On the extension of the polytope model to particular non-affine transformations

The polytope model combines a set of methods and approaches based on a mathematical model to define a new general framework for automatic loop parallelization. The parallelization of n nested loops in this model proceeds in three steps: first modeling the loop nest by a convex polytope, second applying an affine coordinate transformation i.e. finding a schedule " t " and an allocation " a " on the index space I which must be both affine. Mathematically, this is expressed as follows:∀ x∈I, t (x) = λx+α (λ ∈ Z r ×Z n , α∈ Z r) and ∀ x ∈ I, a (x) = σx+β (σ∈Z n-r ×Z n , β∈ Z n-r). T, the matrix constituted by the r rows of λ followed by the n-r rows of σ, is the requested affine transformation. Once the schedule is determined, the allocation must be chosen in a way such that the transformation is invertible (|T|≠ 0). The target polytope is then expressed as (AT-1 , b); A is the integer matrix associated to the source polytope Ax ≤ b denoted (A, b). The third step is translating this polytope back into a nest of target loops. Recently, the basic model [3] was extended to permit the use of both affine by statement and piece-wise affine functions for schedules and allocations, and to deal with singular matrices [1]. In this paper, we propose another extension of the polytope model to permit the use of a particular class of non-affine space/time mapping transformations, we define as follows: T = if i ∈ I 1 then T 1 : if i ∈ I m then T m Where {I 1 , I 2 ,…,I m } is a partition of I and T 1 , T 2 ,…,T m are different affine transformations. Every subset I k (1≤k≤m) is characterized by a logical condition C k. Notice here that a subset I k may not be a polytope, thus it can't be treated by the polytope model. We then propose the following solution: for each k (1≤k≤m), we apply the polytope model on the whole iteration space I, using T k. Then, when generating the target program, we use C k to select only iterations corresponding to the image of I k , i.e. if I k = {x ∈ I / C k (x)} then T k (I k) = {x …