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1. A space and time-efficient index for the compacted colored de Bruijn graph

   https://doi.org/10.1093/bioinformatics/bty292  

   Almodaresi, Fatemeh ; Sarkar, Hirak ; Srivastava, Avi ; Patro, Rob ( June 2018 , Bioinformatics)

   Indexing reference sequences for search—both individual genomes and collections of genomes—is an important building block for many sequence analysis tasks. Much work has been dedicated to developing full-text indices for genomic sequences, based on data structures such as the suffix array, the BWT and the FM-index. However, the de Bruijn graph, commonly used for sequence assembly, has recently been gaining attention as an indexing data structure, due to its natural ability to represent multiple references using a graphical structure, and to collapse highly-repetitive sequence regions. Yet, much less attention has been given as to how to best index such a structure, such that queries can be performed efficiently and memory usage remains practical as the size and number of reference sequences being indexed grows large.

   We present a novel data structure for representing and indexing the compacted colored de Bruijn graph, which allows for efficient pattern matching and retrieval of the reference information associated with each k-mer. As the popularity of the de Bruijn graph as an index has increased over the past few years, so have the number of proposed representations of this structure. Existing structures typically fall into two categories; those that are hashing-based and provide very fast access to the underlying k-mer information, and those that are space-frugal and provide asymptotically efficient but practically slower pattern search. Our representation achieves a compromise between these two extremes. By building upon minimum perfect hashing and making use of succinct representations where applicable, our data structure provides practically fast lookup while greatly reducing the space compared to traditional hashing-based implementations. Further, we describe a sampling scheme for this index, which provides the ability to trade off query speed for a reduction in the index size. We believe this representation strikes a desirable balance between speed and space usage, and allows for fast search on large reference sequences.

   Finally, we describe an application of this index to the taxonomic read assignment problem. We show that by adopting, essentially, the approach of Kraken, but replacing k-mer presence with coverage by chains of consistent unique maximal matches, we can improve the space, speed and accuracy of taxonomic read assignment.

   pufferfish is written in C++11, is open source, and is available at https://github.com/COMBINE-lab/pufferfish.

   Supplementary data are available at Bioinformatics online.

    

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2. μ- PBWT: a lightweight r-indexing of the PBWT for storing and querying UK Biobank data

   https://doi.org/10.1093/bioinformatics/btad552  

   Cozzi, Davide ; Rossi, Massimiliano ; Rubinacci, Simone ; Gagie, Travis ; Köppl, Dominik ; Boucher, Christina ; Bonizzoni, Paola ; Robinson, ed., Peter ( September 2023 , Bioinformatics)

   The Positional Burrows–Wheeler Transform (PBWT) is a data structure that indexes haplotype sequences in a manner that enables finding maximal haplotype matches in h sequences containing w variation sites in O(hw) time. This represents a significant improvement over classical quadratic-time approaches. However, the original PBWT data structure does not allow for queries over Biobank panels that consist of several millions of haplotypes, if an index of the haplotypes must be kept entirely in memory.

   In this article, we leverage the notion of r-index proposed for the BWT to present a memory-efficient method for constructing and storing the run-length encoded PBWT, and computing set maximal matches (SMEMs) queries in haplotype sequences. We implement our method, which we refer to as μ-PBWT, and evaluate it on datasets of 1000 Genome Project and UK Biobank data. Our experiments demonstrate that the μ-PBWT reduces the memory usage up to a factor of 20% compared to the best current PBWT-based indexing. In particular, μ-PBWT produces an index that stores high-coverage whole genome sequencing data of chromosome 20 in about a third of the space of its BCF file. μ-PBWT is an adaptation of techniques for the run-length compressed BWT for the PBWT (RLPBWT) and it is based on keeping in memory only a succinct representation of the RLPBWT that still allows the efficient computation of set maximal matches (SMEMs) over the original panel.

   Our implementation is open source and available at https://github.com/dlcgold/muPBWT. The binary is available at https://bioconda.github.io/recipes/mupbwt/README.html.

    

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3. Fully Functional Parameterized Suffix Trees in Compact Space

   Ganguly, Arnab ; Shah, Rahul ; Thankachan, Sharma V. ( June 2022 , 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022))

   Mikolaj Bojanczyk ; Emanuela Merelli ; David P. Woodruff (Ed.)

   Two equal length strings are a parameterized match (p-match) iff there exists a one-to-one function that renames the symbols in one string to those in the other. The Parameterized Suffix Tree (PST) [Baker, STOC' 93] is a fundamental data structure that handles various string matching problems under this setting. The PST of a text T[1,n] over an alphabet Σ of size σ takes O(nlog n) bits of space. It can report any entry in (parameterized) (i) suffix array, (ii) inverse suffix array, and (iii) longest common prefix (LCP) array in O(1) time. Given any pattern P as a query, a position i in T is an occurrence iff T[i,i+|P|-1] and P are a p-match. The PST can count the number of occurrences of P in T in time O(|P|log σ) and then report each occurrence in time proportional to that of accessing a suffix array entry. An important question is, can we obtain a compressed version of PST that takes space close to the text’s size of nlogσ bits and still support all three functionalities mentioned earlier? In SODA' 17, Ganguly et al. answered this question partially by presenting an O(nlogσ) bit index that can support (parameterized) suffix array and inverse suffix array operations in O(log n) time. However, the compression of the (parameterized) LCP array and the possibility of faster suffix array and inverse suffix array queries in compact space were left open. In this work, we obtain a compact representation of the (parameterized) LCP array. With this result, in conjunction with three new (parameterized) suffix array representations, we obtain the first set of PST representations in o(nlog n) bits (when logσ = o(log n)) as follows. Here ε > 0 is an arbitrarily small constant. - Space O(n logσ) bits and query time O(log_σ^ε n); - Space O(n logσlog log_σ n) bits and query time O(log log_σ n); and - Space O(n logσ log^ε_σ n) bits and query time O(1). The first trade-off is an improvement over Ganguly et al.’s result, whereas our third trade-off matches the optimal time performance of Baker’s PST while squeezing the space by a factor roughly log_σ n. We highlight that our trade-offs match the space-and-time bounds of the best-known compressed text indexes for exact pattern matching and further improvement is highly unlikely. 

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4. Computing the original eBWT faster, simpler, and with less memory

   Christina Boucher, Davide Cenzato ( January 2021 , 28th International Symposium on String Processing and Information Retrieval (SPIRE))

   null (Ed.)

   Given an input string, the Burrows-Wheeler Transform (BWT) can be seen as a reversible permutation of it that allows efficient compression and fast substring queries. Due to these properties, it has been widely applied in the analysis of genomic sequence data, enabling important tasks such as read alignment. Mantaci et al. [TCS2007] extended the notion of the BWT to a collection of strings by defining the extended Burrows-Wheeler Transform (eBWT). This definition requires no modification of the input collection, and has the property that the output is independent of the order of the strings in the collection. However, over the years, the term eBWT has been used more generally to describe any BWT of a collection of strings. The fundamental property of the original definition (i.e., the independence from the input order) is frequently disregarded. In this paper, we propose a simple linear-time algorithm for the construction of the original eBWT, which does not require the preprocessing of Bannai et al. [CPM 2021]. As a byproduct, we obtain the first linear-time algorithm for computing the BWT of a single string that uses neither an end-of-string symbol nor Lyndon rotations. We also combine our new eBWT construction with a variation of prefix-free parsing (PFP) [WABI 2019] to allow for construction of the eBWT on large collections of genomic sequences. We implement this combined algorithm (pfpebwt) and evaluate it on a collection of human chromosomes 19 from the 1,000 Genomes Project, on a collection of Salmonella genomes from GenomeTrakr, and on a collection of SARS-CoV2 genomes from EBI’s COVID-19 data portal. We demonstrate that pfpebwt is the fastest method for all collections, with a maximum speedup of 7.6x on the second best method. The peak memory is at most 2x larger than the second best method. Comparing with methods that are also, as our algorithm, able to report suffix array samples, we obtain a 57.1x improvement in peak. 

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5. Efficient Parallel Output-Sensitive Edit Distance

   https://doi.org/10.4230/LIPIcs.ESA.2023.40  

   Ding, Xiangyun ; Dong, Xiaojun ; Gu, Yan ; Liu, Youzhe ; Sun, Yihan ( September 2023 , 31st Annual European Symposium on Algorithms (ESA 2023))

   Gørtz, Inge Li ; Farach-Colton, Martin ; Puglisi, Simon J. ; Herman, Grzegorz (Ed.)

   In this paper, we study efficient parallel edit distance algorithms, both in theory and in practice. Given two strings A[1..n] and B[1..m], and a set of operations allowed to edit the strings, the edit distance between A and B is the minimum number of operations required to transform A into B. In this paper, we use edit distance to refer to the Levenshtein distance, which allows for unit-cost single-character edits (insertions, deletions, substitutions). Sequentially, a standard Dynamic Programming (DP) algorithm solves edit distance with Θ(nm) cost. In many real-world applications, the strings to be compared are similar to each other and have small edit distances. To achieve highly practical implementations, we focus on output-sensitive parallel edit-distance algorithms, i.e., to achieve asymptotically better cost bounds than the standard Θ(nm) algorithm when the edit distance is small. We study four algorithms in the paper, including three algorithms based on Breadth-First Search (BFS), and one algorithm based on Divide-and-Conquer (DaC). Our BFS-based solution is based on the Landau-Vishkin algorithm. We implement three different data structures for the longest common prefix (LCP) queries needed in the algorithm: the classic solution using parallel suffix array, and two hash-based solutions proposed in this paper. Our DaC-based solution is inspired by the output-insensitive solution proposed by Apostolico et al., and we propose a non-trivial adaption to make it output-sensitive. All of the algorithms studied in this paper have good theoretical guarantees, and they achieve different tradeoffs between work (total number of operations), span (longest dependence chain in the computation), and space. We test and compare our algorithms on both synthetic data and real-world data, including DNA sequences, Wikipedia texts, GitHub repositories, etc. Our BFS-based algorithms outperform the existing parallel edit-distance implementation in ParlayLib in all test cases. On cases with fewer than 10⁵ edits, our algorithm can process input sequences of size 10⁹ in about ten seconds, while ParlayLib can only process sequences of sizes up to 10⁶ in the same amount of time. By comparing our algorithms, we also provide a better understanding of the choice of algorithms for different input patterns. We believe that our paper is the first systematic study in the theory and practice of parallel edit distance. 

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