Hierarchical classification of permutation classes in multistage interconnection networks

This paper explores a new hierarchy among different permutation classes, that has many applications in multistage interconnection networks. The well-known LC (linear-complement) class is shown to be merely a subset of the closure set of the BP (bit-permute) class, known as the BPCL (bit-permute-closure) class; the closure is obtained by applying certain group-transformation rules on the BP-permutations. It indicates that for every permutation P of the LC class, there exists a permutation PI in the BP class, such that the conflict graphs of P and P* are isomorphic, for n-stage MIN´s. This obviates the practice of treating the LC class as a special case; the existing algorithm for optimal routing of BPC class in an n-stage MIN can take care of optimal routing of the LC class as well. Finally, the relationships of BPCL with other classes of permutations, e.g., LIE (linear-input-equivalence), BPIE (bit-permute-input-equivalence), BPOE (bit-permute-output-equivalence) are also exposed. Apart from lending better understanding and an integral view of the universe of permutations, these results are found to be useful in accelerating routability in n-stage MIN´s as well as in (2n-1)-stage Benes and shuffle-exchange networks