Rich and Periodic-Like Words (original) (raw)
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A new characteristic property of rich words
Theoretical Computer Science, 2009
Originally introduced and studied by the third and fourth authors together with J. Justin and S. Widmer (2008), rich words constitute a new class of finite and infinite words characterized by containing the maximal number of distinct palindromes. Several characterizations of rich words have already been established. A particularly nice characteristic property is that all 'complete returns' to palindromes are palindromes. In this note, we prove that rich words are also characterized by the property that each factor is uniquely determined by its longest palindromic prefix and its longest palindromic suffix.
On periodic properties of circular words
Discrete Mathematics, 2016
The conjugacy relation defines a partition of words into equivalence classes. We call these classes circular words. Periodic properties of circular words are investigated in this article. The Periodicity Theorem of Fine and Wilf does not hold for weak periods of circular words; instead we give a strict upper bound on the length of a non-unary circular word that has two given relatively prime weak periods. Weak periods also lead to a way of representing circular words in a more compact form. We investigate in which cases are these representations unique or minimal. We will also analyze weak periods of circular Thue-Morse, Fibonacci and Christoffel words.
Periodic-like words, periodicity, and boxes
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.
Palindromes and local periodicity
2009
In this paper we consider several types of equations on words, motivated by the attempt of characterizing the class of polyominoes that tile the plane by translation in two distinct ways. Words coding the boundary of these polyominoes satisfy an equation whose solutions are in bijection with a subset of the solutions of equations of the form AB A B ≡ XY X Y. It turns out that the solutions are strongly related to local periodicity involving palindromes and conjugate words.
On the least number of palindromes contained in an infinite word
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.
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.
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.
Special factors, periodicity, and an application to Sturmian words
Acta Informatica, 2000
Let w be a finite word and n the least non-negative integer such that w has no right special factor of length n and its right factor of length n is unrepeated. We prove that if all the factors of another word v up to the length n + 1 are also factors of w, then v itself is a factor of w. A similar result for ultimately periodic infinite words is established. As a consequence, some 'uniqueness conditions' for ultimately periodic words are obtained as well as an upper bound for the rational exponents of the factors of uniformly recurrent non-periodic infinite words. A general formula is derived for the 'critical exponent' of a power-free Sturmian word. In particular, we effectively compute the 'critical exponent' of any Sturmian sequence whose slope has a periodic development in a continued fraction.
Rich, Sturmian, and trapezoidal words
Theoretical Computer Science, 2008
In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite Sturmian words. Unfortunately this property does not characterize finite Sturmian words. In this note we show that the only trapezoidal palindromes are Sturmian. More generally we show that Sturmian palindromes can be characterized either in terms of their subword complexity (the trapezoidal property) or in terms of their palindromic complexity. We also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity.
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.