# Average circuit depth and average communication complexity

## Abstract

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 *s* −*t* connectivity by monotone boolean circuits is Ω(log^{2}*n*).

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*(log^{2}*n*).

## Keywords

circuit complexity parallel time communication complexity lower bounds## Preview

Unable to display preview. Download preview PDF.

## References

- [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 - [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 - [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 - [L86]L. Levin,
*Average Case Complexity Problems*, SIAM J. Computing 15, 1986, pp. 285–286.Google Scholar - [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 - [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 - [RW92]R. Raz and A. Wigderson,
*Monotone Circuits for Matching Require Linear Depth*, Journal of the ACM v.39 1992.Google Scholar - [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 - [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 - [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