A simple, practical and complete O-time Algorithm for RNA folding using the Four-Russians Speedup

<p>Abstract</p> <p>Background</p> <p>The problem of computationally predicting the secondary structure (or folding) of RNA molecules was first introduced more than thirty years ago and yet continues to be an area of active research and development. The basic <it>RNA-folding problem </it>of finding a maximum cardinality, non-crossing, matching of complimentary nucleotides in an RNA sequence of length <it>n</it>, has an <it>O</it>(<it>n</it>³)-time dynamic programming solution that is widely applied. It is known that an <it>o</it>(<it>n</it>³) worst-case time solution is possible, but the published and suggested methods are complex and have not been established to be practical. Significant practical improvements to the original dynamic programming method have been introduced, but they retain the <it>O</it>(<it>n</it>³) worst-case time bound when <it>n </it>is the only problem-parameter used in the bound. Surprisingly, the most widely-used, general technique to achieve a worst-case (and often practical) speed up of dynamic programming, the <it>Four-Russians </it>technique, has not been previously applied to the RNA-folding problem. This is perhaps due to technical issues in adapting the technique to RNA-folding.</p> <p>Results</p> <p>In this paper, we give a simple, complete, and practical Four-Russians algorithm for the basic RNA-folding problem, achieving a worst-case time-bound of <it>O</it>(<it>n</it>³/log(<it>n</it>)).</p> <p>Conclusions</p> <p>We show that this time-bound can also be obtained for richer nucleotide matching scoring-schemes, and that the method achieves consistent speed-ups in practice. The contribution is both theoretical and practical, since the basic RNA-folding problem is often solved multiple times in the inner-loop of more complex algorithms, and for long RNA molecules in the study of RNA virus genomes.</p

If the Current Clique Algorithms are Optimal, so is Valiant's Parser

The CFG recognition problem is: given a context-free grammar $\mathcal{G}$ and a string $w$ of length $n$, decide if $w$ can be obtained from $\mathcal{G}$. This is the most basic parsing question and is a core computer science problem. Valiant's parser from 1975 solves the problem in $O(n^{\omega})$ time, where $\omega<2.373$ is the matrix multiplication exponent. Dozens of parsing algorithms have been proposed over the years, yet Valiant's upper bound remains unbeaten. The best combinatorial algorithms have mildly subcubic $O(n^3/\log^3{n})$ complexity. Lee (JACM'01) provided evidence that fast matrix multiplication is needed for CFG parsing, and that very efficient and practical algorithms might be hard or even impossible to obtain. Lee showed that any algorithm for a more general parsing problem with running time $O(|\mathcal{G}|\cdot n^{3-\varepsilon})$ can be converted into a surprising subcubic algorithm for Boolean Matrix Multiplication. Unfortunately, Lee's hardness result required that the grammar size be $|\mathcal{G}|=\Omega(n^6)$. Nothing was known for the more relevant case of constant size grammars. In this work, we prove that any improvement on Valiant's algorithm, even for constant size grammars, either in terms of runtime or by avoiding the inefficiencies of fast matrix multiplication, would imply a breakthrough algorithm for the $k$-Clique problem: given a graph on $n$ nodes, decide if there are $k$ that form a clique. Besides classifying the complexity of a fundamental problem, our reduction has led us to similar lower bounds for more modern and well-studied cubic time problems for which faster algorithms are highly desirable in practice: RNA Folding, a central problem in computational biology, and Dyck Language Edit Distance, answering an open question of Saha (FOCS'14)

Accelerating dynamic programming

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 129-136).Dynamic Programming (DP) is a fundamental problem-solving technique that has been widely used for solving a broad range of search and optimization problems. While DP can be invoked when more specialized methods fail, this generality often incurs a cost in efficiency. We explore a unifying toolkit for speeding up DP, and algorithms that use DP as subroutines. Our methods and results can be summarized as follows. - Acceleration via Compression. Compression is traditionally used to efficiently store data. We use compression in order to identify repeats in the table that imply a redundant computation. Utilizing these repeats requires a new DP, and often different DPs for different compression schemes. We present the first provable speedup of the celebrated Viterbi algorithm (1967) that is used for the decoding and training of Hidden Markov Models (HMMs). Our speedup relies on the compression of the HMM's observable sequence. - Totally Monotone Matrices. It is well known that a wide variety of DPs can be reduced to the problem of finding row minima in totally monotone matrices. We introduce this scheme in the context of planar graph problems. In particular, we show that planar graph problems such as shortest paths, feasible flow, bipartite perfect matching, and replacement paths can be accelerated by DPs that exploit a total-monotonicity property of the shortest paths. - Combining Compression and Total Monotonicity. We introduce a method for accelerating string edit distance computation by combining compression and totally monotone matrices.(cont.) In the heart of this method are algorithms for computing the edit distance between two straight-line programs. These enable us to exploits the compressibility of strings, even if each string is compressed using a different compression scheme. - Partial Tables. In typical DP settings, a table is filled in its entirety, where each cell corresponds to some subproblem. In some cases, by changing the DP, it is possible to compute asymptotically less cells of the table. We show that [theta](n³) subproblems are both necessary and sufficient for computing the similarity between two trees. This improves all known solutions and brings the idea of partial tables to its full extent. - Fractional Subproblems. In some DPs, the solution to a subproblem is a data structure rather than a single value. The entire data structure of a subproblem is then processed and used to construct the data structure of larger subproblems. We suggest a method for reusing parts of a subproblem's data structure. In some cases, such fractional parts remain unchanged when constructing the data structure of larger subproblems. In these cases, it is possible to copy this part of the data structure to the larger subproblem using only a constant number of pointer changes. We show how this idea can be used for finding the optimal tree searching strategy in linear time. This is a generalization of the well known binary search technique from arrays to trees.by Oren Weimann.Ph.D

Safe and Complete Algorithms for Dynamic Programming Problems, with an Application to RNA Folding

Peer reviewe

GENOME ASSEMBLY AND VARIANT DETECTION USING EMERGING SEQUENCING TECHNOLOGIES AND GRAPH BASED METHODS

The increased availability of genomic data and the increased ease and lower costs of DNA sequencing have revolutionized biomedical research. One of the critical steps in most bioinformatics analyses is the assembly of the genome sequence of an organism using the data generated from the sequencing machines. Despite the long length of sequences generated by third-generation sequencing technologies (tens of thousands of basepairs), the automated reconstruction of entire genomes continues to be a formidable computational task. Although long read technologies help in resolving highly repetitive regions, the contigs generated from long read assembly do not always span a complete chromosome or even an arm of the chromosome. Recently, new genomic technologies have been developed that can ''bridge" across repeats or other genomic regions that are difficult to sequence or assemble and improve genome assemblies by ''scaffolding" together large segments of the genome. The problem of scaffolding is vital in the context of both single genome assembly of large eukaryotic genomes and in metagenomics where the goal is to assemble multiple bacterial genomes in a sample simultaneously. First, we describe SALSA2, a method we developed to use interaction frequency between any two loci in the genome obtained using Hi-C technology to scaffold fragmented eukaryotic genome assemblies into chromosomes. SALSA2 can be used with either short or long read assembly to generate highly contiguous and accurate chromosome level assemblies. Hi-C data are known to introduce small inversion errors in the assembly, so we included the assembly graph in the scaffolding process and used the sequence overlap information to correct the orientation errors. Next, we present our contributions to metagenomics. We developed a scaffolding and variant detection method MetaCarvel for metagenomic datasets. Several factors such as the presence of inter-genomic repeats, coverage ambiguities, and polymorphic regions in the genomes complicate the task of scaffolding metagenomes. Variant detection is also tricky in metagenomes because the different genomes within these complex samples are not known beforehand. We showed that MetaCarvel was able to generate accurate scaffolds and find genome-wide variations de novo in metagenomic datasets. Finally, we present EDIT, a tool for clustering millions of DNA sequence fragments originating from the highly conserved 16s rRNA gene in bacteria. We extended classical Four Russians' speed up to banded sequence alignment and showed that our method clusters highly similar sequences efficiently. This method can also be used to remove duplicates or near duplicate sequences from a dataset. With the increasing data being generated in different genomic and metagenomic studies using emerging sequencing technologies, our software tools and algorithms are well timed with the need of the community

Markets, Elections, and Microbes: Data-driven Algorithms from Theory to Practice

Many modern problems in algorithms and optimization are driven by data which often carries with it an element of uncertainty. In this work, we conduct an investigation into algorithmic foundations and applications across three main areas. The first area is online matching algorithms for e-commerce applications such as online sales and advertising. The importance of e-commerce in modern business cannot be overstated and even minor algorithmic improvements can have huge impacts. In online matching problems, we generally have a known offline set of goods or advertisements while users arrive online and allocations must be made immediately and irrevocably when a user arrives. However, in the real world, there is also uncertainty about a user's true interests and this can be modeled by considering matching problems in a graph with stochastic edges that only have a probability of existing. These edges can represent the probability of a user purchasing a product or clicking on an ad. Thus, we optimize over data which only provides an estimate of what types of users will arrive and what they will prefer. We survey a broad landscape of problems in this area, gain a deeper understanding of the algorithmic challenges, and present algorithms with improved worst case performance The second area is constrained clustering where we explore classical clustering problems with additional constraints on which data points should be clustered together. Utilizing these constraints is important for many clustering problems because they can be used to ensure fairness, exploit expert advice, or capture natural properties of the data. In simplest case, this can mean some pairs of points have ``must-link'' constraints requiring that that they must be clustered together. Moving into stochastic settings, we can describe more general pairwise constraints such as bounding the probability that two points are separated into different clusters. This lets us introduce a new notion of fairness for clustering and address stochastic problems such as semi-supervised learning with advice from imperfect experts. Here, we introduce new models of constrained clustering including new notions of fairness for clustering applications. Since these problems are NP-hard, we give approximation algorithms and in some cases conduct experiments to explore how the algorithms perform in practice. Finally, we look closely at the particular clustering problem of drawing election districts and show how constraining the clusters based on past voting data can interact with voter incentives. The third area is string algorithms for bioinformatics and metagenomics specifically where the data deluge from next generation sequencing drives the necessity for new algorithms that are both fast and accurate. For metagenomic analysis, we present a tool for clustering a microbial marker gene, the 16S ribosomal RNA gene. On the more theoretical side, we present a succinct application of the Method of the Four Russians to edit distance computation as well as new algorithms and bounds for the maximum duo-preservation string mapping (MPSM) problem

New Graph Decompositions and Combinatorial Boolean Matrix Multiplication Algorithms

We revisit the fundamental Boolean Matrix Multiplication (BMM) problem. With the invention of algebraic fast matrix multiplication over 50 years ago, it also became known that BMM can be solved in truly subcubic $O(n^\omega)$ time, where $\omega<3$; much work has gone into bringing $\omega$ closer to $2$. Since then, a parallel line of work has sought comparably fast combinatorial algorithms but with limited success. The naive $O(n^3)$-time algorithm was initially improved by a $\log^2{n}$ factor [Arlazarov et al.; RAS'70], then by $\log^{2.25}{n}$ [Bansal and Williams; FOCS'09], then by $\log^3{n}$ [Chan; SODA'15], and finally by $\log^4{n}$ [Yu; ICALP'15]. We design a combinatorial algorithm for BMM running in time $n^3 / 2^{\Omega(\sqrt[7]{\log n})}$ -- a speed-up over cubic time that is stronger than any poly-log factor. This comes tantalizingly close to refuting the conjecture from the 90s that truly subcubic combinatorial algorithms for BMM are impossible. This popular conjecture is the basis for dozens of fine-grained hardness results. Our main technical contribution is a new regularity decomposition theorem for Boolean matrices (or equivalently, bipartite graphs) under a notion of regularity that was recently introduced and analyzed analytically in the context of communication complexity [Kelley, Lovett, Meka; arXiv'23], and is related to a similar notion from the recent work on $3$-term arithmetic progression free sets [Kelley, Meka; FOCS'23]

Efficient approximate string matching techniques for sequence alignment

One of the outstanding milestones achieved in recent years in the field of biotechnology research has been the development of high-throughput sequencing (HTS). Due to the fact that at the moment it is technically impossible to decode the genome as a whole, HTS technologies read billions of relatively short chunks of a genome at random locations. Such reads then need to be located within a reference for the species being studied (that is aligned or mapped to the genome): for each read one identifies in the reference regions that share a large sequence similarity with it, therefore indicating what the read¿s point or points of origin may be. HTS technologies are able to re-sequence a human individual (i.e. to establish the differences between his/her individual genome and the reference genome for the human species) in a very short period of time. They have also paved the way for the development of a number of new protocols and methods, leading to novel insights in genomics and biology in general. However, HTS technologies also pose a challenge to traditional data analysis methods; this is due to the sheer amount of data to be processed and the need for improved alignment algorithms that can generate accurate results quickly. This thesis tackles the problem of sequence alignment as a step within the analysis of HTS data. Its contributions focus on both the methodological aspects and the algorithmic challenges towards efficient, scalable, and accurate HTS mapping. From a methodological standpoint, this thesis strives to establish a comprehensive framework able to assess the quality of HTS mapping results. In order to be able to do so one has to understand the source and nature of mapping conflicts, and explore the accuracy limits inherent in how sequence alignment is performed for current HTS technologies. From an algorithmic standpoint, this work introduces state-of-the-art index structures and approximate string matching algorithms. They contribute novel insights that can be used in practical applications towards efficient and accurate read mapping. More in detail, first we present methods able to reduce the storage space taken by indexes for genome-scale references, while still providing fast query access in order to support effective search algorithms. Second, we describe novel filtering techniques that vastly reduce the computational requirements of sequence mapping, but are nonetheless capable of giving strict algorithmic guarantees on the completeness of the results. Finally, this thesis presents new incremental algorithmic techniques able to combine several approximate string matching algorithms; this leads to efficient and flexible search algorithms allowing the user to reach arbitrary search depths. All algorithms and methodological contributions of this thesis have been implemented as components of a production aligner, the GEM-mapper, which is publicly available, widely used worldwide and cited by a sizeable body of literature. It offers flexible and accurate sequence mapping while outperforming other HTS mappers both as to running time and to the quality of the results it produces.Uno de los avances más importantes de los últimos años en el campo de la biotecnología ha sido el desarrollo de las llamadas técnicas de secuenciación de alto rendimiento (high-throughput sequencing, HTS). Debido a las limitaciones técnicas para secuenciar un genoma, las técnicas de alto rendimiento secuencian individualmente billones de pequeñas partes del genoma provenientes de regiones aleatorias. Posteriormente, estas pequeñas secuencias han de ser localizadas en el genoma de referencia del organismo en cuestión. Este proceso se denomina alineamiento - o mapeado - y consiste en identificar aquellas regiones del genoma de referencia que comparten una alta similaridad con las lecturas producidas por el secuenciador. De esta manera, en cuestión de horas, la secuenciación de alto rendimiento puede secuenciar un individuo y establecer las diferencias de este con el resto de la especie. En última instancia, estas tecnologías han potenciado nuevos protocolos y metodologías de investigación con un profundo impacto en el campo de la genómica, la medicina y la biología en general. La secuenciación alto rendimiento, sin embargo, supone un reto para los procesos tradicionales de análisis de datos. Debido a la elevada cantidad de datos a analizar, se necesitan nuevas y mejoradas técnicas algorítmicas que puedan escalar con el volumen de datos y producir resultados precisos. Esta tesis aborda dicho problema. Las contribuciones que en ella se realizan se enfocan desde una perspectiva metodológica y otra algorítmica que propone el desarrollo de nuevos algoritmos y técnicas que permitan alinear secuencias de manera eficiente, precisa y escalable. Desde el punto de vista metodológico, esta tesis analiza y propone un marco de referencia para evaluar la calidad de los resultados del alineamiento de secuencias. Para ello, se analiza el origen de los conflictos durante la alineación de secuencias y se exploran los límites alcanzables en calidad con las tecnologías de secuenciación de alto rendimiento. Desde el punto de vista algorítmico, en el contexto de la búsqueda aproximada de patrones, esta tesis propone nuevas técnicas algorítmicas y de diseño de índices con el objetivo de mejorar la calidad y el desempeño de las herramientas dedicadas a alinear secuencias. En concreto, esta tesis presenta técnicas de diseño de índices genómicos enfocados a obtener un acceso más eficiente y escalable. También se presentan nuevas técnicas algorítmicas de filtrado con el fin de reducir el tiempo de ejecución necesario para alinear secuencias. Y, por último, se proponen algoritmos incrementales y técnicas híbridas para combinar métodos de alineamiento y mejorar el rendimiento en búsquedas donde el error esperado es alto. Todo ello sin degradar la calidad de los resultados y con garantías formales de precisión. Para concluir, es preciso apuntar que todos los algoritmos y metodologías propuestos en esta tesis están implementados y forman parte del alineador GEM. Este versátil alineador ofrece resultados de alta calidad en entornos de producción siendo varias veces más rápido que otros alineadores. En la actualidad este software se ofrece gratuitamente, tiene una amplia comunidad de usuarios y ha sido citado en numerosas publicaciones científicas

Understanding Quantum Technologies 2022

Understanding Quantum Technologies 2022 is a creative-commons ebook that provides a unique 360 degrees overview of quantum technologies from science and technology to geopolitical and societal issues. It covers quantum physics history, quantum physics 101, gate-based quantum computing, quantum computing engineering (including quantum error corrections and quantum computing energetics), quantum computing hardware (all qubit types, including quantum annealing and quantum simulation paradigms, history, science, research, implementation and vendors), quantum enabling technologies (cryogenics, control electronics, photonics, components fabs, raw materials), quantum computing algorithms, software development tools and use cases, unconventional computing (potential alternatives to quantum and classical computing), quantum telecommunications and cryptography, quantum sensing, quantum technologies around the world, quantum technologies societal impact and even quantum fake sciences. The main audience are computer science engineers, developers and IT specialists as well as quantum scientists and students who want to acquire a global view of how quantum technologies work, and particularly quantum computing. This version is an extensive update to the 2021 edition published in October 2021.Comment: 1132 pages, 920 figures, Letter forma

Efficient homology search for genomic sequence databases

Genomic search tools can provide valuable insights into the chemical structure, evolutionary origin and biochemical function of genetic material. A homology search algorithm compares a protein or nucleotide query sequence to each entry in a large sequence database and reports alignments with highly similar sequences. The exponential growth of public data banks such as GenBank has necessitated the development of fast, heuristic approaches to homology search. The versatile and popular blast algorithm, developed by researchers at the US National Center for Biotechnology Information (NCBI), uses a four-stage heuristic approach to efficiently search large collections for analogous sequences while retaining a high degree of accuracy. Despite an abundance of alternative approaches to homology search, blast remains the only method to offer fast, sensitive search of large genomic collections on modern desktop hardware. As a result, the tool has found widespread use with millions of queries posed each day. A significant investment of computing resources is required to process this large volume of genomic searches and a cluster of over 200 workstations is employed by the NCBI to handle queries posed through the organisation&#039;s website. As the growth of sequence databases continues to outpace improvements in modern hardware, blast searches are becoming slower each year and novel, faster methods for sequence comparison are required. In this thesis we propose new techniques for fast yet accurate homology search that result in significantly faster blast searches. First, we describe improvements to the final, gapped alignment stages where the query and sequences from the collection are aligned to provide a fine-grain measure of similarity. We describe three new methods for aligning sequences that roughly halve the time required to perform this computationally expensive stage. Next, we investigate improvements to the first stage of search, where short regions of similarity between a pair of sequences are identified. We propose a novel deterministic finite automaton data structure that is significantly smaller than the codeword lookup table employed by ncbi-blast, resulting in improved cache performance and faster search times. We also discuss fast methods for nucleotide sequence comparison. We describe novel approaches for processing sequences that are compressed using the byte packed format already utilised by blast, where four nucleotide bases from a strand of DNA are stored in a single byte. Rather than decompress sequences to perform pairwise comparisons, our innovations permit sequences to be processed in their compressed form, four bases at a time. Our techniques roughly halve average query evaluation times for nucleotide searches with no effect on the sensitivity of blast. Finally, we present a new scheme for managing the high degree of redundancy that is prevalent in genomic collections. Near-duplicate entries in sequence data banks are highly detrimental to retrieval performance, however existing methods for managing redundancy are both slow, requiring almost ten hours to process the GenBank database, and crude, because they simply purge highly-similar sequences to reduce the level of internal redundancy. We describe a new approach for identifying near-duplicate entries that is roughly six times faster than the most successful existing approaches, and a novel approach to managing redundancy that reduces collection size and search times but still provides accurate and comprehensive search results. Our improvements to blast have been integrated into our own version of the tool. We find that our innovations more than halve average search times for nucleotide and protein searches, and have no signifcant effect on search accuracy. Given the enormous popularity of blast, this represents a very significant advance in computational methods to aid life science research