O(1) Computation of Legendre polynomials and Gauss-Legendre nodes and weights for parallel computing

A self-contained set of algorithms is proposed for the fast evaluation of Legendre polynomials of arbitrary degree and argument is an element of [-1, 1]. More specifically the time required to evaluate any Legendre polynomial, regardless of argument and degree, is bounded by a constant; i.e., the complexity is O(1). The proposed algorithm also immediately yields an O(1) algorithm for computing an arbitrary Gauss-Legendre quadrature node. Such a capability is crucial for efficiently performing certain parallel computations with high order Legendre polynomials, such as computing an integral in parallel by means of Gauss-Legendre quadrature and the parallel evaluation of Legendre series. In order to achieve the O(1) complexity, novel efficient asymptotic expansions are derived and used alongside known results. A C++ implementation is available from the authors that includes the evaluation routines of the Legendre polynomials and Gauss-Legendre quadrature rules

Accurate determination of node and arc multiplicities in de Bruijn graphs using conditional random fields

Background: De Bruijn graphs are key data structures for the analysis of next-generation sequencing data. They efficiently represent the overlap between reads and hence, also the underlying genome sequence. However, sequencing errors and repeated subsequences render the identification of the true underlying sequence difficult. A key step in this process is the inference of the multiplicities of nodes and arcs in the graph. These multiplicities correspond to the number of times eachk-mer (resp.k+1-mer) implied by a node (resp. arc) is present in the genomic sequence. Determining multiplicities thus reveals the repeat structure and presence of sequencing errors. Multiplicities of nodes/arcs in the de Bruijn graph are reflected in their coverage, however, coverage variability and coverage biases render their determination ambiguous. Current methods to determine node/arc multiplicities base their decisions solely on the information in nodes and arcs individually, under-utilising the information present in the sequencing data. Results: To improve the accuracy with which node and arc multiplicities in a de Bruijn graph are inferred, we developed a conditional random field (CRF) model to efficiently combine the coverage information within each node/arc individually with the information of surrounding nodes and arcs. Multiplicities are thus collectively assigned in a more consistent manner. Conclusions: We demonstrate that the CRF model yields significant improvements in accuracy and a more robust expectation-maximisation parameter estimation. Truek-mers can be distinguished from erroneousk-mers with a higher F(1)score than existing methods. A C++11 implementation is available atunder the GNU AGPL v3.0 license

Scalable parallel computation of the translation operator in three dimensions

We propose a novel algorithm for the parallel, distributed-memory computation of the translation operator in the three-dimensional multilevel fast multipole algorithm (MLFMA). Sequential algorithms can compute the translation operator with L multipoles and O(L-2) sampling points in O(L-2) time. State-of-the-art hierarchical parallelization schemes of the MLFMA rely on the distribution of radiation patterns and associated translation operators among P = O(L-2) parallel processes, necessitating the development of distributed-memory algorithms for the computation of the translation operator. Whereas a baseline parallel algorithm computes this translation operator in O(L) time, we propose an algorithm that achieves this in only O(log L) time. For large translation operators and a high number of parallel processes, our algorithm proves to be roughly ten times faster than the baseline algorithm

BLAMM : BLAS-based algorithm for finding position weight matrix occurrences in DNA sequences on CPUs and GPUs

Background The identification of all matches of a large set of position weight matrices (PWMs) in long DNA sequences requires significant computational resources for which a number of efficient yet complex algorithms have been proposed. Results We propose BLAMM, a simple and efficient tool inspired by high performance computing techniques. The workload is expressed in terms of matrix-matrix products that are evaluated with high efficiency using optimized BLAS library implementations. The algorithm is easy to parallelize and implement on CPUs and GPUs and has a runtime that is independent of the selected p-value. In terms of single-core performance, it is competitive with state-of-the-art software for PWM matching while being much more efficient when using multithreading. Additionally, BLAMM requires negligible memory. For example, both strands of the entire human genome can be scanned for 1404 PWMs in the JASPAR database in 13 min with a p-value of 10(-4) using a 36-core machine. On a dual GPU system, the same task can be performed in under 5 min. Conclusions BLAMM is an efficient tool for identifying PWM matches in large DNA sequences. Its C++ source code is available under the GNU General Public License Version 3 at https://github.com/biointec/blamm

Performing large full-wave simulations by means of a parallel MLFMA implementation

In this paper large full-wave simulations are performed using a parallel Multilevel Fast Multipole Algorithm (MLFMA) implementation. The data structures of the MLFMA-tree are partitioned according to the so-called hierarchical partitioning scheme, while the radiation patterns are partitioned in a blockwise way. To test the implementation of the algorithm, a full-wave simulation of a canonical example with more than 50 millions of unknowns has been performed