Equivalent permutation capabilities between time division optical omega network and non-optical extra stage omega network

Since optical networks must support crosstalk-free (CF) connections, regular N×N multistage interconnection networks (MINs) cannot be directly used to provide connections between N inputs and N outputs. A solution is to use a 2N×2N Cube-Type MIN to provide the N×N connections. C. Qiao (1993) proposed a time domain approach that divides the N optical inputs into several groups such that CF connections can be provided by an N×N regular MIN in several time slots, one for each group. The set of N-permutations realizable in two slots is defined as class θ. Qiao proved that the size of θ is larger than the size of class Ω, where Ω consists of all N-permutations passible through a regular N×N Omega network. The paper first presents an optimal O(NlgN) time algorithm for identifying whether a given permutation belongs to class θ or not. From this algorithm, the paper proves an interesting fact that the class θ is identical to the class Ω+1 which represents the set of N-permutations admissible by a non-optical N×N one-extra stage Omega network