On-Line Maximum Matching in Complete Multipartite Graphs with Implications to the Minimum ADM Problem on a Star Topology

  • Mordechai Shalom
  • Prudence W. H. Wong
  • Shmuel Zaks
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5869)

Abstract

One of the basic problems in optical networks is assigning wavelengths to (namely, coloring of) a given set of lightpaths so as to minimize the number of ADM switches. In this paper we present a connection between maximum matching in complete multipartite graphs and ADM minimization in star networks. A tight 2/3 competitive ratio for finding a maximum matching implies a tight 10/9 competitive ratio for finding a coloring that minimizes the number of ADMs.

Keywords

Online Matching Multi-Partite Graphs Wavelength Assignment Wavelength Division Multiplexing (WDM) Optical Networks Add-Drop Multiplexer (ADM) 
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References

  1. [AC06]
    Azar, Y., Chaiutin, Y.: Optimal node routing. In: The proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science, Marseille, France, February 2006, pp. 596–607 (2006)Google Scholar
  2. [ANR02]
    Azar, Y., Naor, J(S.), Rom, R.: The competitiveness of on-line assignments. In: The proceedings of the 13th SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, January 2002, pp. 203–210 (2002)Google Scholar
  3. [AR05]
    Azar, Y., Richter, Y.: Management of multi-queue switches in QoS networks. Algorithmica 43(1-2), 81–96 (2005)MathSciNetCrossRefMATHGoogle Scholar
  4. [BM08]
    Birnbaum, B., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39(1), 80–87 (2008)CrossRefGoogle Scholar
  5. [CW02]
    Călinescu, G., Wan, P.-J.: Traffic partition in WDM/SONET rings to minimize SONET ADMs. Journal of Combinatorial Optimization 6(4), 425–453 (2002)MathSciNetCrossRefMATHGoogle Scholar
  6. [EL04]
    Epstein, L., Levin, A.: Better bounds for minimizing SONET ADMs. In: Persiano, G., Solis-Oba, R. (eds.) WAOA 2004. LNCS, vol. 3351, pp. 281–294. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  7. [EL09]
    Epstein, L., Levin, A.: Better bounds for minimizing sonet adms. J. Comput. Syst. Sci. 75(2), 122–136 (2009)MathSciNetCrossRefMATHGoogle Scholar
  8. [GLS98]
    Gerstel, O., Lin, P., Sasaki, G.: Wavelength assignment in a WDM ring to minimize cost of embedded SONET rings. In: INFOCOM 1998, Seventeenth Annual Joint Conference of the IEEE Computer and Communications Societies, pp. 69–77 (1998)Google Scholar
  9. [GM08]
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, San Francisco, CA, USA, January 2008, pp. 982–991 (2008)Google Scholar
  10. [GRS98]
    Gerstel, O., Ramaswami, R., Sasaki, G.: Cost effective traffic grooming in WDM rings. In: INFOCOM 1998, Seventeenth Annual Joint Conference of the IEEE Computer and Communications Societies (1998)Google Scholar
  11. [KP93]
    Kalyanasundaram, B., Pruhs, K.: On-line weighted matching. J. Algorithms 14(3), 478–488 (1993)CrossRefMATHGoogle Scholar
  12. [KVV90]
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching (1990)Google Scholar
  13. [Sit96]
    Sitton, D.: Maximum matchings in complete multipartite graphs. Furman University Electronic Journal of Undergraduate Mathematics 2, 6–16 (1996)Google Scholar
  14. [SWZ07]
    Shalom, M., Wong, P.W.H., Zaks, S.: Optimal on-line colorings for minimizing the number of ADMs in optical networks. In: Pelc, A. (ed.) DISC 2007. LNCS, vol. 4731, pp. 435–449. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. [SZ04]
    Shalom, M., Zaks, S.: A 10/7 + ε approximation scheme for minimizing the number of ADMs in SONET rings. In: First Annual International Conference on Broadband Networks, San-José, California, USA, October 2004, pp. 254–262 (2004)Google Scholar
  16. [ZCXG03]
    Zhou, F., Chen, G., Xu, Y., Gu, J.: Minimizing ADMs on WDM directed fiber trees. Journal of Computer Science & Technology 18(8), 725–731 (2003)MathSciNetCrossRefMATHGoogle Scholar
  17. [ZM03]
    Zhu, K., Mukherjee, B.: A review of traffic grooming in WDM optical networks: Architecture and challenges. Optical Networks Magazine 4(2), 55–64 (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Mordechai Shalom
    • 1
  • Prudence W. H. Wong
    • 2
  • Shmuel Zaks
    • 3
  1. 1.TelHai Academic CollegeUpper GalileeIsrael
  2. 2.Department of Computer ScienceThe University of LiverpoolLiverpoolUK
  3. 3.Department of Computer ScienceTechnionHaifaIsrael

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