Average circuit depth and average communication complexity

  • Bruno Codenotti
  • Peter Gemmell
  • Janos Simon
Session 2. Chair: Joseph Diaz
Part of the Lecture Notes in Computer Science book series (LNCS, volume 979)


We use the techniques of Karchmer and Widgerson [KW90] to derive strong lower bounds on the expected parallel time to compute boolean functions by circuits. By average time, we mean the time needed on a self-timed circuit, a model introduced recently by Jakoby, Reischuk, and Schindelhauer, [JRS94] in which gates compute their output as soon as it is determined (possibly by a subset of the inputs to the gate).

More precisely, we show that the average time needed to compute a boolean function on a circuit is always greater than or equal to the average number of rounds required in Karchmer and Widgerson's communication game. We also prove a similar lower bound for the monotone case. We then use these techniques to show that, for a large subset of the inputs, the average time needed to compute st connectivity by monotone boolean circuits is Ω(log2n).

We show, that, unlike the situation for worst case bounds, where the number of rounds characterize circuit depth, in the average case the Karchmer-Widgerson game is only a lower bound. We construct a function g and a set of minterms and maxterms such that on this set the average time needed for any monotone circuit to compute g is polynomial, while the average number of rounds needed in Karchmer and Widgerson's monotone communication game for g is a constant. Related work by Raz and Widgerson [RW89] shows that the monotone probabilistic communication complexity (a model weaker than ours) of the s-t connectivity problem is Ω(log2n).


circuit complexity parallel time communication complexity lower bounds 


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  1. [FKN91]
    T. Feder, E. Kushilevitz, M. Naor, Amortized Communication Complexity, Proc. 32nd Symposium on the Foundations of Computer Science, October 1991, pp. 239–248.Google Scholar
  2. [JRS94]
    A. Jakoby, R. Reischuk, C. Schindelhauer, Circuit Complexity: from the Worst Case to the Average Case, Proc. 26th ACM Symposium on the Theory of Computing, May 1994, pp.58–67.Google Scholar
  3. [KW90]
    M. Karchmer and A. Wigderson, Monotone Circuits for Connectivity Require Superlogarithmic Depth, Siam Journal of Discrete Math, vol. 3, no 2, May 1990, pp.255–265.Google Scholar
  4. [L86]
    L. Levin, Average Case Complexity Problems, SIAM J. Computing 15, 1986, pp. 285–286.Google Scholar
  5. [O91]
    A. Orlitsky, Interactive Communication: Balanced Distributions, Correlated Files, and Average-Case Complexity, Proc. 32nd Symposium on the Foundations of Computer Science, October 1991, pp. 228–238.Google Scholar
  6. [RW89]
    R. Raz and A. Wigderson, Probabilistic Communication Complexity of Boolean Relations, Proc. 30th Symposium on the Foundations of Computer Science, October 1989, pp. 562–567.Google Scholar
  7. [RW92]
    R. Raz and A. Wigderson, Monotone Circuits for Matching Require Linear Depth, Journal of the ACM v.39 1992.Google Scholar
  8. [R85]
    A. A. Razborov, A lower bound on the monotone network complexity of the logical permanent Mat. Zametki v.37 n.6 1985 pp. 887–900.Google Scholar
  9. [RS93]
    R. Reischuk, C. Schindelhauer, Precise Average Case Complexity, Proc. 10th Symposium on Theoretical Aspects of Computer Science, Feb. 1993, pp.650–661.Google Scholar
  10. [W]
    H. S. Wilf, Backtrack: An O(1) expected time algorithm for the graph coloring problem, Information Processing Letters v.18 n.3 1984 pp. 119–121.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Bruno Codenotti
    • 1
  • Peter Gemmell
    • 2
  • Janos Simon
    • 3
  1. 1.Istituto di Matematica Computazionale del CNRPisaItaly
  2. 2.Sandia National Labs.USA
  3. 3.Department of Computer ScienceThe University of ChicagoUSA

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