Supply Voltage Degradation Aware Analytical Placement

1. Supply Voltage Degradation Aware Analytical Placement Andrew B. Kahng, Bao Liu and Qinke Wang UCSD CSE Department {abk, bliu, qiwang}@cs.ucsd.edu

2. Outline • Introduction • Motivation • Related work • Our work • Problem formulation • Analysis and Observations • Voltage Degradation Aware Placement • Experiments • Conclusions

3. Motivation • Increasingly significant voltage degradation along power networks in nanometer VLSI designs • shrinking layout feature sizes • increasing device density • Logic malfunction • Performance degradation • a 10% voltage drop could be responsible for 10% transistor performance degradation, and the effect is super-linear

4. Related Work • Techniques to reduce supply voltage degradation • wiresizing and edge augmentation • decoupling capacitor insertion • circuit de-tuning • Placement and floorplan related techniques • local placement adjustment to allocate whitespace for decoupling capacitor insertion • allocation of power pads: more pads close to current drain hot spots • a floorplan objective for power network construction and supply voltage drop

5. Summary of Existing Works • Existing voltage drop reduction techniques focus on power supply network design • Supply voltage degradation is also a function of supply currents of the circuit • To the best of our knowledge, no analytical placement technique for voltage drop reduction is proposed

6. Our Contributions • We propose voltage degradation aware placement : relocating current drains for voltage drop reduction • represent voltage drop at a power node as a function of current drains and effective resistances • propose voltage drop as placement objective and integrate into an analytical placement framework • test our method on real designs with industry flow

7. Model of Power Network • Power network: modeled as a resistive netlist • parallel metal wires at multiple layers • metal layers connected at crossing points by vias • Power pads on the top layer: modeled as DC voltage sources • Active devices at the bottom layer: modeled as DC current drains • DC currents provide bounds for the actual AC currents

8. Problem Formulation • Given • a power supply network • worst-case current drains for each cell • Find a placement to • reduce supply voltage drop • maintain comparable placement wirelength, area, and timing performance

9. Outline • Introduction • Analysis and Observations • Analysis of voltage drop • Observations on voltage drop optimization • Computation of effective resistance • Voltage Degradation Aware Placement • Experiments • Conclusions

10. Analysis of Voltage Drop • Voltage drop at an observation node t • each current drain Ik has contribution to the voltage drop • : effective resistance for a current drain at node k to inject noise voltage at node t

11. s R1 R3 I1 R2 t I2 I3 An Example Tree-Structure • s : power pad • t : observation node

12. Objectives and Observations • Given a power supply network, find a placement of current drains to minimize: • (a) voltage drop at a given observation node t • (b) average voltage drop of all nodes, or • (c) max voltage drop over all nodes • (a) : greedy algorithm to locate largest current Ik first to have smallest resistance • (b) : greedy algorithm to locate largest current Ik first to have smallest resistance • (c) : NP-hard

13. Effective Resistance (I) • Direct modified nodal analysis • G: conductance matrix • matrix inversion O(n3) • not feasible for practical power networks

14. Effective Resistance (II) • Random walk [Qian et al. DAC 2003] • resistance of the common part of two random walk paths that respectively start from nodes k and t and end at a power pad • a random walk path follows the corresponding current distribution probability: transition probability from node p to q on the random walk path is

15. Effective Resistance (III) • Better scalability and efficiency • contract power netlist by merging bottom-level wires and computing parallel resistances • compute effective resistance between observation nodes • apply bi-linear interpolation for supply voltage drop at any node

16. Outline • Introduction • Analysis and Observations • Voltage Degradation Aware Placement • Introduction of analytical placement • Voltage drop aware placement objectives • Implementation • Experiments • Conclusions

17. Introduction of APlace (I) • APlace: a general analytical placement framework • High solution quality and strong extensibility • Regard Global placement (NP-hard) as a Constrained Nonlinear Optimization Problem: • : density function that equals the total module area in a global cell g • D : average module area over all global cells

18. Introduction of APlace (II) • Apply smooth approximation of placement objectives: wirelength, density function, etc. • Quadratic Penalty method • solve a sequence of unconstrained minimization problems for a sequence of µ ↓ 0 • Conjugate Gradient solver • find an unconstrained minimum of a high-dimensional function • memory required is only linear in the problem size, which makes it adaptable to large-scale placement problems

19. Outline • Introduction • Analysis and Observations • Voltage Degradation Aware Placement • Introduction of analytical placement • Voltage drop aware placement objectives • Implementation • Experiments • Conclusions

20. Average Voltage Drop • N : the number of observation nodes • : effective resistance for a current drain Iv to generate a voltage-drop at node g • function of module v's position during global placement • effective resistance at continuous positions are obtained using bi-linear interpolation • partial differentials are computed accordingly

21. Worst Voltage Drop • LOG-SUM-EXP function • smooth approximation of worst voltage drop • α: smoothing parameter and significance criterion for choosing power network nodes with large voltage drop to minimize • Vworst: strictly convex, continuously differentiable and converges to the worst voltage drop as α converges to 0

22. Outline • Introduction • Analysis and Observations • Voltage Degradation Aware Placement • Introduction of analytical placement • Voltage drop aware placement objectives • Implementation • Experiments • Conclusions

23. Implementation (I) • Integrate voltage drop objectives into the analytical placement framework • Wv : weight of the voltage drop objective • computed according to the gradients derived from the wirelength and voltage drop terms • scaled voltage drop gradients comparable to wirelength gradients

24. Implementation (II) • β : voltage drop ratio • decide the ratio of voltage drop gradients to wirelength gradients • provide a knob to trade-off between voltage drop and wirelength objectives for the placer

25. Outline • Introduction • Analysis and Observations • Voltage Degradation Aware Placement • Experiments • Experimental setup • Results • Conclusions

26. Design #Cells #Rows Tech Utilization AES 13397 129 90nm 0.6 PCI 7128 251 180nm 0.43 Experimental Setup • Two industry circuits • TSMC library • six metal layers • power/ground ring at top 2 layers • 4 power pads at the center of boundaries • AES: 5 stripes at M2 • PCI: 4 stripes at M6 and 5 large fixed macros

27. Experimental Flow • Design inputs: synthesized netlists, technology libraries, timing constraints and floorplans • Power planning and routing, and pad placement in Cadence SoC Encounter • Voltage drop aware and oblivious placements using our placer and wirelength-driven APlace • Fast global and detail routing by Cadence TrialRoute • Steady-state voltage-drop analysis by Cadence VoltageStorm

28. Outline • Introduction • Analysis and Observations • Voltage Degradation Aware Placement • Experiments • Experimental setup • Results • Conclusions

29. Design Placers Vdrop Vdrop Improvements Impact Ratio Avg Vdrop Max Vdrop HPWL CPU (V) (%) (V) (%) (e8) (%) (s) AES APlace 0.00 0.233 0.00% 0.406 0.00% 9.48 0.00% 223.62 our placer 0.05 0.217 6.61% 0.354 12.74% 9.58 -1.10% 286.53 0.10 0.219 6.02% 0.356 12.41% 9.57 -0.94% 265.94 0.15 0.214 8.07% 0.331 18.49% 9.67 -1.95% 239.52 0.20 0.208 10.67% 0.318 21.59% 9.68 -2.09% 227.24 0.25 0.209 10.22% 0.314 22.65% 9.78 -3.17% 217.53 PCI APlace 0.00 0.026 0.00% 0.051 0.00% 19.95 0.00% 120.97 our placer 0.05 0.025 3.18% 0.048 5.54% 20.14 -0.93% 172 0.10 0.025 5.84% 0.046 9.75% 20.25 -1.50% 166 0.15 0.024 9.27% 0.044 13.67% 20.53 -2.92% 156 0.20 0.023 11.52% 0.042 16.65% 20.72 -3.87% 145 0.25 0.023 13.08% 0.041 19.02% 21.01 -5.33% 146 Results (I): Worst Voltage Drop Results of worst voltage-drop aware placements with a variety of voltage drop ratios (β's) (β)

30. Design Placers Vdrop Vdrop Improvements Impact Ratio Avg Vdrop Max Vdrop HPWL CPU (V) (%) (V) (%) (e8) (%) (s) AES APlace 0.00 0.233 0.00% 0.406 0.00% 9.48 0.00% 223.62 our placer 0.05 0.219 6.13% 0.361 11.12% 9.50 -0.23% 284.27 0.10 0.210 9.79% 0.343 15.48% 10.04 -5.88% 273.32 0.15 0.209 10.19% 0.341 16.07% 10.12 -6.76% 319.44 0.20 0.201 13.68% 0.320 21.24% 10.28 -8.46% 311.6 0.25 0.192 17.64% 0.302 25.70% 10.40 -9.74% 285.58 PCI APlace 0.00 0.026 0.00% 0.051 0.00% 19.95 0.00% 120.97 our placer 0.05 0.025 4.94% 0.047 6.75% 20.22 -1.35% 160 -5.44% 0.10 0.024 9.14% 0.044 13.06% 21.04 175 0.15 0.019 26.03% 0.035 29.80% 22.83 -14.45% 206 0.20 0.018 31.02% 0.033 35.14% 23.18 -16.18% 234 0.25 0.016 39.54% 0.028 43.99% 25.16 -26.10% 285 Results (II): Average Voltage Drop Results of average voltage-drop aware placements with a variety of voltage drop ratios (β's) (β)

31. Summary of Results • Improvement • worst voltage drop: 22.7% and 19.0% • average voltage drop: 10.2% and 13.1% • Impact on HPWL: -3.2% and -5.3% • Worst voltage drop objective leads to better results than average voltage drop objective • large voltage drops are among the first to be reduced • benefit the average voltage drop more than trying to reduce all the voltage drops with same efforts

32. HPWL vs. Voltage Drop HPWL, worst-case and average voltage-drop improvements as functions of voltage drop ratio for AES

33. Conclusions • We propose analytical placement for supply voltage drop reduction • We integrate supply voltage drop objective into an analytical placement framework • Our experimental results show on average 20.9% improvement of worst-case voltage drop and 11.7% improvement of average voltage drop with only 4.3% wirelength increase • Ongoing research efforts: supply voltage drop aware timing-driven placement

34. Thank You !