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Arbres digitaux

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Arbres pour l’Algorithmique

Part of the book series: Mathématiques et Applications ((MATHAPPLIC,volume 83))

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Abstract

Dans ce chapitre, nous nous concentrerons uniquement sur la structure de trie. Les autres types d’arbres digitaux ne seront pas abordés sauf en exercice. Les méthodes présentées, ou du moins leurs principes, s’appliquent néanmoins à ces autres arbres. Il est de fait assez courant de confondre arbres digitaux et tries.

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Notes

  1. 1.

    Voir les rappels algorithmiques de l’annexe A.

  2. 2.

    Ce dictionnaire n’est pas celui de la méthode symbolique puisqu’il correspond à la décomposition récursive par préfixes du trie et non à des opérations ensemblistes ou combinatoires.

  3. 3.

    Pour les tries, dans les analyses usuelles de la longueur de cheminement externe et de la taille, cette fonction dépend seulement du cardinal \({\left \lvert \omega \right \rvert }\).

  4. 4.

    La paramétrisation est basée sur un principe analogue à celui utilisé pour le codage arithmétique en compression [227].

  5. 5.

    Nous pouvons penser par analogie aux développements impropres par exemple en base 10 : 0, 999⋯ = 1, 0000….

  6. 6.

    Toujours en admettant que cet intervalle est unique, ce qui est vrai sauf sur l’ensemble de mesure nulle constitué des extrémités des intervalles fondamentaux.

  7. 7.

    De manière un peu surprenante au premier abord, il peut y avoir plusieurs relèvements analytiques d’une même fonction. Cela ne change rien à la validité de la formule de Nörlund-Rice.

  8. 8.

    Selon les auteurs, on parle aussi de série disciplinée ou apprivoisée (en anglais tame). Ces termes signifient simplement que la série admet un domaine où ses pôles et son comportement analytique obéissent à un certain schéma.

  9. 9.

    Bien sûr nous pourrions être plus précis et donner un développement à un ordre supérieur si nous souhaitions évaluer les termes d’erreur ou les termes sous-dominants du développement asymptotique.

  10. 10.

    C’est une propriété de type Perron-Frobenius.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, Université Versailles, Saint-Quentin-en-Yvelines, Versailles Cedex, France

    Brigitte Chauvin

  2. GREYC, CNRS UMR 6072, Normandie Université, Caen Cedex, France

    Julien Clément

  3. Laboratoire DAVID, Université Versailles, Saint-Quentin-en-Yvelines, Versailles Cedex, France

    Danièle Gardy

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  1. Brigitte Chauvin
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  2. Julien Clément
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  3. Danièle Gardy
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Chauvin, B., Clément, J., Gardy, D. (2018). Arbres digitaux. In: Arbres pour l’Algorithmique. Mathématiques et Applications, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-93725-0_7

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