Permutation routing in optical MIN with minimum number of stages

In hybrid optical Multistage Interconnection Networks (MIN’s), optical signals are routed by electronically controlled switches using directional couplers. One important problem of these hybrid optical MIN’s is the path-dependent loss of the optical signal, which is directly proportional to the number of couplers, i.e., the number of switches through which the signal passes. In general, given the network size and the type of the MIN, the number of stages of a MIN is constant. Hence any input signal has to pass through a fixed number of couplers to reach the output. In this paper, we propose that instead of using a fixed-stage N ×N MIN, we may route any arbitrary N × N permutation P with minimum delay and minimum pathdependent loss, if we know the minimum number of stages of the MIN necessary to route P . Here, we present an O(Nn) algorithm (N = 2) that checks whether a given permutation P is admissible in an m stage shuffle-exchange network (SEN), 1 ≤ m ≤ n, and determines in O(Nn log n) time the minimum number of stages m of shuffle-exchange, required to realize P . Furthermore, for n < m ≤ 2n−1, we present a necessary condition for permutation admissibility, in general, which is also a sufficient condition for BPC (bit-permute-complement) class permutations. Hence we find the minimum number of shuffle-exchange stages mmin required to make any arbitrary BPC permutation admissible, 1 ≤ m ≤ 2n − 1, in O(Nn log n) time. By this technique, a BPC permutation can be routed through minimum number of shuffle-exchange stages, that enables us to minimize the path dependent loss of the signal, as well as the communication delay in the optical MIN.