Abstract
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).
The full paper is available as a SUNY at Buffalo Technical Report, no. 95-16, at http://www.ncstrl.org.
Research supported in part by NSF grants CCR-9057486 and CCR-9319093, and an Alfred P. Sloan Fellowship.
Research supported in part by NSF grant CCR-9400229.
Preview
Unable to display preview. Download preview PDF.
References
J. Balcázar and U. Schöning. Bi-immune sets for complexity classes. Mathematical Systems Theory, 18:1–10, 1985.
J. Belanger and J. Wang. Rankable distributions do not provide harder instances than uniform distributions. Manuscript, 1995.
S. Ben-Davíd, B. Chor, O. Goldreich, and M. Luby. On the theory of average case complexity. J. Computer System Sci., 44(2):193–219, 1992.
S. Cook and R. Reckow. Time bounded random access machines. J. Comput. System Sci., 7:354–375, 1973.
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.
J. Geske, D. Huynh, and J. Seiferas. A note on almost-everywhere-complex sets and separating deterministic-time-complexity classes. Inf. and Comput., 92(1):97–104, 1991.
Y. Gurevich. Average case completeness. J. Comput. System Sci., 42:346–398, 1991.
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.
G. Hardy. Properties of logarithmico-exponential functions. Proc. London Math. Soc., (2),10:54–90, 1911.
J. Hartmanis and R. Stearns. On the computational complexity of algorithms. Trans. Amer. Math. Soc., 117:285–306, 1965.
K. Ko. On the definition of some complexity classes of real numbers. Math. Systems Theory, 16:95–109, 1983.
L. Levin. Average case complete problems. SIAM J. of Comput., 15:285–286, 1986.
M. Li and P. Vitányi. Average case complexity under the universal distribution equals worst-case complexity. Inf. Proc. Lett., 42:145–149, 1992.
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.
E. Mayordomo. Almost every set in exponential time is P-bi-immune. Theoret. Comput. Sci., 136:487–506, 1994.
C. Rackoff. Personal communication.
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.
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.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1996 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cai, J.y., Selman, A.L. (1996). Fine separation of average time complexity classes. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_28
Download citation
DOI: https://doi.org/10.1007/3-540-60922-9_28
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60922-3
Online ISBN: 978-3-540-49723-3
eBook Packages: Springer Book Archive