Invariant Properties of the Shuffle-Exchange and a Simplified Cost-Effective Version of the Omega Network

Let Ω denote the set of permutations performable by a pass through the Omega network (an n-stage shuffle-exchange network), and let π denote an arbitrary permutation of N = 2n elements. In this paper we characterize the sets {π|πΩ = Ω} (called left invariants) and {π|Ωπ = Ω} (called right invariants). This characterization provides a useful criterion for determining whether or not the composition of two permutations belongs to Ω; it also allows a number of known results to be generalized and their derivations simplified.