超级交叉立方体互连网络及其拓扑性质

交叉立方体是近年提出的超立方体的一种变种。由于它的许多优越性质（如直径、嵌入性等）,在并行处理领域越来越受到人们的重视。然而,像超立方体一样,它也有一个缺点,即要使交叉立方体升级,就必须成倍地增加其顶点个数。为了解决这一问题,本文将顶点个数的２的次幂的交叉立方体推广到具有任意个顶点的互连网络,提出了超级交叉立方体的定义,并证明它保持了交叉立方体在高速通度、对数级的直径和顶点度数等方面的优良性质,从     

Edge congestion and topological properties of crossed cubes

An n-dimensional crossed cube, CQ_n, is a variation of hypercubes. In this paper, we give a new shortest path routing algorithm based on a new distance measure defined herein. In comparison with Efe's algorithm, which generates one shortest path in O(n²) time, our algorithm can generate more shortest paths in O(n) time. Based on a given shortest path routing algorithm, we consider a new performance measure of interconnection networks called edge congestion. Using our shortest path routing algorithm and assuming that message exchange between all pairs of vertices is equally probable, we show that the edge congestion of crossed cubes is the same as that of hypercubes. Using the result of edge congestion, we can show that the bisection width of crossed cubes is 2^n-1. We also prove that wide diameter and fault diameter are [n/2]+2. Furthermore, we study embedding of cycles in cross cubes and construct more types than previous work of cycles of length at least four     

Optimal Embeddings of Paths with Various Lengths in Twisted Cubes

Twisted cubes are variants of hypercubes. In this paper, we study the optimal embeddings of paths of all possible lengths between two arbitrary distinct nodes in twisted cubes. We use TQ_n to denote the n-dimensional twisted cube and use dist(TQ_n, u, v) to denote the distance between two nodes u and v in TQ_n, where n ges l is an odd integer. The original contributions of this paper are as follows: 1) We prove that a path of length l can be embedded between u and v with dilation 1 for any two distinct nodes u and v and any integer l with dist(TQ_n, u, v) + 2 les l les 2ⁿ - 1 (n ges 3) and 2) we find that there exist two nodes u and v such that no path of length dist(TQ_n, u, v) + l can be embedded between u and v with dilation 1 (n ges 3). The special cases for the nonexistence and existence of embeddings of paths between nodes u and v and with length dist(TQ_n, u, v) + 1 are also discussed. The embeddings discussed in this paper are optimal in the sense that they have dilation 1     

Diagnosability of crossed cubes under the comparison diagnosismodel

Diagnosability of a multiprocessor system is an important study topic in the parallel processing area. As a hypercube variant, the crossed cube has many attractive properties. The diameter, wide diameter and fault diameter of it are all approximately half those of the hypercube. The power with which the crossed cube simulates trees and cycles is stronger than the hypercube. Because of these advantages, the crossed cube has attracted much attention from researchers. In this paper, we show that the n-dimensional crossed cube is n-diagnosable under a major diagnosis model-the comparison diagnosis model proposed by Malek and Maeng (1981) if n ⩾ 4. According to this, the polynomial algorithm presented by Sengupta and Dahbura (1992) may be used to diagnose the n-dimensional crossed cube, provided that the number of the faulty nodes in the n-dimensional crossed cube does not exceed n. The conclusion also indicates that the diagnosability of the n-dimensional crossed cube is the same as that of the n-dimensional hypercube when n ⩾ 5 and better than that of the n-dimensional hypercube when n = 4     

交叉立方体环的Hamilton连通性和Pancyclicity性

交叉立方体是超立方体互连网络的一种变型,它的某些性质优于超立方体。例如,其直径几乎是超立方体的一半;当n≥3,交叉立方体CQn具有Hamilton连通性;当n≥2,所有长度在4到2n之间的圈都能够以扩张1嵌入CQn,即交叉立方体具有Pancyclity性。但是,交叉立方体同超立方体一样,当需要升级时,必须成倍增加结点。交叉立方体环互连网络CRN作为层次环互连网络HRN[8]的一种,可以有效地克服这个缺点,当需要升级时,只需在环上增加一个交叉立方体。在文中,证明了交叉立方体环互连网络仍然保持了交叉立方体具有的Hamilton连通性和Pancyclity性。     

Diagnosability of crossed cubes under the comparison diagnosismodel

Diagnosability of a multiprocessor system is one important study topic in the parallel processing area. As a hypercube variant, the crossed cube has many attractive properties. The diameter, wide diameter and fault diameter of it are all approximately half of those of the hypercube. The power that the crossed cube simulates trees and cycles is stronger than the hypercube. Because of these advantages of the crossed cube, it has attracted much attention from researchers. We show that the n-dimensional crossed cube is n-diagnosable under a major diagnosis model-the comparison diagnosis model proposed by Malek (1980) and Maeng and Malek (1981) if n⩾4. According to this, the polynomial algorithm presented by Sengupta and Dahbura (1992) may be used to diagnose the n-dimensional crossed cube, provided that the number of the faulty nodes in the n-dimensional crossed cube does not exceed n. The conclusion of this paper also indicates that the diagnosability of the n-dimensional crossed cube is the same as that of the n-dimensional hypercube when n>5 and better than that of the n-dimensional hypercube when n=4     

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.     

人工智能神经网络在交叉立方体上的有效映射

Malluhi等人在文献犤1犦中介绍了人工智能神经网络(ANNs)在超立方体上的有效映射,交叉立方体是超立方体的一个重要变型,而且具有比超立方体更优越的性质,如果在交叉立方体上实现ANNs的有效映射,会有更好的意义。论文证明了一个N×NMAT(mesh-of-appendixed-trees)可以嵌入包含4N2个节点的交叉立方体中,其中N是最大一层的长度,并且证明这个嵌入是最优的,从而给出了ANNs在交叉立方体上的一个有效映射。     

On Embedding Hamiltonian Cycles in Crossed Cubes

We study the embedding of Hamiltonian cycle in the Crossed Cube, which is a prominent variant of the classical hypercube, obtained by crossing some straight links of a hypercube, and has been attracting much research interest in literatures since its proposal. We will show that due to the loss of link-topology regularity, generating Hamiltonian cycles in a crossed cube is a more complicated procedure than in its original counterpart. The paper studies how the crossed links affect an otherwise succinct process to generate a host of well-structured Hamiltonian cycles traversing all nodes. The condition for generating these Hamiltonian cycles in a crossed cube is proposed. An algorithm is presented that works out a Hamiltonian cycle for a given link permutation. The useful properties revealed and the algorithm proposed in this paper can find their way when system designers evaluate a candidate network's competence and suitability, balancing regularity and other performance criteria, in choosing an interconnection network.     

One-to-one communication in twisted cubes under restricted connectivity

The dimensions of twisted cubes are only limited to odd integers. In this paper, we first extend the dimensions of twisted cubes to all positive integers. Then, we introduce the concept of the restricted faulty set into twisted cubes. We further prove that under the condition that each node of the n-dimensional twisted cube TQ_n has at least one fault-free neighbor, its restricted connectivity is 2n − 2, which is almost twice as that of TQ_n under the condition of arbitrary faulty nodes, the same as that of the n-dimensional hypercube. Moreover, we provide an O(NlogN) fault-free unicast algorithm and simulations result of the expected length of the fault-free path obtained by our algorithm, where N denotes the node number of TQ_n. Finally, we propose a polynomial algorithm to check whether the faulty node set satisfies the condition that each node of the n-dimensional twisted cube TQ_n has at least one fault-free neighbor.     

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.     

HCH-立方体的Hamilton连通性

新型并行计算系统的研制依赖于对新型互连网络结构及其性质的研究。超立方体及其变型——交叉立方体具有优点,也具有缺点。文献[1]给出了在超立方体与交叉立方体的顶点之间的一种连接——超连接,从而得到了一种称为HCH-立方体的互连网络,文章证明了当n≥4,HCH-立方体任意两个顶点之间存在Hamilton路径,即HCH-立方体是Hamilton连通的,而超立方体不是Hamilton连通的。这表明HCH-立方体具备了交叉立方体在Hamilton连通性方面的性质。文章还给出了在n维HCH-立方体中构造任意两个顶点之间Hamilton路径的算法,该算法的时间复杂度为O(N),其中N=2n,为n维HCH-立方体的顶点个数。     

Embedding a family of disjoint 3D meshes into a crossed cube

Crossed cubes are an important class of hypercube variants. This paper addresses how to embed a family of disjoint 3D meshes into a crossed cube. Two major contributions of this paper are: (1) for n?4, a family of two disjoint 3D meshes of size 2×2×2^n-3 can be embedded in an n-D crossed cube with unit dilation and unit expansion, and (2) for n?6, a family of four disjoint 3D meshes of size 4×2×2^n-5 can be embedded in an n-D crossed cube with unit dilation and unit expansion. These results mean that a family of two or four 3D-mesh-structured parallel algorithms can be executed on a same crossed cube efficiently and in parallel. Our work extends the results recently obtained by Fan and Jia [J. Fan, X. Jia, Embedding meshes into crossed cubes, Information Sciences 177(15) (2007) 3151-3160].     

交叉立方体在两种策略下的可诊断性

互连网络可诊断性度的高低是衡量这种网络性能优劣的重要标志二交叉立方体是近年提出的一类互连网络，它有一些比超立方体更好的性质．本文用PMC模型证明了n维交叉立方体Dn在精确策略和悲观策略下分别是n-可诊断的和（2n－2）／（2n－2）一可诊断的，从而证明民在这两种策略下的可诊断性度与n维超立体的相同．另外，本文在证明Dn是n-可诊断的同时，还得到了Dn中任何两顶点之间的n条互不相交的路径，它们可作为容错远路的依据．     

两类重要网络的传输延迟分析

提出了网络平均距离参数概念,用以度量网络的整体传输性能。与平均距离μ不同,网络平均距离μ′具有较强的网络应用背景。针对叉立方体网络的结构特性,给出了在交叉立方体网络中确定任意两个顶点之间最短路的长度和最短路条数的算法。从最短路、直径、平均距离、网络平均距离方面综合分析比较了超立方体网络和交叉立方体网络的信息传输延迟性能。     

Embedding of generalized Fibonacci cubes in hypercubes with faultynodes

The generalized Fibonacci cubes (abbreviated to GFCs) were recently proposed as a class of interconnection topologies, which cover a spectrum ranging from regular graphs such as the hypercube to semiregular graphs such as the second order Fibonacci cube. It has been shown that the kth order GFC of dimension n+k is equivalent to an n-cube for 0⩽n相似文献   

一种交叉立方体网络的并行路由算法

Efe提出的交叉立方体是超立方体的一种变型，其某些性质优于超立方体。在高性能的并行计算机系统中，信息通过若干条内结点互不交叉的路径并行传输，这些路径的长度将直接影响并行计算的性能。该文提出了一种时间复杂度为o(n2)的交叉立方体网络并行路由算法，可输出源点u到目的点v的3条并行路径P0,P1,P2，并且满足：(1)|P0|= u到v的距离；(2)|Pi|≤u到v的距离+3(i=1,2)。这说明该算法是通信高效的。     

The crossed cube architecture for parallel computation

The construction of a crossed cube which has many of the properties of the hypercube, but has diameter only about half as large, is discussed. This network is self-routing, in the sense that there is a simple distributed routing algorithm which guarantees optimal paths between any pair of vertices. This fact, together with other properties such as regularity, symmetry, high connectivity, and a simple recursive structure, suggests that the crossed cube may be an attractive alternative to the ordinary hypercube for massively parallel architectures, SIMD algorithms, which utilize the architecture are developed, and it is shown that the CQ_n architecture can profitably emulate the ordinary hypercube. It is also shown that addition of simple switches can improve the capabilities of the system significantly. For instance, the dynamic reconfiguration capability allows hypercube algorithms to be executed on the proposed architecture. The use of these switches also improves the embedding properties of the system     

Near-optimal broadcast in all-port wormhole-routed hypercubes usingerror-correcting codes

A new broadcasting method is presented for hypercubes with wormhole routing mechanism. The communication model assumed allows an n-dimensional hypercube to have at most n concurrent 110 communications along its ports. It further assumes a distance insensitivity of (n+1) with no intermediate reception capability for the nodes along the communication path. The approach is based on determination of the set of nodes (called stations) in the hypercube such that for any node in the network there is a station at distance of at most 1. Once stations are identified, parallel disjoint paths are formed from the source to all stations. The broadcasting is accomplished first by sending the message to all stations which will in turn inform the rest of the nodes of the message. To establish node-disjoint paths between the source node and all stations, we introduce a new routing strategy. We prove that multicasting can be done in one routing step as long as the number of destination nodes are at most n in an n-dimensional hypercube. The number of broadcasting steps using our routing is equal to or smaller than that obtained in an earlier work; this number is optimal for all hypercube dimensions n⩽12, except for n=10