Abstract.A new method for computing the discrete Fourier transform (DFT) of data endowed with linear symmetries is presented. The method minimizes operations and memory space requirements by eliminating redundant data and computations induced by the symmetry on the DFT equations. The arithmetic complexity of the new method is always lower, and in some cases significantly lower than that of its predecesor. A parallel version of the new method is also discussed. Symmetry-aware DFTs are crucial in the computer determination of the atomic structure of crystals from x-ray diffraction intensity patterns. (N d log N ) operations. Although this complexity bound cannot be improved for general DFT computations, some attempts to reduce the actual operation count and memory space requirements have been made for problems whose data is endowed with redundancies, such as x-ray crystal diffraction intensity data. In this article we review

Background Long interspersed element 1 (LINE-1 or L1) retrotransposons are mobile elements that constitute 17–20% of the human genome. Strong correlations between abnormal L1 expression and several human diseases have been reported. This has motivated increasing interest in accurate quantification of the number of L1 copies present in any given biologic specimen. A main obstacle toward this aim is that L1s are relatively long DNA segments with regions of high variability, or largely present in the human genome as truncated fragments. These particularities render traditional alignment strategies, such as seed-and-extend inefficient, as the number of segments that are similar to L1s explodes exponentially. This study uses the pattern matching methodology for more accurate identification of L1s. We validate experimentally the superiority of pattern matching for L1 detection over alternative methods and discuss some of its potential applications. Results Pattern matching detected full-length L1 copies with high precision, reasonable computational time, and no prior input information. It also detected truncated and significantly altered copies of L1 with relatively high precision. The method was effectively used to annotate L1s in a target genome and to calculate copy number variation with respect to a reference genome. Crucial to the success of implementation was the selection of a small set of k-mer probes from a set of sequences presenting a stable pattern of distribution in the genome. As in seed-and-extend methods, the pattern matching algorithm sowed these k-mer probes, but instead of using heuristic extensions around the seeds, the analysis was based on distribution patterns within the genome. The desired level of precision could be adjusted, with some loss of recall. Conclusion Pattern matching is more efficient than seed-and-extend methods for the detection of L1 segments whose characterization depends on a finite set of sequences with common areas of low variability. We propose that pattern matching may help establish correlations between L1 copy number and disease states associated with L1 mobilization and evolution.

We introduce a tensor sum which is useful for the design and analysis of digit-index permutations (DIPs) algorithms. Using this operation we obtain a new high-performance algorithm for the family of DIPs. We discuss an implementation in the applicative language Sisal and show how different choices of parameters yield different DIPs. The efficiency of the special case of digit reversal is illustrated with performance results on a Cray C-90.

No abstract

AbstractÐWe propose an algebraic framework for the design and implementation of a large class of data-sorting procedures, including all index-digit permutations used in FFTs. We discuss both old and new algorithms in terms of this framework. We show that the algebraic formulation of the new algorithms can be easily encoded using a functional programming language, and that the resulting code introduces no inefficiencies. We present performance results for implementations of three new algorithms for mixed-radix digitreversal on a Cray C-90 and on a Sun Sparc 5.