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The height of record‐biased trees
Author(s) -
Corsini Benoît
Publication year - 2023
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.21110
Subject(s) - random binary tree , permutation (music) , binary tree , binary search tree , random permutation , ternary search tree , combinatorics , binary number , mathematics , set (abstract data type) , tree (set theory) , weight balanced tree , optimal binary search tree , self balancing binary search tree , node (physics) , k ary tree , interval tree , computer science , tree structure , symmetric group , physics , arithmetic , quantum mechanics , acoustics , programming language
Given a permutationσ $$ \sigma $$ , its corresponding binary search tree is obtained by recursively inserting the valuesσ ( 1 ) , … , σ ( n ) $$ \sigma (1),\dots, \sigma (n) $$ into a binary tree so that the label of each node is larger than the labels of its left subtree and smaller than the labels of its right subtree. In this article, we study the height of binary search trees drawn from the record‐biased model of permutations whose probability measure on the set of permutations is proportional toθ record ( σ )$$ {\theta}^{\mathrm{record}\left(\sigma \right)} $$ , whererecord ( σ ) = | { i ∈ [ n ] : ∀ j < i , σ ( i ) > σ ( j ) } | $$ \mathrm{record}\left(\sigma \right)=\mid \left\{i\in \left[n\right]:\forall j\sigma (j)\right\}\mid $$ . We show that the height of a binary search tree built from a record‐biased permutation of sizen $$ n $$ with parameterθ $$ \theta $$ is of order( 1 + o ℙ ( 1 ) ) max { c ∗ log n , θ log ( 1 + n / θ ) } $$ \left(1+{o}_{\mathbb{P}}(1)\right)\max \left\{{c}^{\ast}\log n,\kern0.3em \theta \log \left(1+n/\theta \right)\right\} $$ , hence extending previous results of Devroye on the height or random binary search trees.