Two optimal parallel algorithms on the commutation class of a word

Article dans revue scientifique avec comité de lecture. internationale.International audienceThe free partially commutative monoid $M(A,\Theta)$ defined by a set of commutation relations $\Theta$ on an alphabet $A$ can be viewed as a model for concurrent computing: indeed, the independence or the simultaneity of two actions can be interpreted by the commutation of two letters that encode them. In this context, the commutation class $C_{\Theta}(w)$ of a word $w$ of the free monoid $A^{*}$ plays a crucial role. In this paper we present an optimal parallel algorithm which computes the minimal automaton of the commutation class of a given word and an optimal parallel algorithm for testing if a word belongs to this commutation class. The prominent original new features of our approach are the notions of $\Theta$-dissection of a word, reference word and the use of an automaton whose construction is based on maps. It differs completely from the methods (based on Foata's normal form) used by C. Cérin and A. Petit \cite{CEa, CEb, CP} for solving similar problems. The second algorithm has a time complexity in $O(log(n))$ if the number of processors is in $O(n)$ where $n$ is the length of the word $w$. The total number of operations of our parallel algorithm is in $O(n)$ and does not depend on the size of the alphabet $A$ as for the classical sequential algorithm. \end{abstract