1. Chih-Hung Lin, Kai-Cheng Wei VLSI CAD 2008
An Efficient Algorithm for Obstacle-Avoiding Rectilinear Steiner Tree Construction
Chih-Hung Lin, Kai-Cheng Wei
VLSI CAD 2008

2. Outline Introduction Problem Formulation Overview Of Algorithm
Detail Of Algorithm
Experimental Results
Conclusions

3. Introduction
Given n points on a plane, a Rectilinear Steiner Minimal Tree (RSMT) connects these points through some extra points called steiner points to achieve a tree with minimal total wire length.
RSMT is a NP-complete problem.

4. Introduction
In fact, today’s design often contains many rectilinear routing blockages, e.g., power nets, macro cells, IP blocks, and pre-routed nets.
The obstacle-avoiding rectilinear Steiner minimal tree (OARSMT) problem is of particular importance for modern VLSI designs.

5. Problem Formulation
Given a set of m pin vertices, P = {p1, p2, …, pm}, and a set of k rectangular obstacles, O={o1, o2, …, ok}.
An OARSMT connects these pin vertices via some Steiner points.

6. Problem Formulation
The main goal is to find an OARSMT with a desired total wirelength in as short as runtime.
All the path are prohibited to run across obstacles.

7. Overview Of Algorithm There are five steps:
Minimum spanning tree (MST) construction
Set of spanning trees (SST) construction
Obstacle-avoiding spanning graphs (OASGs) construction
(A)
(B)
(C)

8. Overview Of Algorithm There are five steps:
Obstacle-avoiding spanning tree (OAST) construction
Obstacle-avoiding rectilinear Steiner minimum tree (OARSMT) construction
(D)
(E)

9. Detail Of Algorithm MST construction (Minimal Spanning Tree)
spanning graph construction
setting bounding boxes and detour regions
weights assignment
MST construction

10. Detail Of Algorithm spanning graph construction
Divide the plane into four search regions of each pin vertex.
Then we search each region and find the closest point in each region and connect it to p.

11. Detail Of Algorithm setting bounding boxes and detour regions
A bounding box is the region which two end points cover, and its boundaries are equally composed with the upper and lower L-shaped paths.
A DTR is the region which any two points can find the shortest path to connect these two end points in this area.

12. Detail Of Algorithm There are three case of setting DTR. A Case1:

13. Detail Of Algorithm Weights assignment
In case 1 and case 2, the shortest distance of two end points is equal to the half-perimeter of DTR.
In case 3, besides the half-perimeter of DTR, an extra length of detour path is summed to the weight.
Case2:
Detour Region 13 6 15 15 6 8 11 9 8
Bounding Box

14. Detail Of Algorithm MST construction
Kruskal’s method is then applied to construct the MST T, which T=(VT , ET).
Among the edges added to ET , the ones intersecting obstacles are recorded simultaneously.

15. Detail Of Algorithm Set of spanning trees (SST) constructionB
Set of spanning trees (SST) construction
Each edge in E is sequentially checked whether it intersects any obstacle or not.
Obstacle is recoded simultaneously when set the DTRs, and it only cost O(m) time to partition the MST into SST.
m is the number of pin vertices

16. Detail Of Algorithm C
OASGs construction (Obstacle-Avoiding Spanning Graph)
An OASG is constructed among the pins and corner vertices in each DTR, and all the edges of OASG are not intersecting obstacles.
The detour paths are iteratively found by breadth-first- search method.
pins
corner vertices

17. Detail Of Algorithm D
OAST construction (Obstacle-Avoiding Spanning Tree)
The partitioned SST are then merged into a complete tree OAST, T = ( V , E ), by shortest detour paths in each DTR.
And there is no edge in E crossing obstacles.
SST
OASG
OAST

18. E
Detail Of Algorithm
OARSMT construction (Obstacle-Avoiding Rectilinear Spanning Minimal Tree)
Then transform the slant edges in E into rectilinear ones by either the upper or the lower L-shaped paths.
Otherwise, this paper have maintained a share-edge- relation table, called “SERT”, to record the adjacent relations of all edge in E.
The composition having the largest share length is selected first and transformed to rectilinear.

19. E
Detail Of Algorithm
When there is no more composition of share length being found, the other edges in E are transformed by the lower L- shaped paths.
Finally, a U-shaped path pushing method is applied to further reduce the wirelength.

20. Experimental Results
This paper collect the 22 test cases used in [11] and randomly generate another 6 larger cases.
Especially in the large cases DT01~DT06, we have 57.94% and 64.9% average time improvement in [10] and [11] respectively.
The second column are the number of pin vertices and obstacles.

21. Experimental Results

22. Experimental Results
Time Improvement : => the number of edges in each OASGs
O(c)
O(n)
c is the number of pin and corner vertices in each DTR
n = P∪{corner vertices in O}
P is the pins, O is the obstacles

23. Conclusions
This paper have proposed an efficient algorithm to construct an OARSMT.
And their algorithm can efficiently construct the OARSMT without degrading the performance of wirelength, especial for the large cases.