Functional and topological relations among banyan multistage networks of differing switch sizes

If two N×N networks W and W ´ have switch sizes r and s, respectively, and if r>s, then W realizes a larger number of permutations than W´. Consequently, the two networks can never be equivalent. However, W may realize all the permutations of W´, in which case W is said to functionally cover W´ in the strict sense. More generally, W is said to functionally cover W´ in the wide sense if the terminals of W can be relabeled so that W realizes all the permutations of W´. Functional covering is topologically characterized, and an optimal algorithm to decide strict functional covering is developed. It is shown that any N-×N-digit permutation network of switch size r functionally covers in the wide sense any other N-×N-digit permutation network of switch size s if and only if r is a perfect power of s, where a digit permutation network is a banyan multistage network such that the interconnections are permutations that permute digits in a specified manner