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Finding Maximal Repetitions in a Word in Linear Time
 In Symposium on Foundations of Computer Science
, 1999
"... A repetition in a word is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in, that is those for which any extended subword of has a bigger period. The set of such repetitions represents in a compact way all repetitions in.We first prove a combi ..."
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Cited by 48 (4 self)
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A repetition in a word is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in, that is those for which any extended subword of has a bigger period. The set of such repetitions represents in a compact way all repetitions in.We first prove a combinatorial result asserting that the sum of exponents of all maximal repetitions of a word of length is bounded by a linear function in. This implies, in particular, that there is only a linear number of maximal repetitions in a word. This allows us to construct a lineartime algorithm for finding all maximal repetitions. Some consequences and applications of these results are discussed, as well as related works. 1.
On Maximal Repetitions in Words
 J. Discrete Algorithms
, 1999
"... A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. ..."
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Cited by 36 (5 self)
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A (fractional) repetition in a word w is a subword with the period of at most half of the subword length. We study maximal repetitions occurring in w, that is those for which any extended subword of w has a bigger period. The set of such repetitions represents in a compact way all repetitions in w. We first study maximal repetitions in Fibonacci words  we count their exact number, and estimate the sum of their exponents. These quantities turn out to be linearlybounded in the length of the word. We then prove that the maximal number of maximal repetitions in general words (on arbitrary alphabet) of length n is linearlybounded in n, and we mention some applications and consequences of this result.
Finding approximate repetitions under Hamming distance
 Theoretical Computer Science
, 2001
"... The problem of computing tandem repetitions with K possible mismatches is studied. Two main definitions are considered, and for both of them an O(nK log K + S) algorithm is proposed (S the size of the output). This improves, in particular, the bound obtained in [LS93]. Finally, other possible defini ..."
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Cited by 25 (1 self)
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The problem of computing tandem repetitions with K possible mismatches is studied. Two main definitions are considered, and for both of them an O(nK log K + S) algorithm is proposed (S the size of the output). This improves, in particular, the bound obtained in [LS93]. Finally, other possible definions are briefly analyzed.
Finding Repeats With Fixed Gap
 IN: PROC. OF THE 7TH INT’L SYMP. ON STRING PROCESSING AND INFORMATION RETRIEVAL (SPIRE). WASHINGTON: IEEE COMPUTER SOCIETY
, 2000
"... We propose an algorithm for finding in a word all pairs of occurrences of the same subword with a given distance r between them. The obtained complexity is O(n log r + S), where S is the size of the output. We also show how the algorithm can be modified in order to find all such pairs of occurrences ..."
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Cited by 7 (2 self)
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We propose an algorithm for finding in a word all pairs of occurrences of the same subword with a given distance r between them. The obtained complexity is O(n log r + S), where S is the size of the output. We also show how the algorithm can be modified in order to find all such pairs of occurrences separated by a given word. The solution uses an algorithm for finding all quasisquares in two strings, a problem that generalizes the known problem of searching for squares.
Euclidean Strings
 University of Newcastle, Australia
, 2000
"... A string p = p0p1 · · · pn−1 of nonnegative integers is a Euclidean string if the string (p0 + 1)p1 · · · (pn−1 − 1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0 + p1 + · · · + pn−1 are relatively prime and that, if they exist ..."
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Cited by 5 (3 self)
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A string p = p0p1 · · · pn−1 of nonnegative integers is a Euclidean string if the string (p0 + 1)p1 · · · (pn−1 − 1) is rotationally equivalent (i.e., conjugate) to p. We show that Euclidean strings exist if and only if n and p0 + p1 + · · · + pn−1 are relatively prime and that, if they exist, they are unique. We show how to construct them using an algorithm with the same structure as the Euclidean algorithm, hence the name. We show that Euclidean strings are Lyndon words and we describe relationships between Euclidean strings and the SternBrocot tree, Fibonacci strings, Beatty sequences, and Sturmian sequences. We also describe an application to a graph embedding problem.
COMP. & STRUCT. APPROACHES TO PERIODICITIES IN STRINGSCOMPUTATIONAL AND STRUCTURAL APPROACHES TO PERIODICITIES IN STRINGS
"... We investigate the function ρd(n) = max { r(x)  x is a (d, n)string} where r(x) is the number of runs in the string x, and a (d, n)string is a string with length n and exactly d distinct symbols. Our investigation is motivated by the conjecture that ρd(n) ≤ n − d. We present and discuss fundam ..."
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We investigate the function ρd(n) = max { r(x)  x is a (d, n)string} where r(x) is the number of runs in the string x, and a (d, n)string is a string with length n and exactly d distinct symbols. Our investigation is motivated by the conjecture that ρd(n) ≤ n − d. We present and discuss fundamental properties of the ρd(n) function. The values of ρd(n) are presented in the (d, n−d)table with rows indexed by d and columns indexed by n − d which reveals the regularities of the function. We introduce the concepts of the rcover and core vector of a string, yielding a novel computational framework for determining ρd(n) values. The computation of the previously intractable instances is achieved via first computing a lower bound, and then using the structural properties to limit our exhaustive search only to strings that can possibly exceed this number of runs. Using this approach, we extended the known maximum number of runs in binary string from 60 to 74. In doing so, we find the first examples of runmaximal strings containing four consecutive identical symbols.
On singularities of extremal periodic strings
, 2012
"... Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a string is at most its length. Similarly, Kolpakov and Kucherov conjectured in 1999 that the number of runs in a string is at most its length. Since then, both conjectures attracted the attention of many researchers and ..."
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Fraenkel and Simpson conjectured in 1998 that the number of distinct squares in a string is at most its length. Similarly, Kolpakov and Kucherov conjectured in 1999 that the number of runs in a string is at most its length. Since then, both conjectures attracted the attention of many researchers and many results have been presented, including asymptotic lower bounds for both, asymptotic upper bounds for runs, and universal upper bounds for distinct squares. We consider the role played by the size of the alphabet of the string in both problems and investigate the functions σd(n) and ρd(n), i.e. the maximum number of distinct primitively rooted squares, respectively runs, over all strings of length n containing exactly d distinct symbols. We revisit earlier results and conjectures and express them in terms of singularities of the two functions where a pair (d, n) is a singularity if σd(n) − σd−1(n − 2) ≥ 2, or ρd(n) − ρd−1(n − 2) ≥ 2 respectively.