Parallelising the Computation of Minimal Absent Words

Barton, Carl; Héliou, Alice; Mouchard, Laurent; Pissis, Solon P.

doi:10.1007/978-3-319-32152-3_23

“…It was also shown that the set of all MAWs of y is sufficient to uniquely reconstruct y [13,15]. 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].…”

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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“…To maintain the LWI across the word y for a pattern x, we need to compute the set of minimal absent words in a sliding window of size m = |x| of y. Several linear-time and linearspace algorithms have been proposed to compute the set of minimal absent words [10,6,3,4,15]. Ota et al presented an on-line algorithm that requires linear time and linear space [28].…”

Section: Introductionmentioning

confidence: 99%

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

Crochemore

¹

,

Héliou

²

,

Kucherov

³

et al. 2017

Fundamentals of Computation Theory

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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

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“…3. Our technique could serve as a basis for a new parallelization scheme for constructing antidictionaries (see also [27]), in which several documents are processed concurrently.…”

Section: Final Remarksmentioning

confidence: 99%

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

Ayad

¹

,

Badkobeh

²

,

Fici

³

et al. 2020

Theory Comput Syst

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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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Parallelising the Computation of Minimal Absent Words

Cited by 17 publications

References 18 publications

Absent words in a sliding window with applications

Absent words in a sliding window with applications

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

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

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