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Topological bandwidth

  • F. S. Makedon
  • C. H. Papadimitriou
  • I. H. Sudborough
Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 159)

Abstract

Let L be a one-to-one function mapping the vertices of an undirected graph G to the positive integers. (L is called a "linear layout" of G.) The "bandwidth of G under the layout L", denoted by b (G,L), is the maximum difference between the integers assigned to vertices in G connected by an edge. The "bandwidth of G", denoted b(G), is the min { b(G,L) | L is a linear layout of G }. The "topological bandwidth of G", denoted by tb(G), is the minimum bandwidth of all graphs obtained from G by subdividing some of G's edges with some number (possibly zero) of degree two vertices.

The "modified cutwidth of G under a layout L", denoted by mcw(G,L), is the maximum, over all integers i, of the number of edges connecting vertices that are assigned to integers smaller than i with vertices assigned to integers larger than i. The "modified cutwidth of G", denoted by mcw(G), is min{ mcw(G,L) | L is a linear layout of G }.

It is shown that, for all graphs G, tb(G)≤mcw(G)+1 and that, for all degree 3 graphs G, tb(G)=mcw(G)+1. This yields an 0(n log n) algorithm for determining the topological bandwidth of an arbitrary degree three tree, using the algorithm in [23] for modified cutwidth. Topological bandwidth is also equated with the "node search number" for degree three graphs.

We give a recursive characterization theorem for topological bandwidth in degree three trees. This yields a description of the set of smallest degree three trees having topological bandwidth k, for all k≥1, and a characterization, by forbidden subtrees, of topological bandwidth k in degree three trees, for all k≥1.

We show that the Topological Bandwidth problem, the Modified Cutwidth problem, the Min Cut Linear Arrangement problem, the Search Number problem, and the Node Search Number problem are all NP-complete even when restricted to graphs that have maximum vertex degree three.

Finally, we give a characterization of graphs with topological bandwidth 2 and thereby derive a linear time algorithm to determine if a graph G satisfies tb(G)=2.

Keywords

Positive Instance Linear Time Algorithm Connection Point Search Number Biconnected Component 
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.

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References

  1. [1]
    I. Arany, L. Szoda, and W.F. Smith, "An Improved Method for Reducing the Bandwidth of Sparse Symetric Matrices", Proc. 1971 IFIP Congress, pp. 1246–1250.Google Scholar
  2. [2]
    F.R.K. Chung, "On the Cutwidth and the Topological Bandwidth of a Tree", Technical Report, Bell Laboratories, Murray Hill, New Jersey, U.S.A. (1982)Google Scholar
  3. [3]
    M.-J. Chung, F. S. Makedon, I. H. Sudborough, and J. Turner, "Polynomial Algorithms for the Min Cut Linear Arrangement Problem on Degree Restricted Trees", Proc. 23rd Annual IEEE Foundations of Computer Science Symp. (1982)Google Scholar
  4. [4]
    J. Chvatalova, A.K. Dewdney, N. E. Gibbs, and R. R. Khorfage, "The Bandwidth Problem for Graphs: A Collection of Recent Results, Research Report 24, Dept. of Computer Science, University of Western Ontario, London, Ontario, Canada.Google Scholar
  5. [5]
    M. Garey, R. L. Graham, D. S. Johnson, and D. Knuth, "Complexity Results for Bandwidth Minimization", SIAM J. on Applied Mathematics 34 (1978), pp. 477–495.CrossRefGoogle Scholar
  6. [6]
    M. Garey and D. S. Johnson, Computers and Intractability, A Guide to the Theory of NP-comleteness, W. H. Freeman and Co., San Francisco (1979).Google Scholar
  7. [7]
    F. Gavril, "Some NP-complete Problems on Graphs", Proc. 11th Annual Conf. on Info. Sciences and Systems, The Johns Hopkins University, Baltimore, Md., U.S.A. (1977), pp. 91–95.Google Scholar
  8. [8]
    E. M. Gurari and I. H. Sudborough, "Improved Dynamic Programming Algorithms for the Bandwidth Minimization Problem and the Min Cut Linear Arrangement problem", Technical Report, Dept. of Electrical Engineering and Computer Science, Northwestern University, Evanston, IL 60201 U. S. A. (1981).Google Scholar
  9. [9]
    A. LaPaugh, "Recontamination Does Not Help", Technical Report, Princeton University, Princeton, New Jersey, U. S. A. (1982).Google Scholar
  10. [10]
    T. Lengauer, "Upper and Lower Bounds on the Complexity of the Min Cut Linear Arrangement Problem on Trees", Technical Report TM-80-1272-9, Bell Laboratories, Murray Hill, New Jersey, U. S. A. (1980).Google Scholar
  11. [11]
    T. Lengauer, "Black-White Pebbles and Graph Separation", Technical Report, Bell Laboratories, Murray Hill, New Jersey, U.S.A. (1980).Google Scholar
  12. [12]
    W. Loui and A.B. Sherman, "Comparative Analysis of the Cuthill-McKee Ordering Algorithms for Sparse Matrices", SIAM J. Numerical Analysis (1976).Google Scholar
  13. [13]
    F. Makedon, Layout Problems and Their Complexity, Ph. D. Thesis, Electrical Engineering and Computer Science Dept., Northwestern University, Evanston, IL U.S.A. (1982).Google Scholar
  14. [14]
    N. Megiddo, L. Hakimi, M.R. Garey, D. S. Johnson, and C. H. Papadimitriou, "The Complexity of Searching a Graph", Proc. 22nd Annual IEEE Foundations of Computer Science Symp. (1981), pp. 376–385.Google Scholar
  15. [15]
    B. Monien and I. H. Sudborough, "On Eliminating Nondeterminism from Turing Machines that Use Less than Logarithm Space", to appear in Theoretical Computer Science.Google Scholar
  16. [16]
    B. Monien and I. H. Sudborough, "Bandwidth Constrained NP-Complete Problems", Proc. 11th Annual ACM Theory of Computer Science SYmp. (1981), pp. 207–217.Google Scholar
  17. [17]
    C. H. Papadimitriou and L. Kyrousis, work in progress.Google Scholar
  18. [18]
    C. H. Papadimitriou, "The NP-Completeness of the Bandwidth Minimization Problem", Computing 16 (1976), pp. 237–267.Google Scholar
  19. [19]
    A. L. Rosenberg and I. H. Sudborough, "Bandwidth and Pebbling", to appear in Computing.Google Scholar
  20. [20]
    J. B. Saxe, "Dynamic Programming Algorithms for Recognizing Small Bandwidth Graphs in Polynomial Time", SIAM J. Algebraic and Discrete Methods (1980).Google Scholar
  21. [21]
    L. J. Stockmeyer, private communication to M. R. Garey and D. S. Johnson, cited in Computers and Intractability, A Guide to the Theory of NP-Completeness, see [6], p. 201.Google Scholar
  22. [22]
    I. H. Sudborough, "Bandwidth Constraints on Problems Complete for Polynomial Time", to appear in Theoretical Computer Science.Google Scholar
  23. [23]
    I. H. Sudborough and J. Turner, "On Computing the Width and Black/White Pebble Demand of Trees", work in progress.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • F. S. Makedon
    • 1
  • C. H. Papadimitriou
    • 2
  • I. H. Sudborough
    • 3
  1. 1.Dept. of Computer Sci.Illinois Institute of Tech.ChicagoUSA
  2. 2.Dept. of Computer Sci.National Tech. UniversityAthensGreece
  3. 3.E.E./C.S. Dept. Tech. InstituteNorthwestern Univ.EvanstonUSA

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