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

We show that any permutation of $1,2,\mathrm{.}\mathrm{.}\mathrm{.},N$ 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 in parallel in time O(1). In the case where the permutation is the $k$-way perfect shuffle we develop two methods for efficiently computing such a pair of involutions. The first method works whenever $N$ is a power of $k$; in this case the time is O(N) and space $O({\log }^{2}N)$. The second method applies to the general case where $N$ is a multiple of $k$; here the time is $O(N\log N)$ and the space is $O({\log }^{2}N)$. If $k=2$ the space usage of the first method can be reduced to $O(\log N)$ on a machine that has a SADD (population count) instruction.