Title:

Interconnection networks for parallel and distributed computing

Parallel computers are generally either sharedmemory machines or distributed memory machines. There are currently technological limitations on sharedmemory architectures and so parallel computers utilizing a large number of processors tend tube distributedmemory machines. We are concerned solely with distributedmemory multiprocessors. In such machines, the dominant factor inhibiting faster global computations is interprocessor 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 distributedmemory 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 tradeoffs 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, fattrees, meshes, cubeconnected cycles, butterflies, Möbius cubes, hypercubes, augmented cubes, kary ncubes, twisted cubes, nstar graphs, (n, k)star graphs, alternating group graphs, de Bruijn networks, and bubblesort graphs, to name but a few. We will mainly focus on kary ncubes and (n, k)star graphs in this thesis. Meanwhile, we propose a new interconnection network called augmented kary n cubes. The following results are given in the thesis.1. Let k ≥ 4 be even and let n ≥ 2. Consider a faulty kary ncube 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 ntuple 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 bipanconnected and edgebipancyclic, when k ≥ 3 and n ≥ 2, and we also show that when k is odd, Q(^k_n) is mpanconnected, for m = (^n(k  1) + 2k  6’ / ‘_2), and (k 1) pancyclic (these bounds are optimal). We introduce a pathshortening 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 kary ncube by extending a kary ncube in a manner analogous to the existing extension of an ndimensional hypercube to an ndimensional augmented cube. We prove that the augmented kary ncube Q(^k_n) has a number of attractive properties (in the context of parallel computing). For example, we show that the augmented kary ncube Q(^k_n)  is a Cayley graph (and so is vertexsymmetric); 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 (n2), 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 nodedisjoint 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)).
