scholarly journals Embedding Complete Binary Trees into Locally Twisted Cubes

2012 ◽  
Vol 6-7 ◽  
pp. 70-75 ◽  
Author(s):  
Yue Juan Han ◽  
Jian Xi Fan ◽  
Lan Tao You ◽  
Yan Wang
Keyword(s):  
Parallel Computing ◽  
Binary Tree ◽  
Interconnection Network ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Twisted Cubes

The locally twisted cube is a newly introduced interconnection network for parallel computing, which possesses many desirable properties. In this paper, the problem of embedding complete binary trees into locally twisted cubes is studied.Let LTQn(V;E) denote the n-dimensional locally twisted cube.We find the following result in this paper: for any integern ≥ 2,we show that a complete binary tree with 2n—1 nodes can be embedded into the LTQn with dilation 2.

Embedding of Binomial Trees in Locally Twisted Cubes with Link Faults

2014 ◽  
Vol 1049-1050 ◽  
pp. 1736-1740
Author(s):  
Lan Tao You
Keyword(s):  
Parallel Computing ◽  
Interconnection Network ◽  
Binomial Tree ◽  
Binomial Trees ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Twisted Cubes

As a newly introduced interconnection network for parallel computing, the locally twisted cube possesses many desirable properties. In this paper, binomial tree embeddings in locally twisted cubes are studied. We present two major results in this paper: (1) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith dilation 1 by randomly choosing any vertex inLTQnas the root. (2) For any integern≥ 2, ann-dimensional binomial treeBncan be embedded inLTQnwith up ton− 1 faulty links inlog(n− 1) steps where dilation = 1. The results are optimal in the sense that the dilations of all embeddings are 1.

The Tightly Super 2-extra Connectivity and 2-extra Diagnosability of Locally Twisted Cubes

2017 ◽  
Vol 17 (02) ◽  
pp. 1750006 ◽  
Author(s):  
YUNXIA REN ◽  
SHIYING WANG
Keyword(s):  
Fault Tolerance ◽  
Fault Diagnosis ◽  
Interconnection Network ◽  
Connected Subgraph ◽  
Minimum Cardinality ◽  
Pmc Model ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Twisted Cubes

Connectivity plays an important role in measuring the fault tolerance of an interconnection network [Formula: see text]. A faulty set [Formula: see text] is called a g-extra faulty set if every component of G − F has more than g nodes. A g-extra cut of G is a g-extra faulty set F such that G − F is disconnected. The minimum cardinality of g-extra cuts is said to be the g-extra connectivity of G. G is super g-extra connected if every minimum g-extra cut F of G isolates one connected subgraph of order g + 1. If, in addition, G − F has two components, one of which is the connected subgraph of order g + 1, then G is tightly [Formula: see text] super g-extra connected. Diagnosability is an important metric for measuring the reliability of G. A new measure for fault diagnosis of G restrains that every fault-free component has at least (g + 1) fault-free nodes, which is called the g-extra diagnosability of G. The locally twisted cube LTQn is applied widely. In this paper, it is proved that LTQn is tightly (3n − 5) super 2-extra connected for [Formula: see text], and the 2-extra diagnosability of LTQn is 3n − 3 under the PMC model ([Formula: see text]) and MM* model ([Formula: see text]).

Binary Trees and the n-Cutset Property

1991 ◽  
Vol 34 (1) ◽  
pp. 23-30 ◽  
Author(s):  
Peter Arpin ◽  
John Ginsburg
Keyword(s):  
Positive Integer ◽  
Partially Ordered Set ◽  
Binary Tree ◽  
Ordered Set ◽  
Maximal Chain ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Maximal Elements ◽  
Extremal Case ◽  
Result Theorem

AbstractA partially ordered set P is said to have the n-cutset property if for every element x of P, there is a subset S of P all of whose elements are noncomparable to x, with |S| ≤ n, and such that every maximal chain in P meets {x} ∪ S. It is known that if P has the n-cutset property then P has at most 2n maximal elements. Here we are concerned with the extremal case. We let Max P denote the set of maximal elements of P. We establish the following result. THEOREM: Let n be a positive integer. Suppose P has the n-cutset property and that |Max P| = 2n. Then P contains a complete binary tree T of height n with Max T = Max P and such that C ∩ T is a maximal chain in T for every maximal chain C of P. Two examples are given to show that this result does not extend to the case when n is infinite. However the following is shown. THEOREM: Suppose that P has the ω-cutset property and that |Max P| = 2ω. If P — Max P is countable then P contains a complete binary tree of height ω

OPTIMUM CONGESTED ROUTING STRATEGY ON TWISTED CUBES

2000 ◽  
Vol 01 (02) ◽  
pp. 115-134 ◽  
Author(s):  
TSENG-KUEI LI ◽  
JIMMY J. M. TAN ◽  
LIH-HSING HSU ◽  
TING-YI SUNG
Keyword(s):  
Shortest Path ◽  
Time Complexity ◽  
Interconnection Network ◽  
Routing Algorithm ◽  
Optimum Time ◽  
Shortest Path Routing ◽  
Routing Strategy ◽  
Bisection Width ◽  
Twisted Cube ◽  
Twisted Cubes

Given a shortest path routing algorithm of an interconnection network, the edge congestion is one of the important factors to evaluate the performance of this algorithm. In this paper, we consider the twisted cube, a variation of the hypercube with some better properties, and review the existing shortest path routing algorithm8. We find that its edge congestion under the routing algorithm is high. Then, we propose a new shortest path routing algorithm and show that our algorithm has optimum time complexity O(n) and optimum edge congestion 2n. Moreover, we calculate the bisection width of the twisted cube of dimension n.

EMBEDDING A COMPLETE BINARY TREE INTO A THREE-DIMENSIONAL GRID

2004 ◽  
Vol 05 (02) ◽  
pp. 111-130
Author(s):  
WOLFGANG W. BEIN ◽  
LAWRENCE L. LARMORE ◽  
CHARLES O. SHIELDS ◽  
I. HAL SUDBOROUGH
Keyword(s):  
Binary Tree ◽  
Three Dimensional ◽  
Expansion Ratio ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Recursive Scheme ◽  
Number Of Layers ◽  
One To One

We describe total congestion 1 embeddings of complete binary trees into three dimensional grids with low expansion ratio r. That is, we give a one-to-one embedding of any complete binary tree into a hexahedron shaped grid such that (a) the number of nodes in the grid is at most r times the number of nodes in the tree, and (b) no tree nodes or edges occupy the same grid positions. The first strategy embeds trees into cube shaped 3D grids. That is, 3D grids in which all dimensions are roughly equal in size, and which thus have no limit in the number of layers. The technique uses a recursive scheme, and we obtain an expansion ratio of r=1.09375. We then give strategies which embed trees into flat 3D grid shapes. That is, we map complete binary trees into 3D grids with a fixed, small number of layers k. Using again a recursive scheme, for k=2, we obtain r=1.25. By a rather different technique, which intricately weaves the branches of various subtrees into each other, we are able to obtain very tight embeddings: We have r=1.171875 for embeddings into five layer grids and r=1.09375 for embeddings into seven layer grids.

Hamiltonian Cycles Passing Through Prescribed Edges in Locally Twisted Cubes

2021 ◽  
Author(s):  
Dongqin Cheng
Keyword(s):  
Hamiltonian Cycle ◽  
Disjoint Paths ◽  
Hamiltonian Cycles ◽  
Induced Subgraph ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Vertex Disjoint ◽  
Cube Formula ◽  
Twisted Cubes

Let [Formula: see text] be a set of edges whose induced subgraph consists of vertex-disjoint paths in an [Formula: see text]-dimensional locally twisted cube [Formula: see text]. In this paper, we prove that if [Formula: see text] contains at most [Formula: see text] edges, then [Formula: see text] contains a Hamiltonian cycle passing through every edge of [Formula: see text], where [Formula: see text]. [Formula: see text] has a Hamiltonian cycle passing through at most one prescribed edge.

Three Types of Two-Disjoint-Cycle-Cover Pancyclicity and Their Applications to Cycle Embedding in Locally Twisted Cubes

2019 ◽  
Author(s):  
Tzu-Liang Kung ◽  
Hon-Chan Chen ◽  
Chia-Hui Lin ◽  
Lih-Hsing Hsu
Keyword(s):  
Disjoint Cycle ◽  
Cycle Cover ◽  
Disjoint Cycles ◽  
Locally Twisted Cubes ◽  
Twisted Cube ◽  
Vertex Disjoint ◽  
Twisted Cubes

Abstract A graph $G=(V,E)$ is two-disjoint-cycle-cover $[r_1,r_2]$-pancyclic if for any integer $l$ satisfying $r_1 \leq l \leq r_2$, there exist two vertex-disjoint cycles $C_1$ and $C_2$ in $G$ such that the lengths of $C_1$ and $C_2$ are $l$ and $|V(G)| - l$, respectively, where $|V(G)|$ denotes the total number of vertices in $G$. On the basis of this definition, we further propose Ore-type conditions for graphs to be two-disjoint-cycle-cover vertex/edge $[r_1,r_2]$-pancyclic. In addition, we study cycle embedding in the $n$-dimensional locally twisted cube $LTQ_n$ under the consideration of two-disjoint-cycle-cover vertex/edge pancyclicity.

Fault-tolerant embedding of complete binary trees in locally twisted cubes

2017 ◽  
Vol 101 ◽  
pp. 69-78 ◽  
Author(s):  
Zhao Liu ◽  
Jianxi Fan ◽  
Jingya Zhou ◽  
Baolei Cheng ◽  
Xiaohua Jia
Keyword(s):  
Fault Tolerant ◽  
Binary Trees ◽  
Locally Twisted Cubes ◽  
Twisted Cubes

A Ratio Inequality for Binary Trees and the Best Secretary

2002 ◽  
Vol 11 (2) ◽  
pp. 149-161 ◽  
Author(s):  
GRZEGORZ KUBICKI ◽  
JENŐ LEHEL ◽  
MICHAŁ MORAYNE
Keyword(s):  
Optimal Stopping ◽  
Binary Tree ◽  
Stopping Time ◽  
Secretary Problem ◽  
Hasse Diagram ◽  
Binary Trees ◽  
Complete Binary Tree ◽  
Maximum Element ◽  
Optimal Stopping Time ◽  
Natural Behaviour

Let Tn be the complete binary tree of height n considered as the Hasse diagram of a poset with its root 1n as the maximum element. Define A(n; T) = [mid ]{S ⊆ Tn : 1n ∈ S, S ≅ T}[mid ], and B(n; T) = [mid ]{S ⊆ Tn : 1n ∉ S, S ≅ T}[mid ]. In this note we prove that for any fixed n and rooted binary trees T1, T2 such that T2 contains a subposet isomorphic to T1. We conjecture that the ratio A/B also increases with T for arbitrary trees. These inequalities imply natural behaviour of the optimal stopping time in a poset extension of the secretary problem.

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