Linear-Size Constant-Depth Polylog-Treshold Circuits

Abstract We present a simple explicit construction giving unbounded fan-in circuits with o(n) gates and depth O(r) for the threshold function of n variables when the threshold is at most (log n)r, for any integer r > 0. This improves a result of Ajtai and Ben-Or, who showed the existence of circuits of size no(1). This is the highest threshold for which polynomial-size, constant-depth circuits are possible.