Loop parallelization in the polytope model

Abstract

During the course of the last decade, a mathematical model for the parallelization of FOR-loops has become increasingly popular. In this model, a (perfect) nest of r FOR-loops is represented by a convex polytope in ℤr. The boundaries of each loop specify the extent of the polytope in a distinct dimension. Various ways of slicing and segmenting the polytope yield a multitude of guaranteed correct mappings of the loops’ operations in space-time. These transformations have a very intuitive interpretation and can be easily quantified and automated due to their mathematical foundation in linear programming and linear algebra. With the recent availability of massively parallel computers, the idea of loop parallelization is gaining significance, since it promises execution speed-ups of orders of magnitude. The polytope model for loop parallelization has its origin in systolic design, but it applies in more general settings and methods based on it will become a part of future parallelizing compilers. This paper provides an overview and future perspective of the polytope model and methods based on it.