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Effective Loop Fusion in Polyhedral Compilation Using Fusion Conflict Graphs

ACM TRANSACTIONS ON ARCHITECTURE AND CODE OPTIMIZATION 2020年第4期17卷 1–26页

作者： Acharya, Aravind Bondhugula, Uday Cohen, Albert Indian Inst Sci Dept Comp Sci & Automat Bengaluru 560012 Karnataka India Google 8 Rue Londres F-75009 Paris France

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 1.8x and 1.07x, respectively, with a maximum performance improvement of 14x over PoCC and 2.6x over pluto.

关键词： Affine transformations pluto algorithm fusion conflict graph

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Polyhedral Auto-transformation with No Integer Linear Programming 2018

Polyhedral Auto-transformation with No Integer Linear Progra...

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39th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI)

作者： Acharya, Aravind Bondhugula, Uday Cohen, Albert Indian Inst Sci Dept Comp Sci & Automat Bangalore Karnataka India INRIA Paris France DI Ecole Normale Super Paris France

ISBN: (纸本)9781450356985

State-of-the-art algorithms used in automatic polyhedral transformation for parallelization and locality optimization typically rely on Integer Linear Programming (ILP). This poses a scalability issue when scaling to tens or hundreds of statements, and may be disconcerting in production compiler settings. In this work, we consider relaxing integrality in the ILP formulation of the pluto algorithm, a popular algorithm used to find good affine transformations. We show that the rational solutions obtained from the relaxed LP formulation can easily be scaled to valid integral ones to obtain desired solutions, although with some caveats. We first present formal results connecting the solution of the relaxed LP to the original pluto ILP. We then show that there are difficulties in realizing the above theoretical results in practice, and propose an alternate approach to overcome those while still leveraging linear programming. Our new approach obtains dramatic compile-time speedups for a range of large benchmarks. While achieving these compile-time improvements, we show that the performance of the transformed code is not sacrificed. Our approach to automatic transformation provides a mean compilation time improvement of 5.6x over state-of-the-art on relevant challenging benchmarks from the NAS PB, SPEC CPU 2006, and PolyBench suites. We also came across situations where prior frameworks failed to find a transformation in a reasonable amount of time, while our new approach did so instantaneously.

关键词： Polyhedral transformation pluto algorithm

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