Constructing Two Edge-Disjoint Hamiltonian Cycles in Locally Twisted Cubes

The $n$-dimensional hypercube network $Q_n$ is one of the most popular interconnection networks since it has simple structure and is easy to implement. The $n$-dimensional locally twisted cube, denoted by $LTQ_n$, an important variation of the hypercube, has the same number of nodes and the same number of connections per node as $Q_n$. One advantage of $LTQ_n$ is that the diameter is only about half of the diameter of $Q_n$. Recently, some interesting properties of $LTQ_n$ were investigated. In this paper, we construct two edge-disjoint Hamiltonian cycles in the locally twisted cube $LTQ_n$, for any integer $n\geqslant 4$. The presence of two edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the locally twisted cube.Comment: 7 pages, 4 figure

Fault-free longest paths in star networks with conditional link faults

AbstractThe star network, which belongs to the class of Cayley graphs, is one of the most versatile interconnection networks for parallel and distributed computing. In this paper, adopting the conditional fault model in which each node is assumed to be incident with two or more fault-free links, we show that an n-dimensional star network can tolerate up to 2n−7 link faults, and be strongly (fault-free) Hamiltonian laceable, where n≥4. In other words, we can embed a fault-free linear array of length n!−1 (n!−2) in an n-dimensional star network with up to 2n−7 link faults, if the two end nodes belong to different partite sets (the same partite set). The result is optimal with respect to the number of link faults tolerated. It is already known that under the random fault model, an n-dimensional star network can tolerate up to n−3 faulty links and be strongly Hamiltonian laceable, for n≥3

Embedding as a method to improve multichannel bus topology parameters

This article attempts to discuss the problems in building new topologies utilizing embedding. We propose to utilize the cartesian product to describe graphs as a right mathematical solution to adjacency matrix. We applied the cartesian product of two and sets of multiple dimensions in analysis. We proposed methodology of building logical topologies based on utilization of regular and symmetric graphs. For three particular products, we performed the analysis to measure traffic intensity in the network

Properties and Algorithms of the KCube Interconnection Networks

The KCube interconnection network was first introduced in 2010 in order to exploit the good characteristics of two well-known interconnection networks, the hypercube and the Kautz graph. KCube links up multiple processors in a communication network with high density for a fixed degree. Since the KCube network is newly proposed, much study is required to demonstrate its potential properties and algorithms that can be designed to solve parallel computation problems. In this thesis we introduce a new methodology to construct the KCube graph. Also, with regard to this new approach, we will prove its Hamiltonicity in the general KC(m; k). Moreover, we will find its connectivity followed by an optimal broadcasting scheme in which a source node containing a message is to communicate it with all other processors. In addition to KCube networks, we have studied a version of the routing problem in the traditional hypercube, investigating this problem: whether there exists a shortest path in a Qn between two nodes 0n and 1n, when the network is experiencing failed components. We first conditionally discuss this problem when there is a constraint on the number of faulty nodes, and subsequently introduce an algorithm to tackle the problem without restrictions on the number of nodes

Interconnection networks for parallel and distributed computing

Parallel computers are generally either shared-memory machines or distributed- memory machines. There are currently technological limitations on shared-memory architectures and so parallel computers utilizing a large number of processors tend tube distributed-memory machines. We are concerned solely with distributed-memory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is inter-processor communication. Communication is dependent upon the topology of the interconnection network, the routing mechanism, the flow control policy, and the method of switching. We are concerned with issues relating to the topology of the interconnection network. The choice of how we connect processors in a distributed-memory multiprocessor is a fundamental design decision. There are numerous, often conflicting, considerations to bear in mind. However, there does not exist an interconnection network that is optimal on all counts and trade-offs have to be made. A multitude of interconnection networks have been proposed with each of these networks having some good (topological) properties and some not so good. Existing noteworthy networks include trees, fat-trees, meshes, cube-connected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, k-ary n-cubes, twisted cubes, n-star graphs, (n, k)-star graphs, alternating group graphs, de Bruijn networks, and bubble-sort graphs, to name but a few. We will mainly focus on k-ary n-cubes and (n, k)-star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented k-ary n- cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty k-ary n-cube Q(^k_n) in which the number of node faults f(_n) and the number of link faults f(_e) are such that f(_n) + f(_e) ≤ 2n - 2. We prove that given any two healthy nodes s and e of Q(^k_n), there is a path from s to e of length at least k(^n) - 2f(_n) - 1 (resp. k(^n) - 2f(_n) - 2) if the nodes s and e have different (resp. the same) parities (the parity of a node Q(^k_n) in is the sum modulo 2 of the elements in the n-tuple over 0, 1, ∙∙∙ , k - 1 representing the node). Our result is optimal in the sense that there are pairs of nodes and fault configurations for which these bounds cannot be improved, and it answers questions recently posed by Yang, Tan and Hsu, and by Fu. Furthermore, we extend known results, obtained by Kim and Park, for the case when n = 2.2. We give precise solutions to problems posed by Wang, An, Pan, Wang and Qu and by Hsieh, Lin and Huang. In particular, we show that Q(^k_n) is bi-panconnected and edge-bipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is m-panconnected, for m = (^n(k - 1) + 2k - 6’ / ‘_2), and (k -1) pancyclic (these bounds are optimal). We introduce a path-shortening technique, called progressive shortening, and strengthen existing results, showing that when paths are formed using progressive shortening then these paths can be efficiently constructed and used to solve a problem relating to the distributed simulation of linear arrays and cycles in a parallel machine whose interconnection network is Q(^k_n) even in the presence of a faulty processor.3. We define an interconnection network AQ(^k_n) which we call the augmented k-ary n-cube by extending a k-ary n-cube in a manner analogous to the existing extension of an n-dimensional hypercube to an n-dimensional augmented cube. We prove that the augmented k-ary n-cube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented k-ary n-cube Q(^k_n) - is a Cayley graph (and so is vertex-symmetric); has connectivity 4n - 2, and is such that we can build a set of 4n - 2 mutually disjoint paths joining any two distinct vertices so that the path of maximal length has length at most max{{n- l)k- (n-2), k + 7}; has diameter [(^k) / (_3)] + [(^k - 1) /( _3)], when n = 2; and has diameter at most (^k) / (_4) (n+ 1), for n ≥ 3 and k even, and at most [(^k)/ (_4) (n + 1) + (^n) / (_4), for n ^, for n ≥ 3 and k odd.4. We present an algorithm which given a source node and a set of n - 1 target nodes in the (n, k)-star graph S(_n,k) where all nodes are distinct, builds a collection of n - 1 node-disjoint paths, one from each target node to the source. The collection of paths output from the algorithm is such that each path has length at most 6k - 7, and the algorithm has time complexity O(k(^3)n(^4))

Aspects of practical implementations of PRAM algorithms

The PRAM is a shared memory model of parallel computation which abstracts away from inessential engineering details. It provides a very simple architecture independent model and provides a good programming environment. Theoreticians of the computer science community have proved that it is possible to emulate the theoretical PRAM model using current technology. Solutions have been found for effectively interconnecting processing elements, for routing data on these networks and for distributing the data among memory modules without hotspots. This thesis reviews this emulation and the possibilities it provides for large scale general purpose parallel computation. The emulation employs a bridging model which acts as an interface between the actual hardware and the PRAM model. We review the evidence that such a scheme crn achieve scalable parallel performance and portable parallel software and that PRAM algorithms can be optimally implemented on such practical models. In the course of this review we presented the following new results: 1. Concerning parallel approximation algorithms, we describe an NC algorithm for finding an approximation to a minimum weight perfect matching in a complete weighted graph. The algorithm is conceptually very simple and it is also the first NC-approximation algorithm for the task with a sub-linear performance ratio. 2. Concerning graph embedding, we describe dense edge-disjoint embeddings of the complete binary tree with n leaves in the following n-node communication networks: the hypercube, the de Bruijn and shuffle-exchange networks and the 2-dimcnsional mesh. In the embeddings the maximum distance from a leaf to the root of the tree is asymptotically optimally short. The embeddings facilitate efficient implementation of many PRAM algorithms on networks employing these graphs as interconnection networks. 3. Concerning bulk synchronous algorithmics, we describe scalable transportable algorithms for the following three commonly required types of computation; balanced tree computations. Fast Fourier Transforms and matrix multiplications