Timing-Driven Placement. Outline

Transcription

1 Tmng-Drven Placement DAC 97 Tutoral 1997 Blaauw, Cong, Tsay Outlne Background + Net-Based Aroach Zero-Slack Algorthm Modfed Zero-Slack Algorthm Path-Based Aroach Analytcal Aroach Fall 99, Prof. Le He 1

2 Cannot Converge!? errors EE Tmes, age u 0.5 u 0.8 u 1.0 u teratons Statc Tmng Analyss Net Delay Arrval Tme/ Requred Arrval Tme 60/96 0/ / 48/7 96/96 1. levelze. Forward swee 3. Backward swee 0/0 / 60/60 84/96 Fall 99, Prof. Le He

3 Early/Late Mode Tmng Volaton FF FF n n+1 n n+1 OK Late Early + Outlne Background Net-Based Aroach Zero-Slack Algorthm Modfed Zero-Slack Algorthm Path-Based Aroach Analytcal Aroach Fall 99, Prof. Le He 3

4 Net Constraned Greedy Aroach mn + [( l x ) + ( x u ) + ] Lower bound Wre length Uer bound 1. Smulated annealng. Parwse nterchange ( x) + x, f x > 0 = 0, otherwse Net Weghtng Aroach [Tsay, DAC 91] mn{ L = 1 c l t d, Under lumed RC delay model t = rc l + ( rc + R c) l Equvalent to mn{ L = 1 c l l, L D, n N} + ξ u, n N} a c a Fall 99, Prof. Le He 4

5 Lagrangan Multler Technque Solve by alyng necessary and suffcent Kuhn- Tucker condtons The Hessan matrx s ostve sem-defnte 1 ( c + ) l λ, The gradent equals zero 1 ( c + λ ) l = 0, The lagrange multler l s non-negatve for any actve constrant and s equal to zero, otherwse l, and λ 0 l, and λ = 0 = u < u Lagrange multler net weghtng ncrease Exact Net Weghtng Wthout erformance constrant Ax Wth erformance constrant ( x x * 1 = b where A B11, ) +, b B x λ kl k < l n β kl ( x k x ) = ( x l * x 1 where β ˆ ˆ ˆ ˆ, [ ˆ kl = a k al a k + a l A = a ] Solve for Lagrange multler δ z δ kl : = ( x : = λ = * kl k< l n x ) ( x x ) ( x β k kl * x ) z kl l * ) Fall 99, Prof. Le He 5

6 Lagrangan Relaxaton Assume that weghtng ncrease of a artcular net has mnor effect on other nets Let l aroaches u f net n s n actve set λ ( x x ) + λ β = 1/ a ( l * u / u ) + 1/ a ( x (1+ λ β x ) = ( x ) l c = l / a * a * x * ) + Outlne Background Net-Based Aroach Zero-Slack Algorthm Modfed Zero-Slack Algorthm Path-Based Aroach Analytcal Aroach A Practcal Aroach Dscusson and Examle Fall 99, Prof. Le He 6

7 Zero-Slack Algorthm [Hauge, ICCAD87] Choose the least non-zero slack node and the ath segment Dstrbute slack on each edge of ath segment Udate slack ath segment: all edges and nodes on the segment have same slack Reeat untl every node has zero slack K Guarantee to meet ath constrant f all net constrants are met. K But t may be too conservatve ZSA Examle (1) Net Delay Arrval Tme/ Requred Arrval Tme 60/96 0/ / 48/7 96/96 0/0 / 60/60 84/96 Fall 99, Prof. Le He 7

8 ZSA Examle () Edge Slack Edge Slack/ delay / ZSA Examle (3) Edge Slack/ delay /6 0/ / Edge Slack/ delay /9 0/9 0/6 0/ / Fall 99, Prof. Le He 8

9 ZSA Examle (4) Edge Slack/ delay 1 0/18 0/9 0/9 0/6 0/ / Edge Slack/ delay 0/1 0/18 0/9 0/9 0/6 0/ / Outlne Background Net-Based Aroach Zero-Slack Algorthm + Modfed Zero-Slack Algorthm Path-Based Aroach Analytcal Aroach Fall 99, Prof. Le He 9

10 Luk s Modfed ZSA [Luk DAC 91] Sort delay ath segments n ncreasng slack For each ath segment, dstrbute slack to each edge udate delay of the node on ath segment from source and snk Slack Udate ubefore slack allocaton: t a ( + ) S = S t a 1 t rn 1 (n-1) n uafter slack allocaton t d d a d ( n 1) n trn trn Arrval tme mrovement t t r a = t = t r a + d + d Requred arrval tme mrovement Fall 99, Prof. Le He 10

11 LMZSA Examle (1) Edge Slack Path segments LMZSA Examle () Sortng Slack dstrbuton 0/0 /6 0/0 0/0 0/0 /6 Fall 99, Prof. Le He 11

12 LMZSA Examle (3) Slack dstrbuton Slack dstrbuton /6 /6 /6 /6 / / /18 LMZSA Examle (4) Slack dstrbuton Slack dstrbuton /6 /6 /6 /6 / / /18 /18 /9 15/9 /9 /9 15/9 /9 15/9 15/9 /1 Fall 99, Prof. Le He

13 Outlne Background Net-Based Aroach Zero-Slack Algorthm Modfed Zero-Slack Algorthm + Path-Based Aroach Analytcal Aroach Path-Based Aroach - TDP SA lacement on arttons MSPT for net delay [Donath DAC 90] Evaluate all affected aths after each move Good artton? Fall 99, Prof. Le He 13

14 Comarson Net Constrants low runtme overhead (P&R s net-based) lnear comlexty O(#nets) no resource sharng no known good constrant generaton algorthm Path Constrants better resource sharng among nets on same ath hgh runtme overhead need exonental number of aths + Outlne Background Net-Based Aroach Zero-Slack Algorthm Modfed Zero-Slack Algorthm Path-Based Aroach Analytcal Aroach Fall 99, Prof. Le He 14

15 s. t. Tmng Quadratc Placement Formulaton mn L = constrant t d a 1 E N t t a a, t a c x, e s y, (( x x ) + ( y y ) ) T, T, = R C + d, ( ( E W ) + C ( N S )) h W S x, y, v n c e ath end onts s ath startng onts 0 [Jackson ICCAD 90] Boundng box Prmal Actve Set Method 1 T T L = w Qw + b w At teraton k, a feasble soluton w (k) (cell locaton, net boundng-box, tmng varables) s known, A (k) s the set of actve constrants Gradent vector g (k) at w (k) ( k ) ( k ) ( k ) g = L( w ) = Qw + b Solve for a correcton vector δ 1 ( k ) T ( k ) ( k ) T ( k) mn{ δ Qδ + δ g } ( k ) subect to A Fall 99, Prof. Le He 15

16 Prmal Actve Set Method (Cont d) Solve by Kuhn-Tucker condtons Q ( k A ) A 0 ( δ ( λ ) g = ) 0 ( k ) T k ( k) k If δ (k) s feasble w.r.t. constrants not n A (k), set w (k+1) = w (k) + δ (k) Otherwse, lne-search for a ste length arameter α (k) such that w (k+1) = w (k) + α (k) δ (k) s feasble. If α (k) < 1, add new constrants If λ q s negatve, remove constrant q from actve set If λ (k) 0, and δ (k) = 0, otmal soluton r mn L s. t. ( x ) ( r f x) Actve Constrant Method mn L( x r ) Check constrant volatons r mn L s. t. ( x) r f ( x) = 0 Fall 99, Prof. Le He 16