Search results for "Combinatorics on word"
showing 10 items of 55 documents
Fine and Wilf's Theorem for Three periods and a Generalization of Sturmian Words
1999
AbstractWe extend the theorem of Fine and Wilf to words having three periods. We then define the set 3-PER of words of maximal length for which such result does not apply. We prove that the set 3-PER and the sequences of complexity 2n + 1, introduced by Arnoux and Rauzy to generalize Sturmian words, have the same set of factors.
DEFECT THEOREMS FOR TREES
2000
We generalize different notions of a rank of a set of words to sets of trees. We prove that almost all of those ranks can be used to formulate a defect theorem. However, as we show, the prefix rank forms an exception.
Equations on trees
1996
We introduce the notion of equation on trees, generalizing the corresponding notion for words, and we develop the first steps of a theory of tree equations. The main result of the paper states that, if a pair of trees is the solution of a tree equation with two indeterminates, then the two trees are both powers of the same tree. As an application, we show that a tree can be expressed in a unique way as a power of a primitive tree. This extends a basic result of combinatorics on words to trees. Some open problems are finally proposed.
Novel Results on the Number of Runs of the Burrows-Wheeler-Transform
2021
The Burrows-Wheeler-Transform (BWT), a reversible string transformation, is one of the fundamental components of many current data structures in string processing. It is central in data compression, as well as in efficient query algorithms for sequence data, such as webpages, genomic and other biological sequences, or indeed any textual data. The BWT lends itself well to compression because its number of equal-letter-runs (usually referred to as $r$) is often considerably lower than that of the original string; in particular, it is well suited for strings with many repeated factors. In fact, much attention has been paid to the $r$ parameter as measure of repetitiveness, especially to evalua…
On the Number of Closed Factors in a Word
2015
A closed word (a.k.a. periodic-like word or complete first return) is a word whose longest border does not have internal occurrences, or, equivalently, whose longest repeated prefix is not right special. We investigate the structure of closed factors of words. We show that a word of length $n$ contains at least $n+1$ distinct closed factors, and characterize those words having exactly $n+1$ closed factors. Furthermore, we show that a word of length $n$ can contain $\Theta(n^{2})$ many distinct closed factors.
Properties of a Class of Toeplitz Words
2021
We study the properties of the uncountable set of Stewart words. These are Toeplitz words specified by infinite sequences of Toeplitz patterns of the form $\alpha\beta\gamma$, where $\alpha,\beta,\gamma$ is any permutation of the symbols 0,1,?. We determine the critical exponent of the Stewart words, prove that they avoid the pattern $xxyyxx$, find all factors that are palindromes, and determine their subword complexity. An interesting aspect of our work is that we use automata-theoretic methods and a decision procedure for automata to carry out the proofs.
Algorithms for Computing Abelian Periods of Words
2012
Constantinescu and Ilie (Bulletin EATCS 89, 167--170, 2006) introduced the notion of an \emph{Abelian period} of a word. A word of length $n$ over an alphabet of size $\sigma$ can have $\Theta(n^{2})$ distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time $O(n^2 \times \sigma)$ using $O(n \times \sigma)$ space. We present an off-line algorithm based on a $\sel$ function having the same worst-case theoretical complexity as the Brute-Force one, but outperforming it in practice. We then present on-line algorithms that also enable to compute all the Abelian periods of all the prefixes of $w$.
Algorithms for Anti-Powers in Strings
2018
Abstract 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 op…
A note on easy and efficient computation of full abelian periods of a word
2016
Constantinescu and Ilie (Bulletin of the EATCS 89, 167-170, 2006) introduced the idea of an Abelian period with head and tail of a finite word. An Abelian period is called full if both the head and the tail are empty. We present a simple and easy-to-implement $O(n\log\log n)$-time algorithm for computing all the full Abelian periods of a word of length $n$ over a constant-size alphabet. Experiments show that our algorithm significantly outperforms the $O(n)$ algorithm proposed by Kociumaka et al. (Proc. of STACS, 245-256, 2013) for the same problem.
Abelian combinatorics on words: A survey
2022
We survey known results and open problems in abelian combinatorics on words. Abelian combinatorics on words is the extension to the commutative setting of the classical theory of combinatorics on words. The extension is based on \emph{abelian equivalence}, which is the equivalence relation defined in the set of words by having the same Parikh vector, that is, the same number of occurrences of each letter of the alphabet. In the past few years, there was a lot of research on abelian analogues of classical definitions and properties in combinatorics on words. This survey aims to gather these results.