Alexander Tiskin

Communication lower bounds for distributed-memory matrix multiplication

We present lower bounds on the amount of communication that matrix multiplication algorithms must perform on a distributed-memory parallel computer. We denote the number of processors by P and the dimension of square matrices by n. We show that the most widely used class of algorithms, the so-called two-dimensional (2D) algorithms, are optimal, in the sense that in any algorithm that only uses O(n2/P) words of memory per processor, at least one processor must send or receive Ω(n2/P1/2) words. We also show that algorithms from another class, the so-called three-dimensional (3D) algorithms, are also optimal. These algorithms use replication to reduce communication. We show that in any algorithm that uses O(n2/P2/3) words of memory per processor, at least one processor must send or receive Ω(n2/P2/3) words. Furthermore, we show a continuous tradeoff between the size of local memories and the amount of communication that must be performed. The 2D and 3D bounds are essentially instantiations of this tradeoff. We also show that if the input is distributed across the local memories of multiple nodes without replication, then Ω(n2) words must cross any bisection cut of the machine. All our bounds apply only to conventional Θ(n3) algorithms. They do not apply to Strassens algorithm or other o(n3) algorithms.

Conserved Noncoding Sequences Highlight Shared Components of Regulatory Networks in Dicotyledonous Plants

This study identifies regions of noncoding DNA in dicot plants that are likely to facilitate complex regulation of genes by binding multiple transcription factors. Regulatory mechanisms that the model organism Arabidopsis is likely to share with crop plants provide a focus for research that has real-world applications. Conserved noncoding sequences (CNSs) in DNA are reliable pointers to regulatory elements controlling gene expression. Using a comparative genomics approach with four dicotyledonous plant species (Arabidopsis thaliana, papaya [Carica papaya], poplar [Populus trichocarpa], and grape [Vitis vinifera]), we detected hundreds of CNSs upstream of Arabidopsis genes. Distinct positioning, length, and enrichment for transcription factor binding sites suggest these CNSs play a functional role in transcriptional regulation. The enrichment of transcription factors within the set of genes associated with CNS is consistent with the hypothesis that together they form part of a conserved transcriptional network whose function is to regulate other transcription factors and control development. We identified a set of promoters where regulatory mechanisms are likely to be shared between the model organism Arabidopsis and other dicots, providing areas of focus for further research.

Semi-local string comparison: algorithmic techniques and applications

Abstract.Given two strings, the longest common subsequence (LCS) problem consists in computing the length of the longest string that is a subsequence of both input strings. Its generalisation, the all semi-local LCS problem, requires computing the LCS length for each string against all substrings of the other string, and for all prefixes of each string against all suffixes of the other string. We survey a number of algorithmic techniques related to the all semi-local LCS problem. We then present a number of algorithmic applications of these techniques, both existing and new. In particular, we obtain a new all semi-local LCS algorithm, with asymptotic running time matching (in the case of an unbounded alphabet) the fastest known global LCS algorithm by Masek and Paterson. We conclude that semi-local string comparison turns out to be a useful algorithmic plug-in, which unifies, and often improves on, a number of previous approaches to various substring- and subsequence-related problems.

Fast distance multiplication of unit-Monge matrices

Monge matrices play a fundamental role in optimisation theory, graph and string algorithms. Distance multiplication of two Monge matrices of size n can be performed in time O(n2). Motivated by applications to string algorithms, we introduced in previous works a subclass of Monge matrices, that we call simple unit-Monge matrices. We also gave a distance multiplication algorithm for such matrices, running in time O(n1.5). Landau asked whether this problem can be solved in linear time. In the current work, we give an algorithm running in time O(nlogn), thus approaching an answer to Landau’s question within a logarithmic factor. The new algorithm implies immediate improvements in running time for a number of algorithms on strings and graphs. In particular, we obtain an algorithm for finding a maximum clique in a circle graph in time O(nlog2n), and a surprisingly efficient algorithm for comparing compressed strings. We also point to potential applications in group theory, by making a connection between unit-Monge matrices and Coxeter monoids. We conclude that unit-Monge matrices are a fascinating object and a powerful tool, that deserve further study from both the mathematical and the algorithmic viewpoints.

Boundary properties of graphs for algorithmic graph problems

The notion of a boundary graph property was recently introduced as a relaxation of that of a minimal property and was applied to several problems of both algorithmic and combinatorial nature. In the present paper, we first survey recent results related to this notion and then apply it to two algorithmic graph problems: Hamiltonian cycle and vertexk-colorability. In particular, we discover the first two boundary classes for the Hamiltonian cycle problem and prove that for any k>3 there is a continuum of boundary classes for vertexk-colorability.

Evolutionary analysis of regulatory sequences (EARS) in plants

Identification of regulatory sequences within non-coding regions of DNA is an essential step towards elucidation of gene networks. This approach constitutes a major challenge, however, as only a very small fraction of non-coding DNA is thought to contribute to gene regulation. The mapping of regulatory regions traditionally involves the laborious construction of promoter deletion series which are then fused to reporter genes and assayed in transgenic organisms. Bioinformatic methods can be used to scan sequences for matches for known regulatory motifs, however these methods are currently hampered by the relatively small amount of such motifs and by a high false-discovery rate. Here, we demonstrate a robust and highly sensitive, in silico method to identify evolutionarily conserved regions within non-coding DNA. Sequence conservation within these regions is taken as evidence for evolutionary pressure against mutations, which is suggestive of functional importance. We test this method on a small set of well characterised promoters, and show that it successfully identifies known regulatory regions. We further show that these evolutionarily conserved sequences contain clusters of transcription binding sites, often described as regulatory modules. A version of the tool optimised for the analysis of plant promoters is available online at http://wsbc.warwick.ac.uk/ears/main.php.

Communication-efficient parallel generic pairwise elimination

The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. In this paper, we consider the parallel complexity of generic pairwise elimination, special cases of which include Gaussian elimination with pairwise pivoting, Gaussian elimination over a finite field, generic Neville elimination and Givens reduction. We develop a new block-recursive, communication-efficient BSP algorithm for generic pairwise elimination.

State-of-the-art in string similarity search and join

String similarity search and its variants are fundamental problems with many applications in areas such as data integration, data quality, computational linguistics, or bioinformatics. A plethora of methods have been developed over the last decades. Obtaining an overview of the state-of-the-art in this field is difficult, as results are published in various domains without much cross-talk, papers use different data sets and often study subtle variations of the core problems, and the sheer number of proposed methods exceeds the capacity of a single research group. In this paper, we report on the results of the probably largest benchmark ever performed in this field. To overcome the resource bottleneck, we organized the benchmark as an international competition, a workshop at EDBT/ICDT 2013. Various teams from different fields and from all over the world developed or tuned programs for two crisply defined problems. All algorithms were evaluated by an external group on two machines. Altogether, we compared 14 different programs on two string matching problems (k-approximate search and k-approximate join) using data sets of increasing sizes and with different characteristics from two different domains. We compare programs primarily by wall clock time, but also provide results on memory usage, indexing time, batch query effects and scalability in terms of CPU cores. Results were averaged over several runs and confirmed on a second, different hardware platform. A particularly interesting observation is that disciplines can and should learn more from each other, with the three best teams rooting in computational linguistics, databases, and bioinformatics, respectively.

Bulk-Synchronous Parallel Gaussian Elimination

The model of bulk-synchronous parallel (BSP) computation is an emerging paradigm of general-purpose parallel computing. We study the BSP complexity of Gaussian elimination and related problems. First, we analyze the Gaussian elimination without pivoting, which can be applied to the LU decomposition of symmetric positive-definite or diagonally dominant real matrices. Then we analyze the Gaussian elimination with Schönhages recursive local pivoting suitable for the LU decomposition of matrices over a finite field, and for the QR decomposition of real matrices by the Givens rotations. Both versions of Gaussian elimination can be performed with an optimal amount of local computation, but optimal communication and synchronization costs cannot be achieved simultaneously. The algorithms presented in the paper allow one to trade off communication and synchronization costs in a certain range, achieving optimal or near-optimal cost values at the extremes. Bibliography: 19 titles.

Towards approximate matching in compressed strings: local subsequence recognition

A grammar-compressed (GC) string is a string generated by a context-free grammar. This compression model includes LZ78 and LZW compression as a special case. We consider the longest common subsequence problem and the local subsequence recognition problem on a GC-text against a plain pattern. We show that, surprisingly, both problems can be solved in time that is within a polylogarithmic factor of the best existing algorithms for the same problems on a plain text. In a wider context presented elsewhere, we use these results as a stepping stone to efficient approximate matching on a GC-text.