# On Special k-Spectra, k-Locality, and Collapsing Prefix Normal Words

The domain of Combinatorics on Words, first introduced by Axel Thue in 1906, covers by now many subdomains. In this work we are investigating scattered factors as a representation of non-complete information and two measurements for words, namely the locality of a word and prefix normality, which have applications in pattern matching. In the first part of the thesis we investigate scattered factors: A word u is a scattered factor of w if u can be obtained from w by deleting some of its letters. That is, there exist the (potentially empty) words u1, u2, . . . , un, and v0,v1,...,vn such that u = u1u2 ̈ ̈ ̈un and w = v0u1v1u2v2 ̈ ̈ ̈unvn. First, we consider the set of length-k scattered factors of a given word w, called the k-spectrum of w and denoted by ScatFactk(w). We prove a series of properties of the sets ScatFactk(w) for binary weakly-0-balanced and, respectively, weakly-c-balanced words w, i.e., words over a two- letter alphabet where the number of occurrences of each letter is the same, or, respectively, one letter has c occurrences more than the other. In particular, we consider the question which cardinalities n = | ScatFactk (w)| are obtainable, for a positive integer k, when w is either a weakly-0- balanced binary word of length 2k, or a weakly-c-balanced binary word of length 2k ́ c. Second, we investigate k-spectra that contain all possible words of length k, i.e., k-spectra of so called k-universal words. We present an algorithm deciding whether the k-spectra for given k of two words are equal or not, running in optimal time. Moreover, we present several results regarding k-universal words and extend this notion to circular universality that helps in investigating how the universality of repetitions of a given word can be determined. We conclude the part about scattered factors with results on the reconstruction problem of words from scattered factors that asks for the minimal information, like multisets of scattered factors of a given length or the number of occurrences of scattered factors from a given set, necessary to uniquely determine a word. We show that a word w P {a, b} ̊ can be reconstructed from the number of occurrences of at most min(|w|a, |w|b) + 1 scattered factors of the form aib, where |w|a is the number of occurrences of the letter a in w. Moreover, we generalise the result to alphabets of the form {1, . . . , q} by showing that at most ∑q ́1 |w|i (q ́ i + 1) scattered factors suffices to reconstruct w. Both results i=1 improve on the upper bounds known so far. Complexity time bounds on reconstruction algorithms are also considered here. In the second part we consider patterns, i.e., words consisting of not only letters but also variables, and in particular their locality. A pattern is called k-local if on marking the pattern in a given order never more than k marked blocks occur. We start with the proof that determining the minimal k for a given pattern such that the pattern is k-local is NP- complete. Afterwards we present results on the behaviour of the locality of repetitions and palindromes. We end this part with the proof that the matching problem becomes also NP-hard if we do not consider a regular pattern - for which the matching problem is efficiently solvable - but repetitions of regular patterns. In the last part we investigate prefix normal words which are binary words in which each prefix has at least the same number of 1s as any factor of the same length. First introduced in 2011 by Fici and Lipták, the problem of determining the index (amount of equivalence classes for a given word length) of the prefix normal equivalence relation is still open. In this paper, we investigate two aspects of the problem, namely prefix normal palindromes and so-called collapsing words (extending the notion of critical words). We prove characterizations for both the palindromes and the collapsing words and show their connection. Based on this, we show that still open problems regarding prefix normal words can be split into certain subproblems.

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