Skip to main content
SpringerLink
Log in

Polynomial-time compression

  • Published:
computational complexity Aims and scope Submit manuscript

Abstract

This paper studies the class ofinfinite sets that have minimal perfect hash functions—one-to-one onto maps between the sets and ∑*-computable in polynomial time. We will call such sets P-compressible. We show that all standard NP-complete sets are P-compressible, and give a structural condition,E = ∑ E2 , sufficient to ensure thatall infinite NP sets are P-compressible. On the other hand, we present evidence that some infinite NP sets, and indeed some infinite P sets, are not P-compressible: if an infinite NP setA is P-compressible, thenA has an infinite sparse NP subset, yet we construct a relativized world in which some infinite NP sets lack infinite sparse NP subsets. This world is built upon a result that is of interest in its own right; we determine optimally—with respect to any relativizable proof technique—the complexity of the easiest infinite sparse subsets that infinite P sets are guaranteed to have.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Aho and D. Lee,Storing a dynamic sparse table, in Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, 1988, pp. 55–60.

  2. E. Allender andR. Rubinstein,P-printable sets, SIAM Journal on Computing, 17 (1988), pp. 1193–1202.

    Google Scholar 

  3. C. Bennett andJ. Gill,Relative to a random oracle A, PA≠NPA≠ coNPA with probability 1, SIAM Journal on Computing, 10 (1981), pp. 96–113.

    Google Scholar 

  4. L. Berman andJ. Hartmanis,On isomorphisms and density of NP and other complete sets, SIAM Journal on Computing, 6 (1977), pp. 305–322.

    Google Scholar 

  5. R. Book,Bounded query machines: On NP and PSPACE, Theoretical Computer Science, 15 (1981), pp. 27–39.

    Google Scholar 

  6. R. Book andC. Wrathall,Bounded query machines: On NP () and NPQUERY () Theoretical Computer Science, 15 (1981), pp. 41–50.

    Google Scholar 

  7. C. Chang,The study of an ordered minimal perfect hashing, Communications of the ACM, 27 (1984), pp. 384–387.

    Google Scholar 

  8. C. Chang andC. Chang,An ordered minimal perfect hashing scheme with single parameter, Information Processing Letters, 27 (1988), pp. 79–83.

    Google Scholar 

  9. M. Dietzfelbinger, A. Karlin, K. Mehlhorn, F. M. Auf Der Heide, H. Rohnert, and R. Tarjan,Dynamic perfect hashing: Upper and lower bounds, in Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Oct. 1988, pp. 524–531.

  10. M. Garey and D. Johnson,Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Company, 1979.

  11. A. Goldberg and M. Sipser,Compression and ranking, in Proceedings of the 17th ACM Symposium on Theory of Computing, 1985, pp. 440–448.

  12. S. Gordon,A mathematical formulation of the requirements of an order preserving hash function, in Computing and Information: Proceedings of the 1989 International Conference on Computing and Information, Canadian Scholars' Press Inc., 1989, pp. 64–68.

  13. J. Grollmann andA. Selman,Complexity measures for public-key cryptosystems, SIAM Journal on Computing, 17 (1988), pp. 309–335.

    Google Scholar 

  14. J. Hartmanis, N. Immerman, andV. Sewelson,Sparse sets in NP-P: EXPTIME versus NEXPTIME Information and Control, 65 (1985), pp. 159–181.

    Google Scholar 

  15. J. Hartmanis andY. Yesha,Computation times of NP sets of different densities, Theoretical Computer Science, 34 (1984), pp. 17–32.

    Google Scholar 

  16. L. Hemachandra,The strong exponential hierarchy collapses, Journal of Computer and System Sciences, 39 (1989), pp. 299–322.

    Google Scholar 

  17. L. Hemachandra,Algorithms from complexity theory: Polynomial-time operations for complex sets, in Proceedings of the 1990 SIGAL International Symposium on Algorithms, Springer-VerlagLecture Notes in Computer Science #450, Aug. 1990, pp. 221–231.

  18. L. Hemachandra, A. Hoene, D. Siefkes, andP. Young,On sets polynomially enumerable by iteration, Theoretical Computer Science, 80 (1991), pp. 203–226.

    Google Scholar 

  19. L. Hemachandra, M. Ogiwara, and O. Watanabe,How hard are sparse sets?, in Proceedings of the 7th Structure in Complexity Theory Conference, IEEE Computer Society Press. To appear.

  20. L. Hemachandra andS. Rudich,On the complexity of ranking, Journal of Computer and System Sciences, 41 (1990), pp. 251–271.

    Google Scholar 

  21. S. Homer andW. Maass,Oracle dependent properties of the lattice of NP sets, Theoretical Computer Science, 24 (1983), pp. 279–289.

    Google Scholar 

  22. D. Huynh The complexity of ranking simple languages, Mathematical Systems Theory, 23 (1990), pp. 1–20.

    Google Scholar 

  23. D. Joseph andP. Young,Some remarks on witness functions for nonpolynomial and non-complete sets in NP, Theoretical Computer Science, 39 (1985), pp. 225–237.

    Google Scholar 

  24. D. Knuth,The Art of Computer Programming: Sorting and Searching, vol. 3 of Computer Science and Information, Addison-Wesley, 1973.

  25. S. Mahaney,Sparse sets and reducibilities, in Studies in Complexity Theory, R. Book, ed., John Wiley and Sons, 1986, pp. 63–118.

  26. S. Mahaney, andP. Young,Reductions among polynomial isomorphism types, Theoretical Computer Science, 39 (1985), pp. 207–224.

    Google Scholar 

  27. A. Meyer and L. Stockmeyer,The equivalence problem for regular expressions with squaring requires exponential space, in Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, 1972, pp. 125–129.

  28. C. Papadimitriou andM. Yannakakis,The complexity of facets (and some facets of complexity), Journal of Computer and System Sciences, 28 (1984), pp. 244–259.

    Google Scholar 

  29. U. Schöning,A low and a high hierarchy in NP, Journal of Computer and System Sciences, 27 (1983), pp. 14–28.

    Google Scholar 

  30. R. Sprugnoli,Perfect hashing functions: A single probe retrieving method for static sets, Communications of the ACM, 20 (1977), pp. 841–850.

    Google Scholar 

  31. L. Stockmeyer,The polynomial-time hierarchy, Theoretical Computer Science 3 (1977), pp. 1–22.

    Google Scholar 

  32. G. Tardos,Query complexity, or why is it difficult to separate NP A∩coNPA from PA by random oracles A, Combinatorica, 9 (1989), pp. 385–392.

    Google Scholar 

  33. L. Valiant,The relative complexity of checking and evaluating, Information Processing Letters 5 (1976), pp. 20–23.

    Google Scholar 

  34. —,The complexity of enumeration and reliability problems, SIAM Journal on Computing, 8 (1979), pp. 410–421.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Manitoba, R3T 2N2, Winnipeg, Manitoba, Canada

    Judy Goldsmith

  2. Computer Sciences Department, University of Wisconsin, 53706, Madison, WI

    Kenneth Kunen

  3. Department of Computer Science, University of Rochester, 14627, Rochester, NY

    Lane A. Hemachandra

Authors
  1. Judy Goldsmith
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kenneth Kunen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Lane A. Hemachandra
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goldsmith, J., Kunen, K. & Hemachandra, L.A. Polynomial-time compression. Comput Complexity 2, 18–39 (1992). https://doi.org/10.1007/BF01276437

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01276437

Subject classifications

Search

Navigation