Quasilinear time complexity theory

  • Ashish V. Naik
  • Kenneth W. Regan
  • D. Sivakumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 775)


This paper furthers the study of quasi-linear time complexity initiated by Schnorr [Sch76] and Gurevich and Shelah [GS89]. We show that the fundamental properties of the polynomial-time hierarchy carry over to the quasilineartime hierarchy. Whereas all previously known versions of the Valiant-Vazirani reduction from NP to parity run in quadratic time, we give a new construction using error-correcting codes that runs in quasilinear time. We show, however, that the important equivalence between search problems and decision problems in polynomial time is unlikely to carry over: if search reduces to decision for SAT in quasi-linear time, then all of NP is contained in quasi-polynomial time. Other connections to work by Stearns and Hunt [SH86, SH90, HS90] on “power indices” of NP languages are made.


Polynomial Time Success Probability Turing Machine Power Index Quadratic Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [ABN+92]
    N. Alon, J. Bruck, J. Naor, M. Naor, and R. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. IEEE Trans. Info. Thy., 38(2):509–512, March 1992.Google Scholar
  2. [AHU74]
    A. Aho, J. Hopcroft, and J. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass., 1974.Google Scholar
  3. [Bar92]
    D. Mix Barrington. Quasipolynomial size circuit classes. In Proc. 7th Structures, pages 86–93, 1992.Google Scholar
  4. [BBFG91]
    R. Beigel, M. Bellare, J. Feigenbaum, and S. Goldwasser. Languages that are easier than their proofs. In Proc. 32nd FOCS, pages 19–28, 1991.Google Scholar
  5. [BD76]
    A. Borodin and A. Demers. Some comments on functional self-reducibility and the NP hierarchy. Technical Report TR 76–284, Cornell Univ. Comp. Sci. Dept., 1976.Google Scholar
  6. [BFLS91]
    L. Babai, L. Fortnow, L. Levin, and M. Szegedy. Checking computations in polylogarithmic time. In Proc. 23rd STOC, pages 21–31, 1991.Google Scholar
  7. [BG93]
    J. Buss and J. Goldsmith. Nondeterminism within P. SIAM J. Comp., 22:560–572, 1993.Google Scholar
  8. [CR73]
    S. Cook and R. Reckhow. Time bounded random access machines. J. Comp. Sys. Sci., 7:354–375, 1973.Google Scholar
  9. [CRS93]
    S. Chari, P. Rohatgi, and A. Srinivasan. Randomness-optimal unique element isolation, with applications to perfect matching and related problems. In Proc. 25th STOC, pages 458–467, 1993.Google Scholar
  10. [CW79]
    J. Carter and M. Wegman. Universal classes of hash functions. J. Comp. Sys. Sci., 18:143–154, 1979.Google Scholar
  11. [GS89]
    Y. Gurevich and S. Shelah. Nearly-linear time. In Proceedings, Logic at Botik'89, volume 363 of LNCS, pages 108–118. Springer Verlag, 1989.Google Scholar
  12. [Gup93]
    S. Gupta. On isolating an odd number of elements and its applications to complexity theory. Technical Report OSU-CISRC-6/93-TR24, Dept. of Comp. Sci., Ohio State University, 1993.Google Scholar
  13. [HS90]
    H. Hunt III and R. Stearns. The complexity of very simple Boolean formulas, with applications. SIAM J. Comp., 19:44–70, 1990.Google Scholar
  14. [HU79]
    J. Hopcroft and J. Ullman. Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading, MA, 1979.Google Scholar
  15. [HW89]
    S. Homer and J. Wang. Absolute results concerning one-way functions and their applications. Math. Sys. Thy., 22:21–35, 1989.Google Scholar
  16. [JLJH92]
    J. Justesen, K. Larsen, H.E. Jensen, and T. Hoholdt. Fast decoding of codes from algebraic plane curves. IEEE Trans. Info. Thy., 38(1):111–119, January 1992.Google Scholar
  17. [JY90]
    D. Joseph and P. Young. Self-reducibility: the effects of internal structure on computational complexity. In A. Selman, editor, Complexity Theory Retrospective, pages 82–107. Springer Verlag, 1990.Google Scholar
  18. [Ko82]
    K. Ko. Some observations on the probabilistic algorithms and NP-hard problems. Inf. Proc. Lett., 14:39–43, 1982.Google Scholar
  19. [LL76]
    R. Ladner and N. Lynch. Relativization of questions about log-space computability. Math. Sys. Thy., 10:19–32, 1976.Google Scholar
  20. [LV90]
    M. Li and P. Vitányi. Applications of Kolmogorov complexity in the theory of computation. In A. Selman, editor, Complexity Theory Retrospective, pages 147–203. Springer Verlag, 1990).Google Scholar
  21. [NN90]
    J. Naor and M. Naor. Small-bias probability spaces. In Proc. 22nd STOC, pages 213-223, 1990.Google Scholar
  22. [NN93]
    J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. SIAM J. Comp., 22:838–856, 1993.Google Scholar
  23. [NOS93]
    A. Naik, M. Ogiwara, and A. Selman. P-selective sets, and reducing search to decision vs. self-reducibility. In Proc. 8th Structures, pages 52–64, 1993.Google Scholar
  24. [PZ83]
    C. H. Papadimitriou and S. Zachos. Two remarks on the power of counting. In The 6th GI Conference on Theoretical Computer Science, Lecture Notes in Computer Science No. 145, pages 269–276. Springer Verlag, 1983.Google Scholar
  25. [Rab80]
    M. Rabin. Probabilistic algorithms in finite fields. SIAM J. Comp., pages 273-280, 1980.Google Scholar
  26. [Sch76]
    C. Schnorr. The network complexity and the Turing machine complexity of finite functions. Acta Informatica, 7:95–107, 1976.Google Scholar
  27. [Sch78]
    C. Schnorr. Satisfiability is quasilinear complete in NQL. J. ACM, 25:136–145, 1978.Google Scholar
  28. [Sel88]
    A. Selman. Natural self-reducible sets. SIAM J. Comp., 17:989–996, 1988.Google Scholar
  29. [SH86]
    R. Stearns and H. Hunt III. On the complexity of the satisfiability problem and the structure of NP. Technical Report 86-21, Dept. of Comp. Sci., SUNY at Albany, 1986.Google Scholar
  30. [SH90]
    R. Stearns and H. Hunt III. Power indices and easier hard problems. Math. Sys. Thy., 23:209–225, 1990.Google Scholar
  31. [She93]
    B.-Z. Shen. A Justesen construction of binary concatenated codes than asymptotically meet the Zyablov bound for low rate. IEEE Trans. Info. Thy., 39(1):239–242, January 1993.Google Scholar
  32. [Sto77]
    L. Stockmeyer. The polynomial time hierarchy. Theor. Comp. Sci., 3:1–22, 1977.Google Scholar
  33. [Sud92]
    M. Sudan. Efficient checking of polynomials and proofs and the hardness of approximation problems. PhD thesis, University of California, Berkeley, 1992.Google Scholar
  34. [Tod91]
    S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comp., 20:865–877, 1991.Google Scholar
  35. [TV91]
    M. Tsfasman and S. Vladut. Algebraic-Geometric Codes, volume 58 of Mathematics and Its Applications (Soviet Series). Kluwer Academic, Dordrecht, 1991.Google Scholar
  36. [VV86]
    L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theor. Comp. Sci., 47:85–93, 1986.Google Scholar
  37. [vzG91]
    J. von zur Gathen. Efficient exponentiation in finite fields. In Proc. 32nd FOCS, pages 384–391, 1991.Google Scholar
  38. [Wra77]
    C. Wrathall. Complete sets and the polynomial-time hierarchy. Theor. Comp. Sci., 3:23–33, 1977.Google Scholar
  39. [Wra78]
    C. Wrathall. Rudimentary predicates and relative computation. SIAM J. Comp., 7:194–209, 1978.Google Scholar
  40. [WW86]
    K. Wagner and G. Wechsung. Computational Complexity. D. Reidel, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Ashish V. Naik
    • 1
  • Kenneth W. Regan
    • 1
  • D. Sivakumar
    • 1
  1. 1.SUNY BuffaloUSA

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