A Quantitative Analysis of Tile Size Selection Algorithms

Abstract

Loop tiling is an effective optimizing transformation to boost the memory performance of a program, especially for dense matrix scientific computations. The magnitude and stability of the achieved performance improvements are heavily dependent on the appropriate selection of tile sizes. Many existing tile selection algorithms try to find tile sizes which eliminate self-interference cache conflict misses, maximize cache utilization, and minimize cross-interference cache conflict misses. These techniques depend heavily on the actual layout of the arrays in memory. Array padding, an effective data layout optimization technique, is therefore incorporated by many algorithms to help loop tiling stabilize its effectiveness by avoiding “pathological” array sizes.

In this paper, we examine several such combined algorithms in terms of cost-benefit trade-offs, and introduce a new algorithm. The preliminary experimental results show that more precise and costly tile selection and array padding algorithms may not be justified by the resulting performance improvements since such improvements may also be achieved by much simpler and therefore less expensive strategies. The key issues in finding a good tiling algorithm are (1) to identify critical performance factors and (2) to develop corresponding performance models that allow predictions at a sufficient level of accuracy. Following this insight, we have developed a new tiling algorithm that performs better than previous algorithms in terms of execution time and stability, and generates code with a performance comparable to the best measured algorithm. Experimental results on two standard benchmark kernels for matrix multiply and LU factorization show that the new algorithm is orders of magnitude faster than the best previous algorithm without sacrificing stability and execution speed of the generated code.