Some Eecient Solutions to the Aane Scheduling Problem Part I One-dimensional Time

Programs and systems of recurrence equations may be represented as sets of actions which are to be executed subject to precedence constraints. In many cases, actions may be labelled by integral vectors in some iteration domain, and precedence constraints may be described by aane 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 nding closed form schedules as aane or piecewise aane functions of the iteration vector. An ee-cient algorithm is presented which reduces the scheduling problem to 1 2 a parametric linear program of small size, which can be readily solved by an eecient algorithm. R esum e De nombreux programmes ou syst emes d' equations de r ecurrence peuvent se repr esenter comme un ensemble d'actions qui doivent ^ etre ex ecut ees en respectant des contraintes de pr ec edence. En g en eral, une action peut ^ etre rep er ee par un vecteur d'it eration a coordonn ees enti eres, et les contraintes de pr ec edence se repr esentent par des relations aanes. Un ordonnancement pour un tel programme est une fonction qui donne la date d'ex ecution de chaque action. La con-naissance d'un ordonnancement permet d'estimer le parall elisme in-trins eque du programme et d'en ecrire tr es facilement une version par-all ele bien adapt ee aux multiprocesseurs ou aux r eseaux systoliques. Le pr esent travail traite de la recherche d'un ordonnancement ex-plicitement repr esent e par des fonctions aanes ou aanes par morceaux. On pr esente une m ethode tr es simple qui ram ene la question a la r esolution d'un programme lin eaire param etrique de petite taille, probl eme qui peut ^ etre eecacement r esolu.