68,996 research outputs found
How 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. 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 - Theta(log n).
We show that a string of length n contains at most 5n/3 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 2n/3
double squares. In addition, the established structural properties provide a
novel proof of Fraenkel and Simpson's result.Comment: 29 pages, 20 figure
Repetition Detection in a Dynamic String
A string UU for a non-empty string U is called a square. Squares have been well-studied both from a combinatorial and an algorithmic perspective. In this paper, we are the first to consider the problem of maintaining a representation of the squares in a dynamic string S of length at most n. We present an algorithm that updates this representation in n^o(1) time. This representation allows us to report a longest square-substring of S in O(1) time and all square-substrings of S in O(output) time. We achieve this by introducing a novel tool - maintaining prefix-suffix matches of two dynamic strings.
We extend the above result to address the problem of maintaining a representation of all runs (maximal repetitions) of the string. Runs are known to capture the periodic structure of a string, and, as an application, we show that our representation of runs allows us to efficiently answer periodicity queries for substrings of a dynamic string. These queries have proven useful in static pattern matching problems and our techniques have the potential of offering solutions to these problems in a dynamic text setting
On the maximal number of cubic subwords in a string
We investigate the problem of the maximum number of cubic subwords (of the
form ) in a given word. We also consider square subwords (of the form
). The problem of the maximum number of squares in a word is not well
understood. Several new results related to this problem are produced in the
paper. We consider two simple problems related to the maximum number of
subwords which are squares or which are highly repetitive; then we provide a
nontrivial estimation for the number of cubes. We show that the maximum number
of squares such that is not a primitive word (nonprimitive squares) in
a word of length is exactly , and the
maximum number of subwords of the form , for , is exactly .
In particular, the maximum number of cubes in a word is not greater than
either. Using very technical properties of occurrences of cubes, we improve
this bound significantly. We show that the maximum number of cubes in a word of
length is between and . (In particular, we improve the
lower bound from the conference version of the paper.)Comment: 14 page
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