K-ary Trees Research Articles

On block-related derived graphs

The cut vertex of a graph G is a vertex whose removal increases the number of components in the graph. A maximal connected subgraph with no cut vertex is called a block of the graph. This paper introduces and analyses the block-degree of a vertex and the cut-degree of a block. The block-degree of a vertex v is the number of blocks containing v. The cut-degree of a block b is the number of cut vertices of G contained in b. Given the block-degree sequence of cut vertices of a graph, the number of blocks of the graph and bounds on the order and size of the graph are studied. The block graph (B(G)) of a graph G is the graph whose vertices are blocks of G in which two vertices are adjacent whenever the corresponding blocks have a vertex in common. The number of cut vertices in B(G) is obtained and the graphs for which B(G) has exactly one cut vertex are characterized. Further, an investigation is carried out on the traversability of B(G). The block cut point graph BC(G) of a graph G is the graph whose vertices are the union of the set of blocks and the set of cut vertices of G and in which two vertices are adjacent if it corresponds to a block b of G and the other to a cut vertex c of G and c belongs to b in G. The properties of BC(G) and its iterations are studied. The graph G for which BC(G) is a perfect m-ary tree is characterised.

Propagation reversal on trees in the large diffusion regime

In this work we study travelling wave solutions to bistable reaction–diffusion equations on bi-infinite k-ary trees in the continuum regime where the diffusion parameter is large. Adapting the spectral convergence method developed by Bates and his coworkers, we find an asymptotic prediction for the speed of travelling front solutions. In addition, we prove that the associated profiles converge to the solutions of a suitable limiting reaction–diffusion PDE. Finally, for the standard cubic nonlinearity we provide explicit formulas to bound the thin region in parameter space where the propagation direction undergoes a reversal.

Optimal L(3,2,1)-labeling of trees

Given a graph G, an L ( 3 , 2 , 1 ) -labeling of G is an assignment f of non-negative integers (labels) to the vertices of G such that | f ( u ) − f ( v ) | ≥ 4 − i if dist ( u , v ) = i (i = 1, 2, 3). For a non-negative integer k, a k- L ( 3 , 2 , 1 ) -labeling is an L ( 3 , 2 , 1 ) -labeling such that no label is greater than k. The L ( 3 , 2 , 1 ) -labeling number of G, denoted by λ 3 , 2 , 1 ( G ) , is the smallest number k such that G has a k- L ( 3 , 2 , 1 ) -labeling. Chia proved that the L ( 3 , 2 , 1 ) -labeling number of a tree T with maximum degree Δ can have one of three values: 2 Δ + 1 , 2 Δ + 2 and 2 Δ + 3 . This paper gives some sufficient conditions for λ 3 , 2 , 1 ( T ) ≥ 2 Δ + 2 and λ 3 , 2 , 1 ( T ) = 2 Δ + 3 , respectively. As a result, the L ( 3 , 2 , 1 ) -labeling numbers of complete m-ary trees, spiders and banana trees are completely determined.

BranchLabelNet: Anatomical Human Airway Labeling Approach using a Dividing-and-Grouping Multi-Label Classification.

Anatomical airway labeling is crucial for precisely identifying airways displaying symptoms such as constriction, increased wall thickness, and modified branching patterns, facilitating the diagnosis and treatment of pulmonary ailments. This study introduces an innovative airway labeling methodology, BranchLabelNet, which accounts for the fractal nature of airways and inherent hierarchical branch nomenclature. In developing this methodology, branch-related parameters, including position vectors, generation levels, branch lengths, areas, perimeters, and more, are extracted from a dataset of 1000 chest computed tomography (CT) images. To effectively manage this intricate branch data, we employ an n-ary tree structure that captures the complicated relationships within the airway tree. Subsequently, we employ a divide-and-group deep learning approach for multi-label classification, streamlining the anatomical airway branch labeling process. Additionally, we address the challenge of class imbalance in the dataset by incorporating the Tomek Links algorithm to maintain model reliability and accuracy. Our proposed airway labeling method provides robust branch designations and achieves an impressive average classification accuracy of 95.94% across fivefold cross-validation. This approach is adaptable for addressing similar complexities in general multi-label classification problems within biomedical systems.

An Inclusive Distance Irregularity Strength of n-ary Tree

An inclusive distance vertex irregular labelling of a simple graph G is a function of the vertex set of to positive integer set such that the sum of its vertex label and the labels of all vertices adjacent to the vertex are distinct. The minimum of maximum label of the vertices is said to be inclusive distance irregularity strength of G, denoted by dis(G). The purpose of this research is showing that dis(T_{n,2})= (n^2+2)/2 where T_{n,2} is a complete n-ary tree to level two.

A distinguished-bit tracking knowledge-based query tree for RFID tag identification

—Many RFID systems own the information of RFID tags in the database, so the reader knows how many tags will be identified and which tags’ IDs might appear. How the RFID anti-collision protocols take advantage of this knowledge is a challenge. Some previous anti-collision protocols, including Query Tree with Shortcutting and Couple Resolution (QTSC) and Bit-tracking Knowledge-based Query Tree (BKQT), solved this task with the knowledge. We propose one novel protocol, Distinguished-bit Tracking Knowledge-based Query Tree (DKQT), which integrates the knowledge with the bit-tracking technique. Utilizing the knowledge and knowing the collided-bit locations, DKQT adopts a distinguishable technique, which can identify the tags that have a distinguished-bit representing that only one tag has a unique “0” or “1” in a bit location of the tag ID, to speed up the tag identification. Innovatively, adopting a distinguishable technique, DKQT forms an n-ary tree where a node has some leaves that are the tags having distinguished-bits. Thus, DKQT using the distinguishable technique can recognize multiple appearing tags in a collision by traversing this n-ary tree. The simulation results show that DKQT can reduce the number of slots by 36.94 % and 15.58 %, and the identification time by 37.47 % and 17.86 %, compared to the previous knowledge-based protocols, QTSC and BKQT, respectively.

[1,2]-dimension of graphs

Let G be a connected graph with vertex set V, where the distance between two vertices is the length of a shortest path between them. A set S⊆V is [1,2]-resolving if each vertex of G is at most distance-two away from a vertex in S and, given a pair of distinct vertices not in S, either there is a vertex in S adjacent to exactly one member of the given pair, or there are two vertices in S each of which is distance-two from exactly one member of the given pair. The [1,2]-dimension of G is the minimum cardinality of a [1,2]-resolving set of G. In this paper, we study the [1,2]-dimension of graphs by proving that the [1,2]-dimension problem is an NP-complete problem, and determine the [1,2]-dimension of some classes of graphs, such as paths, cycles, and full k-ary trees. We also introduce a generalization of metric dimension of which the (original) metric dimension and the [1,2]-dimension, as well as other metric dimension variants in the literature, are special instances.

The one-visibility localization game

We introduce a variant of the Localization game in which the cops only have visibility one, along with the corresponding optimization parameter, the one-visibility localization number ζ1. By developing lower bounds using isoperimetric inequalities, we give upper and lower bounds for ζ1 on k-ary trees with k≥2 that differ by a multiplicative constant, showing that the parameter is unbounded on k-ary trees. We provide a O(n) bound for Kh-minor free graphs of order n, and we show Cartesian grids meet this bound by determining their one-visibility localization number up to four values. We present upper bounds on ζ1 using pathwidth and the domination number and give upper bounds on trees via their depth and order. We conclude with open problems.

Anomaly Prediction over Human Crowded Scenes via Associate-Based Data Mining and K-Ary Tree Hashing

Anomaly detection and behavioral recognition are key research areas widely used to improve human safety. However, in recent times, with the extensive use of surveillance systems and the substantial increase in the volume of recorded scenes, the conventional analysis of categorizing anomalous events has proven to be a difficult task. As a result, machine learning researchers require a smart surveillance system to detect anomalies. This research introduces a robust system for predicting pedestrian anomalies. First, we acquired the crowd data as input from two benchmark datasets (including Avenue and ADOC). Then, different denoising techniques (such as frame conversion, background subtraction, and RGB-to-binary image conversion) for unfiltered data are carried out. Second, texton segmentation is performed to identify human subjects from acquired denoised data. Third, we used Gaussian smoothing and crowd clustering to analyze the multiple subjects from the acquired data for further estimations. The next step is to perform feature extraction to multiple abstract cues from the data. These bag of features include periodic motion, shape autocorrelation, and motion direction flow. Then, the abstracted features are mapped into a single vector in order to apply data optimization and mining techniques. Next, we apply the associate-based mining approach for optimized feature selection. Finally, the resultant vector is served to the k-ary tree hashing classifier to track normal and abnormal activities in pedestrian crowded scenes.

Asymptotic normality for the size of graph tries built from M-ary tree labelings

Graph tries are a new and interesting data structure proposed by Jacquet in 2014. They generalize the classical trie data structure which has found many applications in computer science and is one of the most popular data structure on words. For his generalization, Jacquet considered the size (or space requirement) and derived an asymptotic expansion for the mean and the variance when graph tries are built from n independently chosen random labelings of a rooted M-ary tree. Moreover, he conjectured a central limit theorem for the (suitably normalized) size as the number of labelings tends to infinity. In this paper, we verify this conjecture with the method of moments.

Percolation Problems on N-Ary Trees

Percolation theory is a subject that has been flourishing in recent decades. Because of its simple expression and rich connotation, it is widely used in chemistry, ecology, physics, materials science, infectious diseases, and complex networks. Consider an infinite-rooted N-ary tree where each vertex is assigned an i.i.d. random variable. When the random variable follows a Bernoulli distribution, a path is called head run if all the random variables that are assigned on the path are 1. We obtain the weak law of large numbers for the length of the longest head run. In addition, when the random variable follows a continuous distribution, a path is called an increasing path if the sequence of random variables on the path is increasing. By Stein’s method and other probabilistic methods, we prove that the length of the longest increasing path with a probability of one focuses on three points. We also consider limiting behaviours for the longest increasing path in a special tree.

Eternal Distance-k Domination on Graphs

Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal distance-k domination, guards initially occupy the vertices of a distance-k dominating set. After a vertex is attacked, guards “defend” by each moving up to distance k to form a distance-k dominating set, such that some guard occupies the attacked vertex. The eternal distance-k domination number of a graph is the minimum number of guards needed to defend against any sequence of attacks. The process is well-studied for the situation where $$k=1$$ . We introduce eternal distance-k domination for $$k > 1$$ . Determining whether a given set is an eternal distance-k domination set is in EXP, and in this paper we provide a number of results for paths and cycles, and relate this parameter to graph powers and domination in general. For trees we use decomposition arguments to bound the eternal distance-k domination numbers, and solve the problem entirely in the case of perfect m-ary trees.

Restricted rotation distance between k-ary trees

We study restricted rotation distance between ternary and higher-valence trees using approaches based upon generalizations of Thompson's group $F$. We obtain bounds and a method for computing these distances exactly in linear time, as well as a linear-time algorithm for computing rotations needed to realize these distances. Unlike the binary case, the higher-valence notions of rotation distance do not give Tamari lattices, so there are fewer tools for analysis in the higher-valence settings. Higher-valence trees arise in a range of database and filesystem applications where balance is important for efficient performance.

Counting Power Domination Sets in Complete m-ary Trees

Motivated by the question of computing the probability of successful power domination by placing k monitors uniformly at random, in this paper we give a recursive formula to count the number of power domination sets of size k in a labeled complete m-ary tree. As a corollary we show that the desired probability can be computed in exponential with linear exponent time.

Tree trace reconstruction using subtraces

AbstractTree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, Rácz, and Rashtchian [10] used combinatorial methods to show that $\exp({\mathrm{O}} (k \log_{k} n))$ samples suffice to reconstruct a complete k-ary tree with n nodes with high probability. We provide an alternative proof of this result, which allows us to generalize it to a broader class of tree topologies and deletion models. In our proofs we introduce the notion of a subtrace, which enables us to connect with and generalize recent mean-based complex analytic algorithms for string trace reconstruction.

Weighted Inequalities for Fractional Maximal Functions on the Infinite Rooted k-Ary Tree

In this article, we introduce the fractional Hardy–Littlewood maximal function on the infinite rooted k-ary tree and study its weighted boundedness. We also provide examples of weights for which the fractional Hardy–Littlewood maximal function satisfies strong type (p, q) estimates on the infinite rooted k-ary tree.

Development of an Arabic HQAS-based ASAG to consider an ignored knowledge in misspelled multiple words short answers

Many traditional question answering systems depend on an Automatic Short Answer Grading (ASAG) to evaluate misspelled multiple words short answers in an Arabic Language using common edit-based algorithms such as Hamming, Levenshtein, and Jaro_Winkler, but they ignore and hide a big amount of a significant knowledge of the student answer. In this paper, we have implemented a proposed edit-based Hierarchical question answering system (HQAS) using a traversing by Breadth-First Search (BFS) within an m-ary tree to consider the ignored significant knowledge due to the misspelling at the middle of the dual-ordered incomplete answer, the misspelling at middle and the end of the intra-ordered incomplete answer, and the misspelling due to switching in words of the intra-ordered complete answer. It can differentiate among the students based on their significant hidden knowledge and show a distribution of knowledge content on different depths of the topic to determine which of the topic depths the student has the most significant knowledge.

Hereditary properties of finite ultrametric spaces

A characterization of finite homogeneous ultrametric spaces and finite ultrametric spaces generated by unrooted labeled trees has been made in terms of representing trees. Finite ultrametric spaces having perfect strictly n-ary trees have been characterized in terms of special graphs connected with the space. A detailed survey of some special classes of finite ultrametric spaces, which were considered in the last ten years, has been given and their hereditary properties have been studied. More precisely, we are interested in the following question. Let X be an arbitrary finite ultrametric space from some given class. Does every subspace of X also belong to this class?

Raney numbers, threshold sequences and Motzkin-like paths

We provide new interpretations for a subset of Raney numbers, involving threshold sequences and Motzkin-like paths with long up and down steps.Given three integers n,k,ℓ such that n≥1,k≥2 and 0≤ℓ≤k−2, a (k,ℓ)-threshold sequence of length n is any strictly increasing sequence S=(s1s2…sn) of integers such that ki≤si≤kn+ℓ. These sequences are in bijection with the (ℓ+1)-tuples of k-ary trees with a total of n internal nodes. We prove this result using counting arguments involving Raney numbers, but we also provide an explicit bijection between threshold sequences and tuples of trees. We further show how to represent threshold sequences as Motzkin-like paths with long up and down steps, and deduce that these paths are enumerated by the same Raney numbers. Finally, we illustrate the use of threshold sequences for finding new combinatorial identities.

Theoretical Performance Analysis of Distributed Queue for Massive Machine Type Communications: Throughput, Latency, Energy Consumption

Massive machine type communications (mMTC) is one of main application cases in 5G, which is supposed to support communications of massive number of machine-type devices (MTDs). Distributed queue (DQ) is a variant of tree splitting protocol which combines an m-ary tree splitting algorithm with a set of simple smart rules, organizing every terminal in one out of two virtual queues. Theoretically, DQ allows access to infinite terminals and is stable under any traffic condition, which alleviates the unstable problem of slotted ALOHA, and is especially suitable for mMTC. However, its theoretical comprehensive performance analysis as well as related statistical characteristics is still missing, which severely restricts the full manifestation of its performance advantages. In view of this, the paper proposes a general performance analysis framework for DQ, with which full probability space of DQ evolution process is presented for the first time. To be more specific, probability distribution function (PDF), mean and variance of throughput, latency and energy consumption of DQ is analytically derived to comprehensively evaluate performance. Taking the IEEE 802.15.4 standard for mMTC as example, numerical results validate the accuracy of the proposed analysis framework and the stability of DQ, present effects of number of MTDs, number of contention slots ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> ), and maximum number of transmissions ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula> ) on DQ in terms of aforementioned performance metrics. These results together provide good reference to find appropriate value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${L}$ </tex-math></inline-formula> to balance the performance metrics and enable more practical network optimization.