Network Flow-based Simultaneous Retiming and Slack Budgeting for Low Power Design

Transcription

1 Outline Network Flow-based Simultaneous Retiming and Slack Budgeting for Low Power Design Bei Yu 1 Sheqin Dong 1 Yuchun Ma 1 Tao Lin 1 Yu Wang 1 Song Chen 2 Satoshi GOTO 2 1 Department of Computer Science & Technology Tsinghua University, Beijing, China 2 Graduate School of IPS Waseda University, Kitakyushu, Japan

2 Outline Outline 1 Introduction Previous Works Problem Formulation 2 3

3 Retiming and Slack Budgeting Previous Works Problem Formulation Timing constraint and Low Power become significant requirement. Retiming: relocate flip-flops (FFs) Slack Budgeting: relax the timing constraints of components 6,0 b 6,0 c T=9 3,0 d 6,0 b 6,0 c T=9 3,3 d a e a e 3,0 3,0 v delay, slack 3,0 3,0 v delay, slack

4 Previous Works Introduction Previous Works Problem Formulation Retiming: [C.E.Leiserson et al. Algorithmica 1991]: first work [N. Maheshwari et al. TCAD 1998]: flow based Min-area retiming [H. Zhou, ASPDAC 2005]: incremental Min-period retiming [J. Wang & H. Zhou DAC 2008]: incremental Min-period retiming Slack Budgeting: [R. Nair et al. TCAD 1989]: ZSA, suboptimal heuristic [C. Chen et al. TCAD 2002]: Maximum-Independent-Set, NP-complete [S.Ghiasi et al. ICCAD 2004]: Flow based algorithm

5 Previous Works (Cont.) Previous Works Problem Formulation Retiming + Slack Budgeting: [Y. Hu et al. DAC 2006]: dual-vdd, MILP [S. Liu et al. ASPDAC 2010]: heuristic; MIS based In previous works: A few works consider simultaneous Retiming and Slack Budgeting MILP method or heuristic method In our works: Network-Based Algorithm Speedup

6 Problem Formulation Previous Works Problem Formulation Input: Directed graph G = (V, E, d, w) as synchronous sequential circuit. i V : combinational gate e ij E: signal passing from gate i to j d i : delay of gate i w ij : number of FF on edge e ij period constraint T power-slack tradeoff for each slack level Output: reallocation represented by r, so minimize power consumption under the period constraint

7 Introduction Condition for Φ(G) T a i d i + s i a i T r i r j w ij a j a i + d i + s i Suppose R i = r i + a i /T a i = T R i T r i. i V i V (i, j) E if r i r j = w ij min P( s i ) (II) i V s.t. R i r i s i i V (IIa) R i r i T i V (IIb) r j r i T w ij (i, j) E (IIc) 0 R i, r i N ff i V (IId) s i = { s 1 i,, s k i } i V (IIe) 0 s i T i V (IIf ) R j R i t ij (i, j) E (IIg) t ij s j T w ij (i, j) E (IIh)

8 (cont.) Solved by ILP Solver, but unacceptable runtime Need more effective method Without two constraints, convex cost dual network algorithm [R. K. Ahuja et al. 2003] Removes constraint (IIh), add penalty function P(tij): Generate new problem (III) min P( s i ) (II) i V s.t. R i r i s i i V (IIa) R i r i T i V (IIb) r j r i T w ij (i, j) E (IIc) 0 R i, r i N ff i V (IId) s i = { s 1 i,, s k i } i V (IIe) 0 s i T i V (IIf ) R j R i t ij (i, j) E (IIg) t ij s j T w ij (i, j) E (IIh)

9 (cont.) Solved by ILP Solver, but unacceptable runtime Need more effective method Without two constraints, convex cost dual network algorithm [R. K. Ahuja et al. 2003] Removes constraint (IIh), add penalty function P(tij): Generate new problem (III) min P( s i ) (II) i V s.t. R i r i s i i V (IIa) R i r i T i V (IIb) r j r i T w ij (i, j) E (IIc) 0 R i, r i N ff i V (IId) s i = { s 1 i,, s k i } i V (IIe) 0 s i T i V (IIf ) R j R i t ij (i, j) E (IIg) t ij s j T w ij (i, j) E (IIh)

10 (cont.) min i V P( s i ) + P(t ij ) (i,j) E s.t. (IIa) (IIg) t ij T w ij, (i, j) E (III) min P( s i ) (II) i V s.t. R i r i s i i V (IIa) R i r i T i V (IIb) r j r i T w ij (i, j) E (IIc) 0 R i, r i N ff i V (IId) Given solutions of (III), heuristic generate solution of (II): s j = min(t ij + T w ij, s j ), i FI(j) s i = { s 1 i,, s k i } i V (IIe) 0 s i T i V (IIf ) R j R i t ij (i, j) E (IIg) t ij s j T w ij (i, j) E (IIh)

11 Denote si where P( s i ) is minimum Define Q( s i ): Q( s i ) = { P( s i ) if s i s i P( s i ) if s i > s i Consider new problem (III ), which replaces (IIa) and (IIb) by R i r i = s i min Q( s i ) + P(t ij ) i V (i,j) E (III ) s.t. (IIc) (IIg) R i r i = s i i V t ij T w ij (i, j) E min P( s i ) + P(t ij ) i V (i,j) E s.t. (IIa) (IIg) t ij T w ij, (i, j) E (III) Theorem 1 For every optimal solution ( R, r, s) of problem (III), there is an optimal solution ( R, r, ŝ) of problem (III ), and the converse also holds. Theorem 2 The constraint (IIb) in problem (III) can be removed.

12 Denote si where P( s i ) is minimum Define Q( s i ): Q( s i ) = { P( s i ) if s i s i P( s i ) if s i > s i Consider new problem (III ), which replaces (IIa) and (IIb) by R i r i = s i min Q( s i ) + P(t ij ) i V (i,j) E (III ) s.t. (IIc) (IIg) R i r i = s i i V t ij T w ij (i, j) E min P( s i ) + P(t ij ) i V (i,j) E s.t. (IIa) (IIg) t ij T w ij, (i, j) E (III) Theorem 1 For every optimal solution ( R, r, s) of problem (III), there is an optimal solution ( R, r, ŝ) of problem (III ), and the converse also holds. Theorem 2 The constraint (IIb) in problem (III) can be removed.

13 Primal Network Flow Problem Solve problem (III) by Convex Cost Dual Flow a : tep 1: Transformation to Primal Network Flow Problem 1 Split vertex i into two vertex r i and R i 3 4 ( r i, R i ) Ē1, ( R i, R j ) Ē2, ( r i, r j ) Ē3 Further simplify problem as follow: 2 min (i,j) Ē P(s ij ) (IV ) r1 R1 r4 s.t. µ j µ i s ij (i, j) Ē (IVa) r3 R3 R4 0 µ i N ff i V (IVb) l ij s ij u ij (i, j) Ē (IVc) r2 R2 E1 E2 E3 a Refer to [R. K. Ahuja et al. 2003] for detail about Convex Cost Dual Flow.

14 Primal Network Flow Problem (cont.) Step 1: Transformation to Primal Network Flow Problem (cont.) min (i,j) Ē P(s ij ) (IV ) s.t. µ j µ i s ij (i, j) Ē (IVa) 0 µ i N ff i V (IVb) l ij s ij u ij (i, j) Ē (IVc) Remove constraints by P(s ij ) and B(µ i ) P(s ij ) = P(u ij ) + M(s ij u ij ) s ij > u ij P(s ij ) P(l ij ) M(s ij l ij ) 0 s i T s ij < l ij B(µ i ) = M (µ i N ff ) if µ i > N ff 0 if 0 µ i N ff M µ i if µ i < 0 Get Primal Network Flow Problem: min B(µ i ) (V ) (i,j) Ē P(s ij ) + i V s.t. µ j µ i s ij (i, j) Ē

15 Primal Network Flow Problem (cont.) Step 1: Transformation to Primal Network Flow Problem (cont.) min (i,j) Ē P(s ij ) (IV ) s.t. µ j µ i s ij (i, j) Ē (IVa) 0 µ i N ff i V (IVb) l ij s ij u ij (i, j) Ē (IVc) Remove constraints by P(s ij ) and B(µ i ) P(s ij ) = P(u ij ) + M(s ij u ij ) s ij > u ij P(s ij ) P(l ij ) M(s ij l ij ) 0 s i T s ij < l ij B(µ i ) = M (µ i N ff ) if µ i > N ff 0 if 0 µ i N ff M µ i if µ i < 0 Get Primal Network Flow Problem: min B(µ i ) (V ) (i,j) Ē P(s ij ) + i V s.t. µ j µ i s ij (i, j) Ē

16 Lagrangian Relaxation Step 2: Lagrangian Relaxation Lagrangian relaxation to eliminate constraints Lagrangian sub-problem: L( x) = P(s ij ) + B i (µ i ) i V e(i,j) Ē (µ j µ i s ij )x ij e(i,j) Ē Introduce start node v 0 Final Lagrangian subproblem: L( x) = min [P ij (s ij ) + x ij s ij ] (1) s.t. x ij e(i,j) E j:e(i,j) E j:e(j,i) E x ji = 0 i V x ij 0 (i, j) E 1 E 2 E 3

17 Convex Cost-scaling Approach Step 3: Convex Cost-scaling Approach Define function H ij (x ij ) = min sij {P ij (s ij ) + x ij s ij }: H ij (x ij ) is concave, so C ij (x ij ) = H ij (x ij ) is convex Final problem is a min-cost flow problem: L( x) = min C ij (x ij ) e(i,j) E s.t. x ij x ji = 0 j:e(i,j) E j:e(j,i) E i V (VI) 0 x ij M (i, j) E 1 E 2 E 3 M x ij M (i, j) E 4 For optimal flow x, construct residual network G(x ) In G(x ), solve shortest path distance d(i) Apply µ(i) = d(i) and s ij = µ(i) µ(j) Final solve the problem!!

18 Experiments Setup Implemented in C++ 3.0GHz CPU and 6GB Memory 19 cases from the ISCAS89 Case Name Gate # Edges # Max Output Max Inputs Tmin s27.test s208.1.test s298.test s382.test s386.test s344.test s349.test s444.test s526.test s526n.test s510.test s420.1.test s832.test s820.test s641.test s713.test s838.1.test s1238.test s1488.test

19 Results for Power Consumption and Total Slacks Benchmark T Power Consumption Total Slacks Optimal ILP [18] 1 ours Optimal ILP [18] ours s27.test s208.1.test s298.test s382.test s386.test s344.test s349.test s444.test s526.test s526n.test s510.test s420.1.test s832.test s820.test s641.test s713.test s838.1.test s1238.test s1488.test Avg Diff % +29% 1-31% -16% 1 S.Liu et al., Simultaneous slack budgeting and retiming for synchronous circuits optimization, ASPDAC 2010

20 Results for Runtime: Benchmark T Runtime(s) Optimal ILP [18] ours s27.test s208.1.test s298.test s382.test 44 > s386.test s344.test s349.test s444.test 46 > s526.test s526n.test s510.test 42 > s420.1.test s832.test s820.test s641.test s713.test s838.1.test s1238.test s1488.test 166 > Avg Diff

21 Thank You!