Generalised fine and Wilf's theorem for arbitrary number of periods
Sorin Constantinescu2005, Theoretical Computer Science
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Abstract
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.
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Fine and Wilf words for any periods
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.
Fine and Wilf words for any periods II
Theoretical Computer Science, 2009
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.
Fine and Wilf 's Periodicity on Partial Words and Consequences
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.
Fine and Wilf's theorem for abelian periods in partial words
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.
Computing Weak Periods of Partial Words EXTENDED ABSTRACT
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.
Local periods and binary partial words: an algorithm
Theoretical Computer Science, 2004
The study of the combinatorial properties of strings of symbols from a finite alphabet (also referred to as words) is profoundly connected to numerous fields such as biology, computer science, mathematics, and physics. Research in combinatorics on words goes back roughly a century. There is a renewed interest in combinatorics on words as a result of emerging new application areas such as molecular biology. Partial words were recently introduced in this context. The motivation behind the notion of a partial word is the comparison of genes (or proteins). Alignment of two genes (or two proteins) can be viewed as a construction of partial words that are said to be compatible. While a word can be described by a total function, a partial word can be described by a partial function. More precisely, a partial word of length n over a finite alphabet A is a partial function from {1, . . . , n} into A. Elements of {1, . . . , n} without an image are called holes. A word is just a partial word without holes. The notion of period of a word is central in combinatorics on words. In the case of partial words, there are two notions: one is that of period, the other is that of local period. This paper extends to partial words with one hole the well known result of Guibas and Odlyzko which states that for every word u, there exists a word v of same length as u over the alphabet {0, 1} such that the set of all periods of u coincides with the set of all periods of v. Our result states that for every partial word u with one hole, there exists a partial word v of same length as u with at most one hole over the alphabet {0, 1} such that the set of all periods of u coincides with the set of all periods of v and the set of all local periods of u coincides with the set of all local periods of v. To prove our result, we use the technique of Halava, Harju and Ilie which they used * This material is based upon work supported by the National Science Foundation under Grants CCR-9700228 and CCR-0207673. A Research Assignment from the University of North Carolina at Greensboro is gratefully acknowledged. I thank Phuongchi Thi Le for very valuable comments and suggestions. She received a research assistantship from the University of North Carolina at Greensboro to work with me on this project.
Partial words and a theorem of Fine and Wilf revisited
Theoretical Computer Science, 2002
A word of length n over a ÿnite alphabet A is a map from {0; : : : ; n − 1} into A. A partial word of length n over A is a partial map from {0; : : : ; n − 1} into A. In the latter case, elements of {0; : : : ; n − 1} without image are called holes (a word is just a partial word without holes). In this paper, we extend a fundamental periodicity result on words due to Fine and Wilf to partial words with two or three holes. This study was initiated by Berstel and Boasson for partial words with one hole. Partial words are motivated by molecular biology.
Quasi-Linear Time Computation of the Abelian Periods of a Word
In the last couple of years many research papers have been devoted to Abelian complexity of words. Recently, Constantinescu and Ilie (Bulletin EATCS 89, 167-170, 2006) introduced the notion of Abelian period. In this article we present two quadratic brute force algorithms for computing Abelian periods for special cases and a quasi-linear algorithm for computing all the Abelian periods of a word.
Relationally Periodic Sequences and Subword Complexity
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.
A note on efficient computation of all Abelian periods in a string
We derive a simple efficient algorithm for Abelian periods knowing all Abelian squares in a string. An efficient algorithm for the latter problem was given by Cummings and Smyth in 1997. By the way we show an alternative algorithm for Abelian squares. We also obtain a linear time algorithm finding all "long" Abelian periods. The aim of the paper is a (new) reduction of the problem of all Abelian periods to that of (already solved) all Abelian squares which provides new insight into both connected problems.
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Abelian periods, partial words, and an extension of a theorem of Fine and Wilf
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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.
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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.
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Constantinescu and Ilie (Bulletin EATCS 89, 167-170, 2006) introduced the notion of an Abelian period of a word. A word of length n over an alphabet of size σ can have Θ(n 2 ) distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time O(n 2 × σ) using O(n × σ) space. We present an off-line algorithm based on a select 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.
An Answer to a Conjecture on Overlaps in Partial Words Using Periodicity Algorithms
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.
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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.
Periodicity on partial words
Computers & Mathematics with Applications, 2004
Made available courtesy of Elsevier: http://www.elsevier.com ***Reprinted with permission. No further reproduction is authorized without written permission from Elsevier. This version of the document is not the version of record. Figures and/or pictures may be missing from this format of the document.*** Abstract: A partial word of length n over a finite alphabet A is a partial map from {0, … , n-1} into A. Elements of {0, … , n-1} without image are called holes (a word is just a partial word without holes). A fundamental periodicity result on words due to Fine and Wilf [1] intuitively determines how far two periodic events have to match in order to guarantee a common period. This result was extended to partial words with one hole by Berstel and Boasson [2] and to partial words with two or three holes by Blanchet-Sadri and Hegstrom [3]. In this paper, we give an extension to partial words with an arbitrary number of holes.
Computing abelian periods in words
2011
In the last couple of years many works have been devoted to Abelian complexity of words. Recently, Constantinescu and Ilie (Bulletin EATCS 89, 167-170, 2006) introduced the notion of Abelian period. We show that a word w of length n over an alphabet of size σ can have Θ(n 2) distinct Abelian periods. However, to the best of our knowledge, no efficient algorithm is known for computing these periods. The Brute-Force algorithm computes all the Abelian periods either in time O(n 3 × σ) using O(σ) space or in time O(n 2 × σ) using O(n × σ) space. We present an off-line algorithm running in time O(n 2 ×σ) using O(n+σ) space, thus improving the space complexity. This algorithm is based on a select function. We then present on-line algorithms that also enable to compute all the Abelian periods of all the prefixes of w. Experimental results show that the new off-line algorithm is faster than the Brute-Force one. Moreover, in most cases, one on-line algorithm, though having a worst case time complexity, is also faster than the Brute-Force one.
Fine and Wilf's theorem for three periods and a generalization of Sturmian words
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.
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