Equivalence between functionality and topology of log < e1 > N < /e1 > -stage banyan networks

Existing procedures to decide network equivalence in log < e1 > N < /e1 > -stage banyan networks are based on analysis of permutations, take polynomial but costly time, and do not shed light on or take advantage of the relationship between functionality and topology. This relationship is addressed and it is shown that two log < e1 > N < /e1 > -stage banyan networks of the same switch size are functionally equivalent if and only if they have the same underlying topology. An O( < e1 > N < /e1 > log < e1 > N < /e1 > ) algorithm is derived which decides if two < e1 > N < /e1 > -stage banyan networks of < e1 > N < /e1 > inputs, < e1 > N < /e1 > outputs, and < e1 > r < /e1 > x < e1 > r < /e1 > crossbar switches as building blocks realize the same permutations. The algorithm works by comparing the underlying topologies of the two networks. The algorithm is optimal because the size of the networks is O( < e1 > N < /e1 > log < e1 > N < /e1 > )