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Monoparametric Tiling of Polyhedral Programs

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Abstract

Tiling is a crucial program transformation, adjusting the ops-to-bytes balance of codes to improve locality. Like parallelism, it can be applied at multiple levels. Allowing tile sizes to be symbolic parameters at compile time has many benefits, including efficient autotuning, 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 perform fixed size tiling, or apply parametric tiling in only the final, code generation step. We introduce monoparametric tiling, a restricted parametric tiling transformation. We show that, despite being parametric, it retains the closure properties of the polyhedral model. We first prove that applying monoparametric partitioning (i) to a polyhedron yields a union of polyhedra with modulo conditions, and (ii) to an affine function produces a piecewise-affine function with modulo conditions. We then use these properties to show how to tile an entire polyhedral program. Our monoparametric tiling is general enough to handle tiles with arbitrary tile shapes that can tesselate the iteration space (e.g., hexagonal, trapezoidal, etc). This enables a wide range of polyhedral analyses and transformations to be applied.

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Notes

  1. Available at https://github.com/guillaumeiooss/MPP.

  2. An online demonstration is available at http://foobar.ens-lyon.fr/mppcodegen/index.php.

  3. See the online demonstration at http://foobar.ens-lyon.fr/mppcodegen/index.php.

  4. \(x \in [|a;b|]\) meaning \(x \in [a;b]\) and x is an integer.

  5. Our library is available at https://github.com/guillaumeiooss/MPP.

  6. Our compiler is available at http://foobar.ens-lyon.fr/mppcodegen/index.php and may be tried online.

  7. http://www.cs.colostate.edu/AlphaZsvn/Development/trunk/mde/edu.csu.melange.alphaz.polybench/polybench-alpha-4.0/.

  8. The full exploration framework and logs are available at https://guillaume.iooss.fr/CART/IJPP/explo_experiments.tar.gz.

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Authors and Affiliations

  1. Inria, Paris, France

    Guillaume Iooss

  2. Inria, CNRS, ENS de Lyon, UCBL, Université de Lyon, Lyon, France

    Christophe Alias

  3. Colorado State University, Fort Collins, USA

    Sanjay Rajopadhye

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  1. Guillaume Iooss
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Iooss, G., Alias, C. & Rajopadhye, S. Monoparametric Tiling of Polyhedral Programs. Int J Parallel Prog 49, 376–409 (2021). https://doi.org/10.1007/s10766-021-00694-2

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