Graphes et cycles de de Bruijn dans des langages avec des restrictions

Abstract : Let be a language composed by all words of a given length $n$. A de Bruijn sequence of span $n$ is a cyclic string such that all words in the language appears exactly once as factors of this sequence. One of the algorithms to construct the lexicographically minimal de Bruijn sequence is due to Fredricksen and Maiorana and it use the Lyndon words in the language. This thesis studies how to generalize the concept of de Bruijn sequence for a language composed by a subset of words of length $n$, particularly the languages of all words of length $n$ without factors in a list of forbidden factors. Firstly, we study the case of words without the factor 11. We give a new proof of the algorithm of Fredricksen and Maiorana which allows us to extend this result to the case of words without the factor $1^i$ for any $i$. We characterize for which languages of words of length $n$ exists a de Bruijn sequence, and we also study some symbolic dynamical properties of these languages, particularly of the languages defined by forbidden factors. For these kinds of languages, we present an algorithm to produce a de Bruijn sequence, using the Lyndon words of the language. These results use the notion of de Bruijn graph and reduce the problem to construct an Eulerian cycle in this graph. We study the problem of construct the lexicographically minimal de Bruijn sequence in a language with forbidden factors using the de Bruijn graph. We study two algorithms, a simple and efficient greedy algorithm which works with some sets of languages, and a more complex algorithm which solves this problem for any Eulerian labelled graph.
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Moreno Eduardo. Graphes et cycles de de Bruijn dans des langages avec des restrictions. Informatique [cs]. Université de Marne la Vallée, 2005. Français. ⟨tel-00628709⟩

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