arXiv subjects

Yuto Nakashima

Publications and source records attributed to Yuto Nakashima.

At least 19 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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arXiv2026

Compact Enumeration of Maximal Closed Substrings in Run-Length Encoded Strings

A string $w$ is closed if $|w|=1$, or if $w$ has a non-empty proper border occurring only as its prefix and suffix. A maximal closed substring (MCS) is a maximal occurrence of a closed string; equivalently, it is a maximal closed repeat (MCR). We study the problem of enumerating all MCS occurrences directly from the run-length encoding (RLE) of a string. For a string $T$ of length $n$ with RLE size $m$, we give a compact representation of all MCS occurrences whose worst-case size is $O(m^2)$, and show that this bound is tight for this representation. Our approach is based on a characterization of MCS occurrences in terms of consecutive occurrences of their longest borders, together with data structures built on the RLE of $T$. Denoting the resulting representation by $\mathcal{F}$, we compute it in $O(m\log^2 m + |\mathcal{F}|\log m)$ time using $O(m)$ working space.

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arXiv2026

Online computation of maximal closed substrings

A non-empty string is closed if it has length 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 occurrence of each suffix. 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.

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arXiv2026

Sensitivity of Repetitiveness Measures to String Reversal

We study the impact that string reversal can have on several repetitiveness measures. First, we exhibit an infinite family of strings where the number, $r$, of runs in the run-length encoding of the Burrows--Wheeler transform (BWT) can increase additively by $Θ(n)$ when reversing the string. This substantially improves the known $Ω(\log n)$ lower-bound for the additive sensitivity of $r$ and it is asymptotically tight. We generalize our result to other variants of the BWT, including the variant with an appended end-of-string symbol and the bijective BWT. We show that an analogous result holds for the size $z$ of the Lempel--Ziv 77 (LZ) parsing of the text, and also for some of its variants, including the non-overlapping LZ parsing, and the LZ-end parsing. Moreover, we describe a family of strings for which the ratio $z(w^R)/z(w)$ approaches $3$ from below as $|w|\rightarrow \infty$. We also show an asymptotically tight lower-bound of $Θ(n)$ for the additive sensitivity of the size $v$ of the smallest lexicographic parsing to string reversal. Finally, we show that the multiplicative sensitivity of $v$ to reversing the string is $Θ(\log n)$, and this lower-bound is also tight. Overall, our results expose the limitations of repetitiveness measures that are widely used in practice, against string reversal -- a simple and natural data transformation.

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arXiv2026

Online Computation of Palindromes and Suffix Trees on Tries

We consider the problems of computing maximal palindromes and distinct palindromes in a trie. A trie is a natural generalization of a string, which can be seen as a single-path tree. There is a linear-time offline algorithm to compute maximal palindromes and distinct palindromes in a given (static) trie whose edge-labels are drawn from a linearly-sortable alphabet [Mieno et al., ISAAC 2022]. In this paper, we tackle problems of palindrome enumeration on dynamic tries which support leaf additions and leaf deletions. We propose the first sub-quadratic algorithms to enumerate palindromes in a dynamic trie. For maximal palindromes, we propose an algorithm that runs in $O(N \min(\log h, σ))$ time and uses $O(N)$ space, where $N$ is the maximum number of edges in the trie, $σ$ is the size of the alphabet, and $h$ is the height of the trie. For distinct palindromes, we develop several online algorithms based on different algorithmic frameworks, including approaches using the EERTREE (a.k.a. palindromic tree) and the suffix tree of a trie. These algorithms support leaf insertions and deletions in the trie and achieve different time and space trade-offs. Furthermore, as a by-product, we present online algorithms to construct the suffix tree and the EERTREE of the input trie, which is of independent interest.

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arXiv2026

LZBE: an LZ-style compressor supporting $O(\log n)$-time random access

An LZ-like factorization of a string divides it into factors, each being either a single character or a copy of a preceding substring. While grammar-based compression schemes support efficient random access with space linear in the compressed size, no comparable guarantees are known for general LZ-like factorizations. This limitation motivated restricted variants such as LZ-End [Kreft and Navarro, 2013] and height-bounded LZ (LZHB) [Bannai et al., 2024], which trade off some compression efficiency for faster access. In this paper, we introduce LZ-Begin-End (LZBE), a new LZ-like variant in which every copy factor must refer to a contiguous sequence of preceding factors. This structural restriction ensures that any context-free grammar can be transformed into an LZBE factorization of the same size. We further study the greedy LZBE factorization, which selects each copy factor to be as long as possible while processing the input from left to right, and show that it can be computed in linear time. Moreover, we exhibit a family of strings for which the greedy LZBE factorization is asymptotically smaller than the smallest grammar. These results demonstrate that the LZBE scheme is strictly more expressive than grammar-based compression in the worst case. To support fast queries, we propose a data structure for LZBE-compressed strings that permits O(log n)-time random access within space linear in the compressed size, where n is the length of the input string.

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arXiv2025

Subsequence Matching and LCS under Cartesian-Tree Equivalence

Two strings of the same length are said to Cartesian-tree match (CT-match) if their Cartesian-trees are isomorphic [Park et al., TCS 2020]. Cartesian-tree matching is a natural model that allows for capturing similarities of numerical sequences. Oizumi et al. [CPM 2022] showed that subsequence pattern matching under CT-matching model (CT-MSeq) can be solved in $O(nm \log \log n)$ time, where $n$ and $m$ are text and pattern lengths, respectively. This current article follows this line of research, and gives the following new results: (1) An $O(nm)$-time CT-MSeq algorithm for binary alphabets; (2) An $O((nm)^{1-ε})$-time conditional lower bound for the CT-MSeq problem on alphabets of size 4, for any constant $ε> 0$, under the Orthogonal Vector Hypothesis (OVH). Further, we introduce the new problem of longest common subsequence under CT-matching (CT-LCS) for two given strings $S$ and $T$ of length $n$, and present the following results: (3) An $O(n^6)$-time CT-LCS algorithm for general ordered alphabets; (4) An $O(n^2 / \log n)$-time CT-LCS algorithm for binary alphabets; (5) An $O(n^{2-ε})$-time conditional lower bound for the CT-LCS problem on alphabets of size 5, for any constant $ε> 0$, under OVH.

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arXiv2025

Nyldon Factorization of Thue-Morse Words and Fibonacci Words

The Nyldon factorization is a string factorization that is a non-decreasing product of Nyldon words. Nyldon words and Nyldon factorizations are recently defined combinatorial objects inspired by the well-known Lyndon words and Lyndon factorizations. In this paper, we investigate the Nyldon factorization of several words. First, we fully characterize the Nyldon factorizations of the (finite) Fibonacci and the (finite) Thue-Morse words. Moreover, we show that there exists a non-decreasing product of Nyldon words that is a factorization of the infinite Thue-Morse word.

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arXiv2025

Fast Computation of $k$-Runs, Parameterized Squares, and Other Generalised Squares

A $k$-mismatch square is a string of the form $XY$ where $X$ and $Y$ are two equal-length strings that have at most $k$ mismatches. Kolpakov and Kucherov [Theor. Comput. Sci., 2003] defined two notions of $k$-mismatch repeats, called $k$-repetitions and $k$-runs, each representing a sequence of consecutive $k$-mismatch squares of equal length. They proposed algorithms for computing $k$-repetitions and $k$-runs working in $O(nk \log k + output)$ time for a string of length $n$ over an integer alphabet, where $output$ is the number of the reported repeats. We show that $output=O(nk \log k)$, both in case of $k$-repetitions and $k$-runs, which implies that the complexity of their algorithms is actually $O(nk \log k)$. We apply this result to computing parameterized squares. A parameterized square is a string of the form $XY$ such that $X$ and $Y$ parameterized-match, i.e., there exists a bijection $f$ on the alphabet such that $f(X) = Y$. Two parameterized squares $XY$ and $X'Y'$ are equivalent if they parameterized match. Recently Hamai et al. [SPIRE 2024] showed that a string of length $n$ over an alphabet of size $σ$ contains less than $nσ$ non-equivalent parameterized squares, improving an earlier bound by Kociumaka et al. [Theor. Comput. Sci., 2016]. We apply our bound for $k$-mismatch repeats to propose an algorithm that reports all non-equivalent parameterized squares in $O(nσ\log σ)$ time. We also show that the number of non-equivalent parameterized squares can be computed in $O(n \log n)$ time. This last algorithm applies to squares under any substring compatible equivalence relation and also to counting squares that are distinct as strings. In particular, this improves upon the $O(nσ)$-time algorithm of Gawrychowski et al. [CPM 2023] for counting order-preserving squares that are distinct as strings if $σ= ω(\log n)$.

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arXiv2025

Counting Distinct (Non-)Crossing Substrings in Optimal Time

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 $O(n\log\sigma)$ time in the word RAM model, where $\sigma$ denotes the number of distinct characters occurring in $w$. For $\mathcal{N}(w,k)$, the equality-testing complexity over general unordered alphabets is $\Theta(n^2)$. We also show that our upper bounds are optimal for all of the aforementioned alphabet assumptions and computation models.

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arXiv2025

Tight Additive Sensitivity on LZ-style Compressors and String Attractors

The worst-case additive sensitivity of a string repetitiveness measure $c$ is defined to be the largest difference between $c(w)$ and $c(w')$, where $w$ is a string of length $n$ and $w'$ is a string that can be obtained by performing a single-character edit operation on $w$. We present $O(\sqrt{n})$ upper bounds for the worst-case additive sensitivity of the smallest string attractor size $γ$ and the smallest bidirectional scheme size $b$, which match the known lower bounds $Ω(\sqrt{n})$ for $γ$ and $b$ [Akagi et al. 2023]. Further, we present matching upper and lower bounds for the worst-case additive sensitivity of the Lempel-Ziv family - $Θ(n^{\frac{2}{3}})$ for LZSS and LZ-End, and $Θ(n)$ for LZ78.

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arXiv2025

Computing maximal palindromes in non-standard matching models

Palindromes are popular and important objects in textual data processing, bioinformatics, and combinatorics on words. Let $S = XaY$ be a string where $X$ and $Y$ are of the same length, and $a$ is either a single character or the empty string. Then, there exist two alternative definitions for palindromes: $S$ is said to be a palindrome if $S$ is equal to its reversal $S^R$ (Reversal-based definition); or if its right-arm $Y$ is equal to the reversal of its left-arm $X^R$ (Symmetry-based definition). It is clear that if the ``equality'' ($\approx$) used in both definitions is exact character matching ($=$), then the two definitions are the same. However, if we apply other string-equality criteria $\approx$, including the complementary-matching model for biological sequences, the Cartesian-tree model [Park et al., TCS 2020], the parameterized model [Baker, JCSS 1996], the order-preserving model [Kim et al., TCS 2014], and the palindromic-structure model [I et al., TCS 2013], then are the reversal-based palindromes and the symmetry-based palindromes the same? To the best of our knowledge, no previous work has considered or answered this natural question. In this paper, we first provide answers to this question, and then present efficient algorithms for computing all maximal palindromes under the non-standard matching models in a given string. After confirming that Gusfield's offline suffix-tree-based algorithm for computing maximal symmetry-based palindromes can be readily extended to the aforementioned matching models, we show how to extend Manacher's online algorithm for computing maximal reversal-based palindromes in linear time for all the aforementioned matching models.

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arXiv2025

Tight bounds for the sensitivity of CDAWGs with left-end edits

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 $T$, thus CDAWGs are a compact indexing structure. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation (insertion, deletion, or substitution) is performed at the left-end of the input string $T$, namely, we are interested in the worst-case increase in the size of the CDAWG after a left-end edit operation. We prove that if $e$ is the number of edges of the CDAWG for string $T$, then the number of new edges added to the CDAWG after a left-end edit operation on $T$ does not exceed $e$. Further, we present a matching lower bound on the sensitivity of CDAWGs for left-end insertions, and almost matching lower bounds for left-end deletions and substitutions. We then generalize our lower-bound instance for left-end insertions to leftward online construction of the CDAWG, and show that it requires $Ω(n^2)$ time for some string of length $n$.

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arXiv2024

On the compressiveness of the Burrows-Wheeler transform

The Burrows-Wheeler transform (BWT) is a reversible transform that converts a string $w$ into another string $\mathsf{BWT}(w)$. The size of the run-length encoded BWT (RLBWT) can be interpreted as a measure of repetitiveness in the class of representations called dictionary compression which are essentially representations based on copy and paste operations. In this paper, we shed new light on the compressiveness of BWT and the bijective BWT (BBWT). We first extend previous results on the relations of their run-length compressed sizes $r$ and $r_B$. We also show that the so-called ``clustering effect'' of BWT and BBWT can be captured by measures other than empirical entropy or run-length encoding. In particular, we show that BWT and BBWT do not increase the repetitiveness of the string with respect to various measures based on dictionary compression by more than a polylogarithmic factor. Furthermore, we show that there exists an infinite family of strings that are maximally incompressible by any dictionary compression measure, but become very compressible after applying BBWT. An interesting implication of this result is that it is possible to transcend dictionary compression in some cases by simply applying BBWT before applying dictionary compression.

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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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arXiv2024

Faster Space-Efficient STR-IC-LCS Computation

One of the most fundamental method for comparing two given strings $A$ and $B$ is the longest common subsequence (LCS), where the task is to find (the length) of an LCS of $A$ and $B$. In this paper, we deal with the STR-IC-LCS problem which is one of the constrained LCS problems proposed by Chen and Chao [J. Comb. Optim, 2011]. A string $Z$ is said to be an STR-IC-LCS of three given strings $A$, $B$, and $P$, if $Z$ is a longest string satisfying that (1) $Z$ includes $P$ as a substring and (2) $Z$ is a common subsequence of $A$ and $B$. We present three efficient algorithms for this problem: First, we begin with a space-efficient solution which computes the length of an STR-IC-LCS in $O(n^2)$ time and $O((\ell+1)(n-\ell+1))$ space, where $\ell$ is the length of an LCS of $A$ and $B$ of length $n$. When $\ell = O(1)$ or $n-\ell = O(1)$, then this algorithm uses only linear $O(n)$ space. Second, we present a faster algorithm that works in $O(nr/\log{r}+n(n-\ell+1))$ time, where $r$ is the length of $P$, while retaining the $O((\ell+1)(n-\ell+1))$ space efficiency. Third, we give an alternative algorithm that runs in $O(nr/\log{r}+n(n-\ell'+1))$ time with $O((\ell'+1)(n-\ell'+1))$ space, where $\ell'$ denotes the STR-IC-LCS length for input strings $A$, $B$, and $P$.

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arXiv2024

Edit and Alphabet-Ordering Sensitivity of Lex-parse

We investigate the compression sensitivity [Akagi et al., 2023] of lex-parse [Navarro et al., 2021] for two operations: (1) single character edit and (2) modification of the alphabet ordering, and give tight upper and lower bounds for both operations. For both lower bounds, we use the family of Fibonacci words. For the bounds on edit operations, our analysis makes heavy use of properties of the Lyndon factorization of Fibonacci words to characterize the structure of lex-parse.

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arXiv2024

SwipeGANSpace: Swipe-to-Compare Image Generation via Efficient Latent Space Exploration

Generating preferred images using generative adversarial networks (GANs) is challenging owing to the high-dimensional nature of latent space. In this study, we propose a novel approach that uses simple user-swipe interactions to generate preferred images for users. To effectively explore the latent space with only swipe interactions, we apply principal component analysis to the latent space of the StyleGAN, creating meaningful subspaces. We use a multi-armed bandit algorithm to decide the dimensions to explore, focusing on the preferences of the user. Experiments show that our method is more efficient in generating preferred images than the baseline methods. Furthermore, changes in preferred images during image generation or the display of entirely different image styles were observed to provide new inspirations, subsequently altering user preferences. This highlights the dynamic nature of user preferences, which our proposed approach recognizes and enhances.

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