Abstract voronoi diagrams and their applications
 Rolf Klein
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Abstract
Given a set S of n points in the plane, and for every two of them a separating Jordan curve, the abstract Voronoi diagram V(S) can be defined, provided that the regions obtained as the intersections of all the “halfplanes” containing a fixed point of S are pathconnected sets and together form an exhaustive partition of the plane. This definition does not involve any notion of distance. The underlying planar graph, \(\hat V\) (S), turns out to have O(n) edges and vertices. If S=L ∪ R is such that the set of edges separating Lfaces from Rfaces in \(\hat V\) (S) does not contain loops then \(\hat V\) (L) and \(\hat V\) (R) can be merged within O(n) steps giving \(\hat V\) (S). This result implies that for a large class of metrics d in the plane the dVoronoi diagram of n points can be computed within optimal O(n log n) time. Among these metrics are, for example, the symmetric convex distance functions as well as the metric defined by the city layout of Moscow or Karlsruhe.
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 Title
 Abstract voronoi diagrams and their applications
 Book Title
 Computational Geometry and its Applications
 Book Subtitle
 CG'88, International Workshop on Computational Geometry Würzburg, FRG, March 24–25, 1988 Proceedings
 Pages
 pp 148157
 Copyright
 1988
 DOI
 10.1007/3540503358_31
 Print ISBN
 9783540503354
 Online ISBN
 9783540459750
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 333
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Voronoi diagram
 metric
 computational geometry
 Industry Sectors
 eBook Packages
 Editors
 Authors

 Rolf Klein ^{(1)}
 Author Affiliations

 1. Institut für Informatik, Universität Freiburg, Rheinstr. 1012, 7800, Freiburg, Fed. Rep. of Germany
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