在超大型積體電路設計中，一個直角史坦那樹常被用來作為網路內訊號的連接。在現在的設計內通常都含有許多直角的障礙物，像是動力網路系統、區塊型模組、矽智財模組及預先繞線好的網路，因此，建造一個排除礙障物的直角史坦那樹是一個重要的繞線問題。此外，現今的奈米製程設計提供了多層金屬層作為繞線環境;同時，金屬層上的特定方向條件也限制了每一層的繞線方向。因此，有人提出了「考慮障礙物的特定方向史坦那樹」的議題來應對這些實際的問題。在這篇碩士論文中，我們針對此議題提出了一個新穎、穩健及高品質的演算法。我們的演算法的概念是藉由一個直角史坦那樹的指引，企圖找到一些合適的史坦那頂點。我們也提供了多種最短路徑搜尋演算法來連接這些史坦那頂點和與其相鄰之頂點。此外，我們的方法也可以延伸處理「多層障礙物排除之直角史坦那樹」的問題;這是個排除特定方向條件的問題。實驗數據顯示，我們的演算法在處理「考慮障礙物的特定方向史坦那樹」和「多層障礙物排除之直角史坦那樹」的問題上，都可以得到很好的結果。

In very large scale integration (VLSI) design, a rectilinear Steiner tree (RST) is usually
used to guide router to connect signals for a net. However, In current technology, there
are more and more rectilinear obstacles, such as power networks, macro cells, IP blocks
and pre-routed nets. Therefore, a concept of obstacle-avoiding rectilinear Steiner minimal
tree (OARSMT) has been noticed that can be applied on the routing problem. Besides, in
modern nano-technology designs, there are multiple metal layers can be used for routing,
at the same time, the preferred directions constraint limits the orientations of routing
in each layer for the signal integrity consideration. As a result, the obstacle-avoiding
preferred direction Steiner tree (OAPDST) problem has been proposed to cop with these
practical problems. In this thesis, we present a novel, robust, and high quality OAPDST
construction algorithm for this problem. Our algorithm is based on nding suitable Steiner
points on a graph, and guided by an RST. Several shortest path nding algorithms were
also proposed to connect every Steiner points and their neighbors with minimal wirelength.
Besides, our approach can be easily extended to deal with multi-layer obstacle-avoiding
rectilinear Steiner minimal tree (ML-OARSMT) problem. The experimental results show
our algorithm performs well in total cost, both on OAPDST and ML-OARSMT problems.

[1] M. R. Garey and D. S. Johnson, \The rectilinear steiner tree problem is np-complete,"
SIAM Journal on Applied Mathematics, vol. 32, no. 4, pp. 826{834, 1997.
[2] J. Grith, G. Robins, J. S. Salowe, and T. Zhang, \Closing the gap: Near-optimal
steiner trees in polynomial time," IEEE Trans. Computer-Aided Design, vol. 13, pp.
1351{1365, 1994.
[3] S. Das, S. V. Gosavi, W. H. Hsu, and S. A. Vaze, \An ant colony approach for the
steiner tree problem," in Proc. the Genetic and Evolutionary Computation Conf.,
2002, p. 135.
[4] A. Kahng, I. Mandoiu, and A. Zelikovsky, \Highly scalable algorithms for rectilinear
and octilinear steiner trees," in Proc. Asia and South Pacic Design Automation
Conf., 2003, pp. 827{833.
[5] D. Warme, P. Winter, and M. Zachariasen, "Geosteiner 3.1. package". [Online].
Available: http://www.diku.dk/geosteiner/
[6] C. Chu and Y.-C. Wong, \Flute: Fast lookup table based rectilinear steiner minimal
tree algorithm for vlsi design," IEEE Trans. Computer Aided Design Integr. Circuits
Syst., vol. 27, no. 1, pp. 70{83, Jan 2008.
[7] L. Li, Z. Qian, and E. F. Y. Young, \Generation of optimal obstacle-avoiding recti-
linear steiner minimum tree," in Proc. International Conference on Computer-Aided
Design, 2009, pp. 21{25.
[8] Y. Hu, Z. Feng, T. Jing, X. Hong, Y. Yang, G. Yu, X. Hu, and G. Yan, \Forst: A
3-step heuristic for obstacle-avoiding rectilinear steiner minimum tree construction,"
Journal of Information and Computational Science, pp. 107{116, 2004.
[9] Z. Shen, C. Chu, and Y. Li, \Ecient rectilinear steiner tree construction with rec-
tilinear blockages," in Proc. International Conference on Computer-Aided Design,
2005, pp. 38{44.
[10] Y. Hu, T. Jing, X. Hong, Z. Feng, X. Hu, and G. Yan, \An-oarsman: Obstacle-
avoiding routing tree construction with good length performance," in Proc. Asia and
South Pacic Design Automation Conf., 2005, pp. 7{12.
[11] P. Wu, J. Gao, and T. Wang, \A fast and stable algorithm for obstacle-avoiding
rectilinear steiner minimal tree construction," in Proc. Asia and South Pacic Design
Automation Conf., 2007, pp. 262{267.
[12] J. Long, H. Zhou, and S. O. Memik, \An o(n log n) edge-based algorithm for obstacle-
avoiding rectilinear steiner tree construction," in Proc. International Symposium on
Physical Design, 2008, pp. 126{133.
[13] C.-H. Liu, S.-Y. Yuan, S.-Y. Kuo, and Y.-H. Chou, \An o(n log n) path-based
obstacle-avoiding algorithm for rectilinear steiner tree construction," in Proc. De-
sign Automation Conf., 2009, pp. 314{319.
[14] G. Ajwani, C. Chu, and W.-K. Mak, \Foars: Flute based obstacle-avoiding rectilinear
steiner tree construction," in Proc. International Symposium on Physical Design,
2010, pp. 185{192.
[15] M. C. Yildiz and P. H. Madden, \Preferred direction steiner trees," IEEE Trans.
Computer-Aided Design, vol. 21, no. 11, pp. 1368{1372, 2002.
[16] C.-W. Lin, S.-L. Huang, K.-C. Hsu, M.-X. Li, and Y.-W. Chang, \Ecient multi-
layer obstacle-avoiding rectilinear steiner tree construction," in Proc. International
Conference on Computer-Aided Design, 2007, pp. 380{385.
[17] Y.-S. Lin, \High-quality multi-layer obstacle-avoiding rectilinear steiner tree con-
struction," Master Thesis, National Chiao Tung University, Hsinchu, Taiwan, 2011.
[18] C.-H. Liu, Y.-H. Chou, S.-Y. Yuan, and S.-Y. Kuo, \Ecient multi-layer routing
based on obstacle-avoiding preferred direction steiner tree," in Proc. International
Symposium on Physical Design, 2008, pp. 118{125.
[19] R. Karp, \On the computational complexity of combinatorial problems," Networks,
vol. 5, no. 1, pp. 45{68, 1975.
[20] I. H.-R. Jiang and Y.-T. Yu, \Congurable rectilinear steiner tree construction for
soc and nano technologies," in Proc. International Conference on Computer Design,
2008, pp. 34{39.
[21] J.-R. Chuang and J.-M. Lin, \Ecient multi-layer obstacle-avoiding preferred direc-
tion rectilinear steiner tree construction," in Proc. Asia and South Pacic Design
Automation Conf., 2011, pp. 527{532.
[22] Chris Chu , "FLUTE Library". [Online]. Available:
http://home.eng.iastate.edu/ cnchu/
ute.html
[23] Yagsbpl library. [Online]. Available: https://
ing.seas.upenn.edu/ subhrabh/cgi-
bin/wiki/index.php?n=Projects.IT
[24] Leda package. [Online]. Available: http://www.algorithmicsolutions.com