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

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Abstract

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 palindromic substring 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 \(O(\alpha )\) time after O(n)-time preprocessing, where \(\alpha \) is the number of SUPSs to output (Inoue in J Discrete Algorithms 52-53:122–132, 2018). In this paper, we first show that \(\alpha \) is at most 4, and the upper bound is tight. We also show that the total sum of lengths of minimal unique palindromic 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 \(O(\log \sigma + \log \log W)\) time, where W is the size of the window, and \(\sigma \) 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. 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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Notes

  1. Since MUPSs cannot be nested [23], they are also sorted by their ending positions.

  2. This is inspired by a similar notion defined for minimal unique substrings in [33].

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Acknowledgements

We would like to thank Professor Jeffrey Shallit (University of Waterloo) for his interest in our paper and his advice to simplify our proofs. We would also like to thank the anonymous referees for their helpful comments on the manuscript. This work was partially supported by the JSPS KAKENHI Grant Numbers JP20J11983, JP22K21273 (TM), and JP20J21147 (MF).

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Authors and Affiliations

  1. Department of Computer and Network Engineering, University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, 182-8585, Japan

    Takuya Mieno

  2. Department of Informatics, Kyushu University, 744 Motooka Nishi-ku, Fukuoka, 819-0395, Japan

    Mitsuru Funakoshi

  3. Japan Society for the Promotion of Science, 5-3-1 Kojimachi, Chiyoda-ku, Tokyo, 102-0083, Japan

    Mitsuru Funakoshi

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Mieno, T., Funakoshi, M. Data Structures for Computing Unique Palindromes in Static and Non-Static Strings. Algorithmica 86, 852–873 (2024). https://doi.org/10.1007/s00453-023-01170-8

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