Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

Mieno, Takuya; Kuhara, Yuki; Akagı, Takanori; Fujishige, Yuta; Nakashima, Yuto; Inenaga, Shunsuke; Bannai, Hideo; Takeda, Masayuki

doi:10.1007/978-3-030-38919-2_13

Cited by 11 publications

(8 citation statements)

References 13 publications

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“…A part of the results reported in this article appeared in a preliminary version of this paper [15].…”

Section: Our Contributionmentioning

confidence: 90%

Combinatorics of minimal absent words for a sliding window

Akagı¹,

Kuhara²,

Mieno³

et al. 2021

Preprint

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A string w is called a minimal absent word (MAW) for another string T if w does not occur in T but the proper substrings of w occur in T . For example, let Σ = {a, b, c} be the alphabet. Then, the set of MAWs for string w = abaab is {aaa, aaba, bab, bb, c}.In this paper, we study combinatorial properties of MAWs in the sliding window model, namely, how the set of MAWs changes when a sliding window of fixed length d is shifted over the input string T of length n, where 1 ≤ d < n. We present tight upper and lower bounds on the maximum number of changes in the set of MAWs for a sliding window over T , both in the cases of general alphabets and binary alphabets. Our bounds improve on the previously known best bounds [Crochemore et al., 2020].

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“…A part of the results reported in this article appeared in a preliminary version of this paper [15].…”

Section: Our Contributionmentioning

confidence: 90%

Combinatorics of minimal absent words for a sliding window

Akagı¹,

Kuhara²,

Mieno³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Our Contributionsmentioning

confidence: 99%

“…A part of the results reported in this article appeared in a preliminary version of this paper, [18]. The preliminary paper [18] consists of two parts: (1) efficient computation and combinatorial properties of MUSs for a sliding window, and (2) combinatorial properties of minimal absent words (MAWs) [19] for a sliding window. This current article is a full version of the former part (1) which contains complete proofs and supplemental figures which were omitted in the preliminary version [18].…”

Section: Our Contributionsmentioning

confidence: 99%

“…The preliminary paper [18] consists of two parts: (1) efficient computation and combinatorial properties of MUSs for a sliding window, and (2) combinatorial properties of minimal absent words (MAWs) [19] for a sliding window. This current article is a full version of the former part (1) which contains complete proofs and supplemental figures which were omitted in the preliminary version [18]. We remark that an extended version of the latter part (2) can be found as an independent article [2].…”

Section: Our Contributionsmentioning

confidence: 99%

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Computing Minimal Unique Substrings for a Sliding Window

et al. 2021

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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. In this paper, we study the problem of computing MUSs for a sliding window over a given string T. We first show how the set of MUSs can change when the window slides over T. We then present an $$O(n\log \sigma ')$$ O ( n log σ ′ ) -time and O(d)-space algorithm to compute MUSs for a sliding window of size d over the input string T of length n, where $$\sigma '\le d$$ σ ′ ≤ d is the maximum number of distinct characters in every window.

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“…Recently, several algorithms for computing substrings for a sliding window with certain interesting properties are proposed: For instance, Crochemore et al [4] introduced the problem of computing MAWs for a sliding window, and proposed an O(nσ)-time and O(dσ)-space algorithm using suffix trees for a sliding window. Mieno et al [16] proposed an algorithm for computing minimal unique substrings (MUSs) [11] for a sliding window, in O(n log σ)-time and O(d) space, again based on suffix trees for a sliding window.…”

Section: Introductionmentioning

confidence: 99%

Palindromic Trees for a Sliding Window and Its Applications

Mieno

Watanabe

Nakashima

et al. 2020

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The palindromic tree (a.k.a. eertree) for a string S of length n is a tree-like data structure that represents the set of all distinct palindromic substrings of S, using O(n) space [Rubinchik and Shur, 2018]. It is known that, when S is over an alphabet of size σ and is given in an online manner, then the palindromic tree of S can be constructed in O(n log σ) time with O(n) space. In this paper, we consider the sliding window version of the problem: For a fixed window length d, we propose two algorithms to maintain the palindromic tree of size O(d) for every sliding window S[i..i+d−1] over S, one running in O(n log σ ′ ) time with O(d) space where σ ′ ≤ d is the maximum number of distinct characters in the windows, and the other running in O(n+dσ) time with dσ +O(d) space. We also present applications of our algorithms for computing minimal unique palindromic substrings (MUPS) and for computing minimal absent palindromic words (MAPW) for a sliding window.which enumerate all distinct palindromes in a given sting of length n over an integer alphabet of size σ = n O(1) . For the same problem in the online model, Kosolobov et al. [13] proposed an O(n log σ)-time and O(n)-space algorithm for a general ordered alphabet. Kosolobov et al.'s algorithm is a combination of Manacher's algorithm and Ukkonen's online suffix tree construction algorithm [21]. Rubinchik and Shur [19] proposed a new data structure called eertree, which permits efficient access to distinct palindromes in a string without storing the string itself. Eertrees can be utilized for solving problems related to palindromic structures, such as the palindrome counting problem and the palindromic factorization problem [19]. The size of the eertree of S is linear in the number p S of distinct palindromes in S [19]. It is known that p S is at most |S|+ 1, and that it can be much smaller than the length |S| of the string, e.g., for S = abc n/3 , p S = 4 since all distinct palindromes in S are a, b, c, and the empty string. Thus, the size of the eertree of S can be much smaller than that of the suffix tree of S which is Θ(n). Therefore, it is of significance if one can build eertrees without suffix trees. Rubinchik and Shur [19] indeed proposed an online eertree construction algorithm running in O(n log σ) time without suffix trees.Recently, a concept of palindromic structures called minimal unique palindromic substrings (MUPS ) is introduced by Inoue et al. [12]. A palindromic substring w = S[i..j] of a string S is called a MUPS of S if w occurs in S exactly once, and S[i+1..j −1] occurs at least twice in S. MUPSs are utilized for solving the shortest unique palindromic substring (SUPS ) problem [12], which is motivated by an application in molecular biology. Watanabe et al. [22] proposed an algorithm to solve the SUPS problem based on the run-length encoding (RLE ) version of eertrees, named e 2 rtre 2 .

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Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

Cited by 11 publications

References 13 publications

Combinatorics of minimal absent words for a sliding window

Combinatorics of minimal absent words for a sliding window

Computing Minimal Unique Substrings for a Sliding Window

Palindromic Trees for a Sliding Window and Its Applications

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