Minimal Forbidden Words and Applications

Abstract : This thesis describes the theory and some applications of minimal forbidden words, that are the most little words that do not appear as factors of a given word. In the first part of this thesis, we describe the properties of minimal forbidden words and we show some particular cases, as that of a finite word, a finite set of finite words, and a regular factorial language. We also present the procedures for the computation of the theoretical results. Then we generalize the minimal forbidden words to the case of the existence of a period, which determines the positions of occurrences of the factors modulo a fixed integer. These are called minimal periodic forbidden words. We study their basic properties and give the algorithms for the computation in the case of a finite word and of a finite set of finite words. In the second part we show two applications of minimal forbidden words. The first one is related to constrained systems. We give a polynomial-time construction of the set of sequences that satisfy a constraint defined by a finite list of forbidden blocks, with a specified set of bit positions unconstrained. We also give a linear-time construction of a finite-state presentation of a constrained system defined by a periodic list of forbidden blocks. The second one is a problem issued from biology: the reconstruction of a genomic sequence starting from a set of its fragments. We show that a theoretical formalization of this problem can be solved in linear time using minimal forbidden words. We also prove that our algorithm solves a special case of the Shortest Superstring Problem.
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Gabriele Fici. Minimal Forbidden Words and Applications. Computer Science [cs]. Université de Marne la Vallée, 2006. English. ⟨NNT : 2006MARN0277⟩. ⟨tel-00628628v2⟩

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