Is Your Permutation Algorithm Unbiased for n ≠ 2 m ?

Many papers on parallel random permutation algorithms assume the input size n to be a power of two and imply that these algorithms can be easily generalized to arbitrary n, e.g., by padding the input array to a power of two. We show that this simplifying assumption is not necessarily correct since it may result in a bias (i.e., not all possible permutations are generated with equal likelihood). Many of these algorithms are, however, consistent, i.e., iterating them ultimately converges against an unbiased permutation. We prove this convergence along with proving exponential convergence speed. Furthermore, we present an analysis of iterating applied to a butterfly permutation network, which works inplace and is well-suited for implementation on many-core systems such as GPUs. We also show a method that improves the convergence speed even further and yields a practical implementation of the permutation network on current GPUs.