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Shortest Unique Palindromic Substring Queries in Semi-dynamic Settings

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Combinatorial Algorithms (IWOCA 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13270))

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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 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 \(O(\alpha )\) time after O(n)-time preprocessing, where \(\alpha \) is the number of SUPSs to output [Inoue et al., 2018]. In this paper, we first show that \(\alpha \) 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 \(O(\log \sigma )\) 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. 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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Notes

  1. 1.

    A preliminary version of [19] appeared in IWOCA 2017 [26].

  2. 2.

    Since MUPSs cannot be nested [19], they are also sorted by their ending positions.

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Acknowledgements

We would like to thank Professor Jeffrey Shallit (University of Waterloo), a PC member of IWOCA 2022, 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 supported by the JSPS KAKENHI Grant Numbers JP20J11983 (TM) and JP20J21147 (MF).

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Authors and Affiliations

  1. Faculty of IST, Hokkaido University, Sapporo, Japan

    Takuya Mieno

  2. Department of Informatics, Kyushu University, Fukuoka, Japan

    Mitsuru Funakoshi

  3. Japan Society for the Promotion of Science, Tokyo, Japan

    Mitsuru Funakoshi

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  1. Takuya Mieno
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Correspondence to Takuya Mieno .

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  1. Université Paris-Dauphine, Paris, France

    Cristina Bazgan

  2. Universität Trier, Trier, Germany

    Henning Fernau

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Mieno, T., Funakoshi, M. (2022). Shortest Unique Palindromic Substring Queries in Semi-dynamic Settings. In: Bazgan, C., Fernau, H. (eds) Combinatorial Algorithms. IWOCA 2022. Lecture Notes in Computer Science, vol 13270. Springer, Cham. https://doi.org/10.1007/978-3-031-06678-8_31

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  • DOI: https://doi.org/10.1007/978-3-031-06678-8_31

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