Optimal Slack-Driven Block Shaping Algorithm in Fixed-Outline Floorplanning

Transcription

1 Optmal Slack-Drven Block Sapng Algortm n Fxed-Outlne Floorplannng Jackey Z. Yan Placement Tec. Group Cadence Desgn Systems San Jose, CA 95134, U.S.A. Crs Cu Department of ECE Iowa State Unversty Ames, IA 50010, U.S.A. 1

2 A Quotaton Sometmes te questons are complcated and te answers are smple. --- Dr. Seuss ( ) 2

3 Block Sapng n Fxed-Outlne Floorplan Input n Blocks Area A for block mn Wdt bounds W and W for block mn Hegt bounds and for block I H Constrant graps Fxed-outlne regon G H and G v Output ( x, y ) Block coordnates, wdt All blocks nsde fxed-outlne regon All blocks wtout overlaps w and egt 3

4 Floorplan Representaton Slcng Floorplan Excellent n sapng soft blocks (by sape curve) Can only represent slcng floorplans Non-Slcng Floorplan Can represent non-slcng floorplans (by sequence par) Elegant representatons, effcent manpulaton Muc more complcated n block sapng 4

5 Prevous Work T.C.Wang et al. Optmal floorplan area optmzaton. TCAD 1992 P.Pan et al. Area mnmzaton for floorplans. TCAD 1995 S.Nakatake et al. Module placement on BSG-structure and IC layout applcatons. ICCAD 1996 T.S.Mo et al. Globally optmal floorplannng for a layout problem. TCSI 1996 M.Kang et al. General floorplannng wt L-saped, T-saped and soft blocks based on bounded slcng grd structure. ASP-DAC 1997 H.Murata et al. Sequence-par based placement metod for ard/soft/pre-placed modules. ISPD 1998 F.Y.Young et al. Handlng soft modules n general non-slcng floorplan usng Lagrangan relaxaton. TCAD 2001 S.N.Adya et al. Fxed-outlne floorplannng: Enablng erarccal desgn. TVLSI 2003 C.Ln et al. A revst to floorplan optmzaton by Lagrangan relaxaton. ICCAD

6 SDS Overvew Specfcally formulated for fxed-outlne floorplannng Optmal, effcent and scalable for non-slcng floorplan Man contrbutons Basc Slack-Drven Sapng Tree Optmalty Condtons Slack-Drven Sapng (SDS) Promsng Expermental Results Obtan optmal solutons for bot MCNC & HB bencmarks smply by te basc SDS. For MCNC bencmarks, 253x faster tan Young s, 33x faster tan Ln s, to produce results of smlar qualty. 6

7 Problem Formulaton W Mnmze te layout egt wt a fxed layout wdt upper bound G, G (two dummy vertces 0 & n 1) v 7

8 Noton of Slack n Floorplannng LL( x 0) RL( x W ) 0 n+1 TL( y ) yn 1 y G, Gv Sape of n blocks orzontal slack s (0, ) x x BL( y 0) vertcal slack v s (0, ) y 8

9 Horzontal/Vertcal Crtcal Pat Horzontal Crtcal Pat (HCP) Vertcal Crtcal Pat (VCP) s0 0 0 s s s5 0 s 2 3 v s3 0 v 2 2 v s5 0 5 TL( y ) yn 1 s1 0 s4 2 v s v s4 0 Lengt of VCP = Layout egt yn 1 0 v s0 0 BL( y 0) 9

10 Basc Slack-Drven Sapng Soft blocks are saped teratvely. At eac teraton, apply two operatons: VCP HCP Globally dstrbute te total amount of slack to te ndvdual soft block. Algortm stops wen tere s no dentfed soft block to sape. Layout egt s monotoncally reducng, and layout wdt s bouncng, but always wtn te upper bound. 10

11 Target Soft Blocks I:{ s ard} v II :{ s soft} { s 0, s 0} v III :{ s soft} { s 0, s 0} v IV :{ s soft} { s 0, s 0} { w W } v V:{ s soft} { s 0, s 0} { w W } v v VI :{ s soft} { s 0, s 0} { H } VII :{ s soft} { s 0, s 0} { H }

12 Sapng Sceme IV VI w ' ' w ' w v ' P Basc SDS WHILE (tere s target soft block) IV, VI, s p v v v s p ( W w) W k p k pp s.t. 1, p G s.t. v 1, p G MAX( ( w )) k v 12

13 Dynamc Programmng Approac G n P ( P ) 0 n 0 out P n 1 ( ) 0 MAX( ( W )) k wk pp k V p n out V out P n n ( P ) MAX( ( P )) ( ) n j W w jv out out P P out j W w jv ( P ) ( n ) ( out P P ) ( W w) ( ) MAX( ( )) ( ) = ( P ) 0 n+1 Topologcal sortng Scan (source to snk) Scan (snk to source) Traverse eac block 13

14 A Non-Optmal Case 4 14

15 Optmal Condtons L: a sapng soluton generated by te basc SDS. Hard crtcal pat: all blocks on ts crtcal pat are ard blocks. 1. If tere exsts one ard VCP n L, ten L s optmal. 2. If tere exsts at most one non-ard HCP n L, ten L s optmal. 3. If tere exsts at most one non-ard VCP n L, ten L s optmal. LL( x 0) RL( x W ) TL( y ) yn 1 L BL( y 0) 15

16 Flow of Slack-Drven Sapng Basc slack-drven sapng Any optmalty condton satsfed? Yes No One step of geometrc programmng No Optmal? Yes 16

17 Expermental Results All experments are run on Lnux server wt AMD Opteron 2.59 GHz CPU and 16 GB RAM. Two sets of floorplan bencmarks MCNC: 9~49 soft blocks HB: 500~2000 mxed of ard and soft blocks Aspect rato bound of soft block s [1/3, 3]. Input constrant graps are provded by a floorplanner. In all experments, SDS aceves te optmal solutons smply by te basc SDS. 17

18 Experments on MCNC Bencmarks Compare SDS wt Young s and Ln s algortms Young s s mnmzng te layout area. Ln s s mnmzng te layout alf parameter. SDS s mnmzng te layout egt wt wdt upper bound. Procedure of Experments Conduct two groups of experments: 1) SDS v.s.young s; 2) SDS v.s. Ln s. In eac group, run eac sapng algortm 1000 tmes. Run Young s and Ln s frst, ten use ter resultng wdt as te nput upper-bound wdt of SDS. Compare te fnal result based on Young s and Ln s objectve. 18

19 Compared wt Young s on MCNC (* total sapng tme of 1000 tmes and does not nclude I/O tme) Crcut. #. Soft Blocks Young s [1] ws (%) Sapng Tme*(s) SDS Sapng ws (%) Tme*(s) SDS stops earler ws (%) Sapng Tme*(s) apte xerox p am33a am49a Norm [1] F.Y.Young, C.C.N.Cu, W.S.Luk and Y.C.Wong, Handlng soft modules n general non-slcng floorplan usng Lagrangan relaxaton. TCAD

20 Compared wt Ln s on MCNC (* total sapng tme of 1000 tmes and does not nclude I/O tme) Crcut. #. Soft Blocks Ln s [1] SDS SDS stops earler Half Para. Sapng Tme*(s) Half Para. Sapng Tme*(s) Half Para. Sapng Tme*(s) apte xerox p am33a am49a Norm [1] C.Ln, H.Zou and C.Cu, A revst to floorplan optmzaton by Lagrangan relaxaton. ICCAD

21 Comparson on Runtme Complexty 21

22 Experments on HB Bencmarks Large-scale floorplan desgns (500~2000 blocks) Young s and Ln s algortms cannot andle HB bencmarks Hglgts on SDS s results Average convergence tme: 1.18 second Average total number of teratons: 1901 Average after 5.9%, 9.6%, 22.3% and 47.3%, layout egt s wtn 10%, 5%, 1% and 0.1% dfference from te optmal soluton. 22

23 Convergence Grap (1) [665 soft blks] 23

24 Convergence Grap (2) [1200 soft blks] 24

25 Conclusons SDS effcent, scalable and optmal slack-drven sapng algortm n fxed-outlne floorplannng. Basc Slack-Drven Sapng Optmalty Condtons Slack-Drven Sapng (SDS) Promsng Expermental Results Obtan optmal solutons for bot MCNC & HB bencmarks smply by te basc SDS. For MCNC bencmarks, 253x faster tan Young s, 33x faster tan Ln s, to produce results of smlar qualty. 25

26 Future Work Embed SDS nto a floorplanner. Use te dualty gap as a better stoppng crteron. Propose a more scalable algortm to replace te geometrc programmng metod. Extend SDS to andle non-fxed outlne floorplannng. Appled on buffer/wre szng for tmng optmzaton. 26

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28 p Sapng Sceme : ncrease on w for subset IV LL( x 0) RL( x W ) P p s MAX p( s ) p p s ( p p 1, 0) p s, pp p p s p p W 0 W kp k w ( w ) k ( W w) s W k p k pp MAX( ( w )) k 28

29 Dynamc Programmng Approac P n P out MAX( ( W )) k wk pp kp : pats from source to : pats from to snk ( P ) 0 V n ( P ) n 0 Source out ( P n 1 ) 0 n n ( P ) MAX( ( P )) ( ) n j W w jv out out P P out j W w jv n out P P P W w ( ) MAX( ( )) ( ) ( ) ( ) ( ) ( ) = out V Snk 29

30 Optmal Condtons L: a sapng soluton generated by te basc SDS. In L, te only remanng soft blocks tat can be saped to possbly mprove L are te ones at te ntersecton of HCP&VCP. Hard crtcal pat: all blocks on ts crtcal pat are ard blocks. LEMMA 2. If tere exsts one ard VCP n L, ten L s optmal. LEMMA 3. If tere exsts at most one non-ard HCP or at most one non-ard VCP n L, ten L s optmal. 30

31 Sapng Sceme for blocks n IV w ' LL( x 0) RL( x W ) p w ' w p p p p s MAX p( s ) s s s ( p p 1, 0) p p p W w W kp k ( w ) k p P p s, pp p ( W w) s W k p k pp MAX( ( w )) k 31