Analysis of multidimensional loops with non-uniform dependences

For a parallelizing compiler, mainly based on loop transformations, dependence information that is as complete and precise as possible is required. In this paper, we propose a generalized method for computing, in any multi-dimensional loop, information which proved to be useful in the case of irregular dependences. Firstly, we solve the basic problem of the existence of a dependence with an algorithm composed of a preprocessing phase of reduction and of an integer simplex resolution. If a solution exists, we compute by integer simplex the bounds of the distances associated with loop indices. Depending on the values of these bounds, we finally define problems consisting in evaluating the bounds of slopes of dependence vectors, which we solve by integer linear fractional programming. The amount of computation for each new problem is very low. This algorithm has been implemented as an extension of the Janus Test, which was presented in a previous work