Optimal fine and medium grain parallelism detection in polyhedral reduced dependence graphs

This paper presents an optimal algorithm for detecting fine or medium grain parallelism in nested loops whose dependences are described by an approximation of distance vectors by polyhedra. In particular, this algorithm is optimal for the classical approximation by direction sectors, This result generalizes, to the case of several statements, Wolf and Lam's algorithm which is optimal for a single statement. Our algorithm relies on a dependence uniformization process and on parallelization techniques related to system of uniform recurrence equations. It can also be viewed as a combination of both Alien and Kennedy's algorithm and Wolf and Lam's algorithm.