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We present two novel approaches for the computation of the exact distributionof a pattern in a long sequence. Both approaches take into account the sparsestructure of the problem and are two-part algorithms. The first approach relieson a partial recursion after a fast computation of the second largesteigenvalue of the transition matrix of a Markov chain embedding. The secondapproach uses fast Taylor expansions of an exact bivariate rationalreconstruction of the distribution. We illustrate the interest of bothapproaches on a simple toy-example and two biological applications: thetranscription factors of the Human Chromosome 5 and the PROSITE signatures offunctional motifs in proteins. On these example our methods demonstrate theircomplementarity and their hability to extend the domain of feasibility forexact computations in pattern problems to a new level.

Sparse approaches for the exact distribution of patterns in long state sequences generated by a Markov source