Skip to main content
SpringerLink
Log in

Verifying the determinant in parallel

  • Conference paper
  • First Online:
Algorithms and Computation (ISAAC 1994)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 834))

Included in the following conference series:

Abstract

In this paper we investigate both in the Boolean arithmetic circuit and the Boolean circuit model the complexity of the verification of problems whose computation is equivalent to the determinant. We observe that for a few problems there exist an easy (NC 1) verification algorithm. To characterize the harder ones, we define under two different reductions the class of problems which are reducible to the verification of the determinant and establish a list of complete problems in these classes. In particular we prove that computing the rank is equivalent under AC 0 reduction to verifying the determinant. We show in the Boolean case that none of the complete problems can be recognized in NC 1 unless L=NL. On the other hand we show that even for problems which are hard to verify there exists an NC 1 checker and that they can be extended into problems whose verification is easy.

This research was supported by the ESPRIT Working Group 7097 RAND.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. E. Allender and M. Ogiwara. Relationships Among PL, #L, and the Determinant. to appear in the Proc. 9th IEEE Structure in Complexity Theory, 1994.

    Google Scholar 

  2. M. Ben-Or and R. Cleeve. Computing Algebraic Formulas using a Constant Number of Registers. In Proc. 20th ACM STOC, pp. 254–257, 1988.

    Google Scholar 

  3. S. J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Information Processing Letter, 18:147–150, March 1984.

    Article  Google Scholar 

  4. M. Blum and S. Kannan. Designing Programs That Check Their Work. In Proc. 21st ACM STOC, pp. 249–261, 1979.

    Google Scholar 

  5. A. Borodin, S. Cook, and N. Pippenger. Parallel computation for well-endowed ring and space-Bounded probabilistic machines. Information and Control, 48:113–136, 1983.

    Article  Google Scholar 

  6. A. Borodin, J. von zur Gathen, and J. Hopcroft. Fast parallel matrix and GCD computation. Information and Control, 52:241–256, 1982.

    Article  Google Scholar 

  7. A.L Chistov. Fast parallel calculation of the rank of matrices over a field of arbitrary characteristic. In Proc. Int. Conf. Found. of Comp. Theory, Springer LNCS, vol. 199, pp. 63–69, 1985.

    Google Scholar 

  8. S. A. Cook. A taxonomy of problems with fast parallel algorithms. Inf. and Cont., 64:2–22, 1985.

    Google Scholar 

  9. L. Csanky. Fast Parallel Matrix Inversion Algorithms. SIAM J. COMP., 5(4),pp. 618–623, 1976.

    Google Scholar 

  10. C. Damm. DET=L #L. Informatik-Preprint, Fachbereich Informatik der Humboldt-Universität zu Berlin, 8, 1991.

    Google Scholar 

  11. S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. In Proc. 6th IEEE Structure in Complexity Theory, pp. 30–42, 1991.

    Google Scholar 

  12. J. von zur Gathen. Parallel Linear Algebra. In J. H. Reif, editor, Synthesis of Parallel Algorithms, pp. 574–615. Morgan Kaufmann Publishers, 1993.

    Google Scholar 

  13. O.H. Ibarra, S. Moran, and L.E. Rosier. A note on the parallel complexity of computing the rank of order n matrices. Inf. Proc. Letter, 11:162, 1980.

    Google Scholar 

  14. Sampath Kannan. Program Checkers for algebraic problems. PhD thesis, International Computer Science Institute, Berkeley, CA, 1989.

    Google Scholar 

  15. K. Mulmuley. A fast parallel algorithm to compute the rank of a matrix over an arbitrary field. Combinatorica, 7:101–104, 1987.

    Google Scholar 

  16. S. Toda. Counting Problems Computationally Equivalent to Computing the Determinant. Technical report, CSIM 91-07, 1991.

    Google Scholar 

  17. J. Torán. Complexity classes defined by counting quantifiers. Journal of the Association for Computing Machinery, 38:753–744, 1991.

    Google Scholar 

  18. L.G. Valiant. Completeness classes in algebra. In Proc. 11th ACM STOC, pp. 249–261, 1979.

    Google Scholar 

  19. L.G. Valiant. Why is Boolean complexity theory difficult? In M.S. Paterson ed., Boolean Function Complexity. Cambridge Univ. Press, 1992.

    Google Scholar 

  20. V. Vinay. Counting auxialiary pushdown automata and semi-unbounded arithmetic circuits. In Proc. 6th IEEE Structure in Complexity Theory, pp. 270–284, 1991.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CNRS, URA 410 Université Paris-Sud, 91405, Orsay, France

    Miklos Santha

  2. Ecole Nationale Supérieure de Techniques Avancées, 32 bd Victor, 75015, Paris, France

    Sovanna Tan

Authors
  1. Miklos Santha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sovanna Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Ding-Zhu Du Xiang-Sun Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Santha, M., Tan, S. (1994). Verifying the determinant in parallel. In: Du, DZ., Zhang, XS. (eds) Algorithms and Computation. ISAAC 1994. Lecture Notes in Computer Science, vol 834. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58325-4_167

Download citation

  • DOI: https://doi.org/10.1007/3-540-58325-4_167

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-58325-7

  • Online ISBN: 978-3-540-48653-4

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation