PartitioningBased Approach to Fast OnChip Decap Budgeting and Minimization

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Partitioning-Based Approach to Fast On-Chip Decap
Budgeting and Minimization

• Hang Li, Zhenyu Qi
• Sheldon X.-D. Tan
• Mixed-Signal Nanometer VLSI Research Lab
• University of California, Riverside

• Yici Cai, Xianlong Hong
• Department of Computer Science and Technology
• Tsinghua University, Beijing, China

• This work is sponsored by

• Lifeng Wu
• Cadence Design Systems Inc., San Jose, CA

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Outline

• Introductions
• Review of existing decap budgeting algorithm
• Improved conjugate gradient algorithm
• Partitioning-based decap budgeting strategy
• Experimental results
• Conclusions and future works

3
What is Decap?

• Decap decoupling capacitor

4
Why Adding Decap is Important?

• Power source fluctuations increase significantly

Illustration of voltage drop variation of modern
VLSI chip
From Cadences Voltagestorm product brochure

• Static IR Drop ?V I R (P /Vdd ) R
• Dynamic IR Drop ?V L di/dt noise

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Introduction Voltage Drop Impacts on Timing
10 voltage drop can cause more than 10 delay
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Introduction - Effect of Adding Decaps
Adding decaps is the most effective way to reduce
voltage noises in P/G grids
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Introduction - The Costs of Adding Decaps

• Decaps are mainly made of MOSgate capacitors
• Consuming premium white spaces
• White space can otherwise be used for adding
  buffers, other logic gates for physical
  optimization.
• MOS gates are leaky or become more leaky with
  scaling
• More leakage powers
• Excessive decaps will lead to low yield and low
  circuit resonant frequency, etc.
• Economic use of decaps are important!!

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Review of Previous Decap Budgeting Algorithms

• Charges based methods (Chen LingDAC97, Zhao
  RoyTCAD02)
• Charges for removing the unwanted voltage drops
  are estimated based on simplified P/G grids.
• The decaps added are much more than necessary and
  far from minimum solutions.
• Sensitivity based methods (Bai HajjICCAD00,
  Su SapatnekarTCAD03, WangDAC03, Fu
  TanASPDAC04)
• Decaps are added based on the time-domain
  sensitivity of voltage drop w.r.t. decap area
  (value).
• More precise and accurate on the estimated
  decaps.
• Not very scalable to deal with large circuits.

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Main Contributions of the Proposed New Approach
to Decap Budgeting

• Improved conjugate gradient algorithm
  (efficiency issue)
• Efficient search step during line search phase.
• Partitioned-based merged adjoint network for
  sensitivity computation.
• Localized decap budgeting based on partitioning
  (scalability issue)
• Decap has local effects on the voltage
  fluctuation.
• Partition large circuits into smaller ones and
  optimize each of them individually.

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Problem Definition - Introduction

• Given P/G grids modeled as RC/RLC networks with
  time-varying current sources
• Voltage is less than user specified values. The
  violation becomes the area over time

• Objective use minimum decaps to remove the
  shaded violation area
• Subject to
• Voltage drop limited to user specified values
• Decap area constraints from physical layout
• Other reliability constraints

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Decap Budgeting Problem Formulation

• Optimize decap area subject to IR noise
  constraints presented in power gird network

Objective function
minimize
, where
Constraint
1.
or error bound
where
2.
-- A Nonlinear Optimization Problem!
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Previous Conjugate Gradient Based Method (Fu
TanASPDAC04)

• Transform the above constrained problem into an
  unconstrained one

Penalty function to be minimized

• Solve the unconstrained problem by Conjugate
  Gradient (CG) optimization

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Conjugate Gradient Optimization Gradient
Computation

• Gradient of the penalty function with respect to
  each decap node
• Gradient of decap at a single node

Efficient sensitivity computation is required!
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Conjugate Gradient Optimization Sensitivity
Computation

• Adjoint Network Method
• Two simulations required to calculate sensitivity
  of aSINGLE violation node
• One for original network
• One for adjoint network

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Conjugate Gradient Optimization Sensitivity
Computation (Cont.)

• Merged adjoint network method in time domain
• Compute sensitivity for all the decaps together

Merge all the inputs for AT

• Two simulations to calculate a whole gradient
  vector, i. e. sensitivity for ALL decaps in the
  circuit
• Time Complexity

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Problems for Previous CG Method

• Objective is function of ?
• Difficult to select ?
• Less efficient optimization effects (constraint
  may not be removed completely)
• Line search can be very costly and wasteful for
  CPU time for inappropriate ?

f(x)
?
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Simplified Problem Definitions

• Simplified objective function, which consists of
  the violation area only

minimize
subject to

• Avoid the inherent ambiguous optimization
  objective in previous conjugate gradient method

• Eliminate misleading a in previous objective
  function

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Merged Adjoint Method (FuASPDAC04)

• Merged adjoint network method for efficient
  sensitivity computation

i decap node
j violation node
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Problem with Merged Adjoint Method

• Overestimation of decap budget due to the
  sensitivity loss from merged adjoint network

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Partitioning-Based Merged Adjoint Method

• We observe the positive effect of
  partitioning-based merged adjoint network method
• Create more objective functions for subcircuits
  to reduce quality lose due to the merged
  sensitivity.
• In the extreme case, where every subcircuit has
  one node, we go back to the individual
  sensitivity case (un-converged optimization
  problem).
• Experimental results show that increasing
  partitioning number can improve the decap
  optimization quality.

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Improved Conjugate Gradient Method

• Direct conjugate gradient method requires a
  number of line searches to compute the best
  search step at each optimization step
• Key Idea
• Step size is determined by computing the maximum
  decap value allowed on one or some nodes under
  current search directions (sensitivity)
• Binary search is used to minimize the possibly
  overestimated decap values
• Timing Analysis

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Optimization Algorithm Flow Chart
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Partitioning-Based Decap Budgeting

• For very large circuits, both CG/iCG based
  approaches will be slow
• As circuit simulation time, which is the inter
  loop of the optimization, will be significant.

• Localized adding decap effect also favors the
  partitioning-based strategy
• Basic idea partition the large circuits into
  small ones and optimize each of them
  individually.
• Rational for such localized optimization adding
  decap has local effects on voltage drops in a P/G
  circuits.
• Simulate the rest of the circuit by keeping the
  boundary node voltage waveform recorded from full
  circuit simulation

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Partitioning Algorithm

• General partition algorithm using graph-based
  multilevel minimum cut algorithm (METIS)
• Extremely fast speed ensures the partition phase
  wont bottleneck the entire optimization flow

• Boundary condition for each partition must be
  preserved to avoid overestimation of decap budget
• Boundary node waveform in PWL voltage form serves
  as the boundary condition

G.Karypis, R. Aggarwal, and V.K.S. Shekhar,
Multilevel hypergraph partitioning application
in VLSI domain, IEEE Trans. On Very Large Scale
Integration (VLSI) Systems, vol. 7, no. 1, pp.
69-70, March 1999.
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Noise-Aware Partitioning (NAP)

• Violation nodes can not be at the boundary
• Violation can not be eliminated due to the
  unchanged PWL waveform
• Noise-aware partitioning must be introduced

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Partitioning-Based Decap Budget Flow
Finalize decap value
Decrease previous partition number
N
Violation Criterion Met ?
Y
Solve new circuit with updated decap values
Combine updated decap values Generate new
netlist file
Y
Call iCG solver to do the individual partition
optimization
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Comparison Between CG and iCG

• CG algorithm was modified by explicitly trying to
  bracket the minimum before each line search for
  fair and reasonable comparison to iCG
• Bracketing the minimum avoids useless line
  searches and improve the efficiency of the CG
  algorithm

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Comparison Between Flat and Partitioning-Based
Decap Budgeting

• At least 10X speed-up is achieved with comparable
  decap budget value

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Observation on Partitioned Decap Budget

• Even better decap budget could be achieved from
  partitioning
• Partitioning makes some nodes harder to optimize
• Partitioned merged adjoint method can improve the
  quality of the decap optimization

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Conclusions Future Works

• Applied the time domain merged adjoint network
  method.
• Using improved conjugate gradient techniques for
  decap budgeting.
• The combination of proposed partitioning scheme
  with the merged adjoint network method leads to
  both fast optimization process and better results
  that that given by flat merged adjoint network
  method.
• At least 10X speed-up can be achieved for large
  circuit sizes under the new algorithm.
• Parallel computing will be explored in the future
  to further improve the algorithms efficiency.

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• Thank You