We present the design and implementation of an automatic polyhedral source-to-source transformation framework that can optimize regular programs (sequences of possibly imperfectly nested loops) for parallelism and locality simultaneously. Through this work, we show the practicality of analytical model-driven automatic transformation in the polyhedral model.Unlike previous polyhedral frameworks, our approach is an end-to-end fully automatic one driven by an integer linear optimization framework that takes an explicit view of finding good ways of tiling for parallelism and locality using affine transformations. The framework has been implemented into a tool to automatically generate OpenMP parallel code from C program sections. Experimental results from the tool show very high performance for local and parallel execution on multi-cores, when compared with state-of-the-art compiler frameworks from the research community as well as the best native production compilers. The system also enables the easy use of powerful empirical/iterative optimization for general arbitrarily nested loop sequences.

High-level loop optimizations are necessary to achieve good performance over a wide variety of processors. Their performance impact can be significant because they involve in-depth program transformations that aiming to sustain a balanced workload over the computational, storage, and communication resources of the target architecture. Therefore, it is mandatory that the compiler accurately models the target architecture and the effects of complex code restructuring. However, because optimizing compilers (1) use simplistic performance models that abstract away many of the complexities of modern architectures, (2) rely on inaccurate dependence analysis, and (3) lack frameworks to express complex interactions of transformation sequences, they typically uncover only a fraction of the peak performance available on many applications. We propose a complete iterative framework to address these issues. We rely on the polyhedral model to construct and traverse a large and expressive search space. This space encompasses only legal, distinct versions resulting from the restructuring of any static control loop nest. We first propose a feedback-driven iterative heuristic tailored to the search space properties of the polyhedral model. Though, it quickly converges to good solutions for small kernels, larger benchmarks containing higher dimensional spaces are more challenging and our heuristic misses opportunities for significant performance improvement. Thus, we introduce the use of a genetic algorithm with specialized operators that leverage the polyhedral representation of program dependences. We provide experimental evidence that the genetic algorithm effectively traverses huge optimization spaces, achieving good performance improvements on large loop nests.

Achieving good performance on a modern machine with a multi-level memory hierarchy, and in particular on a machine with software-managed memories, requires precise tuning of programs to the machine’s particular characteristics. A large program on a multi-level machine can easily expose tens or hundreds of inter-dependent parameters which require tuning, and manually searching the resultant large, non-linear space of program parameters is a tedious process of trial-and-error. In this paper we present a general framework for automatically tuning general applications to machines with software-managed memory hierarchies. We evaluate our framework by measuring the performance of benchmarks that are tuned for a range of machines with different memory hierarchy configurations: a cluster of Intel P4 Xeon processors, a single Cell processor, and a cluster of Sony Playstation3’s.

Abstract. Many compute intensive applications spend a significant fraction of their time in nested loops. The polyhedral model provides powerful abstractions to optimize loop nests with regular accesses for parallel execution. Affine transformations in this model capture a complex sequence of execution-reordering loop transformations that can improve performance by parallelization as well as locality enhancement. Although a significant amount of research has addressed affine scheduling and partitioning, the problem of automatically finding good affine transforms for communication-optimized coarse-grained parallelization along with locality optimization for the general case of arbitrarily-nested loop sequences remains a challenging problem. In this paper, we propose an automatic transformation framework to optimize arbitrarily-nested loop sequences with affine dependences for parallelism and locality simultaneously. The approach finds good tiling hyperplanes by embedding a powerful and versatile cost function into an Integer Linear Programming formulation. These tiling hyperplanes are used for communication-minimized coarse-grained parallelization as well as locality optimization. It enables the minimization of inter-tile communication volume in the processor space, and minimization of reuse distances for local execution at each node. Programs requiring one-dimensional versus multi-dimensional time schedules (with schedulingbased approaches) are all handled with the same algorithm. Synchronization-free parallelism, permutable loops or pipelined parallelism at various levels can be detected. Preliminary results from the implemented framework show promising performance and scalability with input size. 1

A long running program often spends most of its time in nested loops. The polyhedral model provides powerful abstractions to optimize loop nests with regular accesses for parallel execution. Affine transforma-tions in this model capture a complex sequence of execution-reordering loop transformations that improve performance by parallelization as well as better locality. Although a significant amount of research has addressed affine scheduling and partitioning, the problem of automatically finding good affine transforms for communication-optimized coarse-grained parallelization along with locality optimization for the general case of arbitrarily-nested loop sequences remains a challenging problem- most frameworks do not treat par-allelization and locality optimization in an integrated manner, and/or do not optimize across a sequence of producer-consumer loops. In this paper, we develop an approach to communication minimization and locality optimization in tiling of arbitrarily nested loop sequences with affine dependences. We address the minimization of inter-tile commu-nication volume in the processor space, and minimization of reuse distances for local execution at each node. The approach can also fuse across a long sequence of loop nests that have a producer/consumer relationship. Programs requiring one-dimensional versus multi-dimensional time schedules are all handled with the same algorithm. Synchronization-free parallelism, permutable loops or pipelined parallelism, and inner parallel loops can be detected. Examples are provided that demonstrate the power of the framework. The algorithm has been incorporated into a tool chain to generate transformations from C/Fortran code in a fully automatic fashion. 1

We present the design and implementation of a fully automatic polyhedral source-to-source transformation framework that can optimize regular programs (sequences of possibly imperfectly nested loops) for parallelism and locality simultaneously. Through this work, we show the practicality of analytical model-driven automatic transformation in the polyhedral model – far beyond what is possible by current production compilers. Unlike previous works, our approach is an end-to-end fully automatic one driven by an integer linear optimization framework that takes an explicit view of finding good ways of tiling for parallelism and locality using affine transformations. We also address generation of tiled code for multiple statement domains of arbitrary dimensionalities under (statement-wise) affine transformations – an issue that has not been addressed previously. Experimental results from the implemented system show very high speedups for local and parallel execution on multi-cores over state-of-the-art compiler frameworks from the research community as well as the best native compilers. The system also enables the easy use of powerful empirical/iterative optimization for general arbitrarily nested loop sequences.

Abstract—Today’s multi-core era places significant demands on an optimizing compiler, which must parallelize programs, exploit memory hierarchy, and leverage the ever-increasing SIMD capabilities of modern processors. Existing model-based heuristics for performance optimization used in compilers are limited in their ability to identify profitable parallelism/locality trade-offs and usually lead to sub-optimal performance. To address this problem, we distinguish optimizations for which effective model-based heuristics and profitability estimates exist, from optimizations that require empirical search to achieve good performance in a portable fashion. We have developed a completely automatic framework in which we focus the empirical search on the set of valid possibilities to perform fusion/code motion, and rely on model-based mechanisms to perform tiling, vectorization and parallelization on the transformed program. We demonstrate the effectiveness of this approach in terms of strong performance improvements on a single target as well as performance portability across different target architectures. I.

Abstract. Iterative optimization has been shown to improve the performance of benchmarks significantly, but its application involves challenges such as the requirement of an expressive search space and the design of efficient search techniques. In this paper, we apply iterative optimization to the problem of optimizing in the polyhedral model, a powerful algebraic representation of any static control program, by using affine multidimensional schedules to represent arbitrarily complex transformation sequences. We propose to study the performance distribution of the search space of affine multidimensional schedules built specifically to guarantee legality and uniqueness of each program version. We extensively study the optimization of 5 representative benchmarks in this representation, and highlight a series of static and dynamic characteristics of the search space. We show how the space can be decoupled into subspaces, which can be statically ordered with respect to their impact on performance. Finally, we present a practical search method leveraging these properties to traverse the search space, yielding a 32.56 % speedup on eight representative kernels.

Loop nest optimization is a combinatorial problem. Due to the growing complexity of modern architectures, it involves two increasingly difficult tasks: (1) analyzing the profitability of sequences of transformations to enhance parallelism, locality, and resource usage, which amounts to a hard problem on a non-linear objective function; (2) the construction and exploration of search space of legal transformation sequences. Practical optimizing and parallelizing compilers decouple these tasks, resorting to a predefined set of enabling transformations to eliminate all sorts of optimization-limiting semantical constraints. State-of-theart optimization heuristics face a hard decision problem on the selection of enabling transformations only remotely related to performance. We propose a new design where optimization heuristics first address the main performance anomalies, then correct potentially illegal loop transformations a posteriori, attempting to minimize the performance impact of the necessary adjustments. We propose a general method to correct any sequence of loop transformations through a combination of loop shifting, code motion and index-set splitting. Sequences of transformations are modeled by compositions of geometric transformations on multidimensional affine schedules. We provide experimental evidence of the scalability of the algorithms on real loop optimizations. 1.

Abstract. In this paper, we describe transformation recipes, which provide a high-level interface to the code transformation and code generation capability of a compiler. These recipes can be generated by compiler decision algorithms or savvy software developers. This interface is part of an auto-tuning framework that explores a set of different implementations of the same computation and automatically selects the best-performing implementation. Along with the original computation, a transformation recipe specifies a range of implementations of the computation resulting from composing a set of high-level code transformations. In our system, an underlying polyhedral framework coupled with transformation algorithms takes this set of transformations, composes them and automatically generates correct code. We first describe an abstract interface for transformation recipes, which we propose to facilitate interoperability with other transformation frameworks. We then focus on the specific transformation recipe interface used in our compiler and present performance results on its application to kernel and library tuning and tuning of key computations in high-end applications. We also show how this framework can be used to generate and auto-tune parallel OpenMP or CUDA code from a high-level specification. 1