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(log2 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 func tion 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 poly nomial, 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 probabil istic communication complexity (a model weaker than ours) of the s-t connectivity problem is Ω(log2 n).