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

TL;DR

This work tackles the problem of bounding the number of non-equivalent parameterized squares in a string under parameterized equivalence. It introduces tighter parameterized periodicity lemmas and a refined counting strategy to prove that, for a string of length $n$ over an alphabet of size $σ$, the number of non-equivalent parameterized squares satisfies $PS(s) < σ n$. This significantly improves the previous upper bound of $2 σ! n$ and tightens our understanding of parameterized repetition structures in strings. The results rely on improved analyses of p-periods, substring alphabet sizes, and overlap behavior, with potential implications for broader pattern-matching and combinatorics-on-words problems in the parameterized setting.

Abstract

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 $σ$ distinct characters is at most $2 σ! n$ [TCS 2016]. In this paper, we show that the maximum number of non-equivalent parameterized squares is less than $σn$, which significantly improves the best-known upper bound by Kociumaka et al.

Figures (2)

• Figure 1: Illustration for the proof of Theorem \ref{['lem:prefix-squares']}: $\sigma+1$ non-equivalent parameterized squares cannot begin at the same position.
• Figure 2: Illustration for Lemma \ref{['lem:lowerbound']}: There are $\sigma$ non-equivalent parameterized square prefixes.

Theorems & Definitions (21)

• Lemma 1: Lemma 3 of Apostolico2008
• Lemma 2: Lemma 5 of Ideguchi2023
• Lemma 3: Lemma 4 of Ideguchi2023
• Theorem 1
• Theorem 2
• Lemma 4
• proof
• Corollary 1
• Lemma 5
• proof