Department of Electrical and Computer Engineering

Transcription

1 LAGRANGIAN RELAXATION FOR GATE IMPLEMENTATION SELECTION Yi-Le Huang, Jiang Hu and Weiping Shi Department of Electrical and Computer Engineering Texas A&M University

2 OUTLINE Introduction and motivation Projection-based descent method for solving Lagrangian dual problem Distribution of Lagrangian multipliers Experimental results Conclusion 2

3 GATE IMPLEMENTATION SELECTION Gate implementation options size P/N ratio threshold voltage (Vt) Large problem size commonly, hundreds of thousands of gates sometimes millions of gates Essential for circuit power and performance 3

4 PREVIOUS WORK (CONTINUOUS) Solved by Linear programming Convex programming Network flow Round fractional solutions to integers Fast Rounding errors Restrictions on delay/power models Difficult to handle P/N ratio unless transistor level 4

5 PREVIOUS WORK (DISCRETE) No rounding error Compatible with different power/delay models Sensitivity based heuristics Simple Quick Greedy Dynamic programming-like search Relatively systematic solution search 5

6 LAGRANGIAN RELAXATION (LR) Handle conflicting objectives or complex constraints With continuous optimization Faster convergence (Chen, Chu and Wong, TCAD 1999) With dynamic programming-like search Alleviate the curse of dimensionality 6

7 OVERVIEW OF LR Original problem Minimize A(x) Subject to: B(x) 0 C(x) 0 LR subproblem Lagrangian multiplier Minimize A(x) + λ B(x) ( ) Subject to: C(x) 0 Lagrangian dual problem Find λ max optimal solution of subproblem 7

8 LR FOR GATE IMPLEMENTATION SELECTION Original problem Min s.t. P ( x ) a j a D i A D a i ij a i j j PO i in( j ) i PI Min LR subproblem P(x) ( a ( a ( D j ij i j A) D ij a i0 i i a ) j ) λ x: implementation decision ij i j P: power D: delay a: arrival time Subgradient == -slack 8

9 LAGRANGIAN DUAL PROBLEM Ideally a piece-wise convex function solved by subgradient method In practice variant of steepest descent no guarantee for optimal subproblem solution dual problem is no longer convex How to solve non-convex dual problem? not well studied λ 9

10 KARUSH-KUHN-TUCKER (KKT) CONDITIONS 1 2 i I II i 2 i ii iii Flow conservation (Chen, Chu and Wong TCAD99)... 10

11 PROBLEM OF SUBGRADIENT METHOD Slack1: -5 Slack3: -5 Slack2: 20 Δλ 1 = 5ρ, Δλ 2 = -20ρ, Δλ 3 = 5ρ ρ: step size in subgradient method Δλ 1 + Δλ 2 Δλ 3 11

12 PROJECTION-BASED DESCENT METHOD Subgradient Projected move direction: smoothed historical gradient λ 1 λ 2 12

13 PROJECTION ESTIMATION Table of (a, λ) in previous iterations Gradient history: (a i -a i-1 )/(λ i - λ i-1 ) Projection direction Weighted average of historical gradients More weight for recent gradients

14 STEP SIZE Δλ = (q a cur ) / η q: required arrival time a cur : current arrival time η: projected move direction 14

15 MULTIPLIER UPDATE FLOW Multipliers at PO are updated by projection They are distributed to entire circuit in reverse- topological order like network flow Alternatively, ti l from PI distribute ib t in topological l order 15

16 MULTIPLIER DISTRIBUTION Ensure flow conservation Try to equalize slack: different slacks imply room for power saving Given outgoing flow, find x x ij ai i in( j) k out( j) jk 1 2 j I II x is the target arrival time Δλ ij = (x-a i )/η ij 16

17 EXPERIMENT SETUP ISCAS85 benchmark Cell library based on 70nm technology Synthesized by SIS Placed by mpl Elmore delay Analytical power model LR subproblem is solved by greedy heuristic Compare our approach (projection+greedy) with baseline (subgradient+greedy) 17

18 RESULTS WITH TIGHT TIMING CONSTRAINTS Initial Sub gradient method Our method testcase # of gates power slack power slack run time power slack run time chain c c c c c c c c c c Sum # of violation

19 RESULTS WITH LOOSE TIMING CONSTRAINTS Initial Sub gradient method Our method testcase # ofgates power slack power slack run time power slack run time chain c c c c c c c c c c Sum # of violation

20 SLACK OVER ITERATIONS (C432) ck Sla Our algorithm Iteration Sub gradient method 20

21 POWER OVER ITERATIONS (C432) ower P Iteration Our algorithm Sub gradient method 21

22 CONCLUSIONS AND FUTURE RESEARCH Drawbacks of subgradient method are investigated New techniques are proposed to solve Lagrangian dual problem for gate implementation selection They lead to better solutions and faster convergence In future, we will integrate them with dynamic programming-like search 22

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