Embedding cube-connected cycles graphs into faulty hypercubes

We consider the problem of embedding a cube-connected cycles graph (CCC) into a hypercube with edge faults. Our main result is an algorithm that, given a list of faulty edges, computes an embedding of the CCC that spans all of the nodes and avoids all of the faulty edges. The algorithm has optimal running time and tolerates the maximum number of faults (in a worst-case setting). Because ascend-descend algorithms can be implemented efficiently on a CCC, this embedding enables the implementation of ascend-descend algorithms, such as bitonic sort, on hypercubes with edge faults. We also present a number of related results, including an algorithm for embedding a CCC into a hypercube with edge and node faults and an algorithm for embedding a spanning torus into a hypercube with edge faults

Parallel Architectures for Planetary Exploration Requirements (PAPER)

The Parallel Architectures for Planetary Exploration Requirements (PAPER) project is essentially research oriented towards technology insertion issues for NASA's unmanned planetary probes. It was initiated to complement and augment the long-term efforts for space exploration with particular reference to NASA/LaRC's (NASA Langley Research Center) research needs for planetary exploration missions of the mid and late 1990s. The requirements for space missions as given in the somewhat dated Advanced Information Processing Systems (AIPS) requirements document are contrasted with the new requirements from JPL/Caltech involving sensor data capture and scene analysis. It is shown that more stringent requirements have arisen as a result of technological advancements. Two possible architectures, the AIPS Proof of Concept (POC) configuration and the MAX Fault-tolerant dataflow multiprocessor, were evaluated. The main observation was that the AIPS design is biased towards fault tolerance and may not be an ideal architecture for planetary and deep space probes due to high cost and complexity. The MAX concepts appears to be a promising candidate, except that more detailed information is required. The feasibility for adding neural computation capability to this architecture needs to be studied. Key impact issues for architectural design of computing systems meant for planetary missions were also identified

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}

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))

On strong Menger-connectivity of star graphs

AbstractMotivated by parallel routing in networks with faults, we study the following graph theoretical problem. Let G be a graph of minimum vertex degree d. We say that G is strongly Menger-connected if for any copy Gf of G with at most d−2 nodes removed, every pair of nodes u and v in Gf are connected by min{degf(u),degf(v)} node-disjoint paths in Gf, where degf(u) and degf(v) are the degrees of the nodes u and v in Gf, respectively. We show that the star graphs, which are a recently proposed attractive alternative to the widely used hypercubes as network models, are strongly Menger-connected. An algorithm of optimal running time is developed that constructs the maximum number of node-disjoint paths of nearly optimal length in star graphs with faults