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Factorizing Strings into Repetitions

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Abstract

A factorization f1,…,fm of a string w is called a repetition factorization of w if each factor fi is a repetition, namely, \(f_{i} = x^{k}x^{\prime }\) for some non-empty string x, an integer k ≥ 2, and \(x^{\prime }\) being a proper prefix of x. Dumitran et al. (Proc. SPIRE 2015) proposed an algorithm which computes an arbitrary repetition factorization of a given string w in O(n) time, where n is the length of w. The number of factors (i.e. repetitions) contained in the output of their algorithm is not known or guaranteed. In this paper, we propose two algorithms for computing smallest/largest repetition factorizations in \(O(n \log n)\) time, which respectively consist of the smallest/largest number of factors. The first algorithm is a simple \(O(n \log n)\)-space algorithm based on a reduction of the problem to the shortest/longest path problem on the DAG of size \(O(n \log n)\). The second one simulates the dynamic programming algorithm for shortest/longest path problem within O(n) space based on the idea of the first algorithm. Moreover, we discuss combinatorial structures of smallest/largest repetition factorizations of Fibonacci strings.

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Notes

  1. A preliminary version of this paper appeared as [16].

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Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers 25240003, 26280003, JP16H02783, JP17H01697, JP18K18002, JP18H04098, and by JST PRESTO Grant Number JPMJPR1922.

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Author notes

  1. Hideo Bannai

    Present address: M&D Data Science Center, Tokyo Medical and Dental University, Tokyo, Japan

Authors and Affiliations

  1. Department of Informatics, Kyushu University, 744, Motooka, Nishiku, Fukuoka, 819-0395, Japan

    Hiroe Inoue, Yoshiaki Matsuoka, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai & Masayuki Takeda

  2. PRESTO, Japan Science and Technology Agency, 4-1-8, Honcho, Kawaguchi, Saitama, 332-0012, Japan

    Shunsuke Inenaga

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  1. Hiroe Inoue
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Correspondence to Yuto Nakashima.

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Inoue, H., Matsuoka, Y., Nakashima, Y. et al. Factorizing Strings into Repetitions. Theory Comput Syst 66, 484–501 (2022). https://doi.org/10.1007/s00224-022-10070-3

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