Parametric Yield Maximization using Gate Sizing based on Efficient Statistical Power and Delay Gradient Computation

Transcription

1 Prmetrc Yeld Mxmzto usg Gte Szg sed o Effcet Sttstcl Power d Dely Grdet Comutto Kvrj Chor, Suml Shh, Ashsh Srvstv, Dvd Bluw, Des Sylvester Uversty of Mchg, EECS Dertmet, A Aror, MI 4809 {kvrj, suml, srvs, Bluw, des}@eecs.umch.edu Astrct Wth the cresed sgfcce of ge ower d erformce vrlty, the yeld of desg s ecomg costred oth y ower d erformce lmts, therey sgfctly comlctg crcut otmzto. I ths er, we roose ew otmzto method for yeld otmzto uder smulteous ge ower d erformce lmts. The otmzto roch uses ovel ge ower d erformce lyss tht s sttstcl ture d cosders the correlto etwee ge ower d erformce to ele ccurte comutto of crcut yeld uder ower d dely lmts. We the roose ew heurstc roch to cremetlly comute the grdet of yeld wth resect to gte szes the crcut wth hgh effcecy d ccurcy. We the show how ths grdet formto c e effectvely used y o-ler otmzer to erform yeld otmzto. We cosder oth ter-de d tr-de vrtos wth correlted d rdom comoets. The roosed roch s mlemeted d tested d we demostrte u to 40% yeld mrovemet comred to determstclly otmzed crcut.. Itroducto d Overvew of Aroch Cotued rocess sclg hs resulted lrge crese rocess vrlty tht leds to lrge fluctutos rocess rmeters from ther oml vlues. These vrtos hve grow due rt to ggressve lthogrhc techques tht re used to tter dmesos smller th the wvelegth of lght. I ddto, smller devce dmesos d smller umer of dog toms crese the fluece of heome such s le edge roughess d rdom dot effects. These rocess vrtos trslte to wde rges erformce metrcs of curret desgs. I rtculr, ge curret whch s extremely sestve to umer of key rocess rmeters hs show huge fluctutos wth [] showg 0X vrto ge ower for 30% vrto erformce cross 000 smles of desg mufctured 80 m techology. I ddto, s the cotruto of ge ower hs grow the fluctuto ower dssto s ow domted y ge ower. Ths results egtve correlto etwee ower dssto d dely of desg. Thus hgh erformce smles of desg re lso exected to hve hgher ower dssto d vce-vers. Ths leds to two-sded costrt o the fesle rego of rmetrc yeld defed y dely d ower lmts [], d cuses sgfct yeld loss uder vrtos rocess rmeters. A umer of lyss techques to cosder the mct of vrlty o tmg [3-7] d ower [8-9] hve ee develoed. However, these roches do ot estmte the true rmetrc yeld of desg cosderg oth ower d erformce correlto. Referece [0] ws the frst to cosder the mct of vrlty o crcut otmzto. The uthors descred the formto of tmg wll due to determstc ower otmzto whch creses the suscetlty of the desg to rocess vrtos, d roosed heurstc roch to revet the uld-u of lrge umer of ths er the crtcl dely of the crcut. Ths ws cheved y ddg elty fucto whch hd egtve mct o the ojectve fucto vlue wheever th hd dely whch ws er crtcl. However, the roch ws determstc ture d dd ot use y sttstcl formto durg otmzto. Recetly, severl sttstcl tmg [-3] d ower [4-6] otmzto roches hve lso ee roosed. However, ll these roches eglect the correlto etwee ower d erformce. Therefore, tmg yeld mrovemets evtly result degrdto ower yeld d vce vers. Moreover, most of these roches suffer from lrge comuttol comlexty d rutmes. Thus, there s crtcl eed to develo roches tht erform true d effcet rmetrc yeld otmzto, where yeld s defed usg oth ower d tmg costrts. Recetly, [7] roosed roch to erform gte-level yeld lyss comuttolly effcet mer whle cosderg the correlto etwee ower d erformce. The roch ws sed o rcl-comoet sed rocess vrto modelg techque to erform tmg d ower lyss usg the sme set of uderlyg rdom vrles (RVs), llowg the correlto ower d erformce to e ctured. Addtolly the roch cosders ll comoets of rocess vrtos: ter-de d tr-de (stlly correlted d rdom) vrlty d c therefore serve s frmework to ele true rmetrc yeld otmzto. I ths work, we roose ovel roch to erform yeld otmzto usg gte szg. The yeld otmzto s formulted s ucostred otmzto rolem, where the ojectve s to mxmze the rmetrc yeld of desg. The otmzto s erformed usg grdet-sed o-ler otmzer. A rute-force grdet comutto roch sed o tertve yeld lyss, however, leds to lrge comuttol overheds. Therefore, we roose effcet heurstc techque to erform the comutto of the yeld grdet. Ths s cheved y erturg the sze of gte the crcut d heurstclly reclcultg the dely d ower rolty dstruto fuctos (dfs) of the ertured crcut. The tmg df of the ertured crcut s clculted sed o ovel cutset roch tht lyzes oly suset of the odes the crcut to estmte the comlete dely df. The ower df of the ertured crcut s clculted wth cremetl ower lyss. Ths volves sutrctg the ower dssto of the ertured gte from the df of totl ower for the comlete crcut d the ddg the ower dssto of the ertured gte whle ccoutg for the gte sze chge. These ertured dfs re the used to comute the yeld of the ertured crcut y tegrtg the ertured vrte Guss dstruto over the rego defed y the ge ower d tmg costrt. Ths grdet comutto techque s the tegrted wth LANCELOT [8], lrge-scle o-ler otmzer, to mrove the rmetrc yeld of the desg, d s foud to rovde 8X mrovemet rutme wth verge error of 0.%. The remder of the er s orgzed s follows. Secto refly revews the rcl comoet sed roch to erform yeld lyss. Secto 3 resets the cremetl tmg d ower lyss techques, whch re the used to comute the grdet of yeld. I Secto 4 we rovde detls regrdg the mlemetto of our yeld otmzto X/05/$ IEEE. 00

2 6 JPDF vlue 4 Fgure. Exmle rttos of crcut usg grd to model the correlted comoet of vrtos roch d reset results cludg comrso of our roch to determstc otmzto. We rovde coclusos Secto 5.. Yeld lyss I ths secto we refly dscuss our modelg ssumtos d yeld lyss roch. We lso defe the yeld otmzto rolem d descre rute-force roch to erform yeld otmzto. The comuttol comlexty of ths rute-force roch motvtes the eed for more effcet grdet comutto techques. Our yeld lyss frmework s sed o the roch [7], d we exress the dely d ge of gte s Dely = d om = Lekge = ex V α om ( P ) = β ( P ) where d om d ex(v om ) re the oml gte dely d ge, resectvely, d α d β ctures the deedece of gte dely d the log of gte ge o the rocess rmeters of terest. The RVs P the ove equto refers to the vrtos these rocess rmeters tht re ssumed to e Guss. The vrto rocess rmeter s the rttoed to correlted d rdom comoet. The correlted comoet s hdled y rttog the desg s show Fgure. For ech rmeter, ech squre the grd s ssged to Guss RV whch ctures the correlted vrto tht rocess rmeter, whch s defed usg correlto mtrx. Ths gves totl of N g RVs for ech rmeter, whch re ssumed to hve jot multorml dstruto. Usg rcl comoets lyss [9] the correlted comoet s exressed s ler comto of N g deedet Guss RVs (z ), d the rdom vrto ll the rocess rmeters s lumed to sgle RV - η for dely d d η for ge, l d the coeffcet of the rdom comoet s clculted y mtchg the vrce of the rdom cotruto. Ths roch gves us cocl exressos for gte dely d ge ower whch re exressed s: Dely = d om α Lekge = ex Vom = = γ z η d R β z R γ ηl = = Tmg lyss s the erformed the srt of [3][7], d the dely s rogted through the crcut, whle mtg the ode delys the sme cocl form wth dfferet coeffcets. The sum oerto s erformed y smly ddg the coeffcets for ech of the RVs, other th the rdom comoet whose coeffcet s oted s the squre root of the sum of the squred coeffcets of the rdom comoets () () Dely(s) log(lekge) Fgure. Jot rolty dstruto fucto for the vrte Guss dstruto for c3540. of the summed dfs. The mx oerto s erformed y mtchg the me, vrce d the correlto of the mx of two RV (whch re oted usg [0]) d the cocl exresso of the mx. Lekge ower lyss s sed o summg logorml RVs usg Wlkso s method [] s roosed [8]. The ge of ech gte s tertvely dded to the sum whch s mted cocl form. I ech ddto the coeffcets of the cocl exresso for the sum re clculted y mtchg the me, vrce d correlto wth the rcl comoets of the sum (oted usg [8]) d the cocl exresso for the sum. At the ed of tmg d ower lyss (tht rovdes the dely d ge ower cocl form) the correlto etwee dely d ge ower s estmted usg: Cov( Dely, Lekge) = α β (3) These fve rmeters (me d vrce of dely d ge ower d ther correlto) re used to defe vrte Guss dstruto for dely d log of ge ower s show Fgure. The rmetrc yeld whch s defed s: = ( D D P ) Y = P 0, P 0 (4) where D s dely of the crcut costred to e less th D 0 d P s the ower of the desg costred to e less th P 0. I ths work, we ssume tht vrtos ower re domted y vrtos ge ower d the dymc ower dssto s ssumed to e fxed umer d sutrcted out of the totl ower udget of the desg to defe the ge ower costrt. Bsed o ths ssumto, we c rewrte (4) s ( D D log P log( P P )) Y = P (5) 0, L 0 D where P L d P D re the ge d dymc ower of the desg. The ove yeld exresso s ow equvlet to the tegrl of vrte Guss RV over rectgulr rego, d c e evluted usg exresso develoed []. Both the tmg d ower comutto stes requre O(N g ) stes, where s the umer of gtes the desg d N g s the umer of regos to whch the desg s rttoed to cture the correlto structure of correlted rocess vrtos. However, the fl yeld comutto ste (5) tself rus costt tme sce the yeld comutto s lwys erformed usg the set of fve rmeters, deedet of the sze of the rolem. Bsed o ths yeld lyss ege, rute-force roch to erform yeld otmzto usg gte szg c e develoed. Ths volves comutg the grdet of yeld to the sze of ech gte, whch c e estmted y reszg ech gte d 0

3 erformg yeld lyss d settg the gte ck to ts orgl sze. After comutg the grdet, we use lrge scle o-ler otmzer to mrove the yeld of the crcut. We ow cosder the comuttol comlexty of oe terto of ths roch. Ech grdet comutto requres yeld lyss rus d thus hs overll comlexty of O( N g ). Note tht the sce the sze of the rttos s fxed, the umer of rttos N g c lso e exected to crese wth the sze of the desg. Thus, the overll comuttol requremets soo ecome utele for lrge desgs. Also, ote tht the rute-force roch seds most of the tme reclcultg the sme formto for most of the crcut d motvtes the eed for effcet grdet comutto roch. 3. Grdet Comutto I ths secto we wll dscuss our ew grdet comutto roch tht clcultes the udted tmg d dely dfs sed o chge gte sze. Both the tmg d ower erturto lyss techques udte the coeffcets of the dely d ge df exresso sed o the chge gte sze. These udted dely d ge ower dfs re the used to comute the yeld of the ertured desg. The chge yeld s used to estmte the grdet of yeld to the sze of ech gte the desg. 3. Tmg Perturto Comutto We wll exl our tmg erturto comutto roch sed o cutsets usg the followg grh reresetto for our crcuts. Defto : A tmg grh s drected cyclc grh hvg exctly oe source d oe sk: G={N,E,s,f}, where N={,,.., k } s set of odes, E={e,e,,el} s set of edges, s N s the source ode d f N s the sk ode d ech edge s ordered r of odes e=(, j ) d ech ode s ssocted wth dely for ech f edge, whch deeds o the chrcterstcs of the fout odes. The odes the tmg grh corresod to gtes d the edges corresod to ets crcut. A rolstc tmg grh s defed s tmg grh where ech ode s ssocted wth RV for the dely for ech f edge. Fgure 3 shows exmle tmg grh wth te odes, eght of whch rereset ctul gtes d odes d 0 rereset the source d sk odes, resectvely. The ltest rrvl tme (AT) d requred rrvl tme (RAT) rolty dstruto fuctos (df) for ech ode the tmg grh re ow defed s: Defto : The ltest rrvl tme (AT) t edge e the rolstc tmg grh s RV whose CDF A e (t) gves the rolty tht determstc smle of ths tmg grh hs rrvl tme less th t. Defto 3: The erlest requred rrvl tme (RAT) t edge e the rolstc tmg grh s RV whose CDF R e (t) gves the rolty tht the determstc smle meets the tmg costrt T crt f the determstc rrvl tme t the ode s less th t. Note tht the sum of the AT d RAT t ode reresets the rtl df of dely sce t does ot tke to ccout the fluece of the edges tht re ot reset ether the f or the fout coe of the ode o the df of crcut dely. To exress the deedece of crcut dely o the dely of oe of the odes let us defe the followg terms. Defto 4: A ler toologcl orderg (LTO) of the odes tmg grh s totl order sed o the reltosh tht the order of y ode x tht les the fout coe of ode s strctly lrger th the order of ode, d tht o two odes tmg grh hve the sme order. A LTO of tmg grh c e esly determed y erformg redth-frst trversl of the tmg grh. Though gve tmg grh c hve my LTOs, fdg the otml LTO s ot the focus of ths er. Fgure 3 llustrtes tmg grh wth odes leled ccordg to LTO of the tmg grh. Note tht swg odes 8 d 9 wll stll mt vld LTO of the odes. Defto 5: A cutset of tmg grh wth gve LTO of ode s defed to e the set of edges (, j ) of the tmg grh whch stsfy LTO( ) LTO() d LTO( j )>LTO(). Defto 6: A ode x of the tmg grh elogs to the cutset-source of ode f there exsts edge (x,*) whch elogs to the cutset of ode. Defto 7: The f-set of ode of tmg grh s the set of mmedte redecessor odes of ode. Defto 8: The rrvl tme set or ATSet of ode s the uo of the f-set of ode d the odes the fout coe of the f-set of ode tht hve order less th or equl to the order of ode. Defto 9: The covoluto-set or CovSet of ode s the tersecto of the ATSet d cutset-source of ode. Ay cutset of the tmg grh dvdes the tmg grh to two dscoected comoets d the sttstcl mxmum of the sum of the AT d RAT of ll edges the cutset gves the comlete df of crcut dely. Now, f we ertur the dely chrcterstcs of ode (y gte szg) we lso chge the cctve lodg of the f gtes, ffectg ther dely chrcterstcs s well. To comute the ew crcut dely we ote tht the RAT of the edges the cutset does ot chge, sce ll the gtes ther fout coe cludes gtes tht hve order strctly greter th the order of ode, d hve uchged dely chrcterstcs. Fgure 3. A tmg grh showg ler toologcl orderg for the odes d cutsets for odes 8 d Fgure 4. A tmg grh showg the ATSet (odes wth the shded ellse) d cutset-source set (odes wth the dshed she) for ode 8 Ths eglects the mct of the chge the sloe of the crcut. However, the roch for tmg lyss usg ckwrd rogto [4] c e used to cosder ther mct wth the sme frmework

4 However, the AT for ll edges tht re the fout coe of the f-set of ode chges. However we re oly terested AT chges for edges tht re drve y odes tht hve order less th the order of gte, sce we eed to comute the AT for the edges the cutset oly. Ths s exctly the set of odes defed y the ATSet of ode. If the AT of edge the cutset chges we eed to recomute the covoluto of the AT d RAT t tht edge. These edges re drve y the odes the tersecto of the ATSet d the cutset whch s defed s the CovSet of ode. Let us revst the exmle tmg grh Fgure 3 d cosder ode 8. The cutset for ths ode s the set of edges (6,9), (8,0) d (5,0) s show y the dshed le. The ATSet for the ode c e detfed s the set of odes 6, 7 d 8 s show Fgure 4. The tersecto of the cutset-source d ATSet defes the CovSet d s the set of odes 6 d 8. Note tht the CovSet detfes tht the AT d RAT hs ot chged o the edge (5,0) d we do ot eed to recomute the covoluto of the AT d RAT for ths edge. However, f we cosder ode 9 the cutset s defed y the edges from odes 5, 8 d 9 to ode 0, s show s the dotted le Fgure 3. The seudo-code to clculte the dely df of the ertured crcut s show elow, where we refer to the edge y the me of the drvg ode. The seudo-code volves the comutto of the AT for ll odes the ATSet, covoluto of the AT d RAT for ll odes the CovSet d the sttstcl mxmum of the covoluto for ll edges the cutset. Note tht ll the comuttos CutSetSst re erformed usg the sme cocl exresso for the dely df. Thus, the fl dely df of the ertured crcut s lso exressed the sme form. Although, the roch s descred seems exct, t s heurstc. Ths results from the fct tht the comutto of the mx fucto of dely dfs s ot exct d forwrd d ckwrd trversls of the grh result tmg delys tht re ot exctly sme. However, ths error s very smll s wll e show lter the results secto. 3. Power Perturto Comutto The sttstcl ower comutto s erformed y summg the ower dssto of ech gte crcut to comute the comlete df of ge ower. To erform ower lyss of crcut wth erturtos the sze of gte, we frst erform sttstcl ower lyss of the uertured crcut s descred Secto. Now the ge ower fter the sze of gte hs ee ertured s exressed s P ert crc = P = P uert crc uert crc\ P P uert gte, ert gte, P ert gte, where P ert d P uert refer to the ertured d uertured ower, resectvely d the suscrt dctes whether the ower refers to the crcut or to the gte. Sce the ge ower s exressed logorml (exoetl of Guss) RV, we c roxmte ther sum usg other logorml. I geerl, f we sum P d c P to ot P, whch s CUTSETSSTA () for ech ode (x ATSET ()) Comute AT(x); for ech ode (x CONVSET ()) CT(x) covoluto (AT(x), RT(x)) for ech edge(x CUTSET ()) T mxmum (T, CT(x)) retur T Fgure 5. Pseudo-code for the comutto of erturtos tmg df of crcut (6) mthemtclly exressed s, P = = ex 0 ex 0 z (7) = = z ex c0 = c z c = P P c the coeffcets the exresso for P c e oted y mtchg the me, vrce d the correlto coeffcet wth the exoetl of the rcl comoets (z s). Ths gves us set of equtos vrles whch c e lytclly solved to ot the followg exresso for the coeffcets ssocted wth the rcl comoets [7] = log E z E( Pe ) z ( P ) E( e ) = log z c z E( P ) ( ) e E Pe c z ( ( ) ( ) ( ) E P E P E e Usg the exressos develoed [3], the remg two coeffcets the exresso for P c e exressed s = 0 log 4 ( E( P ) ( ) E P ( E( P ) ( ) ( ) ( ) ( ) E P Vr P Vr Pc Cov P Pc 0.5 ( ) ( ) ( ) Vr P Vr Pc Cov P Pc ( E( P ) E( P ) = = log (8) (9) (0) Note tht to comute the exresso (6) we eed to use the ove exressos twce to clculte the fl ertured ge. However, whe oe of the logormls s sutrcted the sgs ssocted wth ts exected vlue d covrce terms the ove exressos re reversed. 3.3 Yeld Grdet To ths ot we hve develoed effcet roches to erform sttstcl tmg d ower erturto comutto. Now, we wll use these techques to erform the comutto of the grdet of yeld effcet mer. The comutto of yeld grdet volves the comutto of the erturto yeld for smll chges the sze of gtes the desg. The seudo-code for the comutto of yeld s cluded Fgure 6. After ech reszg move, the o-ler FASTYIELDGRADIENT (CIRCUIT, SIZE ) for ech gte (g CIRCUIT) udte lod c d sze of (g) usg SIZE; for ech gte (g CIRCUIT) comute ew gte dely & ower of (g) T FORWARDSSTA (CIRCUIT) P STATPOWERANALYSIS (CIRCUIT) Y YIELD (P, T) do REVERSESSTA (CIRCUIT) for ech gte(g CIRCUIT) sve curret stte of CIRCUIT s SIZE(g) SIZE(g) comute ew gte dely & ower of g for ech gte( FANIN(g)) udte lod c d dely of T CUTSETSSTA (CIRCUIT) P INCREMSPA (CIRCUIT, P) Y YIELD (P, T ) Y(g) (Y -Y)/ SIZE(g) restore the orgl stte of the CIRCUIT retur Y Fgure 6. Pseudo-code for the comutto of the yeld grdet 03

5 otmzer clls the yeld comutto route FstYeldGrdet. The frst ste s to tlze the crcut so tht ll odes re ssged the correct lod cctce sed o the szes of the gtes ts mmedte fout, d the correct ge ower sed o ts ow sze. Bsed o the lod cctce of the ode ech ut of ode s ssged to dely df, whch reresets the dely of the tmg rc from tht rtculr ut to the outut of the gte. After the tlzto ste, the ext ste volves the rogto of the AT from the source ode to the sk ode the tmg grh. Ths s rereseted s ForwrdSSTA the seudo-code. The ext ste s to erform sttstcl ower lyss d geerte the ge curret df usg SttPowerAlyss. The Yeld fucto s the used to comute the yeld sed o the tmg d ge ower dfs gve ge ower costrt P d dely costrt D, s outled Secto. To erform the comutto of yeld grdet, we frst rogte the RAT from the sk ode to the source ode usg ReverseSSTA. The we go through ech ode the crcut tertvely d ertur the sze of ech gte y smll mout. The lod cctce of the odes the f-set of the ode d the dely df ssged to ech tmg rc of ths ode d the odes the f-set re udted. The usg the sttstcl tmg d ower erturto comutto techques dscussed Sectos we comute the dely d ge ower dfs of ths ertured crcut. The yeld corresodg to the ertured crcut s the clculted d the chge yeld s used to defe the rtculr comoet of the yeld grdet. Let us cosder the comuttol comlexty of our roosed roch d comre t to the rute-force roch, where ech terto hd comlexty of O( N g ). Ech terto, our roosed roch, volves sgle ru of the comlete yeld lyss roch, s dscussed ove, whch hs comlexty of O(N g ). The tmg d ower erturto comutto s reeted O() tmes. The comlexty of the cremetl ower lyss s O(N g ) sce we requre two sum oertos of the logorml RVs. For the sttstcl tmg erturto comutto most of the mx comuttos the cutset c e reused y storg the formto s he. Thus, the tmg erturto comutto hs comlexty of O(N g log()). Thus the overll comlexty of the roch s O(N g log()), whch s lrge mrovemet comred to the rute-force roch. 4. Results d Imlemetto Detls We mlemeted the roosed roch C d comred our yeld mrovemets to determstc crcut otmzto techque. Our roosed roch for the comutto of the yeld grdet s wrtte s suroute whch the otmzer uses to clculte the grdet of the ojectve fucto. The yeld lyss ege serves s the suroute to clculte the ojectve fucto tself. Followg, we reset the ccurcy d rutme results for the roosed roch. If we ssume tht reverse d forwrd SSTA gve exct tmg dstrutos t ech ode, the the rocedure would e exct s well. However, s oted efore, due to the Guss roxmto cosdered whle comutg the mxmum troduces smll ccurcy whle erformg forwrd d reverse SSTA. Now, sce ths ccurcy s fucto of crcut toology d recovergece structure the sestvty of yeld comuted usg oly FORWARDSSTA sed rute-force s eglgle. Tle shows the rutme comrso d ccurcy results of the roosed grdet comutto rocedure s comred to the ïve rute force roch. The crcut sze terms of the umer of gtes d the verge cut-wdth over ll odes the crcut s lso reorted the secod d the thrd colums, resectvely. Rutme er grdet vector comutto Tle.Comrso of Yeld grdet comutto usg FASTYIELDGRADIENT d rute-force roch. Bech. Averge Brute- Crcut #gtes Cut-Sze Force Grdet Seed-u (%) (%) c E-03 c E-03 c E-0 c E-0 c E-03 c E-03 c E-03 c E-03 c E E E E E E E E-03 Fst Mx. Error Avg. Error usg the rute roch d the roosed rocedure re gve Colums 4 d 5, resectvely. The seed u of the roosed method over the rute-force roch s gve Colum 6, d rges etwee 3X to 0X d s foud to e lrger for gger crcuts. The mxmum error, over ll gtes, foud usg grdet comutto ormlzed wth resect to the rute-force method s gve Colum 7, d s foud to e smll most cses wth verge of.4%. The error verged error over ll gtes the crcut s gve the lst colums of Tle d s foud to e extremely smll. 4. Yeld Otmzto The gtes our stdrd cell lrry re chrcterzed for set of szes the rge from mmum sze to mxmum sze d the dely d ge ower for termedte gte szes s oted usg ler terolto. All desgs re the determstclly otmzed for ower uder dely costrts usg ether desg comler or LANCELOT [8]. We use our sttstcl yeld mxmzto roch to mrove the yeld of ths otmzed desg for set of dfferet ower d tmg costrts. Our results dcte tht erformg sttstcl otmzto c sgfctly mrove the tmg yeld of the desg. We comre our results sed o the ISCAS85 [3] echmrks whch were sytheszed 30 m techology. The yeld otmzto results re gve Tle. The frst su-secto cludg colums, 3, 4 d 5 reort the tl tmg d ower sttstcs resultg from determstclly otmzed crcut. The determstc otmzto ws erformed usg oml dely d ower models. We reset yeld otmzto results for two sets of costrts. The frst set cludes yeld otmzto for ggressve oml vlue costrts. As determstc otmzer s uwre of the vrto ower d tmg d ther correlto, the tl yeld t oml costrts s extremely smll. However, the roosed vrlty wre yeld otmzto sgfctly mroves the yeld. For exmle, the yeld for echmrk crcuts c43, c499 d c880 drmtclly mroves from close to 0% to u to 40%. Colum 6, 7, 8 d 9 reort the ost otmzto tmg d ower sttstcs of the crcut. The tl yeld suject to oml vlue costrts d the yeld fter erformg otmzto re gve Colums 0 d, resectvely. The secod set of results reort the erformce of yeld otmzto uder essmstc costrts. For ths cse we use the oml vlue offset y oe stdrd devto s the costrt for oth ower d tmg whle defg the ojectve fucto for otmzto. Ag colums, 3, 4 d 5 lst the ost otmzto sttstcs of the crcut wheres colums 6 d 7 reort the results cheved fter erformg the roosed yeld otmzto. As the costrts re relxed ths cse the tl yeld of the crcut mroves d for the sme reso the mrovemets cheved re reltvely smller s comred to the revous cse. The mxmum mrovemet ths cse s foud to e greter th 0% for the echmrk crcut c

6 Bech. Crcuts Tle : Yeld Otmzto results for dfferet ower d tmg costrts Itl soluto D < Dm, P < Pm D < Dm Ds, P < Pm Ps Dely(s) Power (µw) Dely(s) Power (µw) Yeld(%) Dely(s) Power (µw) Yeld(%) Dm Ds Pm Ps Dm Ds Pm Ps It. Ot. Dm Ds Pm Ps It. Ot. c ~ c ~ c ~ c c c c ~ c c ~ Coclusos To the est of our kowledge, we hve reseted the frst roch to erform gte-level rmetrc yeld otmzto cosderg costrts o ower d erformce, log wth ther correlto. The roch for yeld comutto s show to e comuttolly effcet d s show to rovde 8X mrovemet rutme, s comred to rute-force grdet comutto roch. The yeld grdet s used to gude lrge-scle o ler otmzer to mrove the yeld of desg tht hs ee otmzed determstclly, uder vryg ower d dely costrts. The results show tht we c cheve mrovemets yeld whch re s lrge s 40%. Ackowledgemets Ths work ws suorted rt y NSF, SRC d MARCO/DARPA. Refereces [] T. Krk, S. Borkr, d V. De, Su-90 m techologes chlleges d oortutes for CAD, ACM/IEEE ICCAD, , 00. [] R. Ro et l., Prmetrc yeld lyss d costred-sed suly voltge otmzto, ACM/IEEE ISQED, , 005. [3] H. Chg d S. S. Stekr, Sttstcl tmg lyss cosderg stl correltos usg sgle PERT-lke trversl, ACM/IEEE ICCAD,. 6-65, 003. [4] C. Vsweswerh et l., Frst-order cremetl lock-sed sttstcl tmg lyss, ACM/IEEE DAC, , 004. [5] A. Agrwl et l., Sttstcl tmg lyss usg ouds d selectve eumerto, IEEE Trs. o CAD, , , Set [6] A. Devg d C. Kshy, Block-Bsed sttstcl tmg lyss wth ucertty, ACM/IEEE ICCAD, , 003. [7] M. Orshsky d A. Bdodhyy, Fst sttstcl tmg lyss hdlg rtrry dely correltos, ACM/IEEE DAC, , 004. [8] R.R. Ro, et l., Sttstcl lyss of suthreshold ge curret for VLSI crcuts, IEEE Trs. VLSI Systems,.3-39, Fe [9] S. Nredr et l., Full-ch su-threshold ge ower redcto model for su-0.8µm CMOS, ACM/IEEE ISLPED,. 9-3, 00. [0] X. B, et l., Ucertty wre crcut otmzto, ACM/IEEE DAC,.58-63, 00. [] S. Rj, S. Vrudhul, d J. Wg, A methodology to mrove tmg yeld the resece of rocess vrtos, ACM/IEEE DAC, , 004. [] S. Cho, B. Pul d K. Roy, Novel szg lgorthm for yeld mrovemet uder rocess vrto ometer techology, IEEE/ACM DAC, , 004. [3] A. Agrwl et l., Sttstcl tmg sed otmzto usg gte szg, ACM/IEEE DATE, , 005. [4] A. Srvstv, D. Sylvester, d D. Bluw, Sttstcl otmzto of ge ower cosderg rocess vrtos usg dul-vth d szg, ACM/IEEE DAC, , 004. [5] A. Dvood, V. Khdelwl, d A. Srvstv, Vrlty sred mlemetto selecto rolem, ACM/IEEE ICCAD, , 004. [6] M. M d M. Orshsky, A ew sttstcl lgorthm for gte szg, ACM/IEEE ICCD,. 7-77, 004. [7] A. Srvstv et l., Accurte d effcet gte level rmetrc yeld estmto cosderg correlted vrtos ge ower d erformce, ACM/IEEE DAC, , [8] A. R. Co, N. I. M. Gould, d Ph. L. Tot. LANCELOT: A Fortr ckge for lrge-scle o-ler otmzer (Relese A). Srger, Verlg, 99. [9] D. F. Morrso, Multvrte sttstcl methods, McGrw-Hll Book Comy, 967. [0] C. Clrk, The gretest of fte set of rdom vrles, Oertos Reserch, vol. 9,. 85-9, 96. [] S.C. Schwrtz d Y.S. Yeh, O the dstruto fucto d momets of ower sums wth logorml comoets, Bell Systems Techcl Jourl, vol.6,.44-46, Se. 98. [] J. H. Cdwell, The vrte orml tegrl, Bometrk,. 3-35, Dec. 95. [3] F. Brglez d H. Fujwr, A eutrl etlst of 0 comtol echmrk crcuts d trget trsltor Fortr, Proc. ISCAS, , My 989. [4] D. Lee, V. Zolotov d D. Bluw, Sttc tmg lyss usg ckwrd rogto, ACM/IEEE DAC, ,