Complexity of the parallel evaluation of arithmetic circuits
Complexite de l'evaluation parallele des circuits arithmetiques
Résumé
Algorithms for the parallel evaluation of expressions and arithmetic circuits may be considered as extractors of the intrinsic parallelism contained in sequential programs; far beyond the parallelism that can be read from the dependence graph, this parallelism comes from the meaning of the operators that are employed. The knowledge of their algebraic properties, such as associativity or distributivity, allows the reorganization of the computations without affecting the results. The more the algebraic structure used in the program possesses such properties, the more they can be taken into account to speed up the parallel evaluation of the program. We generalize the algorithms designed for programs over semi-rings in order to propose an algorithm, the complexity of which improves previously known upper bounds for the evaluation of arithmetic circuits over lattices. Simulations of this algorithm highlight its power as an ``automatic complexity predictor''. Furthermore, the explicit reorganization of the computations by means of these evaluation algorithms, by means of a complete compilation, helps to compare real algorithms for distributed memory machines with theoretical parallel ones. A prototype of the compiler has been developed, based on a simplification/extension of the C language. Then, the use of these techniques in the area of automatic parallelization of nested loops is discussed: they can help the detection of hidden reductions in these nested loops in an easy and efficient way, by providing relevant information on the probability of the existence of reductions. This last point proves that the design of theoretical parallel algorithms is related to the search for effective parallelization.
Les algorithmes d'evaluation parallele des expressions et des circuits arithmetiques peuvent etre vus comme des extracteurs du parallelisme intrinseque contenu dans les programmes sequentiels, parallelisme qui depasse celui qui peut etre lu sur le graphe de precedence et qui tient a la semantique des operateurs utilises. La connaissance des proprietes algebriques, comme l'associativite ou la distributivite, permet une reorganisation des calculs qui n'affecte pas les resultats. Plus la structure algebrique utilisee sera riche en proprietes, plus il sera possible d'en tirer parti pour ameliorer les algorithmes d'evaluation. Generalisant les algorithmes concus pour les semi-anneaux, nous proposons un algorithme qui ameliore les majorations precedemment connues pour la contraction de circuits arithmetiques dans un treillis. Des simulations de cet algorithme ont permis de mettre en evidence ses qualites de << predicteur automatique de complexite >>. Reorganiser explicitement les calculs a l'aide de ces algorithmes, c'est-a-dire realiser un compilateur complet, permet de comparer la realite des algorithmes paralleles sur machines a memoire distribuee et la puissance des algorithmes theoriques. Un prototype a ete realise, base sur une simplification/extension du langage C. Enfin, l'interet de ces techniques dans le domaine de la parallelisation des nids de boucles, pour guider la recherche de reductions cachees dans ces nids, semble prometteuse, parce qu'elle est peu couteuse a mettre en oeuvre et fournit des informations de qualite. En cela, les recherches en algorithmique parallele theorique rejoignent les preoccupations de la parallelisation effective.
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