EERTREE: An Efficient Data Structure for Processing Palindromes in Strings

Rubinchik, Mikhail; Shur, Arseny M.

doi:10.1007/978-3-319-29516-9_27

Cited by 20 publications

(33 citation statements)

References 14 publications

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“…; a similar algorithm was presented by I et al [31]. In addition, Rubinchik and Shur [27] devised an O(n)-sized data structure that helps to locate palindromes in a string; they also showed how it can be used to compute a palindromic factorization of X in O(n log n) time. Borozdin et al recently improved this by presenting an O(n)-time algorithm [28].…”

Section: Algorithms For Computing Gd Palindromesmentioning

confidence: 92%

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Comparing Degenerate Strings

Alzamel

Ayad

Bernardini

et al. 2020

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Uncertain sequences are compact representations of sets of similar strings. They highlight common segments by collapsing them, and explicitly represent varying segments by listing all possible options. A generalized degenerate string (GD string) is a type of uncertain sequence. Formally, a GD string Ŝ is a sequence of n sets of strings of total size N, where the ith set contains strings of the same length ki but this length can vary between different sets. We denote by W the sum of these lengths k0, k1, . . . , kn-1. Our main result is an 𝒪(N + M)-time algorithm for deciding whether two GD strings of total sizes N and M, respectively, over an integer alphabet, have a non-empty intersection. This result is based on a combinatorial result of independent interest: although the intersection of two GD strings can be exponential in the total size of the two strings, it can be represented in linear space. We then apply our string comparison tool to devise a simple algorithm for computing all palindromes in Ŝ in 𝒪(min{W, n2}N)-time. We complement this upper bound by showing a similar conditional lower bound for computing maximal palindromes in Ŝ. We also show that a result, which is essentially the same as our string comparison linear-time algorithm, can be obtained by employing an automata-based approach.

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Section: Algorithms For Computing Gd Palindromesmentioning

confidence: 92%

“…The problem that gained significant attention recently is the factorization of a string X of length n into a sequence of palindromes [25,26,27,28,29,30]. We say that X 1 , X 2 , .…”

Section: Algorithms For Computing Gd Palindromesmentioning

confidence: 99%

Comparing Degenerate Strings

Alzamel

Ayad

Bernardini

et al. 2020

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confidence: 99%

“…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].…”

mentioning

confidence: 99%

“…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.…”

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Palindromic Trees for a Sliding Window and Its Applications

Mieno

Watanabe

Nakashima

et al. 2020

Preprint

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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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Shortest Unique Palindromic Substring Queries on Run-Length Encoded Strings

Watanabe

Nakashima

Inenaga

et al. 2019

Lecture Notes in Computer Science

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For a string S, a palindromic substring S[i..j] is said to be a shortest unique palindromic substring (SUPS ) for an interval [s, t] t], and every palindromic substring containing [s, t] which is shorter than S[i..j] occurs at least twice in S. In this paper, we study the problem of answering SUPS queries on run-length encoded strings. We show how to preprocess a given run-length encoded string RLE S of size m in O(m) space and O(m log σ RLE S + m log m/ log log m) time so that all SUPSs for any subsequent query interval can be answered in O( log m/ log log m + α) time, where α is the number of outputs, and σ RLE S is the number of distinct runs of RLE S .

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EERTREE: An Efficient Data Structure for Processing Palindromes in Strings

Cited by 20 publications

References 14 publications

Comparing Degenerate Strings

Comparing Degenerate Strings

Palindromic Trees for a Sliding Window and Its Applications

Shortest Unique Palindromic Substring Queries on Run-Length Encoded Strings

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