Highly Efficient Gradient Computation for Density-Constrained Analytical Placement Methods

Transcription

1 Hghly Effcent Gradent Computaton for Densty-Constraned Analytcal Placement Methods Jason Cong and Guoje Luo UCLA Computer Scence Department { cong, gluo cs.ucla.edu Ths wor s partally supported by NSF CCF and CCF

2 Outlne Related Related Wor, Motvaton & Contrbutons NLP NLP Formulaton and Gradent Computaton Nonlnear programmng formulaton Iteratve methods Densty and smoothng technques Effcent Gradent computaton Expermental Results Summary Summary and Future Wor UCLA VLSICAD LAB 2

3 Related Wor Top Top performng placers n ISPD 06 placement contest Kraftwer [Esenmann & Johannes DAC 98] mpl6 [Chan, Cong, et al., ISPD 06] NTUplace [Chen, Chang, et al. ICCAD 06] Iteratve, Iteratve, densty-drven drven overlap removal Guded by force, or by descend drecton of densty penalty UCLA VLSICAD LAB 3

4 An Example of Analytcal Placement Blue: standard cells Green: movable macros Yellow: fxed macros UCLA VLSICAD LAB 4

5 Motvaton & Contrbutons Common Common strateges n analytcal placers Densty smoothng technques Some smooth operators wthout closed-form formula nvolved Iteratve unconstraned optmzatons Frequent gradent computaton s necessary Our Our contrbutons Unfy a wde range of smoothng technques Derve a hghly effcent gradent computaton method Show good qualty results for mxed-sze placements UCLA VLSICAD LAB 5

6 Nonlnear Programmng Formulaton for Placement Wrelength-drven placement wth densty constrants NLP formulaton wth nequalty constrants mnmze: total wrelength subject to: cell_densty capacty NLP formulaton wth equalty constrants mnmze: total wrelength subject to: cell_densty + fller_densty = capacty UCLA VLSICAD LAB 6

7 Densty Functons (1/3) Orgnal Orgnal Densty UCLA VLSICAD LAB 7

8 Densty Functons (2/3) Bell-shaped Densty (Naylor, APlace, NTUplace) UCLA VLSICAD LAB 8

9 Densty Functons (3/3) Posson/Helmholtz-smoothed smoothed Densty (Kraftwer, mpl) UCLA VLSICAD LAB 9

10 Iteratve Methods Problem Problem formulaton mnmze: wrelength subject to: cell_densty = capacty Unconstraned problems Penalzed objectve functon OBJ μ = wrelength + μ Penalty Penalty functon e.g. squared devaton: Penalty = Iteratve Iteratve method loop { mnmze OBJ μ ; f (converge) brea; else ncrease μ; ; } UCLA VLSICAD LAB 10

11 Densty Penalty Functons (1/3) Bell-shaped densty penalty functon cell other cells Densty Functon (hgh dmensonal) Densty Penalty Functon (projected n x ) UCLA VLSICAD LAB 11

12 Densty Penalty Functons (2/3) Helmholtz-smoothed densty penalty functon cell other cells Densty Penalty Functon (projecton n x ) UCLA VLSICAD LAB 12

13 Densty Penalty Functons (3/3) cell other cells Bell-shaped Helmholtz-smoothed UCLA VLSICAD LAB 13

14 Densty Smoothng Technques Local smoothng Locally smooth the non-dfferentable edges Global smoothng Globally dstrbute the orgnal densty but mantan consstency We unfed these global smoothng technques An ntegral transformaton 1 1 ψ( uv, ) = Du (, v ) Guvu (,,, v ) dudv 0 0 Includng but not lmted to Posson smoothng (Kraftwer) Helmholtz smoothng (mpl) Gaussan smoothng (NTUplace) Keywords: Ellptc PDE, Green s s functon Our gradent computaton wors wth such smoothng technques UCLA VLSICAD LAB 14

15 Gradent Computaton (1/2) Naïve computaton by fnte dfference scheme cell other cells Δ x Penalty Functon Densty Functon Densty Functon Penalty Δx UCLA VLSICAD LAB 15

16 Gradent Computaton (2/2) Our Our effcent method cell other cells Twce Smoothed Densty Densty Penalty = w UCLA VLSICAD LAB 16

17 Comparson of the Gradent Computaton Methods Naïve computaton Penalty Functon Our Our effcent computaton Twce Smoothed Densty Penalty Δx Penalty = w Avod explorng the hgh dmensonal penalty functon Reduce the computatonal tme by a factor of n UCLA VLSICAD LAB 17

18 Comparson wth the mpl Implementaton mpl mpl mplementaton Smoothed Densty (wth operator H) Our Our effcent computaton Twce Smoothed Densty (wth operator H 1/2 ) Penalty Δx A Penalty = w Essentally the same for small cells wth proper scalng A Our method s drectly applcable to large macros UCLA VLSICAD LAB 18

19 Expermental Results on IBM-HB+ Benchmars In average, wrelength s 13% shorter than SCAMPI [Ng, Marov, et al.] And 15% shorter than mpl6 [Chan, Cong, et al.] NTUplace [Chen, Chang, et al.] as the detaled placer UCLA VLSICAD LAB 19

20 Summary & Future Wor We We proposed an effcent gradent computaton method Applcable to a wde range of global smoothng technques Reduce the runtme by a factor of n compared to a nave method Acheved good qualty mxed-sze placement results Future Future wor Apply the computaton to test other teratve methods Such as augmented Lagrangan method Extend the nonlnear programmng framewor to handle complex objectve and constrants such as power densty constrants and TS va constrants n 3D placement UCLA VLSICAD LAB 20

21 Than You! UCLA VLSICAD LAB 21

22 Bacup Slde An An equvalent expresson 1 1 r r ) ) ( r r ( x, y) (, ) (, )) Pxyμ (, ; ) ) D 0 0 a b r r ( x, y) ( u, v) 0 0 μ D u v C u v dudv ) ) = 2 (, ) (, ) = ( r r ( x, y) ) 1 1 r r ( x, y) μ D u v C u v dudv 0 0 y ( x, y ) 2 ( u, v) ( u ', v') G( u, v, u ', v') du ' dv' = Guvx (,, + w /2, v') dv' y - h /2 D ) D + h /2 y + h /2 y - h /2 r r Pxyμ (, ; ) ) ) = 2 μ D ( u', v') C( u', v') dv' ( y + h /2 ( ) ) r r y - h /2 ( x, y) Guvx (,, w /2, v') dv' u' = x w /2 u' = x w /2 r r Pxyμ (, ; ) = Lebnz Integral Rule ) ) r r ( ( x, y) ) 2 μ D ( u', v') C( u', v') v u UCLA VLSICAD LAB 22