Fine separation of average time complexity classes
We extend Levin's theory of average polynomial time to arbitrary time-bounds in accordance with the following general principles: (1) It essentially agrees with Levin's notion when applied to polynomial time-bounds. (2) If a language L belongs to DTIME(T(n)), for some time-bound T(n), then every distributional problem (L,μ) is T on the μ-average. (3) If L does not belong to DTIME(T(n)) almost everywhere, then no distributional problem (L, μ) is T on the μ-average.
We present hierarchy theorems for average-case complexity, for arbitrary time-bounds, that are as tight as the well-known Hartmanis-Stearns [HS65] hierarchy theorem for deterministic complexity. As a consequence, for every time-bound T(n), there are distributional problems (L, μ) that can be solved using only a slight increase in time but that cannot be solved on the μ-average in time T(n).
KeywordsPolynomial Time Turing Machine Complexity Class Time Bound Distributional Problem
Unable to display preview. Download preview PDF.
- [BW95]J. Belanger and J. Wang. Rankable distributions do not provide harder instances than uniform distributions. Manuscript, 1995.Google Scholar
- [CR73]S. Cook and R. Reckow. Time bounded random access machines. J. Comput. System Sci., 7:354–375, 1973.Google Scholar
- [GHS87]J. Geske, D. Huynh, and A. Selman. A hierarchy theorem for almost every-where complex sets with application to polynomial complexity degrees. In STACS 1987, 1987.Google Scholar
- [Har24]G. Hardy. Orders of Infinity, The ‘infinitärcalcül’ of Paul du Bois-Reymond, volume 12 of Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, London, 2nd edition, 1924.Google Scholar
- [Har11]G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., (2),10:54–90, 1911.Google Scholar
- [HS65]J. Hartmanis and R. Stearns. On the computational complexity of algorithms. Trans. Amer. Math. Soc., 117:285–306, 1965.Google Scholar
- [LM94]J. Lutz and E. Mayordomo. Cook versus Karp-Levin: Separating completeness notions if NP is not small. In Proc. 11th Annual Symp. on Theor. Aspects of Com. Sci., Lecture Notes in Computer Science, Springer-Verlag, 775:415–426, 1994.Google Scholar
- [Rac95]C. Rackoff. Personal communication.Google Scholar
- [RS93]R. Reishuk and C. Schindelhauer Precise average case complexity. In Proc. 10th Annual Symp. on Theoretical Aspects of Computer Sci. v. 665 of Lecture Notes in Computer Sci., pages 650–661. Springer Verlag. 1993.Google Scholar
- [SY95]R. Schuler and T. Yamakami Sets computable in polynomial time on the average. In Proc. First Annual Conf. on Computing and Combinatorics v. 959 of Lecture Notes in Computer Sci., pages 400–409. Springer Verlag. 1995.Google Scholar