Complete K-ary Trees Research Articles

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.

Antimagic orientation of subdivided caterpillars

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,…,m} such that no two vertices in D have the same vertex-sum under τ, where the vertex-sum of a vertex v in D under τ is the sum of labels of all arcs entering v minus the sum of labels of all arcs leaving v. Hefetz et al. (2010) conjectured that every connected graph admits an antimagic orientation. The conjecture was confirmed for certain classes of graphs such as regular graphs, graphs with minimum degree at least 33, bipartite graphs with no vertex of degree zero or two, and trees including caterpillars and complete k-ary trees. We prove that every subdivided caterpillar admits an antimagic orientation, where a subdivided caterpillar is a subdivision of a caterpillar T such that the edges of T that are not on the central path of T are subdivided the same number of times.

Weighted Dyck Paths with Special Restrictions on the Levels of Valleys

This paper concentrates on the set $$\mathcal {V}_n$$ of weighted Dyck paths of length 2n with special restrictions on the level of valleys. We first give its explicit formula of the counting generating function in terms of certain weight functions. When the weight functions are specialized, some connections are builded between $$\mathcal {V}_n$$ and other classical combinatorial structures such as (a, b)-Motzkin paths, q-Schröder paths, Delannoy paths and complete k-ary trees. Some bijections are also established between these settings and $$\mathcal {V}_n$$ subject to certain special weight functions.

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.

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.

Search Numbers in Networks with Special Topologies

In this paper, we consider the problem of finding the minimum number of searchers to sweep networks/graphs with special topological structures. Such a number is called the search number. We first study graphs, which contain only one cycle, and present a linear time algorithm to compute the vertex separation and the optimal layout of such graphs; by a linear-time transformation, we can find the search number of this kind of graphs in linear time. We also investigate graphs, in which every vertex lies on at most one cycle and each cycle contains at most three vertices of degree more than two, and we propose a linear time algorithm to compute their search number and optimal search strategy. We prove explicit formulas for the search number of the graphs obtained from complete k-ary trees by replacing vertices by cycles. We also present some results on approximation algorithms.

Global edge alliances in graphs

In the paper we introduce and study a new problem of finding a minimum global edge alliance in a graph which is related to the global defensive alliance (Haynes et al., 2013; Hedetniemi, 2004) and the global defensive set (Lewoń et al., 2016). We proved the NP-completeness of the global edge alliance problem for subcubic graphs and we constructed polynomial time algorithms for trees. We found the exact values of the size of the minimum global edge alliance for certain classes: paths, cycles, wheels, complete k-partite graphs and complete k-ary trees. Moreover, we proved some lower bounds for arbitrary graphs.

Line-broadcasting in complete k -ary trees

A line-broadcasting model in a connected graph G = ( V , E ) , | V | = n , is a model in which one vertex, called the originator of the broadcast, holds a message that has to be transmitted to all vertices of the graph through placement of a series of calls over the graph. In this model, an informed vertex can transmit a message through a path of any length in a single time unit, as long as two transmissions do not use the same edge at the same time. Farley [6] has shown that the process can be completed within at most ⌈ log 2 n ⌉ time units from any originator in a tree (and thus in any connected undirected graph) and that the cumulative cost of broadcasting one message from any vertex, that is, the total number of edges used, is at most ( n − 1 ) ⌈ log 2 n ⌉ . In this article, we present lower and upper bounds for the cumulative cost to broadcast one message in a complete k-ary tree, k ≥ 2 , from any vertex using the line-broadcasting model. We prove that if B(u) is the minimum cumulative cost of line-broadcast in a graph G = ( V , E ) from an originator u ∈ V using the line-broadcasting model, then ( 2 − o ( 1 ) ) n ≤ B ( u ) ≤ ( 2 + o ( 1 ) ) n , where u is any vertex in a complete k-ary tree. Furthermore, for certain conditions, B(u) is about ( 2 − o ( 1 ) ) n . © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 67(2), 139–147 2016

Upper Bounds on Number of Steals in Rooted Trees

Inspired by applications in parallel computing, we analyze the setting of work stealing in multithreaded computations. We obtain tight upper bounds on the number of steals when the computation can be modeled by rooted trees. In particular, we show that if the computation with n processors starts with one processor having a complete k-ary tree of height h (and the remaining n − 1 processors having nothing), the maximum possible number of steals is \({\sum }_{i=1}^{n}(k-1)^{i}\binom {h}{i}\).

ON THE RUPTURE DEGREE OF A GRAPH

In a communication network, vulnerability measures the resistance of the network to disruption of operation after the failure of certain stations or communication links. If we think of a connected graph as model-ing a network, the rupture degree of a graph is one measure of graph vulnerability and it is defined by r(G) = max{w(G − S) − |S| − m(G − S) : S ⊂ V (G);w(G − S) > 1}; where w(G −S) is the number of components of G −S and m(G −S) is the order of a largest component of G − S. In this paper, general results on the rupture degree of a graph are considered. Firstly, some bounds on the rupture degree are given. Further, rupture degree of a complete k-ary tree is calculated. Also several results are given about complete k-ary tree and graph operations. Finally, we give formulas for the rupture degree of the cartesian product of some special graphs.

Some bounds on global alliances in trees

Given a simple graph G=(V,E), a subset S of the vertices is called a global defensive alliance if S is a dominating set and for every vertex v in S at least half of the vertices in the closed neighborhood of v are in S. Similarly, a subset S is called a global offensive alliance if S is a dominating set and for every vertex v not in S at least half of the vertices in the closed neighborhood of v are in S. In this paper, we study the minimum cardinality global defensive and global offensive alliances of complete k-ary trees. We also give a bound on the difference between these two parameters for general trees.

Global defensive alliances of trees and Cartesian product of paths and cycles

Given a graph G, a defensive alliance of G is a set of vertices S⊆V(G) satisfying the condition that for each v∈S, at least half of the vertices in the closed neighborhood of v are in S. A defensive alliance S is called global if every vertex in V(G)−S is adjacent to at least one member of the defensive alliance S. The global defensive alliance number of G, denoted γa(G), is the minimum size around all the global defensive alliances of G. In this paper, we present an efficient algorithm to determine the global defensive alliance numbers of trees, and further give formulas to decide the global defensive alliance numbers of complete k-ary trees for k=2,3,4. We also establish upper bounds and lower bounds for γa(Pm×Pn),γa(Cm×Pn) and γa(Cm×Cn), and show that the bounds are sharp for certain m,n.

Adding Edges between One Node and Every Other Node with the Same Depth in a Complete K-ary Tree

This paper proposes a model of adding relations between members of the same level in a pyramid organization structure which is a complete K-ary tree such that the communication of information between every member in the organization becomes the most efficient. When edges between one node and every other node with the same depth N in a complete K-ary tree of height H are added, an optimal depth N∗ = H is obtained by minimizing the total path length which is the sum of lengths of shortest paths between every pair of all nodes. Keywords—complete K-ary tree, organization structure, shortest path

Adding Edges between One Node and Every Other Node with the Same Depth in a Complete K-ary Tree

This paper proposes a model of adding relations between members of the same level in a pyramid organization structure which is a complete K-ary tree such that the communication of information between every member in the organization becomes the most efficient. When edges between one node and every other node with the same depth N in a complete K-ary tree of height H are added, an optimal depth N∗ = H is obtained by minimizing the total path length which is the sum of lengths of shortest paths between every pair of all nodes. Keywords—complete K-ary tree, organization structure, shortest path

A lower bound for the vertex boundary-width of complete k-ary trees

The vertex boundary-width problem (for short VBWP) is to determine the value of vbw ( G ) = max 1 ⩽ ℓ ⩽ | V | min S ⊆ V , | S | = ℓ | N ( S ) | for a given graph G = ( V , E ) , where N ( S ) = { v ∉ S | v is a neighbor of u for some u ∈ S } . In this paper, we give a lower bound for vertex boundary-width of complete k-ary trees.

Models of adding relations to an organization structure of a complete K-ary tree

This paper proposes three models of adding relations to an organization structure which is a complete K-ary tree of height H: (i) a model of adding an edge between two nodes with the same depth N, (ii) a model of adding edges between every pair of nodes with the same depth N and (iii) a model of adding edges between every pair of siblings with the same depth N. For each of the three models, an optimal depth N ∗ is obtained by maximizing the total shortening path length which is the sum of shortening lengths of shortest paths between every pair of all nodes.

Antibandwidth of Complete k-Ary Trees

Abstract The antibandwidth problem is to label vertices of a n-vertex graph injectively by 1 , 2 , 3 , … n , such that the minimum difference of labels of adjacent vertices is maximised. The problem is motivated by obnoxious facility location problem, radiocolouring, work and game scheduling and is dual to the well known bandwidth problem. We prove exact results for the antibandwidth of complete k-ary trees, k even, and estimate the parameter for odd k up to the second order term. This extends previous results for complete binary trees.

Efficient embeddings of ternary trees into hypercubes

In this paper we present efficient graph embeddings for complete k-ary trees into boolean hypercubes. In particular, we describe an efficient embedding of a complete ternary tree ( k=3) of height h into a hypercube, which achieves dilation 3 and expansion Θ(1.0104… h ). The previously best-known result is dilation 2 and expansion Θ(1.333… h ). Our embedding achieves exponentially better expansion at the cost of an increase of 1 in the dilation. We also describe efficient embeddings of k-ary trees into hypercubes when k=2 p∗3 q for some p, q>0 such that the embeddings achieve small constant dilation.

New results on edge-bandwidth

The edge-bandwidth problem is an analog of the classical bandwidth problem, in which one has to label the edges of a graph by distinct integers such that the maximum difference of labels of any two incident edges is minimized. We prove tight bounds on the edge-bandwidth of hypercube and butterfly graphs, and complete k-ary trees which extends and improves on previous known results.

Embedding complete k-ary trees into k-square 2-D meshes with optimal edge congestion

Wormhole switching is distance insensitive but suffers from blocking in the case of communication conflicts. Embeddings of process graphs into interconnection networks of parallel machines with wormhole switching can ignore dilation, but should minimize edge congestion for a given load or expansion. In this paper, we present an algorithm for embedding complete k-ary trees into k-square 2-D meshes with load 1, asymptotically optimal expansion, and nearly optimal edge congestion.