7 research outputs found
Abelian-Square-Rich Words
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain at most distinct
factors, and there exist words of length containing distinct
abelian-square factors, that is, distinct factors that are abelian squares.
This motivates us to study infinite words such that the number of distinct
abelian-square factors of length grows quadratically with . More
precisely, we say that an infinite word is {\it abelian-square-rich} if,
for every , every factor of of length contains, on average, a number
of distinct abelian-square factors that is quadratic in ; and {\it uniformly
abelian-square-rich} if every factor of contains a number of distinct
abelian-square factors that is proportional to the square of its length. Of
course, if a word is uniformly abelian-square-rich, then it is
abelian-square-rich, but we show that the converse is not true in general. We
prove that the Thue-Morse word is uniformly abelian-square-rich and that the
function counting the number of distinct abelian-square factors of length
of the Thue-Morse word is -regular. As for Sturmian words, we prove that a
Sturmian word of angle is uniformly abelian-square-rich
if and only if the irrational has bounded partial quotients, that is,
if and only if has bounded exponent.Comment: To appear in Theoretical Computer Science. Corrected a flaw in the
proof of Proposition
Words with the Maximum Number of Abelian Squares
An abelian square is the concatenation of two words that are anagrams of one
another. A word of length can contain distinct factors that
are abelian squares. We study infinite words such that the number of abelian
square factors of length grows quadratically with .Comment: To appear in the proceedings of WORDS 201
Hardness of Detecting Abelian and Additive Square Factors in Strings
We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the
3SUM conjecture) of several problems related to finding Abelian square and
additive square factors in a string. In particular, we conclude conditional
optimality of the state-of-the-art algorithms for finding such factors.
Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of
an odd half-length, (b) computing centers of all Abelian square factors, (c)
detecting an additive square factor in a length- string of integers of
magnitude , and (d) a problem of computing a double 3-term
arithmetic progression (i.e., finding indices such that
) in a sequence of integers of
magnitude .
Problem (d) is essentially a convolution version of the AVERAGE problem that
was proposed in a manuscript of Erickson. We obtain a conditional lower bound
for it with the aid of techniques recently developed by Dudek et al. [STOC
2020]. Problem (d) immediately reduces to problem (c) and is a step in
reductions to problems (a) and (b). In conditional lower bounds for problems
(a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it
using several string gadgets that include arbitrarily long Abelian-square-free
strings.
Our reductions also imply conditional lower bounds for detecting Abelian
squares in strings over a constant-sized alphabet. We also show a subquadratic
upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].Comment: Accepted to ESA 202
On k-abelian equivalence and generalized Lagrange spectra
We study the set of -abelian critical exponents of all Sturmian words. It has been proven that in the case this set coincides with the Lagrange spectrum. Thus the sets obtained when can be viewed as generalized Lagrange spectra. We characterize these generalized spectra in terms of the usual Lagrange spectrum and prove that when the spectrum is a dense non-closed set. This is in contrast with the case , where the spectrum is a closed set containing a discrete part and a half-line. We describe explicitly the least accumulation points of the generalized spectra. Our geometric approach allows the study of -abelian powers in Sturmian words by means of continued fractions.</p