The Subspace Model: A Theory of Shapes for Parallel Systems

Abstract

This paper presents a shape based abstraction for compiling to parallel systems. Data layout is often the subject of direct analysis while shape is addressed in ad hoc ways at best. However, a suboptimal shape can be more costly than a suboptimal location. Unnecessary serialization can result if the shape used is too small and unnecessary communication and computation can result if the shape is too large. For each dimension in the space of an object, the object attains that dimension, serially, in parallel or via a parallel prefix operation. These expansion categories are O(N), O(1) and O(log N) respectively, where N is the extent of the dimension. Using an expansion category slower than the natural one is unnecessarily slow. The subspace model addresses these problems. There are three major aspects of this work. First, the subspace model is useful by itself. The subspace abstraction subsumes existing shape related optimizations (such as privatization and invariant code motion) and sha..