Approximating circular arc colouring and bandwidth allocation in all-optical ring networks

  • Vijay Kumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1444)


We present randomized approximation algorithms for the circular arc graph colouring problem and for the problem of bandwidth allocation in all-optical ring networks. We obtain a factor-of-(1+1/e+o(1)) randomized approximation algorithm for the arc colouring problem, an improvement over the best previously known performance ratio of 5/3. For the problem of allocating bandwidth in an all-optical WDM (wavelength division multiplexing) ring network, we present a factor-of-(1.5+1/2e+o(1)) randomized approximation algorithm, improving upon the best previously known performance ratio of 2.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Azuma, K. Weighted Sum of Certain Dependent Random Variables. Tohoku Mathematical Journal, 19:357–367, 1967.MATHMathSciNetGoogle Scholar
  2. 2.
    Cosares, S., Carpenter, T., and Saniee, I. Static Routing and Slotting of Demand in SONET Rings. Presented at the TIMS/ORSA Joint National Meeting, Boston, MA, 1994.Google Scholar
  3. 3.
    Erlebach, T. and Jansen, K. Call Scheduling in Trees, Rings and Meshes. In Proc. 30th Hawaii International Conf. on System Sciences, 1997.Google Scholar
  4. 4.
    Even, S., Itai, A., and Shamir, A. On the complexity of timetable and multicommodity flow problems, SIAM Journal of Computing, 5(1976),691–703.MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S., Miller, G.L. and Papadimitriou, C.H. The Complexity of Coloring Circular Arcs and Chords. SIAM J. Alg. Disc. Meth., 1(2):216–227, 1980.MATHMathSciNetGoogle Scholar
  6. 6.
    Golumbic, M. C, Algorithmic Graph Theory and Perfect Graphs, Academic Press, 1980.Google Scholar
  7. 7.
    Kaklamanis, C. and Persiano, P. Efficient wavelength routing on directed fiber trees. In Proc. 4th Annual European Symposium on Algorithms, 1996.Google Scholar
  8. 8.
    Kumar, V. and Schwabe, E.J. Improved access to optical bandwidth in trees. In Proc 8th Annual ACM-SIAM Symp. on Discrete Algorithms, pp. 437–444, 1997.Google Scholar
  9. 9.
    McDiarmid, C. On the method of bounded differences. In J. Siemons, editor, Surveys in Combinatorics, volume 141 of LMS Lecture Notes Series, pages 148–188. 1989.Google Scholar
  10. 10.
    Mihail, M., Kaklamanis, C, and Rao, S. Efficient Access to Optical Bandwidth. In Proc IEEE Symp on Foundations of Comp Sci, pp. 548–557, 1995.Google Scholar
  11. 11.
    Motwani, R., and Raghavan, P. Randomized Algorithms, Cambridge University Press, 1995.Google Scholar
  12. 12.
    Orlin, J.B., Bonuccelli, M.A., and Bovet, D.P. An O(n 2) Algorithm for Coloring Proper Circular Arc Graphs. SIAM J. Alg. Disc. Meth., 2(2):88–93, 1981.MATHMathSciNetGoogle Scholar
  13. 13.
    Raghavan, P. Randomized Rounding and Discrete Ham-Sandwiches: Provably Good Algorithms for Routing and Packing Problems. PhD Thesis, CS Division, UC Berkeley, 1986.Google Scholar
  14. 14.
    Raghavan, P., and Upfal, E. Efficient Routing in All-Optical Networks. In Proc 26th ACM Symp on Theory of Computing, pp. 134–143, 1994.Google Scholar
  15. 15.
    Shih, W.K. and Hsu, W.L. An Approximation Algorithm for Coloring Circular-Arc Graphs. SIAM Conference on Discrete Mathematics, 1990.Google Scholar
  16. 16.
    Settembre, M. and Matera, F. All-optical implementations of high capacity TDMA networks. Fiber and Integrated Optics, 12:173–186, 1993.Google Scholar
  17. 17.
    Tucker, A. Coloring a Family of Circular Arcs. SIAM J. Appl. Math., 29(3):493–502, 1975.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Vijay Kumar
    • 1
  1. 1.Department of ECENorthwestern UniversityEvanstonUSA

Personalised recommendations