Permutation routing in optical MIN with minimum number of stages

In a hybrid optical multistage interconnection network (MIN), optical signals are routed by electronically controlled switches using directional couplers. A relevant design problem is to minimize 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 has to pass. In general, given the network size and the type of the MIN, the number of stages is a constant. Hence, an input signal has to pass through a fixed number of couplers to reach the output.In this paper, it is shown that the routing delay and path-dependent loss in a fixed-stage N × N MIN, can be significantly reduced on the average by using a variable-stage shuffle-exchange network instead. An arbitrary N × N permutation P can be routed with minimum delay and minimum path-dependent loss, if the minimum number of stages of the MIN necessary to route P is known. An O(Nn) algorithm (N = 2n) is presented here for checking the admissibility of a given permutation P in an m-stage shuffle-exchange network (SEN), where 1 ≤ m ≤ n. The minimum-stage SEN needed to pass P can then be determined in O(Nn log n) time. Furthermore, for n ≤ m ≤ 2n - 1, a necessary condition for permutation admissibility is derived which is shown to be necessary as well as sufficient for the special class of BPC (bit-permute-complement) permutations. It has been shown that, for 1 ≤ m ≤ 2n - 1, the minimum number of stages required to pass a BPC permutation P through a SEN can be determined in O(n2) time, and P can be routed through a variable-stage SEN using the minimum number of stages only. In an optical MIN, this technique helps to reduce the path-dependent loss by limiting the number of stages to be traversed by the optical signal.