K-ary Trees Research Articles

On contact tracing in COVID-19 (SARS-CoV-2) pandemic using lowest common ancestor in m-ary data aggregation tree in the fog-computing enhanced internet of things

PurposeThe purpose of this study is to exploit the lowest common ancestor technique in an m-ary data aggregation tree in the fog computing-enhanced IoT to assist in contact tracing in COVID-19. One of the promising characteristics of the Internet of Things (IoT) that can be used to save the world from the current crisis of COVID-19 pandemic is data aggregation. As the number of patients infected by the disease is already huge, the data related to the different attributes of patients such as patient thermal image record and the previous health record of the patient is going to be gigantic. The authors used the technique of data aggregation to efficiently aggregate the sensed data from the patients and analyse it. Among the various inferences drawn from the aggregated data, one of the most important is contact tracing. Contact tracing in COVID-19 deals with finding out a person or a group of persons who have infected or were infected by the disease.Design/methodology/approachThe authors propose to exploit the technique of lowest common ancestor in an m-ary data aggregation tree in the Fog-Computing enhanced IoT to help the health-care experts in contact tracing in a particular region or community. In this research, the authors argue the current scenario of COVID-19 pandemic, finding the person or a group of persons who has/have infected a group of people is of extreme importance. Finding the individuals who have been infected or are infecting others can stop the pandemic from worsening by stopping the community transfer. In a community where the outbreak has spiked, the samples from either all the persons or the patients showing the symptoms are collected and stored in an m-ary tree-based structure sorted over time.FindingsContact tracing in COVID-19 deals with finding out a person or a group of persons who have infected or were infected by the disease. The authors exploited the technique of lowest common ancestor in an m-ary data aggregation tree in the fog-computing-enhanced IoT to help the health-care experts in contact tracing in a particular region or community. The simulations were carried randomly on a set of individuals. The proposed algorithm given in Algorithm 1 is executed on the samples collected at level-0 of the simulation model, and to aggregate the data and transmit the data, the authors implement Algorithm 2 at the level-1. It is found from the results that a carrier can be easily identified from the samples collected using the approach designed in the paper.Practical implicationsThe work presented in the paper can aid the health-care experts fighting the COVID-19 pandemic by reducing the community transfer with efficient contact tracing mechanism proposed in the paper.Social implicationsFighting COVID-19 efficiently and saving the humans from the pandemic has huge social implications in the current times of crisis.Originality/valueTo the best of the authors’ knowledge, the lowest common ancestor technique in m-ary data aggregation tree in the fog computing-enhanced IoT to contact trace the individuals who have infected or were infected during the transmission of COVID-19 is first of its kind proposed. Creating a graph or an m-ary tree based on the interactions/connections between the people in a particular community like location, friends and time, the authors can attempt to traverse it to find out who infected any two persons or a group of persons or was infected by exploiting the technique of finding out the lowest common ancestor in a m-ary tree.

Antimagic orientation of lobsters

Let m≥1 be an integer and G be a graph with m edges. We say that G has an antimagic orientation if G has an orientation D and a bijection τ:A(D)→{1,2,…,m} such that no two vertices in D have the same vertex-sum under τ, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Hefetz et al. (2010) conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of graphs such as dense graphs, regular graphs, and trees including caterpillars and complete k-ary trees. In this note, we prove that every lobster admits an antimagic orientation.

Computing the topological descriptors of line graph of the complete m-ary trees

Computing the topological descriptors of line graph of the complete m-ary trees

Distance-based indices of complete m-ary trees

Binary and [Formula: see text]-ary trees have extensive applications, particularly in computer science and chemistry. We present exact values of all important distance-based indices for complete [Formula: see text]-ary trees. We solve recurrence relations to obtain the value of the most well-known index called the Wiener index. New methods are used to express the other indices (the degree distance, the eccentric distance sum, the Gutman index, the edge-Wiener index, the hyper-Wiener index and the edge-hyper-Wiener index) as well. Values of distance-based indices for complete binary trees are corollaries of the main results.

Some Domination Parameters of Halin Graph with Perfect K-ary Tree and its Application

Some Domination Parameters of Halin Graph with Perfect K-ary Tree and its Application

Formula omitted]-Ary spanning trees contained in tournaments

A rooted tree is called a k-ary tree, if all non-leaf vertices have exactly k children, except possibly one non-leaf vertex has at most k−1 children. Denote by h(k) the minimum integer such that every tournament of order at least h(k) contains a k-ary spanning tree. It is well-known that every tournament contains a Hamiltonian path, which implies that h(1)=1. Lu et al. (1999) proved the existence of h(k), and showed that h(2)=4 and h(3)=8. The exact values of h(k) remain unknown for k≥4. A result of Erdős on the domination number of tournaments implies h(k)=Ω(klogk). In this paper, we prove that h(4)=10 and h(5)≥13.

The undirected optical indices of complete [formula omitted]-ary trees

The routing and wavelength assignment problem arises from the investigation of optimal wavelength allocation in an optical network that employs Wavelength Division Multiplexing (WDM). Consider an optical network that is represented by a connected, simple graph G. An all-to-all routing R in G is a set of paths connecting all pairs of vertices of G. The undirected optical index of G is the minimum integer k to guarantee the existence of a mapping ϕ:R→{1,2,…,k}, such that ϕ(P)≠ϕ(P′) if P and P′ have common edge(s), over all possible routings R. A natural lower bound of the undirected optical index of G is the (undirected) edge-forwarding index, which is defined to be the minimum of the maximum edge-load over all possible all-to-all routings. In this paper, we first derive the exact value of the optical index of the complete m-ary trees, and then investigate the gap between undirected optical and edge-forwarding indices.

Admissibility versus Ap-Conditions on Regular Trees

Abstract We show that the combination of doubling and (1, p)-Poincaré inequality is equivalent to a version of the Ap-condition on rooted K-ary trees.

Hausdorff Dimensions for Graph-Directed Measures Driven by Infinite Rooted Trees

We give upper and lower bounds for the Hausdorff dimensions for a class of graph-directed measures when its underlying directed graph is the infinite N-ary tree. These measures are different from graph-directed self-similar measures driven by finite directed graphs and are not necessarily Gibbs measures. However our class contains several measures appearing in fractal geometry and functional equations, specifically, measures defined by restrictions of non-constant harmonic functions on the two-dimensional Sierp\'inski gasket, the Kusuoka energy measures on it, and, measures defined by solutions of de Rham's functional equations driven by linear fractional transformations.

Centroid based Tree-Structured Data Clustering Using Vertex/Edge Overlap and Graph Edit Distance

We consider a clustering problem in which the data objects are rooted m-ary trees with known node correspondence. We assume that the nodes of the trees are unweighted, but the edges can be unweighted or weighted. We measure the similarity and distance between two trees using vertex/edge overlap (VEO) and graph edit distance (GED), respectively. For both measures, we first study the problem of finding a centroid tree of a given cluster of trees in both the unweighted and weighted edge cases. We compute the optimal centroid tree of a given cluster for all measures except the weighted VEO for which a heuristic is developed. We then propose k-means based algorithms that repeat cluster assignment and centroid update steps until convergence. The initial centroid trees are constructed based on the properties of the data. The assignment steps utilize unweighted or weighted versions of VEO or GED to assign each tree to the most similar centroid tree. In the update steps, each centroid tree is updated by considering the trees assigned to it. The proposed algorithms are compared with the traditional k-modes and k-means on randomly generated datasets and shown to be more effective and robust (to outliers) in separating trees into clusters. We also apply our algorithms on a real world brain artery data and show that the previously observed age and sex effects on brain artery structures can be revealed better by means of clustering with our algorithms than the traditional k-modes and k-means.

Energy-aware tree network formation among computationally weak nodes

We study the fundamental problem of distributed network formation among mobile agents of limited computational power that aim to achieve energy balance by wirelessly transmitting and receiving energy in a peer-to-peer manner. Specifically, we design simple distributed protocols consisting of a small number of states and interaction rules for the formation of arbitrary and k-ary tree networks. Furthermore, we evaluate (theoretically and also using computer simulations) a plethora of energy redistribution protocols that exploit different levels of knowledge in order to achieve desired energy distributions among the agents which require that every agent has exactly or at least twice the energy of the agents of higher depth, according to the structure of the network. Our study shows that without using any knowledge about the network structure, such energy distributions cannot be achieved in a timely manner, meaning that there might be high energy loss during the redistribution process. On the other hand, only a few extra bits of information seem to be enough to guarantee quick convergence to energy distributions that satisfy particular properties, yielding low energy loss.

Improved algorithms for ranking and unranking (k, m)-ary trees in B-order

Du and Liu (Eur J Comb 28:1312–1321, 2007) introduced (k, m)-ary trees as a generalization of k-ary trees. In a (k, m)-ary tree, every node on even level has degree k (i.e., has k children), and every node on odd level has degree m (which is called a crucial node) or is a leaf. In particular, a (k, m)-ary tree of order n has exactly n crucial nodes. Recently, Amani and Nowzari-Dalini (Bull Iranian Math Soc 45(4):1145–1158, 2019) presented a generation algorithm to produce all (k, m)-ary trees of order n in B-order using Zaks’ encoding, and showed that the generated ordering of this encoding results in a reverse-lexicographical ordering. They also proposed the corresponding ranking and unranking algorithms for (k, m)-ary trees according to such a generated ordering. These algorithms take $$\mathcal {O}(kmn^2)$$ time and space for building a precomputed table in which (k, m)-Catalan numbers (i.e., a kind of generalized Catalan numbers) are stored in advance. Then, each ranking and unranking algorithm can be performed subsequently in $$\mathcal {O}(n)$$ and $$\mathcal {O}(n\log n)$$ time, respectively. In this paper, we revisit the ranking and unranking problems. With the help of an encoding scheme called “right-distance” introduced by Wu et al. (Math Comput Model 53:1331–1335, 2011a; IEICE Trans Inf Syst E94–D:226–232, 2011b), we propose new ranking and unranking algorithms for (k, m)-ary trees of order n in B-order using Zaks’ encoding. We show that both algorithms can be improved in $$\mathcal {O}(kmn)$$ time and $$\mathcal {O}(n)$$ space without building the precomputed table.

Construction of Halin graph with perfect k-ary tree and its independent domination number

Halin graphs can be constructed from a tree non degree 2-vertices, embedded in the plane and pendant vertices are connected by outer cycle. This graph is 3- connected planar graph. In this article we formed some special families of Halin graphs with perfect k ary tree and find the independent domination number of those graphs.

Proof Without Words: Sums of Consecutive Powers of k Via a Perfect k-ary Tree

Summary By counting the nodes of a perfect k-ary tree, an identity involving the sums of consecutive powers of k is proved.

A language hierarchy and Kitchens-type theorem for self-similar groups

We generalize the notion of self-similar groups of infinite tree automorphisms to allow for groups which are defined on a tree but may not act faithfully on it. The elements of such a group correspond to labeled trees which may be recognized by a tree automaton (e.g. Rabin, Büchi, etc.), or considered as elements of a tree shift (e.g. of finite type, sofic) as in symbolic dynamics. We give examples to show how self-similar groups defined in this way can be separated into different tree language hierarchies. As the main result, extending the classical result of Kitchens on one-dimensional group shifts, we provide a sufficient condition for a self-similar group whose elements form a sofic tree shift to be a tree shift of finite type. As an application, we show that the closures of certain self-similar groups of rooted k-ary tree automorphisms that satisfy an algebraic law are not Rabin-recognizable, that is, they can not be described within the second order theory of k successors. In both the main result and in the application a crucial role is played by a distinguished branched subgroup structure of the groups under consideration.

Optimization of Hierarchical Regression Model with Application to Optimizing Multi-Response Regression K-ary Trees

THIS PAPER HAS BEEN RETRACTED The University of California, Merced campus, recently completed a formal investigation into allegations that content in this paper ("Optimization of Hierarchical Regression Model with Application to Optimizing Multi-Response Regression K-ary Trees," in Proceedings of the AAAI Conference on Artificial Intelligence, 2019, vol. 33, pp. 5133– 5142) was not properly attributed to the novel work of Professor Carreira—Perpiñán, reflected in a previously published paper, "Alternating Optimization of Decision Trees, with Application to Learning Sparse Oblique Trees,” published in Advances in Neural Information Processing Systems, 2018, (vol. 31, pp. 1211–1221). The UC Merced investigation was conducted in consultation with NSF, which funded the research reflected in the original Advances in Neural Information Processing Systems paper, and in accordance with UC Merced policies and procedures on research misconduct. The allegations were substantiated by a preponderance of the evidence. The investigation committee found, in pertinent part, that “key novel ideas in the AAAI paper were taken from the NeurIPS paper without appropriate credit." The initial submission of the AAAI paper, upon which the acceptance decision was made, did not include a reference to the NeurIPS paper, and thus represented a clear case of plagiarism. A citation to the NeurIPS paper was added in the final revision, after the reviewers had already decided to accept it, making it impossible for the AAAI peer-review process to consider the contributions of the AAAI article in light of the earlier work reported in the NeurIPS paper. Furthermore, the citation to the NeurIPS paper in this manuscript does not properly credit the original contributions of the NeurIPS paper and how those ideas were reflected in the work reported in this paper. In response to a formal request made by the University of California's Chancellor's office, and after careful review by the AAAI Publications Committee, AAAI has decided to retract this paper. One of the conditions of submission of a paper for publication in AAAI conference proceedings is that authors declare explicitly that their work is original and has not appeared in a publication elsewhere. Reuse of any data should be appropriately cited. As such this paper represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and apologies are offered to readers of this proceedings that this was not detected during the submission process.

Antimagic orientations for the complete k-ary trees

A labeling of a digraph D with m arcs is a bijection from the set of arcs of D to \(\{1,2,\ldots ,m\}\). A labeling of D is antimagic if all vertex-sums of vertices in D are pairwise distinct, where the vertex-sum of a vertex \(u \in V(D)\) for a labeling is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. Hefetz et al. (J Graph Theory 64:219–232, 2010) conjectured that every connected graph admits an antimagic orientation. We support this conjecture for the complete k-ary trees and show that all the complete k-ary trees \(T_k^r\) with height r have antimagic orientations for any k and r.

On the strength of some trees

Let G be a graph of order p. A numbering f of G is a labeling that assigns distinct elements of the set 1,2,…,p to the vertices of G, where each edge uv of G is labeled fu+fv. The strength strfG of a numbering f:VG→1,2,…,p of G is defined by strfG=maxfu+fvuv∈EG,that is, strfG is the maximum edge label of G, and the strength strG of a graph G itself is strG=minstrfGf is a numbering of G.The strengths strT and strTn,k are determined for caterpillars T and k-level complete n-ary trees Tn,k. The strength strG is also given for graphs G obtained by taking the corona of certain graphs and an arbitrary number of isolated vertices. The work of this paper suggests an open problem on the strength of trees.

Some extremal ratios of the distance and subtree problems in binary trees

Among many topological indices of trees the sum of distances σ(T) and the number of subtrees F(T) have been a long standing pair of graph invariants that are well known for their negative correlation. That is, among various given classes of trees, the extremal structures maximizing one usually minimize the other, and vice versa. By introducing the “local” versions of these invariants, σT(v) for the sum of distance from v to all other vertices and FT(v) for the number of subtrees containing v, extremal problems can be raised and studied for vertices within a tree. This leads to the concept of “middle parts” of a tree with respect to different indices. A challenging problem is to find extremal values of the ratios between graph indices and corresponding local functions at middle parts or leaves. This problem also provides new opportunities to further verify the the correlation between different indices such as σ(T) and F(T). Such extremal ratios, along with the extremal structures, were studied and compared for the distance and subtree problems for general trees (Barefoot, 1997; Székely and Wang, 2013, 2014). In this paper, this study is extended to binary trees, a class of trees with numerous practical applications in which the extremal ratio problems appear to be even more complicated. After justifying some basic properties on the distance and subtree problems in trees and binary trees, characterizations are provided for the extremal structures achieving two extremal ratios in binary trees of given order. The generalization of this work to k-ary trees is also briefly discussed. The findings are compared with the previous established extremal structures in general trees. Lastly some potential future work is mentioned.

Algorithmic expedients for the S-labeling problem

Graph labeling problems have been widely studied in the last decades and have a vast area of application. In this work, we study the recently introduced S-labeling problem, in which the nodes get labeled using labels from 1 to |V| and for each edge the contribution to the objective function, called S-labeling number of the graph, is the minimum label of its end-nodes. The goal is to find a labeling with minimum value. The problem is NP-hard for planar subcubic graphs, although for many other graph classes the complexity status is still unknown.In this paper, we present different algorithmic approaches for tackling this problem: We develop an exact solution framework based on Mixed-Integer Programming (MIP) which is enhanced with valid inequalities, starting and primal heuristics and specialized branching rules. We show that our MIP formulation has no integrality gap for paths, cycles and perfect n-ary trees, and, to the best of our knowledge, we give the first polynomial-time algorithm for the problem on n-ary trees as well as a closed formula for the S-labeling number for such trees. Moreover, we also present a Lagrangian heuristic and a constraint programming approach. A computational study is carried out in order to (i) investigate if there may be other special graph classes, where our MIP formulation has no integrality gap, and (ii) assess the effectiveness of the proposed solution approaches for solving the problem on a dataset consisting of general graphs.