Affine transformations have proven to be very powerful for loop restructuring due to their ability to model a very wide range of transformations. A single multi-dimensional affine function can represent a long and complex sequence of simpler transformations. Existing affine transformation frameworks like the Pluto algorithm, that include a cost function for modern multicore architectures where coarse-grained parallelism and locality are crucial, consider only a sub-space of transformations to avoid a combinatorial explosion in finding the transformations. The ensuing practical tradeoffs lead to the exclusion of certain useful transformations, in particular, transformation compositions involving loop reversals and loop skewing by negative factors. In this paper, we propose an approach to address this limitation by modeling a much larger space of affine transformations in conjunction with the Pluto algorithm's cost function. We perform an experimental evaluation of both, the effect on compilation time, and performance of generated codes. The evaluation shows that our new framework, Pluto+, provides no degradation in performance in any of the Polybench benchmarks. For Lattice Boltzmann Method (LBM) codes with periodic boundary conditions, it provides a mean speedup of 1.33× over Pluto. We also show that Pluto+ does not increase compile times significantly. Experimental results on Polybench show that Pluto+ increases overall polyhedral source-to-source optimization time only by 15%. In cases where it improves execution time significantly, it increased polyhedral optimization time only by 2.04×.

The Lattice-Boltzmann method (LBM), a promising new particle-based simulation technique for complex and multiscale fluid flows, has seen tremendous adoption in recent years in computational fluid dynamics. Even with a state-of-the-art LBM solver such as Palabos, a user has to still manually write the program using library-supplied primitives. We propose an automated code generator for a class of LBM computations with the objective to achieve high performance on modern architectures. Few studies have looked at time tiling for LBM codes. We exploit a key similarity between stencils and LBM to enable polyhedral optimizations and in turn time tiling for LBM. We also characterize the performance of LBM with the Roofline performance model. Experimental results for standard LBM simulations like Lid Driven Cavity, Flow Past Cylinder, and Poiseuille Flow show that our scheme consistently outperforms Palabos—on average by up to 3× while running on 16 cores of an Intel Xeon (Sandybridge). We also obtain an improvement of 2.47× on the SPEC LBM benchmark.

Affine transformations have proven to be powerful for loop restructuring due to their ability to model a very wide range of transformations. A single multidimensional affine function can represent a long and complex sequence of simpler transformations. Existing affine transformation frameworks such as the Pluto algorithm, which include a cost function for modern multicore architectures for which coarse-grained parallelism and locality are crucial, consider only a subspace of transformations to avoid a combinatorial explosion in finding transformations. The ensuing practical trade-offs lead to the exclusion of certain useful transformations: in particular, transformation compositions involving loop reversals and loop skewing by negative factors. In addition, there is currently no proof that the algorithm successfully finds a tree of permutable loop bands for all affine loop nests. In this article, we propose an approach to address these two issues (1) by modeling a much larger space of practically useful affine transformations in conjunction with the existing cost function of Pluto, and (2) by extending the Pluto algorithm in a way that allows a proof for its soundness and completeness for all affine loop nests. We perform an experimental evaluation of both, the effect on compilation time, and performance of generated codes. The evaluation shows that our new framework, Pluto+, provides no degradation in performance for any benchmark from Polybench. For the Lattice Boltzmann Method (LBM) simulations with periodic boundary conditions, it provides a mean speedup of 1.33 × over Pluto. We also show that Pluto+ does not increase compilation time significantly. Experimental results on Polybench show that Pluto+ increases overall polyhedral source-to-source optimization time by only 15%. In cases in which it improves execution time significantly, it increased polyhedral optimization time by only 2.04 × .

Polyhedral auto-transformation frameworks are known to find efficient loop transformations that maximize locality and parallelism and minimize synchronization. While complex loop transformations are routinely modeled in these frameworks, they tend to rely on ad hoc heuristics for loop fusion. Although there exist multiple loop fusion models with cost functions to maximize locality and parallelism, these models involve separate optimization steps rather than seamlessly integrating with other loop transformations like loop permutation, scaling, and shifting. Incorporating parallelism-preserving loop fusion heuristics into existing affine transformation frameworks like Pluto, LLVM-Polly, PPCG, and PoCC requires solving a large number of Integer Linear Programming formulations, which increase auto-transformation times significantly. In this work, we incorporate polynomial time loop fusion heuristics into the Pluto-lp-dfp framework. We present a data structure called the fusion conflict graph (FCG), which enables us to efficiently model loop fusion in the presence of other affine loop transformations. We propose a clustering heuristic to group the vertices of the FCG, which further enables us to provide three different polynomial time greedy fusion heuristics, namely, maximal fusion, typed fusion, and hybrid fusion, while maintaining the compile time improvements of Pluto-lp-dfp over Pluto. Our experiments reveal that the hybrid fusion model, in conjunction with Pluto's cost function, finds efficient transformations that outperform PoCC and Pluto by mean factors of 1.8× and 1.07×, respectively, with a maximum performance improvement of 14× over PoCC and 2.6× over Pluto.

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