The parallel complexity of tree embedding problems (extended abstract)

  • Arvind Gupta
  • Naomi Nishimura
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 577)

Abstract

The sequential complexity of various tree embedding problems arises in the recent work by Robertson and Seymour on graph minors; here we consider the parallel complexity of such problems. In particular, we present two CREW PRAM algorithms: an O(n4.5)-processor O(log3n) time randomized algorithm for determining whether there is a topological embedding of one tree in another and an O(n4.5)-processor O(log3n log log n) time randomized algorithm for determining whether or not a tree with a degree constraint is a minor of a general tree. These algorithms are two examples of a general technique that can be used for solving other problems on trees. One by-product of this technique is an NC reduction of tree problems to matching problems.

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References

  1. [BBGSV89]
    O. Berkman, D. Breslauer, Z. Galil, B. Schieber, and U. Vishkin, “Highly parallelizable problems,” Proceedings of the 21st Annual ACM Symposium on the Theory of Computing, pp. 309–319, 1989.Google Scholar
  2. [Bod88]
    H. Bodlaender, “NC-algorithms for graphs with bounded tree-width,” Technical Report RUU-CS-88-4, University of Utrecht, 1988.Google Scholar
  3. [Bre74]
    R. Brent, “The parallel evaluation of general arithmetic expressions,” Journal of the ACM 21, 2, pp. 201–206, 1974.Google Scholar
  4. [CGR86]
    S. Cook, A. Gupta, and V. Ramachandran, “A fast parallel algorithm for formula evaluation,” unpublished manuscript, October 1986.Google Scholar
  5. [Duc91]
    P. Duchet, “Tree Minors,” AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, 1991.Google Scholar
  6. [FL88]
    M. Fellows and M. Langston, “Nonconstructive tools for proving polynomial-time decidability,” Journal of the Association for Computing Machinery 35, 3, pp. 727–739, July 1988.Google Scholar
  7. [FW78]
    S. Fortune and J. Wyllie, “Parallelism in Random Access Machines,” Proceedings of the 10th Annual ACM Symposium on the Theory of Computing, pp. 114–118, 1978.Google Scholar
  8. [GKMS90]
    P. Gibbons, R. Karp, G. Miller, and D. Soroker, “Subtree subtree isomorphism is in random NC,” Discrete Applied Mathematics 29, pp. 35–62, 1990.Google Scholar
  9. [Gup85]
    A. Gupta, “A fast parallel algorithm for recognition of parenthesis languages,” Master's thesis, University of Toronto, 1985.Google Scholar
  10. [Kar75]
    R. Karp, “On the complexity of combinatorial problems,” Networks 5, pp. 45–68, 1975.Google Scholar
  11. [KR88]
    R. Karp and V. Ramachandran, “Parallel algorithms for shared-memory machines,” in Handbook of Theoretical Computer Science, Volume A, ed. J. Van Leeuwen, Elsevier, 1990.Google Scholar
  12. [KUW86]
    R. Karp, E. Upfal, and A. Wigderson, “Constructing a perfect matching is in random NC,” Combinatorica 6, 1, pp. 35–48, 1986.Google Scholar
  13. [KMV89]
    S. Khuller, S. Mitchell, and V. Vazirani, “Processor efficient parallel algorithms for the two disjoint paths problem, and for finding a Kuratowski homeomorph,” Proceedings of the 30th Annual IEEE Symposium on the Foundations of Computer Science, pp. 300–305, 1989.Google Scholar
  14. [Lag90]
    J. Lagergren, “Efficient parallel algorithms for tree-decompositions and related problems,” Proceedings of the 31st Annual IEEE Symposium on the Foundations of Computer Science, pp. 173–181, 1990.Google Scholar
  15. [LK89]
    A. Lingas and M. Karpinski, “Subtree isomorphism is NC reducible to bipartite perfect matching,” Information Processing Letters 30 pp. 27–32, 1989.Google Scholar
  16. [Mat78]
    D. Matula, “Subtree isomorphism in O(n5/2),” Annals of Discrete Mathematics 2, pp. 91–106, North-Holland, 1978.Google Scholar
  17. [MR85]
    G. Miller and J. Reif, “Parallel tree contraction and its application,” Proceedings of the 26th Annual IEEE Symposium on the Foundations of Computer Science, pp. 478–489, 1985.Google Scholar
  18. [MVV87]
    K. Mulmuley, U. Vazirani, and V. Vazirani, “Matching is as easy as matrix inversion,” Proceedings of the 19th Annual ACM Symposium on the Theory of Computing, pp. 345–354, 1987.Google Scholar
  19. [RSa]
    N. Robertson and P. Seymour, “Graph Minors XIII. The disjoint paths problem,” in preparation.Google Scholar
  20. [RSb]
    N. Robertson and P. Seymour, “Graph Minors XV. Wagner's conjecture,” in preparation.Google Scholar
  21. [TV85]
    R. Tarjan and U. Vishkin, “Finding biconnected components and computing tree functions in logarithmic parallel time,” SIAM Journal of Computing 14, pp. 862–874, 1985.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Arvind Gupta
    • 1
  • Naomi Nishimura
    • 2
  1. 1.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  2. 2.Department of Computer ScienceUniversity of WaterlooWaterlooCanada

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