178 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))
Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes
Examples complete our trilogy on the geometric and combinatorial
characterization of global Sturm attractors which consist of a
single closed 3-ball. The underlying scalar PDE is parabolic, on the unit interval with Neumann boundary
conditions. Equilibria are assumed to be hyperbolic. Geometrically, we
study the resulting Thom-Smale dynamic complex with cells defined by the fast
unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a
regular cell complex. In the first two papers we characterized 3-ball Sturm
attractors as 3-cell templates . The
characterization involves bipolar orientations and hemisphere decompositions
which are closely related to the geometry of the fast unstable manifolds. An
equivalent combinatorial description was given in terms of the Sturm
permutation, alias the meander properties of the shooting curve for the
equilibrium ODE boundary value problem. It involves the relative positioning of
extreme 2-dimensionally unstable equilibria at the Neumann boundaries and
, respectively, and the overlapping reach of polar serpents in the
shooting meander. In the present paper we apply these descriptions to
explicitly enumerate all 3-ball Sturm attractors with at most 13
equilibria. We also give complete lists of all possibilities to obtain solid
tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27
equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and
dodecahedra, we indicate a reduction to mere planar considerations as discussed
in our previous trilogy on planar Sturm attractors.Comment: 73+(ii) pages, 40 figures, 14 table; see also parts 1 and 2 under
arxiv:1611.02003 and arxiv:1704.0034
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
Multigrid Methods in Lattice Field Computations
The multigrid methodology is reviewed. By integrating numerical processes at
all scales of a problem, it seeks to perform various computational tasks at a
cost that rises as slowly as possible as a function of , the number of
degrees of freedom in the problem. Current and potential benefits for lattice
field computations are outlined. They include: solution of Dirac
equations; just operations in updating the solution (upon any local
change of data, including the gauge field); similar efficiency in gauge fixing
and updating; operations in updating the inverse matrix and in
calculating the change in the logarithm of its determinant; operations
per producing each independent configuration in statistical simulations
(eliminating CSD), and, more important, effectively just operations per
each independent measurement (eliminating the volume factor as well). These
potential capabilities have been demonstrated on simple model problems.
Extensions to real life are explored.Comment: 4
New Parameters for Beyond-Planar Graphs
Parameters for graphs appear frequently throughout the history of research in this field. They represent very important measures for the properties of graphs and graph drawings, and are often a main criterion for their classification and their aesthetic perception. In this direction, we provide new results for the following graph parameters:
– The segment complexity of trees;
– the membership of graphs of bounded vertex degree to certain graph classes;
– the maximal complete and complete bipartite graphs contained in certain graph classes beyond-planarity;
– the crossing number of graphs;
– edge densities for outer-gap-planar graphs and for bipartite gap-planar graphs with certain properties;
– edge densities and inclusion relationships for 2-layer graphs, as well as characterizations for complete bipartite graphs in the 2-layer setting
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum
Non-abelian Quantum Statistics on Graphs
We show that non-abelian quantum statistics can be studied using certain
topological invariants which are the homology groups of configuration spaces.
In particular, we formulate a general framework for describing quantum
statistics of particles constrained to move in a topological space . The
framework involves a study of isomorphism classes of flat complex vector
bundles over the configuration space of which can be achieved by
determining its homology groups. We apply this methodology for configuration
spaces of graphs. As a conclusion, we provide families of graphs which are good
candidates for studying simple effective models of anyon dynamics as well as
models of non-abelian anyons on networks that are used in quantum computing.
These conclusions are based on our solution of the so-called universal
presentation problem for homology groups of graph configuration spaces for
certain families of graphs.Comment: 50 pages, v3: updated to reflect the published version. Commun. Math.
Phys. (2019
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