Parametric Analysis of Polyhedral Iteration Spaces

In the area of automatic parallelization of programs , analyzing and transforming loop nests with parametric aane 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 rst, computing the para-metric vertices of a polytope deened by a set of para-metric 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. From systolic algorithm synthesis to nested loops parallelization, a common geometric model is used, called the polytope model, leading to common fundamental mathematical problems, dealing with discrete and convex geometry, combinatorics, geometry of numbers and linear programming. One of these fundamental problems is the counting of integer points, i.e. points having integer coordinates , contained in a polytope. Moreover, this counting has to be parameterized, since scientiic algorithms often depend on size parameters. Such a counting is necessary when analyzing a parallel algorithm modeled by a polyhedral iteration space. Many kinds of analysis are reduced to this problem such as nding the number of ops executed by a loop, the number of memory locations touched by a loop, the communication volume, the maximum par-allelism, and so forth. Such information is essential to optimizing compilers when transforming a parallel program. 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. This method consists of rst computing the para-metric vertices of the considered polytope deened by a set of parametric linear constraints, and then computing the Ehrhart polynomial, i.e. a parametric expression of the number of integer points. This second part is based on results due to the mathematician E. Ehrhart, who solved this problem for parametric polytopes that depend on one positive integral parameter. We propose an extension of his results to any number of parameters, showing that this extension is made possible by a new algorithm to nd parametric vertices nding algorithm 12]. This algorithm is based on the classical representation …