Haruki Umezaki

Haruki Umezaki, Hiroki Shibata, Dominik Köppl et al.

Let $w$ be a string of length $n$. The problem of counting factors crossing a position -- Problem 64 from the textbook ``125 Problems in Text Algorithms'' [Crochemore, Lecroq, and Rytter, 2021] -- asks to count the number $\mathcal{C}(w,k)$ (resp. $\mathcal{N}(w,k)$) of distinct substrings in $w$ that have occurrences containing (resp. not containing) a position $k$ in $w$. The solutions provided in their textbook compute $\mathcal{C}(w,k)$ and $\mathcal{N}(w,k)$ in $O(n)$ time for a single position $k$ in $w$, and thus a direct application would require $O(n^2)$ time for all positions $k = 1, \ldots, n$ in $w$. Their solution is designed for constant-size alphabets. In this paper, we present new algorithms which compute $\mathcal{C}(w,k)$ in $O(n)$ total time for general ordered alphabets, and $\mathcal{N}(w,k)$ in $O(n)$ total time for linearly sortable alphabets,for all positions $k = 1, \ldots, n$ in $w$. We further derive model-dependent optimal bounds by separating the algorithms into preprocessing and linear-time postprocessing: for $\mathcal{C}$ the preprocessing is run reporting, and for $\mathcal{N}$ it is preprocessing based on longest previous non-overlapping factors (LPnF) and longest next factors (LNF). In particular, all values $\mathcal{C}(w,k)$ can be computed in $O(n\log n)$ time over general unordered alphabets in which direct accesses to alphabet characters are restricted to equality tests, and in $O(n\logσ)$ time in the word RAM model, where $σ$ denotes the number of distinct characters occurring in $w$. For $\mathcal{N}(w,k)$, the equality-testing complexity over general unordered alphabets is $Θ(n^2)$. We also show that our upper bounds are optimal for all of the aforementioned alphabet assumptions and computation models.

Hiroki Shibata, Haruki Umezaki, Takuya Mieno et al.

A non-empty string is closed if its length is one or its longest border appears exactly twice in the string. An occurrence of a closed substring is a maximal closed substring (MCS) if it cannot be extended to the left or to the right while preserving closedness. MCSs can be regarded as a general class of maximal repetitive structures including runs. In this paper, we study the computation of MCSs of a string given in an online manner, where one character is appended to the string at a time. Our algorithm detects newly formed MCSs after each append operation by using the rightmost previous occurrences of suffixes. To support this efficiently, we introduce the link-cut suffix tree (LCST), a novel data structure combining an online suffix tree with a link-cut tree. The LCST maintains rightmost occurrence information for substrings represented in the suffix tree in $O(n \log n)$ total time and $O(n)$ space, where $n$ is the length of the input string. Using the LCST, we obtain an $O(n \log n)$-time online algorithm for computing all MCSs, which is worst-case optimal. As further direct applications of the LCST, we obtain online algorithms for rightmost LZ77 factorizations and most recent match queries.