33 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))
Properties and algorithms of the hyper-star graph and its related graphs
The hyper-star interconnection network was proposed in 2002 to overcome the
drawbacks of the hypercube and its variations concerning the network cost, which is
defined by the product of the degree and the diameter. Some properties of the graph
such as connectivity, symmetry properties, embedding properties have been studied
by other researchers, routing and broadcasting algorithms have also been designed.
This thesis studies the hyper-star graph from both the topological and algorithmic
point of view. For the topological properties, we try to establish relationships between
hyper-star graphs with other known graphs. We also give a formal equation for the
surface area of the graph. Another topological property we are interested in is the
Hamiltonicity problem of this graph.
For the algorithms, we design an all-port broadcasting algorithm and a single-port
neighbourhood broadcasting algorithm for the regular form of the hyper-star graphs.
These algorithms are both optimal time-wise.
Furthermore, we prove that the folded hyper-star, a variation of the hyper-star, to be
maixmally fault-tolerant
Super edge-connectivity and matching preclusion of data center networks
Edge-connectivity is a classic measure for reliability of a network in the
presence of edge failures. -restricted edge-connectivity is one of the
refined indicators for fault tolerance of large networks. Matching preclusion
and conditional matching preclusion are two important measures for the
robustness of networks in edge fault scenario. In this paper, we show that the
DCell network is super- for and ,
super- for and , or and , and
super- for and . Moreover, as an application of
-restricted edge-connectivity, we study the matching preclusion number and
conditional matching preclusion number, and characterize the corresponding
optimal solutions of . In particular, we have shown that is
isomorphic to the -star graph for .Comment: 20 pages, 1 figur
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
Fault-tolerance embedding of rings and arrays in star and pancake graphs
The star and pancake graphs are useful interconnection networks for connecting processors in a parallel and distributed computing environment. The star network has been widely studied and is shown to possess attactive features like sublogarithmic diameter, node and edge symmetry and high resilience. The star/pancake interconnection graphs, {dollar}S\sb{n}/P\sb{n}{dollar} of dimension n have n! nodes connected by {dollar}{(n-1).n!\over2}{dollar} edges. Due to their large number of nodes and interconnections, they are prone to failure of one or more nodes/edges; In this thesis, we present methods to embed Hamiltonian paths (H-path) and Hamiltonian cycles (H-cycle) in a star graph {dollar}S\sb{n}{dollar} and pancake graph {dollar}P\sb{n}{dollar} in a faulty environment. Such embeddings are important for solving computational problems, formulated for array and ring topologies, on star and pancake graphs. The models considered include single-processor failure, double-processor failure, and multiple-processor failures. All the models are applied to an H-cycle which is formed by visiting all the ({dollar}{n!\over4!})\ S\sb4/P\sb4{dollar}s in an {dollar}S\sb{n}/P\sb{n}{dollar} in a particular order. Each {dollar}S\sb4/P\sb4{dollar} has an entry node where the cycle/path enters that particular {dollar}S\sb4/P\sb4{dollar} and an exit node where the path leaves it. Distributed algorithms for embedding hamiltonian cycle in the presence of multiple faults, are also presented for both {dollar}S\sb{n}{dollar} and {dollar}P\sb{n}{dollar}
Robustness and Stability of Spin Glass Ground States to Perturbed Interactions
Across many scientific and engineering disciplines, it is important to
consider how much the output of a given system changes due to perturbations of
the input. Here, we study the robustness of the ground states of spin
glasses on random graphs to flips of the interactions. For a sparse graph, a
dense graph, and the fully connected Sherrington-Kirkpatrick model, we find
relatively large sets of interactions that generate the same ground state.
These sets can themselves be analyzed as sub-graphs of the interaction domain,
and we compute many of their topological properties. In particular, we find
that the robustness of these sub-graphs is much higher than one would expect
from a random model. Most notably, it scales in the same logarithmic way with
the size of the sub-graph as has been found in genotype-phenotype maps for RNA
secondary structure folding, protein quaternary structure, gene regulatory
networks, as well as for models for genetic programming. The similarity between
these disparate systems suggests that this scaling may have a more universal
origin