Multilayer Obstacle-Avoiding Rectilinear Steiner Tree Construction Based on Spanning Graphs

1. Multilayer Obstacle-Avoiding Rectilinear Steiner Tree Construction Based on Spanning Graphs Chung-Wei Lin, Shih-Lun Huang, Kai-Chi Hsu, Meng-Xiang Lee, and Yao-Wen Chang, Member, IEEE

2. OUTLINE • Introduction • Problem Formulation • Algorithm • Experimental Results • Conclusion

3. Introduction

4. Introduction (Con’t) • From the aforementioned table, direct extensions of these existing methods to ML-OARSMT problem may have limited solution quality or even generate some infeasible solutions.

5. Introduction (Con’t) • So, this paper formulate the ML-OARSMT problem with rectilinear obstacles and then develop an effective and efficient algorithm to deal with the problem. • First work in this problem • More sophisticated method with a more global view to guarantee optimization • Prune the solution space to speed up the processing

6. Problem Formulation • Definition 1 • An obstacle is a rectangle on a layer. No two obstacles overlap with each other, but two obstacles could be point touched at the corner or line touched on the boundary.

7. Problem Formulation (Con’t) • Definition 2 • A pin vertex is a vertex on an arbitrary layer. A pin vertex must not locate inside any obstacle, but it could be at the corner or the boundary of an obstacle.

8. Problem Formulation (Con’t) • Definition 3 • A via on layer z is an edge between (x , y , z) and (x , y , z+1). (x , y , z) and (x , y , z+1) must not locate inside any obstacle, but could be at the corner or on the boundary of an obstacle.

9. Problem Formulation (Con’t) • Moreover, no edge of the ML-OARSMT can intersect with any obstacle, but they could be point touched at the corner or line touched on the boundary of an obstacle.

10. Problem Formulation (Con’t) • Problem: ML-OARSMT • Given constants Cv and Nl, a set P of pins, and a set O of obstacles, construct a multilayer rectilinear Steiner tree to connect the pins in P such that no tree or via intersects an obstacle in O and the total cost of the tree is minimized.

11. Algorithm

12. ML-OASG • Definition 4 • An ML-OASG is an undirected graph connecting all vertices in P U C, and no edge intersects with an obstacle in O.

13. ML-OASG • SL-OASG Connection Rule 1 • The SL-OASG is constructed on all pin and corner vertices. • SL-OASG Connection Rule 2 • Two vertices are connected if 1) there is no other vertex inside or on the boundary of the bounding box of the two vertices and 2) there is no obstacle inside the bounding box of the two vertices

14. ML-OASG • Infeasible solution (extension from the SL-OASG construction)

15. ML-OASG • If an ML-OASG is constructed only on pin and corner vertices

16. ML-OASG • There are two examples (b) directly deals with layers one by one (c) considering multiple layers at the same time.

17. ML-OASG • Theorem 1 • The SL-OASG Connection Rule 2 may result in infeasible solutions for the ML-OARSMT problem • Theorem 2 • SL-OASG Connection Rule 1 cannot guarantee a rectilinear shortest path between two vertices

18. ML-OASG • Vertex Projection Between Layers

19. ML-OASG

20. ML-OASG • Vertex Projection Within a Layer

21. ML-OASG

22. ML-OASG • There is a tradeoff between the completeness of the ML-OASG and the computational efficiency

23. ML-OASG

24. ML-OASG

25. ML-OAST • Definition 7 • An ML-OAST is an undirected tree connecting all pin vertices without intersecting with any obstacle. • Cost

26. ML-OARST • Definition 8 • An ML-OARST is an undirected graph connecting all pin vertices by rectilinear edges within layers and vias between layers without intersecting with any obstacle.

27. ML-OARSMT • For a redundant vertex, we merge the two edges connecting to it. • Moreover, we also remove overlapping edges and mark Steiner vertices.

28. ML-OARSMT (Con’t) • For example

29. Algorithm

30. Experimental Results

31. Experimental Results (Con’t)

32. Experimental Results (Con’t)

33. Conclusion • Connection-Graph-Based approach is better than Construction-by-Correction approach • Global view experience and it’s necessary • Much learning