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Polychromatic Coloring for Half-Planes

  • Shakhar Smorodinsky
  • Yelena Yuditsky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

Abstract

We prove that for every integer k, every finite set of points in the plane can be k-colored so that every half-plane that contains at least 2k − 1 points, also contains at least one point from every color class. We also show that the bound 2k − 1 is best possible. This improves the best previously known lower and upper bounds of \(\frac{4}{3}k\) and 4k − 1 respectively. As a corollary, we also show that every finite set of half-planes can be k colored so that if a point p belongs to a subset H p of at least 4k − 3 of the half-planes then H p contains a half-plane from every color class. This improves the best previously known upper bound of 8k − 3. Another corollary of our first result is a new proof of the existence of small size ε-nets for points in the plane with respect to half-planes.

Keywords

Geometric Hypergraphs Discrete Geometry Polychromatic Coloring ε-Nets 

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References

  1. 1.
    Aloupis, G., Cardinal, J., Collette, S., Imahori, S., Korman, M., Langerman, S., Schwartz, O., Smorodinsky, S., Taslakian, P.: Colorful strips. In: LATIN (to appear, 2010)Google Scholar
  2. 2.
    Aloupis, G., Cardinal, J., Collette, S., Langerman, S., Orden, D., Ramos, P.: Decomposition of multiple coverings into more parts. In: SODA, pp. 302–310 (2009)Google Scholar
  3. 3.
    Aloupis, G., Cardinal, J., Collette, S., Langerman, S., Smorodinsky, S.: Coloring geometric range spaces. Discrete & Computational Geometry 41(2), 348–362 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buchsbaum, A.L., Efrat, A., Jain, S., Venkatasubramanian, S., Yi, K.: Restricted strip covering and the sensor cover problem. In: SODA, pp. 1056–1063 (2007)Google Scholar
  5. 5.
    Fulek, R.: Coloring geometric hypergraph defined by an arrangement of half-planes (manuscript), http://dcg.epfl.ch/page74599.html
  6. 6.
    Gibson, M., Varadarajan, K.R.: Decomposing coverings and the planar sensor cover problem. CoRR, abs/0905.1093 (2009)Google Scholar
  7. 7.
    Haussler, D., Welzl, E.: Epsilon-nets and simplex range queries. Discrete & Computational Geometry 2, 127–151 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Keszegh, B.: Weak conflict-free colorings of point sets and simple regions. In: CCCG, pp. 97–100 (2007)Google Scholar
  9. 9.
    Komlós, J., Pach, J., Woeginger, G.J.: Almost tight bounds for epsilon-nets. Discrete & Computational Geometry 7, 163–173 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Pach, J.: Decomposition of multiple packing and covering. In: 2 Kolloq. über Diskrete Geom., pp. 169–178. Inst. Math. Univ. Salzburg (1980)Google Scholar
  11. 11.
    Pach, J., Mani, P.: Decomposition problems for multiple coverings with unit balls. Unpublished manuscript (1987)Google Scholar
  12. 12.
    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)CrossRefGoogle Scholar
  13. 13.
    Pach, J., Tóth, G.: Decomposition of multiple coverings into many parts. Computational Geometry. Theory and Applications 42(2), 127–133 (2009)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Pach, J., Woeginger, G.: Some new bounds for epsilon-nets. In: SCG, pp. 10–15 (1990)Google Scholar
  15. 15.
    Pálvölgyi, D.: Indecomposable coverings with concave polygons. Discrete & Computational Geometry (2009)Google Scholar
  16. 16.
    Pálvölgyi, D., Tóth, G.: Convex polygons are cover-decomposable. Discrete & Computational Geometry 43(3), 483–496 (2010)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Tardos, G., Tóth, G.: Multiple coverings of the plane with triangles. Discrete & Computational Geometry 38(2), 443–450 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Woeginger, G.J.: Epsilon-nets for halfplanes. In: van Leeuwen, J. (ed.) WG 1988. LNCS, vol. 344, pp. 243–252. Springer, Heidelberg (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Shakhar Smorodinsky
    • 1
  • Yelena Yuditsky
    • 1
  1. 1.Ben-Gurion UniversityBe’er ShevaIsrael

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