Average Circuit Depth and Average Communication Complexity

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 (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 (log 2 n).