Uniform distribution modulo one and binary search trees
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 415-424

Any sequence x=(x k ) k=1 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 ,,x n . Obviously log n log 2 - 1 H n ( x ) 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)clogn as n 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 ={αk},k=1,2,, and α is distributed uniformly at random on [ 0 , 1 ] then H n ( x ) ( 12 / π 2 ) log n log log n as n 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.

On peut construire à partir d’une suite x=(x k ) k=1 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 ,,x n . Il est évident que log n log 2 - 1 H n ( x ) 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)clogn,(n), presque-sûrement. Récemment, Devroye et Goudjil ont montré que si les x sont les suites de Weyl, i.e., x k ={αk},k=1,2,,α est une variable aléatoire uniforme sur [0,1], alors H n ( x ) ( 12 / π 2 ) log n log log n , n , 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.

@article{JTNB_2002__14_2_415_0,
     author = {Dekking, Michel and Van der Wal, Peter},
     title = {Uniform distribution modulo one and binary search trees},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {415-424},
     zbl = {1075.11054},
     mrnumber = {2040685},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_415_0}
}
Dekking, Michel; Van der Wal, Peter. Uniform distribution modulo one and binary search trees. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 415-424. http://www.numdam.org/item/JTNB_2002__14_2_415_0/

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