Embedding graphs with bounded treewidth into optimal hypercubes

  • Volker Heun
  • Ernst W. Mayr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)

Abstract

In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube. This is the first time that embeddings of graphs with a very irregular structure into hypercubes are investigated. The dilation of the presented embedding is bounded by 3 ⌈log((d+1) (t+1))⌉+8, where t denotes the treewidth of the graph and d denotes the maximal degree of a vertex in the graph. Moreover, if the graph has constant treewidth or is represented by a tree-decomposition of width t, this embedding can be efficiently implemented on the optimal hypercube itself.

References

  1. 1.
    N. Alon, D. West: The Borsuk-Ulam Theorem and Bisection of Necklaces, Proc. of the Amer. Math. Soc., Vol. 98 (1986), No. 4, 623–628.Google Scholar
  2. 2.
    S. Arnborg, D. Corneil, A. Proskurowski: Complexity of Finding Embeddings in a k-tree, SIAM J. Alg. Disc. Meth., Vol. 8 (1987), No. 2, 277–284.Google Scholar
  3. 3.
    S. Bhatt, F. Chung, T. Leighton, A. Rosenberg: Efficient Embeddings of Trees in Hypercubes, SIAM J. Comput., Vol. 21 (1992), No. 1, 151–162.CrossRefGoogle Scholar
  4. 4.
    H. Bodlaender, T. Hagerup: Parallel Algorithms with Optimal Speedup for Bounded Treewidth, Utrecht University, Technical Report UU-CS-1995-25, 1995.Google Scholar
  5. 5.
    M.Y. Chan: Embedding of d-Dimensional Grids into Optimal Hypercubes, Proceedings of the 1989 ACM Symposium on Parallel Algorithms and Architectures, 52–57.Google Scholar
  6. 6.
    M.Y. Chan: Embedding of Grids into Optimal Hypercubes, SIAM J. Comput., Vol. 20 (1991), No. 5, 834–864.CrossRefGoogle Scholar
  7. 7.
    M.Y. Chan, F. Chin, C.N. Chu, W.K. Mak: Dilation-5 Embedding of 3-Dimensional Grids into Hypercubes Proceedings of the 5th IEEE Symposium on Parallel and Distributed Processing 1993, 285–289.Google Scholar
  8. 8.
    R. Cypher, G. Plaxton: Deterministic Sorting in Nearly Logarithmic Time on the Hypercube and Related Computers, Proceedings of the 22nd Annual ACM Symposium on Theory of Computing 1990, 193–203.Google Scholar
  9. 9.
    K. Efe: Embedding Mesh of Trees in the Hypercube, J. Parallel and Ditsrib. Comput., Vol. 11 (1991), 222–230.Google Scholar
  10. 10.
    T. Feder, E. Mayr: An Efficient Algorithm for Embedding Complete Binary Trees in the Hypercube, Stanford University, 1987.Google Scholar
  11. 11.
    C. Goldberg, D. West: Bisection of Circle Colorings, SIAM J. Alg. Disc. Meth., 6 (1985), 93–106.Google Scholar
  12. 12.
    I. Havel: On Hamiltonian Circuits and Spanning Trees of Hypercubes, Časopis. Pěst. Mat., Vol. 109 (1984), 145–152 [in Czech.].Google Scholar
  13. 13.
    I. Havel, P. Liebl: Embedding the Polytomic Tree into the n-Cube, Časopis. Pěst. Mat., Vol. 98 (1973), 307–314.Google Scholar
  14. 14.
    V. Heun, E.W. Mayr: A New Efficient Algorithm for Embedding an Arbitrary Binary Tree into Its Optimal Hypercube, Technical Report, TUM-I9321, Technische Universität München, 1993 (to appear in J. Algorithms).Google Scholar
  15. 15.
    T. Kloks: Treewidth: Computations and Approximations, Lecture Notes in Computer Science, Vol. 842, Springer-Verlag, 1994.Google Scholar
  16. 16.
    T. Leighton, M. Newman, A. Ranade, W. Schwabe: Dynamic Tree Embeddings in Butterflies and Hypercubes, SIAM J. Comput., Vol. 21 (1992), No. 4, 639–654.CrossRefGoogle Scholar
  17. 17.
    B. Monien, H. Sudborough: Simulating Binary Trees on Hypercubes, VLSI Algorithms and Architectures, Proceedings of the 3rd Aegean Workshop on Computing 1988, Lecture Notes in Computer Science, Vol. 319, Springer-Verlag, 170–180.Google Scholar
  18. 18.
    D. Nassimi, S. Sahni: Parallel Permutation and Sorting Algorithms and a New Generalized Connection Network, J. ACM, Vol. 29 (1982), No. 3, 642–667.CrossRefGoogle Scholar
  19. 19.
    Y. Saad, M. Schulz: Topological Properties of the Hypercube, Yale University Research Report, RR-389, 1985.Google Scholar
  20. 20.
    X. Sheen, Q. Hu, W. Liang: Embedding k-ary Complete Trees into Hypercubes, J. Parallel Distrib. Comput., Vol. 24 (1995), 100–106.CrossRefGoogle Scholar
  21. 21.
    Q. Stout: Hypercubes and Pyramids, Proceedings of the NATO Advanced Research Workshop on Pyramidal Systems for Computer Vision 1986, Springer-verlag, Series F: Computers and System Science, 75–89.Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Volker Heun
    • 1
  • Ernst W. Mayr
    • 1
  1. 1.Institut für Informatik der TechnischenUniversität MünchenMünchenGermany

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