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

Crochemore, Maxime; Héliou, Alice; Kucherov, Grégory; Mouchard, Laurent; Pissis, Solon P.; Ramusat, Yann

doi:10.1007/978-3-662-55751-8_14

“…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].…”

supporting

confidence: 88%

“…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].…”

Section: Introductionmentioning

confidence: 98%

Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

Mieno

¹

,

Kuhara

²

,

Akagı

³

et al. 2020

SOFSEM 2020: Theory and Practice of Computer Science

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A substring u of a string T is called a minimal unique substring (MUS) of T if u occurs exactly once in T and any proper substring of u occurs at least twice in T . A string w is called a minimal absent word (MAW) of T if w does not occur in T and any proper substring of w occurs in T . In this paper, we study the problems of computing MUSs and MAWs in a sliding window over a given string T . We first show how the set of MUSs can change in a sliding window over T , and present an O(n log σ)-time and O(d)-space algorithm to compute MUSs in a sliding window of width d over T , where σ is the maximum number of distinct characters in every 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].

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“…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).…”

Section: Introductionmentioning

confidence: 99%

Absent words in a sliding window with applications

Crochemore¹,

Héliou²,

Kucherov³

et al. 2020

Information and Computation

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An absent 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. In fact, minimal absent words (MAWs) provide useful information about y and thus have several applications. In this paper, we propose an algorithm that maintains the set of MAWs of a fixed-length window sliding over y online. Our algorithm represents MAWs through nodes of the suffix tree. Specifically, the suffix tree of the sliding window is maintained using modified Senft's algorithm (Senft, 2005), itself generalizing Ukkonen's online algorithm (Ukkonen, 1995). We then apply this algorithm to the approximate pattern-matching problem under the Length Weighted Index distance (Chairungsee and Crochemore, 2012). This results in an online O(σ |y|)-time algorithm for finding approximate occurrences of a word x in y, |x| ≤ |y|, where σ is the alphabet size.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

Ayad

¹

,

Badkobeh

²

,

Fici

³

et al. 2020

Theory Comput Syst

Self Cite

3

0

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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 $\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M { y 1 , … , y k } ℓ of minimal absent words of length at most ℓ of the collection {y1, … , yk}. The set $\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M { y 1 , … , y k } 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 $\mathrm {M}^{\ell }_{y}$ M y ℓ of minimal absent words of a word y is equal to $\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M { y 1 , … , y k } 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 $\mathcal {O}(n)$ 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 $\| \mathrm {M}^{\ell }_{\{y_1,\ldots ,y_N\}}\| =o(n)$ ∥ M { y 1 , … , y N } 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 $\| \mathrm {M}^{12}_{\{y_1,\ldots ,y_k\}}\| \approx 10^{6}$ ∥ M { y 1 , … , y k } 12 ∥ ≈ 1 0 6 . We consider a constant-sized alphabet for stating our results. We show that all$\mathrm {M}^{\ell }_{y_{1}},\ldots ,\mathrm {M}^{\ell }_{\{y_1,\ldots ,y_k\}}$ M y 1 ℓ , … , M { y 1 , … , y k } ℓ can be computed in $\mathcal {O}(kn+{\sum }^{k}_{N=1}\| \mathrm {M}^{\ell }_{\{y_1,\ldots ,y_N\}}\| )$ O ( k n + ∑ N = 1 k ∥ M { y 1 , … , y N } ℓ ∥ ) total time using $\mathcal {O}(\textsc {MaxIn}+\textsc {MaxOut})$ O ( MaxIn + MaxOut ) space, where MaxIn is the length of the longest word in {y1, … , yk} and $\textsc {MaxOut}=\max \limits \{\| \mathrm {M}^{\ell }_{\{y_1,\ldots ,y_N\}}\| :N\in [1,k]\}$ MaxOut = max { ∥ M { y 1 , … , y N } ℓ ∥ : N ∈ [ 1 , k ] } . Proof-of-concept experimental results are also provided confirming our theoretical findings and justifying our contribution.

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

Cited by 9 publications

References 30 publications

Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

Absent words in a sliding window with applications

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

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