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)

Abstract

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.

Keywords

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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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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