495 research outputs found
Quantum Walks, Quantum Gates and Quantum Computers
The physics of quantum walks on graphs is formulated in Hamiltonian language,
both for simple quantum walks and for composite walks, where extra discrete
degrees of freedom live at each node of the graph. It is shown how to map
between quantum walk Hamiltonians and Hamiltonians for qubit systems and
quantum circuits; this is done for both a single- and multi-excitation coding,
and for more general mappings. Specific examples of spin chains, as well as
static and dynamic systems of qubits, are mapped to quantum walks, and walks on
hyperlattices and hypercubes are mapped to various gate systems. We also show
how to map a quantum circuit performing the quantum Fourier transform, the key
element of Shor's algorithm, to a quantum walk system doing the same. The
results herein are an essential preliminary to a Hamiltonian formulation of
quantum walks in which coupling to a dynamic quantum environment is included.Comment: 17 pages, 10 figure
Lower bounds for dilation, wirelength, and edge congestion of embedding graphs into hypercubes
Interconnection networks provide an effective mechanism for exchanging data
between processors in a parallel computing system. One of the most efficient
interconnection networks is the hypercube due to its structural regularity,
potential for parallel computation of various algorithms, and the high degree
of fault tolerance. Thus it becomes the first choice of topological structure
of parallel processing and computing systems. In this paper, lower bounds for
the dilation, wirelength, and edge congestion of an embedding of a graph into a
hypercube are proved. Two of these bounds are expressed in terms of the
bisection width. Applying these results, the dilation and wirelength of
embedding of certain complete multipartite graphs, folded hypercubes, wheels,
and specific Cartesian products are computed
Optimal simulation of full binary trees on faulty hypercubes
The problem of operating full binary tree based algorithms on a hypercube with faulty nodes was investigated. Developing a method for embedding a full binary tree into the faulty hypercube is the solution to this problem. Two outcomes for embedding an (n-1)-tree into an n-cube with unit dilation and load, that were based on a new embedding technique, were presented. For the problem where the root can be mapped to any nonfaulty hypercube node, the optimum toleration of faults was shown. Moreover, it was demonstrated that the algorithm for the variable root embedding problem is maximal within a class algorithms called recursive embedding algorithms as far as the number of tolerable faults is concerned. Lastly, it was demonstrated that when an O(1/√n) fraction of nodes in the hypercube are faulty, a O(1)-load variable root embedding is not always possible regardless of the significance of the dilation.published_or_final_versio
Methods and problems of wavelength-routing in all-optical networks
We give a survey of recent theoretical results obtained for wavelength-routing in all-optical networks. The survey is based on the previous survey in [Beauquier, B., Bermond, J-C., Gargano, L., Hell, P., Perennes, S., Vaccaro, U.: Graph problems arising from wavelength-routing in all-optical networks. In: Proc. of the 2nd Workshop on Optics and Computer Science, part of IPPS'97, 1997]. We focus our survey on the current research directions and on the used methods. We also state several open problems connected with this line of research, and give an overview of several related research directions
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
Fault-Tolerant Ring Embeddings in Hypercubes -- A Reconfigurable Approach
We investigate the problem of designing reconfigurable embedding schemes for a fixed hypercube (without redundant processors and links). The fundamental idea for these schemes is to embed a basic network on the hypercube without fully utilizing the nodes on the hypercube. The remaining nodes can be used as spares to reconfigure the embeddings in case of faults. The result of this research shows that by carefully embedding the application graphs, the topological properties of the embedding can be preserved under fault conditions, and reconfiguration can be carried out efficiently.
In this dissertation, we choose the ring as the basic network of interest, and propose several schemes for the design of reconfigurable embeddings with the aim of minimizing reconfiguration cost and performance degradation. The cost is measured by the number of node-state changes or reconfiguration steps needed for processing of the reconfiguration, and the performance degradation is characterized as the dilation of the new embedding after reconfiguration. Compared to the existing schemes, our schemes surpass the existing ones in terms of applicability of schemes and reconfiguration cost needed for the resulting embeddings
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