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A string s is called a parameterized square when \(s = xy\) for strings x, y and x and y are parameterized equivalent. Kociumaka et al. showed the number of parameterized squares, which are non-equivalent in parameterized equivalence, in a string of length n that contains \(\sigma \) distinct characters is at most \(2 \sigma ! n\) [TCS 2016]. In this paper, we show that the maximum number of non-equivalent parameterized squares is less than \(\sigma n\) , which significantly improves the best-known upper bound by Kociumaka et al.

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On the Number of Non-equivalent Parameterized Squares in a String

  • 会议论文

  • 发表日期：

    2025年

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  • Rikuya Hamai,
  • Kazushi Taketsugu,
  • Yuto Nakashima,
  • Shunsuke Inenaga,
  • Hideo Bannai

摘要

A string s is called a parameterized square when \(s = xy\) for strings x, y and x and y are parameterized equivalent. Kociumaka et al. showed the number of parameterized squares, which are non-equivalent in parameterized equivalence, in a string of length n that contains \(\sigma \) distinct characters is at most \(2 \sigma ! n\) [TCS 2016]. In this paper, we show that the maximum number of non-equivalent parameterized squares is less than \(\sigma n\) , which significantly improves the best-known upper bound by Kociumaka et al.