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Partitioning of complex scenes of geometric objects

  • H. Noltemeier
  • T. Roos
  • C. Zirkelbach
II Mathematical Programming II.1 Computational Geometry
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 180)

Abstract

We are concerned with the problem of partitioning complex scenes of geometric objects in order to support the solutions of proximity problems. We present a data structure called Monotonous Bisector* Tree, which can be regarded as a divisive hierarchical approach of centralized clustering methods (compare [2] and [6]). We analyze some structural properties showing that Monotonous Bisector* Trees are a proper tool for the representation of proximity information in complex scenes of geometric objects, even in general metric spaces.

Given a scene of n convex objects in d-dimensional space and a L p -metric. We show that a Monotonous Bisector* Tree with logarithmic height can be constructed in optimal O(n log n) time using O(n) space. This statement still holds if we demand that the cluster radii, which appear on a path from the root down to a leaf, should generate a geometrically decreasing sequence.

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References

  1. [1]
    L. P. Chew, R. L. Drysdale III, Voronoi Diagrams Based on Convex Distance Functions, 1st ACM Symposium on Computational Geometry, Baltimore, Maryland, S. 235–244, 1985Google Scholar
  2. [2]
    F. Dehne and H. Noltemeier, A Computational Geometry Approach to Clustering Problems, Proceedings of the 1st ACM Symposium on Computational Geometry, Baltimore, Maryland, 1985Google Scholar
  3. [3]
    F.Dehne and H. Noltemeier, Voronoi Trees and Clustering Problems, Information Systems, Vol. 12, No. 2, Pergamon London, 1987Google Scholar
  4. [4]
    H. Heusinger, Clusterverfahren für Mengen geometrischer Objekte, Report, Universität Würzburg, 1989Google Scholar
  5. [5]
    I. Kalantari, G. McDonald, A Data Structure and an Algorithm for the Nearest Point Problem, IEEE Transactions on Software Engineering, Vol. SE-9, No.5, 1983Google Scholar
  6. [6]
    H. Noltemeier, Voronoi Trees and Applications, in H. Imai (ed.): ”Discrete Algorithms and Complexity“ (Proceedings), Fukuoka/Japan, 1989Google Scholar
  7. [7]
    H. Noltemeier, Layout of Flexible Manufacturing Systems — Selected Problems, Proceedings of the Workshop on Applications of Combinatorial Optimization in Science and Technology (COST), New Brunswick, New Jersey, 1991Google Scholar
  8. [8]
    T. Roos, Biscktor*-Bäume und Voronoi*-Bäume für Mengen konvexer Objekte, Techn. Report, Universität Würzburg, 1990Google Scholar
  9. [9]
    C. Zirkelbach, Monotonous Bisector Trees and Clustering Problems, Techn. Report, Universität Würzburg, 1990Google Scholar

Copyright information

© International Federation for Information Processing 1992

Authors and Affiliations

  • H. Noltemeier
  • T. Roos
  • C. Zirkelbach
    • 1
  1. 1.Lehrstuhl für Informatik IUniversität WürzburgWürzburgFed. Rep. of Germany

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