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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.

A string $s$ is called a parameterized square when $s = xy$ for strings $x$, $y$ and $x$ and $y$ are parameterized equivalent. Kociumaka et al. showed the number of parameterized squares, which are non-equivalent in parameterized equivalence, in a string of length $n$ that contains $\sigma$ distinct characters is at most $2 \sigma! n$ [TCS 2016]. 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., Comment: Accepted for SPIRE 2024

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., Comment: 11 pages, 13 figures

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.

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 can be solved in polynomial time. This current article follows and extends this line of research: We present the first polynomial-time algorithm that finds the longest common subsequence under CT-matching of two given strings $S$ and $T$ of length $n$, in $O(n^6)$ time and $O(n^4)$ space for general ordered alphabets. We then show that the problem has a faster solution in the binary case, by presenting an $O(n^2 / \log n)$-time and space algorithm.

In this paper, we present the first study of the computational complexity of converting an automata-based text index structure, called the Compact Directed Acyclic Word Graph (CDAWG), of size $e$ for a text $T$ of length $n$ into other text indexing structures for the same text, suitable for highly repetitive texts: the run-length BWT of size $r$, the irreducible PLCP array of size $r$, and the quasi-irreducible LPF array of size $e$, as well as the lex-parse of size $O(r)$ and the LZ77-parse of size $z$, where $r, z \le e$. As main results, we showed that the above structures can be optimally computed from either the CDAWG for $T$ stored in read-only memory or its self-index version of size $e$ without a text in $O(e)$ worst-case time and words of working space. To obtain the above results, we devised techniques for enumerating a particular subset of suffixes in the lexicographic and text orders using the forward and backward search on the CDAWG by extending the results by Belazzougui et al. in 2015., Comment: The short version of this paper will appear in SPIRE 2023, Pisa, Italy, September 26-28, 2023, Lecture Notes in Computer Science, Springer

We consider labeled directed graphs where each vertex is labeled with a non-empty string. Such labeled graphs are also known as non-linear texts in the literature. In this paper, we introduce a new problem of comparing two given labeled graphs, called the SEQ-IC-LCS problem on labeled graphs. The goal of SEQ-IC-LCS is to compute the the length of the longest common subsequence (LCS) $Z$ of two target labeled graphs $G_1 = (V_1, E_1)$ and $G_2 = (V_2, E_2)$ that includes some string in the constraint labeled graph $G_3 = (V_3, E_3)$ as its subsequence. Firstly, we consider the case where $G_1$, $G_2$ and $G_3$ are all acyclic, and present algorithms for computing their SEQ-IC-LCS in $O(|E_1||E_2||E_3|)$ time and $O(|V_1||V_2||V_3|)$ space. Secondly, we consider the case where $G_1$ and $G_2$ can be cyclic and $G_3$ is acyclic, and present algorithms for computing their SEQ-IC-LCS in $O(|E_1||E_2||E_3| + |V_1||V_2||V_3|\log|\Sigma|)$ time and $O(|V_1||V_2||V_3|)$ space, where $\Sigma$ is the alphabet., Comment: Accepted for PSC 2023

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 $\omega = \log n$. The latter algorithm runs in optimal $O(n + |\mathsf{M}|)$ time for $k = O(\log n)$., Comment: Accepted for SPIRE 2023

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 $\Omega(n^2)$ time for some string of length $n$., Comment: This is a full version of the paper that appeared in the proceedings of WORDS 2023. Accepted for Acta Informatica

LZ-End is a variant of the well-known Lempel-Ziv parsing family such that each phrase of the parsing has a previous occurrence, with the additional constraint that the previous occurrence must end at the end of a previous phrase. LZ-End was initially proposed as a greedy parsing, where each phrase is determined greedily from left to right, as the longest factor that satisfies the above constraint~[Kreft & Navarro, 2010]. In this work, we consider an optimal LZ-End parsing that has the minimum number of phrases in such parsings. We show that a decision version of computing the optimal LZ-End parsing is NP-complete by showing a reduction from the vertex cover problem. Moreover, we give a MAX-SAT formulation for the optimal LZ-End parsing adapting an approach for computing various NP-hard repetitiveness measures recently presented by [Bannai et al., 2022]. We also consider the approximation ratio of the size of greedy LZ-End parsing to the size of the optimal LZ-End parsing, and give a lower bound of the ratio which asymptotically approaches $2$.

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 $P$., Comment: This is a full version of "Space-Efficient STR-IC-LCS Computation" presented at SOFSEM 2023

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: Reversal-based definition: $S$ is equal to its reversal $S^R$; Symmetry-based definition: its left-arm $X$ is equal to the reversal of its right-arm $Y^R$. 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 model for biological sequences, the parameterized model [Baker, JCSS 1996], the order-preserving model [Kim et al., TCS 2014], the Cartesian-tree model [Park et al., TCS 2020], 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 generalized palindromes that occur 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.

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., Comment: Accepted for CPM 2022

The Cartesian-tree pattern matching is a recently introduced scheme of pattern matching that detects fragments in a sequential data stream which have a similar structure as a query pattern. Formally, Cartesian-tree pattern matching seeks all substrings $S'$ of the text string $S$ such that the Cartesian tree of $S'$ and that of a query pattern $P$ coincide. In this paper, we present a new indexing structure for this problem called the Cartesian-tree Position Heap (CPH). Let $n$ be the length of the input text string $S$, $m$ the length of a query pattern $P$, and $\sigma$ the alphabet size. We show that the CPH of $S$, denoted $\mathsf{CPH}(S)$, supports pattern matching queries in $O(m (\sigma + \log (\min\{h, m\})) + occ)$ time with $O(n)$ space, where $h$ is the height of the CPH and $occ$ is the number of pattern occurrences. We show how to build $\mathsf{CPH}(S)$ in $O(n \log \sigma)$ time with $O(n)$ working space. Further, we extend the problem to the case where the text is a labeled tree (i.e. a trie). Given a trie $T$ with $N$ nodes, we show that the CPH of $T$, denoted $\mathsf{CPH}(T)$, supports pattern matching queries on the trie in $O(m (\sigma^2 + \log (\min\{h, m\})) + occ)$ time with $O(N \sigma)$ space. We also show a construction algorithm for $\mathsf{CPH}(T)$ running in $O(N \sigma)$ time and $O(N \sigma)$ working space.

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.

Counting substrings/subsequences that preserve some property (e.g., palindromes, squares) is an important mathematical interest in stringology. Recently, Glen et al. studied the number of Lyndon factors in a string. A string $w = uv$ is called a Lyndon word if it is the lexicographically smallest among all of its conjugates $vu$. In this paper, we consider a more general problem "counting Lyndon subsequences". We show (1) the maximum total number of Lyndon subsequences in a string, (2) the expected total number of Lyndon subsequences in a string, (3) the expected number of distinct Lyndon subsequences in a string.

Pattern matching is the most central task for text indices. Most recent indices leverage compression techniques to make pattern matching feasible for massive but highly-compressible datasets. Within this kind of indices, we propose a new compressed text index built upon a grammar compression based on induced suffix sorting [Nunes et al., DCC'18]. We show that this grammar exhibits a locality sensitive parsing property, which allows us to specify, given a pattern $P$, certain substrings of $P$, called cores, that are similarly parsed in the text grammar whenever these occurrences are extensible to occurrences of $P$. Supported by the cores, given a pattern of length $m$, we can locate all its $occ$ occurrences in a text $T$ of length $n$ within $O(m \lg |\mathcal{S}| + occ_C \lg|\mathcal{S}| \lg n + occ)$ time, where $\mathcal{S}$ is the set of all characters and non-terminals, $occ$ is the number of occurrences, and $occ_C$ is the number of occurrences of a chosen core $C$ of $P$ in the right hand side of all production rules of the grammar of $T$. Our grammar index requires $O(g)$ words of space and can be built in $O(n)$ time using $O(g)$ working space, where $g$ is the sum of the right hand sides of all production rules. We underline the strength of our grammar index with an exhaustive practical evaluation that gives evidence that our proposed solution excels at locating long patterns in highly-repetitive texts., Comment: Our implementation is available at https://github.com/TooruAkagi/GCIS_Index

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 $\Sigma = \{\mathtt{a, b, c}\}$ 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]., Comment: A part of the results of this article appeared in Proc. SOFSEM 2020 (also in arXiv:1909.02804). The results on binary alphabets are the main new material in this article

We consider the communication complexity of the Hamming distance of two strings. Bille et al. [SPIRE 2018] considered the communication complexity of the longest common prefix (LCP) problem in the setting where the two parties have their strings in a compressed form, i.e., represented by the Lempel-Ziv 77 factorization (LZ77) with/without self-references. We present a randomized public-coin protocol for a joint computation of the Hamming distance of two strings represented by LZ77 without self-references. While our scheme is heavily based on Bille et al.'s LCP protocol, our complexity analysis is original which uses Crochemore's C-factorization and Rytter's AVL-grammar. As a byproduct, we also show that LZ77 with/without self-references are not monotonic in the sense that their sizes can increase by a factor of 4/3 when a prefix of the string is removed.

Let $\Sigma$ and $\Pi$ be disjoint alphabets, respectively called the static alphabet and the parameterized alphabet. Two strings $x$ and $y$ over $\Sigma \cup \Pi$ of equal length are said to parameterized match (p-match) if there exists a renaming bijection $f$ on $\Sigma$ and $\Pi$ which is identity on $\Sigma$ and maps the characters of $x$ to those of $y$ so that the two strings become identical. The indexing version of the problem of finding p-matching occurrences of a given pattern in the text is a well-studied topic in string matching. In this paper, we present a state-of-the-art indexing structure for p-matching called the parameterized suffix tray of an input text $T$, denoted by $\mathsf{PSTray}(T)$. We show that $\mathsf{PSTray}(T)$ occupies $O(n)$ space and supports pattern matching queries in $O(m + \log (\sigma+\pi) + \mathit{occ})$ time, where $n$ is the length of $T$, $m$ is the length of a query pattern $P$, $\pi$ is the number of distinct symbols of $|\Pi|$ in $T$, $\sigma$ is the number of distinct symbols of $|\Sigma|$ in $T$ and $\mathit{occ}$ is the number of p-matching occurrences of $P$ in $T$. We also present how to build $\mathsf{PSTray}(T)$ in $O(n)$ time from the parameterized suffix tree of $T$., Comment: Accepted for CIAC 2021

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 $n \log_\phi n \approx 1.441n\log_2 n$, where $\phi$ is the golden ratio, reported by Fraenkel and Simpson [TCS 1999] obtained via the Three Squares Lemma.

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 $\sigma$ and is given in an online manner, then the palindromic tree of $S$ can be constructed in $O(n\log\sigma)$ 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 $O(n\log\sigma')$ time with $O(d)$ space where $\sigma' \leq d$ is the maximum number of distinct characters in the windows, and the second one works in $O(n + d\sigma)$ time with $(d+2)\sigma + O(d)$ 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.

We show that the size $\gamma(t_n)$ of the smallest string attractor of the $n$th Thue-Morse word $t_n$ is 4 for any $n\geq 4$, disproving the conjecture by Mantaci et al. [ICTCS 2019] that it is $n$. We also show that $\delta(t_n) = \frac{10}{3+2^{4-n}}$ for $n \geq 3$, where $\delta(w)$ is the maximum over all $k = 1,\ldots,|w|$, the number of distinct substrings of length $k$ in $w$ divided by $k$, which is a measure of repetitiveness recently studied by Kociumaka et al. [LATIN 2020]. Furthermore, we show that the number $z(t_n)$ of factors in the self-referencing Lempel-Ziv factorization of $t_n$ is exactly $2n$., Comment: accepted to SPIRE 2020

The dynamic time warping (DTW) is a widely-used method that allows us to efficiently compare two time series that can vary in speed. Given two strings $A$ and $B$ of respective lengths $m$ and $n$, there is a fundamental dynamic programming algorithm that computes the DTW distance for $A$ and $B$ together with an optimal alignment in $\Theta(mn)$ time and space. In this paper, we tackle the problem of interactive computation of the DTW distance for dynamic strings, denoted $\mathrm{D^2TW}$, where character-wise edit operation (insertion, deletion, substitution) can be performed at an arbitrary position of the strings. Let $M$ and $N$ be the sizes of the run-length encoding (RLE) of $A$ and $B$, respectively. We present an algorithm for $\mathrm{D^2TW}$ that occupies $\Theta(mN+nM)$ space and uses $O(m+n+\#_{\mathrm{chg}}) \subseteq O(mN + nM)$ time to update a compact differential representation $\mathit{DS}$ of the DP table per edit operation, where $\#_{\mathrm{chg}}$ denotes the number of cells in $\mathit{DS}$ whose values change after the edit operation. Our method is at least as efficient as the algorithm recently proposed by Froese et al. running in $\Theta(mN + nM)$ time, and is faster when $\#_{\mathrm{chg}}$ is smaller than $O(mN + nM)$ which, as our preliminary experiments suggest, is likely to be the case in the majority of instances., Comment: Accepted for SPIRE 2020

We introduce a new class of straight-line programs (SLPs), named the Lyndon SLP, inspired by the Lyndon trees (Barcelo, 1990). Based on this SLP, we propose a self-index data structure of $O(g)$ words of space that can be built from a string $T$ in $O(n \lg n)$ expected time, retrieving the starting positions of all occurrences of a pattern $P$ of length $m$ in $O(m + \lg m \lg n + occ \lg g)$ time, where $n$ is the length of $T$, $g$ is the size of the Lyndon SLP for $T$, and $occ$ is the number of occurrences of $P$ in $T$.

The equidistant subsequence pattern matching problem is considered. Given a pattern string $P$ and a text string $T$, we say that $P$ is an \emph{equidistant subsequence} of $T$ if $P$ is a subsequence of the text such that consecutive symbols of $P$ in the occurrence are equally spaced. We can consider the problem of equidistant subsequences as generalizations of (sub-)cadences. We give bit-parallel algorithms that yield $o(n^2)$ time algorithms for finding $k$-(sub-)cadences and equidistant subsequences. Furthermore, $O(n\log^2 n)$ and $O(n\log n)$ time algorithms, respectively for equidistant and Abelian equidistant matching for the case $|P| = 3$, are shown. The algorithms make use of a technique that was recently introduced which can efficiently compute convolutions with linear constraints.

Two strings $x$ and $y$ over $\Sigma \cup \Pi$ of equal length are said to \emph{parameterized match} (\emph{p-match}) if there is a renaming bijection $f:\Sigma \cup \Pi \rightarrow \Sigma \cup \Pi$ that is identity on $\Sigma$ and transforms $x$ to $y$ (or vice versa). The \emph{p-matching} problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose \emph{parameterized suffix automata} (\emph{p-suffix automata}) and \emph{parameterized directed acyclic word graphs} (\emph{PDAWGs}) which are the p-matching versions of suffix automata and DAWGs. While suffix automata and DAWGs are equivalent for standard strings, we show that p-suffix automata can have $\Theta(n^2)$ nodes and edges but PDAWGs have only $O(n)$ nodes and edges, where $n$ is the length of an input string. We also give an $O(n |\Pi| \log (|\Pi| + |\Sigma|))$-time $O(n)$-space algorithm that builds the PDAWG in a left-to-right online manner. As a byproduct, it is shown that the \emph{parameterized suffix tree} for the reversed string can also be built in the same time and space, in a right-to-left online manner. This duality also leads us to two further efficient algorithms for p-matching: Given the parameterized suffix tree for the reversal of the input string $T$, one can build the PDAWG of $T$ in $O(n)$ time in an offline manner; One can perform \emph{bidirectional} p-matching in $O(m \log (|\Pi|+|\Sigma|) + \mathit{occ})$ time using $O(n)$ space, where $m$ denotes the pattern length and $\mathit{occ}$ is the number of pattern occurrences in the text $T$., Comment: 28 pages, 7 figures

The longest common subsequence (LCS) problem is a central problem in stringology that finds the longest common subsequence of given two strings $A$ and $B$. More recently, a set of four constrained LCS problems (called generalized constrained LCS problem) were proposed by Chen and Chao [J. Comb. Optim, 2011]. In this paper, we consider the substring-excluding constrained LCS (STR-EC-LCS) problem. A string $Z$ is said to be an STR-EC-LCS of two given strings $A$ and $B$ excluding $P$ if, $Z$ is one of the longest common subsequences of $A$ and $B$ that does not contain $P$ as a substring. Wang et al. proposed a dynamic programming solution which computes an STR-EC-LCS in $O(mnr)$ time and space where $m = |A|, n = |B|, r = |P|$ [Inf. Process. Lett., 2013]. In this paper, we show a new solution for the STR-EC-LCS problem. Our algorithm computes an STR-EC-LCS in $O(n|\Sigma| + (L+1)(m-L+1)r)$ time where $|\Sigma| \leq \min\{m, n\}$ denotes the set of distinct characters occurring in both $A$ and $B$, and $L$ is the length of the STR-EC-LCS. This algorithm is faster than the $O(mnr)$-time algorithm for short/long STR-EC-LCS (namely, $L \in O(1)$ or $m-L \in O(1)$), and is at least as efficient as the $O(mnr)$-time algorithm for all cases.

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 $O(n\log\sigma)$-time and $O(d)$-space algorithm to compute MUSs in a sliding window of width $d$ over $T$, where $\sigma$ 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].

We revisit the problem of longest common property preserving substring queries introduced by~Ayad et al. (SPIRE 2018, arXiv 2018). We consider a generalized and unified on-line setting, where we are given a set $X$ of $k$ strings of total length $n$ that can be pre-processed so that, given a query string $y$ and a positive integer $k'\leq k$, we can determine the longest substring of $y$ that satisfies some specific property and is common to at least $k'$ strings in $X$. Ayad et al. considered the longest square-free substring in an on-line setting and the longest periodic and palindromic substring in an off-line setting. In this paper, we give efficient solutions in the on-line setting for finding the longest common square, periodic, palindromic, and Lyndon substrings. More precisely, we show that $X$ can be pre-processed in $O(n)$ time resulting in a data structure of $O(n)$ size that answers queries in $O(|y|\log\sigma)$ time and $O(1)$ working space, where $\sigma$ is the size of the alphabet, and the common substring must be a square, a periodic substring, a palindrome, or a Lyndon word., Comment: minor change from version submitted to SPIRE 2019

We present the first worst-case linear time algorithm that directly computes the parameterized suffix and LCP arrays for constant sized alphabets. Previous algorithms either required quadratic time or the parameterized suffix tree to be built first. More formally, for a string over static alphabet $\Sigma$ and parameterized alphabet $\Pi$, our algorithm runs in $O(n\pi)$ time and $O(n)$ words of space, where $\pi$ is the number of distinct symbols of $\Pi$ in the string., Comment: submitted to SPIRE 2019

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 $T$.

Given a dynamic set $K$ of $k$ strings of total length $n$ whose characters are drawn from an alphabet of size $\sigma$, a keyword dictionary is a data structure built on $K$ that provides locate, prefix search, and update operations on $K$. Under the assumption that $\alpha = w / \lg \sigma$ characters fit into a single machine word $w$, we propose a keyword dictionary that represents $K$ in $n \lg \sigma + \Theta(k \lg n)$ bits of space, supporting all operations in $O(m / \alpha + \lg \alpha)$ expected time on an input string of length $m$ in the word RAM model. This data structure is underlined with an exhaustive practical evaluation, highlighting the practical usefulness of the proposed data structure, especially for prefix searches - one of the most elementary keyword dictionary operations.

Let $\Sigma$ and $\Pi$ be disjoint alphabets of respective size $\sigma$ and $\pi$. Two strings over $\Sigma \cup \Pi$ of equal length are said to parameterized match (p-match) if there is a bijection $f:\Sigma \cup \Pi \rightarrow \Sigma \cup \Pi$ such that (1) $f$ is identity on $\Sigma$ and (2) $f$ maps the characters of one string to those of the other string so that the two strings become identical. We consider the p-matching problem on a (reversed) trie $\mathcal{T}$ and a string pattern $P$ such that every path that p-matches $P$ has to be reported. Let $N$ be the size of the given trie $\mathcal{T}$. In this paper, we propose the parameterized position heap for $\mathcal{T}$ that occupies $O(N)$ space and supports p-matching queries in $O(m \log (\sigma + \pi) + m \pi + \mathit{pocc}))$ time, where $m$ is the length of a query pattern $P$ and $\mathit{pocc}$ is the number of paths in $\mathcal{T}$ to report. We also present an algorithm which constructs the parameterized position heap for a given trie $\mathcal{T}$ in $O(N (\sigma + \pi))$ time and working space.

For a string $S$, a palindromic substring $S[i..j]$ is said to be a \emph{shortest unique palindromic substring} ($\mathit{SUPS}$) for an interval $[s, t]$ in $S$, if $S[i..j]$ occurs exactly once in $S$, the interval $[i, j]$ contains $[s, t]$, and every palindromic substring containing $[s, t]$ which is shorter than $S[i..j]$ occurs at least twice in $S$. In this paper, we study the problem of answering $\mathit{SUPS}$ queries on run-length encoded strings. We show how to preprocess a given run-length encoded string $\mathit{RLE}_{S}$ of size $m$ in $O(m)$ space and $O(m \log \sigma_{\mathit{RLE}_{S}} + m \sqrt{\log m / \log\log m})$ time so that all $\mathit{SUPSs}$ for any subsequent query interval can be answered in $O(\sqrt{\log m / \log\log m} + \alpha)$ time, where $\alpha$ is the number of outputs, and $\sigma_{\mathit{RLE}_{S}}$ is the number of distinct runs of $\mathit{RLE}_{S}$. Additionaly, we consider a variant of the SUPS problem where a query interval is also given in a run-length encoded form. For this variant of the problem, we present two alternative algorithms with faster queries. The first one answers queries in $O(\sqrt{\log\log m /\log\log\log m} + \alpha)$ time and can be built in $O(m \log \sigma_{\mathit{RLE}_{S}} + m \sqrt{\log m / \log\log m})$ time, and the second one answers queries in $O(\log \log m + \alpha)$ time and can be built in $O(m \log \sigma_{\mathit{RLE}_{S}})$ time. Both of these data structures require $O(m)$ space.

Palindromes are important objects in strings which have been extensively studied from combinatorial, algorithmic, and bioinformatics points of views. It is known that the length of the longest palindromic substrings (LPSs) of a given string T of length n can be computed in O(n) time by Manacher's algorithm [J. ACM '75]. In this paper, we consider the problem of finding the LPS after the string is edited. We present an algorithm that uses O(n) time and space for preprocessing, and answers the length of the LPSs in O(\log (\min \{\sigma, \log n\})) time after a single character substitution, insertion, or deletion, where \sigma denotes the number of distinct characters appearing in T. We also propose an algorithm that uses O(n) time and space for preprocessing, and answers the length of the LPSs in O(\ell + \log \log n) time, after an existing substring in T is replaced by a string of arbitrary length \ell.

A maximal repetition, or run, in a string, is a maximal periodic substring whose smallest period is at most half the length of the substring. In this paper, we consider runs that correspond to a path on a trie, or in other words, on a rooted edge-labeled tree where the endpoints of the path must be a descendant/ancestor of the other. For a trie with $n$ edges, we show that the number of runs is less than $n$. We also show an asymptotic lower bound on the maximum density of runs in tries: $\lim_{n\rightarrow\infty}\rho_\mathcal{T}(n)/n \geq 0.993238$ where $\rho_{\mathcal{T}}(n)$ is the maximum number of runs in a trie with $n$ edges. Furthermore, we also show an $O(n\log \log n)$ time and $O(n)$ space algorithm for finding all runs., Comment: an updated version of CPM 2019 paper (10.4230/LIPIcs.CPM.2019.23), submitted to a journal

We analyze the grammar generation algorithm of the RePair compression algorithm and show the relation between a grammar generated by RePair and maximal repeats. We reveal that RePair replaces step by step the most frequent pairs within the corresponding most frequent maximal repeats. Then, we design a novel variant of RePair, called MR-RePair, which substitutes the most frequent maximal repeats at once instead of substituting the most frequent pairs consecutively. We implemented MR-RePair and compared the size of the grammar generated by MR-RePair to that by RePair on several text corpus. Our experiments show that MR-RePair generates more compact grammars than RePair does, especially for highly repetitive texts.

Two strings of equal length are said to parameterized match if there is a bijection that maps the characters of one string to those of the other string, so that two strings become identical. The parameterized pattern matching problem is, given two strings $T$ and $P$, to find the occurrences of substrings in $T$ that parameterized match $P$. Diptarama et al. [Position Heaps for Parameterized Strings, CPM 2017] proposed an indexing data structure called parameterized position heaps, and gave a left-to-right online construction algorithm. In this paper, we present a right-to-left online construction algorithm for parameterized position heaps. For a text string $T$ of length $n$ over two kinds of alphabets $\Sigma$ and $\Pi$ of respective size $\sigma$ and $\pi$, our construction algorithm runs in $O(n \log(\sigma + \pi))$ time with $O(n)$ space. Our right-to-left parameterized position heaps support pattern matching queries in $O(m \log (\sigma + \pi) + m \pi + \mathit{pocc}))$ time, where $m$ is the length of a query pattern $P$ and $\mathit{pocc}$ is the number of occurrences to report. Our construction and pattern matching algorithms are as efficient as Diptarama et al.'s algorithms.

Mauer et al. [A Lempel-Ziv-style Compression Method for Repetitive Texts, PSC 2017] proposed a hybrid text compression method called LZ-LFS which has both features of Lempel-Ziv 77 factorization and longest first substitution. They showed that LZ-LFS can achieve better compression ratio for repetitive texts, compared to some state-of-the-art compression algorithms. The drawback of Mauer et al.'s method is that their LZ-LFS compression algorithm takes $O(n^2)$ time on an input string of length $n$. In this paper, we show a faster LZ-LFS compression algorithm that works in $O(n \log n)$ time. We also propose a simpler version of LZ-LFS that can be computed in $O(n)$ time.