We introduce the concept of cyclic covers, which generalizes the classical notion of covers in strings. Given any string X , a factor W of X is called a cyclic cover if each position of X belongs to an occurrence of a cyclic shift of W in X . Two cyclic covers are distinct if one is not a cyclic shift of the other. The cyclic covers problem asks for all distinct cyclic covers of an input string X . We present an algorithm that solves the cyclic covers problem in O ( n log ⁡ n ) time, where n is the length of X . It is based on finding a well-structured set of standard occurrences of a constant number of factors of a cyclic cover candidate W , computing the regions of X covered by cyclic shifts of W , extending those factors, and taking the union of the results. • We introduce the cyclic cover problem. • Two cyclic covers are distinct if one is not a cyclic shift of the other. • The cyclic cover problem requires finding all distinct cyclic covers of X . • We show that for a string of length n, the cyclic cover problem can be solved in O ( n log ⁡ n ) time.

A degenerate symbol over an alphabet Σ is a non-empty subset of Σ, and a sequence of such symbols is a degenerate string. We investigate the exact computation of maximal degenerate palindromes with gaps and mismatches. We present an algorithm which, given a degenerate string of length n and natural number parameters g and m, efficiently detects exact maximal palindromes with a gap size ≤g, and ≤m permitted mismatches. We show that it can be done in O(k|Σ|(k+log⁡|Σ|)+(k+g+m)n) time and O((g+m)n) space, where k represents an upper bound on the number of degenerate symbols contained in the string. Furthermore, we also show that the problem of factorisation a string into maximal degenerate palindromes with gaps and mismatches can also be done in O(k|Σ|(k+log⁡|Σ|)+(k+g+m)n) time and O((g+m)n) space. An inverted repeat is a specific type of palindrome which refers to a nucleotide sequence followed by its reverse complement. Our results can also be used to find maximal inverted repeated sequences with gaps and mismatches, where changing the structure of palindromes to inverted repeats does not affect the overall running time. Finally we demonstrate our algorithm on several strains of SARS-CoV-2, and quantify the number of inverted repeats found with ≤0,1,2 mismatches and ≤0,10,100 gap size.

Abstract Sequence mappability is an important task in genome resequencing. In the (k, m)-mappability problem, for a given sequence T of length n, the goal is to compute a table whose ith entry is the number of indices $$j \ne i$$ j ≠ i such that the length-m substrings of T starting at positions i and j have at most k mismatches. Previous works on this problem focused on heuristics computing a rough approximation of the result or on the case of $$k=1$$ k = 1 . We present several efficient algorithms for the general case of the problem. Our main result is an algorithm that, for $$k=O(1)$$ k = O ( 1 ) , works in $$O(n)$$ O ( n ) space and, with high probability, in $$O(n \cdot \min \{m^k,\log ^k n\})$$ O ( n · min { m k , log k n } ) time. Our algorithm requires a careful adaptation of the k-errata trees of Cole et al. [STOC 2004] to avoid multiple counting of pairs of substrings. Our technique can also be applied to solve the all-pairs Hamming distance problem introduced by Crochemore et al. [WABI 2017]. We further develop $$O(n^2)$$ O ( n 2 ) -time algorithms to compute all (k, m)-mappability tables for a fixed m and all $$k\in \{0,\ldots ,m\}$$ k ∈ { 0 , … , m } or a fixed k and all $$m\in \{k,\ldots ,n\}$$ m ∈ { k , … , n } . Finally, we show that, for $$k,m = \Theta (\log n)$$ k , m = Θ ( log n ) , the (k, m)-mappability problem cannot be solved in strongly subquadratic time unless the Strong Exponential Time Hypothesis fails. This is an improved and extended version of a paper presented at SPIRE 2018.

BACKGROUND: An inverted repeat is a DNA sequence followed downstream by its reverse complement, potentially with a gap in the centre. Inverted repeats are found in both prokaryotic and eukaryotic genomes and they have been linked with countless possible functions. Many international consortia provide a comprehensive description of common genetic variation making alternative sequence representations, such as IUPAC encoding, necessary for leveraging the full potential of such broad variation datasets. RESULTS: We present IUPACPAL, an exact tool for efficient identification of inverted repeats in IUPAC-encoded DNA sequences allowing also for potential mismatches and gaps in the inverted repeats. CONCLUSION: Within the parameters that were tested, our experimental results show that IUPACPAL compares favourably to a similar application packaged with EMBOSS. We show that IUPACPAL identifies many previously unidentified inverted repeats when compared with EMBOSS, and that this is also performed with orders of magnitude improved speed.

A k-anticover of a string x is a set of pairwise distinct factors of x of equal length k, such that every symbol of x is contained into an occurrence of at least one of those factors. The existence of a k-anticover can be seen as a notion of non-redundancy, which has application in computational biology, where they are associated with various non-regulatory mechanisms. In this paper we address the complexity of the problem of finding a k-anticover of a string x if it exists, showing that the decision problem is NP-complete on general strings for k ≥ 3. We also show that the problem admits a polynomial-time solution for k=2. For unbounded k, we provide an exact exponential algorithm to find a k-anticover of a string of length n (or determine that none exists), which runs in O*(min {3^{(n-k)/3)}, ((k(k+1))/2)^{n/(k+1)) time using polynomial space.

MOTIVATION: Computing the uniqueness of k-mers for each position of a genome while allowing for up to e mismatches is computationally challenging. However, it is crucial for many biological applications such as the design of guide RNA for CRISPR experiments. More formally, the uniqueness or (k, e)-mappability can be described for every position as the reciprocal value of how often this k-mer occurs approximately in the genome, i.e. with up to e mismatches. RESULTS: We present a fast method GenMap to compute the (k, e)-mappability. We extend the mappability algorithm, such that it can also be computed across multiple genomes where a k-mer occurrence is only counted once per genome. This allows for the computation of marker sequences or finding candidates for probe design by identifying approximate k-mers that are unique to a genome or that are present in all genomes. GenMap supports different formats such as binary output, wig and bed files as well as csv files to export the location of all approximate k-mers for each genomic position. AVAILABILITY AND IMPLEMENTATION: GenMap can be installed via bioconda. Binaries and C++ source code are available on https://github.com/cpockrandt/genmap.

Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

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.

We introduce the Longest Common Circular Factor (LCCF) problem in which, given strings $S$ and $T$ of length $n$, we are to compute the longest factor of $S$ whose cyclic shift occurs as a factor of $T$. It is a new similarity measure, an extension of the classic Longest Common Factor. We show how to solve the LCCF problem in $O(n \log^5 n)$ time.

In the Longest Common Factor with k Mismatches (LCF_k) problem, we are given two strings X and Y of total length n, and we are asked to find a pair of maximal-length factors, one of X and the other of Y, such that their Hamming distance is at most k. Thankachan et al. [Thankachan et al. 2016] show that this problem can be solved in O(n log^k n) time and O(n) space for constant k. We consider the LCF_k(l) problem in which we assume that the sought factors have length at least l. We use difference covers to reduce the LCF_k(l) problem with l=Omega(log^{2k+2}n) to a task involving m=O(n/log^{k+1}n) synchronized factors. The latter can be solved in O(m log^{k+1}m) time, which results in a linear-time algorithm for LCF_k(l) with l=Omega(log^{2k+2}n). In general, our solution to the LCF_k(l) problem for arbitrary l takes O(n + n log^{k+1} n/sqrt{l}) time.

A generalised degenerate string (GD string) S^ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length k_i but this length can vary between different sets. We denote the sum of these lengths k_0, k_1,...,k_{n-1} by W. This type of uncertain sequence can represent, for example, a gapless multiple sequence alignment of width W in a compact form. Our first result in this paper is an O(N+M)-time algorithm for deciding whether the intersection of two GD strings of total sizes N and M, respectively, over an integer alphabet, is non-empty. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in only linear space. A similar result can be obtained by employing an automata-based approach but its cost is alphabet-dependent. We then apply our string comparison algorithm to compute palindromes in GD strings. We present an O(min{W,n^2}N)-time algorithm for computing all palindromes in S^. Furthermore, we show a similar conditional lower bound for computing maximal palindromes in S^. Finally, proof-of-concept experimental results are presented using real protein datasets.

In this work, we consider a special type of uncertain sequence called weighted string. In a weighted string every position contains a subset of the alphabet and every letter of the alphabet is associated with a probability of occurrence such that the sum of probabilities at each position equals 1. Usually a cumulative weight threshold 1/z is specified, and one considers only strings that match the weighted string with probability at least 1/z. We provide an 𝒪(nz)-time and 𝒪(nz)-space off-line algorithm, where n is the length of the weighted string and 1/z is the given threshold, to compute a smallest maximal palindromic factorization of a weighted string. This factorization has applications in hairpin structure prediction in a set of closely-related DNA or RNA sequences. Along the way, we provide an 𝒪(nz)-time and 𝒪(nz)-space off-line algorithm to compute maximal palindromes in weighted strings. Finally, we provide an experiment of our proposed algorithm.

Pattern matching on a set of similar texts has received much attention, especially recently, mainly due to its application in cataloguing human genetic variation. In particular, many different algorithms have been proposed for the off-line version of this problem; that is, constructing a compressed index for a set of similar texts in order to answer pattern matching queries efficiently. However, the on-line, more fundamental, version of this problem is a rather undeveloped topic. Solutions to the on-line version can be beneficial for a number of reasons; for instance, efficient on-line solutions can be used in combination with partial indexes as practical trade-offs. We make here an attempt to close this gap via proposing two efficient algorithms for this problem. Notably, one of the algorithms requires time linear in the size of the texts' representation, for short patterns. Furthermore, experimental results confirm our theoretical findings in practical terms.

The observed frequency of the longest proper prefix, the longest proper suffix, and the longest infix of a word w in a given sequence x can be used for classifying w as avoided or overabundant. The definitions used for the expectation and deviation of w in this statistical model were described and biologically justified by Brendel et al. (J Biomol Struct Dyn 1986). We have very recently introduced a time-optimal algorithm for computing all avoided words of a given sequence over an integer alphabet (Algorithms Mol Biol 2017). In this article, we extend this study by presenting an O(n)-time and O(n)-space algorithm for computing all overabundant words in a sequence x of length n over an integer alphabet. Our main result is based on a new non-trivial combinatorial property of the suffix tree T of x: the number of distinct factors of x whose longest infix is the label of an explicit node of T is no more than 3n-4. We further show that the presented algorithm is time-optimal by proving that O(n) is a tight upper bound for the number of overabundant words. Finally, we present experimental results, using both synthetic and real data, which justify the effectiveness and efficiency of our approach in practical terms.

Circular string matching is a problem which naturally arises in many contexts. It consists in finding all occurrences of the rotations of a pattern of lengthmin a text of lengthn. There exist optimal worst- and average-case algorithms for circular string matching. Here, we present a suboptimal average-case algorithm for circular string matching requiring time $\mathcal{O}$ (n) and space $\mathcal{O}$ (m). The importance of our contribution is underlined by the fact that the proposed algorithm can be easily adapted to deal with circular dictionary matching. In particular, we show how the circular dictionary-matching problem can be solved in average-case time $\mathcal{O}$ (n+M) and space $\mathcal{O}$ (M), whereMis the total length of the dictionary patterns, assuming that the shortest pattern is sufficiently long. Moreover, the presented average-case algorithms and other worst-case approaches were also implemented. Experimental results, using real and synthetic data, demonstrate that the implementation of the presented algorithms can accelerate the computations by more than a factor of two compared to the corresponding implementation of other approaches.

Given a set $\mathcal{D}$ ofqdocuments, the Longest Common Substring (LCS) problem asks, for any integer 2 ⩽k⩽q, the longest substring that appears inkdocuments. LCS is a well-studied problem having a wide range of applications in Bioinformatics: from microarrays to DNA sequences alignments and analysis. This problem has been solved by Hui (2000International Journal of Computer Science and Engineering1573–76) by using a famous constant-time solution to the Lowest Common Ancestor (LCA) problem in trees coupled with the use of suffix trees. In this article, we present a simple method for solving the LCS problem by using suffix trees (STs) and classical union-find data structures. In turn, we show how this simple algorithm can be adapted in order to work with other space efficient data structures such as the enhanced suffix arrays (ESA) and the compressed suffix tree.

Strings with don't care symbols, also called partial words, and more general indeterminate strings are a natural representation of strings containing uncertain symbols. A considerable effort has been made to obtain efficient algorithms for pattern matching and periodicity detection in such strings. Among those, a number of algorithms have been proposed that behave well on random data, but still their worst-case running time is Theta(n^2). We present the first truly subquadratic-time solutions for a number of such problems on partial words that can also be adapted to indeterminate strings over a constant-sized alphabet. We show that $n$ longest common compatible prefix queries (which correspond to longest common extension queries in regular strings) can be answered on-line in O(n * sqrt(n * log(n)) time after O(n * sqrt(n * log(n))-time preprocessing. We also present O(n * sqrt(n * log(n))-time algorithms for computing the prefix array and two types of border array of a partial word.

DNA sequencing is the process of determining the exact order of the nucleotide bases of an individual's genome in order to catalogue sequence variation and understand its biological implications. Whole-genome sequencing techniques produce masses of data in the form of short sequences known as reads. Assembling these reads into a whole genome constitutes a major algorithmic challenge. Most assembly algorithms utilise de Bruijn graphs constructed from reads for this purpose. A critical step of these algorithms is to detect typical motif structures in the graph caused by sequencing errors and genome repeats, and filter them out; one such complex subgraph class is a so-called superbubble. In this paper, we propose an O(n+m)-time algorithm to detect all superbubbles in a directed acyclic graph with n vertices and m (directed) edges, improving the best-known O(mlog⁡m)-time algorithm by Sung et al.

The increased usage of Twitter as a medium for reporting news and sharing information between people has caught the attention of researchers from different disciplines. One of the research directions is the analysis of online information from the perspective of its credibility. This paper aims to assess and analyze the credibility of tweets in Arabic language. In order to achieve the stated goal, first we employ the idea of crowdsourcing where users can explicitly express their opinions about credibility of a set of tweets. This information coupled with the data about tweets' features enable us to investigate which features may indicate the credibility level of a tweet, e.g. tweet with attached image and was authored by a person who posts a lot of tweets will be, with high probability, a credible tweet. We distinguish three main groups of features: authority and topical expertise (of the source), data quality (of the content), and popularity (of the content and the source). We argue that content data quality factor based on content linguistic features in addition to source authority is more important than content popularity in identifying credible messages. In addition to this, we identified three experts who also rated the credibility of tweets and based on that we investigate the level of agreement between experts and the crowd, and we identify which expert represents the crowd in the best way. This can allow us to select the most representative expert when it is needed. This study is a pilot of a large study that aims at predicting credibility of Arabic Twitter messages using machine learning approaches.

A weighted biological sequence is a string in which a set of characters may appear at each position with respective probabilities of occurrence. We attempt to locate all the tandem repeats in a weighted sequence. A repeated substring is called a tandem repeat if each occurrence of the substring is directly adjacent to each other. By introducing the idea of equivalence classes in weighted sequences, we identify the tandem repeats of every possible length using an iterative partitioning technique. We also present the algorithm for recording the tandem repeats, and prove that the problem can be solved in O(n²) time.

The constant advances in sequencing technology have redefined the way genome sequencing is performed. They are able to produce tens of millions of short sequences (reads), during a single experiment, and with a much lower cost than previously possible. Due to this massive amount of data, efficient algorithms for mapping these reads to reference sequences are in great demand, and recently, there has been ample work for publishing such algorithms. In this paper, we study a different version of this problem: mapping these reads to a dynamically changing genomic sequence. We propose a new practical algorithm, which employs a suitable data structure that takes into account potential dynamic effects (replacements, insertions, deletions) on the genomic sequence. The presented experimental results demonstrate that the proposed approach can be applied to address the problem of mapping millions of reads to multiple genomic sequences.

The paper aims at tight upper bounds on the size of pattern classification circuits that can be used for a priori parameter settings in a machine learning context. The upper bounds relate the circuit size S(C) to n L := [log 2 m L ], where m L is the number of training samples. In particular, we show that there exist unbounded fan-in threshold circuits with less than (a) S R cc := 2·√2 n L + 3 gates for unbounded depth, (b) S L cc := 34.8 · √2 n L + 14 · n L − 11 · log 2 n L + 2 gates for small bounded depth, where in both cases all m L samples are classified correctly. We note that the upper bounds do not depend on the length n of input (sample) vectors. Since n L << n in real-world problem settings, the upper bounds return values that are suitable for practical applications. We provide experimental evidence that the circuit size estimations work well on a number of pattern classification tasks. As a result, we formulate the conjecture that [1.25 · S R cc or [0.07 · S L cc ] gates are sufficient to achieve a high generalization rate of bounded-depth classification circuits.

In this paper, we study the pattern matching problem in given intervals. Depending on whether the intervals are given a priori for pre-processing, or during the query along with the pattern or, even in both the cases, we develop efficient solutions for different variants of this problem. In particular, we present efficient indexing schemes for each of the above variants of the problem.

Existing symbolic music comparison systems generally consider monophonic music or monophonic reduction of polyphonic music. Adaptation of alignment algorithms to music leads to accurate systems, but their extensions to polyphonic music raise new problems. Indeed, a chord may match several consecutive notes, or the difference between two similar motifs may be a few swapped notes. Moreover, the substitution scores between chords are difficult to set up. In this paper, we propose a general framework for polyphonic music using the substitution score scheme set for monophonic music, which allows new operations by extending the operations proposed by Mongeau and Sankoff [15]. From a practical point of view, the limitation of chord sizes and the number of notes that can be merged consecutively lead to a complexity that remains quadratic.

A repeat in a string is a substring that occurs more than once. A repeat is extendible if every occurrence of the repeat has an identical letter either on the left or on the right; otherwise, it is maximal. A multirepeat is a repeat that occurs at least mmin times (m⩾ 2) in each of at least q ⩾ 1 strings in a given set of strings. In this paper, we describe a family of efficient algorithms based on suffix arrays to compute maximal multirepeats under various constraints. Our algorithms are faster, more flexible and much more space-efficient than algorithms recently proposed for this problem. The results extend recent work by two of the authors computing all maximal repeats in a single string.

The Range Next Value problem (Problem RNV) is a recent interesting variant of the range search problems, where the query is for the immediate next (or equal) value of a given number within a given interval of an array. Problem RNV was introduced and studied very recently by Crochemore et. al [Finding Patterns In Given Intervals, MFCS 2007]. In this paper, we present improved algorithms for Problem RNV. We also show how this problem can be used to achieve optimal query time for a number of interesting variants of the classic pattern matching problems.

A multirepeat in a string is a substring (factor) that appears a predefined number of times. A multirepeat is maximal if it cannot be extended either to the right or to the left and produce a multirepeat. In this paper, we present algorithms for two different versions of the problem of finding maximal multirepeats in a set of strings. In the case of arbitrary gaps, we propose an algorithm with O ( σN2n + α ) time complexity. When the gap is bounded in a small range c , we propose an algorithm with O (( c2 + σ2 ) mN2n log( Nn ) + α ) time complexity. Here, N is the number of strings, n the mean length of each string, m the multiplicity of the multirepeat and α the number of reported occurrences. Our results extend previous work by considering sets of strings as well as by generalizing pairs to multirepeats.

International audience

A partially occluded scene in an image consists of a number of objects that are partially obstructed by others. By validating a partially occluded image one aims to generate a sequence of concatenated and possibly overlapping objects that corresponds to the input image. This is a theoretical study of partially occluded strings (considered as one-dimensional images) allowing for the presence of errors in each occluded object appearing in the input. Using the unit cost edit distance as our measure of errors, for some small integer k ≥ 0, we present a sequential algorithm for validating a k-approximate one-dimensional image x of length n over a dictionary [Formula: see text] of m objects each having equal length τ in O(nd) time where d = mτ is the size of the dictionary.

An $O(s^5 M(s^2 ))$ algorithm for computing the canonical structure of a finite Abelian group represented by an integer matrix of size s (this is the Smith normal form of the matrix) is presented. Moreover, an $O(s^3 M(s^2 ))$ algorithm for computing the Hermite normal form of an integer matrix of size s is given. The upper bounds derived on the computational complexity of the algorithms above improve the upper bounds given by Kannan and Bachem in [SIAM J. Comput., 8 (1979), pp. 499–507] and Chou and Collins in [SIAM J. Comput., 11 (1982), pp. 687–708].

An $O(s^5 M(s^2 ))$ elementary operations algorithm for computing the canonical structure of infinite Abelian groups represented by a matrix of size s is presented. Also given is an algorithm for solving systems of linear Diophantine equations (say, $Ax = b$) in $O(s^{3.376} \log sM(s^2 ) + s^2 M(s^ * ))$ elementary operations, where s is the size of A and $s^ * $ is the size of b. The upper bounds mentioned above improve the results given by Chou and Collins in [SIAM J. Comput., 11 (1982), pp.687–708].