arXiv subjects

Rikuya Hamai

Publications and source records attributed to Rikuya Hamai.

3 recordsLinked to original sources

arXiv2026

The Parameterized Periodicity Lemma

Fine and Wilf [Proc. Amer. Math. Soc. 1965] showed that any string of length at least $p+q-d$ with periods $p$ and $q$ also has period $d=\gcd(p,q)$. For parameterized strings, Apostolico and Giancarlo [Discrete Appl. Math. 2008] proved an analogue with length bound $p+q$, assuming that the two induced bijections commute. Ideguchi et al. [SPIRE 2023] removed this assumption and gave the bound $p+q+\min(p,q)(\sigma-1)$, where $\sigma$ is the number of distinct letters. This was later improved by Hamai et al. [SPIRE 2024] to $p+q+\min(p,q)(\sigma-2)$, which was used to bound the number of non-equivalent parameterized squares. In this paper, we establish the optimal Fine--Wilf type bound for parameterized strings. Namely, if a string $s$ containing $\sigma$ distinct letters has parameterized periods $p$ and $q$ and satisfies $|s| \ge p+q+(\sigma-3)d+1$, where $d=\gcd(p,q)$, then $d$ is also a parameterized period of $s$. We also give matching lower-bound instances, proving that our bound is optimal for any $\sigma \geq 2$.

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arXiv2025

Constant sensitivity on the CDAWGs

Compact directed acyclic word graphs (CDAWGs) [Blumer et al. 1987] are a fundamental data structure on strings with applications in text pattern searching, data compression, and pattern discovery. Intuitively, the CDAWG of a string $T$ is obtained by merging isomorphic subtrees of the suffix tree [Weiner 1973] of the same string $T$, and thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation is performed at an arbitrary position in $T$. We show that the size of the CDAWG after an edit operation on $T$ is asymptotically at most 8 times larger than the original CDAWG before the edit.

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arXiv2024

Tight bounds on the number of non-equivalent parameterized squares in a word

Two words $x,y$ of the same length are said to be \emph{parameterized equivalent} if there exists a character bijection that transforms $x$ into $y$. A word $w$ is called a parameterized square if $w$ is a concatenation of two parameterized equivalent words. Kociumaka et al. [TCS 2016] showed that in a word of length $n$ that contains $\sigma$ distinct characters, the number of \emph{parameterized squares} that are non-equivalent with respect to parameterized equivalence is at most $2 \sigma! n$. 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. Moreover, we construct a family of words containing $\Omega(\sigma n)$ non-equivalent parameterized squares, which demonstrates that the upper bound is asymptotically tight.

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