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Permutation tests are among the simplest and most widely used statisticaltools. Their p-values can be computed by a straightforward sampling ofpermutations. However, this way of computing p-values is often so slow that itis replaced by an approximation, which is accurate only for part of theinteresting range of parameters. Moreover, the accuracy of the approximationcan usually not be improved by increasing the computation time.  We introduce a new sampling-based algorithm which uses the fast Fouriertransform to compute p-values for the permutation test based on Pearson'scorrelation coefficient. The algorithm is practically and asymptotically fasterthan straightforward sampling. Typically, its complexity is logarithmic in theinput size, while the complexity of straightforward sampling is linear. Theidea behind the algorithm can also be used to accelerate the computation ofp-values for many other common statistical tests. The algorithm is easy toimplement, but its analysis involves results from the representation theory ofthe symmetric group.

Fast computation of p-values for the permutation test based on Pearson's correlation coefficient and other statistical tests