Prague Stringology Conference 2014

Frantisek Franek

On the Number of Distinct Squares

Counting the number of types of squares rather than their occurrences, we consider the problem of bounding the maximum number of distinct squares in a string. Fraenkel and Simpson showed in 1998 that a string of length n contains at most 2n distinct squares and indicated that all evidence pointed to n being a natural universal upper bound. Ilie simplified the proof of Fraenkel-Simpson's key lemma in 2005 and presented in 2007 an asymptotic upper bound of 2nΘ(log n). We show that a string of length n contains at most ⌊11n/6⌋ distinct squares for any n. This new universal upper bound is obtained by investigating the combinatorial structure of FS-double squares (named so in honour of Fraenkel and Simpson's pioneering work on the problem), i.e. two rightmost-occurring squares that start at the same position, and showing that a string of length n contains at most ⌊5n/6⌋ FS-double squares. We will also discuss a much more general approach to double-squares, i.e. two squares starting at the same position and satisfying certain size conditions. A complete, so-called canonical factorization of double-squares that was motivated by the work on the number of distinct squares is presented in a separate contributed talk at this conference. The work on the problem of the number of distinct squares is a joint effort with Antoine Deza and Adrien Thierry.

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