Implementation and Evaluation of Algorithmic Skeletons: Parallelisation of Computer Algebra Algorithms

This thesis presents design and implementation approaches for the parallel algorithms of computer algebra. We use algorithmic skeletons and also further approaches, like data parallel arithmetic and actors. We have implemented skeletons for divide and conquer algorithms and some special parallel loops, that we call ‘repeated computation with a possibility of premature termination’. We introduce in this thesis a rational data parallel arithmetic. We focus on parallel symbolic computation algorithms, for these algorithms our arithmetic provides a generic parallelisation approach. The implementation is carried out in Eden, a parallel functional programming language based on Haskell. This choice enables us to encode both the skeletons and the programs in the same language. Moreover, it allows us to refrain from using two different languages—one for the implementation and one for the interface—for our implementation of computer algebra algorithms. Further, this thesis presents methods for evaluation and estimation of parallel execution times. We partition the parallel execution time into two components. One of them accounts for the quality of the parallelisation, we call it the ‘parallel penalty’. The other is the sequential execution time. For the estimation, we predict both components separately, using statistical methods. This enables very confident estimations, although using drastically less measurement points than other methods. We have applied both our evaluation and estimation approaches to the parallel programs presented in this thesis. We haven also used existing estimation methods. We developed divide and conquer skeletons for the implementation of fast parallel multiplication. We have implemented the Karatsuba algorithm, Strassen’s matrix multiplication algorithm and the fast Fourier transform. The latter was used to implement polynomial convolution that leads to a further fast multiplication algorithm. Specially for our implementation of Strassen algorithm we have designed and implemented a divide and conquer skeleton basing on actors. We have implemented the parallel fast Fourier transform, and not only did we use new divide and conquer skeletons, but also developed a map-and-transpose skeleton. It enables good parallelisation of the Fourier transform. The parallelisation of Karatsuba multiplication shows a very good performance. We have analysed the parallel penalty of our programs and compared it to the serial fraction—an approach, known from literature. We also performed execution time estimations of our divide and conquer programs. This thesis presents a parallel map+reduce skeleton scheme. It allows us to combine the usual parallel map skeletons, like parMap, farm, workpool, with a premature termination property. We use this to implement the so-called ‘parallel repeated computation’, a special form of a speculative parallel loop. We have implemented two probabilistic primality tests: the Rabin–Miller test and the Jacobi sum test. We parallelised both with our approach. We analysed the task distribution and stated the fitting configurations of the Jacobi sum test. We have shown formally that the Jacobi sum test can be implemented in parallel. Subsequently, we parallelised it, analysed the load balancing issues, and produced an optimisation. The latter enabled a good implementation, as verified using the parallel penalty. We have also estimated the performance of the tests for further input sizes and numbers of processing elements. Parallelisation of the Jacobi sum test and our generic parallelisation scheme for the repeated computation is our original contribution. The data parallel arithmetic was defined not only for integers, which is already known, but also for rationals. We handled the common factors of the numerator or denominator of the fraction with the modulus in a novel manner. This is required to obtain a true multiple-residue arithmetic, a novel result of our research. Using these mathematical advances, we have parallelised the determinant computation using the Gauß elimination. As always, we have performed task distribution analysis and estimation of the parallel execution time of our implementation. A similar computation in Maple emphasised the potential of our approach. Data parallel arithmetic enables parallelisation of entire classes of computer algebra algorithms. Summarising, this thesis presents and thoroughly evaluates new and existing design decisions for high-level parallelisations of computer algebra algorithms

Architecture aware parallel programming in Glasgow parallel Haskell (GPH)

General purpose computing architectures are evolving quickly to become manycore and hierarchical: i.e. a core can communicate more quickly locally than globally. To be effective on such architectures, programming models must be aware of the communications hierarchy. This thesis investigates a programming model that aims to share the responsibility of task placement, load balance, thread creation, and synchronisation between the application developer and the runtime system. The main contribution of this thesis is the development of four new architectureaware constructs for Glasgow parallel Haskell that exploit information about task size and aim to reduce communication for small tasks, preserve data locality, or to distribute large units of work. We define a semantics for the constructs that specifies the sets of PEs that each construct identifies, and we check four properties of the semantics using QuickCheck. We report a preliminary investigation of architecture aware programming models that abstract over the new constructs. In particular, we propose architecture aware evaluation strategies and skeletons. We investigate three common paradigms, such as data parallelism, divide-and-conquer and nested parallelism, on hierarchical architectures with up to 224 cores. The results show that the architecture-aware programming model consistently delivers better speedup and scalability than existing constructs, together with a dramatic reduction in the execution time variability. We present a comparison of functional multicore technologies and it reports some of the first ever multicore results for the Feedback Directed Implicit Parallelism (FDIP) and the semi-explicit parallelism (GpH and Eden) languages. The comparison reflects the growing maturity of the field by systematically evaluating four parallel Haskell implementations on a common multicore architecture. The comparison contrasts the programming effort each language requires with the parallel performance delivered. We investigate the minimum thread granularity required to achieve satisfactory performance for three implementations parallel functional language on a multicore platform. The results show that GHC-GUM requires a larger thread granularity than Eden and GHC-SMP. The thread granularity rises as the number of cores rises

Parallel evaluation strategies for lazy data structures in Haskell

Conventional parallel programming is complex and error prone. To improve programmer productivity, we need to raise the level of abstraction with a higher-level programming model that hides many parallel coordination aspects. Evaluation strategies use non-strictness to separate the coordination and computation aspects of a Glasgow parallel Haskell (GpH) program. This allows the specification of high level parallel programs, eliminating the low-level complexity of synchronisation and communication associated with parallel programming. This thesis employs a data-structure-driven approach for parallelism derived through generic parallel traversal and evaluation of sub-components of data structures. We focus on evaluation strategies over list, tree and graph data structures, allowing re-use across applications with minimal changes to the sequential algorithm. In particular, we develop novel evaluation strategies for tree data structures, using core functional programming techniques for coordination control, achieving more flexible parallelism. We use non-strictness to control parallelism more flexibly. We apply the notion of fuel as a resource that dictates parallelism generation, in particular, the bi-directional flow of fuel, implemented using a circular program definition, in a tree structure as a novel way of controlling parallel evaluation. This is the first use of circular programming in evaluation strategies and is complemented by a lazy function for bounding the size of sub-trees. We extend these control mechanisms to graph structures and demonstrate performance improvements on several parallel graph traversals. We combine circularity for control for improved performance of strategies with circularity for computation using circular data structures. In particular, we develop a hybrid traversal strategy for graphs, exploiting breadth-first order for exposing parallelism initially, and then proceeding with a depth-first order to minimise overhead associated with a full parallel breadth-first traversal. The efficiency of the tree strategies is evaluated on a benchmark program, and two non-trivial case studies: a Barnes-Hut algorithm for the n-body problem and sparse matrix multiplication, both using quad-trees. We also evaluate a graph search algorithm implemented using the various traversal strategies. We demonstrate improved performance on a server-class multicore machine with up to 48 cores, with the advanced fuel splitting mechanisms proving to be more flexible in throttling parallelism. To guide the behaviour of the strategies, we develop heuristics-based parameter selection to select their specific control parameters

Adaptive structured parallelism

Algorithmic skeletons abstract commonly-used patterns of parallel computation, communication, and interaction. Parallel programs are expressed by interweaving parameterised skeletons analogously to the way in which structured sequential programs are developed, using well-defined constructs. Skeletons provide top-down design composition and control inheritance throughout the program structure. Based on the algorithmic skeleton concept, structured parallelism provides a high-level parallel programming technique which allows the conceptual description of parallel programs whilst fostering platform independence and algorithm abstraction. By decoupling the algorithm specification from machine-dependent structural considerations, structured parallelism allows programmers to code programs regardless of how the computation and communications will be executed in the system platform.Meanwhile, large non-dedicated multiprocessing systems have long posed a challenge to known distributed systems programming techniques as a result of the inherent heterogeneity and dynamism of their resources. Scant research has been devoted to the use of structural information provided by skeletons in adaptively improving program performance, based on resource utilisation. This thesis presents a methodology to improve skeletal parallel programming in heterogeneous distributed systems by introducing adaptivity through resource awareness. As we hypothesise that a skeletal program should be able to adapt to the dynamic resource conditions over time using its structural forecasting information, we have developed ASPara: Adaptive Structured Parallelism. ASPara is a generic methodology to incorporate structural information at compilation into a parallel program, which will help it to adapt at execution

Structured Parallelism by Composition - Design and implementation of a framework supporting skeleton compositionality

This thesis is dedicated to the efficient compositionality of algorithmic skeletons, which are abstractions of common parallel programming patterns. Skeletons can be implemented in the functional parallel language Eden as mere parallel higher order functions. The use of algorithmic skeletons facilitates parallel programming massively. This is because they already implement the tedious details of parallel programming and can be specialised for concrete applications by providing problem specific functions and parameters. Efficient skeleton compositionality is of particular importance because complex, specialised skeletons can be compound of simpler base skeletons. The resulting modularity is especially important for the context of functional programming and should not be missing in a functional language. We subdivide composition into three categories: -Nesting: A skeleton is instantiated from another skeleton instance. Communication is tree shaped, along the call hierarchy. This is directly supported by Eden. -Composition in sequence: The result of a skeleton is the input for a succeeding skeleton. Function composition is expressed in Eden by the ( . ) operator. For performance reasons the processes of both skeletons should be able to exchange results directly instead of using the indirection via the caller process. We therefore introduce the remote data concept. -Iteration: A skeleton is called in sequence a variable number of times. This can be defined using recursion and composition in sequence. We optimise the number of skeleton instances, the communication in between the iteration steps and the control of the loop. To this end, we developed an iteration framework where iteration skeletons are composed from control and body skeletons. Central to our composition concept is remote data. We send a remote data handle instead of ordinary data, the data handle is used at its destination to request the referenced data. Remote data can be used inside arbitrary container types for efficient skeleton composition similar to ordinary distributed data types. The free combinability of remote data with arbitrary container types leads to a high degree of flexibility. The programmer is not restricted by using a predefined set of distributed data types and (re-)distribution functions. Moreover, he can use remote data with arbitrary container types to elegantly create process topologies. For the special case of skeleton iteration we prevent the repeated construction and deconstruction of skeleton instances for each single iteration step, which is common for the recursive use of skeletons. This minimises the parallel overhead for process and channel creation and allows to keep data local on persistent processes. To this end we provide a skeleton framework. This concept is independent of remote data, however the use of remote data in combination with the iteration framework makes the framework more flexible. For our case studies, both approaches perform competitively compared to programs with identical parallel structure but which are implemented using monolithic skeletons - i.e. skeleton not composed from simpler ones. Further, we present extensions of Eden which enhance composition support: generalisation of overloaded communication, generalisation of process instantiation, compositional process placement and extensions of Box types used to adapt communication behaviour

Seeing the natural world: a tension between pupils’ diverse conceptions as revealed by their visual representations and monolithic science lessons

In this paper we report on drawings of the natural environment produced by a sample of 13-14 year-olds. One of our interests is in the extent to which these young people see the world in the way rewarded in science lessons. With rare exceptions, school science generally assumes that for any scientific issue there is a single valid scientific conception so that alternative conceptions are misconceptions. The drawings reveal a plurality of ways in which the natural environment is portrayed and we conclude that there is scientific as well as other worth in this diversity. We argue that schools need to take account of this diversity; many pupils will not be interested in a single, monolithic depiction of the natural world in their school science lessons

Basic completion strategies as another application of the Maude strategy language

The two levels of data and actions on those data provided by the separation between equations and rules in rewriting logic are completed by a third level of strategies to control the application of those actions. This level is implemented on top of Maude as a strategy language, which has been successfully used in a wide range of applications. First we summarize the Maude strategy language design and review some of its applications; then, we describe a new case study, namely the description of completion procedures as transition rules + control, as proposed by Lescanne.Comment: In Proceedings WRS 2011, arXiv:1204.531

GUMSMP: a scalable parallel Haskell implementation

The most widely available high performance platforms today are hierarchical, with shared memory leaves, e.g. clusters of multi-cores, or NUMA with multiple regions. The Glasgow Haskell Compiler (GHC) provides a number of parallel Haskell implementations targeting different parallel architectures. In particular, GHC-SMP supports shared memory architectures, and GHC-GUM supports distributed memory machines. Both implementations use different, but related, runtime system (RTS) mechanisms and achieve good performance. A specialised RTS for the ubiquitous hierarchical architectures is lacking. This thesis presents the design, implementation, and evaluation of a new parallel Haskell RTS, GUMSMP, that combines shared and distributed memory mechanisms to exploit hierarchical architectures more effectively. The design evaluates a variety of design choices and aims to efficiently combine scalable distributed memory parallelism, using a virtual shared heap over a hierarchical architecture, with low-overhead shared memory parallelism on shared memory nodes. Key design objectives in realising this system are to prefer local work, and to exploit mostly passive load distribution with pre-fetching. Systematic performance evaluation shows that the automatic hierarchical load distribution policies must be carefully tuned to obtain good performance. We investigate the impact of several policies including work pre-fetching, favouring inter-node work distribution, and spark segregation with different export and select policies. We present the performance results for GUMSMP, demonstrating good scalability for a set of benchmarks on up to 300 cores. Moreover, our policies provide performance improvements of up to a factor of 1.5 compared to GHC- GUM. The thesis provides a performance evaluation of distributed and shared heap implementations of parallel Haskell on a state-of-the-art physical shared memory NUMA machine. The evaluation exposes bottlenecks in memory management, which limit scalability beyond 25 cores. We demonstrate that GUMSMP, that combines both distributed and shared heap abstractions, consistently outper- forms the shared memory GHC-SMP on seven benchmarks by a factor of 3.3 on average. Specifically, we show that the best results are obtained when shar- ing memory only within a single NUMA region, and using distributed memory system abstractions across the regions

Transparent fault tolerance for scalable functional computation

Reliability is set to become a major concern on emergent large-scale architectures. While there are many parallel languages, and indeed many parallel functional languages, very few address reliability. The notable exception is the widely emulated Erlang distributed actor model that provides explicit supervision and recovery of actors with isolated state. We investigate scalable transparent fault tolerant functional computation with automatic supervision and recovery of tasks. We do so by developing HdpH-RS, a variant of the Haskell distributed parallel Haskell (HdpH) DSL with Reliable Scheduling. Extending the distributed work stealing protocol of HdpH for task supervision and recovery is challenging. To eliminate elusive concurrency bugs, we validate the HdpH-RS work stealing protocol using the SPIN model checker. HdpH-RS differs from the actor model in that its principal entities are tasks, i.e. independent stateless computations, rather than isolated stateful actors. Thanks to statelessness, fault recovery can be performed automatically and entirely hidden in the HdpH-RS runtime system. Statelessness is also key for proving a crucial property of the semantics of HdpH-RS: fault recovery does not change the result of the program, akin to deterministic parallelism. HdpH-RS provides a simple distributed fork/join-style programming model, with minimal exposure of fault tolerance at the language level, and a library of higher level abstractions such as algorithmic skeletons. In fact, the HdpH-RS DSL is exactly the same as the HdpH DSL, hence users can opt in or out of fault tolerant execution without any refactoring. Computations in HdpH-RS are always as reliable as the root node, no matter how many nodes and cores are actually used. We benchmark HdpH-RS on conventional clusters and an HPC platform: all benchmarks survive Chaos Monkey random fault injection; the system scales well e.g. up to 1,400 cores on the HPC; reliability and recovery overheads are consistently low even at scale

Light intensity, photoperiod duration, daily light flux and coral growth of Galaxea fascicularis in an aquarium setting: a matter of photons?

Light is one of the most important abiotic factors influencing the (skeletal) growth of scleractinian corals. Light stimulates coral growth by the process of light-enhanced calcification, which is mediated by zooxanthellar photosynthesis. However, the quantity of light that is available for daily coral growth is not only determined by light intensity (i.e. irradiance), but also by photoperiod (i.e. the light duration time). Understanding and optimizing conditions for coral growth is essential for sustainable coral aquaculture. Therefore, in this study, the question was explored whether more light (i.e. more photons), presented either as irradiance or as light duration, would result in more growth. A series of nine genetically identical coral colonies of Galaxea fascicularis L. were cultured for a period of 18 weeks at different light duration times (8 hours 150 µE m-2 s-1:16 hours dark, 12 hours 150 µE m-2 s-1:12 hours dark, 16 hours 150 µE m-2 s-1:8 hours dark, 24 hours 150 µE m-2 s-1:0 hours dark) and different irradiance levels (8 hours 150 µE m-2 s-1:16 hours dark, 8 hours 225 µE m-2 s-1:16 hours dark and 8 hours 300 µE m-2 s-1:16 hours dark). Growth was determined every two weeks by measuring buoyant weight. Temperature, salinity and feeding levels were kept constant during the experiment. To detect possible acclimation of the corals to an increased light duration, rates of net photosynthesis and dark respiration were measured, hereby comparing coral colonies grown under an 8:16 hours light (150 µE m-2 s-1):dark cycle with corals grown under a 16:8 hours light (150 µE m-2 s-1):dark cycle. No increase in growth was detected with either increasing photoperiod or irradiance. Continuous lighting (24 hours 150 µE m-2 s-1:0 hours dark) resulted in immediate bleaching and the corals died after 14 weeks. Hourly photosynthetic rates were significantly reduced in the 16 hour light treatment compared to the 8 hour light treatment. As a result, daily net photosynthetic rates were not significantly different, which may explain the observed specific growth rates. Acclimation to photoperiod duration appeared neither to be mediated by changes in chlorophyll-a concentration nor zooxanthellae density. Based on the results of this study, we can conclude that the enhancing effect of light on coral growth is not only a matter of photons. Obviously, the availability of light was not limiting growth in these experiments and was probably in excess (i.e. stressful amounts). Other factors are discussed that play a role in determining growth rates and might explain our results