Two-Pattern strings

This paper introduces a new class of strings on {a, b}, called two-pattern strings, that constitute a substantial generalization of Sturmian strings while at the same time sharing many of their nice properties. In particular, we show that, in common with Sturmian strings, only time linear in the string length is required to recognize a two-pattern string as well as to compute all of its repetitions. We also show that two-pattern strings occur in some sense frequently in the class of all strings on {a,b}

On the maximal number of cubic subwords in a string

We investigate the problem of the maximum number of cubic subwords (of the form $www$) in a given word. We also consider square subwords (of the form $ww$). The problem of the maximum number of squares in a word is not well understood. Several new results related to this problem are produced in the paper. We consider two simple problems related to the maximum number of subwords which are squares or which are highly repetitive; then we provide a nontrivial estimation for the number of cubes. We show that the maximum number of squares $xx$ such that $x$ is not a primitive word (nonprimitive squares) in a word of length $n$ is exactly $\lfloor \frac{n}{2}\rfloor - 1$, and the maximum number of subwords of the form $x^k$, for $k\ge 3$, is exactly $n-2$. In particular, the maximum number of cubes in a word is not greater than $n-2$ either. Using very technical properties of occurrences of cubes, we improve this bound significantly. We show that the maximum number of cubes in a word of length $n$ is between $(1/2)n$ and $(4/5)n$. (In particular, we improve the lower bound from the conference version of the paper.)Comment: 14 page

Quasiperiodic Sturmian words and morphisms

We characterize all quasiperiodic Sturmian words: a Sturmian word is not quasiperiodic if and only if it is a Lyndon word. Moreover, we study links between Sturmian morphisms and quasiperiodicity

Permutation Complexity Related to the Letter Doubling Map

Given a countable set X (usually taken to be the natural numbers or integers), an infinite permutation, \pi, of X is a linear ordering of X. This paper investigates the combinatorial complexity of infinite permutations on the natural numbers associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula for the complexity is established for the Sturmian words and the Thue-Morse word.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Algorithms for Computing Abelian Periods of Words

Constantinescu and Ilie (Bulletin EATCS 89, 167--170, 2006) introduced the notion of an \emph{Abelian period} of a word. A word of length $n$ over an alphabet of size $\sigma$ can have $\Theta(n^{2})$ distinct Abelian periods. The Brute-Force algorithm computes all the Abelian periods of a word in time $O(n^2 \times \sigma)$ using $O(n \times \sigma)$ space. We present an off-line algorithm based on a \sel 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 $w$.Comment: Accepted for publication in Discrete Applied Mathematic

Sorting suffixes of two-pattern strings

Recently, several authors presented linear recursive algorithms for sorting suffixes of a string. All these algorithms employ a similar three-step approach, based on an initial division of the suffixes of x into two sets: in step 1 sort the first set using recursive reduction of the problem, in step 2 determine the order of the suffixes in the second set based on the order of the suffixes in the first set, and in step 3 merge the two sets together. To optimize such an algorithm either for space or time, it may not be sufficient to optimize one of the three steps, since in doing so, one might increase the resources required for the others to an unacceptable extent. Franek, Lu, and Smyth introduced two-pattern strings as a generalization of Sturmian strings. Like Sturmian strings, two-pattern strings are generated by iterated morphisms, but they exhibit a much richer structure. In this paper we show that the suffixes of two-pattern strings can be sorted in linear time using a variant of the three step approach outlined above. It turns out that, given the order of the suffixes in a two-pattern string, one can almost directly list in linear time all the suffixes of its expansion under a two-pattern morphism

Sorting suffixes of two-pattern strings

Recently, several authors presented linear recursive algorithms for sorting suffixes of a string. All these algorithms employ a similar three-step approach, based on an initial division of the suffixes of x into two sets: in step 1 sort the first set using recursive reduction of the problem, in step 2 determine the order of the suffixes in the second set based on the order of the suffixes in the first set, and in step 3 merge the two sets together. To optimize such analgorithm either for space or time, it may not be sufficient to optimize one of the three steps, since in doing so, one might increase the resources required for the others to an unacceptable extent. Franek, Lu, and Smyth introduced two-pattern strings as a generalization of Sturmian strings. Like Sturmian strings, two-pattern strings are generated by iterated morphisms, but they exhibit a much richer structure. In this paper we show that the suffixes of two-pattern strings can be sorted in linear time using a variant of the three step approach outlined above. It turns out that, given the order of the suffixes in a two-pattern string, one can almost directly list in linear time all the suffixes of its expansion under a two-pattern morphism