1. SLIP 2000April 9, 2000 --1-- Wiring Layer Assignments with Consistent Stage Delays Andrew B. Kahng (UCLA) Dirk Stroobandt (Ghent University) Supported by Cadence Design Systems, Inc. and the MARCO Gigascale Silicon Research Center

2. SLIP 2000April 9, 2000 --2-- Outline Introduction: wiring layer assignment Problem and models Optimization objective function Our layer assignment method Discussion and results Conclusion

3. SLIP 2000April 9, 2000 --3-- Introduction DSM design routing tools have to account for –delay constraints –yield –power –… Conventional technique: –router assigns wires to layers –wire sizing, repeater insertion/sizing applied More interesting approach: –wire sizing etc. used by router to assign wires

4. SLIP 2000April 9, 2000 --4-- Our Layer Assignment Concept Search for optimal layer for a wire with –optimal wire size, number and size of repeaters for each wire –meeting consistent stage delay constraints –taking total repeater area constraint into account –accounting for impact of vias A priori estimation techniques make it useful for application both before / after placement Potential applications –improving CAD layout tools –studying effects of technological parameters –optimizing fabrication process

5. SLIP 2000April 9, 2000 --5-- Problem and Models Optimization objective: # of layers needed Degrees of freedom (for each wire) –choice of layer parameters –wire width –number of repeaters –size of the repeaters Find the optimal assignment of wires to wiring layers subject to delay constraints and total repeater area constraints

6. SLIP 2000April 9, 2000 --6-- Layer Assignment Assumptions –layer pairs form tiers (H and V) –tiers grouped in tier types (equal parameters) –(even) number of layers/type to be determined –wires are routed on 1 tier –inputs to the method: number of tier types and their layer parameters order (bottom-to-top) user-defined –output: optimal number of layers / tier type layer to which wires are assigned H H V V } } tier

7. SLIP 2000April 9, 2000 --7-- Delay Constraint Model Sakurai’s [IEEE TED, 1993] delay equation –depends on wire length/width through wire R, C Delay can be reduced by –increasing wire width (for fixed length and layer) consider uniform wire sizing (no tapering) continuous wire sizing (no discrete set of widths) –optimal gate sizing –repeater insertion and sizing repeaters at equal distances number of repeaters even Trade-off between delay and area

8. SLIP 2000April 9, 2000 --8-- Via Impact Model Sai-Halasz [1995]: every layer blocks 15% Newer models: Chong [1999], Chen [1999] –terminal vias and turn vias –each wire uses 2 via stacks –number of terminal vias defined by layer assignment model Via impact factor Chong: Chen: H H V V } } tier H H V V } }

9. SLIP 2000April 9, 2000 --9-- Wire Length Distribution All wires classified according to their length Wire length distribution needed –measuring distances between placed gates –applying a priori wirelength estimation

10. SLIP 2000April 9, 2000 --10-- Cost Function = Number of Layers Non-integer by considering area needed/tier A = available area/layer Area parts: wiring area + area “lost” to vias Tier type i W(2) Wire 0 Wire 2 Wire 1 SiSi l(2)

11. SLIP 2000April 9, 2000 --11-- Cost Function (cont.) Via area assumptions –square area with side = minimal wire width/layer –line of vias for wider wires –via sizes scale with minimal wire widths for lower layers W W min Tier type 2 Tier type 1 Tier type 0

12. SLIP 2000April 9, 2000 --12-- Cost Function (cont.) Number of vias –each repeater adds 2 vias on layer of tier below –each repeater adds 1 via on layer of own tier –no repeaters: as if 1 repeater Gate Repeater Wire on type 2 Tier type 1 Tier type 2

13. SLIP 2000April 9, 2000 --13-- Via Impact Limits Number of Layers Via impact factor must be < 1 For maximum number of layers (e.g., 10) –8 layers for wires: f<0.2 –number of wires < 300,000 (250nm, 10M trans., logic area 54mm, 4  m wire pitch on tier, all wires minimum width, no repeaters) 2

14. SLIP 2000April 9, 2000 --14-- Layer Assignment method 2 phases: optimize, then round to integers Phase 1: Calculate minimal delay T min. T min <T target ? A min <A limit ? Sort costs in increasing order. Repeat for all wires.  C min < 0? Calculate cost  C for moving wire to other tier while optimizing W, N r. Calculate minimal repeater area A min. Y T target =T min. N A limit =A min Solution found! N Solution found! N Move K wires with smallest  C<0. Repeat. Y Create initial solution on “fattest” tier. Y

15. SLIP 2000April 9, 2000 --15-- A Typical Example Tier type 2 Tier type 1 Tier type 0 Wirelength (mm) Delay (ps) Wire width (  m) 02400224Number of repeaters

16. SLIP 2000April 9, 2000 --16-- Target Delay Influence Tier type 2 Tier type 1 Tier type 0 Wirelength (mm) Delay (ps) Wire width (  m) Wirelength (mm)

17. SLIP 2000April 9, 2000 --17-- Uniform Versus Non-uniform Stacks Uniform still 3 numbers: via impact = L i f i 6.3134 7.1053 Tier 2 Tier 1 Tier 0 10 9 8 7 6 5 4 3 2 1 0 9 8 7 6 5 4 3 2 1 0 Number of layers per tier type 8.0251 Total

18. SLIP 2000April 9, 2000 --18-- Optimal Layer Stack Monotonic? Results depend on delay constraint 5.4426 7.0002 19.3665 Tier 2 Tier 1 Tier 0 7.1285 20 18 16 14 12 10 8 6 4 2 0 Number of layers per tier type 20 18 16 14 12 10 8 6 4 2 0

19. SLIP 2000April 9, 2000 --19-- Conclusions Layer assignment is becoming more critical Our proposal: use stage delay constraints Current work: 2-D length-delay distribution Wire/repeater sizing + via impact + area limit Via impact severely limits number of wires Interesting conclusions: –maximum wire width on tier type not dependent on delay constraint –monotonic non-uniform layer stack (fat-wires-on-top) better than uniform –non-monotonic worse for tight delay constraints but “non-fat” tier on top can be beneficial Useful to search for optimal layer stack parameters Find threshold values to ensure optimality of layer stack and make layer assignment more “trivial”