Partial words and the critical factorization theorem

Information and Computation, 2008

The concept of periodicity has played over the years a centra1 role in the development of combinatorics on words and has been a highly valuable too1 for the design and analysis of algorithms. Fine and Wilf's famous periodicity result, which is one of the most used and known results on words, has extensions to partia1 words, or sequences that may have a number of "do not know" symbols. These extensions fal1 into two categories: the ones that relate to strong periodicity and the ones that relate to weak periodicity. In this paper, we obtain consequences by generalizing, in particular, the combinatoria1 property that "for any word u over {a, b}, ua or ub is primitive," which proves in some sense that there exist very many primitive partia1 words.

The concept of periodicity has played over the years a central role in the development of combinatorics on words and has been a highly valuable tool for the design and analysis of algorithms. There are many fundamental periodicity results on words. Among them is the famous result of Fine and Wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period. This result states that for positive integers p and q, if the word u has periods p and q and the length of u is not less than p + q − gcd(p, q), then u has also period gcd(p, q). Fine and Wilf's result, which is one of the most used and known results on words, has extensions to partial words, or sequences that may have a number of "do not know" symbols. These extensions fall into two categories: The ones that relate to strong periodicity and the ones that relate to weak periodicity. In this paper, we study some consequences of these results.

2012

We propose an algorithm that given as input a full word w of length n, and positive integers p and d, outputs, if any exists, a maximal p-periodic partial word contained in w with the property that no two holes are within distance d (so-called d-valid). Our algorithm runs in O (nd) time and is used for the study of repetition-freeness of partial words.

Lecture Notes in Computer Science, 2009

We propose an algorithm that given as input a full word w of length n, and positive integers p and d, outputs (if any exists) a maximal p-periodic partial word contained in w with the property that no two holes are within distance d. Our algorithm runs in O(nd) time and is used for the study of freeness of partial words. Furthermore, we construct an infinite word over a five-letter alphabet that is overlapfree even after the insertion of an arbitrary number of holes, answering affirmatively a conjecture from Blanchet-Sadri, Mercaş, and Scott.

Information and Computation, 2008

Acta Informatica, 2001

We introduce the notion of periodic-like word. It is a word whose longest repeated prefix is not right special. Some different characterizations of this concept are given. In particular, we show that a word w is periodic-like if and only if it has a period not larger than |w| − R w , where R w is the least non-negative integer such that any prefix of w of length ≥ R w is not right special. We derive that if a word w has two periods p, q ≤ |w| − R w , then also the greatest common divisor of p and q is a period of w. This result is, in fact, an improvement of the theorem of Fine and Wilf. We also prove that the minimal period of a word w is equal to the sum of the minimal periods of its components in a suitable canonical decomposition in periodic-like subwords. Moreover, we characterize periodic-like words having the same set of proper boxes, in terms of the important notion of root-conjugacy. Finally, some new uniqueness conditions for words, related to the maximal box theorem are given.

2012

Partial words, or sequences over a finite alphabet that may have do-not-know symbols or holes, have been recently the subject of much investigation. Several interesting combinatorial properties have been studied such as the periodic behavior and the counting of distinct squares in partial words. In this paper, we extend the three-squares lemma on words to partial words with one hole.

Lecture Notes in Computer Science, 2008

By the famous theorem of Morse and Hedlund, a word is ultimately periodic if and only if it has bounded subword complexity, i.e., for sufficiently large n, the number of factors of length n is constant. In this paper we consider relational periods and relationally periodic sequences, where the relation is a similarity relation on words induced by a compatibility relation on letters. We investigate what would be a suitable definition for a relational subword complexity function such that it would imply a Morse and Hedlund-like theorem for relationally periodic words. We consider strong and weak relational periods and two candidates for subword complexity functions.

2008

Combinatorics on words, or sequences or strings of symbols over a finite alphabet, is a rather new field although the first papers were published at the beginning of the 20th century [120, 121]. The interest in the study of combinatorics on words has been increasing since it finds applications in various research areas of mathematics, computer science, and biology where the data can be easily represented as words over some alphabet. Such areas may be concerned with algorithms on strings [38, 48, 50, 51, 52, 69, 72, 84, 102, 118], semigroups, automata and languages [2, 45, 55, 75, 82, 92, 93], molecular genetics [78], or codes [5, 73, 79]. Motivated by molecular biology of nucleic acids, Berstel and Boasson introduced in 1999 the notion of partial words which are sequences that may contain a number of “do not know” symbols or “holes” [4]. DNA molecules are the carriers of the genetic information in almost all organisms. Let us look into the structure of such a molecule. A single stra...

RAIRO - Theoretical Informatics and Applications, 2009

A well known result of Fraenkel and Simpson states that the number of distinct squares in a full word of length n is bounded by 2n since at each position there are at most two distinct squares whose last occurrence start. In this paper, we investigate squares in partial words with one hole, or sequences over a finite alphabet that have a "do not know" symbol or "hole." A square in a partial word over a given alphabet has the form uv where u is compatible with v, and consequently, such square is compatible with a number of full words over the alphabet that are squares. Recently, it was shown that for partial words with one hole, there may be more than two squares that have their last occurrence starting at the same position. Here, we prove that if such is the case, then the length of the shortest square is at most half the length of the third shortest square. As a result, we show that the number of distinct full squares compatible with factors of a partial word with one hole of length n is bounded by 7n 2 .

RAIRO - Theoretical Informatics and Applications, 2013

Recently, Constantinescu and Ilie proved a variant of the well-known periodicity theorem of Fine and Wilf in the case of two relatively prime abelian periods and conjectured a result for the case of two non-relatively prime abelian periods. In this paper, we answer some open problems they suggested. We show that their conjecture is false but we give bounds, that depend on the two abelian periods, such that the conjecture is true for all words having length at least those bounds and show that some of them are optimal. We also extend their study to the context of partial words, giving optimal lengths and describing an algorithm for constructing optimal words.

Information and Computation/information and Control, 2011

One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson-Crick complement, denoted here as θ(u). Thus, any expression consisting of repetitions of u and θ(u) can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger's classical result about equations of the form u l = v n w m , to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if l 5, n, m 3, then all three words involved can be expressed in terms of a common word t and its complement θ(t). Moreover, if l 5, then n = m = 3 is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement θ(u), which is also obtained in this paper. Crown

Indagationes Mathematicae, 2003

Let w = wt . ..w., be a word of maximal length n, and with a maximal number of distinct letters for this length, such that w has periods pt, . . ..pr but not period gcd(pt, . . ..pr). We provide a fast algorithm to compute n and w. We show that w is uniquely determined apart from isomorphism and that it is a palindrome. Furthermore we give lower and upper bounds for n as explicit functions of pt, . . ..pr. For I = 2 the exact value of n is due to Fine and Wilf. In case the number of distinct letters in the extremal word equals r a formula for n had been given by Castelli, Mignosi and Restivo in case Y = 3 and by Justin if r > 3.

Theoretical Computer Science, 1999

We 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 Amoux and Rauzy to generalize Sturmian words, have the same set of factors.

Proceedings of the 13th …, 2010

Recently, Constantinescu and Ilie proved a variant of the well-known periodicity theorem of Fine and Wilf in the case of two relatively prime abelian periods, and conjectured a result for the case of two non-relatively prime abelian periods. More precisely, they proved that any full word having two coprime abelian periods p, q and length at least 2pq − 1 has also gcd(p, q) = 1 as a period. In this paper, we answer some open problems they suggested by proving that the length 2pq − 1 is optimal and by answering affirmatively their conjecture. We also extend their study in the context of partial words, giving optimal lengths and describing an algorithm for constructing optimal words.