Coloring Geometric Range Spaces

  • Greg Aloupis
  • Jean Cardinal
  • Sébastien Collette
  • Stefan Langerman
  • Shakhar Smorodinsky
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)


Given a set of points in \({\mathbb R}^2\) or \({\mathbb R}^3\), we aim to color them such that every region of a certain family (for instance disks) containing at least a certain number of points contains points of many different colors. Using k colors, it is not always possible to ensure that every region containing k points contains all k colors. Thus, we introduce two relaxations: either we allow the number of colors to increase to c(k), or we require that the number of points in each region increases to p(k). We give upper bounds on c(k) and p(k) for halfspaces, disks, and pseudo-disks. We also consider the dual question, where we want to color regions instead of points. This is related to previous results of Pach, Tardos and Tóth on decompositions of coverings.


Unit Circle Chromatic Number Distinct Color Range Space Projective Duality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Aigner, M., Ziegler, G.M.: Proofs from the book. Springer, Heidelberg (1998)zbMATHGoogle Scholar
  2. 2.
    Buchsbaum, A., Efrat, A., Jain, S., Venkatasubramanian, S., Yi, K.: Restricted strip covering and the sensor cover problem. In: ACM-SIAM Symposium on Discrete Algorithms (SODA 2007) (2007)Google Scholar
  3. 3.
    Chan, T.M.: Low-dimensional linear programming with violations. SIAM Journal on Computing 34(4), 879–893 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chen, X., Pach, J., Szegedy, M., Tardos, G.: Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles (manuscript, 2006)Google Scholar
  5. 5.
    Lick, D.R., White, A.T.: k-degenerate graphs. Canadian Journal on Mathematics 12, 1082–1096 (1970)MathSciNetGoogle Scholar
  6. 6.
    Pach, J.: Decomposition of multiple packing and covering. In: 2. Kolloq. über Diskrete Geom., Inst. Math. Univ. Salzburg, pp. 169–178 (1980)Google Scholar
  7. 7.
    Pach, J.: Personal communication (2007)Google Scholar
  8. 8.
    Pach, J., Tardos, G.: Personal communication (2006)Google Scholar
  9. 9.
    Pach, J., Tardos, G., Tóth, G.: Indecomposable coverings. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds.) CJCDGCGT 2005. LNCS, vol. 4381, pp. 135–148. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Pach, J., Tóth, G.: Decomposition of multiple coverings into many parts. In: Proc. of the 23rd ACM Symposium on Computational Geometry, pp. 133–137 (2007)Google Scholar
  11. 11.
    Sharir, M.: On k-sets in arrangement of curves and surfaces. Discrete & Computational Geometry 6, 593–613 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Smorodinsky, S.: On the chromatic number of some geometric hypergraphs. SIAM Journal on Discrete Mathematics (to appear)Google Scholar
  13. 13.
    Smorodinsky, S., Sharir, M.: Selecting points that are heavily covered by pseudo-circles, spheres or rectangles. Combinatorics, Probability and Computing 13(3), 389–411 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tucker, A.: Coloring a family of circular arcs. SIAM Journal of Applied Mathematics 229(3), 493–502 (1975)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Tukey, J.: Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, vol. 2, pp. 523–531 (1975)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Greg Aloupis
    • 1
  • Jean Cardinal
    • 1
  • Sébastien Collette
    • 1
  • Stefan Langerman
    • 1
  • Shakhar Smorodinsky
    • 2
  1. 1.Université Libre de Bruxelles, CP212, Bld. du Triomphe, 1050 Brussels, Belgium. Partially supported by the Communauté française de Belgique - ARC 
  2. 2.Institute of MathematicsHebrew University, Givat-RamJerusalemIsrael

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