Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)
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)
Chan, J., Chin, F., Ye, D., Zhang, Y., Zhu, H.: Frequency allocation problems for linear cellular networks, full version (2011) (Personal communication)
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)
Durrett, R.: Probability: Theory and Examples. Dixbury Press, Belmont (1991)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966)
Hoffmann-Jørgensen, J.: Probability with a View towards Statistics, Chapman & Hall Probability Series, vol. I. Chapman & Hall, London (1994)
Janssen, J., Krizanc, D., Narayanan, L., Shende, S.: Distributed online frequency assignment in cellular networks. J. Algorithms 36(2), 119–151 (2000)
Jensen, T., Toft, B.: Graph Coloring Problems. Wiley, London (1995)
Karlin, A., Manasse, M., Rudolph, L., Sleator, D.: Competitive snoopy caching. Algorithmica 3, 79–119 (1988)
Sleator, D., Tarjan, R.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)
Sparl, P., Zerovnik, J.: 2-local 4/3-competitive algorithm for multicoloring hexagonal graphs. J Algorithms 55(1), 29–41 (2005)
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)
Yao, A.C.: Probabilistic computations: toward a unified measure of complexity. In: 18th FOCS, pp. 222–227 (1977)
Acknowledgments
This work was supported in part by the Danish Council for Independent Research.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Christ, M.G., Favrholdt, L.M. & Larsen, K.S. Online multi-coloring on the path revisited. Acta Informatica 50, 343–357 (2013). https://doi.org/10.1007/s00236-013-0184-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00236-013-0184-4