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The technique of Finite Markov Chain Imbedding (FMCI) is a classical approach to complex combinatorial problems related to sequences. In order to get efficient algorithms, it is known that such approaches need to be first rewritten using recursive relations. We propose here to give here a general recursive algorithms allowing to compute in a numerically stable manner exact Cumulative Distribution Function (CDF) or complementary CDF (CCDF). These algorithms are then applied in two particular cases: the local score of one sequence and pattern statistics. In both cases, asymptotic developments are derived. For the local score, our new approach allows for the very first time to compute exact p-values for a practical study (finding hydrophobic segments in a protein database) where only approximations were available before. In this study, the asymptotic approximations appear to be completely unreliable for 99.5% of the considered sequences. Concerning the pattern statistics, the new FMCI algorithms dramatically outperform the previous ones as they are more reliable, easier to implement, faster and with lower memory requirements.

Effective p-value computations using Finite Markov Chain Imbedding (FMCI): application to local score and to pattern statistics