How many double squares can a string contain?

Abstract

Counting the types of squares rather than their occurrences, we consider the problem of bounding the 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. Ilie presented in 2007 an asymptotic upper bound of 2n−Θ(log n). We show that a string of length n contains at most 11n/6 distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most 5n/6 particular double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.

On a lemma of Crochemore and Rytter

“…Moreover, as shown in [4], for a balanced double square (u, v : u 1 , u 2 , e 1 , e 2 ) 0 ≤ lcs(u 2 u 2 , u 2 u 2 ) + lcp(u 2 u 2 , u 2 u 2 ) ≤ |u 1 |−2.…”

“…A right shift by one position of a substring The notion of double squares and their factorization can be traced to Lam [10], and was further investigated and generalized by Deza, Franek, and Thierry [4] and by Bai, Franek, and Smyth [1]. Given a balanced double square (u, v), the unique 4-tuple (u 1 , u 2 , e 1 , e 2 ) yielded by Lemma 5 is referred to as the canonical factorization of the double square (u, v) and is denoted by (u, v : u 1 , u 2 , e 1 , e 2 ).…”

“…The inversion factor is defined as u 2 u 2 u 2 u 2 . As shown in [4], the inversion factor has only two occurrences in v 2 as indicated in bold below:…”

Efficient enumeration of non-equivalent squares in partial words with few holes

“…Obviously, a word W of length n may contain Θ(n 2 ) square factors (e.g. W = a n ), however, it is known that such a word contains only O(n) distinct square factors (Fraenkel and Simpson 1998;Ilie 2005); currently the best known upper bound is 11 6 n (Deza et al 2015). Moreover, all distinct square factors of a word over an integer alphabet can be listed in O(n) time using the suffix tree (Gusfield and Stoye 2004;Bannai et al 2017) or the suffix array and the structure of runs (maximal repetitions) in the word (Crochemore et al 2014).…”

Constructing Words with High Distinct Square Densities

“…This is due to their fundamental importance in algorithms and combinatorics on words. Different notions and techniques such as primitively or non-primitively-rooted squares [16,32], positions starting a square [25], frequencies of occurrences of squares [33,39], three-squares property [18,31], overlapping squares [21], distinct squares [17,19,20,[26][27][28]38], double squares [17], non-standard squares [29], etc., have been studied and extended to partial words [2][3][4][5][6][7][8]24].…”