Linear-time Computation of Minimal Absent Words Using Suffix Array

An absent word of a word y of length n is a word that does not occur in y. It is a minimal absent word if all its proper factors occur in y. Minimal absent words have been computed in genomes of organisms from all domains of life; their computation provides a fast alternative for measuring approximation in sequence comparison. There exists an O(n)-time and O(n)-space algorithm for computing all minimal absent words on a fixed-sized alphabet based on the construction of suffix automata (Crochemore et al., 1998). No implementation of this algorithm is publicly available. There also exists an O(n^2)-time and O(n)-space algorithm for the same problem based on the construction of suffix arrays (Pinho et al., 2009). An implementation of this algorithm was also provided by the authors and is currently the fastest available. In this article, we bridge this unpleasant gap by presenting an O(n)-time and O(n)-space algorithm for computing all minimal absent words based on the construction of suffix arrays. Experimental results using real and synthetic data show that the respective implementation outperforms the one by Pinho et al

Space-efficient detection of unusual words

Detecting all the strings that occur in a text more frequently or less frequently than expected according to an IID or a Markov model is a basic problem in string mining, yet current algorithms are based on data structures that are either space-inefficient or incur large slowdowns, and current implementations cannot scale to genomes or metagenomes in practice. In this paper we engineer an algorithm based on the suffix tree of a string to use just a small data structure built on the Burrows-Wheeler transform, and a stack of $O(\sigma^2\log^2 n)$ bits, where $n$ is the length of the string and $\sigma$ is the size of the alphabet. The size of the stack is $o(n)$ except for very large values of $\sigma$. We further improve the algorithm by removing its time dependency on $\sigma$, by reporting only a subset of the maximal repeats and of the minimal rare words of the string, and by detecting and scoring candidate under-represented strings that $\textit{do not occur}$ in the string. Our algorithms are practical and work directly on the BWT, thus they can be immediately applied to a number of existing datasets that are available in this form, returning this string mining problem to a manageable scale.Comment: arXiv admin note: text overlap with arXiv:1502.0637

Minimal Forbidden Factors of Circular Words

Minimal forbidden factors are a useful tool for investigating properties of words and languages. Two factorial languages are distinct if and only if they have different (antifactorial) sets of minimal forbidden factors. There exist algorithms for computing the minimal forbidden factors of a word, as well as of a regular factorial language. Conversely, Crochemore et al. [IPL, 1998] gave an algorithm that, given the trie recognizing a finite antifactorial language $M$, computes a DFA recognizing the language whose set of minimal forbidden factors is $M$. In the same paper, they showed that the obtained DFA is minimal if the input trie recognizes the minimal forbidden factors of a single word. We generalize this result to the case of a circular word. We discuss several combinatorial properties of the minimal forbidden factors of a circular word. As a byproduct, we obtain a formal definition of the factor automaton of a circular word. Finally, we investigate the case of minimal forbidden factors of the circular Fibonacci words.Comment: To appear in Theoretical Computer Scienc

Optimal Computation of Avoided Words

The deviation of the observed frequency of a word $w$ from its expected frequency in a given sequence $x$ is used to determine whether or not the word is avoided. This concept is particularly useful in DNA linguistic analysis. The value of the standard deviation of $w$, denoted by $std(w)$, effectively characterises the extent of a word by its edge contrast in the context in which it occurs. A word $w$ of length $k>2$ is a $\rho$-avoided word in $x$ if $std(w) \leq \rho$, for a given threshold $\rho < 0$. Notice that such a word may be completely absent from $x$. Hence computing all such words na\"{\i}vely can be a very time-consuming procedure, in particular for large $k$. In this article, we propose an $O(n)$-time and $O(n)$-space algorithm to compute all $\rho$-avoided words of length $k$ in a given sequence $x$ of length $n$ over a fixed-sized alphabet. We also present a time-optimal $O(\sigma n)$-time and $O(\sigma n)$-space algorithm to compute all $\rho$-avoided words (of any length) in a sequence of length $n$ over an alphabet of size $\sigma$. Furthermore, we provide a tight asymptotic upper bound for the number of $\rho$-avoided words and the expected length of the longest one. We make available an open-source implementation of our algorithm. Experimental results, using both real and synthetic data, show the efficiency of our implementation

On finding minimal absent words

<p>Abstract</p> <p>Background</p> <p>The problem of finding the shortest absent words in DNA data has been recently addressed, and algorithms for its solution have been described. It has been noted that longer absent words might also be of interest, but the existing algorithms only provide generic absent words by trivially extending the shortest ones.</p> <p>Results</p> <p>We show how absent words relate to the repetitions and structure of the data, and define a new and larger class of absent words, called minimal absent words, that still captures the essential properties of the shortest absent words introduced in recent works. The words of this new class are minimal in the sense that if their leftmost or rightmost character is removed, then the resulting word is no longer an absent word. We describe an algorithm for generating minimal absent words that, in practice, runs in approximately linear time. An implementation of this algorithm is publicly available at <url>ftp://www.ieeta.pt/~ap/maws</url>.</p> <p>Conclusion</p> <p>Because the set of minimal absent words that we propose is much larger than the set of the shortest absent words, it is potentially more useful for applications that require a richer variety of absent words. Nevertheless, the number of minimal absent words is still manageable since it grows at most linearly with the string size, unlike generic absent words that grow exponentially. Both the algorithm and the concepts upon which it depends shed additional light on the structure of absent words and complement the existing studies on the topic.</p

R-enum: Enumeration of Characteristic Substrings in BWT-runs Bounded Space

Enumerating characteristic substrings (e.g., maximal repeats, minimal unique substrings, and minimal absent words) in a given string has been an important research topic because there are a wide variety of applications in various areas such as string processing and computational biology. Although several enumeration algorithms for characteristic substrings have been proposed, they are not space-efficient in that their space-usage is proportional to the length of an input string. Recently, the run-length encoded Burrows-Wheeler transform (RLBWT) has attracted increased attention in string processing, and various algorithms for the RLBWT have been developed. Developing enumeration algorithms for characteristic substrings with the RLBWT, however, remains a challenge. In this paper, we present r-enum (RLBWT-based enumeration), the first enumeration algorithm for characteristic substrings based on RLBWT. R-enum runs in O(n log log (n/r)) time and with O(r log n) bits of working space for string length n and number r of runs in RLBWT. Here, r is expected to be significantly smaller than n for highly repetitive strings (i.e., strings with many repetitions). Experiments using a benchmark dataset of highly repetitive strings show that the results of r-enum are more space-efficient than the previous results. In addition, we demonstrate the applicability of r-enum to a huge string by performing experiments on a 300-gigabyte string of 100 human genomes

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

A word x that is absent from a word y is called minimal if all its proper factors occur in y. Given a collection of k words y1, … , yk over an alphabet Σ, we are asked to compute the set M{y1,…,yk}ℓ of minimal absent words of length at most ℓ of the collection {y1, … , yk}. The set M{y1,…,yk}ℓ contains all the words x such that x is absent from all the words of the collection while there exist i,j, such that the maximal proper suffix of x is a factor of yi and the maximal proper prefix of x is a factor of yj. In data compression, this corresponds to computing the antidictionary of k documents. In bioinformatics, it corresponds to computing words that are absent from a genome of k chromosomes. Indeed, the set Myℓ of minimal absent words of a word y is equal to M{y1,…,yk}ℓ for any decomposition of y into a collection of words y1, … , yk such that there is an overlap of length at least ℓ − 1 between any two consecutive words in the collection. This computation generally requires Ω(n) space for n = |y| using any of the plenty available O(n) -time algorithms. This is because an Ω(n)-sized text index is constructed over y which can be impractical for large n. We do the identical computation incrementally using output-sensitive space. This goal is reasonable when ∥M{y1,…,yN}ℓ∥=o(n), for all N ∈ [1,k], where ∥S∥ denotes the sum of the lengths of words in set S. For instance, in the human genome, n ≈ 3 × 109 but ∥M{y1,…,yk}12∥≈106. We consider a constant-sized alphabet for stating our results. We show that allMy1ℓ,…,M{y1,…,yk}ℓ can be computed in O(kn+∑N=1k∥M{y1,…,yN}ℓ∥) total time using O(MaxIn+MaxOut) space, where MaxIn is the length of the longest word in {y1, … , yk} and MaxOut=max{∥M{y1,…,yN}ℓ∥:N∈[1,k]}. Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution