Peaker Guo

Kaisei Kishi, Peaker Guo, Cristian Urbina et al.

A \emph{morphism} is a mapping that transforms words through letter-wise substitution, where each symbol is consistently replaced by a fixed word. In the field of combinatorics on words, one topic that has attracted considerable attention is the characterization of morphisms that preserve specific properties, such as overlap-freeness, square-freeness, lexicographic order, and primitivity. Continuing this direction, we initiate the study on \emph{occurrence-preserving morphisms}, which address the following fundamental question: given a morphism $ϕ$, two words $u$ and $v$, and $k \geq 1$, under what conditions does the number of occurrences of $u$ in $v$ equal the number of occurrences of $ϕ^k(u)$ in $ϕ^k(v)$? To answer this question, we introduce the notion of \emph{interference-free morphisms}, examine their properties, develop an efficient algorithm for deciding interference-freeness, and uncover a connection to \emph{recognizable morphisms}. We then present a precise characterization of occurrence-preserving morphisms in terms of interference-freeness. As applications of our characterization, we first show that there exists a bijection between the starting positions of the occurrences of $u$ in $v$ and those of $ϕ^k(u)$ in $ϕ^k(v)$. We then apply the characterization to the Fibonacci and Thue-Morse words to identify their \emph{minimal unique substrings~(MUSs)}. Finally, we exploit the connection between MUSs and \emph{net occurrences} to simplify existing proofs on net occurrences in these words.

Peaker Guo, Rayne Holland, Hao Wu

Given a dataset of $n$ user-contributed strings, each of length at most $\ell$, a key problem is how to identify all frequent substrings while preserving each user's privacy. Recent work by Bernardini et al. (PODS'25) introduced a $\varepsilon$-differentially private algorithm achieving near-optimal error, but at the prohibitive cost of $O(n^2\ell^4)$ space and processing time. In this work, we present a new $\varepsilon$-differentially private algorithm that retains the same near-optimal error guarantees while reducing space complexity to $O(n \ell+ |Σ| )$ and time complexity to $O(n \ell\log |Σ| + |Σ| )$, for input alphabet $Σ$. Our approach builds on a top-down exploration of candidate substrings but introduces two new innovations: (i) a refined candidate-generation strategy that leverages the structural properties of frequent prefixes and suffixes, and (ii) pruning of the search space guided by frequency relations. These techniques eliminate the quadratic blow-ups inherent in prior work, enabling scalable frequent substring mining under differential privacy.