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Hierarchical Classification of Permutation Classes in Multistage Interconnection Networks
December 1994 (vol. 43 no. 12)
pp. 1439-1444

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.

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Index Terms:
multistage interconnection networks; hierarchical classification; permutation classes; multistage interconnection networks; linear-complement class; bit-permute class; bit-permute-closure; shuffle-exchange networks; Benes networks.
N. Das, B.B. Bhattacharya, J. Dattagupta, "Hierarchical Classification of Permutation Classes in Multistage Interconnection Networks," IEEE Transactions on Computers, vol. 43, no. 12, pp. 1439-1444, Dec. 1994, doi:10.1109/12.338106
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