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Periodicity on Partial Words
 Computers and Mathematics with Applications 47
, 2004
"... Codes play an important role in the study of combinatorics on words. Recently, we introduced pcodes that play a role in the study of combinatorics on partial words. Partial words are strings over a finite alphabet that may contain a number of “do not know ” symbols. In this paper, the theory of code ..."
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Cited by 25 (9 self)
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Codes play an important role in the study of combinatorics on words. Recently, we introduced pcodes that play a role in the study of combinatorics on partial words. Partial words are strings over a finite alphabet that may contain a number of “do not know ” symbols. In this paper, the theory of codes of words is revisited starting from pcodes of partial words. We present some important properties of pcodes. We give several equivalent definitions of pcodes and the monoids they generate. We investigate in particular the Defect Theorem for partial words. We describe an algorithm to test whether or not a finite set of partial words is a pcode. We also discuss twoelement pcodes, complete pcodes, maximal pcodes, and the class of circular pcodes. A World Wide Web server interface has been established at
Rotation of Periodic Strings and Short Superstrings
, 1996
"... This paper presents two simple approximation algorithms for the shortest superstring problem, with approximation ratios 2 2 3 ( 2:67) and 2 25 42 ( 2:596), improving the best previously published 2 3 4 approximation. The framework of our improved algorithms is similar to that of previous a ..."
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Cited by 22 (0 self)
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This paper presents two simple approximation algorithms for the shortest superstring problem, with approximation ratios 2 2 3 ( 2:67) and 2 25 42 ( 2:596), improving the best previously published 2 3 4 approximation. The framework of our improved algorithms is similar to that of previous algorithms in the sense that they construct a superstring by computing some optimal cycle covers on the distance graph of the given strings, and then break and merge the cycles to finally obtain a Hamiltonian path, but we make use of new bounds on the overlap between two strings. We prove that for each periodic semiinfinite string ff = a1a2 \Delta \Delta \Delta of period q, there exists an integer k, such that for any (finite) string s of period p which is inequivalent to ff, the overlap between s and the rotation ff[k] = ak ak+1 \Delta \Delta \Delta is at most p+ 1 2 q. Moreover, if p q, then the overlap between s and ff[k] is not larger than 2 3 (p+q). In the previous shortes...
Partial Words and the Critical Factorization Theorem
 J. Combin. Theory Ser. A
, 2007
"... The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial word ..."
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Cited by 11 (6 self)
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The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial words that are sequences that may have a number of “do not know ” symbols also called “holes”. This paper is devoted to a fundamental result on periods of words, the Critical Factorization Theorem, which states that the period of a word is always locally detectable in at least one position of the word resulting in a corresponding critical factorization. Here, we describe precisely the class of partial words w with one hole for which the weak period is locally detectable in at least one position of w. Our proof provides an algorithm which computes a critical factorization when one exists. A World Wide Web server interface at
Efficient String Algorithmics
, 1992
"... Problems involving strings arise in many areas of computer science and have numerous practical applications. We consider several problems from a theoretical perspective and provide efficient algorithms and lower bounds for these problems in sequential and parallel models of computation. In the sequ ..."
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Cited by 9 (6 self)
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Problems involving strings arise in many areas of computer science and have numerous practical applications. We consider several problems from a theoretical perspective and provide efficient algorithms and lower bounds for these problems in sequential and parallel models of computation. In the sequential setting, we present new algorithms for the string matching problem improving the previous bounds on the number of comparisons performed by such algorithms. In parallel computation, we present tight algorithms and lower bounds for the string matching problem, for finding the periods of a string, for detecting squares and for finding initial palindromes.
Saving Comparisons in the CrochemorePerrin String Matching Algorithm
 IN PROC. OF 1ST EUROPEAN SYMP. ON ALGORITHMS
, 1992
"... Crochemore and Perrin discovered an elegant lineartime constantspace string matching algorithm that makes at most 2n \Gamma m symbol comparison. This paper shows how to modify their algorithm to use fewer comparisons. Given any fixed ffl ? 0, the modified algorithm takes linear time, uses constant ..."
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Cited by 9 (1 self)
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Crochemore and Perrin discovered an elegant lineartime constantspace string matching algorithm that makes at most 2n \Gamma m symbol comparison. This paper shows how to modify their algorithm to use fewer comparisons. Given any fixed ffl ? 0, the modified algorithm takes linear time, uses constant space and makes at most n+ b 1+ffl 2 (n \Gamma m)c comparisons. If O(log m) space is available, then the algorithm makes at most n + b 1 2 (n \Gamma m)c comparisons. The pattern preprocessing step also takes linear time and uses constant space. These are the first string matching algorithms that make fewer than 2n \Gamma m comparisons and use sublinear space.
Periodicity and unbordered words
 In STACS 2004 (Montpellier
, 2004
"... The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. Let µ(w) denote the maximum length of its unbordered factors, and let ∂(w) denote the period of w. Clearly, µ(w) ≤ ∂(w). We establish t ..."
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Cited by 5 (4 self)
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The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. Let µ(w) denote the maximum length of its unbordered factors, and let ∂(w) denote the period of w. Clearly, µ(w) ≤ ∂(w). We establish that µ(w) = ∂(w), if w has an unbordered prefix of length µ(w) and n ≥ 2µ(w) − 1. This bound is tight and solves a 21 year old conjecture by Duval. It follows from this result that, in general, n ≥ 3µ(w) − 2 implies µ(w) = ∂(w) which gives an improved bound for the question asked by Ehrenfeucht and Silberger in 1979. Keywords: combinatorics on words, periodicity, unbordered factors, Duval’s conjecture TUCS Laboratory Periodicity and borderedness are two properties of words—the most basic
Periodicity and unbordered words: A proof of the extended Duval conjecture
 J. ACM
"... The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered if it has a proper prefix which is also a suffix of that word. Let µ(w) denote the maximum length of all unborde ..."
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The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this paper. Consider a finite word w of length n. We call a word bordered if it has a proper prefix which is also a suffix of that word. Let µ(w) denote the maximum length of all unbordered factors of w, and let ∂(w) denote the period of w. Clearly, µ(w) ≤ ∂(w). We establish that µ(w) = ∂(w), if w has an unbordered prefix of length µ(w) and n ≥ 2µ(w) − 1. This bound is tight and solves the stronger version of an old conjecture by Duval (1983). It follows from this result that, in general, n ≥ 3µ(w) − 3 implies µ(w) = ∂(w) which gives an improved bound for the question raised by Ehrenfeucht and Silberger in 1979. 1
Equations on partial words
 MFCS 2006 31st International Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Computer Science
, 2006
"... It is well known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation x m y ..."
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It is well known that some of the most basic properties of words, like the commutativity (xy = yx) and the conjugacy (xz = zy), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation x m y n = z p has only periodic solutions in a free monoid, that is, if x m y n = z p holds with integers m, n, p ≥ 2, then there exists a word w such that x, y, z are powers of w. This result, which received a lot of attention, was first proved by Lyndon and Schützenberger for free groups. In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of “do not know ” symbols. When we speak about equations on partial words, we replace the notion of equality (=) with compatibility (↑). Among other equations, we solve xy ↑ yx, xz ↑ zy, and special cases of x m y n ↑ z p for integers m, n, p ≥ 2.