On the Computational Complexity of Optimal Sorting Network Verification

  • Ian Parberry
Part of the Lecture Notes in Computer Science book series (LNCS, volume 505)


A sorting network is a combinational circuit for sorting, constructed from comparison-swap units. The depth of such a circuit is a measure of its running time. It is reasonable to hypothesize that only the fastest (that is, the shallowest) networks are likely to be fabricated. It is shown that the problem of verifying that a given sorting network actually sorts is Co-NP complete even for sorting networks of depth only 4 [log n] + O(1) greater than optimal. This is shallower than previous depth bounds by a factor of two.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Ajtai, J. Komlós, and E. Szemerédi. An O(n. log n) sorting network. Proc. 15th Ann. ACM Symp. on Theory of Computing, pages 1–9, April 1983.Google Scholar
  2. [2]
    M. Ajtai, J. Komlós, and E. Szemerédi. Sorting in clog n parallel steps. Combinatorica, 3: 1–48, 1983.MathSciNetCrossRefGoogle Scholar
  3. [3]
    K. E. Batcher. Sorting networks and their applications. In Proc. AFIPS Spring Joint Computer Conference, volume 32, pages 307–314, April 1968.Google Scholar
  4. [4]
    R. C. Bose and R. J. Nelson. A sorting problem. J. Assoc. Comput. Mach., 9: 282–296, 1962.MathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Chung and B. Ravikumar. On the size of test sets for sorting and related problems. In Proc. 1987 International Conference on Parallel Processing. Penn State Press, August 1987.Google Scholar
  6. [6]
    M. J. Chung and B. Ravikumar. Strong nondeterministic Turing reduction — a technique for proving intractability. J. Comput. System Sci., 39 (1): 2–20, 1989.MathSciNetCrossRefGoogle Scholar
  7. [7]
    R. L. Drysdale. Sorting networks which generalize batcher’s odd-even merge. Honors Paper, Knox College, May 1973.Google Scholar
  8. [8]
    R. W. Floyd and D. E. Knuth. Improved constructions for the Bose-Nelson sorting problem (preliminary report). Notices of the AMS, 14: 283, 1967.Google Scholar
  9. [9]
    R. W. Floyd and D. E. Knuth. The Bose-Nelson sorting problem. In J. N. Srivastava, editor, A Survey of Combinatorial Theory. North-Holland, 1973.CrossRefGoogle Scholar
  10. [10]
    M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979.Google Scholar
  11. [11]
    D. E. Knuth. Sorting and Searching, volume 3 of The Art of Computer Programming. Addison-Wesley, 1973.Google Scholar
  12. [12]
    I. Parberry. The alternating sorting network. Technical Report CS-87–26, Dept. of Computer Science, Penn. State Univ., September 1987.Google Scholar
  13. [13]
    I. Parberry. Parallel Complexity Theory. Research Notes in Theoretical Computer Science. Pitman Publishing, London, 1987.MATHGoogle Scholar
  14. [14]
    I. Parberry. Sorting networks. Technical Report CS-88–08, Dept. of Computer Science, Penn. State Univ., March 1988.Google Scholar
  15. [15]
    I. Parberry. A computer-assisted optimal depth lower bound for sorting networks with nine inputs. In Proceedings of Supercomputing ‘89, pages 152–161, 1989.Google Scholar
  16. [16]
    I. Parberry. Single-exception sorting networks and the computational complexity of optimal sorting network verification. Mathematical Systems Theory, 23: 81–93, 1990.MathSciNetCrossRefGoogle Scholar
  17. [17]
    M. S. Paterson. Improved sorting networks with O(log n) depth. Algorithmica, 5 (4): 75–92, 1990.MathSciNetCrossRefGoogle Scholar
  18. [18]
    T. J. Schaefer. The complexity of satisfiability problems. In Proc. 10th Annual ACM Symposium on Theory of Computing, pages 216–226. Association for Computing Machinery, 1978.Google Scholar
  19. [19]
    D. C. Van Voorhis. An economical construction for sorting networks. In Proc. AFIPS National Computer Conference, volume 43, pages 921–926, 1974.Google Scholar
  20. [20]
    A. Yao. Bounds on selection networks. SIAM Journal on Computing, 9, 1980.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Ian Parberry
    • 1
  1. 1.Department of Computer ScienceThe Pennsylvania State UniversityUSA

Personalised recommendations