Practical algorithms to rank necklaces, Lyndon words, and de Bruijn sequences

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چکیده

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Practical algorithms to rank necklaces, Lyndon words, and de Bruijn sequences

We present practical algorithms for ranking k-ary necklaces and Lyndon words of length n. The algorithms are based on simple counting techniques. By repeatedly applying the ranking algorithms, both necklaces and Lyndon words can be efficiently unranked. Then, explicit details are given to rank and unrank the length n substrings of the lexicographically smallest de Bruijn sequence of order n.

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This work generalizes the concepts of Lyndon words and de Bruijn sequence to the language of subshifts of finite type, extending its properties to this more general case. It is also studied the validity of Fredricksen and Maiorana theorem in this context and it is presented an algorithm to construct a de Bruijn sequence using Lyndon words.

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This paper creates a simple Gray code for fixed-density binary necklaces and Lyndon words. The Gray code is simultaneously a left-shift Gray code and a 2-transposition Gray code and is cyclic with these properties. It also has the advantage of using canonical representations (lexicographically minimum rotations) yielding the first known Gray code for Lyndon words ordered by density. This work s...

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Gray codes for necklaces and Lyndon words of arbitrary base

Recently, a Gray code for unrestricted binary necklaces and their relatives was discovered by Vajnovszki [Discrete Mathematics & Theoretical Computer Science, to appear]. The Gray code is constructed by modifying the classical FKM algorithm for generating necklaces in lexicographic order. We present a generalisation of Vajnovszki’s algorithm, giving a Gray code for necklaces and their relatives...

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ژورنال

عنوان ژورنال: Journal of Discrete Algorithms

سال: 2017

ISSN: 1570-8667

DOI: 10.1016/j.jda.2017.01.003