Graph Connectivity, Partial Words, and a Theorem of Fine and Wilf (original) (raw)
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Periodicity algorithms and a conjecture on overlaps in partial words
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.
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.
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.
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.
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.
A periodicity result of partial words with one hole
Computers & Mathematics with Applications, 2003
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, All rights reserved.
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.
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.
Periodicity properties on partial words
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.
Partial words and the critical factorization theorem
Journal of Combinatorial Theory, Series A, 2005
The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial words that are sequences that may have a number of "do not know" symbols also called "holes". This paper is devoted to a fundamental result on periods of words, the Critical Factorization Theorem, which states that the period of a word is always locally detectable in at least one position of the word resulting in a corresponding critical factorization. Here, we describe precisely the class of partial words w with one hole for which the weak period is locally detectable in at least one position of w. Our proof provides an algorithm which computes a critical factorization when one exists. A World Wide Web server interface at http://www.uncg.edu/mat/cft/ has been established for automated use of the program. We thank Ajay Chriscoe for very valuable comments and suggestions, and for implementing Algorithm 2 and creating a World Wide Web site for this research. We also thank the referee of a preliminary version of this paper for his/her very valuable comments and suggestions. 1 sequences that may have a number of "do not know" symbols. Such sequences are referred to as partial words and appear, for instance, when genes or proteins are compared. Another area of current interest for the study of the combinatorics on partial words is data communication where some information may be missing, lost, or unknown. While a word can be described by a total function, a partial word can be described by a partial function.