Google’s MapReduce programming model serves for processing large data sets in a massively parallel manner. We deliver the first rigorous description of the model including its advancement as Google’s domain-specific language Sawzall. To this end, we reverse-engineer the seminal papers on MapReduce and Sawzall, and we capture our findings as an executable specification. We also identify and resolve some obscurities in the informal presentation given in the seminal papers. We use typed functional programming (specifically Haskell) as a tool for design recovery and executable specification. Our development comprises three components: (i) the basic program skeleton that underlies MapReduce computations; (ii) the opportunities for parallelism in executing MapReduce computations; (iii) the fundamental characteristics of Sawzall’s aggregators as an advancement of the MapReduce approach. Our development does not formalize the more implementational aspects of an actual, distributed execution of MapReduce computations.

The problems involved in developing efficient parallel programs have proved harder than those in developing efficient sequential ones, both for programmers and for compilers. Although program calculation has been found to be a promising way to solve these problems in the sequential world, we believe that it needs much more effort to study its effective use in the parallel world. In this paper, we propose a calculational framework for the derivation of efficient parallel programs with two main innovations: - We propose a novel inductive synthesis lemma based on which an elementary but powerful parallelization theorem is developed. - We make the first attempt to construct a calculational algorithm for parallelization, deriving associative operators from data type definition and making full use of existing fusion and tupling calculations. Being more constructive, our method is not only helpful in the design of efficient parallel programs in general but also promising in the construc...

Abstract program schemes, such as scan or homomorphism, can capture a wide range of data parallel programs. While versatile, these schemes are of limited practical use on their own. A key problem is that the more natural sequential specifications may not have associative combine operators required by these schemes. As a result, they often fail to be immediately identified. To resolve this problem, we propose a method to systematically derive parallel programs from sequential definitions. This method is special in that it can automatically invent auxiliary functions needed by associative combine operators. Apart from a formalisation, we also provide new theorems, based on the notion of context preservation, to guarantee parallelization for a precise class of sequential programs. 1 Introduction It is well-recognised that a key problem of parallel computing remains the development of efficient and correct parallel software. This task is further complicated by the variety of parallel arc...

A Functional Abstract Notation (FAN) is proposed for the specification and design of parallel algorithms by means of skeletons - high-level patterns with parallel semantics. The main weakness of the current programming systems based on skeletons is that the user is still responsible for finding the most appropriate skeleton composition for a given application and a given parallel architecture. We describe a transformational framework for the development of skeletal programs which is aimed at filling this gap. The framework makes use of transformation rules which are semantic equivalences among skeleton compositions. For a given problem, an initial, possibly inefficient skeleton specification is refined by applying a sequence of transformations. Transformations are guided by a set of performance prediction models which forecast the behavior of each skeleton and the performance benefits of different rules. The design process is supported by a graphical tool which locates applicable transformations and provides performance estimates, thereby helping the programmer in navigating through the program refinement space. We give an overview of the FAN framework and exemplify its use with performance-directed program derivations for simple case studies. Our experience can be viewed as a first feasibility study of methods and tools for transformational, performance-directed parallel programming using skeletons.

Homomorphisms are functions that match the divide-and-conquer paradigm and thus can be computed in parallel. Two problems are studied for homomorphisms on lists: (1) parallelism extraction: finding a homomorphic representation of a given function; (2) parallelism implementation: deriving an efficient parallel program that computes the function. A systematic approach to parallelism extraction proceeds by generalization of two sequential representations based on traditional cons lists and dual snoc lists. For some non-homomorphic functions, e.g., the maximum segment sum problem, our method provides an embedding into a homomorphism. The implementation is addressed by introducing a subclass of distributable homomorphisms and deriving for them a parallel program schema, which is time optimal on the hypercube architecture. The derivation is based on equational reasoning in the Bird-Meertens formalism, which guarantees the correctness of the parallel target program. The approach is illustrated with function...

A structured approach to parallel programming allows to construct applications by composing skeletons, i.e., recurring patterns of task- and data-parallelism. First academic and commercial experience with skeleton-based systems has demonstrated both the benefits of the approach and the lack of a special methodology for algorithm design and performance prediction. In the paper, we take a first step toward such a methodology, by developing a general transformational framework named FAN, and integrating it with an existing skeleton-based programming system, P3L. The framework includes a new functional abstract notation for expressing parallel algorithms, a set of semantics-preserving transformation rules, and analytical estimates of the rules' impact on the program performance. The use of FAN is demonstrated on a case study: we design a parallel algorithm for the maximum segment sum problem, translate the algorithm in P3L, and experiment with the target C+MPI code on a Fujitsu AP1000 parallel machine.

this paper, we study functions called list homomorphisms, which represent a particular pattern of parallelism.

Parallel skeletons intend to encourage programmers to build...

Introduction We study two popular programming schemas: scan (also known as prefix sums, parallel prefix, etc.) and reduction (also known as fold). Originally from the functional world [3], they are becoming increasingly popular as primitives of parallel programming. The reasons are that, first, such higher-order combinators are adequate and useful for a broad class of applications [4], second, they encourage well-structured, coarse-grained parallel programming and, third, their implementation in the MPI standard [14] makes the target programs portable across different parallel architectures with predictable performance. Our contributions are as follows: -- We formally prove two optimization rules: the first rule transforms a sequential composition of scan and reduction into a single reduction, the second rule transforms a composition of two scans into a single scan. -- We apply the first rule in the formal derivation of a parallel algorithm for the

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.