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Solving the Chromatic Cone Clustering Problem via Minimum Spanning Sphere

  • Hu Ding
  • Jinhui Xu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

In this paper, we study the following Chromatic Cone Clustering (CCC) problem: Given n point-sets with each containing k points in the first quadrant of the d-dimensional space R d , find k cones apexed at the origin such that each cone contains at least one distinct point (i.e., different from other cones) from every point-set and the total size of the k cones is minimized, where the size of a cone is the angle from any boundary ray to its center line. CCC is motivated by an important biological problem and finds applications in several other areas. Our approaches for solving the CCC problem relies on solutions to the Minimum Spanning Sphere (MinSS) problem for point-sets. For the MinSS problem, we present two (1 + ε)-approximation algorithms based on core-sets and ε-net respectively. With these algorithms, we then show that the CCC problem admits (1 + ε)-approximation solutions for constant k. Our results are the first solutions to these problems.

Keywords

High Dimension Chromatic Clustering Core-Set 

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References

  1. 1.
    Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristan, V.: The Farthest Color Voronoi Diagram and Related Problems. In: Abstracts 17th European Workshop Comput. Geom. (2001)Google Scholar
  2. 2.
    Agarwal, P.K., Har-Peled, S., Varadarajan, K.R.: Geometric Approximation via Coresets. Combinatorial and Computational Geometry 52, 1–30 (2005)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Badoiu, M., Clarkson, K.: Smaller core-sets for balls. In: Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 801–802 (2003)Google Scholar
  4. 4.
    Badoiu, M., Har-Peled, S., Indyk, P.: Approximate clustering via core-sets. In: Proceedings of the 34th Symposium on Theory of Computing, pp. 250–257 (2002)Google Scholar
  5. 5.
    Cheong, O., Everett, H., Glisse, M., Gudmundsson, J., Hornus, S., Lazard, S., Lee, M., Na, H.: Farthest-Polygon Voronoi Diagrams. In: Proceedings of the 15th Annual European Conference on Algorithms, pp. 407-418 (2007)Google Scholar
  6. 6.
    Clarkson, K.: Coresets, sparse greedy approximation, and the Frank-Wolfe algorithm. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 922–931 (2008)Google Scholar
  7. 7.
    Huttenlocher, D.P., Kedem, K., Sharir, M.: The Upper Envelope of Voronoi Surfaces and its Application. In: Proceedings of the Seventh Annual Symposium on Computational Geometry, pp. 194–203 (1991)Google Scholar
  8. 8.
    Kumar, P., Mitchell, J., Yildirim, A.: Computing Core-Sets and Approximate Smallest Enclosing Hyperspheres in High Dimensions (2002) (manuscript)Google Scholar
  9. 9.
    Megiddo, N.: On the Complexity of Some Geometric Problems in Unbounded Dimension. J. Symb. Comput. 10, 327–334 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Panigrahy, R.: Minimum enclosing polytope in high dimensions, CoRR cs.CG/0407020 (2004)Google Scholar
  11. 11.
    Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|}|E|)\) algorithm for finding maximum matching in general graphs. In: Proceedings of 21st IEEE Symp. Foundations of Computer Science, pp. 17–27 (1980), doi:10.1109/SFCS Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Hu Ding
    • 1
  • Jinhui Xu
    • 1
  1. 1.Department of Computer Science and EngineeringState University of New York at BuffaloUSA

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