On Infinite Prefix Normal Words

Cicalese, Ferdinando; Lipták, Zsuzsanna; Rossi, Massimiliano

doi:10.1007/978-3-030-10801-4_11

Cited by 3 publications

(2 citation statements)

References 26 publications

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“…Enumeration and language-theoretic results were given by Burcsi et al in [12], and Balister and Gerke strengthened some results in [5]: in particular, they proved a conjecture about the asymptotic growth behaviour of the number of prefix normal words and gave a new result about the size of the equivalence classes. Cicalese et al gave a generation algorithm in [15], with linear running time per word, and studied infinite prefix normal words in [16]. Prefix normal words and prefix normal forms have been applied to a certain family of graphs by Blondin-Massé et al [7], and were shown to pertain to a new class of languages connected to the Reflected Binary Gray Code by Sawada et al [29].…”

Section: Related Workmentioning

confidence: 99%

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

Burcsi

Fici²,

Lipták

et al. 2020

Preprint

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A prefix normal word is a binary word with the property that no substring has more 1s than the prefix of the same length. By proving that the set of prefix normal words is a bubble language, we can exhaustively list all prefix normal words of length n as a combinatorial Gray code, where successive strings differ by at most two swaps or bit flips. This Gray code can be generated in O(log 2 n) amortized time per word, while the best generation algorithm hitherto has O(n) running time per word. We also present a membership tester for prefix normal words, as well as a novel characterization of bubble languages.

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Section: Related Workmentioning

confidence: 99%

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

Burcsi

Fici²,

Lipták

et al. 2020

Preprint

Self Cite

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show abstract

“…In "OEIS" [61] the number of prefix normal words of length n (A194850), a list of binary prefix normal words (A238109), the number of prefix normal palindromes of length n (A308465), and the maximum size of a class of binary words of length n having the same prefix normal form (A238110), can be found. An extension to infinite words is presented in [20].…”

Section: Chapter Introductionmentioning

confidence: 99%

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

Fleischmann¹

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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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On Collapsing Prefix Normal Words

Fleischmann

Kulczynski

Nowotka

et al. 2020

Language and Automata Theory and Applications

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Prefix normal words are binary words in which each prefix has at least the same number of 1s as any factor of the same length. Firstly introduced by Fici and Lipták in 2011, the problem of determining the index of the prefix 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.

show abstract

On Infinite Prefix Normal Words

Cited by 3 publications

References 26 publications

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

Generating a Gray code for prefix normal words in amortized polylogarithmic time per word

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

On Collapsing Prefix Normal Words

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