Diverse Palindromic Factorization is NP-Complete

Bannai, Hideo; Gagie, Travis; Inenaga, Shunsuke; Kärkkäinen, Juha; Kempa, Dominik; Piatkowski, Marcin; Sugimoto, Shiho

doi:10.1142/s0129054118400014

Cited by 8 publications

(2 citation statements)

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“…A substring S[i..j] is called a palindromic substring of S if S[i..j] and the reversed string of S[i..j] is the same string. The study of combinatorial properties and structures on palindromes is still an important and well studied topic in stringology [1,3,4,5,8,14]. Droubay et al [3] showed a string of length n can contain at most n + 1 distinct palindromes.…”

Section: Introductionmentioning

confidence: 99%

Shortest Unique Palindromic Substring Queries in Optimal Time

Nakashima

Inoue

Mieno

et al. 2018

Lecture Notes in Computer Science

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A palindrome is a string that reads the same forward and backward. A palindromic substring P of a string S is called a shortest unique palindromic substring (SUPS ) for an interval [s, t] in S, if P occurs exactly once in S, this occurrence of P contains interval [s, t], and every palindromic substring of S which contains interval [s, t] and is shorter than P occurs at least twice in S. The SUPS problem is, given a string S, to preprocess S so that for any subsequent query interval [s, t] all the SUPS s for interval [s, t] can be answered quickly. We present an optimal solution to this problem. Namely, we show how to preprocess a given string S of length n in O(n) time and space so that all SUPS s for any subsequent query interval can be answered in O(α + 1) time, where α is the number of outputs.

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Section: Introductionmentioning

confidence: 99%

Shortest Unique Palindromic Substring Queries in Optimal Time

Nakashima

Inoue

Mieno

et al. 2018

Lecture Notes in Computer Science

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“…To derive our result we consider a specific factorization of a rich word into distinct rich palindromes, here called UPS-factorization (Unioccurrent Palindromic Suffix factorization), see Definition 3.2. Let us mention that another palindromic factorizations have already been studied, see [3,7]: Minimal (minimal number of palindromes), maximal (every palindrome cannot be extended on the given position) and diverse (all palindromes are distinct). Note that only the minimal palindromic factorization has to exist for every word.…”

Section: Introductionmentioning

confidence: 99%

On the Number of Rich Words

Rukavicka

2017

Developments in Language Theory

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Any finite word w of length n contains at most n + 1 distinct palindromic factors. If the bound n + 1 is reached, the word w is called rich. The number of rich words of length n over an alphabet of cardinality q is denoted R n (q). For binary alphabet, Rubinchik and Shur deduced that R n (2) ≤ c1.605 n for some constant c. We prove that lim n→∞ n R n (q) = 1 for any q, i.e. R n (q) has a subexponential growth on any alphabet.

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