Polychromatic coloring for half-planes
Journal of Combinatorial Theory Series A
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Polychromatic coloring for half-planes
SWAT'10 Proceedings of the 12th Scandinavian conference on Algorithm Theory
Coloring planar homothets and three-dimensional hypergraphs
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Proceedings of the twenty-eighth annual symposium on Computational geometry
On coloring points with respect to rectangles
Journal of Combinatorial Theory Series A
Coloring planar homothets and three-dimensional hypergraphs
Computational Geometry: Theory and Applications
Coloring hypergraphs induced by dynamic point sets and bottomless rectangles
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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We study several coloring problems for geometric range-spaces. In addition to their theoretical interest, some of these problems arise in sensor networks. Given a set of points in ℝ2 or ℝ3, we want to color them so that every region of a certain family (e.g., every disk containing at least a certain number of points) contains points of many (say, k) different colors. In this paper, we think of the number of colors and the number of points as functions of k. Obviously, for a fixed k using k colors, it is not always possible to ensure that every region containing k points has all colors present. Thus, we introduce two types of relaxations: either we allow the number of colors used to increase to c(k), or we require that the number of points in each region increases to p(k). Symmetrically, given a finite set of regions in ℝ2 or ℝ3, we want to color them so that every point covered by a sufficiently large number of regions is contained in regions of k different colors. This requires the number of covering regions or the number of allowed colors to be greater than k. The goal of this paper is to bound these two functions for several types of region families, such as halfplanes, halfspaces, disks, and pseudo-disks. This is related to previous results of Pach, Tardos, and Tóth on decompositions of coverings.