The binary string matching problem consists in finding all the occurrences of a pattern in a text where both strings are built on a binary alphabet. This is an interesting problem in computer science, since binary data are omnipresent in telecom and computer network applications. Moreover the problem finds applications also in the field of image processing and in pattern matching on compressed texts. Recently it has been shown that adaptations of classical exact string matching algorithms are not very efficient on binary data. In this paper we present two efficient algorithms for the problem adapted to completely avoid any reference to bits allowing to process pattern and text byte by byte. Experimental results show that the new algorithms outperform existing solutions in most cases.
Exact string matching has been a fundamental problem in computer science for decades because of many practical applications. Some are related to common procedures, such as searching in files and text editors, or, more recently, to more advanced problems such as pattern detection in Artificial Intelligence and Bioinformatics. Tens of algorithms and methodologies have been developed for pattern matching and several programming languages, packages, applications and online systems exist that can perform exact string matching in biological sequences. These techniques, however, are limited to search…
String matching is the problem of finding all the substrings of a text which match a given pattern. It is one of the most investigated problems in computer science, mainly due to its very diverse applications in several fields. Recently, much research in the string matching field has focused on the efficiency and flexibility of the searching procedure and quite effective techniques have been proposed for speeding up the existing solutions. In this context, algorithms based on factors recognition are among the best solutions. In this paper, we present a simple and very efficient algorithm for s…
The abelian pattern matching problem consists in finding all substrings of a text which are permutations of a given pattern. This problem finds application in many areas and can be solved in linear time by a naive sliding window approach. In this short communication we present a new class of algorithms based on a new efficient fingerprint computation approach, called Heap-Counting, which turns out to be fast, flexible and easy to be implemented. It can be proved that our solutions have a linear worst case time complexity and, in addition, we present an extensive experimental evaluation which s…
String matching is the problem of finding all the occurrences of a pattern in a text. We propose improved versions of the fast family of string matching algorithms based on hashing $q$-grams. The improvement consists of considering minimal values $q$ such that each $q$-grams of the pattern has a unique hash value. The new algorithms are fastest than algorithm of the HASH family for short patterns on large size alphabets.
In this short note we present a comprehensive bibliography for the online exact string matching problem. The problem consists in finding all occurrences of a given pattern in a text. It is an extensively studied problem in computer science, mainly due to its direct applications to such diverse areas as text, image and signal processing, speech analysis and recognition, data compression, information retrieval, computational biology and chemistry. Since 1970 more than 120 string matching algorithms have been proposed. In this note we present a comprehensive list of (almost) all string matching a…
Approximate string matching is the problem of finding all factors of a text t of length n that are at a distance at most k from a pattern x of length m. Approximate circular string matching is the problem of finding all factors of t that are at a distance at most k from x or from any of its rotations. In this article, we present a new algorithm for approximate circular string matching under the edit distance model with optimal average-case search time O(n(k + log m)/m). Optimal average-case search time can also be achieved by the algorithms for multiple approximate string matching (Fredriksson…
We present an online algorithm to deal with pattern matching in strings. The problem we investigate is commonly known as string matching with mismatches in which the objective is to report the number of characters that match when a pattern is aligned with every location in the text. The novel method we propose is based on the frequencies of individual characters in the pattern and the text. The algorithm consumes minimal space in the form of simple arrays, which reduces the cost overhead to maintain the complex data structures such as suffix trees or automaton.
String matching is a fundamental problem in algorithm. This study examines the development and construction of two reversible string-matching algorithms: a naive string-matching algorithm and the Rabin-Karp algorithm. The algorithms are used to introduce reversible computing concepts, beginning from basic reversible programming techniques to more advanced considerations about the injectivization of the polynomial hash-update function employed by the Rabin-Karp algorithm. The results are two clean input-preserving reversible algorithms that require no additional space and have the same asymptot…
A degenerate or indeterminate string on an alphabet $Σ$ is a sequence of non-empty subsets of $Σ$. Given a degenerate string $t$ of length $n$, we present a new method based on the Burrows--Wheeler transform for searching for a degenerate pattern of length $m$ in $t$ running in $O(mn)$ time on a constant size alphabet $Σ$. Furthermore, it is a hybrid pattern-matching technique that works on both regular and degenerate strings. A degenerate string is said to be conservative if its number of non-solid letters is upper-bounded by a fixed positive constant $q$; in this case we show that the search…
Given a pattern string $P$ of length $n$ consisting of $δ$ distinct characters and a query string $T$ of length $m$, where the characters of $P$ and $T$ are drawn from an alphabet $Σ$ of size $Δ$, the {\em exact string matching} problem consists of finding all occurrences of $P$ in $T$. For this problem, we present a randomized heuristic that in $O(nδ)$ time preprocesses $P$ to identify $sparse(P)$, a rarely occurring substring of $P$, and then use it to find all occurrences of $P$ in $T$ efficiently. This heuristic has an expected search time of $O( \frac{m}{min(|sparse(P)|, Δ)})$, where $|sp…
More than 120 algorithms have been developed for exact string matching within the last 40 years. We show by experiments that the \naive{} algorithm exploiting SIMD instructions of modern CPUs (with symbols compared in a special order) is the fastest one for patterns of length up to about 50 symbols and extremely good for longer patterns and small alphabets. The algorithm compares 16 or 32 characters in parallel by applying SSE2 or AVX2 instructions, respectively. Moreover, it uses loop peeling to further speed up the searching phase. We tried several orders for comparisons of pattern symbols a…
The approximate string matching is a fundamental and recurrent problem that arises in most computer science fields. This problem can be defined as follows: Let $D=\{x_1,x_2,\ldots x_d\}$ be a set of $d$ words defined on an alphabet $Σ$, let $q$ be a query defined also on $Σ$, and let $k$ be a positive integer. We want to build a data structure on $D$ capable of answering the following query: find all words in $D$ that are at most different from the query word $q$ with $k$ errors. In this thesis, we study the approximate string matching methods in dictionaries, texts, and indexes, to propose pr…
The genome rearrangement problem computes the minimum number of operations that are required to sort all elements of a permutation. A block-interchange operation exchanges two blocks of a permutation which are not necessarily adjacent and in a prefix block-interchange, one block is always the prefix of that permutation. In this paper, we focus on applying prefix block-interchanges on binary and ternary strings. We present upper bounds to group and sort a given binary/ternary string. We also provide upper bounds for a different version of the block-interchange operation which we refer to as the…
We generalise a multiple string pattern matching algorithm, recently proposed by Fredriksson and Grabowski [J. Discr. Alg. 7, 2009], to deal with arbitrary dictionaries on an alphabet of size $s$. If $r_m$ is the number of words of length $m$ in the dictionary, and $ϕ(r) = \max_m \ln(s\, m\, r_m)/m$, the complexity rate for the string characters to be read by this algorithm is at most $κ_{{}_\textrm{UB}}\, ϕ(r)$ for some constant $κ_{{}_\textrm{UB}}$. On the other side, we generalise the classical lower bound of Yao [SIAM J. Comput. 8, 1979], for the problem with a single pattern, to deal with…
我们考虑字符串匹配与可变长度的差距。给定一个字符串$T$和一个由可变长度间隔（指定范围内的任意长度字符串）分隔的字符串组成的模式$P$，问题是找到$T$中与$P$匹配的子串的所有结束位置。这个问题是计算生物学应用中的一个基本问题。令$m$和$n$分别为$P$和$T$的长度，令$k$为$P$中的字符串数。本文提出了一种新的时间$O(n\log k + m +α)$和空间$O(m + A)$的计算方法，其中$A$是$P$中间隔长度的下界之和，$α$是$P$中字符串在$T$中出现的总次数.与以前的结果相比，该界基本上同时实现了已知的最佳时间复杂度和空间复杂度。因此，我们的算法获得了几乎所有$m$、$n$、$k$、$A$和$α$组合的最佳已知边界。我们的算法是令人惊讶的简单和直接实现。我们还提出了算法，发现和编码的位置，所有字符串在$P$的模式的每一个匹配。
Given an indeterminate string pattern $p$ and an indeterminate string text $t$, the problem of order-preserving pattern matching with character uncertainties ($μ$OPPM) is to find all substrings of $t$ that satisfy one of the possible orderings defined by $p$. When the text and pattern are determinate strings, we are in the presence of the well-studied exact order-preserving pattern matching (OPPM) problem with diverse applications on time series analysis. Despite its relevance, the exact OPPM problem suffers from two major drawbacks: 1) the inability to deal with indetermination in the text, t…
In this paper, we extend the notion of gapped strings to elastic-degenerate strings. An elastic-degenerate string can been seen as an ordered collection of k > 1 seeds (substrings/subpatterns) interleaved by elastic-degenerate symbols such that each elastic-degenerate symbol corresponds to a set of two or more variable length strings. Here, we present an algorithm for solving the pattern matching problem with (solid) pattern and elastic-degenerate text, running in O(N+αγnm) time; where m is the length of the given pattern; n and N are the length and total size of the given elastic-degenerate t…
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 seq…
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 $σ$ the alphabet size. We show…
In this paper we study a variant of string pattern matching which deals with tuples of strings known as \textit{multi-track strings}. Multi-track strings are a generalisation of strings (or \textit{single-track strings}) that have primarily found uses in problems related to searching multiple genomes and music information retrieval. A multi-track string $\mathcal{T} = (t_1, t_2, t_3, \ldots , t_N)$ of length $n$ and track count $N$ is a multi-set of $N$ strings of length $n$ with characters drawn from a common alphabet of size $σ_U$. Given two multi-track strings $\mathcal{T} = (t_1, t_2, t_3,…
Park et al. [TCS 2020] observed that the similarity between two (numerical) strings can be captured by the Cartesian trees: The Cartesian tree of a string is a binary tree recursively constructed by picking up the smallest value of the string as the root of the tree. Two strings of equal length are said to Cartesian-tree match if their Cartesian trees are isomorphic. Park et al. [TCS 2020] introduced the following Cartesian tree substring matching (CTMStr) problem: Given a text string $T$ of length $n$ and a pattern string of length $m$, find every consecutive substring $S = T[i..j]$ of a text…
Let $G = (V, E)$ be a digraph where each vertex is unlabeled, each edge is labeled by a character in some alphabet $Ω$, and any two edges with both the same head and the same tail have different labels. The powerset construction gives a transform of $G$ into a weakly connected digraph $G' = (V', E')$ that enables solving the decision problem of whether there is a walk in $G$ matching an arbitrarily long query string $q$ in time linear in $|q|$ and independent of $|E|$ and $|V|$. We show $G$ is uniquely determined by $G'$ when for every $v_\ell \in V$, there is some distinct string $s_\ell$ on …
The problem of String Matching to Labeled Graphs (SMLG) asks to find all the paths in a labeled graph $G = (V, E)$ whose spellings match that of an input string $S \in Σ^m$. SMLG can be solved in quadratic $O(m|E|)$ time [Amir et al., JALG], which was proven to be optimal by a recent lower bound conditioned on SETH [Equi et al., ICALP 2019]. The lower bound states that no strongly subquadratic time algorithm exists, even if restricted to directed acyclic graphs (DAGs). In this work we present the first parameterized algorithms for SMLG in DAGs. Our parameters capture the topological structure …
Searching for all occurrences of a pattern in a text is a fundamental problem in computer science with applications in many other fields, like natural language processing, information retrieval and computational biology. Sampled string matching is an efficient approach recently introduced in order to overcome the prohibitive space requirements of an index construction, on the one hand, and drastically reduce searching time for the online solutions, on the other hand. In this paper we present a new algorithm for the sampled string matching problem, based on a characters distance sampling approa…
