Exact analysis of the cache behavior of nested loops

Abstract

We develop from first principles an exact model of the behavior of loop nests executing in a memory hierarchy, by using a nontraditional classification of misses that has the key property of composability. We use Presburger formulas to express various kinds of misses as well as the state of the cache at the end of the loop nest. We use existing tools to simplify these formulas and to count cache misses. The model is powerful enough to handle imperfect loop nests and various flavors of non-linear array layouts based on bit interleaving of array indices. We also indicate how to handle modest levels of associativity, and how to perform limited symbolic analysis of cache behavior. The complexity of the formulas relates to the static structure of the loop nest rather than to its dynamic trip count, allowing our model to gain efficiency in counting cache misses by exploiting repetitive patterns of cache behavior. Validation against cache simulation confirms the exactness of our formulation. Our method can serve as the basis for a static performance predictor to guide program and data transformations to improve performance.

Bounding Worst-Case Data Cache Behavior by Analytically Deriving Cache Reference Patterns

“…They include the Cache Miss Equations by Ghosh et al [6], which we have built upon, a probabilistic method of analysis as proposed by Fraguella et al [5] and another analytical method by Chatterjee et al [3]. The basic idea behind all these methods is the same -to characterize data cache behavior by means of a set of mathematical equations.…”

“…The formula-based method [3] is a method that models cache behavior exactly using Presburger formulae to specify cache misses. This method can also deal with multiple loop nests and conditionals.…”

Automatic Tuning of Discrete Fourier Transforms Driven by Analytical Modeling

“…Models are particularly useful for larger problem sizes, where the performance is strongly impacted by the memory hierarchy. Several cache models have been proposed, including the Cache Miss Equations [9], the Presburger formula based approach in [10], the Stack Distances Model [11], and the Probabilistic Miss Equations [12]. All of these models support loop nests with regular access patterns similar to the ones considered in this paper.…”

Tightening the Bounds on Feasible Preemption Points

“…They include the Cache Miss Equations by Ghosh et al [7], a probabilistic analysis method proposed by Fraguella et al [6] and another analytical method by Chatterjee et al [5]. The common idea behind these methods is to characterize data cache behavior by means of a set of mathematical equations.…”