Fine separation of average time complexity classes

  • Jin -yi Cai
  • Alan L. Selman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1046)


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).


Polynomial Time Turing Machine Complexity Class Time Bound Distributional Problem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Jin -yi Cai
    • 1
  • Alan L. Selman
    • 1
  1. 1.Department of Computer ScienceState University of New York at BuffaloBuffalo

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