Algorithmica

, Volume 70, Issue 4, pp 732–749 | Cite as

Strong Conflict-Free Coloring for Intervals

  • Panagiotis Cheilaris
  • Luisa Gargano
  • Adele A. Rescigno
  • Shakhar Smorodinsky
Article

Abstract

We consider the \(k\)-strong conflict-free (\(k\)-SCF) coloring of a set of points on a line with respect to a family of intervals: Each point on the line must be assigned a color so that the coloring is conflict-free in the following sense: in every interval \(I\) of the family there are at least \(k\) colors each appearing exactly once in \(I\). We first present a polynomial-time approximation algorithm for the general problem; the algorithm has approximation ratio 2 when \(k=1\) and \(5-\frac{2}{k}\) when \(k\ge 2\). In the special case of a family that contains all possible intervals on the given set of points, we show that a 2-approximation algorithm exists, for any \(k \ge 1\). We also provide, in case \(k=O({{\mathrm{polylog}}}(n))\), a quasipolynomial time algorithm to decide the existence of a \(k\)-SCF coloring that uses at most \(q\) colors.

Keywords

Conflict-free coloring Interval hypergraph Wireless networks 

Notes

Acknowledgments

We are grateful to the anonymous reviewers for their comments and suggestions, which significantly helped us improve the quality of the paper.

References

  1. 1.
    Abam, M.A., de Berg, M., Poon, S.H.: Fault-tolerant conflict-free coloring. In: Proc. 20th Canadian Conference on Computational Geometry (CCCG) (2008)Google Scholar
  2. 2.
    Abellanas, M., Bose, P., Garcia, J., Hurtado, F., Nicolas, M., Ramos, P.A.: On properties of higher order Delaunay graphs with applications. In: Proc. 21st European Workshop on Computational Geometry (EWCG), pp. 119–122 (2005)Google Scholar
  3. 3.
    Bar-Noy, A., Cheilaris, P., Olonetsky, S., Smorodinsky, S.: Online conflict-free colouring for hypergraphs. Comb. Probab. Comput. 19, 493–516 (2010)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bar-Noy, A., Cheilaris, P., Smorodinsky, S.: Deterministic conflict-free coloring for intervals: from offline to online. ACM Trans. Algorithms 4(4), 44 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chen, K., Fiat, A., Levy, M., Matoušek, J., Mossel, E., Pach, J., Sharir, M., Smorodinsky, S., Wagner, U., Welzl, E.: Online conflict-free coloring for intervals. SIAM J. Comput. 36, 545–554 (2006)CrossRefGoogle Scholar
  6. 6.
    Chen, K., Kaplan, H., Sharir, M.: Online conflict free coloring for congruent disks, and axis-parallel rectangles. ACM Trans. Algorithms 5(2) (2009). doi: 10.1145/1497290.1497292
  7. 7.
    Cui, Z., Hu, Z.C.: \(k\)-conflict-free coloring and \(k\)-strong-conflict-free coloring for one class of hypergraphs and online \(k\)-conflict-free coloring (2011). arXiv:1107.0138
  8. 8.
    Even, G., Lotker, Z., Ron, D., Smorodinsky, S.: Conflict-free colorings of simple geometric regions with applications to frequency assignment in cellular networks. SIAM J. Comput. 33, 94–136 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Horev, E., Krakovski, R., Smorodinsky, S.: Conflict-free coloring made stronger. In: Proc. 12th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pp. 105–117 (2010)Google Scholar
  10. 10.
    Katz, M., Lev-Tov, N., Morgenstern, G.: Conflict-free coloring of points on a line with respect to a set of intervals. Comput. Geom. 45, 508–514 (2012)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lev-Tov, N., Peleg, D.: Conflict-free coloring of unit disks. Discrete Appl. Math. 157(7), 1521–1532 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Nguyen, H.L., Nguyen, U.T.: Algorithms for bandwidth efficient multicast routing in multi-channel multi-radio wireless mesh networks. In: Proc. IEEE Wireless Communications and Networking Conference (WCNC), pp. 1107–1112 (2011)Google Scholar
  13. 13.
    Papadimitriou, C.: Computational Complexity. Addison Wesley, Boston (1993)Google Scholar
  14. 14.
    Smorodinsky, S.: Conflict-free coloring and its applications (2010). arXiv:1005.3616
  15. 15.
    Zeng, G., Wang, B., Ding, Y., Xiao, L., Mutka, M.: Efficient multicast algorithms for multichannel wireless mesh networks. IEEE Trans. Parallel Distrib. Syst. 21, 86–99 (2010)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Panagiotis Cheilaris
    • 1
  • Luisa Gargano
    • 2
  • Adele A. Rescigno
    • 2
  • Shakhar Smorodinsky
    • 3
  1. 1.Faculty of InformaticsUniversità della Svizzera italianaLuganoSwitzerland
  2. 2.Dipartimento di InformaticaUniversity of SalernoFiscianoItaly
  3. 3.Mathematics DepartmentBen-Gurion UniversityBe’er ShevaIsrael

Personalised recommendations