Equivalent permutation capabilities between time-division optical Omega networks and non-optical extra-stage Omega networks

Because signals carried by two waveguides entering a common switch element would generate crosstalk, a regular N×N multistage interconnection network (MIN) cannot be directly used as an optical switch between N inputs and N outputs in an optical network. A simple solution is to use a 2N×2N cube-type MIN to provide the N×N connections, which needs a much larger hardware cost. A previous research proposed another solution, called the time-domain approach, that divides the N optical inputs into several groups such that crosstalk-free connections can be provided by an N×N regular MIN in several time slots, one for each group. Researchers studied this approach on Omega networks and defined the class set θ to be the set of N-permutations realizable in two time slots on an Omega network. They proved that the size of θ is larger than the size of class Ω, where Ω consists of all N-permutations admissible to a regular N×N (nonoptical) Omega network. This paper first presents an optimal O(NlogN) time algorithm for identifying whether a given permutation belongs to class θ or not. Using this algorithm, this paper then proves an interesting result that the class θ is identical to the class Ω+1 which represents the set of N-permutations admissible to a nonoptical N×N one-extra stage Omega network