Conflict-free coloring for rectangle ranges using O ( n .382 ) colors
Deepak Ajwani, Khaled Elbassioni, Sathish Govindarajan, Saurabh Ray · 2007
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 non-empty axis-parallel rectangle T in the plane contains a point whose color is distinct from all other points in P ∩ T. 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 bases 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 Õ(nβ+ε) colors in expected polynomial time, for any arbitrarily small ε > 0 and β = 3?√5 2 < 0.382. This improves upon the previously known bound of O(√nlog log n/ log n).