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On Rent’s Rule for Rectangular Regions

On Rent’s Rule for Rectangular Regions. J . Dambre, P. Verplaetse, D. Stroobandt and J. Van Campenhout, Ghent University, Electronics and Information Systems Department. Presentation o v erview. Rent’s Rule and it’s applications Does Rent’s rule apply for rectangular regions?

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On Rent’s Rule for Rectangular Regions

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  1. On Rent’s Rule for Rectangular Regions J. Dambre, P. Verplaetse, D. Stroobandt and J. Van Campenhout, Ghent University, Electronics and Information Systems Department

  2. Presentation overview • Rent’s Rule and it’s applications • Does Rent’s rule apply for rectangular regions? • Model derivation • Results for rectangular regions • Future model extensions? • Conclusions

  3. Region III Region II tG t Rent’s Rule p: Rent exponent t: Rent coefficient • Empirical formula • Relates the average number of terminals T coming out of a subcircuit to the average number of gates G in that subcircuit

  4. Intrinsic vs.layout Rent properties Intrinsic Rent parameters (ti, pi) Sub-circuits are obtained by hierarchically partitioning the circuit according to a pin-minimization criterion Layout Rent parameters (tl, pl) Sub-circuits correspond to square or diamond-shaped regions from the circuit layout

  5. ? ? ? ? Can we use the same Rent parameters for all aspect ratios? Is Rent’s rule at all applicable? Dependence on region shape? Rectangular regions appear when modelling anisotropic architectures, routing congestion or pin-limitation

  6. Dependence on region shape?square grid circuit example

  7. Dependence on region shape?square grid circuit example

  8. Dependence on region shape?square grid circuit example

  9. Dependence on region shape?square grid circuit example

  10. Dependence on region shape?square grid circuit example

  11. Dependence on region shape?square grid circuit example

  12. Dependence on region shape?square grid circuit example

  13. Dependence on region shape?square grid circuit example

  14. Dependence on region shape?square grid circuit example

  15. Dependence on region shape?square grid circuit example

  16. New model Theoretical model for the average number of terminals T, required to connect a sub-region of a circuit layout with the rest of the circuit, as a function of the sub-region’s shape Circuit assumptions : • homogeneous circuits • disregarding disruptions from the circuit boundary • two-pin nets only

  17. Model principle For circuits with only two-pin connections:  Count internal connections

  18. Model principle For circuits with only two-pin connections:  Count internal connections

  19. Model principle For circuits with only two-pin connections:  Count internal connections

  20. Model principle For circuits with only two-pin connections:  Count internal connections

  21. Model principle For circuits with only two-pin connections:  Count internal connections

  22. Model principle For circuits with only two-pin connections:  Count internal connections

  23. Model principle For circuits with only two-pin connections:  Count internal connections

  24. Model principle For circuits with only two-pin connections:  Count internal connections

  25. Model principle For circuits with only two-pin connections:  Count internal connections

  26. Model principle For circuits with only two-pin connections:  Count internal connections

  27. Model derivation For circuits with only two-pin connections:  Count internal connections For each internal gate { For each pin { For each possible wirelength { } } }

  28. = Average number terminals per gate = Normalized wirelength distribution = Site density function for region under consideration Model derivation Total number of internal connections:

  29. Model derivation

  30.  From circuit  With enumeration techniques or by counting  Simplified theoretical model based on intrinsic Rent exponent Model parameters

  31. w h Model application to rectangular regions • Rectangular regions defined by h (longest side) and w (shortest side) • Distances measured in gate pitches • Aspect ratio 

  32. p = 0.15 p = 0.65 Model resultsverification of Rent’s Rule Model is conform with Rent’s rule = 9 = 4 = 1

  33. Model results for pl • Practically independent of  • Lower limit of 0.5 for simple designs • Converges to pi for complex designs

  34. = 10 Model results for tl (normalized to ti) • Significant dependence on  • Converges towards tG for complex circuits (large pi) = 1

  35. Explanation for model results Perimeter region of gates with external terminals Small p Larger p

  36. Explanation for model results For small pi, width of this region << w T scales ~ to rectangle boundary For 2D embedding: Perimeter region of gates with external terminals Small p

  37. Explanation for model results Perimeter region of gates with external terminals For larger pi, this approximation no longer holds Larger p

  38. Model verification 10 synthetic benchmarks ( gnl ) with 240 000 gates and controlled values of pi Only 2-pin connections

  39. Model verification (p) Model results in slight underestimation of p

  40. = 8 = 8 = 1 = 1 Model verification (t)

  41. = 8 = 1 Model verification (t) • Irregular fluctuations due to fitting procedure on logarithmic scale • Convergence towards 1 (high pi) in accordance with model • Experimental values much higher than model results

  42. Empirical distribution Impact of wirelength distribution Real placement results in sub-optimal wirelength distribution g8 g2

  43. Impact of wirelength distribution (p) Model with empirical wirelength distribution is very accurate for p

  44. = 8 = 8 = 1 = 1 Impact of wirelength distribution (t) Model with empirical wirelength distribution is a lot better for t

  45. Impact of multi-pin connections • Results for p remain practically unchanged • Results for t: similar trends but different scaling • Future work: try to include multi-pin connections in model: not trivial (see presentation by D. Stroobandt) = 8 = 1

  46. Conclusionsmodel New model for pins at the edge of a layout region with given shape: • Relates layout Rent parameters to intrinsic Rent parameters (pre-layout) • Shows a lower limit for the 2D layout Rent exponent of 0.5, due to perimeter effect • Gives good results for rectangular regions

  47. Conclusionsmodel Future model improvements include: • 3D as well as 2D architectures • Multi-pin connections • A more detailed study of wirelength distributions resulting from different placement algorithms (see presentation by P. Verplaetse)

  48. Conclusionsfor rectangular regions • Rent’s rule is still a good approximation for rectangular layout regions • Layout Rent exponent is practically independent of aspect ratio  • Layout Rent coefficient strongly depends on aspect ratio 

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