arXiv subjects

Takuya Mieno

Publications and source records attributed to Takuya Mieno.

36 records · Page 2Linked to original sources

arXiv2023

Data structures for computing unique palindromes in static and non-static strings

A palindromic substring $T[i.. j]$ of a string $T$ is said to be a shortest unique palindromic substring (SUPS) in $T$ for an interval $[p, q]$ if $T[i.. j]$ is a shortest palindromic substring such that $T[i.. j]$ occurs only once in $T$, and $[i, j]$ contains $[p, q]$. The SUPS problem is, given a string $T$ of length $n$, to construct a data structure that can compute all the SUPSs for any given query interval. It is known that any SUPS query can be answered in $O(α)$ time after $O(n)$-time preprocessing, where $α$ is the number of SUPSs to output [Inoue et al., 2018]. In this paper, we first show that $α$ is at most $4$, and the upper bound is tight. We also show that the total sum of lengths of minimal unique palindromic substrings of string $T$, which is strongly related to SUPSs, is $O(n)$. Then, we present the first $O(n)$-bits data structures that can answer any SUPS query in constant time. Also, we present an algorithm to solve the SUPS problem for a sliding window that can answer any query in $O(\log\log W)$ time and update data structures in amortized $O(\logσ+ \log\log W)$ time, where $W$ is the size of the window, and $σ$ is the alphabet size. Furthermore, we consider the SUPS problem in the after-edit model and present an efficient algorithm. Namely, we present an algorithm that uses $O(n)$ time for preprocessing and answers any $k$ SUPS queries in $O(\log n\log\log n + k\log\log n)$ time after single character substitution. Finally, as a by-product, we propose a fully-dynamic data structure for range minimum queries (RmQs) with a constraint where the width of each query range is limited to poly-logarithmic. The constrained RmQ data structure can answer such a query in constant time and support a single-element edit operation in amortized constant time.

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arXiv2023

Linear-time computation of generalized minimal absent words for multiple strings

A string $w$ is called a minimal absent word (MAW) for a string $S$ if $w$ does not occur as a substring in $S$ and all proper substrings of $w$ occur in $S$. MAWs are well-studied combinatorial string objects that have potential applications in areas including bioinformatics, musicology, and data compression. In this paper, we generalize the notion of MAWs to a set $\mathcal{S} = \{S_1, \ldots, S_k\}$ of multiple strings. We first describe our solution to the case of $k = 2$ strings, and show how to compute the set $\mathsf{M}$ of MAWs in optimal $O(n + |\mathsf{M}|)$ time and with $O(n)$ working space, where $n$ denotes the total length of the strings in $\mathcal{S}$. We then move on to the general case of $k > 2$ strings, and show how to compute the set $\mathsf{M}$ of MAWs in $O(n \lceil k / \log n \rceil + |\mathsf{M}|)$ time and with $O(n (k + \log n))$ bits of working space, in the word RAM model with machine word size $ω= \log n$. The latter algorithm runs in optimal $O(n + |\mathsf{M}|)$ time for $k = O(\log n)$.

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arXiv2023

Finding Top-k Longest Palindromes in Substrings

Palindromes are strings that read the same forward and backward. Problems of computing palindromic structures in strings have been studied for many years with a motivation of their application to biology. The longest palindrome problem is one of the most important and classical problems regarding palindromic structures, that is, to compute the longest palindrome appearing in a string $T$ of length $n$. The problem can be solved in $O(n)$ time by the famous algorithm of Manacher [Journal of the ACM, 1975]. This paper generalizes the longest palindrome problem to the problem of finding top-$k$ longest palindromes in an arbitrary substring, including the input string $T$ itself. The internal top-$k$ longest palindrome query is, given a substring $T[i..j]$ of $T$ and a positive integer $k$ as a query, to compute the top-$k$ longest palindromes appearing in $T[i.. j]$. This paper proposes a linear-size data structure that can answer internal top-$k$ longest palindromes query in optimal $O(k)$ time. Also, given the input string $T$, our data structure can be constructed in $O(n\log n)$ time. For $k = 1$, the construction time is reduced to $O(n)$.

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arXiv2022

Computing palindromes on a trie in linear time

A trie $\mathcal{T}$ is a rooted tree such that each edge is labeled by a single character from the alphabet, and the labels of out-going edges from the same node are mutually distinct. Given a trie $\mathcal{T}$ with $n$ edges, we show how to compute all distinct palindromes and all maximal palindromes on $\mathcal{T}$ in $O(n)$ time, in the case of integer alphabets of size polynomial in $n$. This improves the state-of-the-art $O(n \log h)$-time algorithms by Funakoshi et al. [PCS 2019], where $h$ is the height of $\mathcal{T}$. Using our new algorithms, the eertree with suffix links for a given trie $\mathcal{T}$ can readily be obtained in $O(n)$ time. Further, our trie-based $O(n)$-space data structure allows us to report all distinct palindromes and maximal palindromes in a query string represented in the trie $\mathcal{T}$, in output optimal time. This is an improvement over an existing (naïve) solution that precomputes and stores all distinct palindromes and maximal palindromes for each and every string in the trie $\mathcal{T}$ separately, using a total $O(n^2)$ preprocessing time and space, and reports them in output optimal time upon query.

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arXiv2022

Online algorithms for finding distinct substrings with length and multiple prefix and suffix conditions

Let two static sequences of strings $P$ and $S$, representing prefix and suffix conditions respectively, be given as input for preprocessing. For the query, let two positive integers $k_1$ and $k_2$ be given, as well as a string $T$ given in an online manner, such that $T_i$ represents the length-$i$ prefix of $T$ for $1 \leq i \leq |T|$. In this paper we are interested in computing the set $\mathit{ans_i}$ of distinct substrings $w$ of $T_i$ such that $k_1 \leq |w| \leq k_2$, and $w$ contains some $p \in P$ as a prefix and some $s \in S$ as a suffix. More specifically, the counting problem is to output $|\mathit{ans_i}|$, whereas the reporting problem is to output all elements of $\mathit{ans_i}$, for each iteration $i$. Let $σ$ denote the alphabet size, and for a sequence of strings $A$, $\Vert A\Vert=\sum_{u\in A}|u|$. Then, we show that after $O((\Vert P\Vert +\Vert S\Vert)\logσ)$-time preprocessing, the solutions for the counting and reporting problems for each iteration up to $i$ can be output in $O(|T_i| \logσ)$ and $O(|T_i| \logσ+ |\mathit{ans_i}|)$ total time. The preprocessing time can be reduced to $O(\Vert P\Vert +\Vert S\Vert)$ for integer alphabets of size polynomial with regard to $\Vert P\Vert +\Vert S\Vert$. Our algorithms have possible applications to network traffic classification.

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arXiv2022

Combinatorics of minimal absent words for a sliding window

A string $w$ is called a minimal absent word (MAW) for another string $T$ if $w$ does not occur in $T$ but the proper substrings of $w$ occur in $T$. For example, let $Σ= \{\mathtt{a, b, c}\}$ be the alphabet. Then, the set of MAWs for string $w = \mathtt{abaab}$ is $\{\mathtt{aaa, aaba, bab, bb, c}\}$. In this paper, we study combinatorial properties of MAWs in the sliding window model, namely, how the set of MAWs changes when a sliding window of fixed length $d$ is shifted over the input string $T$ of length $n$, where $1 \leq d < n$. We present \emph{tight} upper and lower bounds on the maximum number of changes in the set of MAWs for a sliding window over $T$, both in the cases of general alphabets and binary alphabets. Our bounds improve on the previously known best bounds [Crochemore et al., 2020].

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arXiv2022

Cartesian Tree Subsequence Matching

Park et al. [TCS 2020] observed that the similarity between two (numerical) strings can be captured by the Cartesian trees: The Cartesian tree of a string is a binary tree recursively constructed by picking up the smallest value of the string as the root of the tree. Two strings of equal length are said to Cartesian-tree match if their Cartesian trees are isomorphic. Park et al. [TCS 2020] introduced the following Cartesian tree substring matching (CTMStr) problem: Given a text string $T$ of length $n$ and a pattern string of length $m$, find every consecutive substring $S = T[i..j]$ of a text string $T$ such that $S$ and $P$ Cartesian-tree match. They showed how to solve this problem in $\tilde{O}(n+m)$ time. In this paper, we introduce the Cartesian tree subsequence matching (CTMSeq) problem, that asks to find every minimal substring $S = T[i..j]$ of $T$ such that $S$ contains a subsequence $S'$ which Cartesian-tree matches $P$. We prove that the CTMSeq problem can be solved efficiently, in $O(m n p(n))$ time, where $p(n)$ denotes the update/query time for dynamic predecessor queries. By using a suitable dynamic predecessor data structure, we obtain $O(mn \log \log n)$-time and $O(n \log m)$-space solution for CTMSeq. This contrasts CTMSeq with closely related order-preserving subsequence matching (OPMSeq) which was shown to be NP-hard by Bose et al. [IPL 1998].

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arXiv2022

RePair Grammars are the Smallest Grammars for Fibonacci Words

Grammar-based compression is a loss-less data compression scheme that represents a given string $w$ by a context-free grammar that generates only $w$. While computing the smallest grammar which generates a given string $w$ is NP-hard in general, a number of polynomial-time grammar-based compressors which work well in practice have been proposed. RePair, proposed by Larsson and Moffat in 1999, is a grammar-based compressor which recursively replaces all possible occurrences of a most frequently occurring bigrams in the string. Since there can be multiple choices of the most frequent bigrams to replace, different implementations of RePair can result in different grammars. In this paper, we show that the smallest grammars generating the Fibonacci words $F_k$ can be completely characterized by RePair, where $F_k$ denotes the $k$-th Fibonacci word. Namely, all grammars for $F_k$ generated by any implementation of RePair are the smallest grammars for $F_k$, and no other grammars can be the smallest for $F_k$. To the best of our knowledge, Fibonacci words are the first non-trivial infinite family of strings for which RePair is optimal.

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arXiv2022

Minimal Absent Words on Run-Length Encoded Strings

A string $w$ is called a minimal absent word (MAW) for another string $T$ if $w$ does not occur (as a substring) in $T$ and any proper substring of $w$ occurs in $T$. State-of-the-art data structures for reporting the set $\mathsf{MAW}(T)$ of MAWs from a given string $T$ of length $n$ require $O(n)$ space, can be built in $O(n)$ time, and can report all MAWs in $O(|\mathsf{MAW}(T)|)$ time upon a query. This paper initiates the problem of computing MAWs from a compressed representation of a string. In particular, we focus on the most basic compressed representation of a string, run-length encoding (RLE), which represents each maximal run of the same characters $a$ by $a^p$ where $p$ is the length of the run. Let $m$ be the RLE-size of string $T$. After categorizing the MAWs into five disjoint sets $\mathcal{M}_1$, $\mathcal{M}_2$, $\mathcal{M}_3$, $\mathcal{M}_4$, $\mathcal{M}_5$ using RLE, we present matching upper and lower bounds for the number of MAWs in $\mathcal{M}_i$ for $i = 1,2,4,5$ in terms of RLE-size $m$, except for $\mathcal{M}_3$ whose size is unbounded by $m$. We then present a compact $O(m)$-space data structure that can report all MAWs in optimal $O(|\mathsf{MAW}(T)|)$ time.

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arXiv2021

Minimal unique palindromic substrings after single-character substitution

A palindrome is a string that reads the same forward and backward. A palindromic substring $w$ of a string $T$ is called a minimal unique palindromic substring (MUPS) of $T$ if $w$ occurs only once in $T$ and any proper palindromic substring of $w$ occurs at least twice in $T$. MUPSs are utilized for answering the shortest unique palindromic substring problem, which is motivated by molecular biology [Inoue et al., 2018]. Given a string $T$ of length $n$, all MUPSs of $T$ can be computed in $O(n)$ time. In this paper, we study the problem of updating the set of MUPSs when a character in the input string $T$ is substituted by another character. We first analyze the number $d$ of changes of MUPSs when a character is substituted, and show that $d$ is in $O(\log n)$. Further, we present an algorithm that uses $O(n)$ time and space for preprocessing, and updates the set of MUPSs in $O(\logσ+ (\log\log n)^2 + d)$ time where $σ$ is the alphabet size. We also propose a variant of the algorithm, which runs in optimal $O(1+d)$ time when the alphabet size is constant.

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arXiv2021

On the approximation ratio of LZ-End to LZ77

A family of Lempel-Ziv factorizations is a well-studied string structure. The LZ-End factorization is a member of the family that achieved faster extraction of any substrings (Kreft & Navarro, TCS 2013). One of the interests for LZ-End factorizations is the possible difference between the size of LZ-End and LZ77 factorizations. They also showed families of strings where the approximation ratio of the number of LZ-End phrases to the number of LZ77 phrases asymptotically approaches 2. However, the alphabet size of these strings is unbounded. In this paper, we analyze the LZ-End factorization of the period-doubling sequence. We also show that the approximation ratio for the period-doubling sequence asymptotically approaches 2 for the binary alphabet.

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arXiv2021

A Separation of $γ$ and $b$ via Thue--Morse Words

We prove that for $n\geq 2$, the size $b(t_n)$ of the smallest bidirectional scheme for the $n$th Thue--Morse word $t_n$ is $n+2$. Since Kutsukake et al. [SPIRE 2020] show that the size $γ(t_n)$ of the smallest string attractor for $t_n$ is $4$ for $n \geq 4$, this shows for the first time that there is a separation between the size of the smallest string attractor $γ$ and the size of the smallest bidirectional scheme $b$, i.e., there exist string families such that $γ= o(b)$.

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arXiv2020

Palindromic Trees for a Sliding Window and Its Applications

The palindromic tree (a.k.a. eertree) for a string $S$ of length $n$ is a tree-like data structure that represents the set of all distinct palindromic substrings of $S$, using $O(n)$ space [Rubinchik and Shur, 2018]. It is known that, when $S$ is over an alphabet of size $σ$ and is given in an online manner, then the palindromic tree of $S$ can be constructed in $O(n\logσ)$ time with $O(n)$ space. In this paper, we consider the sliding window version of the problem: For a sliding window of length at most $d$, we present two versions of an algorithm which maintains the palindromic tree of size $O(d)$ for every sliding window $S[i..j]$ over $S$, where $1 \leq j-i+1 \leq d$. The first version works in $O(n\logσ')$ time with $O(d)$ space where $σ' \leq d$ is the maximum number of distinct characters in the windows, and the second one works in $O(n + dσ)$ time with $(d+2)σ+ O(d)$ space. We also show how our algorithms can be applied to efficient computation of minimal unique palindromic substrings (MUPS) and minimal absent palindromic words (MAPW) for a sliding window.

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arXiv2020

Space-Efficient Algorithms for Computing Minimal/Shortest Unique Substrings

Given a string $T$ of length $n$, a substring $u = T[i..j]$ of $T$ is called a shortest unique substring (SUS) for an interval $[s,t]$ if (a) $u$ occurs exactly once in $T$, (b) $u$ contains the interval $[s,t]$ (i.e. $i \leq s \leq t \leq j$), and (c) every substring $v$ of $T$ with $|v| < |u|$ containing $[s,t]$ occurs at least twice in $T$. Given a query interval $[s, t] \subset [1, n]$, the interval SUS problem is to output all the SUSs for the interval $[s,t]$. In this article, we propose a $4n + o(n)$ bits data structure answering an interval SUS query in output-sensitive $O(\mathit{occ})$ time, where $\mathit{occ}$ is the number of returned SUSs. Additionally, we focus on the point SUS problem, which is the interval SUS problem for $s = t$. Here, we propose a $\lceil (\log_2{3} + 1)n \rceil + o(n)$ bits data structure answering a point SUS query in the same output-sensitive time. We also propose space-efficient algorithms for computing the minimal unique substrings of $T$.

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arXiv2020

Lyndon Words, the Three Squares Lemma, and Primitive Squares

We revisit the so-called "Three Squares Lemma" by Crochemore and Rytter [Algorithmica 1995] and, using arguments based on Lyndon words, derive a more general variant which considers three overlapping squares which do not necessarily share a common prefix. We also give an improved upper bound of $n\log_2 n$ on the maximum number of (occurrences of) primitively rooted squares in a string of length $n$, also using arguments based on Lyndon words. To the best of our knowledge, the only known upper bound was $n \log_ϕn \approx 1.441n\log_2 n$, where $ϕ$ is the golden ratio, reported by Fraenkel and Simpson [TCS 1999] obtained via the Three Squares Lemma.

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arXiv2019

Minimal Unique Substrings and Minimal Absent Words in a Sliding Window

A substring $u$ of a string $T$ is called a minimal unique substring (MUS) of $T$ if $u$ occurs exactly once in $T$ and any proper substring of $u$ occurs at least twice in $T$. A string $w$ is called a minimal absent word (MAW) of $T$ if $w$ does not occur in $T$ and any proper substring of $w$ occurs in $T$. In this paper, we study the problems of computing MUSs and MAWs in a sliding window over a given string $T$. We first show how the set of MUSs can change in a sliding window over $T$, and present an $O(n\logσ)$-time and $O(d)$-space algorithm to compute MUSs in a sliding window of width $d$ over $T$, where $σ$ is the maximum number of distinct characters in every window. We then give tight upper and lower bounds on the maximum number of changes in the set of MAWs in a sliding window over $T$. Our bounds improve on the previous results in [Crochemore et al., 2017].

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arXiv2017

Shortest unique palindromic substring queries in optimal time

A palindrome is a string that reads the same forward and backward. A palindromic substring $P$ of a string $S$ is called a shortest unique palindromic substring ($\mathit{SUPS}$) for an interval $[x, y]$ in $S$, if $P$ occurs exactly once in $S$, this occurrence of $P$ contains interval $[x, y]$, and every palindromic substring of $S$ which contains interval $[x, y]$ and is shorter than $P$ occurs at least twice in $S$. The $\mathit{SUPS}$ problem is, given a string $S$, to preprocess $S$ so that for any subsequent query interval $[x, y]$ all the $\mathit{SUPS}\mbox{s}$ for interval $[x, y]$ can be answered quickly. We present an optimal solution to this problem. Namely, we show how to preprocess a given string $S$ of length $n$ in $O(n)$ time and space so that all $\mathit{SUPS}\mbox{s}$ for any subsequent query interval can be answered in $O(k+1)$ time, where $k$ is the number of outputs.

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arXiv2017

Tight bounds on the maximum number of shortest unique substrings

A substring Q of a string S is called a shortest unique substring (SUS) for interval [s,t] in S, if Q occurs exactly once in S, this occurrence of Q contains interval [s,t], and every substring of S which contains interval [s,t] and is shorter than Q occurs at least twice in S. The SUS problem is, given a string S, to preprocess S so that for any subsequent query interval [s,t] all the SUSs for interval [s,t] can be answered quickly. When s = t, we call the SUSs for [s,t] as point SUSs, and when s \leq t, we call the SUSs for [s,t] as interval SUSs. There exist optimal O(n)-time preprocessing scheme which answers queries in optimal O(k) time for both point and interval SUSs, where n is the length of S and k is the number of outputs for a given query. In this paper, we reveal structural, combinatorial properties underlying the SUS problem: Namely, we show that the number of intervals in S that correspond to point SUSs for all query positions in S is less than 1.5n, and show that this is a matching upper and lower bound. Also, we consider the maximum number of intervals in S that correspond to interval SUSs for all query intervals in S.

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