How Many Square Occurrences Must a Binary Sequence Contain?
نویسندگان
چکیده
منابع مشابه
How Many Square Occurrences Must a Binary Sequence Contain?
Every binary word with at least four letters contains a square. A. Fraenkel and J. Simpson showed that three distinct squares are necessary and sufficient to construct an infinite binary word. We study the following complementary question: how many square occurrences must a binary word contain? We show that this quantity is, in the limit, a constant fraction of the word length, and prove that t...
متن کاملHow Many Squares Must a Binary Sequence Contain?
Let g(n) be the length of a longest binary string containing at most n distinct squares (two identical adjacent substrings). Then g(0) = 3 (010 is such a string), g(1) = 7 (0001000) and g(2) = 18 (010011000111001101). How does the sequence { g(n) } behave? We give a complete answer.
متن کاملHow many runs can a string contain?
Given a string x = x[1..n], a repetition of period p in x is a substring ur = x[i+1..i+rp], p = |u|, r ≥ 2, where neither u = x[i+1..i+p] nor x[i+1..i+(r+1)p+1] is a repetition. The maximum number of repetitions in any string x is well known to be Θ(n log n). A run or maximal periodicity of period p in x is a substring urt = x[i+1..i+rp+ |t|] of x, where ur is a repetition, t a proper prefix of...
متن کاملHow Many Squares Can a String Contain?
All our words (strings) are over a fixed alphabet. A square is a subword of the form uu=u, where u is a nonempty word. Two squares are distinct if they are of different shape, not just translates of each other. A word u is primitive if u cannot be written in the form u=v j for some j 2. A square u with u primitive is primitive rooted. Let M(n) denote the maximum number of distinct squares, P(n)...
متن کاملHow many double squares can a string contain?
Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. In 1998 Fraenkel and Simpson showed that a string of length n contains at most 2n distinct squares. In 2007 Ilie provided an asymptotic upper bound of 2n−Θ(log n). We show that a string of length n contains at most b5n/3c distinct squares. This new upper b...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2003
ISSN: 1077-8926
DOI: 10.37236/1705