Improved processor bounds for combinatorial problems in RNC


Our main result improves the known processor bound by a factor ofn 4 (maintaining the expected parallel running time,O(log3 n)) for the following important problem:find a perfect matching in a general or in a bipartite graph with n vertices. A solution to that problem is used in parallel algorithms for several combinatorial problems, in particular for the problems of finding i) a (perfect) matching of maximum weight, ii) a maximum cardinality matching, iii) a matching of maximum vertex weight, iv) a maximums-t flow in a digraph with unit edge capacities. Consequently the known algorithms for those problems are substantially improved.

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  1. [1]

    S. Berkowitz, On Computing the Determinant in Small Parallel Time Using a Small Number of Processors,Information Processing Letters.18 (1984), 147–150.

    Google Scholar 

  2. [2]

    A. Borodin, S. Cook andN. Pippenger, Parallel Computation for Well-Endowed Rings and and Space-Bounded Probabilistic Machines,Inform, and Control,58 (1983), 113–136.

    Google Scholar 

  3. [3]

    A. Borodin, J. von zur Gathen andJ. Hopcroft, Fast Parallel Matrix and GCD Computations,Inform. and Control,53 (1982), 241–256.

    Google Scholar 

  4. [4]

    A.Broder,private communication, 1985.

  5. [5]

    D.Coppersmith and S.Winograd, Matrix Multiplication via Arithmetic Progressions,Proc. 19th Ann. ACM Symp. on Theory of Computing, 1987, 1–6.

  6. [6]

    H.Gabow,private communication, 1985.

  7. [7]

    R. M.Karp, E.Uffal and A.Wigderson, Constructing a Perfect Matching Is in Random NC,Proc. 17-th Ann. ACM Symp. on Theory of Computing, (1985), 22–32.

  8. [8]

    M. Marcus andH. Minc,A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Boston, 1964.

    Google Scholar 

  9. [9]

    K. Mulmuley, U. Vazirani andV. Vazirani, Matching Is As Easy As Matrix Inversion,Combinatorica,7 (1987), 105–114.

    Google Scholar 

  10. [10]

    V. Pan, Fast and Efficient Parallel Algorithms for Exact Inversion of Integer Matrices,Proc. Fifth Conf. on Software Technology and Theoretical Computer Science, Computer Science,206 (1985), 504–521; Complexity of Parallel Matrix Computations,Theoretical Computer Science,54 (1987) (to appear).

    Google Scholar 

  11. [11]

    F. P. Preparata andD. V. Sarwate, An Improved Parallel Processor Bound in Fast Matrix Inversion,Inform. Proc. Letters,7 (1978), 148–149.

    Google Scholar 

  12. [12]

    M. O.Rabin and V.Vazirani, Maximum Matchings in Graphs through RandomizationTech. Report TR-I5-84, Center for Research in Computer Technology, Aiken Computation Laboratory, Harvard University, 1984.

  13. [13]

    J. E. Savage,The Complexity of Computing, John Wiley and Sons, N.Y., 1976.

    Google Scholar 

  14. [14]

    J. T. Schwartz, Fast Probabilistic Algorithms for Verification of Polynomial Identities,J. of ACM,27 (1980), 701–717.

    Google Scholar 

  15. [15]

    D. Y. Y.Yun, Algebraic Algorithms Usingp-adic Constructions,Proc. 1976 ACM Symp, Symbolic and Algebraic Computation, (1976), 248–259.

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Partially supported by NSF Grants MCS 8303139 and DCR 8511713.

Supporeted by NSF Grants MCS 8203232 and DCR 8507573.

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Galil, Z., Pan, V. Improved processor bounds for combinatorial problems in RNC. Combinatorica 8, 189–200 (1988).

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AMS subject classification(1980)

  • 05 C 68