Tuilage monoparamétré de programmes polyédriques

Tiling is a crucial program transformation, adjusting the ops-to-bytes balance of codes to improvelocality. Like parallelism, it can be applied at multiple levels. Allowing tile sizes to be symbolicparameters at compile time has many benefits, including ecient autotuning, and run-time adaptability tosystem 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 thereforeeither perform fixed size tiling, or apply parametric tiling in only the final, code generation step. We introducemonoparametric tiling, a restricted parametric tiling transformation. We show that, despite beingparametric, it retains the closure properties of the polyhedral model. We first prove that applying monoparametricpartitioning (i) to a polyhedron yields a union of polyhedra, and (ii) to an ane function produces apiecewise-ane function. 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 theiteration space (e.g., hexagonal, trapezoidal, etc). This enables a wide range of polyhedral analyses andtransformations to be applied