Discrete & Computational Geometry

, Volume 48, Issue 1, pp 39-52

First online:

Conflict-Free Coloring for Rectangle Ranges Using O(n .382) Colors

  • Deepak AjwaniAffiliated withCentre for Unified Computing, University College Cork Email author 
  • , Khaled ElbassioniAffiliated withMax-Planck-Institut für Informatik
  • , Sathish GovindarajanAffiliated withComputer Science Department, Indian Institute of Science
  • , Saurabh RayAffiliated withMax-Planck-Institut für Informatik

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Given a set of points P⊆ℝ2, a conflict-free coloring of P w.r.t. rectangle ranges is an assignment of colors to points of P, such that each nonempty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in PT. This notion has been the subject of recent interest and is motivated by frequency assignment in wireless cellular networks: one naturally would like to minimize the number of frequencies (colors) assigned to base stations (points) such that within any range (for instance, rectangle), there is no interference. We show that any set of n points in ℝ2 can be conflict-free colored with \(O(n^{\beta^{*}+o(1)})\) colors in expected polynomial time, where \(\beta^{*}=\frac{3-\sqrt{5}}{2} < 0.382\).


Frequency assignment in wireless networks Conflict-free coloring Axis-parallel rectangles Boundary sets Monotone sequences