Random Permutations

Abstract

Summary The problem of generating random permutations of the integers 1, 2, …, n arises, for example, when sampling a randomization distribution and when tables of permutations are required for application to experimental design. An algorithm is described which minimizes the amount of randomization necessary to generate a random permutation, although in practice longer but simpler procedures are likely to be preferable. We discuss the probability distributions for the number of digits required by standard methods of generation from a sequence of random digits. Particular attention is given to the asymptotically efficient multistage randomization procedure. We obtain approximate formulae for the mean and variance of the number of digits and compare them with the exact results. An argument of a partly conjectural nature is put forward for the asymptotic normality of this number. The possibility that the permutations have unequal probabilities is briefly considered.

Generating Random Permutations by Coin Tossing

“…It turns out that exactly the same binomial splitting idea was already developed in the early 1960s in the statistical literature in Rao [1961] and independently in Sandelius [1962] and analyzed later in Plackett [1968]. The articles by Rao and by Sandelius also propose other variants, which have their modern algorithmic interests per se.…”

“…From a historical perspective, such a clarification through analytic means was first worked out by Flajolet and his co-authors in the early 1980s; see again Fuchs et al [2014] for a brief account. However, the periodic oscillations had already been observed in the 1960s by Plackett [1968] based on heuristic arguments and figures, which seems less expected because of the limited computer power at that time and of the proper normalization needed to visualize the fluctuations; see Figures 2 and 3 for the subtleties involved. Unlike Algorithm FY, Algorithm RS is more easily adapted to a distributed or parallel computing environment because the random bits needed can be generated simultaneously.…”

“…Continue this procedure until the last element is removed. A direct implementation of this algorithm results in a two-loop procedure; a simple one-loop procedure was devised in Robson [1969] and is shown on the right; see also Plackett [1968] and Devroye's book [Devroye 1986, Section XIII.1] for variants, implementation details, and references. This algorithm is mostly useful when n is small, say, less than 20, because n! grows very fast and the large number arithmetics involved reduce its efficiency for large n. Also the generation of the uniform distribution is better realized by the coin-tossing algorithms (essentially Knuth-Yao's algorithm [Knuth and Yao 1976]), described in Section 2; this generation algorithm will be referred to as LLKY.…”

“…However, all these algorithms have remained little known not only in computer science but also in statistics (see Berry et al [2014] and Devroye [1986]), partly because they rely on tables of random digits instead of more modern computer-generated random bits, although the underlying principle remains the same. Since a complete and rigorous analysis of the bit complexity of these algorithms remains open, for historical reasons and for completeness, we will provide a detailed analysis of the algorithms proposed in Rao [1961] and Sandelius [1962] (and partially analyzed in Plackett [1968]) and two versions of Fisher-Yates with different implementations of the underlying uniform Unif[0, n − 1] by coin tossing: one relying on von Neumann's rejection method [Devroye and Gravel 2016;von Neumann 1951] and the other on Knuth-Yao's tree method [Devroye 1986;Knuth and Yao 1976].…”

Breaking the barriers

“…There are a variety of algorithms in the literature which could be used to determine a permutation based on random numbers [5,8,10,11,13,14]. It is necessary to analyze any possible algorithm in terms of the security and complexity metrics introduced in Section 3.1.…”