The Complexity of Computing Symmetric Functions Using Threshold Circuits

Beame, R., E. Brisson and R. Ladner, The complexity of computing symmetric functions using threshold circuits, Theoretical Computer Science 100 (1992) 2533265. This paper considers size-depth tradeoffs for threshold circuits computing symmetric functions. The size measure used is the number of connections or edges in the threshold circuits as opposed to the number of gates in the circuits. The main result is that for all d > 2 and n > 82d there is a threshold circuit to compute any n-input symmetric function which has size 0 (J 1+ log n ~. ,I + l/(2*1, 2*-1 > and depth bounded by 6d+ 8. As a consequence, there is a threshold circuit for any n-input symmetric function which has size O(n) and depth bounded by O(log log n). A somewhat simpler construction that contains many features of the general solution shows that for all d> 1 and n >2*‘-’ there is a threshold circuit for the n-input parity function which has size bounded by (27/2$)ul+ rWd1) and depth bounded by 2d.