Abstract
The computational delay of a circuit can be described by the natural concept of time [Jakoby et al. STOC94]. We show that for a given input x and circuit C the computation of timeC(x) is P-complete. Moreover, we show that it is NP-complete to decide whether there exists an input x such that timeC (x)≤t for a given time bound t.
We introduce the notion of worst time of a circuit and show that to decide whether a given time bound is the worst time of a circuit is BH 2-complete. We also prove that the computation of an arbitrary worst case input is FP NPtt -hard, whereas the search of the lexicographically minimal worst case input is FP NP -complete and of the lex. middle worst case input is FP #P-complete.
Computation of the expected time E μD (timeC) of a circuit C with respect to a distribution μ D generated by circuit D is #P-complete under metric reducibility. Nevertheless we show that a polynomial time bounded probabilistic Turing machine approximates E μD (timeC) up to an arbitrary additive constant with high probability.
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References
S. Ben-David, B. Chor, O. Goldreich, M. Luby, On the Theory of Average Case Complexity, J. CSS 44, 1992, 193–219.
B. Bollig, M. Hühne, S. Pölt, P. Savický, On the Average Case Circuit Delay of Disjunction, Technical Report, University of Dortmund, 1994.
S. A. Cook, The Complexity of Theorem-Proving Procedures, Proc. of the 3rd IEEE Symp. on the Foundations of Computer Science, 151–158, 1971.
I. David, R. Ginosar, M. Yoelli, An Efficient Implementation of Boolean Functions and Finite State Machines as Self-Timed Circuits, ACM SIGARCH, 1989, 91–104.
R. Greenlaw, H. J. Hoover, W. L. Ruzzo, A Compendium of Problems Complete for P, Technical Report 91-05-01, University of Washington, 1991.
A. Jakoby, R. Reischuk, C. Schindelhauer, Circuit Complexity: from the Worst Case to the Average Case, Proc. 26th SToC, 1994, 58–67; see also Technical Report, TH Darmstadt, 1993.
A. Jakoby, R. Reischuk, C. Schindelhauer, Malign Distributions for Average Case Circuit Complexity, Proc. 12th STACS, 1995, 628–639.
A. Jakoby, R. Reischuk, C. Schindelhauer, S. Weis The Average Case Complexity of the Parallel Prefix Problem, Proc. 21st ICALP, 1994, 593–604; Technical Report, TH Darmstadt, 1993.
V. Krapchenko, Depth and Delay in a Network, Soviet Math. Dokl. 19, 1978, 1006–1009.
M. W. Krentel The complexity of optimization problems, Proc. 18th SToC, 1986, 79–86; see also J. CSS 36, 1988, 490–509.
W. Lam, R. Brayton, A. Sangiovanni-Vincentelli, Circuit Delay Models and Their Exact Computation Using Timed Boolean Functions, ACM/IEEE, Design Automation Conference, 1993, 128–133.
M. Li, P. Vitanyi, Average Case Complexity under the Universal Distribution Equals Worst-Case Complexity, IPL 42, 1992, 145–149.
P. Miltersen, The Complexity of Malign Ensembles, SIAM. J. Comput. 22, 1993, 147–156.
C. H. Papadimitriou, M. Yannakakis, The Complexity of Facets (and some facets of complexity), Proc. 14th SToC, 1982, 255–260; also, J.CSS 28, 1984, 244–259.
C. H. Papadimitriou, Computational Complexity, Addison-Wesley, 1994.
J. Reif, Probabilistic Parallel Prefix Computation, Comp. Math. Applic. 26, 1993, 101–110.
S. Toda, The Complexity of Finding Medians, 31th FoCS, 1990, 778–787.
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© 1996 Springer-Verlag Berlin Heidelberg
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Jakoby, A., Schindelhauer, C. (1996). On the complexity of worst case and expected time in a circuit. 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_25
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DOI: https://doi.org/10.1007/3-540-60922-9_25
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