Embeddings into the Pancake Interconnection Network

Article paru en 2002 dans Parallel Processing LettersInternational audienceOwing to its nice properties, the pancake is one of the Cayley graphs that were proposed as alternatives to the hypercube for interconnecting processors in parallel computers. In this paper, we present embeddings of rings, grids and hypercubes into the pancake with constant dilation and congestion. We also extend the results to similar efficient embeddings into the star graph

Recursive circulants and their embeddings among hypercubes

AbstractWe propose an interconnection structure for multicomputer networks, called recursive circulant. Recursive circulant G(N,d) is defined to be a circulant graph with N nodes and jumps of powers of d. G(N,d) is node symmetric, and has some strong hamiltonian properties. G(N,d) has a recursive structure when N=cdm, 1⩽c<d. We develop a shortest-path routing algorithm in G(cdm,d), and analyze various network metrics of G(cdm,d) such as connectivity, diameter, mean internode distance, and visit ratio. G(2m,4), whose degree is m, compares favorably to the hypercube Qm. G(2m,4) has the maximum possible connectivity, and its diameter is ⌈(3m−1)/4⌉. Recursive circulants have interesting relationship with hypercubes in terms of embedding. We present expansion one embeddings among recursive circulants and hypercubes, and analyze the costs associated with each embedding. The earlier version of this paper appeared in Park and Chwa (Proc. Internat. Symp. Parallel Architectures, Algorithms and Networks ISPAN’94, Kanazawa, Japan, December 1994, pp. 73–80)

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

Graph Coverings with Few Eigenvalues or No Short Cycles

This thesis addresses the extent of the covering graph construction. How much must a cover X resemble the graph Y that it covers? How much can X deviate from Y? The main statistics of X and Y which we will measure are their regularity, the spectra of their adjacency matrices, and the length of their shortest cycles. These statistics are highly interdependent and the main contribution of this thesis is to advance our understanding of this interdependence. We will see theorems that characterize the regularity of certain covering graphs in terms of the number of distinct eigenvalues of their adjacency matrices. We will see old examples of covers whose lack of short cycles is equivalent to the concentration of their spectra on few points, and new examples that indicate certain limits to this equivalence in a more general setting. We will see connections to many combinatorial objects such as regular maps, symmetric and divisible designs, equiangular lines, distance-regular graphs, perfect codes, and more. Our main tools will come from algebraic graph theory and representation theory. Additional motivation will come from topological graph theory, finite geometry, and algebraic topology

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))

Problems in the Theory of Convergence Spaces

We investigate several problems in the theory of convergence spaces: generalization of Kolmogorov separation from topological spaces to convergence spaces, representation of reflexive digraphs as convergence spaces, construction of differential calculi on convergence spaces, mereology on convergence spaces, and construction of a universal homogeneous pretopological space. First, we generalize Kolmogorov separation from topological spaces to convergence spaces; we then study properties of Kolmogorov spaces. Second, we develop a theory of reflexive digraphs as convergence spaces, which we then specialize to Cayley graphs. Third, we conservatively extend the concept of differential from the spaces of classical analysis to arbitrary convergence spaces; we then use this extension to obtain differential calculi for finite convergence spaces, finite Kolmogorov spaces, finite groups, Boolean hypercubes, labeled graphs, the Cantor tree, and real and binary sequences. Fourth, we show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete, and consequently, any model of general extensional mereology is indistinguishable from a model of set theory; we then generalize these results to the cartesian closed category of convergence spaces. Finally, we show that every convergence space can be embedded into a homogeneous convergence space; we then use this result to construct a universal homogeneous pretopological space