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Fine separation of average time complexity classes

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STACS 96 (STACS 1996)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1046))

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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.

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Authors and Affiliations

  1. Department of Computer Science, State University of New York at Buffalo, 14260, Buffalo, NY

    Jin -yi Cai & Alan L. Selman

Authors
  1. Jin -yi Cai
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  2. Alan L. Selman
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Claude Puech Rüdiger Reischuk

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© 1996 Springer-Verlag Berlin Heidelberg

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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

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  • DOI: https://doi.org/10.1007/3-540-60922-9_28

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-60922-3

  • Online ISBN: 978-3-540-49723-3

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