Code Generation Methods for tiling transformations

Tiling or supernode transformation has been widely used to improve locality in multi-level memory hierarchies, as well as to efficiently execute loops onto parallel architectures. However, automatic code generation for tiled loops can be a very complex compiler work due to non-rectangular tile shapes and arbitrary iteration space bounds. In this paper, we first survey code generation methods for nested loops which are transformed using non-unimodular transformations. All methods are based on Fourier-Motzkin (FM) elimination. Next, we consider and enhance previous work on rewriting tiled loops by considering parallelepiped tiles and arbitrary iteration space shapes. In order to generate tiled code, all methods first enumerate the tiles containing points within the iteration space, and second, sweep the points within each tile. For the first, we extend previous work in order to access all tile origins correctly, while for the latter, we propose the transformation of the initial parallelepiped tile iteration space into a rectangular one, so as to generate code efficiently with the aid of a non-unimodular transformation matrix and its Hermite Normal Form (HNF). The resulting systems of inequalities are much simpler than those appeared in bibliography; thus their solutions are more efficiently determined using the FM elimination. Experimental results which compare all presented approaches, show that the proposed method for generating tiled code is significantly accelerated, thus rewriting any $n$-D tiled loop in a much more efficient and direct way