Embedding a family of disjoint multi-dimensional meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint multi-dimensional meshes into a crossed cube. We prove that for n?4 and 1?m?⌊n/2⌋−1, a family of m² disjoint k-dimensional meshes of size t₁²×t₂²×?×tk² each can be embedded in an n-dimensional crossed cube with unit dilation, where and max1?i?k{ti}?n−2m−1. This result means that a family of mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends some recently obtained results.     

Embedding meshes into crossed cubes

Crossed cubes are important variants of hypercubes. In this paper, we consider embeddings of meshes in crossed cubes. The major research findings in this paper are: (1) For any integer n ? 1, a 2 × 2ⁿ⁻¹ mesh can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 1. (2) For any integer n ? 4, two node-disjoint 4 × 2ⁿ⁻³ meshes can be embedded in the n-dimensional crossed cube with dilation 1 and expansion 2. The obtained results are optimal in the sense that the dilations of the embeddings are 1. The embedding of the 2 × 2ⁿ⁻¹ mesh is also optimal in terms of expansion because it has the smallest expansion 1.     

Embedding a family of meshes into twisted cubes

The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement. The twisted cube is an important variation of the hypercube. Let TQn denote the n-dimensional twisted cube. In this paper, we consider embedding a family of 2-dimensional meshes into a twisted cube. The main results obtained in this paper are: (1) For any odd integer n?1, there exists a mesh of size 2ײn−1 that can be embedded in the TQn with unit dilation and unit expansion. (2) For any odd integer n?5, there exists a mesh of size 4ײn−2 that can be embedded in the TQn with dilation 2 and unit expansion. (3) For any odd integer n?5, a family of two disjoint meshes of size 4ײn−3 can be embedded into the TQn with unit dilation and unit expansion. Results (1) and (3) are optimal in the sense that the dilations and expansions of the embeddings are unit values.     

Embedding a long fault-free cycle in a crossed cube with more faulty nodes

The crossed cube is an important variant of the most popular hypercube network for parallel computing. In this paper, we consider the problem of embedding a long fault-free cycle in a crossed cube with more faulty nodes. We prove that for n?5 and f?2n−7, a fault-free cycle of length at least n²−f−(n−5) can be embedded in an n-dimensional crossed cube with f faulty nodes. Our work extends some previously known results in the sense of the maximum number of faulty nodes tolerable in a crossed cube.     

Embedding multi-dimensional meshes into twisted cubes

The twisted cube is an important variant of the most popular hypercube network for parallel processing. In this paper, we consider the problem of embedding multi-dimensional meshes into twisted cubes in a systematic way. We present a recursive method for embedding a family of disjoint multi-dimensional meshes into a twisted cube with dilation 1 and expansion 1. We also prove that a single multi-dimensional mesh can be embedded into a twisted cube with dilation 2 and expansion 1. Our work extends some previously known results.     

Complete path embeddings in crossed cubes

Crossed cubes are popular variants of hypercubes. In this paper, we study path embeddings between any two distinct nodes in crossed cubes. We prove two important results in the n-dimensional crossed cube: (a) for any two nodes, all paths whose lengths are greater than or equal to the distance between the two nodes plus 2 can be embedded between the two nodes with dilation 1; (b) for any two integers n ? 2 and l with , there always exist two nodes x and y whose distance is l, such that no path of length l + 1 can be embedded between x and y with dilation 1. The obtained results are optimal in the sense that the dilations of path embeddings are all 1. The results are also complete, because the embeddings of paths of all possible lengths between any two nodes are considered.     

Embedding meshes into locally twisted cubes

As a newly introduced interconnection network for parallel computing, the locally twisted cube possesses many desirable properties. In this paper, mesh embeddings in locally twisted cubes are studied. Let LTQ_n(V, E) denote the n-dimensional locally twisted cube. We present three major results in this paper: (1) For any integer n ? 1, a 2 × 2ⁿ⁻¹ mesh can be embedded in LTQ_n with dilation 1 and expansion 1. (2) For any integer n ? 4, two node-disjoint 4 × 2ⁿ⁻³ meshes can be embedded in LTQ_n with dilation 1 and expansion 2. (3) For any integer n ? 3, a 4  × (2ⁿ⁻² − 1) mesh can be embedded in LTQ_n with dilation 2. The first two results are optimal in the sense that the dilations of all embeddings are 1. The embedding of the 2 × 2ⁿ⁻¹ mesh is also optimal in terms of expansion. We also present the analysis of 2p × 2q mesh embedding in locally twisted cubes.     

Embedding meshes into twisted-cubes

The n-dimensional twisted-cube, TN_n, is a variation of the hypercube. In this paper, we study embedding of meshes into TN_n. We prove three major results in this paper: (1) For any integer n ? 1, a 2 × 2ⁿ⁻¹ mesh can be embedded into TN_n with dilation 1 and expansion 1. (2) For any integer n ? 4, an m × k(m ? 3, k ? 3) mesh cannot be embedded into TN_n with dilation 1. (3) For any integer n ? 4, two node-disjoint 4 × 2ⁿ⁻³ meshes can be embedded into TN_n with dilation 2 and expansion 1.     

The twisted crossed cube

The topology of interconnection networks plays an important role in the performance of parallel and distributed computing systems. In this paper, we propose a new interconnection network called twisted crossed cube (TCQ_n) and investigate its basic network properties in terms of the regularity, connectivity, fault tolerance, recursiveness, hamiltonicity and ability to simulate other architectures, and so on. Then, we develop an effective routing algorithm Route (u, v) for TCQ_n that takes no more than d(u, v) + 1 steps for any two nodes (u, v) to communicate with each other, and the routing process shows that the diameter, wide diameter, and fault‐tolerant diameter of TCQ_n are about half of the corresponding diameters of the equivalent hypercube with the same dimension. In the end, by combining TCQ_n with crossed cube (CQ_n), we propose a preferable dynamic network structure, that is, the dynamic crossed cube, which has the same network diameter as TCQ_n/CQ_n and better properties in other respects, for example, its connection complexity is half of that of TCQ_n/CQ_n when the network scale is large enough, and the number of its average routing steps is also much smaller than that in TCQ_n/CQ_n. Copyright © 2015 John Wiley & Sons, Ltd.     

The connectivity of exchanged crossed cube

The exchanged crossed cube, proposed by Li et al., is a new interconnection network having better properties than other variations of hypercube in the area of the fewer diameter, smaller links and lower cost factor. In this work we will show that the connectivity of exchanged crossed cube is equal to its minimum degree. Moreover each smallest cut in exchanged crossed cube is a neighborhood of some vertex.     

A dynamic programming algorithm for simulation of a multi-dimensional torus in a crossed cube

The torus is a popular interconnection topology and several commercial multicomputers use a torus as the basis of their communication network. Moreover, there are many parallel algorithms with torus-structured and mesh-structured task graphs have been developed. If one network can embed a mesh or torus network, the algorithms with mesh-structured or torus-structured can also be used in this network. Thus, the problem of embedding meshes or tori into networks is meaningful for parallel computing. In this paper, we prove that for n ? 6 and 1 ? m ? ⌈n/2⌉ − 1, a family of 2^m disjoint k-dimensional tori of size 2^s₁×2^s₂×?×2^s_k each can be embedded in an n-dimensional crossed cube with unit dilation, where each s_i ? 2, , and max_1?i?k{s_i} ? 3 if n is odd and ; otherwise, max_1?i?k{s_i} ? n − 2m − 1. A new concept, cycle skeleton, is proposed to construct a dynamic programming algorithm for embedding a desired torus into the crossed cube. Furthermore, the time complexity of the algorithm is linear with respect to the size of desired torus. As a consequence, a family of disjoint tori can be simulated on the same crossed cube efficiently and in parallel.     

Embedding of Cycles in Twisted Cubes with Edge-Pancyclic

In this paper, we study the embedding of cycles in twisted cubes. It has been proven in the literature that, for any integer l, 4≤l≤2 ⁿ , a cycle of length l can be embedded with dilation 1 in an n-dimensional twisted cube, n≥3. We obtain a stronger result of embedding of cycles with edge-pancyclic. We prove that, for any integer l, 4≤l≤2 ⁿ , and a given edge (x,y) in an n-dimensional twisted cube, n≥3, a cycle C of length l can be embedded with dilation 1 in the n-dimensional twisted cube such that (x,y) is in C in the twisted cube. Based on the proof of the edge-pancyclicity of twisted cubes, we further provide an O(llog l+n ²+nl) algorithm to find a cycle C of length l that contains (u,v) in TQ _n for any (u,v)∈E(TQ _n ) and any integer l with 4≤l≤2 ⁿ .     

Node-pancyclicity and edge-pancyclicity of crossed cubes

Crossed cubes are important variants of the hypercubes. It has been proven that crossed cubes have attractive properties in Hamiltonian connectivity and pancyclicity. In this paper, we study two stronger features of crossed cubes. We prove that the n-dimensional crossed cube is not only node-pancyclic but also edge-pancyclic for n?2. We also show that the similar property holds for hypercubes.     

Fault-free Hamiltonian cycles in crossed cubes with conditional link faults

The crossed cube, which is a variation of the hypercube, possesses some properties superior to the hypercube. In this paper, assuming that each node is incident with at least two fault-free links, we show that an n-dimensional crossed cube contains a fault-free Hamiltonian cycle, even if there are up to 2n − 5 link faults. The result is optimal with respect to the number of link faults tolerated. We also verify that the assumption is practically meaningful by evaluating its occurrence probability, which is very close to 1.     

Topological properties of the crossed cube architecture

Crossed cube is a variant obtained from the hypercube by redirecting a subset of the edges to span two or more dimensions. As a result, the diameter is reduced by half without increasing the link complexity. The use of the crossed cube as a parallel architecture, and in a reconfigurable system has been investigated earlier. The topological properties of the crossed cube are investigated in this paper. The main results of this paper include: an analysis of the number of isomorphic subgraphs, a formal proof for the diameter, and some new embedding properties.     

Mapping Between 2-D Meshes of the Same Size

1IntroductionInrece11tyears,n1a11yparallelalgoritlImshavebeendesignedtosolvedifferentproblemso1lvario1ls11etworktopologics.Bi11arytrees,meshesandhypercubesarethethreeimportal1tl1etworktop()logieswllicllhaterpcoivedintensivestlldy.WiththeadvanceofVLSI,manyllewl1etworkssuchasstargrapl1[1]havebeenorwiIlbeintroduced.Inor相似文献   

Fault-tolerant cycle-embedding of crossed cubes

The crossed cube CQ_n introduced by Efe has many properties similar to those of the popular hypercube. However, the diameter of CQ_n is about one half of that of the hypercube. Failures of links and nodes in an interconnection network are inevitable. Hence, in this paper, we consider the hybrid fault-tolerant capability of the crossed cube. Letting f_e and f_v be the numbers of faulty edges and vertices in CQ_n, we show that a cycle of length l, for any 4?l?|V(CQ_n)|−f_v, can be embedded into a wounded crossed cube as long as the total number of faults (f_v+f_e) is no more than n−2, and we say that CQ_n is (n−2)-fault-tolerant pancyclic. This result is optimal in the sense that if there are n−1 faults, there is no guarantee of having a cycle of a certain length in it.     

模型导向的矩形建筑物三维重建

针对以航空影像为数据源的矩形建筑物半自动重建进行了研究。采用模型导向的方法,用CSG与B-Rep相结合描述的矩形体基本模型,提取航空影像中建筑物边缘线,人机交互建立建筑物的初始模型;将初始模型反投到影像上,根据建筑物边缘与模型投影线之间套合程度,通过基于广义点摄影测量理论的模型影像匹配算法计算建筑物的精确参数。该方法利用影像上的直线信息进行广义点平差,减少运算量,提高建模效率与建模精度。实验结果证明该方法有较高的准确性。     

交叉立方体中嵌入超立方体的研究

Efe提出的交叉立方体(crossedcube)是超立方体(hypercube)的一种变型。但是,交叉立方体的某些性质却优于超立方体,其直径几乎是超立方体的一半。在本文中,研究了用交叉立方体互连网络来模拟超立方体互连网络,其实质是图嵌入问题,得出了以下结论:当n≤2,2n维交叉立方体CQ2n可同构嵌入两个n 1维立方体Qn 1。当n≥3,2n维交叉立方体CQ2n可同胚嵌入n 1维超立方体Qn 1。     

Optimally Embedding 3-Ary n-Cubes into Grids

Journal of Computer Science and Technology - The 3-ary n-cube, denoted as $$ {Q}_n^3 $$ , is an important interconnection network topology proposed for parallel computers, owing to its many...