event).

Partially supported by a grant from Hughes Research Laboratories, Malibu, CA, and by NSF Grant ECSE-8857642.

Work on this paper by Thomas Roos was supported by the Deutsche Forschungsgemeinschaft (DFG) under contract (No 88/10-1).

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Author information

Authors and Affiliations

  1. DEC Systems Research Center and Computer Science Department, Stanford University, USA

    Leonidas J. Guibas

  2. Operations Research, Cornell University, USA

    Joseph S. B. Mitchell

  3. Computer Science Department, University of Würzburg, FRG

    Thomas Roos

Authors
  1. Leonidas J. Guibas
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  2. Joseph S. B. Mitchell
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  3. Thomas Roos
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Gunther Schmidt Rudolf Berghammer

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© 1992 Springer-Verlag Berlin Heidelberg

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Guibas, L.J., Mitchell, J.S.B., Roos, T. (1992). Voronoi diagrams of moving points in the plane. In: Schmidt, G., Berghammer, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 1991. Lecture Notes in Computer Science, vol 570. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-55121-2_11

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  • DOI: https://doi.org/10.1007/3-540-55121-2_11

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  • Print ISBN: 978-3-540-55121-8

  • Online ISBN: 978-3-540-46735-9

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