COUNTING NETWORKS WITH ARBITRARY FAN-OUT

It is shown that an acyclic smoothing network (and hence counting network) with fan-out n cannot be constructed from balancers of fan-out b(1),..., b(k), if there exists a prime factor p of 12, such that p does not divide b(i), for all i, 1 less than or equal to i less than or equal to k. This holds regardless of the depth, fan-in or size of the network, as long as they are finite. On the positive side, a simple construction of cyclic counting networks with fan-out n, for arbitrary n, is presented. An acyclic counting network with fan-in and fan-out p2(k), for any integer k greater than or equal to 0, is constructed out of 2-balancers and p-balancers.