Malign distributions for average case circuit complexity
In contrast to machine models like Turing machines or random access machines, circuits are a rigid computational model. The internal information flow of a computation is fixed in advance, independent of the actual input. Therefore, in complexity theory only worst case complexity measures have been used to analyse this model. In [JRS94] we have defined an average case measure for the time complexity of circuits. Using this notion tight upper and lower bounds could be obtained for the average case complexity of several basic Boolean functions.
In this paper we will examine the asymptotic average case complexity of the set of n-input Boolean functions. In contrast to the worst case a simple counting argument does not work. We prove that almost all Boolean function require at least n— log n— log n expected time even for the uniform probability distribution. On the other hand, we show that there are significant subsets of functions that can be computed with a constant average delay.
Finally, the worst case and average case complexity of a Boolean function will be compared. We show that for each function that requires circuit depth d, the expected time complexity will be at least d— log n— log d with respect to an explicitely defined probability distribution. A nontrivial bound on the complexity of such a distribution is obtained.
Key wordsaverage case complexity circuit complexity delay distribution
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- [BCGL92]S. Ben-David, B. Chor, O. Goldreich, M. Luby, On the Theory of Average Case Complexity, J. CSS 44, 1992, 193–219.Google Scholar
- [BHPS94]B. Bollig, M. Hühne, S. Pölt, P. Savický, On the Average Case Circuit Delay of Disjunction, Technical Report, University of Dortmund, 1994.Google Scholar
- [DGY89]I. David, R. Ginosar, M. Yoelli, An Efficient Implementation of Boolean Functions and Finite State Machines as Self-Timed Circuits, ACM SIG-ARCH, 1989, 91–104.Google Scholar
- [Gask78]Gaskov, The Depth of Boolean Functions, Prob. Kybernet. 34, 1978, 265–268.Google Scholar
- [Grap90]P. Grape, Complete Problems with L-sampleable Distributions, Proc. 2. SWAT 90, 360–367.Google Scholar
- [Gure91]Y. Gurevich Average Case Completeness, J. CSS 42, 1991, 346–398.Google Scholar
- [JRS94]A. Jakoby, R. Reischuk, C. Schindelhauer, Circuit Complexity: from the Worst Case to the Average Case, Proc. 26. STOC, 1994, 58–67.Google Scholar
- [JRSW94]A. Jakoby, R. Reischuk, C. Schindelhauer, S. Weis The Average Case Complexity of the Parallel Prefix Problem, Proc. 21. ICALP, 1994, 593–604.Google Scholar
- [Koba93]K. Kobayashi, On Malign Input Distributions for Algorithms, IEICE Trans. Inf. & Syst. 76, 1993, 634–640.Google Scholar
- [Krap78]V. Krapchenko, Depth and Delay in a Network, Soviet Math. Dokl. 19, 1978, 1006–1009.Google Scholar
- [LBS93]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.Google Scholar
- [Levi86]L. Levin, Average Case Complete Problems, SIAM J. Computing 15, 1986, 285–286.Google Scholar
- [LiVi89]M. Li, P. Vitanyi, Inductive Reasoning and Kolmogorov Complexity, Proc. 4. Structure, 1989, 165–185.Google Scholar
- [LiVi92]M. Li, P. Vitanyi, Average Case Complexity under the Universal Distribution Equals Worst-Case Complexity, IPL 42, 1992, 145–149.Google Scholar
- [Milt91]P. Miltersen, The Complexity of Malign Ensembles, Proc. 6. Structure in Complexity Theory, 1991, 164–171, see also SIAM. J. Comput. 22, 1993, 147–156.Google Scholar
- [ReSc93]R. Reischuk, C. Schindelhauer, Precise Average Case Complexity, Proc. 10. GI-AFCET Symposium on Theoretical Aspects of Computer Science, STACS 1993, Springer Lecture Notes, 650–661.Google Scholar
- [Weg87]I. Wegener, The Complexity of Boolean Functions, Wiley-Teubner, 1987.Google Scholar
- [WaBe92]J. Wang, J. Belanger, On Average P vs. Average NP, Proc. 7. Structure, 1992, 318–326.Google Scholar