Dirk Nowotka

Height-deterministic pushdown automata

We define the notion of height-deterministic pushdown automata, a model where for any given input string the stack heights during any (nondeterministic) computation on the input are a priori fixed. Different subclasses of height-deterministic pushdown automata, strictly containing the class of regular languages and still closed under boolean language operations, are considered. Several such language classes have been described in the literature. Here, we suggest a natural and intuitive model that subsumes all the formalisms proposed so far by employing height-deterministic pushdown automata. Decidability and complexity questions are also considered.

Periodicity and unbordered words: A proof of the extended duval conjecture

The relationship between the length of a word and the maximum length of its unbordered factors is investigated in this article. Consider a finite word <i>w</i> of length <i>n</i>. We call a word <i>bordered</i> if it has a proper prefix, which is also a suffix of that word. Let μ(<i>w</i>) denote the maximum length of all unbordered factors of <i>w</i>, and let ∂(<i>w</i>) denote the period of <i>w</i>. Clearly, μ(<i>w</i>) ≤ ∂(<i>w</i>). We establish that μ(<i>w</i>) = ∂(<i>w</i>), if <i>w</i> has an unbordered prefix of length μ(<i>w</i>) and <i>n</i> ≥ 2μ(<i>w</i>) − 1. This bound is tight and solves the stronger version of an old conjecture by Duval [1983]. It follows from this result that, in general, <i>n</i> ≥ 3μ(<i>w</i>) − 3 implies μ(<i>w</i>) = ∂(<i>w</i>), which gives an improved bound for the question raised by Ehrenfeucht and Silberger in 1979.

MINIMAL DUVAL EXTENSIONS

A word v=wu is a (nontrivial) Duval extension of the unbordered word w, if (u is not a prefix of v and) w is an unbordered factor of v of maximum length. After a short survey of the research topic related to Duval extensions, we show that, if wu is a minimal Duval extension, then u is a factor of w. We also show that finite, unbordered factors of Sturmian words are Lyndon words.

Density of Critical Factorizations

We investigate the density of critical factorizations of infinte sequences of words. The density of critical factorizations of a word is the ratio between the number of positions that permit a critical factorization, and the number of all positions of a word. We give a short proof of the Critical Factorization Theorem and show that the maximal number of noncritical positions of a word between two critical ones is less than the period of that word. Therefore, we consider only words of index one, that is words where the shortest period is larger than one half of their total length, in this paper. On one hand, we consider words with the lowest possible number of critical points and show, as an example, that every Fibonacci word longer than five has exactly one critical factorization and every palindrome has at least two critical factorizations. On the other hand, sequences of words with a high density of critical points are considered. We show how to construct an infinite sequence of words in four letters where every point in every word is critical. We construct an infinite sequence of words in three letters with densities of critical points approaching one, using square-free words, and an infinite sequence of words in two letters with densities of critical points approaching one half, using Thue-Morse words. It is shown that these bounds are optimal.

Testing Generalised Freeness of Words

Pseudo-repetitions are a natural generalisation of the classical notion of repetitions in sequences: they are the repeated concatenation of a word and its encoding under a certain morphism or antimorphism (anti-/morphism, for short). We approach the problem of deciding eciently, for a word w and a literal anti-/morphism f, whether w contains an instance of a given pattern involving a variablex and its image underf, i.e., f(x). Our results generalise both the problem of finding fixed repetitive structures (e.g., squares, cubes) inside a word and the problem of finding palindromic structures inside a word. For instance, we can detect eciently a factor of the form xx R xxx R , or any other pattern of such type. We also address the problem of testing eciently, in the same setting, whether the wordw contains an arbitrary pseudo-repetition of a given exponent. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems

Fine and wilf's theorem and pseudo-repetitions

The notion of repetition of factors in words is central to considerations on sequences. One of the recent generalizations regarding this concept was introduced by Czeizler et al. (2010) and investigates a restricted version of that notion in the context of DNA computing and bioinformatics. It considers a word to be a pseudo-repetition if it is the iterated concatenation of one of its prefixes and the image of this prefix through an involution. We present here a series of results in the fashion of the Fine and Wilf Theorem in a more general setting where we consider the periods of some word given by a prefix of it and images of that prefix through some arbitrary morphism or antimorphism.

Unary patterns with involution

An infinite word w avoids a pattern p with the involution if there is no substitution for the variables in p and no involution on substituted variables such that the resulting word is a factor of w. An avoidance index of pattern p is the smallest alphabet size for which a word exists such that p is avoided. A pattern is called unary, if only one variable occurs in it. In this paper, we give the avoidance indices for all unary patterns with involution.

On the independence of equations in three variables

We prove that an independent system of equations in three variables with a nonperiodic solution and at least two equations consists of balanced equations only. For that, we show that the intersection of two different entire systems contains only balanced equations, where an entire system is the set of all equations solved by a given morphism. Furthermore, we establish that two equations which have a common nonperiodic solution have the same set of periodic solutions or are not independent.

The avoidability of cubes under permutations

In this paper we consider the avoidance of patterns in infinite words. Generalising the traditional problem setting, functional dependencies between pattern variables are allowed here, in particular, patterns involving permutations. One of the remarkable facts is that in this setting the notion of avoidability index (the smallest alphabet size for which a pattern is avoidable) is meaningless since a pattern with permutations that is avoidable in one alphabet can be unavoidable in a larger alphabet. We characterise the (un-)avoidability of all patterns of the form πi(x) πj(x) πk(x), called cubes under permutations here, for all alphabet sizes in both the morphic and antimorphic case.

Estimation of the click volume by large scale regression analysis

How could one estimate the total number of clicks a new advertisement could potentially receive in the current market? This question, called the click volume estimation problem is investigated in this paper. This constitutes a new research direction for advertising engines. We propose a model of computing an estimation of the click volume. A key component of our solution is the application of linear regression to a large (but sparse) data set. We propose an iterative method in order to achieve a fast approximation of the solution. We prove that our algorithm always converges to optimal parameters of linear regression. To the best of our knowledge, it is the first time when linear regression is considered in such a large scale context