Construction of Do Loops From Systems of Affine Constraints

Most parallelization techniques for [Formula: see text] loop nests are based on reindexation. Reindexation yields a new iteration space, which is a convex integer polyhedron defined by a set of affine constraints. Parallel code generation thus needs to scan all the integer points of this convex, thereby requiring the construction of a new [Formula: see text] loop nest. We detail an algorithm to this purpose, which relies on a parametrized version of the Dual Simplex. We show how the resulting loop nest and especially the loop bounds can be kept simple, thus reducing the control overhead of parallelization to a minimum.