On fast and memory-efficient construction of an antidictionary array

Fukae, Hirotada; Ota, Tetsuji; Morita, Hiroyuki

doi:10.1109/isit.2012.6283021

“…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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“…State-of-the-art algorithms compute all minimal absent words of y in O(σ n) time [2,6,7] or in O(n + M y ) time [8,9] for integer alphabets. There also exist space-efficient data structures based on the Burrows-Wheeler transform of y that can be applied for this computation [10,11].…”

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

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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“…The computation of minimal absent words based on the construction of suffix arrays was considered in [9]; although this algorithm has a linear-time performance in practice, the worst-case time complexity is O(n 2 ). New O(n)-time and O(n)-space suffix-array-based algorithms were presented in [10,11,12] to bridge this unpleasant gap. An implementation of the algorithm presented in [11] is currently, to the best of our knowledge, the fastest available for the computation of minimal absent words.…”

Section: Introductionmentioning

confidence: 99%

Alignment-free sequence comparison using absent words

Charalampopoulos

¹

,

Fici

²

,

Mercaş

³

et al. 2018

Information and Computation

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Sequence comparison is a prerequisite to virtually all comparative genomic analyses. It is often realised by sequence alignment techniques, which are computationally expensive. This has led to increased research into alignment-free techniques, which are based on measures referring to the composition of sequences in terms of their constituent patterns. These measures, such as q-gram distance, are usually computed in time linear with respect to the length of the sequences. In this paper, we focus on the complementary idea: how two sequences can be efficiently compared based on information that does not occur in the sequences. A word is an absent word of some sequence if it does not occur in the sequence. An absent word is minimal if all its proper factors occur in the sequence. Here we present the first linear-time and linear-space algorithm to compare two sequences by considering all their minimal absent words. In the process, we present results of combinatorial interest, and also extend the proposed techniques to compare circular sequences. We also present an algorithm that, given a word x of length n, computes the largest integer for which all factors of x of that length occur in some minimal absent word of x in time and space O(n). Finally, we show that the known asymptotic upper bound on the number of minimal absent words of a word is tight.

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On fast and memory-efficient construction of an antidictionary array

Cited by 7 publications

References 7 publications

Absent words in a sliding window with applications

Absent words in a sliding window with applications

Constructing Antidictionaries of Long Texts in Output-Sensitive Space

Alignment-free sequence comparison using absent words

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