The Relation Between Diamond Tiling and Hexagonal Tiling

Iterative stencil computations are important in scientific computing and more and more also in the embedded and mobile domain. Recent publications have shown that tiling schemes that ensure concurrent start provide efficient ways to execute these kernels. Diamond tiling and hybrid-hexagonal tiling are two successful tiling schemes that enable concurrent start. Both have different advantages: diamond tiling is integrated in a general purpose optimization framework and uses a cost function to choose among tiling hyperplanes, whereas the more flexible tile sizes of hybrid-hexagonal tiling have proven to be effective for the generation of GPU code. We show that these two approaches are even more interesting when combined. We revisit the formalization of diamond and hexagonal tiling, present the effects of tile size and wavefront choices on tile-level parallelism, and formulate constraints for optimal diamond tile shapes. We then extend the diamond tiling formulation into a hexagonal tiling one, combining the benefits of both. The paper closes with an outlook of hexagonal tiling in higher dimensional spaces, an important generalization suitable for massively parallel architectures.