SMO: an integrated approach to intra-array and inter-array storage optimization

Abstract

The polyhedral model provides an expressive intermediate representation that is convenient for the analysis and subsequent transformation of affine loop nests. Several heuristics exist for achieving complex program transformations in this model. However, there is also considerable scope to utilize this model to tackle the problem of automatic memory footprint optimization. In this paper, we present a new automatic storage optimization technique which can be used to achieve both intra-array as well as inter-array storage reuse with a predetermined schedule for the computation. Our approach works by finding statement-wise storage partitioning hyperplanes that partition a unified global array space so that values with overlapping live ranges are not mapped to the same partition. Our heuristic is driven by a fourfold objective function which not only minimizes the dimensionality and storage requirements of arrays required for each high-level statement, but also maximizes interstatement storage reuse. The storage mappings obtained using our heuristic can be asymptotically better than those obtained by any existing technique. We implement our technique and demonstrate its practical impact by evaluating its effectiveness on several benchmarks chosen from the domains of image processing, stencil computations, and high-performance computing.

Extended lattice-based memory allocation

“…Can we guarantee that we obtain the mapping that is dimension-wise optimal? • While Heuristic 1 can be generalized to inter-array optimizations where different statements have different mappings, as done by Bhaskaracharya for their intra-array mapping [3], it is unclear how to extend our lattice-based method with reuse vectors. • Although we have focused on compactness of the allocation in this paper, it is not always the best for performance.…”

“…The first difference is that we assume a single σ, i.e., σ T = σ S , so that we can work with the difference j − i. But we could also apply Heuristic 1 to map different arrays in the same memory space, each with a possibly different mappings, as explored in SMO [3]. The second difference is that, at each step, we remove all pairs such that σ( j) − σ( i) = 0 (as in the search for maximal parallelism [12]) while, in Pluto, dependences need to be kept for defining other dimensions for tiling.…”

DeLICM: scalar dependence removal at zero memory cost

“…There is the possibility to contract array space after scheduling [6,7,11,19,25,34]. However, the re-scheduled program may inherently require more memory than the source program, especially if the scheduler is unaware that its decisions may increase the memory footprint.…”

“…A conflict set as presented in [7,11] can be useful to determine whether a scalar is conflicting with an array (Listing 6 line 13). For the purpose of DeLICM we additionally need to analyze the stored content.…”

PCOT: Cache Oblivious Tiling of Polyhedral Programs

“…Memory allocation for polyhedral programs is a well studied problem, and there are two main approaches. One either does memory allocation after the schedule is chosen [4,5,19,20,43,52,64,66] since it often leads to a smaller memory footprint, or else uses a schedule independent memory allocation, based on the so called universal occupancy vectors (UOV). This problem is solved when the program has uniform dependences, i.e., when each dependence can be described by a constant vector, and for some simple extensions of this [57,72].…”