Ke Chen
ABSTRACT
We study the problem of conflict-free (CF) coloring of a set of points in the plane, in an online fashion, with respect to halfplanes, nearly-equal axis-parallel rectangles, and congruent disks. As a warm-up exercise, the online CF coloring of points on the line with respect to intervals is also considered. We present randomized algorithms in the oblivious adversary model, where the adversary does not see the colors used. For the problems considered, the algorithms always produce valid CF colorings, and use O(logn) colors with high probability (these bounds are optimal in the worst case). Our randomized online algorithms are considerably simpler than previous algorithms for this problem and use fewer colors.We also present a deterministic algorithm for the CF coloring of points in the plane with respect to nearly-equal axis-parallel rectangles, using O(polylog(n)) colors. This is the first efficient deterministic online CF coloring algorithm for this problem.
- N. Alon and S. Smorodinsky. Conflict-free colorings of shallow discs. In Proc. 22nd Annu. ACM Sympos. Comput. Geom., 2006. To appear. Google Scholar
Digital Library
- A. Borodin and R. El-Yaniv. Online computation and competitive analysis. Cambridge University Press, New York, NY, USA, 1998. Google ScholarDigital Library
- K. Elbassioni and N. H. Mustafa. Conflict-free colorings of rectangle ranges. In B. Durand and W. Thomas, editors, Proc. 23rd International Sympos. Theoretical Aspects of Comp. Sci., volume 3884 of LNCS, pages 254--263, 2006. Google ScholarDigital Library
- G. Even, Z. Lotker, D. Ron, and S. Smorodinsky. Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM J. Comput., 33(1):94--136, 2003. Google ScholarDigital Library
- A. Fiat, M. Levy, J. Matoušek, E. Mossel, J. Pach, M. Sharir, S. Smorodinsky, U. Wagner, and E. Welzl. Online conflict-free coloring for intervals. In Proc. 16th ACM-SIAM Sympos. Discrete Algo., pages 545--554, 2005. Google ScholarDigital Library
- S. Har-Peled and S. Smorodinsky. Conflict-free coloring of points and simple regions in the plane. Discrete Comput. Geom., 34(1):47--70, 2005. Google ScholarDigital Library
- H. Kaplan and M. Sharir. Online CF coloring for halfplanes, congruent disks, and axis-parallel rectangles, 2004.Google Scholar
- J. Pach and G. Toth. Conflict-free colorings. Discrete Comput. Geom., The Goodman-Pollack Festschrift, pages 665--671, 2003.Google ScholarCross Ref
- M. Sharir, 2005. Personal communication.Google Scholar
- S. Smorodinsky. On the chromatic number of some geometric hypergraphs. In Proc. 17th ACM-SIAM Sympos. Discrete Algo., pages 316--323, 2006. Google ScholarDigital Library
Index Terms
- How to play a coloring game against a color-blind adversary
Recommendations
Online Conflict-Free Coloring for Intervals
We consider an online version of the conflict-free coloring of a set of points on the line, where each newly inserted point must be assigned a color upon insertion, and at all times the coloring has to be conflict-free, in the sense that in every ...
On coloring points with respect to rectangles
In a coloring of a set of points P with respect to a family of geometric regions one requires that in every region containing at least two points from P, not all the points are of the same color. Perhaps the most notorious open case is coloring of n ...
Conflict-free coloring for rectangle ranges using O(n.382) colors
SPAA '07: Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architecturesGiven 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 ...
Comments