# Node-disjoint paths on the mesh and a new trade-off in VLSI layout

@article{Aggarwal1996NodedisjointPO, title={Node-disjoint paths on the mesh and a new trade-off in VLSI layout}, author={Alok Aggarwal and Jon M. Kleinberg and David P. Williamson}, journal={SIAM J. Comput.}, year={1996}, volume={29}, pages={1321-1333} }

A number of basic models for VLSI layout are based on the construction of node- disjoint paths between terminals on a multilayer grid. In this setting, one is interested in minimizing both the number of layers required and the area of the underlying grid. Building on work of Cutler and Shiloach (Networks, 8 (1978), pp. 253{278), Aggarwal et al. (Proc. 26th IEEE Symposium on Foundations of Computer Science, Portland, OR, 1985; Algorithmica, 6 (1991), pp. 241{255), and Aggarwal, Klawe, and Shor…

## 37 Citations

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The algorithm shows that when all demand pairs are of the latter type, the integrality gap of the multicommodity flow LP-relaxation is at most O(n^{1/4} * log(n), and it is complemented by proving that NDP is APX-hard on grid graphs.

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An eecient algorithm for the n-cube pairwise node disjoint shortest paths problem in the presence of faulty nodes and an O(m 2:5) time algorithm, where m is the input length, to construct a set of such paths.

### A Set-to-Set Disjoint Paths Routing Algorithm in Tori

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It is proved that the paths selected by the proposed algorithm have lengths at most 2( k +1) n and can be obtained with a time complexity of O ( kn 3 n 3 log n ).

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This work uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods, resulting in an efficient randomized $2^{O(\sqrt{\log n} \cdot \log\log n)}$-approximation algorithm for this problem.

### Set-to-Set Disjoint Paths in Tori

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This paper proposes an algorithm that constructs 2n mutually node-disjoint paths from a set S of 2n source nodes to a set D of2n destination nodes in an n-dimensional k-ary torus Tn,k (n¡Ã 1, k ¡Ã 3).

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Robertson and Seymour have proven that the disjoint paths problem can be solved in polynomial time if k is fixed, and give single exponential lower bounds both for the tree-width of planar graphs with vital linkages, and for the size of the grid necessary for finding irrelevant vertices.

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