Abstract
The novel octilinear routing paradigm (X-architecture) in VLSI design requires new approaches for the construction of Steiner trees. In this paper, we consider two versions of the shortest octilinear Steiner tree problem for a given point set K of terminals in the plane: (1) a version in the presence of hard octilinear obstacles, and (2) a version with rectangular soft obstacles.
The interior of hard obstacles has to be avoided completely by the Steiner tree. In contrast, the Steiner tree is allowed to run over soft obstacles. But if the Steiner tree intersects some soft obstacle, then no connected component of the induced subtree may be longer than a given fixed length L. This kind of length restriction is motivated by its application in VLSI design where a large Steiner tree requires the insertion of buffers (or inverters) which must not be placed on top of obstacles.
For both problem types, we provide reductions to the Steiner tree problem in graphs of polynomial size with the following approximation guarantees. Our main results are (1) a 2–approximation of the octilinear Steiner tree problem in the presence of hard rectilinear or octilinear obstacles which can be computed in O(n log2 n) time, where n denotes the number of obstacle vertices plus the number of terminals, (2) a (2+ ε)–approximation of the octilinear Steiner tree problem in the presence of soft rectangular obstacles which runs in O(n 3) time, and (3) a (1.55 + ε)–approximation of the octilinear Steiner tree problem in the presence of soft rectangular obstacles.
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References
Teig, S.L.: The X architecture: not your father’s diagonal wiring. In: SLIP 2002: Proceedings of the 2002 International Workshop on System-Level Interconnect Prediction, pp. 33–37. ACM Press, New York (2002)
Chen, H., Cheng, C.K., Kahng, A.B., Mǎndoiu, I., Wang, Q.: Estimation of wirelength reduction for λ-geometry vs. Manhattan placement and routing. In: Proceedings of SLIP 2003, pp. 71–76. ACM Press, New York (2003)
Paluszewski, M., Winter, P., Zachariasen, M.: A new paradigm for general architecture routing. In: Proceedings of the 14th ACM Great Lakes Symposium on VLSI (GLSVLSI), pp. 202–207 (2004)
Garey, M.R., Johnson, D.S.: The rectilinear Steiner tree problem is NP-complete. SIAM Journal on Applied Mathematics 32, 826–834 (1977)
Garey, M.R., Graham, R.L., Johnson, D.S.: The complexity of computing Steiner minimal trees. SIAM Journal on Applied Mathematics 32, 835–859 (1977)
Müller-Hannemann, M., Schulze, A.: Hardness and approximation of octilinear Steiner trees. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 256–265. Springer, Heidelberg (2005)
Nielsen, B.K., Winter, P., Zachariasen, M.: An exact algorithm for the uniformly-oriented Steiner tree problem. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 760–772. Springer, Heidelberg (2002)
Coulston, C.: Constructing exact octagonal Steiner minimal trees. In: ACM Great Lakes Symposium on VLSI, pp. 1–6 (2003)
Zachariasen, M., Winter, P.: Obstacle-avoiding Euclidean Steiner trees in the plane: An exact approach. In: Goodrich, M.T., McGeoch, C.C. (eds.) ALENEX 1999. LNCS, vol. 1619, pp. 282–295. Springer, Heidelberg (1999)
Althaus, E., Polzin, T., Daneshmand, S.V.: Improving linear programming approaches for the Steiner tree problem. Research Report MPI-I-2003-1-004, Max-Planck-Institut für Informatik, Saarbrücken, Germany (2003)
Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. In: Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 770–779 (2000)
Du, D.Z., Hwang, F.K.: Reducing the Steiner problem in a normed space. SIAM Journal on Computing 21, 1001–1007 (1992)
Lee, D.T., Shen, C.F.: The Steiner minimal tree problem in the λ-geometry plane. In: Nagamochi, H., Suri, S., Igarashi, Y., Miyano, S., Asano, T. (eds.) ISAAC 1996. LNCS, vol. 1178, pp. 247–255. Springer, Heidelberg (1996)
Lin, G.H., Xue, G.: Reducing the Steiner problem in four uniform orientations. Networks 35, 287–301 (2000)
Arora, S.: Polynomial time approximation schemes for the Euclidean traveling salesman and other geometric problems. Journal of the ACM 45, 753–782 (1998)
Mitchell, J.S.B.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing 28, 1298–1309 (1999)
Kahng, A.B., Mǎndoiu, I.I., Zelikovsky, A.Z.: Highly scalable algorithms for rectilinear and octilinear Steiner trees. In: Proceedings 2003 Asia and South Pacific Design Automation Conference (ASP-DAC), pp. 827–833 (2003)
Zhu, Q., Zhou, H., Jing, T., Hong, X., Yang, Y.: Efficient octilinear Steiner tree construction based on spanning graphs. In: Proceedings 2004 Asia and South Pacific Design Automation Conference (ASP-DAC), pp. 687–690 (2004)
Müller-Hannemann, M., Peyer, S.: Approximation of rectilinear Steiner trees with length restrictions on obstacles. In: Dehne, F., Sack, J.-R., Smid, M. (eds.) WADS 2003. LNCS, vol. 2748, pp. 207–218. Springer, Heidelberg (2003)
Mehlhorn, K.: A faster approximation algorithm for the Steiner problem in graphs. Information Processing Letters 27, 125–128 (1988)
Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 633–701. Elsevier, Amsterdam (2000)
Lee, D.T., Yang, C.D., Wong, C.K.: Rectilinear paths among rectilinear obstacles. Discrete Applied Mathematics 70, 185–215 (1996)
Wu, Y.F., Widmayer, P., Schlag, M.D.F., Wong, C.K.: Rectilinear shortest paths and minimum spanning trees in the presence of rectilinear obstacles. IEEE Transactions on Computing, 321–331 (1987)
Clarkson, K.L., Kapoor, S., Vaidya, P.M.: Rectilinear shortest paths through polygonal obstacles in O(n (logn)2) time. In: Proceedings of the 3rd Annual ACM Symposium on Computational Geometry, pp. 251–257 (1987)
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Müller-Hannemann, M., Schulze, A. (2006). Approximation of Octilinear Steiner Trees Constrained by Hard and Soft Obstacles. In: Arge, L., Freivalds, R. (eds) Algorithm Theory – SWAT 2006. SWAT 2006. Lecture Notes in Computer Science, vol 4059. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11785293_24
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DOI: https://doi.org/10.1007/11785293_24
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