Binary search trees of permuton samples

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Advances in Applied Mathematics Pub Date : 2024-09-11 DOI:10.1016/j.aam.2024.102774

Benoît Corsini , Victor Dubach , Valentin Féray

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引用次数: 0

Abstract

Binary search trees (BST) are a popular type of structure when dealing with ordered data. They allow efficient access and modification of data, with their height corresponding to the worst retrieval time. From a probabilistic point of view, BSTs associated with data arriving in a uniform random order are well understood, but less is known when the input is a non-uniform permutation.

We consider here the case where the input comes from i.i.d. random points in the plane with law μ, a model which we refer to as a permuton sample. Our results show that the asymptotic proportion of nodes in each subtree only depends on the behavior of the measure μ at its left boundary, while the height of the BST has a universal asymptotic behavior for a large family of measures μ. Our approach involves a mix of combinatorial and probabilistic tools, namely combinatorial properties of binary search trees, coupling arguments, and deviation estimates.

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permuton 样本的二元搜索树

二叉搜索树（BST）是处理有序数据时常用的一种结构类型。它们允许高效访问和修改数据，其高度与最短检索时间相对应。从概率论的角度来看，与以均匀随机顺序到达的数据相关的 BST 已广为人知，但对于输入为非均匀排列时的 BST 却知之甚少。我们在此考虑的情况是，输入来自平面上具有 μ 规律的 i.i.d. 随机点，我们将这种模型称为排列样本。我们的结果表明，每个子树中节点的渐近比例只取决于其左边界上的度量 μ 的行为，而 BST 的高度对于一大系列度量 μ 具有普遍的渐近行为。我们的方法涉及组合工具和概率工具的混合，即二叉搜索树的组合属性、耦合参数和偏差估计。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Advances in Applied Mathematics 数学-应用数学

CiteScore

2.00

自引率

9.10%

发文量

审稿时长

85 days

期刊介绍： Interdisciplinary in its coverage, Advances in Applied Mathematics is dedicated to the publication of original and survey articles on rigorous methods and results in applied mathematics. The journal features articles on discrete mathematics, discrete probability theory, theoretical statistics, mathematical biology and bioinformatics, applied commutative algebra and algebraic geometry, convexity theory, experimental mathematics, theoretical computer science, and other areas. Emphasizing papers that represent a substantial mathematical advance in their field, the journal is an excellent source of current information for mathematicians, computer scientists, applied mathematicians, physicists, statisticians, and biologists. Over the past ten years, Advances in Applied Mathematics has published research papers written by many of the foremost mathematicians of our time.