Random Binary Trees Research Articles

An Improved Lower Bound on the Largest Common Subtree of Random Leaf-Labeled Binary Trees

An Improved Lower Bound on the Largest Common Subtree of Random Leaf-Labeled Binary Trees

Expected Number of Induced Subtrees Shared by Two Independent Copies of a Random Tree

Consider a rooted tree with leaf-set and with all nonleaf vertices having out-degree 2, at least. A rooted tree with leaf-set is induced by in if is the lowest common ancestor subtree for , with all its degree-2 vertices suppressed. A “maximum agreement subtree” (MAST) for a pair of two trees and is a tree with a largest leaf-set such that is induced by both in and . Bryant, McKenzie, and Steel [BioConsensus, AMS, Providence, RI, 2003, pp. 55–65] and Bernstein et al. [SIAM J. Discrete Math., 29 (2015), pp. 2065–2074] proved, among other results, that for and being two independent copies of a random binary (uniform or Yule–Harding distributed) tree , the likely magnitude order of is . We prove this bound for a wide class of random rooted trees: is a terminal tree of a branching, Galton–Watson, process with an ordered-offspring distribution of mean 1, conditioned on “total number of leaves is .”

The height of record‐biased trees

AbstractGiven a permutation , its corresponding binary search tree is obtained by recursively inserting the values 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 to , where . We show that the height of a binary search tree built from a record‐biased permutation of size with parameter is of order , hence extending previous results of Devroye on the height or random binary search trees.

Guest editorial: Smart communications and networking: architecture, applications, and future challenges

Guest editorial: Smart communications and networking: architecture, applications, and future challenges

On the Largest Common Subtree of Random Leaf-Labeled Binary Trees

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 22 June 2020Accepted: 29 October 2021Published online: 27 January 2022Keywordsmaximum agreement subtree, random tree, stochastic fragmentationAMS Subject Headings60C05, 05C80Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec

Weakly protected nodes in random binary search trees

Here, we derive the exact mean and variance of the number of weakly protected nodes (the nodes that are not leaves and at least one of their children is not a leaf) in binary search trees grown from random permutations. Furthermore, by using contraction method, we prove normal limit law for a properly normalized version of this tree parameter.

Adaptive clustering-based approach for forgery detection in images containing similar appearing but authentic objects

Copy-move forgery is one of the well-known image forgery technique which exploits regions of the same image to create forged image by replicating or hiding authentic content of the original image. Original images can also contain similar looking but authentic objects. In such cases, identification of authentic and tampered images is a complicated task. To tackle this problem, we propose a method in which Stationary Wavelet Transform (SWT) and spatial-constrained edge preserving watershed segmentation are applied over input image in preprocessing step. Keypoint extraction and descriptor computation are performed using Cascaded Features from Accelerated Segment Test (Cascaded FAST) and Binary Robust Invariant Scalable Keypoint (BRISK) descriptor, respectively. Approximate nearest neighbor search is performed using Random Binary Search Tree (RBST) method. For keypoint clustering, Adaptive Density Peak Clustering (ADPC) technique is employed. Outlier removal is performed using Random Sample Consensus (RANSAC) technique. Further, forged regions are localized using correlation map generation. Experimental results display that the proposed approach can effectively distinguish between forged and original images containing similar appearing but authentic objects. It is also able to detect forged images sustaining different post-processing attacks. For COVERAGE dataset, proposed technique achieves high F-Measure = 86.901% and low False Positive Rate (FPR) = 15.241% in comparison to state-of-the-art techniques.

The height of Mallows trees

Random binary search trees are obtained by recursively inserting the elements σ(1),σ(2),…,σ(n) of a uniformly random permutation σ of [n]={1,…,n} into a binary search tree data structure. Devroye (J. Assoc. Comput. Mach. 33 (1986) 489–498) proved that the height of such trees is asymptotically of order c∗logn, where c∗=4.311… is the unique solution of clog((2e)/c)=1 with c≥2. In this paper, we study the structure of binary search trees Tn,q built from Mallows permutations. A Mallows(q) permutation is a random permutation of [n]={1,…,n} whose probability is proportional to qInv(σ), where Inv(σ)=|{i<j:σ(i)>σ(j)}|. This model generalizes random binary search trees, since Mallows(q) permutations with q=1 are uniformly distributed. The laws of Tn,q and Tn,q−1 are related by a simple symmetry (switching the roles of the left and right children), so it suffices to restrict our attention to q≤1. We show that, for q∈[0,1], the height of Tn,q is asymptotically (1+o(1))(c∗logn+n(1−q)) in probability. This yields three regimes of behaviour for the height of Tn,q, depending on whether n(1−q)/logn tends to zero, tends to infinity or remains bounded away from zero and infinity. In particular, when n(1−q)/logn tends to zero, the height of Tn,q is asymptotically of order c∗logn, like it is for random binary search trees. Finally, when n(1−q)/logn tends to infinity, we prove stronger tail bounds and distributional limit theorems for the height of Tn,q.

Anti‐collision algorithm based on slotted random regressive‐style binary search tree in RFID technology

In recent years, the rapid development of the Internet of Things (IoT) technology has provided a strong technical support for the technological transformation of the logistics industry. The informatization development of logistics industry increasingly relies on the Internet of Things technology represented by Radio Frequency Identification (RFID) technology. These technologies lead the whole business process to optimize the business process in the direction of accurate, efficient and real-time. In order to solve the problem that the reader cannot identify the label information correctly due to the phenomenon of data collision in the application of RFID technology, this paper proposes an anti-collision algorithm based on Slotted Random Regressive-style Binary Search Tree (SR-RBST). Based on Slotted ALOHA (SA) the method proposed in this paper uses the Regressive-style Binary Search Tree (RBST) to process the RFID labels in the collision time slot. With the same size of tags, the SR-RBST algorithm needs less total time slot and has higher efficiency and shorter identification time, while with the increase of the number of tags, the SR-RBST anti-collision algorithm has more obvious advantages. The SR-RBST algorithm effectively improves the time slot utilization efficiency of the system.

Objective-Based Hierarchical Clustering of Deep Embedding Vectors

We initiate a comprehensive experimental study of objective-based hierarchical clustering methods on massive datasets consisting of deep embedding vectors from computer vision and NLP applications. This includes a large variety of image embedding (ImageNet, ImageNetV2, NaBirds), word embedding (Twitter, Wikipedia), and sentence embedding (SST-2) vectors from several popular recent models (e.g. ResNet, ResNext, Inception V3, SBERT). Our study includes datasets with up to 4.5 million data points with embedding dimensions up to 2048. In order to address the challenge of scaling up hierarchical clustering to such large datasets we propose a new practical hierarchical clustering algorithm B++&amp;C. It gives a 5%/20% improvement on average for the popular Moseley-Wang (MW)/ Cohen-Addad et al. (CKMM) objectives (normalized) compared to a wide range of classic methods and recent heuristics. We also introduce a theoretical algorithm B2SAT&amp;C which achieves a 0.74-approximation for the CKMM objective in polynomial time. This is the first substantial improvement over the trivial 2/3-approximation achieved by a random binary tree. Prior to this work, the best poly-time approximation of ≈2/3 + 0.0004 was due to Charikar et al. (SODA’19)

A note on the independence number, domination number and related parameters of random binary search trees and random recursive trees

We identify the mean growth of the independence number of random binary search trees and random recursive trees and show normal fluctuations around their means. Similarly we also show normal limit laws for the domination number and variations of it for these two cases of random tree models. Our results are an application of a recent general theorem of Holmgren and Janson on fringe trees in these two random tree models.

Random Partitioning Forest for Point-Wise and Collective Anomaly Detection—Application to Network Intrusion Detection

In this paper, we propose DiFF-RF, an ensemble approach composed of random partitioning binary trees to detect point-wise and collective (as well as contextual) anomalies. Thanks to a distance-based paradigm used at the leaves of the trees, this semi-supervised approach solves a drawback that has been identified in the isolation forest (IF) algorithm. Moreover, taking into account the frequencies of visits in the leaves of the random trees allows to significantly improve the performance of DiFF-RF when considering the presence of collective anomalies. DiFF-RF is fairly easy to train, and excellent performance can be obtained by using a simple semi-supervised procedure to setup the extra hyper-parameter that is introduced. We first evaluate DiFF-RF on a synthetic data set to i) verify that the limitation of the IF algorithm is overcome, ii) demonstrate how collective anomalies are actually detected and iii) to analyze the effect of the meta-parameters it involves. We assess the DiFF-RF algorithm on a large set of datasets from the UCI repository, as well as two benchmarks related to intrusion detection applications. Our experiments show that DiFF-RF almost systematically outperforms the IF algorithm, but also challenges the one-class SVM baseline and a deep learning variational auto-encoder architecture. Furthermore, our experience shows that DiFF-RF can work well in the presence of small-scale learning data, which is conversely difficult for deep neural architectures. Finally, DiFF-RF is computationally efficient and can be easily parallelized on multi-core architectures.

Quantum fast hitting on glued trees mapped on a photonic chip

Quantum walks on graphs play an important role in the field of quantum algorithms. Fast hitting is one of the properties that quantum walk algorithms can utilize to outperform classical random walk algorithms. Fast hitting refers to a particle starting from the entrance node on a graph and trying to hit the exit node quickly. Especially, continuous-time quantum walks on random glued binary trees have been investigated in theories extensively for their exponentially faster hitting speed over classical random walks. Here, using heralded single photons to represent quantum walkers and laser-written waveguide arrays to simulate the theoretical graph, we are able to demonstrate the hitting efficiency of quantum walks with tree depth as high as 16 layers for the first time. Furthermore, we expand the graph’s branching rate from 2 to 5, revealing that quantum walks can exhibit more superiority over classical random walks as the branching rate increases. Our results may shed light on the physical implementation of quantum walk algorithms as well as quantum computation and quantum simulation.

Verified Analysis of Random Binary Tree Structures

This work is a case study of the formal verification and complexity analysis of some famous probabilistic algorithms and data structures in the proof assistant Isabelle/HOL. In particular, we consider the expected number of comparisons in randomised quicksort, the relationship between randomised quicksort and average-case deterministic quicksort, the expected shape of an unbalanced random Binary Search Tree, the randomised binary search trees described by Martínez and Roura, and the expected shape of a randomised treap. The last three have, to our knowledge, not been analysed using a theorem prover before and the last one is of particular interest because it involves continuous distributions.

Large Deviation Theorem for Branches of the Random Binary Tree in the Horton--Strahler Analysis

The Horton-Strahler analysis is a graph-theoretic method to measure the bifurcation complexity of branching patterns, by defining a number called the order to each branch. The main result of this paper is a large deviation theorem for the number of branches of each order in a random binary tree. The rate function associated with a large deviation cannot be derived in a closed form; instead, asymptotic forms of the rate function are given.

Random Binary Search Trees for approximate nearest neighbour search in binary spaces

Approximate nearest neighbour (ANN) search is one of the most important problems in numerous computer science applications, including data mining, machine learning and computer vision. In this paper, we address the problem of ANN for high-dimensional binary vectors and we propose a simple yet powerful search method that is based on Random Binary Search Trees (RBST). Our method is validated on a dataset of 1.25M binary local feature descriptors obtained from a real-life image-based localisation system provided by Google as a part of Project Tango (now known as ARCore). The results of an extensive evaluation of our method performed on this dataset against the state-of-the-art ANN algorithms, such as Locality Sensitive Hashing, Uniform LSH and Multi-probe LSH, show the superiority of our method in terms of retrieval precision with a performance boost of over 20%.

Non-criticality criteria for Abelian sandpile models with sources and sinks

We prove that the Abelian sandpile model on a random binary and binomial tree, as introduced in Redig, Ruszel, and Saada [J. Stat. Phys. 147, 653–677 (2012)], is not critical for all branching probabilities p &amp;lt; 1; by estimating the tail of the annealed survival time of a random walk on the binary tree with randomly placed traps, we obtain some more information about the exponential tail of the avalanche radius. Next we study the sandpile model on Zd with some additional dissipative sites: we provide examples and sufficient conditions for non-criticality; we also make a connection with the parabolic Anderson model. Finally we initiate the study of the sandpile model with both sources and sinks and give a sufficient condition for non-criticality in the presence of a finite number of sources, using a connection with the homogeneous pinning model.

And/or trees: A local limit point of view

We present here a new and universal approach for the study of random and/or trees, unifying in one framework many different models, including some novel ones not yet understood in the literature. An and/or tree is a Boolean expression represented in (one of) its tree shapes. Fix an integer k, take a sequence of random (rooted) trees of increasing size, say , and label each of these random trees uniformly at random in order to get a random Boolean expression on k variables. We prove that, under rather weak local conditions on the sequence of random trees , the distribution induced on Boolean functions by this procedure converges as n tends to infinity. In particular, we characterize two different behaviors of this limit distribution depending on the shape of the local limit of : a degenerate case when the local limit has no leaves; and a non‐degenerate case, which we are able to describe in more details under stronger conditions. In this latter case, we provide a relationship between the probability of a given Boolean function and its complexity. The examples covered by this unified framework include trees that interpolate between models with logarithmic typical distances (such as random binary search trees) and other ones with square root typical distances (such as conditioned Galton–Watson trees).

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

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 &gt; 1 form a single interval, from 2 to the largest prime not exceeding 2k+1 + 1.

Profile of Random Exponential Binary Trees

We introduce the random exponential binary tree (EBT) and study its profile. As customary, the tree is extended by padding each leaf node (considered internal), with the appropriate number of external nodes, so that the outdegree of every internal node is made equal to 2. In a random EBT, at every step, each external node is promoted to an internal node with probability p, stays unchanged with probability 1 - p, and the resulting tree is extended. We study the internal and external profiles of a random EBT and get exact expectations for the numbers of internal and external nodes at each level. Asymptotic analysis shows that the average external profile is richest at level \(\frac {2p}{p+1}n\), and it experiences phase transitions at levels a n, where the a’s are the solutions to an algebraic equation. The rates of convergence themselves go through an infinite number of phase changes in the sublinear range, and then again at the nearly linear levels.