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.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Aigner, M., Ziegler, G.M.: Proofs from the book. Springer, Heidelberg (1998)
Buchsbaum, A., Efrat, A., Jain, S., Venkatasubramanian, S., Yi, K.: Restricted strip covering and the sensor cover problem. In: ACM-SIAM Symposium on Discrete Algorithms (SODA 2007) (2007)
Chan, T.M.: Low-dimensional linear programming with violations. SIAM Journal on Computing 34(4), 879–893 (2005)
Chen, X., Pach, J., Szegedy, M., Tardos, G.: Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles (manuscript, 2006)
Lick, D.R., White, A.T.: k-degenerate graphs. Canadian Journal on Mathematics 12, 1082–1096 (1970)
Pach, J.: Decomposition of multiple packing and covering. In: 2. Kolloq. über Diskrete Geom., Inst. Math. Univ. Salzburg, pp. 169–178 (1980)
Pach, J.: Personal communication (2007)
Pach, J., Tardos, G.: Personal communication (2006)
Pach, J., Tardos, G., Tóth, G.: Indecomposable coverings. In: Akiyama, J., Chen, W.Y.C., Kano, M., Li, X., Yu, Q. (eds.) CJCDGCGT 2005. LNCS, vol. 4381, pp. 135–148. Springer, Heidelberg (2007)
Pach, J., Tóth, G.: Decomposition of multiple coverings into many parts. In: Proc. of the 23rd ACM Symposium on Computational Geometry, pp. 133–137 (2007)
Sharir, M.: On k-sets in arrangement of curves and surfaces. Discrete & Computational Geometry 6, 593–613 (1991)
Smorodinsky, S.: On the chromatic number of some geometric hypergraphs. SIAM Journal on Discrete Mathematics (to appear)
Smorodinsky, S., Sharir, M.: Selecting points that are heavily covered by pseudo-circles, spheres or rectangles. Combinatorics, Probability and Computing 13(3), 389–411 (2004)
Tucker, A.: Coloring a family of circular arcs. SIAM Journal of Applied Mathematics 229(3), 493–502 (1975)
Tukey, J.: Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, vol. 2, pp. 523–531 (1975)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Aloupis, G., Cardinal, J., Collette, S., Langerman, S., Smorodinsky, S. (2008). Coloring Geometric Range Spaces. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_13
Download citation
DOI: https://doi.org/10.1007/978-3-540-78773-0_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78772-3
Online ISBN: 978-3-540-78773-0
eBook Packages: Computer ScienceComputer Science (R0)