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We revisit the complexity of approximate pattern matching in an elastic-degenerate string. Such a string is a sequence of $n$ finite sets of strings of total length $N$, and compactly describes a collection of strings obtained by first…

Data Structures and Algorithms · Computer Science 2025-03-04 Paweł Gawrychowski , Adam Górkiewicz , Pola Marciniak , Solon P. Pissis , Karol Pokorski

Given a text $T$ and a pattern $P$ over alphabet $\Sigma$, the classic exact matching problem searches for all occurrences of pattern $P$ in text $T$. Unlike exact matching problem, order-preserving pattern matching (OPPM) considers the…

Data Structures and Algorithms · Computer Science 2017-05-29 Davaajav Jargalsaikhan , Diptarama , Ryo Yoshinaka , Ayumi Shinohara

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…

Data Structures and Algorithms · Computer Science 2020-04-14 Satoshi Kobayashi , Diptarama Hendrian , Ryo Yoshinaka , Ayumi Shinohara

In the $k$-mismatch problem, given a pattern and a text of length $n$ and $m$ respectively, we have to find if the text has a sub-string with a Hamming distance of at most $k$ from the pattern. This has been studied in the classical setting…

Quantum Physics · Physics 2026-01-13 Ruhan Habib , Shadman Shahriar

We show how to determine whether a given pattern p of length m occurs in a given text t of length n in ${\tilde O}(\sqrt{n}+\sqrt{m})$\footnote{${\tilde O}$ allows for logarithmic factors in m and $n/m$} time, with inverse polynomial…

Quantum Physics · Physics 2007-05-23 H. Ramesh , V. Vinay

We consider the approximate pattern matching problem under the edit distance. Given a text $T$ of length $n$, a pattern $P$ of length $m$, and a threshold $k$, the task is to find the starting positions of all substrings of $T$ that can be…

Data Structures and Algorithms · Computer Science 2022-04-08 Panagiotis Charalampopoulos , Tomasz Kociumaka , Philip Wellnitz

We present a new streaming algorithm for the $k$-Mismatch problem, one of the most basic problems in pattern matching. Given a pattern and a text, the task is to find all substrings of the text that are at the Hamming distance at most $k$…

Data Structures and Algorithms · Computer Science 2019-04-24 Jakub Radoszewski , Tatiana Starikovskaya

A central task in string processing is text indexing, where the goal is to preprocess a text (a string of length $n$) into an efficient index (a data structure) supporting queries about the text. Cole, Gottlieb, and Lewenstein (STOC 2004)…

Data Structures and Algorithms · Computer Science 2025-10-31 Tomasz Kociumaka , Jakub Radoszewski

We consider a class of pattern matching problems where a normalising transformation is applied at every alignment. Normalised pattern matching plays a key role in fields as diverse as image processing and musical information processing…

Data Structures and Algorithms · Computer Science 2015-03-19 Ayelet Butman , Peter Clifford , Raphael Clifford , Markus Jalsenius , Noa Lewenstein , Benny Porat , Ely Porat , Benjamin Sach

This article introduces a new optimization method to improve mergesort's runtime complexity, when sorting sequences that have equal keys to $O(n log_2 k)$, where $k$ is the number of distinct keys in the sequence. When $k$ is constant, it…

Data Structures and Algorithms · Computer Science 2020-12-17 Albert Tedja

We introduce a new metric of match, called Cartesian tree matching, which means that two strings match if they have the same Cartesian trees. Based on Cartesian tree matching, we define single pattern matching for a text of length n and a…

Data Structures and Algorithms · Computer Science 2019-05-23 Sung Gwan Park , Amihood Amir , Gad M. Landau , Kunsoo Park

Encoding data structures store enough information to answer the queries they are meant to support but not enough to recover their underlying datasets. In this paper we give the first encoding data structure for the challenging problem of…

Data Structures and Algorithms · Computer Science 2017-02-21 Travis Gagie , Giovanni Manzini , Rossano Venturini

We revisit the complexity of one of the most basic problems in pattern matching. In the k-mismatch problem we must compute the Hamming distance between a pattern of length m and every m-length substring of a text of length n, as long as…

Data Structures and Algorithms · Computer Science 2015-08-28 Raphaël Clifford , Allyx Fontaine , Ely Porat , Benjamin Sach , Tatiana Starikovskaya

We present a sorting algorithm for the case of recurrent random comparison errors. The algorithm essentially achieves simultaneously good properties of previous algorithms for sorting $n$ distinct elements in this model. In particular, it…

Data Structures and Algorithms · Computer Science 2017-09-22 Barbara Geissmann , Stefano Leucci , Chih-Hung Liu , Paolo Penna

Given an indeterminate string pattern $p$ and an indeterminate string text $t$, the problem of order-preserving pattern matching with character uncertainties ($\mu$OPPM) is to find all substrings of $t$ that satisfy one of the possible…

Data Structures and Algorithms · Computer Science 2019-05-08 Diogo Costa , Luís M. S. Russo , Rui Henriques , Hideo Bannai , Alexandre P. Francisco

Permutation patterns and pattern avoidance have been intensively studied in combinatorics and computer science, going back at least to the seminal work of Knuth on stack-sorting (1968). Perhaps the most natural algorithmic question in this…

Data Structures and Algorithms · Computer Science 2019-04-17 László Kozma

We revisit a fundamental problem in string matching: given a pattern of length m and a text of length n, both over an alphabet of size $\sigma$, compute the Hamming distance between the pattern and the text at every location. Several…

Data Structures and Algorithms · Computer Science 2020-01-03 Timothy M. Chan , Shay Golan , Tomasz Kociumaka , Tsvi Kopelowitz , Ely Porat

Approximate pattern matching is a natural and well-studied problem on strings: Given a text $T$, a pattern $P$, and a threshold $k$, find (the starting positions of) all substrings of $T$ that are at distance at most $k$ from $P$. We…

Data Structures and Algorithms · Computer Science 2020-11-17 Panagiotis Charalampopoulos , Tomasz Kociumaka , Philip Wellnitz

We present new algorithms for the problem of multiple string matching of gapped patterns, where a gapped pattern is a sequence of strings such that there is a gap of fixed length between each two consecutive strings. The problem has…

Data Structures and Algorithms · Computer Science 2014-07-08 Emanuele Giaquinta , Kimmo Fredriksson , Szymon Grabowski , Alexandru I. Tomescu , Esko Ukkonen

In this work, we revisit the fundamental and well-studied problem of approximate pattern matching under edit distance. Given an integer $k$, a pattern $P$ of length $m$, and a text $T$ of length $n \ge m$, the task is to find substrings of…

Data Structures and Algorithms · Computer Science 2021-06-14 Tomasz Kociumaka , Ely Porat , Tatiana Starikovskaya