An Approach for Finding Permutations Quickly: Fusion and Dimension matching

Polyhedral compilers can perform complex loop optimizations that improve parallelism and cache behaviour of loops in the input program. These transformations result in significant performance gains on modern processors which have large compute power and deep memory hierarchies. The paper, "Polyhedral Auto-transformation with No Integer Linear Programming", identifies issues that adversely affect scalability of polyhedral transformation frameworks; in particular the Pluto algorithm. The construction and solving of a complex Integer Linear Programming (ILP) problem increases the time taken by a polyhedral compiler significantly. The paper presents two orthogonal ideas, which together overcome the scalability issues in the affine scheduling problem. It first relaxes the ILP to a Linear Programming (LP) problem, thereby solving a cheaper algorithm. To overcome the sub-optimalities that arise due to this relaxation, the affine scheduling problem is decomposed into following three components: (1) Fusion and dimension matching, (2) Loop scaling and shifting, and (3) Loop skewing. This new auto-transformation framework, pluto-lp-dfp, significantly improves the time taken by the Pluto algorithm without sacrificing performance of the generated code. This report first provides proofs for the theoretical claims made in the paper surrounding relaxed LP formulation of the Pluto algorithm. The second part of the report describes an approach to find good loop fusion (or distribution) and loop permutations that enable tileability. This short report serves as the supplementary material for the paper.