5,507 research outputs found
Efficient generation of random derangements with the expected distribution of cycle lengths
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
with and the length of the
derangement. We prove that the sequential importance sampling algorithm
generates random derangements in time with probability of
failing.Comment: This version corrected and updated; 14 pages, 2 algorithms, 2 tables,
4 figure
Random Latin squares and Sudoku designs generation
Uniform random generation of Latin squares is a classical problem. In this
paper we prove that both Latin squares and Sudoku designs are maximum cliques
of properly defined graphs. We have developed a simple algorithm for uniform
random sampling of Latin squares and Sudoku designs. It makes use of recent
tools for graph analysis. The corresponding SAS code is annexed
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