Tiling is a critical loop transformation for performance optimization. 
However, despite significant advances in developing integrated affine 
transformation frameworks, tiling has not been as tightly integrated as other 
transformations such as permutation, skewing, reversal, fusion, and fission. 
The main reason for the somewhat isolated treatment of tiling is that tiled 
code has more loops than an equivalent untiled version, creating a mismatch 
in the dimensionalities of their iteration spaces. Therefore, direct modeling 
of tiling in an affine transformation framework is problematic. Consequently, 
in affine frameworks, tiling has generally been modeled indirectly, in 
terms of identifying permutable bands of loops in the transformed space; 
tiling is performed as a subsequent step after other transformations have 
already been determined.

In this talk, we consider the benefits of taking a dual view of affine loop 
transforms: as partitioners in the iteration spaces, in addition to the 
usual view as generators of dependence-preserving time schedules. This 
enables a more integrated treatment of tiling with other loop 
transformations.  We show how direct modeling of concurrency in tiled 
execution and communication minimization for coarse-grained parallelization 
are facilitated in an affine transformation framework.