Abstract not found

Abstract not found

A weak repetition in a string consists of two or more adjacent substrings which are permutations of each other. We describe a straightforward \Theta(n 2 ) algorithm which computes all the weak repetitions in a given string of length n defined on an arbitrary alphabet A. Using results on Fibonacci and other simple strings, we prove that this algorithm is asymptotically optimal over all known encodings of the output. 1 INTRODUCTION Interest in the periodic behaviour of strings dates back to Thue [T06] at the turn of the century. Thue considered what we call here strong repetitions (equal adjacent substrings) and showed how to construct an infinitely long string on an alphabet of only three letters with no strong repetitions. (Other constructions on three letters have been discovered several times since, most recently by Dekking [D79] and Pleasants [P70] --- the latter lists several references to earlier constructions.) More recently, Erdos [E61, p. 240] considered "Abelian squares" (w...

this paper, it is shown that the subword complexity of a D0L language is bounded by cn (resp. cn log n, cn) if the morphism that generates the languages is arbitrary (resp. growing, uniform). This result was extended in [Pan84a]: Theorem 6.7. The subword complexity of an in nite word generated by iterating a morphism is of one of the following types: (n), (n log n), (n log n log n), (n ), or (1)

A sequence a = a1a2...an is said to be non-repetitive if no two adjacent blocks of a are exactly the same. For instance the sequence 1232321 contains a repetition 2323, while 123132123213 is non-repetitive. A theorem of Thue asserts that, using only three symbols, one can produce arbitrarily long non-repetitive sequences. In this paper we consider a natural generalization of Thue’s sequences for colorings of graphs. A coloring of the set of edges of a given graph G is non-repetitive if the sequence of colors on any path in G is non-repetitive. We call the minimal number of colors needed for such a coloring the Thue number of G and denote it by π(G). The main problem we consider is the relation between the numbers π(G) and ∆(G). We show, by an application of the Lovász Local Lemma, that the Thue number stays bounded for graphs with bounded maximum degree, in particular, π(G) ≤ c∆(G) 2 for some absolute constant c. For certain special classes of graphs we obtain linear upper bounds on π(G), by giving explicit colorings. For instance, the Thue number of the complete graph Kn is at most 2n − 3, and π(T) ≤ 4(∆(T) − 1) for any tree T with at least two edges. We conclude by discussing some generalizations and proposing several problems and conjectures.

We count the number of occurrences of certain patterns in given words. We choose these words to be the set of all finite approximations of a sequence generated by a morphism with certain restrictions. The patterns in our considerations are either classical patterns 1-2, 2-1, 1-1- · · ·-1, or arbitrary generalized patterns without internal dashes, in which repetitions of letters are allowed. In particular, we find the number of occurrences of the patterns 1-2, 2-1, 12, 21, 123 and 1-1- · · ·-1 in the words obtained by iterations of the morphism 1 → 123, 2 → 13, 3 → 2, which is a classical example of a morphism generating a nonrepetitive sequence.

Abstract. The paper contains all the definitions and notations needed to understand the results concerning the field dealing with occurrences of patterns in permutations and words. Also, this paper includes a historical overview on the results obtained in this subject. The authors tried to collect all the currently existing references to the papers directly related to the subject. Moreover, a number of basic approaches to study the pattern problems are discussed.

A sequence u = u 1 u 2 :::u n is said to be nonrepetitive if no two adjacent blocks of u are exactly the same. For instance, the sequence abcbcba contains a repetition bcbc, while abcacbabcbac is nonrepetitive. A well known theorem of Thue asserts that there are arbitrarily long nonrepetitive sequences over the set fa; b; cg. This fact implies, via König's Infinity Lemma, the existence of an infinite ternary sequence without repetitions of any length. In this

A string is Abelian square-free if it contains no Abelian squares; that is, adjacent substrings which are permutations of each other. An Abelian square-free string is maximal if it cannot be extended to the left or right by concatenating alphabet symbols without introducing an Abelian square. We construct Abelian square-free finite strings which are maximal by modifying a construction of Zimin. The new construction produces maximal strings whose length as a function of alphabet size is much shorter than that in the construction described by Zimin. 1

In this paper some of the work done on repetitions in strings is surveyed, especially that of an algorithmic nature. Several open problems are described and conjectures formulated about some of them. KEYWORDS: string, word, repetition, repeat, cover 1 INTRODUCTION Repetitions in strings are usually thought of as adjacent or \tandem"; that is, the string uvu is counted as a repetition of u if and only if v = , the empty string. However, in certain contexts | for example, DNA sequence analysis [S98], data compression [IS98], analysis of musical texts [CIR96] | this denition may be too narrow. Here therefore we take a wider view and regard uvu as a repetition of a nonempty string u for any nite string v. Even more generally, we also count as repetitions cases where the string u overlaps itself; for example, abaabaab is accepted as a repetition of abaab. In order to make sure that these ideas are clear, we express them more formally. Throughout this paper x will denote a string of l...