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Parallel computation of discrete Voronoi diagrams

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STACS 89 (STACS 1989)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 349))

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Abstract

Consider a discrete universe {U := {(x,y) ∈ Z 2 | 1 ≤ x,y ≤ n}. We give a natural definition for Voronoi diagrams in such a universe. This Discrete Voronoi Diagram may be considered as the digitization of the well-known Voronoi diagram in the plane.

We give an O(log n) algorithm to compute the discrete Voronoi diagram for the L 1-metric on the mesh of trees architecture and we give some evidence from number theory that leads us to the conjecture that it is not possible to compute the Discrete Voronoi diagram in the Euclidean metric in polylogarithmic time on that architecture.

Instead, we give an O(log3 n) algorithm to compute an approximation for any L κ-metric, 1 ≤ κ ≤ ∞.

Using a result by Miller and Stout, it is easy to show that there exist polynomial lower bounds for this problem on the pyramid architecture, which is currently the most popular architecture in the image processing community.

Finally, we give an O(log2 n) algorithm to compute the Delaunay Triangulation of points in a discrete universe, and use this to build a space-time efficient VLSI-circuit for the computation of Delaunay Triangulations.

This research was supported by the DFG under Grant Al 253/1-1

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Authors and Affiliations

  1. Freie Universität Berlin, FB Mathematik, WE 03, Arnimallee 2–6, D-1000, Berlin 33, West Germany

    Otfried Schwarzkopf

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  1. Otfried Schwarzkopf
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B. Monien R. Cori

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© 1989 Springer-Verlag Berlin Heidelberg

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Schwarzkopf, O. (1989). Parallel computation of discrete Voronoi diagrams. In: Monien, B., Cori, R. (eds) STACS 89. STACS 1989. Lecture Notes in Computer Science, vol 349. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028984

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  • DOI: https://doi.org/10.1007/BFb0028984

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-50840-3

  • Online ISBN: 978-3-540-46098-5

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