Polly's Polyhedral Scheduling in the Presence of Reductions

The polyhedral model provides a powerful mathematical abstraction to enable effective optimization of loop nests with respect to a given optimization goal, e.g., exploiting parallelism. Unexploited reduction properties are a frequent reason for polyhedral optimizers to assume parallelism prohibiting dependences. To our knowledge, no polyhedral loop optimizer available in any production compiler provides support for reductions. In this paper, we show that leveraging the parallelism of reductions can lead to a significant performance increase. We give a precise, dependence based, definition of reductions and discuss ways to extend polyhedral optimization to exploit the associativity and commutativity of reduction computations. We have implemented a reduction-enabled scheduling approach in the Polly polyhedral optimizer and evaluate it on the standard Polybench 3.2 benchmark suite. We were able to detect and model all 52 arithmetic reductions and achieve speedups up to 2.21$\times$ on a quad core machine by exploiting the multidimensional reduction in the BiCG benchmark.Comment: Presented at the IMPACT15 worksho

Nested-Loops Tiling for Parallelization and Locality Optimization

Data locality improvement and nested loops parallelization are two complementary and competing approaches for optimizing loop nests that constitute a large portion of computation times in scientific and engineering programs. While there are effective methods for each one of these, prior studies have paid less attention to address these two simultaneously. This paper proposes a unified approach that integrates these two techniques to obtain an appropriate locality conscious loop transformation to partition the loop iteration space into outer parallel tiled loops. The approach is based on the polyhedral model to achieve a multidimensional affine scheduling as a transformation that result the largest groups of tilable loops with maximum coarse grain parallelism, as far as possible. Furthermore, tiles will be scheduled on processor cores to exploit maximum data reuse through scheduling tiles with high volume of data sharing on the same core consecutively or on different cores with shared cache at around the same time

Optimizing I/O for Big Array Analytics

Big array analytics is becoming indispensable in answering important scientific and business questions. Most analysis tasks consist of multiple steps, each making one or multiple passes over the arrays to be analyzed and generating intermediate results. In the big data setting, I/O optimization is a key to efficient analytics. In this paper, we develop a framework and techniques for capturing a broad range of analysis tasks expressible in nested-loop forms, representing them in a declarative way, and optimizing their I/O by identifying sharing opportunities. Experiment results show that our optimizer is capable of finding execution plans that exploit nontrivial I/O sharing opportunities with significant savings.Comment: VLDB201

Compiler Optimization Techniques for Scheduling and Reducing Overhead

Exploiting parallelism in loops in programs is an important factor in realizing the potential performance of processors today. This dissertation develops and evaluates several compiler optimizations aimed at improving the performance of loops on processors. An important feature of a class of scientific computing problems is the regularity exhibited by their access patterns. Chapter 2 presents an approach of optimizing the address generation of these problems that results in the following: (i) elimination of redundant arithmetic computation by recognizing and exploiting the presence of common sub-expressions across different iterations in stencil codes; and (ii) conversion of as many array references to scalar accesses as possible, which leads to reduced execution time, decrease in address arithmetic overhead, access to data in registers as opposed to caches, etc. With the advent of VLIW processors, the exploitation of fine-grain instruction-level parallelism has become a major challenge to optimizing compilers. Fine-grain scheduling of inner loops has received a lot of attention, little work has been done in the area of applying it to nested loops. Chapter 3 presents an approach to fine-grain scheduling of nested loops by formulating the problem of finding theminimum iteration initiation interval as one of finding a rational affine schedule for each statement in the body of a perfectly nested loop which is then solved using linear programming. Frequent synchronization on multiprocessors is expensive due to its high cost. Chapter 4 presents a method for eliminating redundant synchronization for nested loops. In nested loops, a dependence may be redundant in only a portion of the iteration space. A characterization of the non-uniformity of the redundancy of a dependence is developed in terms of the relation between the dependences and the shape and size of the iteration space. Exploiting locality is critical for achieving high level of performance on a parallel machine. Chapter 5 presents an approach using the concept of affinity regions to find transformations such that a suitable iteration-to-processor mapping can be found for a sequence of loop nests accessing shared arrays. This not only improves the data locality but significantly reduces communication overhead

Some advances in the polyhedral model

Department Head: L. Darrell Whitley.2010 Summer.Includes bibliographical references.The polyhedral model is a mathematical formalism and a framework for the analysis and transformation of regular computations. It provides a unified approach to the optimization of computations from different application domains. It is now gaining wide use in optimizing compilers and automatic parallelization. In its purest form, it is based on a declarative model where computations are specified as equations over domains defined by "polyhedral sets". This dissertation presents two results. First is an analysis and optimization technique that enables us to simplify---reduce the asymptotic complexity---of such equations. The second is an extension of the model to richer domains called Ƶ-Polyhedra. Many equational specifications in the polyhedral model have reductions---application of an associative and commutative operator to collections of values to produce a collection of answers. Moreover, expressions in such equations may also exhibit reuse where intermediate values that are computed or used at different index points are identical. We develop various compiler transformations to automatically exploit this reuse and simplify the computational complexity of the specification. In general, there is an infinite set of applicable simplification transformations. Unfortunately, different choices may result in equivalent specifications with different asymptotic complexity. We present an algorithm for the optimal application of simplification transformations resulting in a final specification with minimum complexity. This dissertation also presents the Ƶ-Polyhedral model, an extension to the polyhedral model to more general sets, thereby providing a transformation framework for a larger set of regular computations. For this, we present a novel representation and interpretation of Ƶ-Polyhedra and prove a number of properties of the family of unions of Ƶ-Polyhedra that are required to extend the polyhedral model. Finally, we present value based dependence analysis and scheduling analysis for specifications in the Ƶ-Polyhedral model. These are direct extensions of the corresponding analyses of specifications in the polyhedral model. One of the benefits of our results in the Ƶ-Polyhedral model is that our abstraction allows the reuse of previously developed tools in the polyhedral model with straightforward pre- and post-processing

Beyond shared memory loop parallelism in the polyhedral model

2013 Spring.Includes bibliographical references.With the introduction of multi-core processors, motivated by power and energy concerns, parallel processing has become main-stream. Parallel programming is much more difficult due to its non-deterministic nature, and because of parallel programming bugs that arise from non-determinacy. One solution is automatic parallelization, where it is entirely up to the compiler to efficiently parallelize sequential programs. However, automatic parallelization is very difficult, and only a handful of successful techniques are available, even after decades of research. Automatic parallelization for distributed memory architectures is even more problematic in that it requires explicit handling of data partitioning and communication. Since data must be partitioned among multiple nodes that do not share memory, the original memory allocation of sequential programs cannot be directly used. One of the main contributions of this dissertation is the development of techniques for generating distributed memory parallel code with parametric tiling. Our approach builds on important contributions to the polyhedral model, a mathematical framework for reasoning about program transformations. We show that many affine control programs can be uniformized only with simple techniques. Being able to assume uniform dependences significantly simplifies distributed memory code generation, and also enables parametric tiling. Our approach implemented in the AlphaZ system, a system for prototyping analyses, transformations, and code generators in the polyhedral model. The key features of AlphaZ are memory re-allocation, and explicit representation of reductions. We evaluate our approach on a collection of polyhedral kernels from the PolyBench suite, and show that our approach scales as well as PLuTo, a state-of-the-art shared memory automatic parallelizer using the polyhedral model. Automatic parallelization is only one approach to dealing with the non-deterministic nature of parallel programming that leaves the difficulty entirely to the compiler. Another approach is to develop novel parallel programming languages. These languages, such as X10, aim to provide highly productive parallel programming environment by including parallelism into the language design. However, even in these languages, parallel bugs remain to be an important issue that hinders programmer productivity. Another contribution of this dissertation is to extend the array dataflow analysis to handle a subset of X10 programs. We apply the result of dataflow analysis to statically guarantee determinism. Providing static guarantees can significantly increase programmer productivity by catching questionable implementations at compile-time, or even while programming

Array-OL Revisited, Multidimensional Intensive Signal Processing Specification

This paper presents the Array-OL specification language. It is a high-level visual language dedicated to multidimensional intensive signal processing applications. It allows to specify both the task parallelism and the data parallelism of these applications on focusing on their complex multidimensional data access patterns. This presentation includes several extensions and tools developed around Array-OL during the last few years and discusses the mapping of an Array-OL specification onto a distributed heterogeneous hardware architecture