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### Longest common factor of two words [chapter]

Maxime Crochemore
1987 Lecture Notes in Computer Science
The LCF of two words u and v is the maximal length of a common factor of u and v.  ...  A linear time algorithm to compute LCF is given, based on a linear time algorithm to build the minimal suffix automaton of a word.  ...  For instance c lassicalk string-matching algorithms often use a distance related to the longest common prefix of the two strings [KMP 77] .  ...

### Avoiding approximate repetitions with respect to the longest common subsequence distance

2016 Involve. A Journal of Mathematics
In this paper, we show the existence of words avoiding approximate repetitions, where the measure of similarity between adjacent factors is based on the length of the longest common subsequence.  ...  An approximate repetition is a factor of the form xx', where x and x' are close to being identical.  ...  Here, we measure the closeness of two words based on the length of their longest common subsequence.  ...

### Longest Lyndon Substring After Edit

Yuki Urabe, Yuto Nakashima, Shunsuke Inenaga, Hideo Bannai, Masayuki Takeda, Marc Herbstritt
2018 Annual Symposium on Combinatorial Pattern Matching
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 .  ...  We can modify our algorithm so as to output all the longest Lyndon substrings of T for both problems.  ...  C P M Longest Common Extension For any string w, the longest common extension query is, given two positions 1 ≤ i, j ≤ |w|, to answer LCE w (i, j) = max{k | w[i..i + k − 1] = w[j..j + k − 1], i + k −  ...

### An Optimal Algorithm for Detecting Pseudo-squares [chapter]

Srečko Brlek, Xavier Provençal
2006 Lecture Notes in Computer Science
We consider the problem of determining if a given word, which encodes the boundary of a discrete figure, tiles the plane by translation.  ...  These words have been characterized by the Beauquier-Nivat condition, for which we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic algorithm of  ...  Our algorithms borrow from Lothaire  that the Longest-Common-Factor, the Longest-Common-Prefix and the Longest-Common-Suffix in two words may be computed in linear time.  ...

### A note on the longest common substring with k-mismatches problem [article]

Szymon Grabowski
2014 arXiv   pre-print
The recently introduced longest common substring with k-mismatches (k-LCF) problem is to find, given two sequences S_1 and S_2 of length n each, a longest substring A_1 of S_1 and A_2 of S_2 such that  ...  We first present two output-dependent algorithms solving the k-LCF problem and show that for k = O(^1-ε n), where ε > 0, at least one of them works in subquadratic time, using O(n) words of space.  ...  Introduction The longest common substring (or factor) problem (LCF) is to find the longest contiguous string shared by two strings S 1 and S 2 , of length n and m, m ≤ n, respectively. W.l.o.g.  ...

### Searching for gapped palindromes

Roman Kolpakov, Gregory Kucherov
2009 Theoretical Computer Science
We study the problem of finding, in a given word, all maximal gapped palindromes verifying two types of constraints, that we call long-armed and length-constrained palindromes.  ...  For each of the two classes, we propose an algorithm that runs in time O(n + S) for a constantsize alphabet, where S is the number of output palindromes.  ...  Acknowledgments Part of this work was done during the stay of R. Kolpakov  ...

### A note on the longest common Abelian factor problem [article]

Szymon Grabowski
2015 arXiv   pre-print
AILR2015 presented the first solution for the longest common Abelian factor problem for a pair of strings, reaching O(σ n^2) time with O(σ n n) bits of space, where n is the length of the strings and σ  ...  In this note we show how the time complexity can be preserved while the space is reduced by a factor of σ, and then how the time complexity can be improved, if the alphabet is not too small, when superlinear  ...  We prefer the word "factor" in the problem name, to avoid confusion with the abbreviation for the longest common subsequence.  ...

### Lyndon Words versus Inverse Lyndon Words: Queries on Suffixes and Bordered Words [chapter]

Paola Bonizzoni, Clelia De Felice, Rocco Zaccagnino, Rosalba Zizza
2020 Lecture Notes in Computer Science
As a main result, we prove an upper bound on the length of the longest common extension (or longest common prefix) for two factors of a word w.  ...  This property suggests that, given two words sharing a common overlap, their Lyndon factorizations could be used to capture the common overlap of these two words.  ...  Let w ∈ Σ + be a word which is not an inverse Lyndon word and let ICFL(w) = A Bound on the Length of the Longest Common Prefix Given a word w and two factors x, y of w, we denote by lcp(x, y) the longest  ...

### On the tiling by translation problem

S. Brlek, X. Provençal, Jean-Marc Fédou
2009 Discrete Applied Mathematics
By using the constant time algorithms for computing the longest common extensions in two words, we provide a linear time algorithm in the case of pseudo-square polyominoes, improving the previous quadratic  ...  algorithm of Gambini and Vuillon.  ...  Our algorithms borrow from Lothaire  (for instance) that the Longest-Common-Factor, the Longest-Common-Prefix and the Longest-Common-Suffix in two words may be computed in linear time.  ...

### Construction Of A Rich Word Containing Given Two Factors [article]

Josef Rukavicka
2019 arXiv   pre-print
It is known that there are pairs of rich words that cannot be factors of a common rich word.  ...  A finite word w with w=n contains at most n+1 distinct palindromic factors. If the bound n+1 is attained, the word w is called rich. Let (w) be the set of factors of the word w.  ...  The author acknowledges support by the Czech Science Foundation grant GAČR 13-03538S and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS14/205/OHK4/3T/14.  ...

### Finite-Repetition threshold for infinite ternary words

2011 Electronic Proceedings in Theoretical Computer Science
It is the smallest rational number FRt(a) for which there exists an infinite word whose finite factors have exponent at most FRt(a) and that contains a finite number of factors with exponent r(a).  ...  We show that FRt(3)=r(3)=7/4 and that the bound is achieved with an infinite word containing only two 7/4-exponent words, the smallest number.  ...  Acknowledgments We warmly thank Pascal Ochem for pointing his powerful result on word morphisms and for profitable discussions.  ...

### A string-matching interpretation of the equation xmyn = zp

J. Néraud, M. Crochemore
1992 Theoretical Computer Science
Question: find all maximal factors of \I' which are prefixes of a word of X*.  ...  Crochemore, A string-matching interpretation of the equation x"y" = zD, Theoretical Computer Science 92 (1992) 145-164.  ...  Let us mention the classical Knuth-Morris-Pratt (KMP) algorithm  , generalized in [l] , and an algorithm computing the longest common factor of two words [S] ).  ...

### Longest (Sub-)Periodic Subsequence [article]

Hideo Bannai and Tomohiro I and Dominik Köppl
2022 arXiv   pre-print
We present an algorithm computing the longest periodic subsequence of a string of length n in O(n^7) time with O(n^4) words of space.  ...  We obtain improvements when restricting the exponents or extending the search allowing the reported subsequence to be subperiodic down to O(n^3) time and O(n^2) words of space.  ...   generalized this setting to consider the longest such subsequence common of two texts T and S of length n, and gave an algorithm computing this sequence in O(n 6 ) time using O(n 4 ) space, also  ...

### A Quasilinear-Time Algorithm for Tiling the Plane Isohedrally with a Polyomino [article]

Stefan Langerman, Andrew Winslow
2016 arXiv   pre-print
A plane tiling consisting of congruent copies of a shape is isohedral provided that for any pair of copies, there exists a symmetry of the tiling mapping one copy to the other.  ...  This improves on the O(n^18)-time algorithm of Keating and Vince and generalizes recent work by Brlek, Provençal, Fédou, and the second author.  ...  Acknowledgements The authors wish to thank anonymous reviewers for comments that improved the correctness of the paper.  ...

### Paper Similarity Detection Method Based on Distance Matrix Model with Row-Column Order Penalty Factor

Jun Li, Yaqing Han, Junshan Pan
2014 Journal of Multimedia
This model integrates the characteristics of vector detection, hamming distance and the longest common substring and carries out detection specific to near-synonyms, word deletion and changes in word order  ...  In the aspect of similarity comparison, a distance matrix model with row-column order penalty factor is proposed, which recognizes new words through search engine exponent.  ...  Otherwise it is the longest common substring affected by order penalty factor.  ...
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