On context constrained squares and repetitions in a string.

*(English)*Zbl 0543.68067Summary: Some combinatorial and computational problems concerning repetitions and repetition roots in a string x on a finite alphabet - that are characterized in general by an O(n log n) bound in terms of the length n of x - are shown to admit of a linear bound when approached in particular contexts. More precisely, it is shown that the number of distinct repetition roots u that are bond to the occurrence of their cube \(u^ 3\) somewhere along the textstring is bounded by n, whence this same bound an be drawn for the number of distinct cube substrings appearing in a generic string. Constraints of similar nature are also discussed that guarantee linear time square-free recognition, and a linear time strategy is proposed to detect, in correspondence with each primitive root u that meets such conditions on x, and for all possible forms of u-rooted repetitions in x, the leftmost occurring repetition in this form.

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