Automatic Collapsing of Non-Rectangular Loops

International audienceLoop collapsing is a well-known loop transformation which combines some loops that are perfectly nested into one single loop. It allows to take advantage of the whole amount of parallelism exhibited by the collapsed loops, and provides a perfect load balancing of iterations among the parallel threads. However, in the current implementations of this loop optimization , as the ones of the OpenMP language, automatic loop collapsing is limited to loops with constant loop bounds that define rectangular iteration spaces, although load imbalance is a particularly crucial issue with non-rectangular loops. The OpenMP language addresses load balance mostly through dynamic runtime scheduling of the parallel threads. Nevertheless, this runtime schedule introduces some unavoidable execution-time overhead, while preventing to exploit the entire parallelism of all the parallel loops. In this paper, we propose a technique to automatically collapse any perfectly nested loops defining non-rectangular iteration spaces, whose bounds are linear functions of the loop iterators. Such spaces may be triangular, tetrahedral, trapezoidal, rhomboidal or parallelepiped. Our solution is based on original mathematical results addressing the inversion of a multi-variate polynomial that defines a ranking of the integer points contained in a convex polyhedron. We show on a set of non-rectangular loop nests that our technique allows to generate parallel OpenMP codes that outperform the original parallel loop nests, parallelized either by using options " static " or " dynamic " of the OpenMP-schedule clause

Algebraic Tiling

International audienceIn this paper, we present an ongoing work whose aim is to propose a new loop tiling technique where tiles are characterized by their volumes-the number of embedded iterations-instead of their sizes-the lengths of their edges. Tiles of quasi-equal volumes are dynamically generated while the tiled loops are running, whatever are the original loop bounds, which may be constant or depending linearly of surrounding loop iterators. The adopted strategy is to successively and hierarchically slice the iteration domain in parts of quasi-equal volumes, from the outermost to the innermost loop dimensions. Since the number of such slices can be exactly chosen, quasi-perfect load balancing is reached by choosing, for each parallel loop, the number of slices as being equal to the number of parallel threads, or to a multiple of this number. Moreover, the approach avoids partial tiles by construction, thus yielding a perfect covering of the iteration domain minimizing the loop control cost. Finally, algebraic tiling makes dynamic scheduling of the parallel threads fairly purposeless for the handled parallel tiled loops