scholarly journals On a random search tree: asymptotic enumeration of vertices by distance from leaves

2017 ◽  
Vol 49 (3) ◽  
pp. 850-876
Author(s):  
Miklós Bóna ◽  
Boris Pittel
Keyword(s):  
Random Search ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Asymptotic Enumeration ◽  
Prime Divisors ◽  
Fixed Size ◽  
Single Interval ◽  
Fixed Rank

Abstract A random binary search tree grown from the uniformly random permutation of [n] is studied. We analyze the exact and asymptotic counts of vertices by rank, the distance from the set of leaves. The asymptotic fraction ck of vertices of a fixed rank k ≥ 0 is shown to decay exponentially with k. We prove that the ranks of the uniformly random, fixed size sample of vertices are asymptotically independent, each having the distribution {ck}. Notoriously hard to compute, the exact fractions ck have been determined for k ≤ 3 only. We present a shortcut enabling us to compute c4 and c5 as well; both are ratios of enormous integers, the denominator of c5 being 274 digits long. Prompted by the data, we prove that, in sharp contrast, the largest prime divisor of the denominator of ck is at most 2k+1 + 1. We conjecture that, in fact, the prime divisors of every denominator for k > 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.

scholarly journals On the Number of Descendants and Ascendants in Random Search Trees

10.37236/1358 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Conrado Martínez ◽  
Alois Panholzer ◽  
Helmut Prodinger
Keyword(s):  
Generating Functions ◽  
Random Search ◽  
Natural Extension ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Higher Moments ◽  
Binary Search Trees ◽  
Search Trees ◽  
Explicit Formulae

The number of descendants of a node in a binary search tree (BST) is the size of the subtree having this node as a root; the number of ascendants is the number of nodes on the path connecting this node with the root. Using a purely combinatorial approach (generating functions and differential equations) we are able to extend previous results. For the number of descendants we get explicit formulæ for all moments; for the number of ascendants, which is harder, we get the variance. A natural extension of binary search trees occurs when performing local reorganisations. Poblete and Munro have already analyzed some aspects of these locally balanced binary search trees (LBSTs). Here, we relate these structures with the performance of median–of–three Quicksort. We get as new results the variances for ascendants and descendants in this setting. If the rank of the node itself is picked at random ("grand averages"), the corresponding parameters only depend on the size $n$. In this instance, we get all the moments for the descendants (BST and LBST), as well as the probabilities. For ascendants (LBST), we get the variance and (in principle) the higher moments, as well as the (normal) limiting distribution. The emphasis is on explicit formulæ, and these are sometimes quite involved. Thus, in some instances, we have decided to state abridged versions in the paper and collect the long forms into an appendix that can be downloaded from the URLs http://info.tuwien.ac.at/theoinf/abstract/abs_120.htm and http://www.lsi.upc.es/~conrado/research/.

scholarly journals An almost sure result for path lengths in binary search trees

2003 ◽  
Vol 35 (02) ◽  
pp. 363-376
Author(s):  
F. M. Dekking ◽  
L. E. Meester
Keyword(s):  
Path Length ◽  
Random Variable ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Total Path Length ◽  
Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variableZ. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜mof finite binary trees with at mostmnodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector timesZ.

scholarly journals An almost sure result for path lengths in binary search trees

2003 ◽  
Vol 35 (2) ◽  
pp. 363-376 ◽  
Author(s):  
F. M. Dekking ◽  
L. E. Meester
Keyword(s):  
Path Length ◽  
Random Variable ◽  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Binary Search Trees ◽  
Search Trees ◽  
Total Path Length ◽  
Path Lengths

This paper studies path lengths in random binary search trees under the random permutation model. It is known that the total path length, when properly normalized, converges almost surely to a nondegenerate random variable Z. The limit distribution is commonly referred to as the ‘quicksort distribution’. For the class 𝒜m of finite binary trees with at most m nodes we partition the external nodes of the binary search tree according to the largest tree that each external node belongs to. Thus, the external path length is divided into parts, each part associated with a tree in 𝒜m. We show that the vector of these path lengths, after normalization, converges almost surely to a constant vector times Z.

scholarly journals Almost sure asymptotics for the random binary search tree

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Matthew Roberts
Keyword(s):  
Random Permutation ◽  
Search Tree ◽  
Binary Search ◽  
Saturation Level ◽  
Binary Search Tree ◽  
International Audience ◽  
Permutation Model ◽  
Random Permutation Model

International audience We consider a (random permutation model) binary search tree with $n$ nodes and give asymptotics on the $\log$ $\log$ scale for the height $H_n$ and saturation level $h_n$ of the tree as $n \to \infty$, both almost surely and in probability. We then consider the number $F_n$ of particles at level $H_n$ at time $n$, and show that $F_n$ is unbounded almost surely.

scholarly journals E-ART: A New Encryption Algorithm Based on the Reflection of Binary Search Tree

Cryptography ◽  
2021 ◽  
Vol 5 (1) ◽  
pp. 4
Author(s):  
Bayan Alabdullah ◽  
Natalia Beloff ◽  
Martin White
Keyword(s):  
Data Storage ◽  
Statistical Tests ◽  
Data Encryption ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Security Level ◽  
Avalanche Effect ◽  
High Security Level ◽  
Tree Data Structure

Data security has become crucial to most enterprise and government applications due to the increasing amount of data generated, collected, and analyzed. Many algorithms have been developed to secure data storage and transmission. However, most existing solutions require multi-round functions to prevent differential and linear attacks. This results in longer execution times and greater memory consumption, which are not suitable for large datasets or delay-sensitive systems. To address these issues, this work proposes a novel algorithm that uses, on one hand, the reflection property of a balanced binary search tree data structure to minimize the overhead, and on the other hand, a dynamic offset to achieve a high security level. The performance and security of the proposed algorithm were compared to Advanced Encryption Standard and Data Encryption Standard symmetric encryption algorithms. The proposed algorithm achieved the lowest running time with comparable memory usage and satisfied the avalanche effect criterion with 50.1%. Furthermore, the randomness of the dynamic offset passed a series of National Institute of Standards and Technology (NIST) statistical tests.

Improving the Red-Black tree delete algorithm

2021 ◽  
Author(s):  
ZEGOUR Djamel Eddine
Keyword(s):  
Data Structure ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Search Performance ◽  
Running Time ◽  
Standard Algorithm ◽  
Color Changes ◽  
Red Black Tree ◽  
Symbol Tables

Abstract Today, Red-Black trees are becoming a popular data structure typically used to implement dictionaries, associative arrays, symbol tables within some compilers (C++, Java …) and many other systems. In this paper, we present an improvement of the delete algorithm of this kind of binary search tree. The proposed algorithm is very promising since it colors differently the tree while reducing color changes by a factor of about 29%. Moreover, the maintenance operations re-establishing Red-Black tree balance properties are reduced by a factor of about 11%. As a consequence, the proposed algorithm saves about 4% on running time when insert and delete operations are used together while conserving search performance of the standard algorithm.

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