Parametric analysis of polyhedral iteration spaces

In the area of automatic parallelization of programs, analyzing and transforming loop nests with parametric affine loop bounds requires fundamental mathematical results. The most common geometrical model of iteration spaces, called the polytope model, is based on mathematics dealing with convex and discrete geometry, linear programming, combinatorics and geometry of numbers. In this paper, we present an automatic method for computing the number of integer points contained in a convex polytope or in a union of convex polytopes. The procedure consists of first, computing the parametric vertices of a polytope defined by a set of parametric linear constraints, and then computing the Ehrhart polynomial, i.e. a parametric expression of the number of integer points. The paper is illustrated with the computation of the maximum available parallelism of a given loop nest