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High-Speed Circuit-Tuning Techniques Based on Lagrangian Relaxation Charlie Chung-Ping Chen ICCAD 99’ Embedded Tutorial Session 12A email@example.com (608)2651145
People Involved • Joint work Charlie Chen, University of Wisconsin at Madison Chris Chu, Iowa State University D. F. Wong, University of Texas at Austin • Publication “Fast and Exact Simultaneous Gate and Wire Sizing by Lagrangian Relaxation”, IEEE Transactions on Computer-Aided Design, July 1999
Acknowledgement • Strategic CAD Labs, Intel Corp. Steve Burns, Prashant Sawkar, N. Sherwani, and Noel Menezes • IBM T. J. Watson Center Chandu Visweswariah • C. Kime, L. He (UWisc-Madison)
Outline • Motivation • Overview of Circuit Tuning Techniques • Lagrangian Relaxation Based Circuit Tuning
Motivation • Double the work load and design complexity every 18 months
Motivation • Trends • Increased custom design • Aggressive tuning for performance improvement • Shorter time to market • Interconnect effects severe • Signal integrity issues emerging • Circuit Tuning • Can significantly improve circuit performance and signal integrity without major modification
Manual Sizing 1000+ iterations • Pros • Takes advantage of human experience • Reliable • Simultaneously combines with other optimization techniques directly • Cons • Slow, tedious, limited, and error-prone procedure • Rely too much on experience, requires solid training • Optimality not guaranteed (don’t know when to stop) Change Simulate Satisfy?
Automatic Circuit Tuning • Pros • Fast • Achieves the best performance with interconnect considerations • Explores alternatives (power/delay/noise tradeoff) • Boosts productivity • Optimality guaranty (for convex problems) • Insures timing and reliability • Cons • Complicated tool development and support ($$) • Tool testing, integration, and training
Good Tuning Algorithm • Fast • Optimality guaranteed (for convex problem) • Versatile • Easy to use • Solution quality index (error bound to the optimal solution) • Simple (Easy to develop and maintain)
Static vs. Dynamic Sizing • Static Sizing • Stage Based • Nature circuit decomposition, large scale tuning capability • Very reasonable accuracy (when using good model) • No need for sensitization vectors • Solves for all critical paths in a polynomial formulation • False paths; Potentially inaccurate modeling of slopes of input excitation • Dynamic Sizing • Simulation based • More accurate • No false path problems • Need good input vectors; good for circuits for which critical paths are known and limited • Takes care of a few scenario only • Relatively slower
A Simple Sizing Problem • Minimize the maximum delay Dmax by changing w1,…,wn w7 w9 w4 w1 D1<Dmax a w5 w10 D2<Dmax b w2 w6 w11 w3 w8
Existing Sizing Works • Algorithm: fast, non-optimal for general problem formulation • TILOS (J. Fishburn, A. Dunlop, ICCAD 85’) • Weight Delay Optimization (J. Cong et al., ICCAD 95’) • Mathematical Programming: slower, optimal • Geometrical Programming (TILOS) • Augmented Lagrangian (D. P. Marple et al., 86’) • Sequential Linear Programming (S. Sapatnekar et al.) • Interior Point Method (S. Sapatnekar et al., TCAD 93’) • Sequential Quadratic Programming (N. Menezes et al., DAC 95’) • Augmented Lagrangian + Adjoin Sensitivity (C. Visweswariah et al., ICCAD 96’, ICCAD97’) • Is there any method that is fast and optimal?
Converge? Augmented Lagrangian Weighted Delay SQP Fast Optimal TILOS SLP ? Mathematical Programming Algorithm
Heuristic Approach • TILOS: (J. Fishburn etc ICCAD 85’) • Find all the sensitivities associated with each gate • Up-Size one gate only with the maximum sensitivity • To minimize the object function Minimize Dmax w2 w1 w3 a w4 D1<Dmax w5 w6 D2<Dmax b w11 w9 w7 w8 w10
Weighted Delay Optimization • J. Cong ICCAD 95’ • Size one wire at a time in DFS order • To minimize the weighted delay • best weight? Minimize l1D1 +l2D2 Drivers Loads w1 w2 w3 l1D1 l2D2 w5 w4
Mathematical Programming • Problem Formulation: • Lagrangian: • Optimality (Necessary) Condition: (Kuhn-Tucker Condition)
PSLP v.s. SQP • Penalty Sequential Linear Programming • Sequential Quadratic Programming
Lagrangian Methods • Augmented Lagrangian • Lagrangian Relaxation
Lagrangian Relaxation Theory • LRS (Lagrangian Relaxation Subproblem) • There exist Lagrangian multipliers will lead LRS to find the optimal solution for convex programming problem • The optimal solution for any LRS is a lower bound of the original problem for any type of problem
Lagrangian Relaxation Lagrangian Relaxation Weighted Delay!
Lagrangian Relaxation Augmented Lagrangian Weighted Delay SQP TILOS SLP Lagrangian Relaxation Sink Weights=Multipliers Mathematical Programming Algorithm
Lagrangian Relaxation Framework Update Multipliers Weighted Delay Optimization Converge?
Lagrangian Relaxation Framework Dmax l1 l2 D1 D2 Dmax l1 l2 D1 D2 More Critical -> More Resource -> More Weight D1 D2
Weighted Minimization • Traverse the circuit in topological order • Resize each component to minimize Lagrangian during visit Minimize l1D1 +l2D2 w1 a D1 D2 b w2 w3
Multipliers Adjustmenta subgradient approach • Subgradient: An extension definition of gradient for non-smooth function • Experience: Simple heuristic implementation can achieve very good convergence rate • Reference: Non-smooth function optimization: N. Z. Shor
Path Delay Formulation d1 d2 Aa D1 Ab d3 D2 Ac • Exponential growing • More accurate • Can exclude false paths
Stage Delay Formulation d1 d2 Ae Aa D1 Ab d3 D2 Ac • Polynomial size • Less accurate • Contains false paths
Compatible? ? Path Based Stage Based
Both Multipliers Satisfy KCL(Flow Conservation) l41 l51 l42 l52 Path Based Stage Based l43 1 l31 1 4 4 3 2 2 3 l32 5 5 l53 l3,in =l3,out l43 +l53=l31 +l32
Mixed Delay Formulation Stage Based Stage Based Path Based
Compatible? Lagrangian Relaxation Both Multipliers Satisfy KCL Stage Based Path Based
Hierarchical Objective Function Decomposition • Divide the Lagrangian into who terms (containing or not containing variable wi ) • Hierarchically update the Lagrangian during resizing
Intermediate Variables Cancellation Ae Aa D1 Ab D2 Ac lae lae +lbe =le1 +le2 lbe le1 + le2 lc2 lae (Aa+ d1 )+lbe (Ab+ d1)+le1 (d2 - D1 )+ le2 (d3 - D2 )
Decomposition and Pruning • Flow Decomposition • Prune out all the gates with zero multipliers
Complimentary Condition Implications • li (Di-Dmax )= 0 • Optimal Solution • Critical Path, weight l i >= 0.0, path delay=Dmax • Non-critical path, weight l i = 0.0, path delay < Dmax
Convergence Sequence Optimal Solution Max Delay Any Feasible Maximum Delay= Upper Bound Lagrangian=Lower Bound Weighted Delay<=Maximum Delay # Iteration
Conclusion • Lagrangian Relaxation • General mathematical programming algorithm • Optimality guarantee for convex programming problem • Versatile • No extra complication (no quadratic penalty function) • Lagrangian multiplier provides connections between mathematical programming and algorithmic approaches • Multipliers satisfy KCL (flow conservation) • Hierarchical update objective function provides extreme efficiency • Solution quality guaranteed (by providing lower bound)