Palindromic trees for a sliding window and its applications

Mieno, Takuya; Watanabe, Kiichi; Nakashima, Yuto; Inenaga, Shunsuke; Bannai, Hideo; Takeda, Masayuki

doi:10.1016/j.ipl.2021.106174

Cited by 6 publications

(7 citation statements)

References 24 publications

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“…Also, the sliding-window suffix tree can be applied to compute minimal absent words [10] and minimal unique substrings [24], which are significant concepts for bioinformatics, in the sliding-window model. Recently, the sliding-window palindromic tree was proposed, and it can be applied to compute MUPSs in the slidingwindow model [25].…”

Section: Related Workmentioning

confidence: 99%

“…It is shown in [25] that the number of changes of MUPSs is constant when we append a character or delete the first character, and we can detect the changes in amortized O(log σ) time. Further, predecessor and successor data structures on the MUPSs can be updated dynamically in O(log log n) time using van Emde Boas trees [11].…”

Section: Sliding-window Data Structuresmentioning

confidence: 99%

“…Edit operations are given as queries, and they are discarded after finishing to process for the query. As related work, the set of minimal unique palindromic substrings (MUPSs) can be maintained efficiently in the sliding-window model [25]. Also, the set of MUPSs can be updated efficiently in the after-edit model [14].…”

Section: Introductionmentioning

confidence: 99%

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Shortest Unique Palindromic Substring Queries in Semi-dynamic Settings

Mieno¹,

Funakoshi²

2022

Preprint

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A palindromic substring T [i..j] of a string T is said to be a shortest unique palindromic substring (SUPS) in T for an interval [p, q] if T [i..j] is a shortest one such that T [i..j] occurs only once in T , and [i, j] contains [p, q]. The SUPS problem is, given a string T of length n, to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in O(α) time after O(n)-time preprocessing, where α is the number of SUPSs to output [Inoue et al., 2018]. In this paper, we first show that α is at most 4, and the upper bound is tight. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in O(log log W ) time and update data structures in amortized O(log σ) time, where W is the size of the window, and σ is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses O(n) time for preprocessing and answers any k SUPS queries in O(log n log log n + k log log n) time after single character substitution. As a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to polylogarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time.

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Section: Related Workmentioning

confidence: 99%

Section: Sliding-window Data Structuresmentioning

confidence: 99%

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Shortest Unique Palindromic Substring Queries in Semi-dynamic Settings

Mieno¹,

Funakoshi²

2022

Preprint

Self Cite

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“…Edit operations are given as queries, and they are discarded after finishing to process for the query. As related work, the set of minimal unique Data structures for computing unique palindromes in (non-)static strings palindromic substrings (MUPSs) can be maintained efficiently in the slidingwindow model [29]. Also, the set of MUPSs can be updated efficiently in the after-edit model [16].…”

Section: Introductionmentioning

confidence: 99%

Data Structures for Computing Unique Palindromes in Static and Non-Static Strings

Mieno,

Funakoshi

2023

Algorithmica

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A palindromic substring T [i..j] of a string T is said to be a shortest unique palindromic substring (SUPS) in T for an interval [p, q] if T [i..j] is a shortest one such that T [i..j] occurs only once in T , and [i, j] contains [p, q]. The SUPS problem is, given a string T of length n, to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in O(α) time after O(n)-time preprocessing, where α is the number of SUPSs to output [Inoue et al., 2018]. In this paper, we first show that α is at most 4, and the upper bound is tight.We also show that the total sum of lengths of minimal unique substrings of string T , which is strongly related to SUPSs, is O(n). Then, we present the first O(n)-bits data structures that can answer any SUPS query in constant time. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in O(log log W ) time and update data structures in amortized O(log σ) time, where W is the size of the window, and σ is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses O(n) time for preprocessing and answers any k SUPS queries Data structures for computing unique palindromes in (non-)static strings in O(log n log log n + k log log n) time after single character substitution. Finally, as a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to poly-logarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time.

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“…They showed that there are no more than m MUPSs in a string whose RLE size is m. Also, they proposed an O(m log σ R )-time and O(m)-space algorithm to compute all MUPSs of a string given in RLE, where m is the RLE size of the string, and σ R is the number of distinct single-character runs in the RLE string. Recently, Mieno et al [18] considered the problems of computing palindromic structures in the sliding window model. They showed that the set of MUPSs in a sliding window can be maintained in a total of O(n log σ W ) time and O(D) space while a window of size D shifts over a string of length n from the left-end to the right-end, where σ W is the maximum number of distinct characters in the windows.…”

Section: Introductionmentioning

confidence: 99%

Minimal unique palindromic substrings after single-character substitution

Funakoshi¹,

Mieno²

2021

Preprint

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A palindrome is a string that reads the same forward and backward. A palindromic substring w of a string T is called a minimal unique palindromic substring (MUPS ) of T if w occurs only once in T and any proper palindromic substring of w occurs at least twice in T . MUPSs are utilized for answering the shortest unique palindromic substring problem, which is motivated by molecular biology [Inoue et al., 2018]. Given a string T of length n, all MUPSs of T can be computed in O(n) time. In this paper, we study the problem of updating the set of MUPSs when a character in the input string T is substituted by another character. We first analyze the number d of changes of MUPSs when a character is substituted, and show that d is in O(log n). Further, we present an algorithm that uses O(n) time and space for preprocessing, and updates the set of MUPSs in O(log σ + (log log n) 2 + d) time where σ is the alphabet size. We also propose a variant of the algorithm, which runs in optimal O(d) time when the alphabet size is constant.

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Palindromic trees for a sliding window and its applications

Cited by 6 publications

References 24 publications

Shortest Unique Palindromic Substring Queries in Semi-dynamic Settings

Shortest Unique Palindromic Substring Queries in Semi-dynamic Settings

Data Structures for Computing Unique Palindromes in Static and Non-Static Strings

Minimal unique palindromic substrings after single-character substitution

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