Parallel and sequential in-place permuting and perfect shuffling using involutions

Every permutation of {1, 2, . . . , N} can be written as the product of two involutions. As a consequence, any permutation of the elements of an array can be performed in-place using simultaneous swaps in two rounds of swaps. In the case where the permutation is the k-way perfect shuffle we develop two methods for efficiently computing the pair of involutions that accomplishes these swaps. The first method works whenever N is a power of k; in this case the time is O(N) and space O(log N). The second method applies to the general case where N is a multiple of k; here the time is O(N logN) and the space is O(log N). If k = 2 the space usage of the first method can be reduced to O(logN) on a machine that has a SADD (population count) instruction. keywords: Perfect shuffle, merging, sorting, permutation, involution.