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| → ∞. 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.

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, and thus CDAWGs are a compact indexing structure. Indeed, the CDAWG size 𝖾 can be sublinear in n for some highly repetitive strings. Of its various applications, the CDAWG allows for computing pattern occurrences, maximal exact matches (MEMs), minimal absent words (MAWs), and minimal unique substrings (MUSs) in optimal time using O(𝖾) space. For designing space-efficient data storage, it is crucial that the underlying data structure is robust against data edits and errors. As a mathematical measure for this, the notion of compression sensitivity [Akagi et al. 2023] was introduced as the maximum of the size increase in the compressed data structures after edits operations. In this paper, we investigate the sensitivity of CDAWGs when a single character edit operation is performed at an arbitrary position in the input string T. We show that the size of the CDAWG after an edit operation on T is asymptotically at most 8 times larger than the original CDAWG before the edit. This O(1) upper bound significantly improves on the only known upper bound O(n/log n) for the problem.

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.

Palindromes are popular and important objects in textual data processing, bioinformatics, and combinatorics on words. Let S = X a Y 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” (≈) used in both definitions is exact character matching (=), then the two definitions are the same. However, if we apply other string-equality criteria ≈, 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.

Abstract A bit catastrophe, loosely defined, is when a change in just one character of a string causes a significant change in the size of the compressed string. We study this phenomenon for the Burrows-Wheeler Transform (BWT), a string transform at the heart of several of the most popular compressors and aligners today. The parameter determining the size of the compressed data is the number of equal-letter runs of the BWT, commonly denoted r . We exhibit infinite families of strings in which insertion, deletion, resp. substitution of one character increases r from constant to $$\Theta (\log n)$$ Θ ( log n ) , where n is the length of the string. These strings can be interpreted both as examples for an increase by a multiplicative or an additive $$\Theta (\log n)$$ Θ ( log n ) -factor. As regards the multiplicative factor, they attain the upper bound given by Akagi, Funakoshi, and Inenaga [Inf & Comput. 2023] of $$\mathcal{O}(\log n \log r)$$ O ( log n log r ) , since here $$r=\mathcal{O}(1)$$ r = O ( 1 ) . We then give examples of strings in which insertion, deletion, resp. substitution of a character increases r by a $$\Theta (\sqrt{n})$$ Θ ( n ) additive factor. These strings significantly improve the best known lower bound for an additive factor of $$\Omega (\log n)$$ Ω ( log n ) [Giuliani et al., SOFSEM 2021].

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.

The sensitivity of a string compression algorithm C asks how much the output size C(T) for an input string T can increase when a single character edit operation is performed on T. This notion enables one to measure the robustness of compression algorithms in terms of errors and/or dynamic changes occurring in the input string. In this paper, we analyze the worst-case multiplicative sensitivity of string compression algorithms, which is defined by maxT∈Σn⁡{C(T′)/C(T):ed(T,T′)=1}, where ed(T,T′) denotes the edit distance between T and T′. In particular, for the most common versions of the Lempel-Ziv 77 compressors, we prove that the worst-case multiplicative sensitivity is only a small constant (2 or 3, depending on the version of the Lempel-Ziv 77 and the edit operation type), i.e., the size of the Lempel-Ziv 77 factorizations can be larger by only a small constant factor. We strengthen our upper bound results by presenting matching lower bounds on the worst-case sensitivity for all these major versions of the Lempel-Ziv 77 factorizations. We generalize these results to the smallest bidirectional scheme b. In addition, we show that the sensitivity of a grammar-based compressor called GCIS (Grammar Compression by Induced Sorting) is also a small constant. Further, we extend the notion of the worst-case sensitivity to string repetitiveness measures such as the smallest string attractor size γ and the substring complexity δ, and show that the worst-case sensitivity of δ is also a small constant. These results contrast with the previously known related results such that the size z78 of the Lempel-Ziv 78 factorization can increase by a factor of Ω(n1/4) (shown by Lagarde and Perifel), and the number r of runs in the Burrows-Wheeler transform can increase by a factor of Ω(log⁡n) (shown by Giuliani et al.) when a character is prepended to an input string of length n. By applying our sensitivity bounds of δ or the smallest grammar to known results (cf. Navarro's survey) some non-trivial upper bounds for the sensitivities of important string compressors and repetitiveness measures including γ, r, LZ-End, RePair, LongestMatch, and AVL-grammar, are derived. We also exhibit the worst-case additive sensitivity maxT∈Σn⁡{C(T′)−C(T):ed(T,T′)=1}, which allows one to observe more details in the changes of the output sizes.

A string w is called a minimal absent word for another string T if w does not occur (as a substring) in T and all proper substrings of w occur in T. State-of-the-art data structures for reporting the set 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(|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 ℳ₁, ℳ₂, ℳ₃, ℳ₄, ℳ₅ using RLE, we present matching upper and lower bounds for the number of MAWs in ℳ_i for i = 1,2,4,5 in terms of RLE-size m, except for ℳ₃ whose size is unbounded by m. We then present a compact O(m)-space data structure that can report all MAWs in optimal O(|MAW(T)|) time.

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.

Abstract 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. In this paper, we study the problem of computing MUSs for a sliding window over a given string T. We first show how the set of MUSs can change when the window slides over T. We then present an $$O(n\log \sigma ')$$ O ( n log σ ′ ) -time and O(d)-space algorithm to compute MUSs for a sliding window of size d over the input string T of length n, where $$\sigma '\le d$$ σ ′ ≤ d is the maximum number of distinct characters in every window.

Abstract The similarity between a pair of time series, i.e., sequences of indexed values in time order, is often estimated by the dynamic time warping (DTW) distance, instead of any in the well-studied family of measures including the longest common subsequence (LCS) length and the edit distance. Although it may seem as if the DTW and the LCS(-like) measures are essentially different, we reveal that the DTW distance can be represented by the longest increasing subsequence (LIS) length of a sequence of integers, which is the LCS length between the integer sequence and itself sorted. For a given pair of time series of lengthnsuch that the dissimilarity between any elements is an integer between zero andc, we propose an integer sequence that represents any substring-substring DTW distance as its band-substring LIS length. The length of the produced integer sequence is $$O(c n^2)$$ O(cn2) , which can be translated to $$O(n^2)$$ O(n2) for constant dissimilarity functions. To demonstrate that techniques developed under the LCS(-like) measures are directly applicable to analysis of time series via our reduction of DTW to LIS, we present time-efficient algorithms for DTW-related problems utilizing the semi-local sequence comparison technique developed for LCS-related problems.

Given a dynamic set K of k strings of total length n whose characters are drawn from an alphabet of size σ , a keyword dictionary is a data structure built on K that provides lookup, prefix search, and update operations on K . Under the assumption that α = w / lg ⁡ σ characters fit into a single machine word of w bits, we propose a keyword dictionary that represents K in either n lg ⁡ σ + Θ ( k lg ⁡ n ) or | T | lg ⁡ σ + Θ ( k w ) bits of space, where | T | is the number of nodes of a trie representing K . It supports all operations in O ( m / α + lg ⁡ α ) expected time on an input string of length m in the word RAM model. An evaluation of our implementation highlights the practical usefulness of the proposed data structure, especially for prefix searches — one of the most essential keyword dictionary operations.

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(nlog⁡σ) 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≤j−i+1≤d. The first version works in O(nlog⁡σ′) time with O(d) space where σ′≤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.

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={a,b,c} be the alphabet. Then, the set of MAWs for string w=abaab is {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≤d

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 $T$.

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 each edge is labeled with a single symbol, and 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: limn→∞⁡ρT(n)/n>0.9932348 where ρT(n) is the maximum number of runs in a trie with n edges. Furthermore, we also show an O(nlog⁡log⁡n) time and O(n) space algorithm for finding all runs.

In a seminal paper, Charikar et al. derive upper and lower bounds on the approximation ratios for several grammar-based compressors, but in all cases there is a gap between the lower and upper bound. Here the gaps for LZ78 and BISECTION are closed by showing that the approximation ratio of LZ78 is Θ((n/log n)2/3), whereas the approximation ratio of BISECTION is Θ(√(n/log n)). In addition, the lower bound for RePair is improved from Ω(√(log n)) to Ω(log n/log log n). Finally, results of Arpe and Reischuk relating grammar-based compression for arbitrary alphabets and binary alphabets are improved.

The equidistant subsequence pattern matching problem is considered. Given a pattern string P and a text string T, we say that P is an 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(n²) time algorithms for finding k-(sub-)cadences and equidistant subsequences. Furthermore, O(nlog² n) and O(nlog 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 Σ ∪ Π of equal length are said to parameterized match (p-match) if there is a renaming bijection f:Σ ∪ Π → Σ ∪ Π that is identity on Σ and transforms x to y (or vice versa). The p-matching problem is to look for substrings in a text that p-match a given pattern. In this paper, we propose parameterized suffix automata (p-suffix automata) and parameterized directed acyclic word graphs (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 Θ(n²) nodes and edges but PDAWGs have only O(n) nodes and edges, where n is the length of an input string. We also give O(n |Π| log (|Π| + |Σ|))-time O(n)-space algorithm that builds the PDAWG in a left-to-right online manner. As a byproduct, it is shown that the parameterized suffix tree for the reversed string can also be built in the same time and space, in a right-to-left online manner.

Given a dynamic set K of k strings of total length n whose characters are drawn from an alphabet of size σ, a keyword dictionary is a data structure built on K that provides locate, prefix search, and update operations on K. Under the assumption that α = w / lg σ characters fit into a single machine word w, we propose a keyword dictionary that represents K in n lg σ + Θ(k lg n) bits of space, supporting all operations in Θ(m / α + lg α) 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.

The similarity between a pair of time series, i.e., sequences of indexed values in time order, is often estimated by the dynamic time warping (DTW) distance, instead of any in the well-studied family of measures including the longest common subsequence (LCS) length and the edit distance. Although it may seem as if the DTW and the LCS(-like) measures are essentially different, we reveal that the DTW distance can be represented by the longest increasing subsequence (LIS) length of a sequence of integers, which is the LCS length between the integer sequence and itself sorted. For a given pair of time series of n integers between zero and c, we propose an integer sequence that represents any substring-substring DTW distance as its band-substring LIS length. The length of the produced integer sequence is O(c⁴ n²) or O(c² n²) depending on the variant of the DTW distance used, both of which can be translated to O(n²) for constant cost functions. To demonstrate that techniques developed under the LCS(-like) measures are directly applicable to analysis of time series via our reduction of DTW to LIS, we present time-efficient algorithms for DTW-related problems utilizing the semi-local sequence comparison technique developed for LCS-related problems.

Palindromes are important objects in strings which have been extensively studied from combinatorial, algorithmic, and bioinformatics points of views. Manacher [J. ACM 1975] proposed a seminal algorithm that computes the longest substring palindromes (LSPals) of a given string in O(n) time, where n is the length of the string. In this paper, we consider the problem of finding the LSPal after the string is edited. We present an algorithm that uses O(n) time and space for preprocessing, and answers the length of the LSPals in O(l + log log n) time, after a substring in T is replaced by a string of arbitrary length l. This outperforms the query algorithm proposed in our previous work [CPM 2018] that uses O(l + log n) time for each query.

The suffix trees are fundamental data structures for various kinds of string processing. The suffix tree of a string T of length n has O(n) nodes and edges, and the string label of each edge is encoded by a pair of positions in T. Thus, even after the tree is built, the input text T needs to be kept stored and random access to T is still needed. The linear-size suffix tries (LSTs), proposed by Crochemore et al. [Linear-size suffix tries, TCS 638:171-178, 2016], are a "stand-alone" alternative to the suffix trees. Namely, the LST of a string T of length n occupies O(n) total space, and supports pattern matching and other tasks in the same efficiency as the suffix tree without the need to store the input text T. Crochemore et al. proposed an offline algorithm which transforms the suffix tree of T into the LST of T in O(n log sigma) time and O(n) space, where sigma is the alphabet size. In this paper, we present two types of online algorithms which "directly" construct the LST, from right to left, and from left to right, without constructing the suffix tree as an intermediate structure. Both algorithms construct the LST incrementally when a new symbol is read, and do not access to the previously read symbols. The right-to-left construction algorithm works in O(n log sigma) time and O(n) space and the left-to-right construction algorithm works in O(n (log sigma + log n / log log n)) time and O(n) space. The main feature of our algorithms is that the input text does not need to be stored.

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 O(n sqrt{log n}log log n) time and O(n) space algorithm for counting and finding the shallower endpoint of all runs. We further show an O(n log n) time and O(n) space algorithm for finding both endpoints of all runs. We also discuss how to improve the running time even more.

Lempel-Ziv (LZ) factorization and Lyndon factorization are well-known factorizations of strings. Recently, Kärkkäinen et al. studied the relation between the sizes of the two factorizations, and showed that the size of the Lyndon factorization is always smaller than twice the size of the non-overlapping LZ factorization [STACS 2017]. In this paper, we consider a similar problem for the overlapping version of the LZ factorization. Since the size of the overlapping LZ factorization is always smaller than the size of the non-overlapping LZ factorization and, in fact, can even be an O(log n) factor smaller, it is not immediately clear whether a similar bound as in previous work would hold. Nevertheless, in this paper, we prove that the size of the Lyndon factorization is always smaller than four times the size of the overlapping LZ factorization.

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 corpora. Our experiments show that MR-RePair generates more compact grammars than RePair does, especially for highly repetitive texts.

For a set D of documents and a positive integer d, a string w is said to be d-left-right maximal, if (1) w occurs in at least d documents in D, and (2) any proper superstring of w occurs in less than d documents. The left-right-maximal generic words problem is, given a set D of documents, to preprocess D so that for any string p and for any positive integer d, all the superstrings of p that are d-left-right maximal can be answered quickly. In this paper, we present an O(n log m) space data structure (in words) which answers queries in O(|p| + o log log m) time, where n is the total length of documents in D, m is the number of documents in D and o is the number of outputs. Our solution improves the previous one by Nishimoto et al. (PSC 2015), which uses an O(n log n) space data structure answering queries in O(|p|+ r * log n + o * log^2 n) time, where r is the number of right-extensions q of p occurring in at least d documents such that any proper right extension of q occurs in less than d documents.

An Elastic-Degenerate String [Iliopoulus et al., LATA 2017] is a sequence of sets of strings, which was recently proposed as a way to model a set of similar sequences. We give an online algorithm for the Elastic-Degenerate String Matching (EDSM) problem that runs in O(nm sqrt{m log m} + N) time and O(m) working space, where n is the number of elastic degenerate segments of the text, N is the total length of all strings in the text, and m is the length of the pattern. This improves the previous algorithm by Grossi et al. [CPM 2017] that runs in O(nm^2 + N) time.

It is known that the length of the longest substring palindromes (LSPals) 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 LSPal after the string is edited. We present an algorithm that uses O(n) time and space for preprocessing, and answers the length of the LSPals in O(log (min {sigma, log n })) time after 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 LSPals in O(l + log n) time, after an existing substring in T is replaced by a string of arbitrary length l.

We revisit the problem of computing the Lyndon factorization of a string w of length N which is given as a straight line program (SLP) of size n. For this problem, we show a new algorithm which runs in O(P(n, N) + Q(n, N)n log log N) time and O(n log N + S(n, N)) space where P(n, N), S(n,N), Q(n,N) are respectively the pre-processing time, space, and query time of a data structure for longest common extensions (LCE) on SLPs. Our algorithm improves the algorithm proposed by I et al. (TCS '17), and can be more efficient than the O(N)-time solution by Duval (J. Algorithms '83) when w is highly compressible.

A square is a non-empty string of form YY. The longest common square subsequence (LCSqS) problem is to compute a longest square occurring as a subsequence in two given strings A and B. We show that the problem can easily be solved in O(n^6) time or O(|M|n^4) time with O(n^4) space, where n is the length of the strings and M is the set of matching points between A and B. Then, we show that the problem can also be solved in O(sigma |M|^3 + n) time and O(|M|^2 + n) space, or in O(|M|^3 log^2 n log log n + n) time with O(|M|^3 + n) space, where sigma is the number of distinct characters occurring in A and B. We also study lower bounds for the LCSqS problem for two or more strings.

The longest Lyndon substring of a string T is the longest substring of T which is a Lyndon word. LLS(T) denotes the length of the longest Lyndon substring of a string T. In this paper, we consider computing LLS(T') where T' is an edited string formed from T. After O(n) time and space preprocessing, our algorithm returns LLS(T') in O(log n) time for any single character edit. We also consider a version of the problem with block edits, i.e., a substring of T is replaced by a given string of length l. After O(n) time and space preprocessing, our algorithm returns LLS(T') in O(l log sigma + log n) time for any block edit where sigma is the number of distinct characters in T. We can modify our algorithm so as to output all the longest Lyndon substrings of T' for both problems.

An α-gapped repeat (α ≥ 1) in a word w is a factor uvu of w such that |u v| ≤ α|u|; the two occurrences of u are called arms of this α-gapped repeat. An α-gapped repeat is called maximal if its arms cannot be extended simultaneously with the same character to the right nor to the left. We show that the number of all maximal α-gapped repeats occurring in words of length n is upper bounded by 18α n. In the case of α-gapped palindromes, i.e., factors $uv{{u}^{\intercal }}$ with |u v|≤ α|u|, we show that the number of all maximal α-gapped palindromes occurring in words of length n is upper bounded by 28α n + 7n. Both upper bounds allow us to construct algorithms finding all maximal α-gapped repeats and/or all maximal α-gapped palindromes of a word of length n on an integer alphabet of size $n^{\mathcal {O}(1)}$ in ${\mathcal {O}(\alpha n)}$ time. The presented running times are optimal since there are words that have Θ(α n) maximal α-gapped repeats/palindromes.

Given a string on an integer alphabet, we present an algorithm that computes the set of all distinct squares belonging to this string in time linear to the string length. As an application, we show how to compute the tree topology of the minimal augmented suffix tree in linear time. Asides from that, we elaborate an algorithm computing the longest previous table in a succinct representation using compressed working space.

The constrained LCS problem asks one to find a longest common subsequence of two input strings A and B with some constraints. The STR-IC-LCS problem is a variant of the constrained LCS problem, where the solution must include a given constraint string C as a substring. Given two strings A and B of respective lengths M and N, and a constraint string C of length at most min{M, N}, the best known algorithm for the STR-IC-LCS problem, proposed by Deorowicz (Inf. Process. Lett., 11:423-426, 2012), runs in O(MN) time. In this work, we present an O(mN + nM)-time solution to the STR-IC-LCS problem, where m and n denote the sizes of the run-length encodings of A and B, respectively. Since m <= M and n <= N always hold, our algorithm is always as fast as Deorowicz's algorithm, and is faster when input strings are compressible via RLE.

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 <= 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.

We consider the problem of computing all maximal repetitions contained in a string that is given in run-length encoding. Given a run-length encoding of a string, we show that the maximum number of maximal repetitions contained in the string is at most m+k-1, where m is the size of the run-length encoding, and k is the number of run-length factors whose exponent is at least 2. We also show an algorithm for computing all maximal repetitions in O(m \alpha(m)) time and O(m) space, where \alpha denotes the inverse Ackermann function.

The longest common extension (LCE) problem is to preprocess a given string w of length n so that the length of the longest common prefix between suffixes of w that start at any two given positions is answered quickly. In this paper, we present a data structure of O(z \tau^2 + \frac{n}{\tau}) words of space which answers LCE queries in O(1) time and can be built in O(n \log \sigma) time, where 1 \leq \tau \leq \sqrt{n} is a parameter, z is the size of the Lempel-Ziv 77 factorization of w and \sigma is the alphabet size. The proposed LCE data structure not access the input string w when answering queries, and thus w can be deleted after preprocessing. On top of this main result, we obtain further results using (variants of) our LCE data structure, which include the following: - For highly repetitive strings where the z\tau^2 term is dominated by \frac{n}{\tau}, we obtain a constant-time and sub-linear space LCE query data structure. - Even when the input string is not well compressible via Lempel-Ziv 77 factorization, we still can obtain a constant-time and sub-linear space LCE data structure for suitable \tau and for \sigma \leq 2^{o(\log n)}. - The time-space trade-off lower bounds for the LCE problem by Bille et al. [J. Discrete Algorithms, 25:42-50, 2014] and by Kosolobov [CoRR, abs/1611.02891, 2016] do not apply in some cases with our LCE data structure.

We give a new characterization of maximal repetitions (or runs) in strings based on Lyndon words. The characterization leads to a proof of what was known as the “runs” conjecture [R. M. Kolpakov and G. Kucherov, Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, Los Alamitos, CA, 1999, pp. 596--604]), which states that the maximum number of runs $\rho(n)$ in a string of length $n$ is less than $n$. The proof is remarkably simple, considering the numerous endeavors to tackle this problem in the last 15 years, and significantly improves our understanding of how runs can occur in strings. In addition, we obtain an upper bound of 3n for the maximum sum of exponents $\sigma(n)$ of runs in a string of length $n$, improving on the best known bound of 4.1n by Crochemore et al. [ J. Discrete Algorithms, 14 (2012), pp. 29--36], as well as other improved bounds on related problems. The characterization also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel--Ziv factorization of the string. We also establish a relationship between runs and nodes of the Lyndon tree, which gives a simple optimal solution to the 2-period query problem that was recently solved by Kociumaka et al. [ Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA) 2015, San Diego, CA, SIAM, Philadelphia, 2015, pp. 532--551].

A square factorization of a string w is a factorization of w in which each factor is a square. Dumitran et al. [SPIRE 2015, pp. 54-66] showed how to find a square factorization of a given string of length n in O(n log n) time, and they posed a question whether it can be done in O(n) time. In this paper, we answer their question positively, showing an O(n)-time algorithm for square factorization in the standard word RAM model with machine word size omega = Omega(log n). We also show an O(n + (n log^2 n) / omega)-time (respectively, O(n log n)-time) algorithm to find a square factorization which contains the maximum (respectively, minimum) number of squares.

We consider fully-online construction of indexing data structures for multiple texts. Let T = {T_1, ..., T_K} be a collection of texts. By fully-online, we mean that a new character can be appended to any text in T at any time. This is a natural generalization of semi-online construction of indexing data structures for multiple texts in which, after a new character is appended to the kth text T_k, then its previous texts T_1, ..., T_k-1 will remain static. Our fully-online scenario arises when we maintain dynamic indexes for multi-sensor data. Let N and sigma denote the total length of texts in T and the alphabet size, respectively. We first show that the algorithm by Blumer et al. [Theoretical Computer Science, 40:31-55, 1985] to construct the directed acyclic word graph (DAWG) for T can readily be extended to our fully-online setting, retaining O(N log sigma)-time and O(N)-space complexities. Then, we give a sophisticated fully-online algorithm which constructs the suffix tree for T in O(N log sigma) time and O(N) space. A key idea of this algorithm is synchronized maintenance of the DAWG and the suffix tree.

Given a string S of n symbols, a longest common extension query LCE(i,j) asks for the length of the longest common prefix of the $i$th and $j$th suffixes of S. LCE queries have several important applications in string processing, perhaps most notably to suffix sorting. Recently, Bille et al. (J. Discrete Algorithms 25:42-50, 2014, Proc. CPM 2015:65-76) described several data structures for answering LCE queries that offers a space-time trade-off between data structure size and query time. In particular, for a parameter 1 <= tau <= n, their best deterministic solution is a data structure of size O(n/tau) which allows LCE queries to be answered in O(tau) time. However, the construction time for all deterministic versions of their data structure is quadratic in n. In this paper, we propose a deterministic solution that achieves a similar space-time trade-off of O(tau * min(log(tau),log(n/tau)) query time using O(n/tau) space, but significantly improve the construction time to O(n*tau).

The directed acyclic word graph (DAWG) of a string y is the smallest (partial) DFA which recognizes all suffixes of y and has only O(n) nodes and edges. We present the first O(n)-time algorithm for computing the DAWG of a given string y of length n over an integer alphabet of polynomial size in n. We also show that a straightforward modification to our DAWG construction algorithm leads to the first O(n)-time algorithm for constructing the affix tree of a given string y over an integer alphabet. Affix trees are a text indexing structure supporting bidirectional pattern searches. As an application to our O(n)-time DAWG construction algorithm, we show that the set MAW(y) of all minimal absent words of y can be computed in optimal O(n + |MAW(y)|) time and O(n) working space for integer alphabets.

We consider the problem of answering shortest unique substring (SUS) queries on run-length encoded strings. For a string S, a unique substring u = S[i..j] is said to be a shortest unique substring (SUS) of S containing an interval [s, t] (i <= s <= t <= j) if for any i' <= s <= t <= j' with j-i > j'-i', S[i'..j'] occurs at least twice in S. Given a run-length encoding of size m of a string of length N, we show that we can construct a data structure of size O(m+pi_s(N, m)) in O(m log m + pi_c(N, m)) time such that queries can be answered in O(pi_q(N, m) + k) time, where k is the size of the output (the number of SUSs), and pi_s(N,m), pi_c(N,m), pi_q(N,m) are, respectively, the size, construction time, and query time for a predecessor/successor query data structure of m elements for the universe of [1,N]. Using the data structure by Beam and Fich (JCSS 2002), this results in a data structure of O(m) space that is constructed in O(m log m) time, and answers queries in O(sqrt(log m/loglog m)+k) time.

A Longest Common Extension (LCE) query on a text T of length N asks for the length of the longest common prefix of suffixes starting at given two positions. We show that the signature encoding G of size w = O(min(z log N log^* M, N)) [Mehlhorn et al., Algorithmica 17(2):183-198, 1997] of T, which can be seen as a compressed representation of T, has a capability to support LCE queries in O(log N + log ell log^* M) time, where ell is the answer to the query, z is the size of the Lempel-Ziv77 (LZ77) factorization of T, and M >= 4N is an integer that can be handled in constant time under word RAM model. In compressed space, this is the fastest deterministic LCE data structure in many cases. Moreover, G can be enhanced to support efficient update operations: After processing G in O(w f_A) time, we can insert/delete any (sub)string of length y into/from an arbitrary position of T in O((y + log Nlog^* M) f_A) time, where f_A = O(min{ (loglog M loglog w)/(logloglog M), sqrt(log w/loglog w)}). This yields the first fully dynamic LCE data structure working in compressed space. We also present efficient construction algorithms from various types of inputs: We can construct G in O(N f_A) time from uncompressed string T; in O(n loglog (n log^* M) log N log^* M) time from grammar-compressed string T represented by a straight-line program of size n; and in O(z f_A log N log^* M) time from LZ77-compressed string T with z factors. On top of the above contributions, we show several applications of our data structures which improve previous best known results on grammar-compressed string processing.

For alpha >=1, an alpha-gapped repeat in a word w is a factor uvu of w such that |uv| <= alpha * |u|; the two occurrences of a factor u in such a repeat are called arms. Such a repeat is called maximal if its arms cannot be extended simultaneously with the same symbol to the right nor to the left. We show that the number of all maximal alpha-gapped repeats occurring in words of length n is upper bounded by 18 * alpha * n, allowing us to construct an algorithm finding all maximal alpha-gapped repeats of a word on an integer alphabet of size n^{O}(1)} in {O}(alpha * n) time. This result is optimal as there are words that have Theta(alpha * n) maximal alpha-gapped repeats. Our techniques can be extended to get comparable results in the case of alpha-gapped palindromes, i.e., factors uvu^{T} with |uv| <= alpha |u|.

We give a new characterization of maximal repetitions (or runs) in strings, using a tree defined on recursive standard factorizations of Lyndon words, called the Lyndon tree. The characterization leads to a remarkably simple novel proof of the linearity of the maximum number of runs ρ(n) in a string of length n. Furthermore, we show an upper bound of ρ(n) < 1.5n, which improves on the best upper bound 1.6n (Crochemore & Ilie 2008) that does not rely on computational verification. The proof also gives rise to a new, conceptually simple linear-time algorithm for computing all the runs in a string. A notable characteristic of our algorithm is that, unlike all existing linear-time algorithms, it does not utilize the Lempel-Ziv factorization of the string.

We present a new on-line algorithm for computing the Lempel-Ziv factorization of a string that runs in O(N.log(N)) time and uses only O(N.log(s)) bits of working space, where N is the length of the string and s is the size of the alphabet. This is a notable improvement compared to the performance of previous on-line algorithms using the same order of working space but running in either O(N.log^3(N)) time [Okanohara and Sadakane, 2009] or O(N.log^2(N)) time [Starikovskaya, 2012]. The key to our new algorithm is in the utilization of an elegant but less popular index structure called Directed Acyclic Word Graphs, or DAWGs [Blumer et al., 1985]. We also present an opportunistic variant of our algorithm, which, given the run length encoding of size m of a string of length N, computes the Lempel-Ziv factorization of the string on-line, in O(m.min{log(log(m)).log(log(N))/(log(log(log(N)))), (log(m))^{1/2}/(log(log(m)))^{1/2})}) time and O(m.log(N)) bits of space.

In this paper, we present efficient algorithms for interconversion between Lempel-Ziv 78 (LZ78) encoding and run length encoding (RLE). We show how, given an RLE of size n for a string S, we can compute the corresponding LZ78 encoding of size m for S in O((n + m) log σ) time, where σ is the number of distinct characters appearing in S. We also show how, given an LZ78 encoding of size m for a string S, we can compute the corresponding RLE of size n in O(n + m) time. Both algorithms use O(m) extra working space.

The convolution between a text string S of length N and a pattern string P of length m can be computed in O(N log m) time by FFT. It is known that various types of approximate string matching problems are reducible to convolution. In this paper, we assume that the input text string is given in a compressed form, as a straight-line program (SLP), which is a context free grammar in the Chomsky normal form that derives a single string. Given an SLP S of size n describing a text S of length N, and an uncompressed pattern P of length m, we present a simple O(nm log m)-time algorithm to compute the convolution between S and P. We then show that this can be improved to O(min{nm, N - α} log m) time, where α ≥ 0 is a value that represents the amount of redundancy that the SLP captures with respect to the length-m substrings. The key of the improvement is our new algorithm that computes the convolution between a trie of size r and a pattern string P of length m in O(r log m) time.

We consider grammar based text compression with longest-first substitution, where non-overlapping occurrences of a longest repeating substring of the input text are replaced by a new non-terminal symbol. We present a new text compression algorithm by simplifying the algorithm presented in S. Inenaga et al., (2003). We give a new formulation of the correctness proof introducing the sparse lazy suffix tree data structure. We also present another type of longest-first substitution strategy that allows better compression. We show results of preliminary experiments comparing grammar sizes of the two versions of the longest-first strategy and the most frequent strategy

The Compact Directed Acyclic Word Graph (CDAWG) is a space-eflcient data structure that supports indices of a string. The Symmetric Directed Acyclic Word Graph (SCDAWG) for a string w is a dual structure that supports indices of both w and the reverse of w simultaneously. Blumer et al. gave the first algorithm to construct an SCDAWG from a given string, that works in an of-line manner. In this papec we show an on-line algorithm that constructs an SCDAWGfiom a given string directly.