A New Measure for the Bandwidth Minimization Problem

  • Jose Torres-Jimenez
  • Eduardo Rodriguez-Tello
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1952)

Abstract

The Bandwidth Minimization Problem for Graphs (BMPG) can be defined as finding a labeling for the vertices of a graph, where the maximum absolute difference between labels of each pair of connected vertices is minimum. The most used measure for the BMPG algorithms isβ, that indicates only the maximum of all absolute differences.

After analyzing some drawbacks of β, a measure, calledγ, which uses a positional numerical system with variable base and takes into account all the absolute differences of a graph is given.

In order to test the performance of γ and β a stochastic search procedure based on a Simulated Annealing (SA) algorithm has been applied to solve the BMPG. The experiments show that the SA that uses γ has better results for many classes of graphs than the one that uses β.

Keywords

Bandwidth Metric Bandwidth Minimization Problem Graphs Simulated Annealing 

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References

  1. 1.
    Kosko, E.: Matrix Inversion by Partitioning. The Aeronautical Quart. 8 (1956) 157MathSciNetGoogle Scholar
  2. 2.
    Livesley, R.R.: The Analysis of Large Structural Systems. Computer J. 3 (1960)34MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chinn, P.Z., Chvátalová, J., Dewdney, A.K., Gibbs, N.E.: The Bandwidth Problem for Graphs and Matrices-A Survey. J. of Graph Theory 6 (1982) 223–254MATHCrossRefGoogle Scholar
  4. 4.
    Harper, L.H.: Optimal assignment of numbers to vertices. J. of SIAM 12 (1964) 131–135MATHMathSciNetGoogle Scholar
  5. 5.
    Harary, F.: Theory of graphs and its aplications. Czechoslovak Academy of Science, Prague (1967) 161Google Scholar
  6. 6.
    Torres-Jiménez, J.: Minimización del Ancho de Banda de un Grafo Usando un Algoritmo Genético. Ph. D. ITESM-Mor., México (1997)Google Scholar
  7. 7.
    Papadimitriou, C.H.: The NP-Completeness of the bandwidth minimization problem. J. of Comp. 16 (1976) 263–270MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM J. of App. Math. 34 (1978) 477–495MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gurari, E.M., Sudborough, I.H.: Improved dynamic programming algorithms for bandwidth minimization and the min-cut linear arrangement problem. J. of Algorithms 5 (1984) 531–546MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. Proceedings 24th National of the ACM (1969) 157–172Google Scholar
  11. 11.
    Gibbs, N.E., et. al.: An algorithm for reducing the bandwidth and profile of a sparse matrix. SIAM J. on Numerical Analysis 13 (1976) 235–251MathSciNetCrossRefGoogle Scholar
  12. 12.
    George, A., Liu, W.: Computer Solution of Large Sparse Positive Definite Systems. Prentice Hall, Englewood Cliffs, NJ (1981)Google Scholar
  13. 13.
    Haralambides, J., Makedon, F., Monien, B.: An aproximation algorithm for cater-pillars. J. of Math. Systems Theory 24 (1991) 169–177MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dueck, G., Jeffs, J.: A Heuristic Bandwidth Reduction Algorithm. J. of Comb. Math. and Comp. 18 (1995)Google Scholar
  15. 15.
    Esposito, A., Fiorenzo, S., Malucelli, F., Tarricone, L.: A wonderful bandwidth matrix reduction algorithm. Submitted to Operations Research LettersGoogle Scholar
  16. 16.
    Kirkpatrick, S., Gelatt, C.D., Vecchi, M.P.: Optimization by Simulated Annealing. Science 220 (1983) 671–680CrossRefMathSciNetGoogle Scholar
  17. 17.
    Spears, W.M.: Simulated Annealing for hard satisfiability problems. AI Center, Naval Research Laboratory, Washington, DC 20375 AIC-93-015 (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Jose Torres-Jimenez
    • 1
  • Eduardo Rodriguez-Tello
    • 1
  1. 1.ITESM, Campus MorelosLomas de CuernavacaMEXICO

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