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We show how to generate random derangements efficiently by two differenttechniques: random restricted transpositions and sequential importancesampling. The algorithm employing restricted transpositions can also be used togenerate random fixed-point-free involutions only, a.k.a. random perfectmatchings on the complete graph. Our data indicate that the algorithms generaterandom samples with the expected distribution of cycle lengths, which wederive, and for relatively small samples, which can actually be very large inabsolute numbers, we argue that they generate samples indistinguishable fromthe uniform distribution. Both algorithms are simple to understand andimplement and possess a performance comparable to or better than those ofcurrently known methods. Simulations suggest that the mixing time of thealgorithm based on random restricted transpositions (in the total variancedistance with respect to the distribution of cycle lengths) is$O(n^{a}\log{n}^{2})$ with $a \simeq \frac{1}{2}$ and $n$ the length of thederangement. We prove that the sequential importance sampling algorithmgenerates random derangements in $O(n)$ time with probability $O(1/n)$ offailing.

Efficient generation of random derangements with the expected distribution of cycle lengths