A Succinct Four Russians Speedup for Edit Distance Computation and One-against-many Banded Alignment

The classical Four Russians speedup for computing edit distance (a.k.a. Levenshtein distance), due to Masek and Paterson [Masek and Paterson, 1980], involves partitioning the dynamic programming table into k-by-k square blocks and generating a lookup table in O(psi^{2k} k^2 |Sigma|^{2k}) time and O(psi^{2k} k |Sigma|^{2k}) space for block size k, where psi depends on the cost function (for unit costs psi = 3) and |Sigma| is the size of the alphabet. We show that the O(psi^{2k} k^2) and O(psi^{2k} k) factors can be improved to O(k^2 lg{k}) time and O(k^2) space. Thus, we improve the time and space complexity of that aspect compared to Masek and Paterson [Masek and Paterson, 1980] and remove the dependence on psi. We further show that for certain problems the O(|Sigma|^{2k}) factor can also be reduced. Using this technique, we show a new algorithm for the fundamental problem of one-against-many banded alignment. In particular, comparing one string of length m to n other strings of length m with maximum distance d can be performed in O(n m + m d^2 lg{d} + n d^3) time. When d is reasonably small, this approaches or meets the current best theoretic result of O(nm + n d^2) achieved by using the best known pairwise algorithm running in O(m + d^2) time [Myers, 1986][Ukkonen, 1985] while potentially being more practical. It also improves on the standard practical approach which requires O(n m d) time to iteratively run an O(md) time pairwise banded alignment algorithm. Regarding pairwise comparison, we extend the classic result of Masek and Paterson [Masek and Paterson, 1980] which computes the edit distance between two strings in O(m^2/log{m}) time to remove the dependence on psi even when edits have arbitrary costs from a penalty matrix. Crochemore, Landau, and Ziv-Ukelson [Crochemore, 2003] achieved a similar result, also allowing for unrestricted scoring matrices, but with variable-sized blocks. In practical applications of the Four Russians speedup wherein space efficiency is important and smaller block sizes k are used (notably k < |Sigma|), Kim, Na, Park, and Sim [Kim et al., 2016] showed how to remove the dependence on the alphabet size for the unit cost version, generating a lookup table in O(3^{2k} (2k)! k^2) time and O(3^{2k} (2k)! k) space. Combining their work with our result yields an improvement to O((2k)! k^2 lg{k}) time and O((2k)! k^2) space

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

Models for Parallel Computation in Multi-Core, Heterogeneous, and Ultra Wide-Word Architectures

Multi-core processors have become the dominant processor architecture with 2, 4, and 8 cores on a chip being widely available and an increasing number of cores predicted for the future. In addition, the decreasing costs and increasing programmability of Graphic Processing Units (GPUs) have made these an accessible source of parallel processing power in general purpose computing. Among the many research challenges that this scenario has raised are the fundamental problems related to theoretical modeling of computation in these architectures. In this thesis we study several aspects of computation in modern parallel architectures, from modeling of computation in multi-cores and heterogeneous platforms, to multi-core cache management strategies, through the proposal of an architecture that exploits bit-parallelism on thousands of bits. Observing that in practice multi-cores have a small number of cores, we propose a model for low-degree parallelism for these architectures. We argue that assuming a small number of processors (logarithmic in a problem's input size) simplifies the design of parallel algorithms. We show that in this model a large class of divide-and-conquer and dynamic programming algorithms can be parallelized with simple modifications to sequential programs, while achieving optimal parallel speedups. We further explore low-degree-parallelism in computation, providing evidence of fundamental differences in practice and theory between systems with a sublinear and linear number of processors, and suggesting a sharp theoretical gap between the classes of problems that are efficiently parallelizable in each case. Efficient strategies to manage shared caches play a crucial role in multi-core performance. We propose a model for paging in multi-core shared caches, which extends classical paging to a setting in which several threads share the cache. We show that in this setting traditional cache management policies perform poorly, and that any effective strategy must partition the cache among threads, with a partition that adapts dynamically to the demands of each thread. Inspired by the shared cache setting, we introduce the minimum cache usage problem, an extension to classical sequential paging in which algorithms must account for the amount of cache they use. This cache-aware model seeks algorithms with good performance in terms of faults and the amount of cache used, and has applications in energy efficient caching and in shared cache scenarios. The wide availability of GPUs has added to the parallel power of multi-cores, however, most applications underutilize the available resources. We propose a model for hybrid computation in heterogeneous systems with multi-cores and GPU, and describe strategies for generic parallelization and efficient scheduling of a large class of divide-and-conquer algorithms. Lastly, we introduce the Ultra-Wide Word architecture and model, an extension of the word-RAM model, that allows for constant time operations on thousands of bits in parallel. We show that a large class of existing algorithms can be implemented in the Ultra-Wide Word model, achieving speedups comparable to those of multi-threaded computations, while avoiding the more difficult aspects of parallel programming

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum