# The parallel complexity of tree embedding problems (extended abstract)

## 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*(*n*^{4.5})-processor *O*(log^{3}*n*) time randomized algorithm for determining whether there is a topological embedding of one tree in another and an *O*(*n*^{4.5})-processor *O*(log^{3}*n* 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.

## Keywords

Parallel Algorithm Disjoint Path Parallel Complexity Annual IEEE Symposium Embedding Problem## Preview

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## References

- [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 - [Bod88]H. Bodlaender, “NC-algorithms for graphs with bounded tree-width,” Technical Report RUU-CS-88-4, University of Utrecht, 1988.Google Scholar
- [Bre74]R. Brent, “The parallel evaluation of general arithmetic expressions,”
*Journal of the ACM***21**, 2, pp. 201–206, 1974.Google Scholar - [CGR86]S. Cook, A. Gupta, and V. Ramachandran, “A fast parallel algorithm for formula evaluation,” unpublished manuscript, October 1986.Google Scholar
- [Duc91]P. Duchet, “Tree Minors,” AMS-IMS-SIAM Joint Summer Research Conference on Graph Minors, 1991.Google Scholar
- [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 - [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 - [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 - [Gup85]A. Gupta, “A fast parallel algorithm for recognition of parenthesis languages,” Master's thesis, University of Toronto, 1985.Google Scholar
- [Kar75]
- [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 - [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 - [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 - [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 - [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 - [Mat78]D. Matula, “Subtree isomorphism in
*O(n5/2)*,”*Annals of Discrete Mathematics***2**, pp. 91–106, North-Holland, 1978.Google Scholar - [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 - [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 - [RSa]N. Robertson and P. Seymour, “Graph Minors XIII. The disjoint paths problem,” in preparation.Google Scholar
- [RSb]N. Robertson and P. Seymour, “Graph Minors XV. Wagner's conjecture,” in preparation.Google Scholar
- [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