Paul Feautrier

Fine-Grain Scheduling under Resource Constraints

Many present-day microprocessors have fine grain parallelism, be it in the form of a pipeline, of multiple functional units, or replicated processors. The efficient use of such architectures depends on the capability of the compiler to schedule the execution of the object code in such a way that most of the available hardware is in use while complying with data dependences. In the case of one simple loop, the schedule may be expressed as an affine form in the loop counter. The coefficient of the loop counter in the schedule is the initiation interval, and gives the mean rate at which loop bodies may be executed. The dependence constraints may be converted to linear inequalities in the coefficients of a closed form schedule, and then solved by classical linear programming algorithms. The resource constraints, however, translate to non-linear constraints. These constraints become linear if the initiation interval is known. This leads to a fast searching algorithm, in which the initiation interval is increased until a feasible solution is found.

On the Equivalence of Two Systems of Affine Recurrence Equations

This paper deals with the problem of deciding whether two Systems of Affine Recurrence Equations are equivalent or not. A solution to this problem would be a step toward algorithm recognition, an important tool in program analysis, optimization and parallelization. We first prove that in the general case, the problem is undecidable. We then show that there nevertheless exists a semi-decision procedure, in which the key ingredient is the computation of transitive closures of affine relations. This is a non-effective process which has been extensively studied. Many partial solutions are known. We then report on a pilot implementation of the algorithm, describe its limitations, and point to unsolved problems.

Forward communication only placements and their use for parallel program construction

The context of this paper is automatic parallelization by the space-time mapping method. One key issue in that approach is to adjust the granularity of the derived parallelism. For that purpose, we use tiling in the space and time dimensions. While space tiling is always legal, there are constraints on the possibility of time tiling, unless the placement is such that communications always go in the same direction (forward communications only). We derive an algorithm that automatically constructs an FCO placement – if it exists. We show that the method is applicable to many familiar kernels and that it gives satisfactory speedups.

Detection of Recurrences in Sequential Programs with Loops

To improve the performances of parallelizing compilers, one must detect recurrences in scientific programs and subject them to special parallelization methods. We present a method for detecting recurrences which is based on the analysis of Systems of Recurrence Equations. This method identifies recurrences on arrays, recurrences of arbitrary order and multi-equations recurrences. We explain how to associate a SRE to a restricted class of imperative programs. We present a normalization of such SRE that allows the detection of recurrences by simple inspection of equations. When detected, a recurrence may be replaced by a symbolic expression of its solution. To iterate the process can lead to the identification of multi-dimensional recurrences.