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.

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.

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.

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.