1. Toward Better Wireload Models in the Presence of Obstacles* Chung-Kuan Cheng, Andrew B. Kahng, Bao Liu and Dirk Stroobandt† UC San Diego CSE Dept. †Ghent University ELIS Dept. e-mail: {kuan,abk,bliu}@cs.ucsd.edu, dstr@elis.rug.ac.be *This work was supported in part by the MARCO Gigascale Silicon Research Center and a grant from Cadence Design Systems, Inc.

2. 2 Presentation Outline  Motivation and Background  Wirelengths and Obstacles  Two-terminal Nets with a Single Obstacle  Two-terminal Nets with Multiple Obstacles  Model Applications  Conclusion

3. 3 Motivation and Background  Impasse of interconnect delay and placement  To break impasse: use wireload models  Wireload models benefit from wirelength estimation techniques  IP blocks in SOC design form routing obstacles  Wirelength estimation cannot be blind to routing obstacles

4. 4 A Priori or Online Wirelength Estimation A Priori WLE Placement Global Routing Detailed Routing OK? done Online WLE Synthesis

5. 5 Presentation Outline  Motivation and Background  Wire Lengths and Obstacles  Two-terminal Nets with a Single Obstacle  Two-terminal Nets with Multiple Obstacles  Model Applications  Conclusion

6. 6 Problem Formulation  Given obstacles and n terminals uniformly distributed in a rectangular routing region that lie outside the obstacles  Find the expected rectilinear Steiner minimal length of the n-terminal net M N

7. 7 Effects of Routing Obstacles on Expected Wirelength  Detours that have to be made around the obstacles  Changes due to redistribution of interconnect terminals M N

8. 8 Definitions of Wirelength Components  Intrinsic wirelength L i is expected wirelength without any obstacle  Point redistribution wirelength L p is expected wirelength with transparent obstacles  Resultant wirelength L r is expected wirelength with opaque obstacles

9. 9 Wirelength Components P1P1 Intrinsic L i P2P2 M N P2’P2’ P1’P1’ Point Redistribution L p W H Resultant L r

10. 10 Summary of Wirelength Components  Redistribution effect equals L p -L i (in the presence of transparent obstacles)  Blockage effect equals L r -L p (in the presence of opaque obstacles) Resultant Lr Redistribution Lp Intrinsic Li Blockage effect Redistribution effect  Blockage effect is ALWAYS POSITIVE  Redistribution effect CAN BE NEGATIVE

11. 11 Presentation Outline  Motivation and Background  Wire Lengths and Obstacles  Two-terminal Nets with a Single Obstacle  Two-terminal Nets with Multiple Obstacles  Model Applications  Conclusion

12. 12 Intrinsic Wirelength of Two-terminal Nets M N  Average expected wirelength between two terminals is one third of the half perimeter of the layout region without obstacles

13. 13 Point Redistribution WL of Two-terminal Nets  Observation 1: The redistribution effect L p -L i (the difference of average expected wirelength with and without transparent obstacles) mainly increases with the obstacle area M N W H where  =f(M,N,a,b) (a,b)

14. 14 Detour Wirelength of Two-terminal Nets N M W H p1p1 p2p2 p2p2 p2p2  Detour WL dependence on position of, e.g., P 2  Linear for P 2 with y coordinate b-H/2 < y P2 < y P1 and 2b-y P1 < y P2 < b+H/2  Constant for all P 2 with y coordinate y P1 < y P2 < 2b-y P1 Detour WL y P2 (a,b)

15. 15 Resultant Wirelength of Two-terminal Nets  Observation 2: The blockage effect L r -L p (the difference of average expected wirelength with transparent and opaque obstacles) mainly increases with the largest obstacle dimension where  =f(M,N,W,H,a) and  =g(M,N,W,H,b)

16. 16 Experimental Setup  Random point generator  Visibility graph  each terminal and obstacle corner is a vertex  each “visible” pair of vertices is connected by an edge  Graph Steiner minimal tree heuristic* 1 1 p1p1 p2p2 p3p3 * L.Kou, G.Markowsky and L.Berman,“A Fast Algorithm for Steiner Trees”, Acta Informatica, 15(2), 1981, pp.141-145

17. 17 Redistribution Effect vs. Obstacle Dimension  Observation 1: The redistribution effect L p -L i mainly increases with the obstacle area

18. 18 Blockage Effect vs. Obstacle Dimension  Observation 2: The blockage effect L r -L p mainly increases with the largest obstacle dimension

19. 19 Redistribution Effect of Ten-terminal Nets  Observation 3: For multi-terminal nets the redistribution effect increases with the number of terminals and with the obstacle area

20. 20 Blockage Effect of Ten-terminal Nets  Observation 3: For multi-terminal nets the blockage effect increases with the number of terminals and with the difference between obstacle dimensions

21. 21 Experiment Setting for Obstacle Displacement 1 1 0.5 0.2 0.5 0.2 0.5 0.2 0.5 0.2 0.5 0.2

22. 22 Redistribution Wirelength vs. Obstacle Displacement  Observation 4: The closer the obstacle is to the routing region boundary the smaller is the redistribution effect

23. 23 Blockage Effect vs. Obstacle Displacement  The closer the obstacle is to the routing region boundary the smaller is the blockage effect  Observation 5: Displacement along the longest obstacle side has little effect on blockage effect

24. 24 Effect of Layout Region Aspect Ratio  Observation 6: The redistribution effect does not depend on the aspect ratios of the region and the obstacle: it dominates when the aspect ratios are similar  The blockage effect is very dependent on the aspect ratios: it dominates when the aspect ratios are different

25. 25  Observation 7: L-shaped region has negative redistribution effect (L p < 0.67) and no blockage effect (L r = L p ) Experimental Setting for L-shaped Routing Region W H 1 1 P1P1 P2P2

26. 26 Effect of L-shaped Routing Region  Observation 8: The more the L-shaped region deviates from a rectangle the less its total wirelength

27. 27 Experimental Setting for C-shaped Routing Region W H 1 1

28. 28 Blockage Effect in a C-shaped Region Comparing with:  The blockage effect doubles for a very oblong obstacle that is on the routing region boundary

29. 29 Blockage Effect in a C-shaped Region  The blockage effect mainly increases with the obstacle dimension that is perpendicular to the routing region boundary that the obstacle is on

30. 30 Redistribution Effect in a C-shaped Region  The redistribution effect does not generally increase with the obstacle area when the obstacle is on the routing region boundary

31. 31 Presentation Outline  Motivation and Background  Wire Lengths and Obstacles  Two-terminal Nets with a Single Obstacle  Two-terminal Nets with Multiple Obstacles  Model Applications  Conclusion

32. 32 Additive Property for Multiple Obstacles  Redistribution effect can be obtained by “polynomial expansion” = x - x = x - x - x + x = x - 2 ( x + x ) + x + 2 x x + xx

33. 33 Additive Property for Multiple Obstacles  Blockage effect for m W i xH i obstacles with non- overlapping x- and y-spans: x+x

34. 34 Experiment Setting for Additive Property 1 1 Region 1 Region 2 Region 2'

35. 35 Additive Property in Region 1  Observation 9: The redistribution effect is additive for obstacles with small areas  Observation 10: The blockage effect is additive if there is no x- or y-span overlap between any obstacle pair

36. 36 Non-Additive Property in Region 2  Observation 9: The redistribution effect is additive for obstacles with small areas  Observation 10: The blockage effect is not additive for obstacles with overlapping x- or y-spans

37. 37 Effect of Obstacle Number  Randomly generating a given number m of obstacles with a prescribed total obstacle area A  Observation 11: The total wirelength increases as the number of obstacles increases while the total obstacle area remains the same

38. 38 Presentation Outline  Motivation and Background  Wire Lengths and Obstacles  Two-terminal Nets with a Single Obstacle  Two-terminal Nets with Multiple Obstacles  Model Applications  Conclusion

39. 39 Analyze Individual Wires  Redistribution effect is an average effect over all wires  Blockage effect is different for each wire with different length  Which wire of what length suffers blockage effect the most?

40. 40 Blockage Distribution  Blockage effect makes a lot of differences for medium-sized wires (~30% wires make detour, up to a 60% increase in wirelength)  Can be combined with different wirelength distribution models

41. 41 Presentation Outline  Motivation and Background  Wire Lengths and Obstacles  Two-terminal Nets with a Single Obstacle  Two-terminal Nets with Multiple Obstacles  Model Applications  Conclusion

42. 42 Conclusion  The first work to consider routing obstacle effect in wirelength estimation  Distinguish two routing obstacle effects  Theoretical expressions for 2-terminal nets and a single obstacle  Lookup table for multi-terminal nets and additive property for multiple obstacles  Help to guide SOC design and improve wireload models

43. 43 Future Directions  Continuous study on multi-obstacle cases for finding equivalent obstacle relationships  Combination with different wirelength distributions which count placement optimization effect  Effects of channel capacity and routing sequence  Wirelength estimation for skew-balanced clock trees

44. 44 Discrete Analysis Approach  Site density function f(l) is the number of wires with length l  generating polynomial V(x)=  f(l)x l  Complete expression for intrinsic, redistribution and resultant wirelenghts

45. 45 Multiple Obstacle Analysis  Two obstacles with disjoint spans  Two obstacles with identical x- or y-spans  Two obstacles with covering x- or y-spans

46. 46 Two obstacles with covering x- or y-spans  Number of medium- length wires decreases as any of the obstacle widths increases.