Finding the Most Probable String and the Consensus String: an Algorithmic Study

The problem of finding the most probable string for a distribution generated by a weighted finite automaton or a probabilistic grammar is related to a number of important questions: computing the distance between two distributions or finding the best translation (the most probable one) given a probabilistic finite state transducer. The problem is undecidable with general weights and is NP-hard if the automaton is probabilistic. We give a pseudo-polynomial algorithm which computes the most probable string in time polynomial in the inverse of the probability of the most probable string itself, both for probabilistic finite automata and probabilistic context-free grammars. We also give a randomised algorithm solving the same problem.