48,424 research outputs found

    On pole-swapping algorithms for the eigenvalue problem

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    Pole-swapping algorithms, which are generalizations of the QZ algorithm for the generalized eigenvalue problem, are studied. A new modular (and therefore more flexible) convergence theory that applies to all pole-swapping algorithms is developed. A key component of all such algorithms is a procedure that swaps two adjacent eigenvalues in a triangular pencil. An improved swapping routine is developed, and its superiority over existing methods is demonstrated by a backward error analysis and numerical tests. The modularity of the new convergence theory and the generality of the pole-swapping approach shed new light on bi-directional chasing algorithms, optimally packed shifts, and bulge pencils, and allow the design of novel algorithms

    Complexity of Token Swapping and its Variants

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    In the Token Swapping problem we are given a graph with a token placed on each vertex. Each token has exactly one destination vertex, and we try to move all the tokens to their destinations, using the minimum number of swaps, i.e., operations of exchanging the tokens on two adjacent vertices. As the main result of this paper, we show that Token Swapping is W[1]W[1]-hard parameterized by the length kk of a shortest sequence of swaps. In fact, we prove that, for any computable function ff, it cannot be solved in time f(k)no(k/logk)f(k)n^{o(k / \log k)} where nn is the number of vertices of the input graph, unless the ETH fails. This lower bound almost matches the trivial nO(k)n^{O(k)}-time algorithm. We also consider two generalizations of the Token Swapping, namely Colored Token Swapping (where the tokens have different colors and tokens of the same color are indistinguishable), and Subset Token Swapping (where each token has a set of possible destinations). To complement the hardness result, we prove that even the most general variant, Subset Token Swapping, is FPT in nowhere-dense graph classes. Finally, we consider the complexities of all three problems in very restricted classes of graphs: graphs of bounded treewidth and diameter, stars, cliques, and paths, trying to identify the borderlines between polynomial and NP-hard cases.Comment: 23 pages, 7 Figure

    Three-dimensional unstructured grid refinement and optimization using edge-swapping

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    This paper presents a three-dimensional (3-D) 'edge-swapping method based on local transformations. This method extends Lawson's edge-swapping algorithm into 3-D. The 3-D edge-swapping algorithm is employed for the purpose of refining and optimizing unstructured meshes according to arbitrary mesh-quality measures. Several criteria including Delaunay triangulations are examined. Extensions from two to three dimensions of several known properties of Delaunay triangulations are also discussed

    Swapping algorithm and meta-heuristic solutions for combinatorial optimization n-queens problem

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    This research proposes the swapping algorithm a new algorithm for solving the n-queens problem, and provides data from experimental performance results of this new algorithm. A summary is also provided of various meta-heuristic approaches which have been used to solve the n-queens problem including neural networks, evolutionary algorithms, genetic programming, and recently Imperialist Competitive Algorithm (ICA). Currently the Cooperative PSO algorithm is the best algorithm in the literature for finding the first valid solution. Also the research looks into the effect of the number of hidden nodes and layers within neural networks and the effect on the time taken to find a solution. This paper proposes a new swapping algorithm which swaps the position of queens

    Locality-Aware Qubit Routing for the Grid Architecture

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    Due to the short decohorence time of qubits available in the NISQ-era, it is essential to pack (minimize the size and or the depth of) a logical quantum circuit as efficiently as possible given a sparsely coupled physical architecture. In this work we introduce a locality-aware qubit routing algorithm based on a graph theoretic framework. Our algorithm is designed for the grid and certain \u27grid-like\u27 architectures. We experimentally show the competitiveness of algorithm by comparing it against the approximate token swapping algorithm, which is used as a primitive in many state-of-the-art quantum trans pilers. Our algorithm produces circuits of comparable depth (better on random permutations) while being an order of magnitude faster than a typical implementation of the approximate token swapping algorithm
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