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- ItemA framework for space-efficient read clustering in metagenomic samples(BioMed Central, 2017-03-14) Alanko, Jarno; Cunial, Fabio; Belazzougui, Djamal; Mäkinen, Veli
Show more Background: A metagenomic sample is a set of DNA fragments, randomly extracted from multiple cells in an environment, belonging to distinct, often unknown species. Unsupervised metagenomic clustering aims at partitioning a metagenomic sample into sets that approximate taxonomic units, without using reference genomes. Since samples are large and steadily growing, space-efficient clustering algorithms are strongly needed. Results: We design and implement a space-efficient algorithmic framework that solves a number of core primitives in unsupervised metagenomic clustering using just the bidirectional Burrows-Wheeler index and a union-find data structure on the set of reads. When run on a sample of total length n, with m reads of maximum length ℓ each, on an alphabet of total size σ, our algorithms take O(n(t+logσ)) time and just 2n+o(n)+O(max{ℓ σlogn,K logm}) bits of space in addition to the index and to the union-find data structure, where K is a measure of the redundancy of the sample and t is the query time of the union-find data structure. Conclusions: Our experimental results show that our algorithms are practical, they can exploit multiple cores by a parallel traversal of the suffix-link tree, and they are competitive both in space and in time with the state of the art.Show more - ItemA Framework for Space-Efficient String Kernels(Springer, 2017-02-17) Belazzougui, Djamal; Cunial, Fabio
Show more String kernels are typically used to compare genome-scale sequences whose length makes alignment impractical, yet their computation is based on data structures that are either space-inefficient, or incur large slowdowns. We show that a number of exact kernels on pairs of strings of total length n, like the k-mer kernel, the substrings kernels, a number of length-weighted kernels, the minimal absent words kernel, and kernels with Markovian corrections, can all be computed in O(nd) time and in o(n) bits of space in addition to the input, using just a rangeDistinct data structure on the Burrows–Wheeler transform of the input strings that takes O(d) time per element in its output. The same bounds hold for a number of measures of compositional complexity based on multiple values of k, like the k-mer profile and the k-th order empirical entropy, and for calibrating the value of k using the data. All such algorithms become O(n) using a suitable implementation of the rangeDistinct data structure, and by concatenating them to a suitable BWT construction algorithm, we can compute all the mentioned kernels and complexity measures, directly from the input strings, in O(n) time and in O(n log σ) bits of space in addition to the input, where σ is the size of the alphabet. Using similar data structures, we also show how to build a compact representation of the variable-length Markov chain of a string T of length n, that takes just 3n log σ+o(n log σ) bits of space, and that can be learnt in randomized O(n) time using O(n log σ) bits of space in addition to the input. Such model can then be used to assign a probability to a query string S of length m in O(m) time and in 2m+o(m) bits of additional space, thus providing an alternative, compositional measure of the similarity between S and T that does not require alignment.Show more - ItemFast Label Extraction in the CDAWG(Springer, 2017-09-06) Belazzougui, Djamal; Cunial, Fabio
Show more The compact directed acyclic word graph (CDAWG) of a string T of length n takes space proportional just to the number e of right extensions of the maximal repeats of T, and it is thus an appealing index for highly repetitive datasets, like collections of genomes from similar species, in which e grows significantly more slowly than n. We reduce from O(m log log n) to O(m) the time needed to count the number of occurrences of a pattern of length m, using an existing data structure that takes an amount of space proportional to the size of the CDAWG. This implies a reduction from O(m log log n+occ) to O(m+occ) in the time needed to locate all the occocc occurrences of the pattern. We also reduce from O(k log log n) to O(k) the time needed to read the k characters of the label of an edge of the suffix tree of T, and we reduce from O(m log log n) to O(m) the time needed to compute the matching statistics between a query of length m and T, using an existing representation of the suffix tree based on the CDAWG. All such improvements derive from extracting the label of a vertex or of an arc of the CDAWG using a straight-line program induced by the reversed CDAWG.Show more - ItemFast matching statistics in small space(Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2018-06-27) Belazzougui, Djamal; Cunial, Fabio; Denas, Olgert
Show more Computing the matching statistics of a string S with respect to a string T on an alphabet of size sigma is a fundamental primitive for a number of large-scale string analysis applications, including the comparison of entire genomes, for which space is a pressing issue. This paper takes from theory to practice an existing algorithm that uses just O(|T|log{sigma}) bits of space, and that computes a compact encoding of the matching statistics array in O(|S|log{sigma}) time. The techniques used to speed up the algorithm are of general interest, since they optimize queries on the existence of a Weiner link from a node of the suffix tree, and parent operations after unsuccessful Weiner links. Thus, they can be applied to other matching statistics algorithms, as well as to any suffix tree traversal that relies on such calls. Some of our optimizations yield a matching statistics implementation that is up to three times faster than a plain version of the algorithm, depending on the similarity between S and T. In genomic datasets of practical significance we achieve speedups of up to 1.8, but our fastest implementations take on average twice the time of an existing code based on the LCP array. The key advantage is that our implementations need between one half and one fifth of the competitor's memory, and they approach comparable running times when S and T are very similar.Show more - ItemFlexible Indexing of Repetitive Collections(Springer, 2017-06-07) Belazzougui, Djamal; Cunial, Fabio; Gagie, Travis; Prezza, Nicola; Raffinot, Mathieu
Show more Highly repetitive strings are increasingly being amassed by genome sequencing experiments, and by versioned archives of source code and webpages. We describe practical data structures that support count- ing and locating all the exact occurrences of a pattern in a repetitive text, by combining the run-length encoded Burrows-Wheeler transform (RLBWT) with the boundaries of Lempel-Ziv 77 factors. One such vari- ant uses an amount of space comparable to LZ77 indexes, but it answers count queries between two and four orders of magnitude faster than all LZ77 and hybrid index implementations, at the cost of slower lo- cate queries. Combining the RLBWT with the compact directed acyclic word graph answers locate queries for short patterns between four and ten times faster than a version of the run-length compressed suffix ar- ray (RLCSA) that uses comparable memory, and with very short pat- terns our index achieves speedups even greater than ten with respect to RLCSA.Show more - ItemA framework for space-efficient variable-order Markov models(Oxford University Press, 2019-11-15) Cunial, Fabio; Alanko, Jarno; Belazzougui, Djamal
Show more Motivation: Markov models with contexts of variable length are widely used in bioinformatics for representing sets of sequences with similar biological properties. When models contain many long contexts, existing implementations are either unable to handle genome-scale training datasets within typical memory budgets, or they are optimized for specific model variants and are thus inflexible. Results: We provide practical, versatile representations of variable-order Markov models and of interpolated Markov models, that support a large number of context-selection criteria, scoring functions, probability smoothing methods, and interpolations, and that take up to four times less space than previous implementations based on the suffix array, regardless of the number and length of contexts, and up to ten times less space than previous trie-based representations, or more, while matching the size of related, state-of-the-art data structures from Natural Language Processing. We describe how to further compress our indexes to a quantity related to the redundancy of the training data, saving up to 90% of their space on very repetitive datasets, and making them become up to sixty times smaller than previous implementations based on the suffix array. Finally, we show how to exploit constraints on the length and frequency of contexts to further shrink our compressed indexes to half of their size or more, achieving data structures that are a hundred times smaller than previous implementations based on the suffix array, or more. This allows variable-order Markov models to be used with bigger datasets and with longer contexts on the same hardware, thus possibly enabling new applications. Availability and implementation: https://github.com/jnalanko/VOMMShow more - ItemFully-Functional Bidirectional Burrows-Wheeler Indexes and Infinite-Order De Bruijn Graphs(Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2019-06-18) Belazzougui, Djamal; Cunial, Fabio
Show more Given a string T on an alphabet of size σ, we describe a bidirectional Burrows-Wheeler index that takes O(|T| log σ) bits of space, and that supports the addition and removal of one character, on the left or right side of any substring of T, in constant time. Previously known data structures that used the same space allowed constant-time addition to any substring of T, but they could support removal only from specific substrings of T. We also describe an index that supports bidirectional addition and removal in O(log log |T|) time, and that takes a number of words proportional to the number of left and right extensions of the maximal repeats of T. We use such fully-functional indexes to implement bidirectional, frequency-aware, variable-order de Bruijn graphs with no upper bound on their order, and supporting natural criteria for increasing and decreasing the order during traversal.Show more - ItemLinear-time string indexing and analysis in small space(Association for Computing Machinery, 2020-03-09) Belazzougui, Djamal; Cunial, Fabio; Kärkkäinen, Juha; Mäkinen, Veli
Show more The field of succinct data structures has flourished over the last 16 years. Starting from the compressed suffix array by Grossi and Vitter (STOC 2000) and the FM-index by Ferragina and Manzini (FOCS 2000), a number of generalizations and applications of string indexes based on the Burrows-Wheeler transform (BWT) have been developed, all taking an amount of space that is close to the input size in bits. In many large-scale applications, the construction of the index and its usage need to be considered as one unit of computation. For example, one can compare two genomes by building a common index for their concatenation, and by detecting common substructures by querying the index. Efficient string indexing and analysis in small space lies also at the core of a number of primitives in the data-intensive field of high-throughput DNA sequencing. We report the following advances in string indexing and analysis. We show that the BWT of a string T ∈ {1, . . . ,σ }^n can be built in deterministic O (n) time using just O (n log σ ) bits of space, where σ ≤ n. Deterministic linear time is achieved by exploiting a new partial rank data structure that supports queries in constant time, and that might have independent interest. Within the same time and space budget, we can build an index based on the BWT that allows one to enumerate all the internal nodes of the suffix tree of T . Many fundamental string analysis problems, such as maximal repeats, maximal unique matches, and string kernels, can be mapped to such enumeration, and can thus be solved in deterministic O (n) time and in O (n log σ ) bits of space from the input string, by tailoring the enumeration algorithm to some problem-specific computations. We also show how to build many of the existing indexes based on the BWT, such as the compressed suffix array, the compressed suffix tree, and the bidirectional BWT index, in randomized O (n) time and in O (n log σ ) bits of space. The previously fastest construction algorithms for BWT, compressed suffix array and compressed suffix tree, which used O (n log σ ) bits of space, took O (n log log σ ) time for the first two structures, and O (n log^ϵ n) time for the third, where ϵ is any positive constant smaller than one. Alternatively, the BWT could be previously built in linear time if one was willing to spend O (n log σ log log_σ n) bits of space. Contrary to the state of the art, our bidirectional BWT index supports every operation in constant time per element in its output.Show more - ItemRepresenting the Suffix Tree with the CDAWG(Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2017-07-04) Belazzougui, Djamal; Cunial, Fabio
Show more Given a string T, it is known that its suffix tree can be represented using the compact directed acyclic word graph (CDAWG) with e_T arcs, taking overall O(e_T+e_REV(T)) words of space, where REV(T) is the reverse of T, and supporting some key operations in time between O(1) and O(log(log(n))) in the worst case. This representation is especially appealing for highly repetitive strings, like collections of similar genomes or of version-controlled documents, in which e_T grows sublinearly in the length of T in practice. In this paper we augment such representation, supporting a number of additional queries in worst-case time between O(1) and O(log(n)) in the RAM model, without increasing space complexity asymptotically. Our technique, based on a heavy path decomposition of the suffix tree, enables also a representation of the suffix array, of the inverse suffix array, and of T itself, that takes O(e_T) words of space, and that supports random access in O(log(n)) time. Furthermore, we establish a connection between the reversed CDAWG of T and a context-free grammar that produces T and only T, which might have independent interest.Show more - ItemSmaller Fully-Functional Bidirectional BWT Indexes(Springer Nature, 2020-09-17) Belazzougui,, Djamal; Cunial, Fabio
Show more Burrows-Wheeler indexes that support both extending and contracting any substring of the text $T$ of length $n$ on which they are built, in any direction, provide substantial flexibility in traversing the text and can be used to implement several algorithms. The practical appeal of such indexes is contingent on them being compact, and current designs that are sensitive to the compressibility of the input take either $O(e+\REV{e})$ words of space, where $e$ and $\REV{e}$ are the number of right and left extensions of the maximal repeats of $T$, or $O(r\log(n/r)+\REV{r}\log(n/\REV{r}))$ words, where $r$ and $\REV{r}$ are the number of runs in the Burrows-Wheeler transform of $T$ and of its reverse. In this paper we describe a fully-functional bidirectional index that takes $O(m+r+\REV{r})$ words, where $m$ is the number of maximal repeats of $T$, as well as a variant that takes $O(r+\REV{r})$ words.Show more