On the Scalability of Loop Tiling Techniques

The Polyhedral model has proven to be a valuable tool for improving memory locality and exploiting parallelism for optimizing dense array codes. This model is expressive enough to describe transformations of imperfectly nested loops, and to capture a variety of program transformations, including many approaches to loop tiling. Tools such as the highly successful PLuTo automatic parallelizer have provided empirical confirmation of the success of polyhedral-based optimization, through experiments in which a number of benchmarks have been executed on machines with smallto medium-scale parallelism. In anticipation of ever higher degrees of parallelism, we have explored the impact of various loop tiling strategies on the asymptotic degree of available parallelism. In our analysis, we consider “weak scaling” as described by Gustavson, i.e., in which the data set size grows linearly with the number of processors available. Some, but not all, of the approaches to tiling provide weak scaling. In particular, the tiling currently performed by PLuTo does not scale in this sense. In this article, we review approaches to loop tiling in the published literature, focusing on both scalability and implementation status. We find that fully scalable tilings are not available in general-purpose tools, and call upon the polyhedral compilation community to focus on questions of asymptotic scalability. Finally, we identify ongoing work that may resolve this issue.