Discrete & Computational Geometry

, Volume 15, Issue 1, pp 73–105 | Cite as

A compact piecewise-linear voronoi diagram for convex sites in the plane

  • M. McAllister
  • D. Kirkpatrick
  • J. Snoeyink
Article

Abstract

In the plane the post-office problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classically solved by computing a Voronoi diagram. When the sites arek disjoint convex sets we give a compact representation of the Voronoi diagram, usingO (k) line segments, that is sufficient for logarithmic time post-office location queries and motion planning. If these sets are polygons withn total vertices given in standard representations, we compute this diagram optimally in Θ (k logn) deterministic time for the Euclidean metric and inO (k logn logm) deterministic time for the convex distance function defined by a convexm-gon.

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Copyright information

© Springer-Verlag New York Inc 1996

Authors and Affiliations

  • M. McAllister
    • 1
  • D. Kirkpatrick
    • 1
  • J. Snoeyink
    • 1
  1. 1.Department of Computer ScienceUniversity of British ColumbiaVancouverCanada

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