Differential Equations Solutions 96

Differential Equations Solutions 96 - The program KRS.m by...

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106 Chapter 17. Solutions: Case Study: Monte-Carlo Minimization and Counting A third alternative is to keep track of both edges and nodes. Think of it as a matching problem: each node can be matched with any of its four neighbors in a dimer, or it can be a monomer. We maintain an array of nodes, where the j th value is 0 if the node is a monomer, and equal to k ,if( k, j ) is a dimer. We store the edges in an n 2 × 4 array, where the row index indicates the node at the beginning of the edge, and the nonzero entries in the row record the indices of the neighboring nodes. Thus, each physical edge has two entries in the array (in rows corresponding to its two nodes), and a few of the entries at the edges are 0, since some nodes have fewer than 4 edges. We can generate a KRS change by picking an edge from this array, and we update the node array after we decide whether an addition, deletion, or swap should be considered.
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Unformatted text preview: The program KRS.m , by Sungwoo Park, on the website, is an efficient imple-mentation of the second alternative. Sample results are shown in Figure 17.9. Please refer to the original paper [2] for information on how to set the parame-ters to KRS. Kenyon, Randall, and Sinclair showed that the algorithm samples well if both the number of steps and the interval between records are very large, but in practice the algorithm is considerably less sensitive than the analysis predicts. [1] D. Bertsimas and J. Tsitsiklis, “Simulated annealing,” Statistical Science 8(1):10-15, 1993. [2] C. Kenyon, D. Randall, and A. Sinclair, “Approximating the number of monomer-dimer coverings of a lattice,” J. Stat. Phys. 83(3-4):637-659, 1996....
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