仿射程序参数数据移动下界的自动推导

Auguste Olivry, J. Langou, L. Pouchet, P. Sadayappan, F. Rastello
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引用次数: 19

摘要

长期以来,研究人员和实践者一直致力于提高算法的计算复杂性,重点是减少执行计算所需的操作数量。然而,如今的硬件趋势清楚地表明,与计算相比,数据移动的性能和能源成本更高:高质量的算法必须尽可能地减少数据移动。算法的理论操作复杂性是必须执行的操作总数的函数,而不管它们实际执行的顺序如何。但理论上的数据移动(或I/O)复杂性是根本不同的:必须考虑所有可能的合法操作时间表,以确定可实现的最小数据移动数量,这是一个主要的理论挑战。I/O复杂度已经通过复杂的手工证明进行了研究,例如,Hong & Kung从Ω(n3/√S)对缓存大小为S的矩阵乘法进行了改进,到Smith等人的2n3/√S。虽然渐近复杂性可能足以比较广泛不同算法之间的I/O潜力,但推理的准确性取决于这些I/O下界的紧密性。准确地说,暴露常数对于实现不同算法之间的精确比较至关重要:例如,2n3/√S下界允许演示面板-面板平铺矩阵乘法的最优性。我们提出了第一个静态分析,以自动导出具有缩放常数的数据移动下界的非渐近参数表达式,用于任意仿射计算。我们的方法是全自动的,帮助算法设计者推断I/O复杂性,并对算法替代方案做出明智的决策。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Automated derivation of parametric data movement lower bounds for affine programs
Researchers and practitioners have for long worked on improving the computational complexity of algorithms, focusing on reducing the number of operations needed to perform a computation. However the hardware trend nowadays clearly shows a higher performance and energy cost for data movements than computations: quality algorithms have to minimize data movements as much as possible. The theoretical operational complexity of an algorithm is a function of the total number of operations that must be executed, regardless of the order in which they will actually be executed. But theoretical data movement (or, I/O) complexity is fundamentally different: one must consider all possible legal schedules of the operations to determine the minimal number of data movements achievable, a major theoretical challenge. I/O complexity has been studied via complex manual proofs, e.g., refined from Ω(n3/√S) for matrix-multiply on a cache size S by Hong & Kung to 2n3/√S by Smith et al. While asymptotic complexity may be sufficient to compare I/O potential between broadly different algorithms, the accuracy of the reasoning depends on the tightness of these I/O lower bounds. Precisely, exposing constants is essential to enable precise comparison between different algorithms: for example the 2n3/√S lower bound allows to demonstrate the optimality of panel-panel tiling for matrix-multiplication. We present the first static analysis to automatically derive non-asymptotic parametric expressions of data movement lower bounds with scaling constants, for arbitrary affine computations. Our approach is fully automatic, assisting algorithm designers to reason about I/O complexity and make educated decisions about algorithmic alternatives.
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