Optimal Parallelogram Selection for Hierarchical Tiling

Abstract

Loop tiling is an effective optimization to improve performance of multiply nested loops, which are the most time-consuming parts in many programs. Most massively parallel systems today are organized hierarchically, and different levels of the hierarchy differ in the organization of parallelism and the memory models they adopt. To make better use of these machines, it is clear that loop nests should be tiled hierarchically to fit the hierarchical organization of the machine; however, it is not so clear what should be the exact form of these hierarchical tiles. In particular, tile shape selection is of critical importance to expose parallelism of the tiled loop nests. Although loop tiling is a well-known optimization, not much is known about tile shape selection. In this article, we study tile shape selection when the shapes are any type of parallelograms and introduce a model to relate the tile shape of the hierarchy to the execution time. Using this model, we implement a system that automatically finds the tile shapes that minimize the execution time in a hierarchical system. Our experimental results show that in several cases, the tiles automatically selected by our system outperform the most intuitive tiling schemes usually adopted by programmers because of their simplicity.

Flextended Tiles

“…We do not believe that parallelogram tiles perform better than any of these in general, and the diamond tiling evaluation provides some evidence of this [6]. Our results motivate further work in surveying all known tile shapes, revisiting the Zhou et al [36] shape selection results; we are still missing a comprehensive understanding of the relative merits of each shape, depending on the application, dataset, and target architecture.…”

Tiling and Scheduling of Three-level Perfectly Nested Loops with Dependencies on Heterogeneous Systems

“…These problems are either computationally intensive, or working on large-scale multidimensional data or both [1,2]. Nested loops are one of the most time-consuming parts and the largest sources of parallelism in these problems [3,4]. In order to meet the ever-increasing computing requirement of scientific applications, it is necessary to use high-level computational capacity and optimization techniques.…”

“…Loop optimization and parallelization have always been an important role to achieve higher performance [4]. A lot of loop optimization techniques have been developed to decrease the execution time of the nested loops and improve the performance.…”