Experiences with Enumeration of Integer Projections of Parametric Polytopes

Abstract

Abstract. Many compiler optimization techniques depend on the ability to calculate the number of integer values that satisfy a given set of linear constraints. This count (the enumerator of a parametric polytope) is a function of the symbolic parameters that may appear in the constraints. In an extended problem (the "integer projection" of a parametric polytope), some of the variables that appear in the constraints may be existentially quantified and then the enumerated set corresponds to the projection of the integer points in a parametric polytope. This paper shows how to reduce the enumeration of the integer projection of parametric polytopes to the enumeration of parametric polytopes. Two approaches are described and experimentally compared. Both can solve problems that were considered very difficult to solve analytically.

Integer Polyhedra for Program Analysis

“…This work has been extended to count points in the projection of a constrained space [21]. The results of these systems are impressive, but are still slower than sampling based techniques for estimating the number of integer points in a polytope as used in the approximation thread of this work.…”

Signal Assignment Model for the Memory Management of Multidimensional Signal Processing Applications

“…If the columns are concatenated decreasingly ((A[index 1 ][index 2 ], index 1 = 0, 18), index 2 = 9, 0), there are 9 pairs of elements simultaneously alive, mapped at a maximum distance. Two such elements are, for instance, A [14][0]-produced in the iteration (i = 14, j = 0) and consumed in (i = 24, j = 0), and A [4] [9]-produced in the iteration (i = 4, j = 9) and consumed in (i = 14, j = 9), the distance between them in the linearization being 9 × 19 + 10 = 181. , index 2 = 0, 9), index 1 = 18, 0), there are 9 pairs of elements simultaneously alive, as well (e.g., A [14][0] and A [4] [9]), placed at the maximum distance 10 × 10 + 9 = 109.…”

“…Figure 4 displays an illustrative code and the index (array) space of the array reference A [i][ j], all the points (A-elements) being black (alive) after the execution of the loop nest. According to the mapping model [5], the problem is to compute the integer projections [14] of this lattice on the m = 2 coordinate axes, the elements of the m-dimensional memory window being the sizes of these m projections. For this example, W = (12 − 2 + 1, 10 − 4 + 1) = (11, 7).…”

System-level exploration tool for energy-awarememory management in the design of multidimensional signal processing systems

“…In addition, this decomposition allows to obtain lifetime information on entire groups of array elements (which is sufficient for most memory management tasks), without the need of operating with individual array elements (which is a strategy prohibitively time-consuming [5]). The decomposition into disjoint lattices is analytically performed, using intersections and differences of lattices -operations quite complex [2] involving computations of Hermite Normal Forms, solving Diophantine linear systems [12], computing the vertices of Z-polytopes [1] and their supporting polyhedral cones, counting integral points in Z-polyhedra [3], and computing integer projections of polytopes [14].…”

“…In this way, the problem reduces to the computation of the projection of a Z-polytope, this latter problem being well-studied (e.g., [14]). …”