Synthesizing transformations for locality enhancement of imperfectly-nested loop nests

Abstract

We present an approach for synthesizing transformations to enhance locality in imperfectly-nested loops. The key idea is to embed the iteration space of every statement in a loop nest into a special iteration space called the product space. The product space can be viewed as a perfectly-nested loop nest, so embedding generalizes techniques like code sinking and loop fusion that are used in ad hoc ways in current compilers to produce perfectly-nested loops from imperfectly-nested ones. In contrast to these ad hoc techniques however, our embeddings are chosen carefully to enhance locality. The product space is then transformed further to enhance locality, after which fully permutable loops are tiled, and code is generated. We evaluate the effectiveness of this approach for dense numerical linear algebra benchmarks, relaxation codes, and the tomcatv code from the SPEC benchmarks. BACKGROUND AND PREVIOUS WORKSophisticated algorithms based on polyhedral algebra have been developed for determining good sequences of linear loop transformations (permutation, skewing, reversal and scaling) for enhancing locality in perfectly-nested loops 1 . Highlights of this technology are the following. The iterations of the loop nest are modeled as points in an integer lattice, and linear loop transformations are modeled as nonsingular matrices mapping one lattice to another. A sequence of loop transformations is modeled by the product of matrices representing the individual transformations; since the set of nonsingular matrices is closed under matrix product, this means that a sequence of linear loop transformations can be represented by a nonsingular matrix. The problem of finding an optimal sequence of linear loop transformations is thus reduced to the problem of finding an integer matrix that satisfies some desired property, permitting the full machinery of matrix methods and lattice theory to ¢ This work was supported by NSF grants CCR-9720211, EIA-9726388, ACI-9870687,EIA-9972853. £A perfectly-nested loop is a set of loops in which all assignment statements are contained in the innermost loop.for t = 1,T for i1 = 2,N-1 for j1 = 2,N-1 S1:L(i1,j1) = (A(i1,j1+1) + A(i1,j1-1) + A(i1+1,j1) + A(i1-1,j1)) / 4 end end for i2 = 2,N-1 for j2 = 2,N-1 S2:A(i2,j2) = L(i2,j2) end end end This technology is fairly mature, and it has been incorporated into production compilers, enabling these compilers to produce good code for perfectly-nested loop nests. In most programs however, most loop nests are imperfectly-nested because one or more assignment statements are contained in some but not all of the loops of the loop nest. For example, important matrix factorizations like Cholesky, LU and QR factorizations [9] are all imperfectly-nested loop nests. An entire procedure, which usually is a sequence of perfectly-or imperfectly-nested loop nests, can itself be considered to be imperfectly-nested loop nest. As an example, consider the Jacobi code fragment in Figure 1 which is typical of programs that solve partial differential equations (p...

Tiling Imperfectly-nested Loop Nests

“…In principle, this technology can be brought to bear on any program without restriction to problem domain. In practice, most of the work in this area has focused on perfectly-nested loop nests that manipulate arrays 1 . Highlights of the restructuring technology for perfectly-nested loop nests are the following.…”

Specializing Compiler Optimizations through Programmable Composition for Dense Matrix Computations

“…Our work integrated these optimizations and aims to enhance their aggregated effectiveness through collective customization and coordinated composition of the optimizations. A number of general frameworks [18], [1], [21], e.g., the polyhedral model [7], has been shown to be effective at collectively considering a large set of the loop optimizations and generating efficient code. However, this body of work is limited in that they consider only loop optimizations.…”

Semi-Automatic Composition of Loop Transformations for Deep Parallelism and Memory Hierarchies

“…We expect an additional level of tiling and more aggressive unrolling will be profitable (due to the improved 11 Athlon XP: -m32 -Ofast -OPT:ro=2:Olimit=0:div split=on:alias=typed -LNO:fusion=2:prefetch=2 -fno-math-errno; Athlon 64 (in 64 bits mode): -march=athlon64 -LNO:fusion=2:prefetch=2 -m64 -Ofast -msse2 -lmpath; pathf90 always outperformed Intel ICC by a small percentage. 12 With optimization flag -Ofast. 13 Notice we consider the SPEC 2000 version of swim, much harder to optimize through loop fusion than the SPEC 95 version.…”

“…7. This framework improves on classical polyhedral representations (8)(9)(10)(11)(12)(13) to support a large array of useful and efficient program transformations (loop fusion, tiling, array forward substitution, statement reordering, software pipelining, array padding, etc. ), as well as compositions of these transformations.…”

“…Compared to the attempts at expressing a large array of program transformations as matrix operations within the polyhedral model, (9,10,14) the distinctive asset of our representation lies in the simplicity of the formalism to compose non-unimodular transformations across long, flexible sequences. Existing formalisms have been designed for black-box optimization, (8,11,12) and applying a classical loop transformation within themas proposed in Ref. 3, 9, 10 -requires a syntactic form of the program to anchor the transformation to existing statements.…”