This paper introduces a new approach to optimising array algorithms in functional languages. We are specifically aiming at an efficient implementation of irregular array algorithms that are hard to implement in conventional array languages such as Fortran. We optimise the storage layout of arrays containing complex data structures and reduce the running time of functions operating on these arrays by meansofequationalprogramtransformations. Inparticular, this paper discusses a novel form of combinator loop fusion, whichbyremovingintermediatestructuresoptimisestheuse of the memory hierarchy. We identify a combinator named loopP that provides a general scheme for iterating over an array and that in conjunction with an array constructor replicateP is sufficient to express a wide range of array algorithms. On this basis, we define equational transformation rules that combine traversals of loopP and replicateP as well as sequences of applications of loopP into a single loopP traversal. Our approach naturally generalises to a parallel implementation and includes facilities for optimising load balancing and communication. A prototype implementation based on the rewrite rule pragma of the Glasgow Haskell Compiler is significantly faster than standard Haskell arrays and approaches the speed of hand coded C for simple examples. 1.

This paper discusses an extension of Haskell by support for nested data-parallel programming in the style of the special-purpose language Nesl. More precisely, the extension consists of a parallel array type, array comprehensions, and a set of primitive parallel array operations. This extension brings a hitherto unsupported style of parallel programming to Haskell. Moreover, nested data parallelism should receive wider attention when available in a standardised language like Haskell. This paper outlines the language extension and demonstrates its usefulness with two case studies.

ABSTRACT. If you want to program a parallel computer, a purely functional language like Haskell is a promising starting point. Since the language is pure, it is by-default safe for parallel evaluation, whereas imperative languages are by-default unsafe. But that doesn’t make it easy! Indeed it has proved quite difficult to get robust, scalable performance increases through parallel functional programming, especially as the number of processors increases. A particularly promising and well-studied approach to employing large numbers of processors is data parallelism. Blelloch’s pioneering work on NESL showed that it was possible to combine a rather flexible programming model (nested data parallelism) with a fast, scalable execution model (flat data parallelism). In this paper we describe Data Parallel Haskell, which embodies nested data parallelism in a modern, general-purpose language, implemented in a state-of-the-art compiler, GHC. We focus particularly on the vectorisation transformation, which transforms nested to flat data parallelism. 1

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Parallel primitives (skeletons) intend to encourage programmers to build a parallel program from ready-made components for which efficient implementations are known to exist, making the parallelization process easier. However, programmers often suffer from the difficulty to choose a combination of proper parallel primitives so as to construct efficient parallel programs. To overcome this difficulty, we shall propose a new transformation, called diffusion, which can efficiently decompose a recursive definition into several functions such that each function can be described by some parallel primitive. This allows programmers to describe algorithms in a more natural recursive form. We demonstrate our idea with several interesting examples. Our diffusion transformation should be significant not only in development of new parallel algorithms, but also in construction of parallelizing compilers.

Portable, efficient, parallel programming requires cost models to compare different possible implementations. In turn, these require knowledge of the shapes of the data structures being used, as well as knowledge of the hardware parameters. This paper shows how shape analysis techniques developed in the FISh programming language could be exploited to produce a data parallel language with an accurate, portable cost model. 1 Introduction The problem of constructing portable efficient parallel programs is still unsolved. It originates in the observation that an algorithm that executes efficiently in one setting may be extremely inefficient in another. Hence, the challenge is to automatically adapt the algorithm to match the circumstances. To do this during compilation requires a cost model that is able to identify which of two alternative algorithms is faster. To date, most work has focussed on measuring the impact of changes to hardware as observed through a small suite of hardwar...

Abstract. Vectorisation for functional programs, also called the flattening transformation, relies on drastically reordering computations and restructuring the representation of data types. As a result, it only applies to the purely functional core of a fully-fledged functional language, such as Haskell or ML. A concrete implementation needs to apply vectorisation selectively and integrate vectorised with unvectorised code. This is challenging, as vectorisation alters the data representation, which must be suitably converted between vectorised and unvectorised code. In this paper, we present an approach to partial vectorisation that selectively vectorises sub-expressions and data types, and also, enables linking vectorised with unvectorised modules.

Data parallelism is currently one of the most successful models for programming massively parallel computers. The central idea is to evaluate a uniform collection of data in parallel by simultaneously manipulating each data element in the collection. Despite many of its promising features, the current approach suffers from two problems. First, the main parallel data structures that most data parallel languages currently support are restricted to simple collection data types like lists, arrays or similar structures. But other useful data structures like trees have not been well addressed. Second, parallel programming relies on a set of parallel primitives that capture parallel skeletons of interest. However, these primitives are not well structured, and efficient parallel programming with these primitives is difficult. In this paper, we propose a polytypic framework for developing efficient parallel programs on most data structures. We showhow a set of polytypic parallel primitives can be formally defined for manipulating most data structures, how these primitives can be successfully structured into a uniform recursive definition, and how an efficient combination of primitives can be derived from a naive specification program. Our framework should be significant not only in development of new parallel algorithms, but also in construction of parallelizing compilers.

. Modern dialects of Fortran enjoy wide use and good support on highperformance computers as performance-oriented programming languages. By providing the ability to express nested data parallelism, modern Fortran dialects enable irregular computations to be incorporated into existing applications with minimal rewriting and without sacrificing performance within the regular portions of the application. Since performance of nested data-parallel computation is unpredictable and often poor using current compilers, we investigate threading and flattening, two source-to-source transformation techniques that can improve performance and performance stability. For experimental validation of these techniques, we explore nested data-parallel implementations of the sparse matrixvector product and the Barnes-Hut n-body algorithm by hand-coding threadbased (using OpenMP directives) and flattening-based versions of these algorithmsandevaluatingtheirperformanceonanSGIOrigin2000andanNEC SX-...

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