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 Random Records and Cuttings in Split Trees: Extended Abstract

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Cecilia Holmgren
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Search Trees ◽  
Digital Search Trees ◽  
International Audience ◽  
Split Trees ◽  
Digital Search

International audience We study the number of records in random split trees on $n$ randomly labelled vertices. Equivalently the number of random cuttings required to eliminate an arbitrary random split tree can be studied. After normalization the distributions are shown to be asymptotically $1$-stable. This work is a generalization of our earlier results for the random binary search tree which is one specific case of split trees. Other important examples of split trees include $m$-ary search trees, quadtrees, median of $(2k+1)$-trees, simplex trees, tries and digital search trees.

Multiway Trees of Maximum and Minimum Probability under the Random Permutation Model

1996 ◽  
Vol 5 (4) ◽  
pp. 351-371 ◽  
Author(s):  
Robert P. Dobrow ◽  
James Allen Fill
Keyword(s):  
Probability Model ◽  
Mass Function ◽  
Random Permutation ◽  
Search Tree ◽  
Probability Mass Function ◽  
Search Trees ◽  
Asymptotic Expressions ◽  
Minimum Probability ◽  
Permutation Model ◽  
Random Permutation Model

Multiway trees, also known as m–ary search trees, are data structures generalising binary search trees. A common probability model for analysing the behaviour of these structures is the random permutation model. The probability mass function Q on the set of m–ary search trees under the random permutation model is the distribution induced by sequentially inserting the records of a uniformly random permutation into an initially empty m–ary search tree. We study some basic properties of the functional Q, which serves as a measure of the ‘shape’ of the tree. In particular, we determine exact and asymptotic expressions for the maximum and minimum values of Q and identify and count the trees achieving those values.

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 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 A hooray for Poisson approximation

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Rudolf Grübel
Keyword(s):  
Analysis Of Algorithms ◽  
Poisson Approximation ◽  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Variation Distance ◽  
International Audience ◽  
Election Algorithm ◽  
Bernoulli Variables ◽  
Quantities Of Interest

International audience We give several examples for Poisson approximation of quantities of interest in the analysis of algorithms: the distribution of node depth in a binary search tree, the distribution of the number of losers in an election algorithm and the discounted profile of a binary search tree. A simple and well-known upper bound for the total variation distance between the distribution of a sum of independent Bernoulli variables and the Poisson distribution with the same mean turns out to be very useful in all three cases.

scholarly journals Average depth in a binary search tree with repeated keys

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Margaret Archibald ◽  
Julien Clément
Keyword(s):  
Search Tree ◽  
Binary Search ◽  
Binary Search Tree ◽  
Average Depth ◽  
Binary Search Trees ◽  
Search Trees ◽  
Random Sequences ◽  
International Audience ◽  
The Right

International audience Random sequences from alphabet $\{1, \ldots,r\}$ are examined where repeated letters are allowed. Binary search trees are formed from these, and the average left-going depth of the first $1$ is found. Next, the right-going depth of the first $r$ is examined, and finally a merge (or 'shuffle') operator is used to obtain the average depth of an arbitrary node, which can be expressed in terms of the left-going and right-going depths. The variance of each of these parameters is also found.

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.

Sign in / Sign up

Export Citation Format

Share Document