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Let O denote the set of permutations performable by a pass through the Omega network (an n-stage shuffle-exchange network), and let p denote an arbitrary permutation of N = 2n elements. In this paper we characterize the sets {p|pO = O} (called left invariants) and {p|Op = O} (called right invariants). This characterization provides a useful criterion for determining whether or not the composition of two permutations belongs to O; it also allows a number of known results to be generalized and their derivations simplified.
triangular permutation, Admissible permutation, invariance, linear permutation, lower triangular shuffle-exchange network, Omega network, parallel processing, permutation network, shuffle-exchange, SIMD array
D. Steinberg, "Invariant Properties of the Shuffle-Exchange and a Simplified Cost-Effective Version of the Omega Network", IEEE Transactions on Computers, vol. 32, no. , pp. 444-450, May 1983, doi:10.1109/TC.1983.1676255
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