arXiv subjects

Diptarama Hendrian

Publications and source records attributed to Diptarama Hendrian.

At least 19 recordsLinked to original sources

arXiv2024

Height-bounded Lempel-Ziv encodings

We introduce height-bounded LZ encodings (LZHB), a new family of compressed representations that are variants of Lempel-Ziv parsings with a focus on bounding the worst-case access time to arbitrary positions in the text directly via the compressed representation. An LZ-like encoding is a partitioning of the string into phrases of length $1$ which can be encoded literally, or phrases of length at least $2$ which have a previous occurrence in the string and can be encoded by its position and length. An LZ-like encoding induces an implicit referencing forest on the set of positions of the string. An LZHB encoding is an LZ-like encoding where the height of the implicit referencing forest is bounded. An LZHB encoding with height constraint $h$ allows access to an arbitrary position of the underlying text using $O(h)$ predecessor queries. While computing the smallest LZHB encoding efficiently seems to be difficult [Cicalese \& Ugazio 2024, arxiv], we give the first linear time algorithm for strings over a constant size alphabet that computes the greedy LZHB encoding, i.e., the string is processed from beginning to end, and the longest prefix of the remaining string that can satisfy the height constraint is taken as the next phrase. Our algorithms significantly improve both theoretically and practically, the very recently and independently proposed algorithms by Lipták et al. (arxiv, to appear at CPM 2024). We also analyze the size of height bounded LZ encodings in the context of repetitiveness measures, and show for some constant $c$, the size $z_{HB}$ of the optimal LZHB encoding with height bound $c\log n$ is $O(g_{rl})$, where $g_{rl}$ is the size of the smallest run-length grammar. We also show $z_{HB} = o(g_{rl})$ for some family of strings, making $z_{HB}$ one of the smallest known repetitiveness measures for which $O({\sf polylog} n)$ time access is possible using linear space.

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arXiv2024

Algorithms for Galois Words: Detection, Factorization, and Rotation

Lyndon words are extensively studied in combinatorics on words -- they play a crucial role on upper bounding the number of runs a word can have [Bannai+, SIAM J. Comput.'17]. We can determine Lyndon words, factorize a word into Lyndon words in lexicographically non-increasing order, and find the Lyndon rotation of a word, all in linear time within constant additional working space. A recent research interest emerged from the question of what happens when we change the lexicographic order, which is at the heart of the definition of Lyndon words. In particular, the alternating order, where the order of all odd positions becomes reversed, has been recently proposed. While a Lyndon word is, among all its cyclic rotations, the smallest one with respect to the lexicographic order, a Galois word exhibits the same property by exchanging the lexicographic order with the alternating order. Unfortunately, this exchange has a large impact on the properties Galois words exhibit, which makes it a nontrivial task to translate results from Lyndon words to Galois words. Up until now, it has only been conjectured that linear-time algorithms with constant additional working space in the spirit of Duval's algorithm are possible for computing the Galois factorization or the Galois rotation. Here, we affirm this conjecture as follows. Given a word $T$ of length $n$, we can determine whether $T$ is a Galois word, in $O(n)$ time with constant additional working space. Within the same complexities, we can also determine the Galois rotation of $T$, and compute the Galois factorization of $T$ online. The last result settles Open Problem~1 in [Dolce et al., TCS 2019] for Galois words.

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arXiv2023

Linear Time Online Algorithms for Constructing Linear-size Suffix Trie

The suffix trees are fundamental data structures for various kinds of string processing. The suffix tree of a text 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 string $T$ needs to be kept stored and random access to $T$ is still needed. The \emph{linear-size suffix tries} (\emph{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 an input text string $T$ of length $n$ occupies $O(n)$ total space, and supports pattern matching and other tasks with the same efficiency as the suffix tree without the need to store the input text string $T$. Crochemore et al. proposed an \emph{offline} algorithm which transforms the suffix tree of $T$ into the LST of $T$ in $O(n \log σ)$ time and $O(n)$ space, where $σ$ is the alphabet size. In this paper, we present two types of \emph{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 the previously read symbols. Both of the right-to-left construction algorithm and the left-to-right construction algorithm work in $O(n \log σ)$ time and $O(n)$ space. The main feature of our algorithms is that the input text string does not need to be stored.

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arXiv2023

Breaking a Barrier in Constructing Compact Indexes for Parameterized Pattern Matching

A parameterized string (p-string) is a string over an alphabet $(Σ_{s} \cup Σ_{p})$, where $Σ_{s}$ and $Σ_{p}$ are disjoint alphabets for static symbols (s-symbols) and for parameter symbols (p-symbols), respectively. Two p-strings $x$ and $y$ are said to parameterized match (p-match) if and only if $x$ can be transformed into $y$ by applying a bijection on $Σ_{p}$ to every occurrence of p-symbols in $x$. The indexing problem for p-matching is to preprocess a p-string $T$ of length $n$ so that we can efficiently find the occurrences of substrings of $T$ that p-match with a given pattern. Extending the Burrows-Wheeler Transform (BWT) based index for exact string pattern matching, Ganguly et al. [SODA 2017] proposed the first compact index (named pBWT) for p-matching, and posed an open problem on how to construct it in compact space, i.e., in $O(n \lg |Σ_{s} \cup Σ_{p}|)$ bits of space. Hashimoto et al. [SPIRE 2022] partially solved this problem by showing how to construct some components of pBWTs for $T$ in $O(n \frac{|Σ_{p}| \lg n}{\lg \lg n})$ time in an online manner while reading the symbols of $T$ from right to left. In this paper, we improve the time complexity to $O(n \frac{\lg |Σ_{p}| \lg n}{\lg \lg n})$. We remark that removing the multiplicative factor of $|Σ_{p}|$ from the complexity is of great interest because it has not been achieved for over a decade in the construction of related data structures like parameterized suffix arrays even in the offline setting. We also show that our data structure can support backward search, a core procedure of BWT-based indexes, at any stage of the online construction, making it the first compact index for p-matching that can be constructed in compact space and even in an online manner.

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arXiv2023

Efficient Parameterized Pattern Matching in Sublinear Space

The parameterized matching problem is a variant of string matching, which is to search for all parameterized occurrences of a pattern $P$ in a text $T$. In considering matching algorithms, the combinatorial natures of strings, especially periodicity, play an important role. In this paper, we analyze the properties of periods of parameterized strings and propose a generalization of Galil and Seiferas's exact matching algorithm (1980) into parameterized matching, which runs in $O(π|T|+|P|)$ time and $O(\log{|P|}+|{\rmΠ}|)$ space in addition to the input space, where ${\rmΠ}$ is the parameter alphabet and $π$ is the number of parameter characters appearing in $P$ plus one.

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arXiv2022

Inferring Strings from Position Heaps in Linear Time

Position heaps are index structures of text strings used for the string matching problem. They are rooted trees whose edges and nodes are labeled and numbered, respectively. This paper is concerned with variants of the inverse problem of position heap construction and gives linear-time algorithms for those problems. The basic problem is to restore a text string from a rooted tree with labeled edges and numbered nodes. In the variant problems, the input trees may miss edge labels or node numbers which we must restore as well.

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arXiv2022

Parameterized DAWGs: efficient constructions and bidirectional pattern searches

Two strings $x$ and $y$ over $Σ\cup Π$ of equal length are said to \emph{parameterized match} (\emph{p-match}) if there is a renaming bijection $f:Σ\cup Π\rightarrow Σ\cup Π$ that is identity on $Σ$ 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 $Θ(n^2)$ 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 $O(n |Π| \log (|Π| + |Σ|))$-time $O(n)$-space algorithm that builds the PDAWG in a left-to-right online manner. As a byproduct, it is shown that the \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 $O(m \log (|Π|+|Σ|) + \mathit{occ})$ time using $O(n)$ space, where $m$ denotes the pattern length and $\mathit{occ}$ is the number of pattern occurrences in the text $T$.

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arXiv2022

Computing the Parameterized Burrows--Wheeler Transform Online

Parameterized strings are a generalization of strings in that their characters are drawn from two different alphabets, where one is considered to be the alphabet of static characters and the other to be the alphabet of parameter characters. Two parameterized strings are a parameterized match if there is a bijection over all characters such that the bijection transforms one string to the other while keeping the static characters (i.e., it behaves as the identity on the static alphabet). Ganguly et al. [SODA 2017] proposed the parameterized Burrows--Wheeler transform (pBWT) as a variant of the Burrows--Wheeler transform for space-efficient parameterized pattern matching. In this paper, we propose an algorithm for computing the pBWT online by reading the characters of a given input string one-by-one from right to left. Our algorithm works in $O(|Π| \log n / \log \log n)$ amortized time for each input character, where $n$ and $Π$ denote the size of the input string and the alphabet of the parameter characters, respectively.

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arXiv2022

Parallel algorithm for pattern matching problems under substring consistent equivalence relations

Given a text and a pattern over an alphabet, the pattern matching problem searches for all occurrences of the pattern in the text. An equivalence relation $\approx$ is called a substring consistent equivalence relation (SCER), if for two strings $X$ and $Y$, $X \approx Y$ implies $|X| = |Y|$ and $X[i:j] \approx Y[i:j]$ for all $1 \le i \le j \le |X|$. In this paper, we propose an efficient parallel algorithm for pattern matching under any SCER using the"duel-and-sweep" paradigm. For a pattern of length $m$ and a text of length $n$, our algorithm runs in $O(ξ_m^\mathrm{t} \log^2 m)$ time and $O(ξ_m^\mathrm{w} \cdot n \log^2 m)$ work, with $O(τ_n^\mathrm{t} + ξ_m^\mathrm{t} \log^2 m)$ time and $O(τ_n^\mathrm{w} + ξ_m^\mathrm{w} \cdot m \log^2 m)$ work preprocessing on the Priority Concurrent Read Concurrent Write Parallel Random-Access Machines (P-CRCW PRAM).

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arXiv2020

Computing Covers under Substring Consistent Equivalence Relations

Covers are a kind of quasiperiodicity in strings. A string $C$ is a cover of another string $T$ if any position of $T$ is inside some occurrence of $C$ in $T$. The shortest and longest cover arrays of $T$ have the lengths of the shortest and longest covers of each prefix of $T$, respectively. The literature has proposed linear-time algorithms computing longest and shortest cover arrays taking border arrays as input. An equivalence relation $\approx$ over strings is called a substring consistent equivalence relation (SCER) iff $X \approx Y$ implies (1) $|X| = |Y|$ and (2) $X[i:j] \approx Y[i:j]$ for all $1 \le i \le j \le |X|$. In this paper, we generalize the notion of covers for SCERs and prove that existing algorithms to compute the shortest cover array and the longest cover array of a string $T$ under the identity relation will work for any SCERs taking the accordingly generalized border arrays.

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arXiv2020

In-Place Bijective Burrows-Wheeler Transforms

One of the most well-known variants of the Burrows-Wheeler transform (BWT) [Burrows and Wheeler, 1994] is the bijective BWT (BBWT) [Gil and Scott, arXiv 2012], which applies the extended BWT (EBWT) [Mantaci et al., TCS 2007] to the multiset of Lyndon factors of a given text. Since the EBWT is invertible, the BBWT is a bijective transform in the sense that the inverse image of the EBWT restores this multiset of Lyndon factors such that the original text can be obtained by sorting these factors in non-increasing order. In this paper, we present algorithms constructing or inverting the BBWT in-place using quadratic time. We also present conversions from the BBWT to the BWT, or vice versa, either (a) in-place using quadratic time, or (b) in the run-length compressed setting using $O(n \lg r / \lg \lg r)$ time with $O(r \lg n)$ bits of words, where $r$ is the sum of character runs in the BWT and the BBWT.

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arXiv2020

Fast and linear-time string matching algorithms based on the distances of $q$-gram occurrences

Given a text $T$ of length $n$ and a pattern $P$ of length $m$, the string matching problem is a task to find all occurrences of $P$ in $T$. In this study, we propose an algorithm that solves this problem in $O((n + m)q)$ time considering the distance between two adjacent occurrences of the same $q$-gram contained in $P$. We also propose a theoretical improvement of it which runs in $O(n + m)$ time, though it is not necessarily faster in practice. We compare the execution times of our and existing algorithms on various kinds of real and artificial datasets such as an English text, a genome sequence and a Fibonacci string. The experimental results show that our algorithm is as fast as the state-of-the-art algorithms in many cases, particularly when a pattern frequently appears in a text.

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arXiv2020

Grammar compression with probabilistic context-free grammar

We propose a new approach for universal lossless text compression, based on grammar compression. In the literature, a target string $T$ has been compressed as a context-free grammar $G$ in Chomsky normal form satisfying $L(G) = \{T\}$. Such a grammar is often called a \emph{straight-line program} (SLP). In this paper, we consider a probabilistic grammar $G$ that generates $T$, but not necessarily as a unique element of $L(G)$. In order to recover the original text $T$ unambiguously, we keep both the grammar $G$ and the derivation tree of $T$ from the start symbol in $G$, in compressed form. We show some simple evidence that our proposal is indeed more efficient than SLPs for certain texts, both from theoretical and practical points of view.

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arXiv2019

Generalized Dictionary Matching under Substring Consistent Equivalence Relations

Given a set of patterns called a dictionary and a text, the dictionary matching problem is a task to find all occurrence positions of all patterns in the text. The dictionary matching problem can be solved efficiently by using the Aho-Corasick algorithm. Recently, Matsuoka et al. [TCS, 2016] proposed a generalization of pattern matching problem under substring consistent equivalence relations and presented a generalization of the Knuth-Morris-Pratt algorithm to solve this problem. An equivalence relation $\approx$ is a substring consistent equivalence relation (SCER) if for two strings $X,Y$, $X \approx Y$ implies $|X| = |Y|$ and $X[i:j] \approx Y[i:j]$ for all $1 \le i \le j \le |X|$. In this paper, we propose a generalization of the dictionary matching problem and present a generalization of the Aho-Corasick algorithm for the dictionary matching under SCER. We present an algorithm that constructs SCER automata and an algorithm that performs dictionary matching under SCER by using the automata. Moreover, we show the time and space complexity of our algorithms with respect to the size of input strings.

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arXiv2019

Efficient computation of longest single-arm-gapped palindromes in a string

In this paper, we introduce new types of approximate palindromes called single-arm-gapped palindromes (shortly SAGPs). A SAGP contains a gap in either its left or right arm, which is in the form of either $wguc u^R w^R$ or $wuc u^Rgw^R$, where $w$ and $u$ are non-empty strings, $w^R$ and $u^R$ are respectively the reversed strings of $w$ and $u$, $g$ is a string called a gap, and $c$ is either a single character or the empty string. Here we call $wu$ and $u^R w^R$ the arm of the SAGP, and $|uv|$ the length of the arm. We classify SAGPs into two groups: those which have $ucu^R$ as a maximal palindrome (type-1), and the others (type-2). We propose several algorithms to compute type-1 SAGPs with longest arms occurring in a given string, based on suffix arrays. Then, we propose a linear-time algorithm to compute all type-1 SAGPs with longest arms, based on suffix trees. Also, we show how to compute type-2 SAGPs with longest arms in linear time. We also perform some preliminary experiments to show practical performances of the proposed methods.

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arXiv2019

Query Learning Algorithm for Residual Symbolic Finite Automata

We propose a query learning algorithm for residual symbolic finite automata (RSFAs). Symbolic finite automata (SFAs) are finite automata whose transitions are labeled by predicates over a Boolean algebra, in which a big collection of characters leading the same transition may be represented by a single predicate. Residual finite automata (RFAs) are a special type of non-deterministic finite automata which can be exponentially smaller than the minimum deterministic finite automata and have a favorable property for learning algorithms. RSFAs have both properties of SFAs and RFAs and can have more succinct representation of transitions and fewer states than RFAs and deterministic SFAs accepting the same language. The implementation of our algorithm efficiently learns RSFAs over a huge alphabet and outperforms an existing learning algorithm for deterministic SFAs. The result also shows that the benefit of non-determinism in efficiency is even larger in learning SFAs than non-symbolic automata.

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arXiv2019

An Extension of Linear-size Suffix Tries for Parameterized Strings

In this paper, we propose a new indexing structure for parameterized strings which we call PLSTs, by generalizing linear-size suffix tries for ordinary strings. Two parameterized strings are said to match if there is a bijection on the symbol set that makes the two coincide. PLSTs are applicable to the parameterized pattern matching problem, which is to decide whether the input parameterized text has a substring that matches the input parameterized pattern. The size of PLSTs is linear in the text size, with which our algorithm solves the parameterized pattern matching problem in linear time in the pattern size. PLSTs can be seen as a compacted version of parameterized suffix tries and a combination of linear-size suffix tries and parameterized suffix trees. We experimentally show that PLSTs are more space efficient than parameterized suffix trees for highly repetitive strings.

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arXiv2019

Online Algorithms for Constructing Linear-size Suffix Trie

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 $O(n \log σ)$ time and $O(n)$ space, where $σ$ 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 $O(n \log σ)$ time and $O(n)$ space and the left-to-right construction algorithm works in $O(n (\log σ+ \log n / \log \log n))$ time and $O(n)$ space. The main feature of our algorithms is that the input text does not need to be stored.

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