Our program provides three different methods to initialize the centroids described in detail below. Divide the whole dataset into subsets S equal to the number of compute nodes M participating in the algorithm. Send each data subset S , the number of clusters k , and the initial centroid vectors to each compute node for processing. On each compute node, individual data elements in a subset S are assigned to one of k clusters based on the shortest distance using either Euclidean distance or Pearson Correlation of each element to each cluster's centroid.
Note that each data subset S may contain elements from all, some or only one cluster and that the calculation of the cluster centroids is based on the entire dataset, not the subsets sent to each node. Each compute node then calculates the components of the sufficient statistic SS for each cluster based on the data subset S assigned to each cluster on that node.
The components of the SS for each cluster k on each compute node are the first moment FM , the second moment SecM and the number of elements n c in each cluster:. The components of the Sufficient Statistics FM, SecM, and n c for each cluster, c , for each data subset on each compute node are sent back to the master computer to calculate the new global cluster centroids gCC c for each cluster.
The Sufficient Statistics are then used to calculate the Performance Function used to measure the quality of the clusters [ 9 ]. The global Performance Function is simply the sum of all Performance Functions calculated for each cluster Perf c. The new global cluster centroids are sent back to each compute node and replace the previous iterations centroids. The algorithm then loops between steps 4 and 7 above.
When the Performance Function reaches a minimum or doesn't change between iterations, the algorithm execution stops and the clustered data are retrieved and collated from the compute nodes. This parallel implementation of the k -means algorithm does not require expensive hardware and the number of compute nodes do not depend on the number of clusters. In fact, any number of inexpensive desktop computers connected by a network can be used.
The data partitioning scheme is not restricted and is entirely dependent on the number of compute nodes participating in the algorithm. Additionally, the data subsets S are sent only once from the master computer to the compute nodes. Only the data necessary to calculate the sufficient statistics is sent between nodes, dramatically reducing communication latency. ParaKMeans implements two different metrics for assigning a gene vector to a cluster. The first is the common Euclidian Distance and the second is Pearson Correlation. For the Pearson Correlation, we use 1-r for the distance calculation.
While we can use either metric, all the data generated for this manuscript uses Euclidian Distance. The execution time and cluster quality of k -means clustering algorithms are very sensitive to the values of the initial centroids. We provide three different methods to determine the initial cluster centroids:.
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This method is a common method used by most k -means algorithm implementations. Random Initial Assignments RIA : All genes are randomly assigned to one of the k clusters, and the mean of these randomly assigned clusters are used as the starting centroids. This method works by first randomly selecting one gene gene0 from the data. The gene gene1 that is the greatest distance based on either Euclidean Distance or Pearson Correlation from that initial gene gene0 is selected, becoming the first centroid.
The gene gene2 that is the greatest distance from gene1 becomes the second centroid, and the third centroid is the gene that is the farthest distance from both gene1 AND gene2. This process continues until all centroids have been initialized. While this initialization scheme can be time consuming, it provides more stable and consistent clusters. ParaKMeans is a high performance multithreaded application. We designed ParaKMeans with an easy and manageable client-server application model that can be easily deployed in most laboratories.
The system can be deployed on a single computer or across many computers nodes. All the software was written using the. NET Framework v1. The application was designed in a modular fashion to provide both deployment flexibility and flexibility in the user interface, and is made of three software components Figure 1 :. ParaKMeans software components and deployment strategy. The GUI is installed on the local machine while the ParallelCluster web services is installed on one or more laboratory computers.
Installation of both the GUI and web service is done by double clicking on the. A web service is used to perform the distance calculations and cluster assignments, allowing for parallel computation. Web services servers are not needed to run web services are a distributed computing technology that makes computing resources hardware and software available over the Internet. The technology behind web services is based on common standards of communication, data representation and service description allowing for interoperability between different computers.
The web service is responsible for assigning the vectors e. This library is compiled into and used by the graphical user interfaces. The API provides the methods to load the data, initialize the centroids, partition the data and orchestrate the asynchronous multithreaded connections to the ParallelCluster web services to perform the parallel k -means algorithm. The stand-alone GUI can be installed on any Windows machine and provides easy file management, compute node management, program options and a results window for data viewing and saving.
Ajax is a technique for creating interactive web applications that are more responsive by exchanging small amounts of data with the server behind the scenes. Using Ajax results in only the relevant portions of the web page needing to be posted and reloaded each time the user makes a change.
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This technique increases the web page's interactivity, speed, and usability. We implemented Ajax using an open source. The web GUI provides the same functionality as the stand alone program. ParaKMeans user interfaces. The interface provides information on the data and program options being used. B Home page of the web based ParaKMeans application, providing an overview of the current data and program options being used.
Essentially, the ParallelCluster web service is installed on each machine that will be a compute node, followed by the GUI being installed on the computer to be directly used by the user. The intended installation is to have one master computer and multiple compute nodes. However, technically, both the GUI and web service could be installed on a single machine.
You can simulate multiple compute nodes by adding the IP address of the local machine multiple times. The program will spawn a separate thread for each "node". However, a single machine install will lose any advantage gained by distributing the algorithm and data across multiple machines and will impact performance. We simulated clusters using separate multivariate normal distributions as the basis for each cluster. The relative size of the two variances controls the degree of separation between the clusters.
The covariance matrix was defined as follows. We also analyzed a dataset that utilized cDNA printed on glass slides to examine expression differences in peripheral blood lymphocytes PBL between healthy controls and type 1 diabetic patients [ 11 ]. To increase the reliability of the data, 2—3 replicate hybridizations were completed for each RNA sample, with the average of the replicates being used in all analyses. Analyses of data for differential expression between controls and patients identified cDNA clones with differences between the controls and patients.
We evaluated the time taken to cluster these data by the various methods and examined the stability of the clusters produced using these data. The length of time to perform the parallel k -means clustering algorithm will not only depend on the efficiency of the algorithm, but also the computing hardware used to perform the algorithm.
The performance, accuracy and stability of ParaKMeans were evaluated using one master computer with between 1 to 7 compute nodes. For comparisons, we installed Michael Eisen's Cluster program [ 12 ] on the master computer. We did not compare our program with other parallel software versions of k-means clustering because our goal was to develop a user-friendly version for general laboratory use. The master computer and the seven compute nodes were all identical machines: Dell Poweredge with Dual 3.
For all analyses, we used at least twelve replicate runs under each condition to evaluate ParaKMeans, and recorded run time and cluster assignments. Performance was measured using run time, the accuracy of the identified clusters the extent to which the clusters identified reflected the clusters used in the simulation , and stability of identified clusters extent of consistency between replicate runs without regard to the actual clusters based on simulation conditions. Run time was log-transformed for all analyses, and factorial analyses of variance ANOVAs were used to analyze run time for all comparisons.
We observed heterogeneity in the variance of log-transformed run time across the conditions, which we accommodated by fitting a heterogeneous variance model using the Proc Mixed procedure in the SAS statistical system SAS We fit multiple models to evaluate which conditions showed the greatest heterogeneity in the variances, and used the model that had the best Aikaike's information criterion AIC.
All significant effects were subsequently examined using Tukey's HSD. We used the adjusted Rand index [ 13 , 14 ] to measure both stability and accuracy. For accuracy, the ARI was calculated for each run relative to the actual cluster identity. The ARI was calculated only between two pairs of replicate runs to measure stability. Comparisons of ARI were based on plots because the ARI showed no variance in several conditions and differences were quite large when they existed. A four way ANOVA that included number of nodes, number of clusters, number of genes, and number of arrays was used to examine the differences in run times.
This ANOVA model included heterogeneity in the variance among the levels for the number of genes in an array. For this comparison, we used only those datasets that had and genes. Cluster , number of clusters, and number of genes was used to examine differences in run times.
This ANOVA model included heterogeneity in the variance among the levels for the number of genes in an array with these variances also differing between algorithms. We then compared initialization methods for ParaKMeans. For this comparison, we ran ParaKMeans using both 1 and 7 nodes for each initialization method, and we analyzed only those datasets having arrays, and 10, genes with 4 and 20 clusters. We used factorial ANOVA to compare the run times, using a model that included heterogeneity in the variance among the levels for the combination of number of genes and initialization method.
The average time taken to cluster each dataset ranged from 0. The long average time for large complex datasets was reduced to 3. Execution times were evaluated for ParaKMeans three ways: 1 the effect of the parallel algorithm with multiple computers on time to completion; 2 the point at which adding more compute nodes did not provide any further decrease in time to completion; 3 the effect of the number of arrays, genes or clusters on the execution time of the algorithm. Each of these effects was evaluated using ANOVA to assess the interactions between the numbers of genes, arrays, clusters and compute nodes on the length of time to run the program using the simulated datasets 45 separate files, each run 12 times.
We defined speedup as the average time for execution for each test divided by the average time when using only one compute node. All the individual plots of speedup relative to the number of genes, arrays and clusters versus the number of nodes can be found in the online supplement [see Additional files 1 , 2 , 3 ]. Detected interactions that affect the time of execution using ParaKMeans. The column charts plot the speedup fold increase relative to a single node configuration versus the number of compute nodes used in the analysis.
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Parallel computing is everywhere, on smartphones, laptops; at online shopping sites, universities, computing centers; behind the search engines. Efficiency and productivity at these scales and contexts are only possible by scalable parallel algorithms using efficient communication schemes, routing and networks. Global chair: Christos Zaroliagis. The need for high performance computations is driven by the need for large-scale simulations in science and engineering, finance, life sciences etc.
This demand goes hand in hand with the necessity to develop highly scalable numerical methods and algorithms that are able to efficiently exploit modern computer architectures. The scalability of these algorithms and methods and their suitability to efficiently utilize the available high performance, but in general heterogeneous, computer resources, is a key point to improve the performance of Computational Science and Engineering applications. Global chair: Elisabeth Larsson. Hardware accelerators of various kinds offer a potential for achieving massive performance in applications that can leverage their high degree of parallelism and customization.
Examples include graphics processors GPUs , manycore co-processors, as well as more customizable devices, such as FPGA-based systems, and streaming data-flow architectures. Global chair: Angeles Navarro. Euro-Par Topic list 1. Support Tools and Environments Despite an impressive body of research, parallel and distributed programming remains a complex task prone to subtle software issues that can affect both the correctness and the performance of the application. Global chair: Siegfried Benkner Know more.
Performance and Power Modeling, Prediction and Evaluation In recent years, a range of novel methods and tools have been developed for the evaluation, design, and modeling of parallel and distributed systems and applications. Global chair: Leonel Sousa Know more. Scheduling and Load Balancing New computer systems supply an opportunity to improve the performance and the energy consumption of the applications by the exploitation of several parallelism levels. Global chair: Anne Benoit Know more. High Performance Architectures and Compilers This topic deals with architecture design, languages, and compilation for parallel high performance systems.
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