Advertisement

Online Multi-Coloring with Advice

  • Marie G. Christ
  • Lene M. Favrholdt
  • Kim S. Larsen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8952)

Abstract

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.

Keywords

Bipartite Graph Cellular Network Competitive Ratio Online Algorithm Color Class 
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.

Notes

Acknowledgments

We would like to thank anonymous referees for comments, improving the presentation of our results.

References

  1. 1.
    Albers, S., Favrholdt, L.M., Giel, O.: On paging with locality of reference. J. Comput. Syst. Sci. 70(2), 145–175 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barhum, K., Böckenhauer, H.-J., Forišek, M., Gebauer, H., Hromkovič, J., Krug, S., Smula, J., Steffen, B.: On the power of advice and randomization for the disjoint path allocation problem. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Tjoa, A.M. (eds.) SOFSEM 2014. LNCS, vol. 8327, pp. 89–101. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  3. 3.
    Bianchi, M.P., Böckenhauer, H.-J., Hromkovič, J., Keller, L.: Online coloring of bipartite graphs with and without advice. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 519–530. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  4. 4.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R.: On the advice complexity of the \(k\)-server problem. ICALP. LNCS 6755, 207–218 (2011)Google Scholar
  5. 5.
    Böckenhauer, H.-J., Komm, D., Královič, R., Královič, R., Mömke, T.: On the advice complexity of online problems. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 331–340. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  6. 6.
    Böckenhauer, H.-J., Komm, D., Královič, R., Rossmanith, P.: On the advice complexity of the Knapsack problem. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 61–72. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  7. 7.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998) zbMATHGoogle Scholar
  8. 8.
    Borodin, A., Irani, S., Raghavan, P., Schieber, B.: Competitive paging with locality of reference. J. Comput. Syst. Sci. 50(2), 244–258 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Boyar, J., Favrholdt, L.M., Larsen, K.S., Nielsen, M.N.: Extending the accommodating function. Acta Informatica 40(1), 3–35 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Boyar, J., Gupta, S., Larsen, K.S.: Access graphs results for LRU versus FIFO under relative worst order analysis. In: Fomin, F.V., Kaski, P. (eds.) SWAT 2012. LNCS, vol. 7357, pp. 328–339. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  11. 11.
    Boyar, J., Kamali, S., Larsen, K.S., López-Ortiz, A.: Online bin packing with advice. In Thirty-First International Symposium on Theoretical Aspects of Computer Science (STACS), vol. 25 of LIPIcs, pp. 174–186. Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH, German (2014)Google Scholar
  12. 12.
    Boyar, J., Larsen, K.S.: The seat reservation problem. Algorithmica 25(4), 403–417 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Boyar, J., Larsen, K.S., Nielsen, M.N.: The accommodating function: a generalization of the competitive ratio. SIAM J. Comput. 31(1), 233–258 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Chan, J.W.-T., Chin, F.Y.L., Ye, D., Zhang, Y.: Absolute and asymptotic bounds for online frequency allocation in cellular networks. Algorithmica 58(2), 498–515 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Chan, J.W.-T., Chin, F.Y.L., Ye, D., Zhang, Y., Zhu, H.: Frequency allocation problems for linear cellular networks. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 61–70. Springer, Heidelberg (2006) CrossRefGoogle Scholar
  16. 16.
    Christ, M.G., Favrholdt, L.M., Larsen, K.S.: Online multi-coloring on the path revisited. Acta Informatica 50(5–6), 343–357 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Christ, M.G., Favrholdt, L.M., Larsen, K.S.: Online multi-coloring with advice. (2014) arXiv:1409.1722 [cs.DS]
  18. 18.
    Chrobak, M., Jez, L., Sgall, J.: Better bounds for incremental frequency allocation in bipartite graphs. Theor. Comput. Sci. 514, 75–83 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Chrobak, M., Sgall, J.: Three results on frequency assignment in linear cellular networks. Theor. Comput. Sci. 411(1), 131–137 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Dobrev, S., Královič, R., Pardubská, D.: Measuring the problem-relevant information in input. RAIRO Theor. Inf. Appl. 43(3), 585–613 (2009)CrossRefzbMATHGoogle Scholar
  21. 21.
    Dorrigiv, R., He, M., Zeh, N.: On the advice complexity of buffer management. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 136–145. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  22. 22.
    Elias, P.: Universal codeword sets and representations of the integers. IEEE Trans. Inf. Theory 21(2), 194–203 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Emek, Y., Fraigniaud, P., Korman, A., Rosén, A.: Online computation with advice. Theor. Comput. Sci. 412(24), 2642–2656 (2011)CrossRefzbMATHGoogle Scholar
  24. 24.
    Forišek, M., Keller, L., Steinová, M.: Advice complexity of online coloring for paths. In: Dediu, A.-H., Martín-Vide, C. (eds.) LATA 2012. LNCS, vol. 7183, pp. 228–239. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  25. 25.
    Hromkovič, J., Královič, R., Královič, R.: Information complexity of online problems. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 24–36. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  26. 26.
    Janssen, J., Kilakos, K., Marcotte, O.: Fixed preference channel assignment for cellular telephone systems. IEEE Trans. Veh. Technol. 48(2), 533–541 (1999)CrossRefGoogle Scholar
  27. 27.
    Janssen, J., Krizanc, D., Narayanan, L., Shende, S.M.: Distributed online frequency assignment in cellular networks. J. Algorithms 36(2), 119–151 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Karlin, A.R., Manasse, M.S., Rudolph, L., Sleator, D.D.: Competitive snoopy caching. Algorithmica 3, 79–119 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Komm, D., Královič, R.: Advice complexity and barely random algorithms. RAIRO Theor. Inf. Appl 45(2), 249–267 (2011)CrossRefzbMATHGoogle Scholar
  30. 30.
    Komm, D., Královič, R., Mömke, T.: On the advice complexity of the set cover problem. In: Hirsch, E.A., Karhumäki, J., Lepistö, A., Prilutskii, M. (eds.) CSR 2012. LNCS, vol. 7353, pp. 241–252. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  31. 31.
    Bianchi, M.P., Böckenhauer, H.-J., Hromkovič, J., Krug, S., Steffen, B.: On the advice complexity of the online L(2,1)-coloring problem on paths and cycles. In: Du, D.-Z., Zhang, G. (eds.) COCOON 2013. LNCS, vol. 7936, pp. 53–64. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  32. 32.
    McDiarmid, C., Reed, B.A.: Channel assignment and weighted coloring. Networks 36(2), 114–117 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Narayanan, L.: Channel Assignment and Graph Multicoloring, pp. 71–94. Wiley, New York (2002) Google Scholar
  34. 34.
    Narayanan, L., Shende, S.M.: Static frequency assignment in cellular networks. Algorithmica 29(3), 396–409 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    Narayanan, L., Shende, S.M.: Corrigendum: static frequency assignment in cellular networks. Algorithmica 32(4), 679 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Seibert, S., Sprock, A., Unger, W.: Advice complexity of the online coloring problem. In: Spirakis, P.G., Serna, M. (eds.) CIAC 2013. LNCS, vol. 7878, pp. 345–357. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  37. 37.
    Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Sparl, P., Zerovnik, J.: 2-local 4/3-competitive algorithm for multicoloring hexagonal graphs. J. Algorithms 55(1), 29–41 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Witkowski, R., Žerovnik, J.: 1-local 33/24-competitive algorithm for multicoloring hexagonal graphs. In: Frieze, A., Horn, P., Prałat, P. (eds.) WAW 2011. LNCS, vol. 6732, pp. 74–84. Springer, Heidelberg (2011) CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Marie G. Christ
    • 1
  • Lene M. Favrholdt
    • 1
  • Kim S. Larsen
    • 1
  1. 1.University of Southern DenmarkOdenseDenmark

Personalised recommendations