1. Efficient Multi-Layer Obstacle- Avoiding Rectilinear Steiner Tree Construction Chung-Wei Lin, Shih-Lun Huang, Kai-Chi Hsu,Meng-Xiang Li, Yao-Wen Chang ICCAD ’ 07

2. Outline Introduction Introduction Problem Formulation Problem Formulation Algorithm Algorithm SL-OARSMT SL-OARSMT ML-OARSMT ML-OARSMT Experimental Result Experimental Result Conclusion Conclusion

3. Introduction Multi-layer obstacle-avoiding rectilinear Steiner minimal tree (ML-OARSMT) connects pins which locate on routing layers by rectilinear edges, and avoids running through any obstacle. Multi-layer obstacle-avoiding rectilinear Steiner minimal tree (ML-OARSMT) connects pins which locate on routing layers by rectilinear edges, and avoids running through any obstacle. ML-OARSMT can facilitate the cell placement. ML-OARSMT can facilitate the cell placement. Router should connect all pins, no matter on which layer and consider routing obstacle Router should connect all pins, no matter on which layer and consider routing obstacle

4. Introduction This is first paper for ML-OARSMT problem. This is first paper for ML-OARSMT problem. Previous work only handle single-layer, can be classified into two categories Previous work only handle single-layer, can be classified into two categories construct-and-correction approach construct-and-correction approach Construct Steiner tree not consider obstacle then replace edges overlapping obstacle. Construct Steiner tree not consider obstacle then replace edges overlapping obstacle. connect graph based approach connect graph based approach Construct connect graph using pins and obstacle corners to find minimum spanning tree. Construct connect graph using pins and obstacle corners to find minimum spanning tree.

5. Problem Formulation Legal and Illegal Legal and Illegal P={p 1,p 2, …,p m }: set of pin-vertices for m-pin net P={p 1,p 2, …,p m }: set of pin-vertices for m-pin net O={o 1,o 2, …,o k }: set of k obstacle O={o 1,o 2, …,o k }: set of k obstacle n: size of P ∪ {corners in O}, n ≤ m+4k n: size of P ∪ {corners in O}, n ≤ m+4k

6. Problem Formulation C v : via cost ; N l : number of layers C v : via cost ; N l : number of layers Cost (v i, v j )=|x i -x j |+|y i -y j |+|z i -z j |*C v Cost (v i, v j )=|x i -x j |+|y i -y j |+|z i -z j |*C v Give constants Cv and Nl, a set P of pins, a set O of obstacles, construct amulti-layer rectilinear Steiner tree to connect the pins in P, such that no tree edge or via intersects an obstacle in O and total cost of the tree is minimized. Give constants Cv and Nl, a set P of pins, a set O of obstacles, construct amulti-layer rectilinear Steiner tree to connect the pins in P, such that no tree edge or via intersects an obstacle in O and total cost of the tree is minimized.

7. Algorithm This paper base on “ Efficient Obstacle – Avoiding Rectilinear Steiner Tree Construction ” which is a single layer Steiner tree construct method announce in ISPD ’ 07 by same lab. This paper base on “ Efficient Obstacle – Avoiding Rectilinear Steiner Tree Construction ” which is a single layer Steiner tree construct method announce in ISPD ’ 07 by same lab. This paper use connect graph approach method. This paper use connect graph approach method.

8. SL-OARSMT

9. Algorithm 1. Construct multi-layer obstacle-avoiding spanning graph. 1. Construct multi-layer obstacle-avoiding spanning graph. 2. Prim ’ s algorithm construct multi-layer obstacle-avoiding spanning tree. 2. Prim ’ s algorithm construct multi-layer obstacle-avoiding spanning tree. 3. Rectilinear to construct multi-layer obstacle- avoiding rectilinear spanning tree. 3. Rectilinear to construct multi-layer obstacle- avoiding rectilinear spanning tree. 4. Reduce to multi-layer obstacle-avoiding rectilinear Steiner minimal tree. 4. Reduce to multi-layer obstacle-avoiding rectilinear Steiner minimal tree.

10. Algorithm

11. ML-OASG It ’ s not feasible to direct extend the single-layer obstacle-avoiding spanning graph. It ’ s not feasible to direct extend the single-layer obstacle-avoiding spanning graph. SL-OASG Connection rule: SL-OASG Connection rule: 1. The SL-OASG is constructed on all pin-vertices and corner-vertices. 1. The SL-OASG is constructed on all pin-vertices and corner-vertices. 2. Two vertices connected if there is no other vertex inside or on the boundary of the bounding box or obstacle inside the bounding box. 2. Two vertices connected if there is no other vertex inside or on the boundary of the bounding box or obstacle inside the bounding box. So using projection to obtain more feasible solution. So using projection to obtain more feasible solution.

12. Trivial Extend Error

13. Vertex Projection between layers Project point on other layer if is inside obstacle then insert some vertices to extend solution space. Project point on other layer if is inside obstacle then insert some vertices to extend solution space.

14. Vertex Projection within a Layer Other method to extend solution space, but this projection not effective to improve solution when number of vertices is large. Other method to extend solution space, but this projection not effective to improve solution when number of vertices is large.

15. ML-OASG Construct Algorithm T n is a user define number to trade of runtime and solution. T n is a user define number to trade of runtime and solution.

16. OARST Construction It rectilinear method define three case: It rectilinear method define three case:

17. Experimental Result Tn=10, so this method not use projection within layer. Tn=10, so this method not use projection within layer.

18. Experimental Result Compare with construction-by-correction approach Compare with construction-by-correction approach

19. Experimental Result Compare using projection within layer or not Compare using projection within layer or not

20. Conclusion This is a contest problem, a good project method extend can get good result. This is a contest problem, a good project method extend can get good result. Faster then my previous method, might using some good method to improve. Faster then my previous method, might using some good method to improve.