Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Coloring Geometric Range Spaces

Abstract

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.

References

  1. 1.

    Aigner, M., Ziegler, G.M.: Proofs from The Book. Springer, Berlin (1998)

  2. 2.

    Aloupis, G.: Geometric measures of data depth. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pp. 147–158. Am. Math. Soc., Providence (2006)

  3. 3.

    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’07) (2007)

  4. 4.

    Chan, T.M.: Low-dimensional linear programming with violations. SIAM J. Comput. 34(4), 879–893 (2005)

  5. 5.

    Chazelle, B.: The Discrepancy Method: Randomness and Complexity. Cambridge University Press, Cambridge (2000)

  6. 6.

    Chen, X., Pach, J., Szegedy, M., Tardos, G.: Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles. Manuscript (2006)

  7. 7.

    Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987)

  8. 8.

    Kříž, I., Nešetřil, J.: Chromatic number of Hasse diagrams, eyebrows and dimension. Order, 8(1), 41–48 (1991)

  9. 9.

    Lick, D.R., White, A.T.: k-degenerate graphs. Can. J. Math. 12, 1082–1096 (1970)

  10. 10.

    Mani, P., Pach, J.: Decomposition problems for multiple coverings with unit balls. Manuscript (1986)

  11. 11.

    Pach, J.: Decomposition of multiple packing and covering. In: 2. Kolloq. über Diskrete Geom., pp. 169–178. Inst. Math. Univ. Salzburg (1980)

  12. 12.

    Pach, J.: Personal communication (2007)

  13. 13.

    Pach, J., Tardos, G.: Personal communication (2006)

  14. 14.

    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)

  15. 15.

    Pach, J., Tardos, G., Tóth, G.: Indecomposable coverings. In: The China–Japan Joint Conference on Discrete Geometry, Combinatorics and Graph Theory (CJCDGCGT 2005). Lecture Notes in Computer Science, pp. 135–148. Springer, Berlin (2007)

  16. 16.

    Sharir, M.: On k-sets in arrangement of curves and surfaces. Discrete Comput. Geom. 6, 593–613 (1991)

  17. 17.

    Smorodinsky, S.: On the chromatic number of some geometric hypergraphs. SIAM J. Discrete Math. (to appear)

  18. 18.

    Smorodinsky, S., Sharir, M.: Selecting points that are heavily covered by pseudo-circles, spheres or rectangles. Comb. Probab. Comput. 13(3), 389–411 (2004)

  19. 19.

    Tucker, A.: Coloring a family of circular arcs. SIAM J. Appl. Math. 229(3), 493–502 (1975)

  20. 20.

    Tukey, J.: Mathematics and the picturing of data. In: Proceedings of the International Congress of Mathematicians, vol. 2, pp. 523–531 (1975)

Download references

Author information

Correspondence to Greg Aloupis.

Additional information

The research of G. Aloupis, J. Cardinal, S. Collette and S. Langerman was supported by the Communauté française de Belgique—Actions de Recherche Concertées (ARC).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Aloupis, G., Cardinal, J., Collette, S. et al. Coloring Geometric Range Spaces. Discrete Comput Geom 41, 348–362 (2009). https://doi.org/10.1007/s00454-008-9116-4

Download citation

Keywords

  • Coloring
  • Covering
  • Decompositions
  • Geometric hypergraphs