Hamiltonian cycles in the shuffle-exchange network

The usefulness of the shuffle-exchange network in parallel processing applications is well established. The optimal embedding of a shuffle-exchange network of a given size depends on the number of cycles of the shuffle permutation of that size. The cost of one method of adding fault-tolerance through reconfigurability depends upon the number of such cycles, and the manner in which they can be connected to form larger cycles. An exact equation for the number of cycles of a shuffle of size 2^W is presented. That result is used to demonstrate that it is always possible to form a Hamiltonian cycle on all processors in a shuffle-exchange connected array. From this, it is apparent that there are a large number of ways of sharing spare processors among the members of many cycles. Thus, redundancy can be supplied in any strength, from adding on a spare processor, to adding one for each cycle of the shuffle