 
Summary: On the Expected Depth of Random Circuits
Sunil Arya
Mordecai J. Golin
Kurt Mehlhorn
April 3, 1998
Abstract: In this paper we analyze the expected depth of random circuits of fixed
fanin f. Such circuits are built a gate at a time, with the f inputs of each new gate
being chosen randomly from among the previously added gates. The depth of the new
gate is defined to be one more than the maximal depth of its input gates. We show
that the expected depth of a random circuit with n gates is bounded from above by
eflnn and from below by 2.04 . . . flnn.
1 Introduction
Recently, Diaz, Serna, Spirakis and Toran [1], motivated by the problem of dis
covering how quickly a circuit could be evaluated in parallel, posed the question
of calculating the depth of a random circuit. They described a model for the
generation of random circuits in which each gate has fanin two and showed an
upper bound of O(log3
n) on the circuit's expected depth, where n is the number
of nodes in the circuit. In this paper we extend their problem by investigating
random circuits of fixed fanin f where f can be any integer greater than 0 and
