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Section: New Results

Mono-parametric Tiling

Participants : Guillaume Iooss, Sanjay Rajopadhye [Colorado State University] , Christophe Alias, Yun Zou [Colorado State University] .

Tiling is a crucial program transformation with many benefits. It improves locality, exposes parallelism, allows for adjusting the ops-to-bytes balance of codes, and can be applied at multiple levels. Allowing tile sizes to be symbolic parameters at compile time has many benefits, including efficient auto-tuning, and run-time adaptability to system variations. For polyhedral programs, parametric tiling in its full generality is known to be non-linear, breaking the mathematical closure properties of the polyhedral model. Most compilation tools therefore either avoid it by only performing fixed size tiling, or apply it only in the final, code generation step. Both strategies have limitations.

We first introduced mono-parametric partitioning, a restricted parametric, tiling-like transformation that can be used to express a tiling. We showed that, despite being parametric, it is a polyhedral transformation. We first proved that applying mono-parametric partitioning (i) to a polyhedron yields a union of polyhedra, and (ii) to an affine function produces a piecewise-affine function. We then used these properties to show how to partition an entire polyhedral program, including one with reductions. Next, we generalized this transformation to tiles with arbitrary tile shapes that can tessellate the iteration space (e.g., hexagonal, trapezoidal, etc). We showed how mono-parametric tiling can be applied at multiple levels, and how it enables a wide range of polyhedral analyses and transformations to be applied.

This work has been published as an Inria research report [14] and will be submitted to a journal. It is the extended version of the work published at IMPACT'14 [26] .