Wavelengths requirement for permutation routing in all-optical multistage interconnection networks

Previous studies showed that the cross-talk problem on the all-optical networks exists at both links and switches of the networks. To solve the cross-talk problem at both links and switches, one approach is to assign the wavelengths to the communication paths so that the paths which receive the same wavelength are node-disjoint. Our goal is to minimize the number of wavelengths required for permutation routings by node-disjoint paths on all-optical MINs which consists of n stages of 2×2 switches connecting N = 2n inputs and outputs. We prove that the problem of finding the minimum number of wavelengths for arbitrary partial permutation routings on the MINs is NP-complete. We show that any partial permutation routing can be realized by 2n/2 wavelengths and there exist permutation routings that require at least 2n/2 wavelengths. Although the general problem is NP-complete, we give an efficient algorithm for computing the minimum number of wavelengths for the class of BPC (bit permute-complement) permutations.