We show how to generate random derangements efficiently by two different techniques: random restricted transpositions and sequential importance sampling. The algorithm employing restricted transpositions can also be used to generate random fixed-point-free involutions only, a. k. a. random perfect matchings on the complete graph. Our data indicate that the algorithms generate random samples with the expected distribution of cycle lengths, which we derive, and for relatively small samples, which can actually be very large in absolute numbers, we argue that they generate samples indistinguishable from the uniform distribution. Both algorithms are simple to understand and implement and possess a performance comparable to or better than those of currently known methods. Simulations suggest that the mixing time of the algorithm based on random restricted transpositions (in the total variance distance 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 the derangement. We prove that the sequential importance sampling algorithm generates random derangements in O(n) time with probability O(1/n) of failing.

中文翻译：

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高效生成随机周期，并具有预期的周期长度分布

我们展示了如何通过两种不同的技术有效地生成随机排列：随机受限换位和顺序重要性采样。采用受限换位的算法也可以仅用于生成随机无定点对合。k。一种。完整图上的随机完美匹配。我们的数据表明，该算法生成的随机样本具有预期的周期长度分布（由我们得出），而对于相对较小的样本（实际上绝对数可能非常大），我们认为它们生成的样本与均匀分布没有区别。两种算法都易于理解和实现，并且具有与当前已知方法相当或更好的性能。\（O（n ^ {a} \ log {n} ^ {2}）\）与\（a \ simeq \ frac {1} {2} \）和n排列的长度。我们证明了顺序重要性抽样算法会在O（n）时间内以O（1 / n）失败的概率生成随机排列。