Mathematical systems theory

, Volume 24, Issue 1, pp 169–177

Bandwidth Minimization: An approximation algorithm for caterpillars

  • J. Haralambides
  • F. Makedon
  • B. Monien


The Bandwidth Minimization Problem (BMP) is the problem, given a graphG and an integerk, to map the vertices ofG to distinct positive integers, so that no edge ofG has its endpoints mapped to integers that differ by more thank. There is no known approximation algorithm for this problem, even for the case of trees. We present an approximation algorithm for the BMP for the case of special graphs, called caterpillars. The BMP arises from many different engineering applications which try to achieve efficient storage and processing and has been studied extensively, especially with relation to other graph layout problems. In particular, the BMP for caterpillars is related to multiprocessor scheduling. It has been shown to be NP-complete, even for degree-3 trees. Our algorithm, gives a logn times optimal algorithm, wheren is the number of nodes of the caterpillar. It is based on the idea of level algorithms.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    C. H. Papadimitriou, The NP-completeness of the bandwidth minimization problem,Computing,16 (1976), 263–270.Google Scholar
  2. [2]
    M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to the Theory of NP-completeness, Freeman, San Francisco, 1979.Google Scholar
  3. [3]
    M. R. Garey, R. L. Graham, D. S. Johnson, and D. E. Knuth, Complexity results for bandwidth minimization,SIAM J. Appl. Math.,34 (1978), 477–495.Google Scholar
  4. [4]
    J. B. Saxe, Dynamic programming algorithms for recognizing small bandwidth graphs in polynomial time,SIAM J. Algebraic Discrete Methods (1980).Google Scholar
  5. [5]
    E. M. Gurari and I. H. Sudborough, Improved dynamic programming algorithms for bandwidth minimization and the min-cut linear arrangement problems,J. Algorithms,5 (1984), 531–546.Google Scholar
  6. [6]
    B. Monien, The Bandwidth-Minimization Problem for Caterpillars with Hairlength 3 is NP Complete,SIAM J. Algebraic Discrete Methods,7 (1986), 505–512.Google Scholar
  7. [7]
    S. F. Assman, G. W. Peck, M. M. Syslo, and J. Zak, The bandwidth of caterpillars with hairs of length 1 and 2,SIAM J. Algebraic Discrete Methods (1981), 387–391.Google Scholar
  8. [8]
    J. Turner, Bandwidth and Probabilistic Complexity, Thesis, Northwestern University, Evanston, IL.Google Scholar
  9. [9]
    E. Cuthill and J. McKee, Reducing the bandwidth of sparse symmetric matrices,ACM National Conference Proc.,24 (1969), 137–172.Google Scholar
  10. [10]
    K. Y. Cheng, Minimizing the bandwidth of sparse symmetric matrices,Computing,11 (1973), 103–110.Google Scholar
  11. [11]
    P. Z. Chinn, J. Chvatalova, A. K. Dewdney, and N. E. Gibbs, The bandwidth problem for graphs and matrices—a survey,J. Graph Theory,6 (1982), 223–254.Google Scholar
  12. [12]
    I. Arany, L. Szoda, and W. F. Smith, An improved method for reducing the bandwidth of sparse symmetric matrices,Proc. 1971 IFIP Congress, pp. 1246–1250.Google Scholar
  13. [13]
    F. R. K. Chung, Some problems and results in labeling of graphs, inThe Theory and Applications of Graphs (G. Chartrand, ed.), Wiley, New York, 1981, pp. 255–263.Google Scholar
  14. [14]
    W. Lin and A. B. Sherman, Comparative analysis of the Cuthill-McKee ordering algorithms for sparse matrices,SIAM J. Numer. Anal.,13 (1976), 198–213.Google Scholar
  15. [15]
    I. H. Sudborough, Bandwidth constraints on problems complete for polynomial time,Theoret. Comput. Sci.,26 (1983), 25–52.Google Scholar
  16. [16]
    F. S. Makedon, Graph Layout Problems and Their Complexity, Ph.D. Thesis, August 1982, Electrical Engineering and Computer Science Department, Northwestern University, Evanston, IL.Google Scholar
  17. [17]
    F. S. Makedon and I. H. Sudborough, Graph layout problems, inSurveys in Computer Science, (H. Maurer, ed.), Bibliographisches Institut, Zurich, 1984, pp. 145–192.Google Scholar
  18. [18]
    M. M. Syslo and J. Zak, The Bandwidth Problem for Ordered Caterpillars, Report CS-80-065, Computer Science Department, Washington State University, 1980.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1991

Authors and Affiliations

  • J. Haralambides
    • 1
  • F. Makedon
    • 1
  • B. Monien
    • 2
  1. 1.Computer Science ProgramThe University of Texas at DallasRichardsonUSA
  2. 2.Department of Mathematics and Computer ScienceUniversity of Paderborn, Warburger StrassePaderbornFederal Republic of Germany

Personalised recommendations