Uniform distribution modulo one and binary search trees

Michel Dekking; Peter Van der Wal

doi:10.5802/jtnb.366

Michel Dekking; Peter Van der Wal

Journal de Théorie des Nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 415-424.

Abstract
Résumé

Any sequence $x = {(x_{k})}_{k = 1}^{\infty}$ of distinct numbers from [0,1] generates a binary tree by storing the numbers consecutively at the nodes according to a left-right algorithm (or equivalently by sorting the numbers according to the Quicksort algorithm). Let $H_{n} (x)$ be the height of the tree generated by $x_{1}, \dots, x_{n} .$ Obviously $\frac{log n}{log 2} - 1 \leq H_{n} (x) \leq n - 1 .$ If the sequences $x$ are generated by independent random variables having the uniform distribution on [0, 1], then it is well known that there exists $c > 0$ such that $H_{n} (x) \sim c log n$ as $n \to \infty$ for almost all sequences $x$ . Recently Devroye and Goudjil have shown that if the sequences $x$ are Weyl sequences, i.e., defined by $x_{k} = {α k}, k = 1, 2, \dots,$ and $α$ is distributed uniformly at random on $[0, 1]$ then $H_{n} (x) \sim (12 / π^{2}) log n log log n$ as $n \to \infty$ in probability. In this paper we consider the class of all uniformly distributed sequences $x$ , and we show that the only behaviour that is excluded by the equidistribution of $x$ is that of the worst case, i.e., for these $x$ we necessarily have $H_{n} (x) = o (n)$ as $n \to \infty$ .

On peut construire à partir d’une suite $x = {(x_{k})}_{k = 1}^{\infty}$ de nombres distincts de l’intervalle [0,1] un arbre binaire en plaçant successivement ces nombres sur les noeuds selon un algorithme “gauche-droite” (cela revient à classer les nombres selon l’algorithme Quicksort). On note $H_{n} (x)$ la hauteur de l’arbre obtenu à partir des nombres $x_{1}, \dots, x_{n} .$ Il est évident que $\frac{log n}{log 2} - 1 \leq H_{n} (x) \leq n - 1 .$ Si la suite $x$ est obtenue comme valeurs de variables aléatoires indépendantes uniformes sur [0,1], alors on sait qu’il existe $c > 0$ tel que $H_{n} (x) \sim c log n, (n \to \infty)$ , presque-sûrement. Récemment, Devroye et Goudjil ont montré que si les $x$ sont les suites de Weyl, i.e., $x_{k} = {α k}, k = 1, 2, \dots,$ où $α$ est une variable aléatoire uniforme sur [0,1], alors $H_{n} (x) \sim (12 / π^{2}) log n log log n, n \to \infty,$ en probabilité. Dans ce papier nous considérons la classe de toutes les suites $x$ uniformément réparties pour lesquelles nous montrons que l’on a nécessairement $H_{n} (x) = o (n)$ quand $n \to \infty$ .

MR: 2040685 | Zbl: 1075.11054

DOI: 10.5802/jtnb.366

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     author = {Michel Dekking and Peter Van der Wal},
     title = {Uniform distribution modulo one and binary search trees},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {415--424},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
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     year = {2002},
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     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.366/}
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Michel Dekking; Peter Van der Wal. Uniform distribution modulo one and binary search trees. Journal de Théorie des Nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 415-424. doi : 10.5802/jtnb.366. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.366/

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