Abstract
Parameterized tiled loops-where the tile sizes are not fixed at compile time, but remain symbolic parameters until later--are quite useful for iterative compilers and auto-tuners that produce highly optimized libraries and codes. Tile size parameterization could also enable optimizations such as register tiling to become dynamic optimizations. Although it is easy to generate such loops for (hyper) rectangular iteration spaces tiled with (hyper) rectangular tiles, many important computations do not fall into this restricted domain. Parameterized tile code generation for the general case of convex iteration spaces being tiled by (hyper) rectangular tiles has in the past been solved with bounding box approaches or symbolic Fourier Motzkin approaches. However, both approaches have less than ideal code generation efficiency and resulting code quality. We present the theoretical foundations, implementation, and experimental validation of a simple, unified technique for generating parameterized tiled code. Our code generation efficiency is comparable to all existing code generation techniques including those for fixed tile sizes, and the resulting code is as efficient as, if not more than, all previous techniques. Thus the technique provides parameterized tiled loops for free! Our one-size-fits-all solution, which is available as open source software can be adapted for use in production compilers.
Highlights
The algorithm implemented on a parallel computer with distributed memory has, as a rule, a tiled structure: a set of operations is divided into subsets, called tiles
One of the modern approaches to obtaining tiled versions of algorithms is a tiling transformation based on information sections of the iteration space, resulting in macro-operations
Compared to one-dimensional structures, the use of two-dimensional structures is possible in a smaller number of cases, but it can have advantages when implementing algorithms on parallel computers with distributed memory
Summary
Реализуемый на параллельном компьютере с распределенной памятью, имеет, как правило, зернистую структуру: множество операций разбито на подмножества, называемые зернами вычислений. В настоящей работе для алгоритмов, заданных вложенными многомерными циклами, предложен способ построения зернистых вычислительных процессов, логически организованных в двумерную структуру. По сравнению с одномерными структурами, использование двумерных структур возможно в меньшем числе случаев, но может иметь преимущества при реализации алгоритмов на параллельных компьютерах с распределенной памятью. Представленные исследования обобщают на случай двумерной структуры некоторые аспекты метода построения параллельных вычислительных процессов, организованных в одномерную структуру. В более ранних теоретических исследованиях параллельность зернистых вычислений гарантировалась при наличии информационных разрезов по всем координатам итерационного пространства, а для более простого случая одномерной структуры – по двум координатам. Ключевые слова: параллельные вычисления, распараллеливание алгоритмов, параллельный компьютер с распределенной памятью, уменьшение числа обменов данными.
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