Acta Informatica

, Volume 50, Issue 5–6, pp 343–357 | Cite as

Online multi-coloring on the path revisited

  • Marie G. Christ
  • Lene M. Favrholdt
  • Kim S. Larsen
Original Article


Multi-coloring on the path is a model for frequency assignment in linear cellular networks. Two models have been studied in previous papers: calls may either have finite or infinite duration. For hexagonal networks, a variation of the models where limited frequency reassignment is allowed has also been studied. We add the concept of frequency reassignment to the models of linear cellular networks and close these problems by giving matching upper and lower bounds in all cases. We prove that no randomized algorithm can have a better competitive ratio than the best deterministic algorithms. In addition, we give an exact characterization of the natural greedy algorithms for these problems. All of the above results are with regard to competitive analysis. Taking steps towards a more fine-grained analysis, we consider the case of finite calls and no frequency reassignment and prove that, even though randomization cannot bring the competitive ratio down to one, it seems that the greedy algorithm is expected optimal on uniform random request sequences. We prove this for small paths and indicate it experimentally for larger graphs.


Greedy Algorithm Competitive Ratio Online Algorithm Competitive Analysis Maximal Color 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work was supported in part by the Danish Council for Independent Research.


  1. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  2. Chan, J., Chin, F., Ye, D., Zhang, Y., Zhu, H.: Frequency allocation problems for linear cellular networks. In: 17th International Symposium on Algorithms and Computation, LNCS, vol. 4288, pp. 61–70. Springer, Berlin (2006)Google Scholar
  3. Chan, J., Chin, F., Ye, D., Zhang, Y., Zhu, H.: Frequency allocation problems for linear cellular networks, full version (2011) (Personal communication)Google Scholar
  4. Chrobak, M., Sgall, J.: Three results on frequency assignment in linear cellular networks. In: 5th International Conference on Algorithmic Aspects in Information and Management, LNCS, vol. 5564, pp. 129–139. Springer, Berlin (2009)Google Scholar
  5. Durrett, R.: Probability: Theory and Examples. Dixbury Press, Belmont (1991)zbMATHGoogle Scholar
  6. Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)CrossRefGoogle Scholar
  7. Hoffmann-Jørgensen, J.: Probability with a View towards Statistics, Chapman & Hall Probability Series, vol. I. Chapman & Hall, London (1994)Google Scholar
  8. Janssen, J., Krizanc, D., Narayanan, L., Shende, S.: Distributed online frequency assignment in cellular networks. J. Algorithms 36(2), 119–151 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. Jensen, T., Toft, B.: Graph Coloring Problems. Wiley, London (1995)zbMATHGoogle Scholar
  10. Karlin, A., Manasse, M., Rudolph, L., Sleator, D.: Competitive snoopy caching. Algorithmica 3, 79–119 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  11. Sleator, D., Tarjan, R.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)MathSciNetCrossRefGoogle Scholar
  12. Sparl, P., Zerovnik, J.: 2-local 4/3-competitive algorithm for multicoloring hexagonal graphs. J Algorithms 55(1), 29–41 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  13. Witkowski, R., Zerovnik, J.: 1-local 33/24-competitive algorithm for multicoloring hexagonal graphs. In: 8th International Workshop on Algorithms and Models for the Web Graph, LNCS, vol. 6732, pp. 74–84 (2011)Google Scholar
  14. Yao, A.C.: Probabilistic computations: toward a unified measure of complexity. In: 18th FOCS, pp. 222–227 (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Marie G. Christ
    • 1
  • Lene M. Favrholdt
    • 1
  • Kim S. Larsen
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdense MDenmark

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