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Cyclic versions of covers and roots of a string are considered in this paper. A prefix V of a string S is a cyclic root of S if S is a concatenation of cyclic rotations of V . A prefix V of S is a cyclic cover of S if the occurrences of the cyclic rotations of V cover all positions of S. We present O(n)-time algorithms computing all cyclic roots (using number-theoretic tools) and all cyclic covers (using tools related to seeds) of a length-n string over an integer alphabet. Our results improve upon O(n log log n) and O(n log n) time complexities of recent algorithms of Grossi et al. (WALCOM 2023) for the respective problems and provide novel approaches to the problems. As a by-product, we obtain an optimal data structure for Internal Circular Pattern Matching queries that generalize Internal Pattern Matching and Cyclic Equivalence queries of Kociumaka et al. (SODA 2015).

We introduce the concept of cyclic covers, which generalizes the classical notion of covers in strings. Given any nonempty string X of length n, a factor W of X is called a cyclic cover if every position of X belongs to an occurrence of a cyclic shift of W. 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 present an algorithm that solves the cyclic cover problem in time. This 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.

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.

Motivated by applications in bioinformatics and image searching, in what follows, we study the classic pattern matching problem in the context of elastic-degenerate strings: the generalised notion of gapped strings. An elastic-degenerate string can be seen as an ordered collection of k strings interleaved by k−1 elastic-degenerate symbols, where each such elastic-degenerate symbol corresponds to a set of two or more variable-length strings. We present efficient algorithms for two variants of the pattern matching problem on elastic-degenerate strings: first, for a solid pattern and an elastic-degenerate text; second, for an elastic-degenerate pattern and a solid text. A proof-of-concept implementation of the former is provided.

We introduce a new family of string processing problems. Given two or more strings, we are asked to compute a factor common to all strings that preserves a specific property and has maximal length. We consider three fundamental string properties: square-free factors, periodic factors, and palindromic factors under three different settings, one per property. In the first setting, we are given a string x and we are asked to construct a data structure over x answering the following type of online queries: given a string y, find a longest square-free factor common to x and y. In the second setting, we are given k strings and an integer 1

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 S 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 (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 S in (min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in S. 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 closed string contains a proper factor occurring as both a prefix and a suffix but not elsewhere in the string. Closed strings were introduced by Fici (WORDS 2011) as objects of combinatorial interest. This paper addresses a new problem by extending the closed string problem to the k-closed string problem, for which a level of approximation is permitted up to a number of Hamming distance errors, set by the parameter k. We address the problem of deciding whether or not a given string of length n over an integer alphabet is k-closed and additionally specifying the border resulting in the string being k-closed. Specifically, we present an (kn)-time and (n)-space algorithm to achieve this along with the pseudocode of an implementation and proof-of-concept experimental results.

In the k-mappability problem, we are given a string x of length n and integers m and k, and we are asked to count, for each length-m factor y of x, the number of other factors of length m of x that are at Hamming distance at most k from y. We focus here on the version of the problem where k=1. There exists an algorithm to solve this problem for k=1 requiring time O(mnlog⁡n/log⁡log⁡n) using space O(n). Here we present two new algorithms that require worst-case time O(mn) and O(nlog⁡nlog⁡log⁡n), respectively, and space O(n), thus greatly improving the previous result. Moreover, we present another algorithm that requires average-case time and space O(n) for integer alphabets of size σ if m=Ω(logσ⁡n). Notably, we show that this algorithm is generalizable for arbitrary k, requiring average-case time O(kn) and space O(n) if m=Ω(klogσ⁡n), assuming that the letters are independent and uniformly distributed random variables. Finally, we provide an experimental evaluation of our average-case algorithm demonstrating its competitiveness to the state-of-the-art implementation.

A string X = X[1..n], n > 1, is said to be closed if it has a nonempty proper prefix that is also a suffix, but that otherwise occurs nowhere else in X; for n = 1, every X is closed. Closed strings were introduced by Fici in [1] as objects of combinatorial interest. Recently Badkobeh et al. [2] described a variety of algorithms to factor a given string into closed factors. In particular, they studied the Longest Closed Factorization (LCF) problem, which greedily computes the decomposition X = X1X2・・・Xk, where X1 is the longest closed prefix ofX, X2 the longest closed prefix ofX with prefixX1 removed, and so on. In this paper we present an O(log n) amortized per character algorithm to compute LCF on-line, where n is the length of the string. We also introduce the Minimum Closed Factorization (MCF) problem, which identifies the minimum number of closed factors that cover X. We first describe an off-line O(n log2 n)-time algorithm to compute MCF(X), then we present an on-line algorithm for the same problem. In fact, we show that MCF(X) can be computed in O(Llog n) time from MCF(X_), where X_ = X[1..n−1], and L is the largest integer such that the suffix X[n−L+1..n] is a substring of X_.

Edited Book

In this paper we describe algorithms for factoring words over sets of strings known as circ-UMFFs, generalizations of the well-known Lyndon words based on lexorder, whose properties were first studied in 1958 by Chen, Fox and Lyndon. In 1983 Duval designed an elegant linear-time sequential (RAM) Lyndon factorization algorithm; a corresponding parallel (PRAM) algorithm was described in 1994 by Daykin, Iliopoulos and Smyth. In 2003 Daykin and Daykin introduced various circ-UMFFs, including one based on V-words and V-ordering; in 2011 linear string comparison and sequential factorization algorithms based on V-order were given by Daykin, Daykin and Smyth. Here we first describe generic RAM and PRAM algorithms for factoring a word over any circ-UMFF; then we show how to customize these generic algorithms to yield optimal parallel Lyndon-like V-word factorization.

A factor u of a string y is a cover of y if every letter of y lies within some occurrence of u in y; thus every cover u is also a border—both prefix and suffix—of y. If u is a cover of a superstring of y then u is a seed of y. Covers and seeds are two formalisations of quasiperiodicity, and there exist linear-time algorithms for computing all the covers and seeds of y. A string y covered by u thus generalises the idea of a repetition; that is, a string composed of exact concatenations of u. Even though a string is coverable somewhat more frequently than it is a repetition, still a string that can be covered by a single u is rare. As a result, seeking to find a more generally applicable and descriptive notion of cover, many articles were written on the computation of a minimum k-cover of y; that is, the minimum cardinality set of strings of length k that collectively cover y. Unfortunately, this computation turns out to be NP-hard. Therefore, in this article, we propose new, simple, easily-computed, and widely applicable notions of string covering that provide an intuitive and useful characterisation of a string: the enhanced cover; the enhanced left cover; and the enhanced left seed.

This special issue contains full versions of selected papers presented at the 21st International Workshop on Combinatorial Algorithms (IWOCA) held in July 2010 at the Strand campus of Kingʼs College London...

A fundamental problem in music is to classify songs according to their rhythm. A rhythm is represented by a sequence of “Quick” (Q) and “Slow” (S) symbols, which correspond to the (relative) duration of notes, such that S = 2Q. In this paper, we present an efficient algorithm for locating the maximum-length substring of a music text t that can be covered by a given rhythm r.

The number of segregating sites provides an indicator of the degree of DNA sequence variation that is present in a sample, and has been of great interest to the biological, pharmaceutical and medical professions. In this paper, we first provide linear- and expected-sublinear-time algorithms for finding all the segregating sites of a given set of DNA sequences. We also describe a data structure for tracking segregating sites in a set of sequences, such that every time the set is updated with the insertion of a new sequence or removal of an existing one, the segregating sites are updated accordingly without the need to re-scan the entire set of sequences.

In this paper we have presented new algorithms to handle the pattern matching problem where the pattern can contain variable length gaps. Given a pattern P with variable length gaps and a text T our algorithm works in O(n + m + α log(max 1<=i<=l(b i –a i ))) time where n is the length of the text, m is the summation of the lengths of the component subpatterns, α is the total number of occurrences of the component subpatterns in the text and a i and b i are, respectively, the minimum and maximum number of don’t cares allowed between the ith and (i+1)st component of the pattern. We also present another algorithm which, given a suffix array of the text, can report whether P occurs in T in O(m + α loglogn) time. Both the algorithms record information to report all the occurrences of P in T. Furthermore, the techniques used in our algorithms are shown to be useful in many other contexts.

The study of approximately periodic strings is relevant to diverse applications such as molecular biology, data compression, and computer-assisted music analysis. Here we study different forms of approximate periodicity under a variety of distance functions. We consider three related problems, for two of which we derive polynomial-time algorithms; we then show that the third problem is NP-complete.

The study of approximately periodic strings is relevant to diverse applications such as molecular biology, data compression, and computer-assisted music analysis. Here we study different forms of approximate periodicity under a variety of distance rules.We consider three related problems, for two of which we derive polynomial-time algorithms; we then show that the third problem is NP-complete.

Fibonacci strings turn out to constitute worst cases for a number of computer algorithms which find generic patterns in strings. Examples of such patterns are repetitions, Abelian squares, and "covers". In particular, we characterize in this paper the covers of a circular Fibonacci string C(F k ) and show that they are \Theta(jF k j 2 ) in number. We show also that, by making use of an appropriate encoding, these covers can be reported in \Theta(jF k j) time. By contrast, the fastest known algorithm for computing the covers of an arbitrary circular string of length n requires time O(n log n).