Given a program written in a simple imperative language (assignment statements, ./or loops, affine indices and loop limits), this paper presents an algorithm for analyzing the patterns along which values flow as the execution proceeds. For each array or scalar reference, the result is the name and iteration vector of the source statement as a function of the iteration vector of the referencing statement. The paper discusses several applications of the method: conversion of a program to a set of recurrence equations, array and scalar expansion, program verification and parallel program construction.

AffineThis paper extends the algorithms which were developed in Part I to cases in which there is no affine schedule, i.e. to problems whose parallel complexity is polynomial but not linear. The natural generalization is to multidimensional schedules with lexicographic ordering as temporal succession. Multidimensional affine schedules, are, in a sense, equivalent to polynomial schedules, and are much easier to handle automatically. Furthermore, there is a strong connection between multidimensional schedules and loop nests, which allows one to prove that a static control program always has a multidimensional schedule. Roughly, a larger dimension indicates less parallelism. In the algorithm which is presented here, this dimension .is computed dynamically, and is just sufficient for scheduling the source program. The algorithm lends itself to a "divide and conquer" strategy. The paper gives some experimental evidence for the applicability, performances and limitations of the algorithm.

Programs and systems of recurrence equations may be represented as sets of actions which are to be executed subject to precedence constraints. In may cases, actions may be labelled by integral vectors in some iterations domains, and precedence constraints may be described by affine relations. A schedule for such a program is a function which assigns an execution date to each action. Knowledge of such a schedule allows one to estimate the intrinsic degree of parallelism of the program and to compile a parallel version for multiprocessor architectures or systolic arrays. This paper deals with the problem of finding closed form schedules as affine or piecewise affine functions of the iteration vector. An algorithm is presented which reduces the scheduling problem to a parametric linear program of small size, which can be readily solved by an efficient algorithm.

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Revue française d'automatique, d'informatique et de recherche opérationnelle. Recherche opérationnelle, tome 22, n o 3 (1988), p. 243-268. © AFCET, 1988, tous droits réservés. L'accès aux archives de la revue « Revue française d'automatique, d'informatique et de recherche opérationnelle. Recherche opérationnelle » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/ legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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Asynchronous CALL statements are necessary in order to use more than one processor in current multiprocessors. Detecting CALL statements that may be executed in parallel is one way to fill this need. This approach requires accurate approximations of called procedure effects. This is achieved by using new objects called Region and Ezecution Contezt. An algorithm to find asynchronous CALL statements is given. It involves a new dependence test to compute data dependence graphs, which provides better results than previous ones even when no CALL statements are involved. This method has been implemented in Parafrase and preliminary results are encouraging.