Parallel computation of discrete Voronoi diagrams

Extended abstract
  • Otfried Schwarzkopf
Contributed Papers Parallel Algorithms

DOI: 10.1007/BFb0028984

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

Abstract

Consider a discrete universe {U := {(x,y) ∈ Z2 | 1 ≤ x,yn}. 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 L1-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(log3n) algorithm to compute an approximation for any Lκ-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(log2n) 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.

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Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Otfried Schwarzkopf
    • 1
  1. 1.Freie Universität Berlin, FB Mathematik, WE 03Berlin 33West Germany

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