6 research outputs found
Algorithms for anti-powers in strings
A string S[1,n] is a power (or tandem repeat) of order k and period n/k if it can be decomposed into k consecutive equal-length blocks of letters. Powers and periods are fundamental to string processing, and algorithms for their efficient computation have wide application and are heavily studied. Recently, Fici et al. (Proc. ICALP 2016) defined an anti-power of order k to be a string composed of k pairwise-distinct blocks of the same length (n/k, called anti-period). Anti-powers are a natural converse to powers, and are objects of combinatorial interest in their own right. In this paper we initiate the algorithmic study of anti-powers. Given a string S, we describe an optimal algorithm for locating all substrings of S that are anti-powers of a specified order. The optimality of the algorithm follows form a combinatorial lemma that provides a lower bound on the number of distinct anti-powers of a given order: we prove that a string of length n can contain Θ(n2/k) distinct anti-powers of order k.Peer reviewe
Anti-Power -fixes of the Thue-Morse Word
Recently, Fici, Restivo, Silva, and Zamboni introduced the notion of a
-anti-power, which is defined as a word of the form , where are distinct words of the
same length. For an infinite word and a positive integer , define
to be the set of all integers such that is a -anti-power, where denotes the -th letter of .
Define also ,
where denotes the Thue-Morse word. For all ,
is a well-defined positive integer,
and for sufficiently large, is a well-defined odd positive
integer. In his 2018 paper, Defant shows that and
grow linearly in . We generalize Defant's methods to prove that
and grow linearly in for any nonnegative
integer . In particular, we show that and . Additionally, we show
that and
.Comment: 19 page
Powers and Anti-Powers in Binary Words
Fici et al. recently introduced the notion of anti-powers in the context of combinatorics on words. A power (also called tandem repeat) is a sequence of consecutive identical blocks. An anti-power is a sequence of consecutive distinct blocks of the same length. Fici et al. showed that the existence of powers or anti-powers is an unavoidable regularity for sufficiently long words. In this thesis we explore this notion further in the context of binary words and obtain new results