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Item Range Majorities and Minorities in Arrays(Springer Nature, 2021-03-19) Belazzougui, Djamal; Gagie, Travis; Munro, J. Ian; Navarro, Gonzalo; Nekrich, YakovThe problem of parameterized range majority asks us to preprocess a string of length n such that, given the endpoints of a range, one can quickly find all the distinct elements whose relative frequencies in that range are more than a threshold τ. This is a more tractable version of the classical problem of finding the range mode, which is unlikely to be solvable in polylogarithmic time and linear space. In this paper we give the first linear-space solution with optimal 𝓞(1/τ ) query time, even when τ can be specified with the query. We then consider data structures whose space is bounded by the entropy of the distribution of the symbols in the sequence. For the case when the alphabet size is polynomial on the computer word size, we retain the optimal time within optimally compressed space (i.e., with sublinear redundancy). Otherwise, either the compressed space is increased by an arbitrarily small constant factor or the time rises to any function in (1/τ ) · ω(1). We obtain the same results on the complementary problem of parameterized range minority.Item Weighted Ancestors in Suffix Trees Revisited(Schloss Dagstuhl - Leibniz-Zentrum für Informatik 2021, 2021-06-30) Belazzougui, Djamal; Kosolobov, Dmitry; Puglisi, Simon J.; Raman, RajeevThe weighted ancestor problem is a well-known generalization of the predecessor problem to trees. It is known to require Ω(log log n) time for queries provided 𝒪(n polylog n) space is available and weights are from [0..n], where n is the number of tree nodes. However, when applied to suffix trees, the problem, surprisingly, admits an 𝒪(n)-space solution with constant query time, as was shown by Gawrychowski, Lewenstein, and Nicholson (Proc. ESA 2014). This variant of the problem can be reformulated as follows: given the suffix tree of a string s, we need a data structure that can locate in the tree any substring s[p..q] of s in 𝒪(1) time (as if one descended from the root reading s[p..q] along the way). Unfortunately, the data structure of Gawrychowski et al. has no efficient construction algorithm, limiting its wider usage as an algorithmic tool. In this paper we resolve this issue, describing a data structure for weighted ancestors in suffix trees with constant query time and a linear construction algorithm. Our solution is based on a novel approach using so-called irreducible LCP values.Item Block Trees(ELSEVIER, 2021-05) Belazzougui, Djamal; Cáceres, Manuel; Gagie, Travis; Gawrychowski, Paweł; Kärkkäinen, Juha; Navarro, Gonzalo; Ordóñez, Alberto; Puglisi, Simon J.; Tabei, YasuoLet string S[1..n] be parsed into z phrases by the Lempel-Ziv algorithm. The corresponding compression algorithm encodes S in 𝒪(z) space, but it does not support random access to S. We introduce a data structure, the block tree, that represents S in 𝒪(z log(n/z)) space and extracts any symbol of T in time 𝒪(log(n/z)), among other space-time tradeoffs. By multiplying the space by the alphabet size, we also support rank and select queries, which are useful for building compressed data structures on top of S. Further, block trees can be built in a scalable manner. Our experiments show that block trees offer relevant space-time tradeoffs compared to other compressed string representations for highly repetitive strings.Item DIAG a diagnostic web application based on lung CT Scan images and deep learning(IOS Press Ebooks, 2021-05-29) Hadj Bouzid, Amel Imene; Yahiaoui, Saïd; Lounis, Anis; Berrani, Sid-Ahmed; Belbachir, Hacène; Naili, Qaid; Abdi, Mohamed El Hafedh; Bensalah, Kawthar; Belazzougui, DjamalCoronavirus disease is a pandemic that has infected millions of people around the world. Lung CT-scans are effective diagnostic tools, but radiologists can quickly become overwhelmed by the flow of infected patients. Therefore, automated image interpretation needs to be achieved. Deep learning (DL) can support critical medical tasks including diagnostics, and DL algorithms have successfully been applied to the classification and detection of many diseases. This work aims to use deep learning methods that can classify patients between Covid-19 positive and healthy patient. We collected 4 available datasets, and tested our convolutional neural networks (CNNs) on different distributions to investigate the generalizability of our models. In order to clearly explain the predictions, Grad-CAM and Fast-CAM visualization methods were used. Our approach reaches more than 92% accuracy on 2 different distributions. In addition, we propose a computer aided diagnosis web application for Covid-19 diagnosis. The results suggest that our proposed deep learning tool can be integrated to the Covid-19 detection process and be useful for a rapid patient management.Item Space-Efficient Representation of Genomic k-Mer Count Tables(Schloss Dagstuhl -- Leibniz-Zentrum für Informatik, 2021-07-22) Shibuya, Yoshihiro; Belazzougui, Djamal; Kucherov, Gregoryk-mer counting is a common task in bioinformatic pipelines, with many dedicated tools available. Output formats could rely on quotienting to reduce the space of k-mers in hash tables, however counts are not usually stored in space-efficient formats. Overall, k-mer count tables for genomic data take a considerable space, easily reaching tens of GB. Furthermore, such tables do not support efficient random-access queries in general. In this work, we design an efficient representation of k-mer count tables supporting fast random-access queries. We propose to apply Compressed Static Functions (CSFs), with space proportional to the empirical zero-order entropy of the counts. For very skewed distributions, like those of k-mer counts in whole genomes, the only currently available implementation of CSFs does not provide a compact enough representation. By adding a Bloom Filter to a CSF we obtain a Bloom-enhanced CSF (BCSF) effectively overcoming this limitation. Furthermore, by combining BCSFs with minimizer-based bucketing of k-mers, we build even smaller representations breaking the empirical entropy lower bound, for large enough k. We also extend these representations to the approximate case, gaining additional space. We experimentally validate these techniques on k-mer count tables of whole genomes (E.Coli and C.Elegans) as well as on k-mer document frequency tables for 29 E.Coli genomes. In the case of exact counts, our representation takes about a half of the space of the empirical entropy, for large enough k’s.Item Efficient tree-structured categorical retrieval(Leibniz International Proceedings in Informatics (LIPIcs), 2020-06-09) Belazzougui, Djamal; Kucherov, GregoryWe study a document retrieval problem in the new framework where D text documents are organized in a category tree with a predefined number h of categories. This situation occurs e.g. with taxomonic trees in biology or subject classification systems for scientific literature. Given a string pattern p and a category (level in the category tree), we wish to efficiently retrieve the t categorical units containing this pattern and belonging to the category. We propose several efficient solutions for this problem. One of them uses n(log σ(1+o(1))+log D + O(h)) + O(∆) bits of space and O(|p| + t) query time, where n is the total length of the documents, σ the size of the alphabet used in the documents and ∆ is the total number of nodes in the category tree. Another solution uses n(log σ(1+o(1))+O(log D))+O(∆)+O(D log n) bits of space and O(|p| + t log D) query time. We finally propose other solutions which are more space-efficient at the expense of a slight increase in query time.Item Linear-time string indexing and analysis in small space(Association for Computing Machinery, 2020-03-09) Belazzougui, Djamal; Cunial, Fabio; Kärkkäinen, Juha; Mäkinen, VeliThe 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.Item A framework for space-efficient variable-order Markov models(Oxford University Press, 2019-11-15) Cunial, Fabio; Alanko, Jarno; Belazzougui, DjamalMotivation: 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/VOMMItem Fully-Functional Bidirectional Burrows-Wheeler Indexes and Infinite-Order De Bruijn Graphs(Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2019-06-18) Belazzougui, Djamal; Cunial, FabioGiven 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.Item Computing the Antiperiod(s) of a String(Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik, 2019-06-18) Alamro, Hayam; Badkobeh, Golnaz; Belazzougui, Djamal; Iliopoulos, Costas S.; Puglisi, Simon J.A string S[1, n] is a power (or repetition or tandem repeat) of order k and period n/k, if it can be decomposed into k consecutive identical blocks of length n/k. Powers and periods are fundamental structures in the study of strings and algorithms to compute them efficiently have been widely studied. Recently, Fici et al. (Proc. ICALP 2016) introduced an antipower of order k to be a string composed of k distinct blocks of the same length, n/k, called the antiperiod. An arbitrary string will have antiperiod t if it is prefix of an antipower with antiperiod t. In this paper, we describe efficient algorithm for computing the smallest antiperiod of a string S of length n in O(n) time. We also describe an algorithm to compute all the antiperiods of S that runs in O(n log n) time.
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