Fine and Wilf words for any periods II

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, 2013

We investigate the least number of palindromic factors in an infinite word. We first consider general alphabets, and give answers to this problem for periodic and non-periodic words, closed or not under reversal of factors. We then investigate the same problem when the alphabet has size two.

Theoretical Computer Science, 2005

The well known Fine and Wilf's theorem for words states that if a word has two periods and its length is at least as long as the sum of the two periods minus their greatest common divisor, then the word also has as period the greatest common divisor. We generalise this result for an arbitrary number of periods. Our bound is strictly better in some cases than previous generalisations. Moreover, we prove it optimal. We show also that any extremal word is unique up to letter renaming and give an algorithm to compute both the bound and a corresponding extremal word.

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.

Fine and Wilf's well-known theorem states that any word having periods p, q and length at least p + q − gcd(p, q) also has gcd(p, q), the greatest common divisor of p and q, as a period. Moreover, the length p + q − gcd(p, q) is critical since counterexamples can be provided for shorter words. This result has since been extended to partial words, or finite sequences that may contain a number of "do not know" symbols or "holes." More precisely, any partial word u with H holes having weak periods p, q and length at least the so-denoted l H (p, q) also has strong period gcd(p, q) provided u is not (H,(p, q))-special. This extension was done for one hole by Berstel and Boasson (where the class of (1,(p, q))-special partial words is empty), for two or three holes by Blanchet-Sadri and Hegstrom, and for an arbitrary number of holes by Blanchet-Sadri. In this paper, we further extend these results, allowing an arbitrary number of weak periods. In addition to speciality, the concepts of intractable period sets and interference between periods play a role. * This material is based upon work supported by the National Science Foundation under Grant No. DMS-0452020. We thank the referees of a preliminary version of this paper for their very valuable comments and suggestions.

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.

Theoretical Computer Science, 2019

We investigate the least number of distinct palindromic sub-arrays in two-dimensional words over a finite alphabet = {a 1 , a 2 • • • , a q } for a given alphabet size q. We discuss the case for both periodic as well as aperiodic words.

Discrete Applied Mathematics, 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.

Discrete Mathematics, 2010

The palindrome complexity function pal w of a word w attaches to each n ∈ N the number of palindromes (factors equal to their mirror images) of length n contained in w. The number of all the nonempty palindromes in a finite word is called the total palindrome complexity of that word. We present exact bounds for the total palindrome complexity and construct words which have any palindrome complexity between these bounds, for binary alphabets as well as for alphabets with the cardinal greater than 2. Denoting by M q (n) the average number of palindromes in all words of length n over an alphabet with q letters, we present an upper bound for M q (n) and prove that the limit of M q (n)/n is 0. A more elaborate estimation leads to M q (n) = O( √ n).

Journal of Combinatorial Theory, Series A, 2015

We regard a finite word u = u 1 u 2 · · · u n up to word isomorphism as an equivalence relation on {1, 2, . . . , n} where i is equivalent to j if and only if u i = u j . Some finite words (in particular all binary words) are generated by palindromic relations of the form k ∼ j +i−k for some choice of 1 ≤ i ≤ j ≤ n and k ∈ {i, i + 1, . . . , j}. That is to say, some finite words u are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function µ(u) defined as the least number of palindromic relations required to generate u. We show that if x is an infinite word such that µ(u) ≤ 2 for each factor u of x, then x is ultimately periodic. On the other hand we establish the existence of non-ultimately periodic words for which µ(u) ≤ 3 for each factor u of x, and obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast, for the Thue-Morse word, we show that the function µ is unbounded.