Abstract
The maze routing problem is to find an optimal path between a given pair of cells on a grid plane. Lee's algorithm and its variants, probably the most widely used maze routing method, fails to work in the 4-geometry of the grid plane. Our algorithm solves this problem by using a suitable data structure for uniform wave propagation in the 4-geometry, 8-geometry, etc. The algorithm guarantees finding an optimal path if it exists and has linear time and space complexities. Next, to solve the obstacle-avoiding rectilinear and 4-geometry Steiner tree problems, a heuristic algorithm is presented. The algorithm utilizes a cost accumulation scheme based on the maze router to determine the Torricelli vertices (points) for improving the quality of multiterminal nets. Our experimental results show that the algorithm works well in practice. Furthermore, using the 4-geometry router, path lengths can be significantly reduced up to 12% compared to those in the rectilinear router.
- Ahuja, R. K., Mehlhorn, K., Orlin, J. B., and Tarjan, R. E. 1990. Faster algorithms for the shortest path problem. J. Assoc. Comput. Mach. 37, 2 (April), 213--223.]] Google ScholarDigital Library
- Barraquand, J. and Latombe, J. C. 1991. Robot motion planning: A distributed representation approach. Int. J. Robotics Res. 10, 6 (Dec.), 628--649.]] Google ScholarDigital Library
- Brimberg, J. 1995. The Fermat-Weber location problem revisited. Math. Programm. B 71, 1, 71--76.]] Google ScholarDigital Library
- Cieslik, D. 1991. The 1-Steiner-minimal-tree problem in Minkowski-spaces, Optimization 22, 2, 291--296.]]Google ScholarCross Ref
- Cieslik, D. 1998. Steiner Minimal Trees, Kluwer Academic Publishers, Dordrecht, The Netherlands.]]Google Scholar
- Dí-az de León S., J. L., and Sossa A., J. H. 1998. Automatic path planning for a mobile robot among obstacles of arbitrary shape. IEEE Trans. Syst. Man. Cybernet. B. 28, 3 (June), 467--472.]] Google Scholar
- Dijkstra, E. W. 1959. A note on two problems in connexion with graphs. Numer. Math. 1, 269--271.]]Google ScholarDigital Library
- Dinic, E. A. 1978. Economical algorithms for finding shortest paths in a network, In Transportation Modeling Systems, Y. S. Popkov and B. L. Shmulyian, Eds. Institute for System Studies, Moscow, Russia, 36--44.]]Google Scholar
- Dreyer, D. R. and Overton, M. L. 1998. Two heuristics for the Euclidean Steiner tree problem. J. Glob. Opt. 13, 1, 95--106.]] Google ScholarCross Ref
- Fawcett, J. and Robinson, P. 2000. Adaptive routing for road traffic. IEEE Comput. Graph. Appl. 20, 3 (May/June), 46--53.]] Google ScholarDigital Library
- Ganley, J. L. and Cohoon, J. P. 1994. Routing a multi-terminal critical net: Steiner tree construction in the presence of obstacles. Proceedings of the IEEE International Symposium on Circuits and Systems, vol. 1. 113--116.]]Google Scholar
- Georgakopolulos, G. and Papadimitrious, C. H. 1987. The 1-Steiner tree problem. J. Algorith. 8, 1, 122--130.]] Google ScholarDigital Library
- Goldberg, A. V. 2001. Simple shortest path algorithm with linear average time. Proceedings of the European Symposium on Algorithms (ESA). Lecture Notes in Computer Science, Vol. 2161. Springer-Verlag, Berlin Germeny, 230--241.]] Google Scholar
- Hart, P. E., Nilsson, N. J., and Raphael, B. 1968. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Syst. Sci. Cybernet. 4, 2, 100--107.]]Google ScholarCross Ref
- Henzinger, M. R., Klein, P., Rao, S., and Subramanian, S. 1997. Faster shortest-path algorithms for planar graphs. J. Comput. Syst. Sci. 55, 1 (Aug.), 3--23.]] Google ScholarDigital Library
- Joel, J. H. 1976. Some variations of Lee's algorithm. IEEE Trans. Comput. C-25, 1 (Jan.), 19--24.]]Google Scholar
- Hwang, F. K. 1979. An O(n log n) algorithm for suboptimal rectilinear Steiner trees. IEEE Trans. Circ. Syst. CAS-26, 75--77.]]Google ScholarCross Ref
- Jan, G. E., Lin, M. B., and Chen, Y. Y. 1997. Computerized shortest path searching for vessels. J. Marine Sci. Tech. 5, 1, 95--99.]]Google Scholar
- Kahng, A. and Robins, G. 1992. A new class of iterative Steiner tree heuristics with good performance. IEEE Trans. Comput.-Aided Des. Integrat. Circ. Syst. 11, 7, 893--902.]]Google ScholarDigital Library
- Kimmel, R., Amir, A., and Bruckstein, A. M. 1995. Finding shortest paths on surfaces using level sets propagation. IEEE Trans. Patt. Anal. Mach. Intell. 17, 6 (June), 635--640.]] Google ScholarDigital Library
- Kuhn, H. W. 1973. A note on Fermt's problem. Math. Programm. 4, 98--107.]]Google ScholarCross Ref
- Lee, C. Y. 1961. An algorithm for path connections and its applications. IRE Trans. Electron. Comput. EC-10, Sept., 346--365.]]Google ScholarCross Ref
- Lee, J. H., Bose, N. K., and Hwang, F. K. 1976. Use of Steiner's problem in suboptimal routing in rectilinear metric. IEEE Trans. Circ. Syst. CAS-23, 470--476.]]Google ScholarCross Ref
- Lin, Y. L., Hsu, Y. C., and Tsai, F. S. 1990. Hybrid routing. IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. 9, 2 (Feb.), 151--157.]]Google ScholarDigital Library
- Ogniewicz, R. L. and Kubler, O. 1995. Voronoi tessellation of points with integer coordinates: Time-efficient implementation and online edge-list generation. Patt. Recogn. 28, 12 (Dec.), 1839--1844.]] Google ScholarDigital Library
- Rubin, F. 1974. The Lee path connection algorithm. IEEE Trans. Comput. C-23, 9 (Sept.), 907--914.]]Google ScholarDigital Library
- Sherwani, N. A. 1999. Algorithms for VLSI Physical Design Automation, 3rd. ed. Kulwer Academic Publishers, Boston, MA, 260--279.]]Google Scholar
- Smith, J. M.-G., Lee, D. T., and Liebman, J. S. 1981. An O(n log n) heuristic for Steiner minimal tree problems on the Euclidean metric. Networks 11, 1, 23--39.]]Google ScholarCross Ref
- Teig, S. L. 2002. The X architecture: Not your father's diagonal wiring. In Proceedings of the 2002 International Workshop on System-Level Interconnect Prediction (San Diego, CA, Apr. 6--7), ACM Press, New York, NY, 33--37.]] Google Scholar
- Thorup, M. 1999. Undirected single-source shortest paths with positive integer weights in linear time. J. Assoc. Comput. Mach. 46, 3, 362 -- 394.]] Google ScholarDigital Library
- Winter, P. 1993. Euclidean Steiner minimal trees with obstacles and Steiner visibility graphs. Discrete App. Math. 47, 2, 187--206.]] Google ScholarDigital Library
- Xing, Z. and Kao, R. 2002. Shortest path search using tiles and piecewise linear cost propagation. IEEE Trans. Comput.-Aided Des. Integr. Circ. Syst. 21, 2 (Feb.), 145--158.]] Google Scholar
Index Terms
- A 4-geometry maze router and its application on multiterminal nets
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