Keywords and Synonyms
Powers; Runs; Tandem repeats
Problem Definition
Periodicities and repetitions in strings have been extensively studied and are important both in theory and practice (combinatorics of words, pattern‐matching, computational biology). The words of the type ww and www, where w is a nonempty primitive (not of the form u k for an integer \( { k > 1 } \)) word, are called squares and cubes, respectively. They are well‐investigated objects in combinatorics on words [16] and in string‐matching with small memory [5].
A string w is said to be periodic iff \( { period(w)\le |w|/2 } \), where period(w) is the smallest positive integer p for which \( { w[i]=w[i+p] } \) whenever both sides of the equality are defined. In particular each square and cube is periodic.
A repetition in a string \( { x = x_1 x_2 \dots x_n } \) is an interval \( { [i\mathinner{\ldotp\ldotp} j]\subseteq[1\mathinner{\ldotp\ldotp} n] } \) for which the associated factor \( { x[i\mathinner{\ldotp\ldotp} j]...
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Recommended Reading
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Crochemore, M., Rytter, W. (2008). Squares and Repetitions. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_392
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