Minimal Absent Words in a Sliding Window and Applications to On-Line Pattern Matching

Abstract

International audienceAn absent (or forbidden) word of a word y is a word that does not occur in y. It is then called minimal if all its proper factors occur in y. There exist linear-time and linear-space algorithms for computing all minimal absent words of y (Crochemore et al. in Inf Process Lett 67:111–117, 1998; Belazzougui et al. in ESA 8125:133–144, 2013; Barton et al. in BMC Bioinform 15:388, 2014). Minimal absent words are used for data compression (Crochemore et al. in Proc IEEE 88:1756–1768, 2000, Ota and Morita in Theoret Comput Sci 526:108–119, 2014) and for alignment-free sequence comparison by utilizing a metric based on minimal absent words (Chairungsee and Crochemore in Theoret Comput Sci 450:109–116, 2012). They are also used in molecular biology; for instance, three minimal absent words of the human genome were found to play a functional role in a coding region in Ebola virus genomes (Silva et al. in Bioinformatics 31:2421–2425, 2015). In this article we introduce a new application of minimal absent words for on-line pattern matching. Specifically, we present an algorithm that, given a pattern x and a text y, computes the distance between x and every window of size |x| on y. The running time is O(σ|y|)O(σ|y|) , where σσ is the size of the alphabet. Along the way, we show an O(σ|y|)O(σ|y|) -time and O(σ|x|)O(σ|x|) -space algorithm to compute the minimal absent words of every window of size |x| on y, together with some new combinatorial insight on minimal absent words

Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

“…We then give tight upper and lower bounds on the maximum number of changes in the set of MAWs in a sliding window over T . Our bounds improve on the previous results in [ Crochemore et al, 2017].…”

“…MAWs in a sliding window have already been studied by Crochemore et al [4]. They studied the number of MAWs to be added / deleted when the current window is shifted, and we improve some of these results (Section 5): For any string T over an alphabet of size σ, let MAW(T [i..j]) be the set of all MAWs in the substring T [i..j].…”

Absent words in a sliding window with applications

“…Several linear-time and linear-space algorithms have been proposed to compute the set of MAWs for constant-sized [13,26,17,5,2,3] or integer [16,8] alphabets. These algorithms are based on text indexing data structures such as suffix tree, suffix array or ✩ A preliminary version of this paper was presented at the 21st International Symposium on Fundamentals of Computation Theory (FCT 2017) [12]. directed acyclic word graph (DAWG and Factor automaton).…”

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

“…There also exist space-efficient data structures based on the Burrows-Wheeler transform of y that can be applied for this computation [10,11]. In many real-world applications of minimal absent words, such as in data compression [12][13][14][15], in sequence comparison [3,9], in on-line pattern matching [16], or in identifying pathogen-specific signatures [17], only a subset of minimal absent words may be considered, and, in particular, the minimal absent words of length (at most) . Since, in the worst case, the number of minimal absent words of y is Θ(σ n), Ω(σ n) space is required to represent them explicitly.…”