47 research outputs found
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))
Interconnection Networks Embeddings and Efficient Parallel Computations.
To obtain a greater performance, many processors are allowed to cooperate to solve a single problem. These processors communicate via an interconnection network or a bus. The most essential function of the underlying interconnection network is the efficient interchanging of messages between processes in different processors. Parallel machines based on the hypercube topology have gained a great respect in parallel computation because of its many attractive properties. Many versions of the hypercube have been introduced by many researchers mainly to enhance communications. The twisted hypercube is one of the most attractive versions of the hypercube. It preserves the important features of the hypercube and reduces its diameter by a factor of two. This dissertation investigates relations and transformations between various interconnection networks and the twisted hypercube and explore its efficiency in parallel computation. The capability of the twisted hypercube to simulate complete binary trees, complete quad trees, and rings is demonstrated and compared with the hypercube. Finally, the fault-tolerance of the twisted hypercube is investigated. We present optimal algorithms to simulate rings in a faulty twisted hypercube environment and compare that with the hypercube
Embedding long paths in k-ary n-cubes with faulty nodes and links
Let be even and let . Consider a faulty k-ary n-cube in which the number of node faults and the number of link faults are such that . We prove that given any two healthy nodes s and e of , there is a path from s to e of length at least (resp. ) if the nodes s and e have different (resp. the same) parities (the parity of a node 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
Fault-tolerant Hamiltonian laceability of Cayley graphs generated by transposition trees
AbstractA bipartite graph is Hamiltonian laceable if there exists a Hamiltonian path joining every pair of vertices that are in different parts of the graph. It is well known that Cay(Sn,B) is Hamiltonian laceable, where Sn is the symmetric group on {1,2,…,n} and B is a generating set consisting of transpositions of Sn. In this paper, we show that for any F⊆E(Cay(Sn,B)), if |F|≤n−3 and n≥4, then there exists a Hamiltonian path in Cay(Sn,B)−F joining every pair of vertices that are in different parts of the graph. The result is optimal with respect to the number of edge faults
A survey of the theory of hypercube graphs
We present a comprehensive survey of the theory of hypercube graphs. Basic properties related to distance, coloring, domination and genus are reviewed. The properties of the n-cube defined by its subgraphs are considered next, including thickness, coarseness, Hamiltonian cycles and induced paths and cycles. Finally, various embedding and packing problems are discussed, including the determination of the cubical dimension of a given cubical graph.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27522/1/0000566.pd
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Resource placement, data rearrangement, and Hamiltonian cycles in torus networks
Many parallel machines, both commercial and experimental, have been/are being designed with toroidal interconnection networks. For a given number of nodes, the torus has a relatively larger diameter, but better cost/performance tradeoffs, such as higher channel bandwidth, and lower node degree, when compared to the hypercube. Thus, the torus is becoming a popular topology for the interconnection network of a high performance parallel computers.
In a multicomputer, the resources, such as I/O devices or software packages, are distributed over the networks. The first part of the thesis investigates efficient methods of distributing resources in a torus network. Three classes of placement methods are studied. They are (1) distant-t placement problem: in this case, any non-resource node is at a distance of at most t from some resource nodes, (2) j-adjacency problem: here, a non-resource node is adjacent to at least j resource nodes, and (3) generalized placement problem: a non-resource node must be a distance of at most t from at least j resource nodes.
This resource placement technique can be applied to allocating spare processors to provide fault-tolerance in the case of the processor failures. Some efficient
spare processor placement methods and reconfiguration schemes in the case of processor failures are also described.
In a torus based parallel system, some algorithms give best performance if the data are distributed to processors numbered in Cartesian order; in some other cases, it is better to distribute the data to processors numbered in Gray code order. Since the placement patterns may be changed dynamically, it is essential to find efficient methods of rearranging the data from Gray code order to Cartesian order and vice versa. In the second part of the thesis, some efficient methods for data transfer from Cartesian order to radix order and vice versa are developed.
The last part of the thesis gives results on generating edge disjoint Hamiltonian cycles in k-ary n-cubes, hypercubes, and 2D tori. These edge disjoint cycles are quite useful for many communication algorithms