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A Faster Algorithm for Finding Disjoint Paths in Grids

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Algorithms and Computation (ISAAC 1999)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1741))

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Abstract

Given a set of sources and a set of sinks in the two dimensional grid of size n, the disjoint paths (DP) problem is to connect every source to a distinct sink by a set of edge-disjoint paths. Let v be the total number of sources and sinks. In [3], Chan and Chin showed that without loss of generality we can assume v ≤ n ≤ 4v 2. They also showed how to compress the grid optimally to a dynamic network (structure of the network may change depending on the paths found currently) of size \( O(\sqrt {nv} ) \) , and solve the problem in \( O(\sqrt n v^{3/2} ) \) time using augmenting path method in maximum flow. In this paper, we improve the time complexity of solving the DP problem to O(n 3/4 v 3/4). The factor of improvement is as large as \( \sqrt v \) when n is ⊝(itv), while it is at least \( \sqrt[4]{v} \) for n is ⊝(v 2).

The research is partially supported by a Hong Kong RGC grant 338/065/0022.

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References

  1. Y. Birk and J. B. Lotspiech. On finding non-intersecting straightline connections of grid points to the boundary. J. Algorithms, 13(4):636–656, Dec. 1992.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Bruck and V. P. Roychowdhury. How to play bowling in parallel on the grid. J. Algorithms, 12(3):516–529, Sept. 1991.

    Article  MATH  MathSciNet  Google Scholar 

  3. W.-T. Chan and F. Y. L. Chin. Efficient algorithms for finding maximum number of disjoint paths in grids. J. Algorithms, To appear. (A preliminary version of this paper appeared in SODA’97.).

    Google Scholar 

  4. T. H. Cormen, C. E. Leiserson, and R. L. Rivest. Introduction to Algorithms. MIT Press, Cambridge, MA, 1991.

    Google Scholar 

  5. S. Even and R. E. Tarjan. Network flow and testing graph connectivity. SIAM J. Comput., 4(4):507–518, Dec. 1975.

    Article  MATH  MathSciNet  Google Scholar 

  6. L. R. Ford and D. R. Fulkerson. Maximal flow through a network. Canad. J. Math., 8:399–404, 1956.

    MATH  MathSciNet  Google Scholar 

  7. A. V. Goldberg and R. Kennedy. Global price updates help. SIAM J. Discrete Math., 10:551–572, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. Hershberger and S. Suri. Efficient breakout routing in printed circuit boards. In Algorithms and Data Structures, 5th International Workshop, Lecture Notes in Computer Science, pages 462–471. Springer, 1997.

    Google Scholar 

  9. D. R. Karger and M. S. Levine. Finding maximum flows in undirected graphs seems easier than bipartite matching. In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, 1998.

    Google Scholar 

  10. L. Palios. Connecting the maximum number of nodes in the grid to the boundary with nonintersecting line segments. J. Algorithms, 22(1):57–92, Jan. 1997.

    Article  MATH  MathSciNet  Google Scholar 

  11. V. P. Roychowdhury and J. Bruck. On finding non-intersecting paths in grids and its application in reconfiguring VLSI/WSI arrays. In Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 454–464, 1990.

    Google Scholar 

  12. V. P. Roychowdhury, J. Bruck, and T. Kailath. Efficient algorithms for reconfiguration in VLSI/WSI arrays. IEEE Trans. Comput., 39(4):480–489, Apr. 1990.

    Article  Google Scholar 

  13. T. A. Varvarigou, V. P. Roychowdhury, and T. Kailath. Reconfigurating processor arrays using multiple-track models: The 3-track-1-spare-approach. IEEE Trans. Comput., 42(11):1281–1292, Nov. 1993.

    Article  Google Scholar 

  14. M.-F. Yu and W. W.-M. Dai. Pin assignment and routing on a single-layer pin grid array. In Proceedings of 1st Asia and South Pacific Design Automation Conference, pages 203–208. IEEE, 1995.

    Google Scholar 

  15. M.-F. Yu and W. W.-M. Dai. Single-layer fanout routing and routability analysis for ball grid arrays. In Proceedings of IEEE/ACM International Conference on Computer-aided Design, pages 581–586, 1995.

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Information Systems, The University of Hong Kong, Pokfulam Road, Hong Kong

    Wun-Tat Chan, Francis Y. L. Chin & Hing-Fung Ting

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  1. Wun-Tat Chan
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  2. Francis Y. L. Chin
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  3. Hing-Fung Ting
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© 1999 Springer-Verlag Berlin Heidelberg

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Chan, WT., Chin, F.Y.L., Ting, HF. (1999). A Faster Algorithm for Finding Disjoint Paths in Grids. In: Algorithms and Computation. ISAAC 1999. Lecture Notes in Computer Science, vol 1741. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46632-0_40

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  • DOI: https://doi.org/10.1007/3-540-46632-0_40

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66916-6

  • Online ISBN: 978-3-540-46632-1

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