1. Maze Routing with Buffer Insertion and Wire sizing Minghorng Lai, D.F. Wong DAC 2000

2. Elmore delay model

3. R wire =1.2 C wire =2.4F

4. Elmore delay model R wire =1.2 C wire =2.4F R buffer =0.2 C buffer =1F

5. Problem s t

6. Solution Previously Proposed Methods  Dynamic programming Waste time and space With wire sizing, the sizes of the sub-solution sets for dynamic programming increase significantly. Proposed method  Shortest Path Formulation Time :O(|V| 2 log(|V|)) Space:O(|B| 2 |V| 2)

7. s tu v x Shortest Path Formulation Graph G={V,E} SP Graph BG ={V BG,E BG } s b-1 t b-1

8. Construct SP Graph Step 1.Compute shortest distance Step 2.Create new vertices and edges Step 3.Assign every edges’ weight Shortest Path Formulation Find the shortest path from source to sink Wire sizing

9. Shortest Path Formulation Step 1.Compute shortest distance Step 2.Create new vertices and edges Step 3.Assign every edges’ weight Construct SP Graph

10. Shortest Path Formulation Step 2.Create new vertices & edges v b0 v b1 u b0 u b1 vu Buffer Library B={b0,b1}

11. Shortest Path Formulation Step 2.Create new vertices & edges Graph G={V,E} s t v u bo b1 u bo b1 t bo b1 v x bo b1 x s bo b1

12. Shortest Path Formulation Step 2.Create new vertices & edges Add pseudo node s b-1 & t bi-1 s b-1 t b-1 SP Graph

13. Shortest Path Formulation Step 1.Compute shortest distance Step 2.Create new vertices and edges Step 3.Assign every edges’ weight Step 4.Find the shortest path from source to sink

14. Shortest Path Formulation Step 3.Assign every edges’ weight C. Chu and D. F.Wong, “ A New Approach to Simultaneous Buffer Insertion and Wire Sizing," IEEE Trans. on CAD, 1997 use (bi, bj, d(u, v)) as index to check look-up table d(u, v) l1l1 l2l2 l3l3 lnln bibj h1h1 h2h2 h3h3 hnhn

15. Shortest Path Formulation Step 4.Find the shortest path from source to sink Step 1.Compute shortest distance Step 2.Create new vertices and edges Step 3.Assign every edges’ weight

16. Time Complexity  Shortest path O(|V GV |log|V GV |)  at most |B||V| new vertices are created in the BP-Graph Space Complexity  O(|V BG | 2 )=>O(|B| 2 |V| 2 ). Shortest Path Formulation

17. Experimental Results DP-Routing SP-Routing Memory(Mb) Time(s) Name SRL1 SRL2 SRL3 SRL4 SRL5 SRL6 SRL7 SRL8 SRL9 SRL10 2.88 2.71 2. 3.09 3.08 2.69 2.82 2.88 2.85 3.48 741 919 827 1044 1306 961 969 767 868 1243 0.524 0.318 0.355 0.740 0.672 0.572 0.767 0.384 0.479 0.869 35.3 18.2 19.5 51.0 60.2 38.1 46.1 25.5 22.6 71.0

18. DP-Routing SP-Routing Memory(Mb) Time(s) Name Experimental Results TRL1 TRL2 TRL3 TRL4 TRL5 TRL6 TRL7 TRL8 TRL9 TRL10 1.70 1.66 1.61 1.88 2.01 1.72 1.77 1.68 1.72 1.66 292 387 322 432 522 406 420 337 393 374 0.190 0.126 0.151 0.327 0.355 0.250 0.384 0.110 0.226 0.197 11.3 5.1 6.1 15.2 18.2 13.4 17.2 5.6 9.6 7.3

19. Conclusion The lookup-table construction only needs to be done once and can be reused in multi-net maze routing. congestion avoidance