A Framework for Block-Based Timing Sensitivity Analysis

Transcription

1 39.3 Framework for Block-Based Tmg Sestvty alyss Sajay V. Kumar Chadramoul V. Kashyap Sach S. Sapatekar Uversty of Mesota Itel Corporato Uversty of Mesota Meapols MN Hllsboro OR 973 Meapols MN BSTRCT Sce process ad evrometal varatos ca o loger be gored hgh-performace mcroprocessor desgs, t s ecessary to develop techques for computg the sestvtes of the tmg slacks to parameter varatos. Ths addtoal slack formato eables desgers to exame paths that have large sestvtes to varous parameters: such paths are ot robust, eve though they may have large omal slacks ad may hece be gored tradtoal tmg aalyss. We preset a framework for block-based tmg aalyss, where the parameters are specfed as rages rather tha statstcal dstrbutos whch are hard to kow practce. We show that our approach whch scales well wth the umber of processors s accurate at all values of the parameters wth the specfed bouds, ad ot just at the worstcase corer. Ths allows the desgers to quatfy the robustess of the desg at ay desg pot. We valdate our approach o crcut blocks extracted from a commercal 45m mcroprocessor. Categores ad Subject Descrptors B.8. Performace alyss ad Desg ds Geeral Terms lgorthms, Performace, Desg Keywords Varatos, rrval tmes, Slacks, Prug, Reorderg. Itroducto Mcroprocessors are desged uder the omal or typcal codtos where the process parameters, such as chael legth (L e ), threshold voltage (V t ) etc, whch affect the trasstor drve stregths, ad evrometal parameters, such as supply voltage (V dd ), are assumed to be at fxed values. Ulke SICs, whch are desged uder worst case assumptos, mcroprocessor desgers have reled upo at-speed testg of maufactured parts to grade parts by frequecy, wth hgher frequecy parts sellg at hgher prces. However, due to creasg levels of parameter varatos, as well as aggressve desg styles strvg for the best performace at the lowest power, desgg at the omal pot causes surprses slco []. Ofte paths wth large postve slacks tur out to be speed lmtg slco. Therefore, from a desger s perspectve, orderg paths by omal slack, as s customary, does ot provde a complete prortzato of paths to work o. For example, a represetatve slack dstrbuto of paths a moder mcroprocessor s show Fg (a). Due to power performace tradeoffs, a steep tmg wall s created, where a large umber of paths have the same slack. Whe the frst slco arrves, t may so happe that the drve stregth of certa kds of devces for Ths work was supported part by the SRC uder cotract 007-TJ-57. Permsso to make dgtal or hard copes of all or part of ths work for persoal or classroom use s grated wthout fee provded that copes are ot made or dstrbuted for proft or commercal advatage ad that copes bear ths otce ad the full ctato o the frst page. To copy otherwse, or republsh, to post o servers or to redstrbute to lsts, requres pror specfc permsso ad/or a fee. DC 08, Jue 8-3, 008, ahem, C, 008. Copyrght 008 CM /08/008 $5.00. example, low power devces turs out to be lower tha was assumed durg desg. s a result, paths that are more susceptble to varatos ths devce type are lkely to show up as speed lmtg slco, ecesstatg costly desg re-sps. I the above example, however, f addto to omal slacks the slack sestvty of paths to the drve stregth of the low power devces were avalable, t would have bee possble to fx paths that are very sestve to the drve curret varatos of ths partcular devce before tape-out. Here, by slack sestvty we mea the chage slack for a gve chage a parameter. For example, the slack sestvty dstrbuto of the same set of paths, whe the drve curret of all low-power devces s weaker by 0% s show Fg (b). Note that the steep tmg wall ow has a fte slope whe vewed from a sestvty perspectve paths that appeared equvalet terms of slack appear dfferet terms of sestvtes. Thus, by takg to accout the omal slack ad the sestvty of the slack to parameters cojucto wth the amout of varatos the parameters ofte specfed as a rage, rather tha a statstcal dstrbuto a more effectve prortzato of paths to work o ca be provded to the desger. case for sestvty aalyss was also made [4]. However, the paper does ot provde ay algorthmc detals of how sestvtes ca be propagated a block-based maer a o-statstcal settg. % of Paths 00% 80% 60% 40% 0% 0% Normalzed Slack 0% Fg : cdf of top 000 paths of a block showg: (a) omal slack (b) slack sestvty whe I d of all low power devces s 0% weaker I ths paper, we propose a block-based algorthmc framework for solvg the followg problem: Gve a set of parameters ad ther rages (bouds), compute accurate arrval tmes (slacks) for all settgs of the parameters wth the specfed rages a sgle tmg ru. Ths allows us to compute the arrval tme (slack) sestvty at ay gve pot wth the rage of varatos by smply queryg for tmg formato the eghborhood of the pot. For stace, referrg to Fg (b), ths framework allows us to compute the chage slack slack sestvty for ay varato drve curret wth the 0% boud. It s geerally accepted that block-based techques have certa advatages over path-based methods fast ru tmes, cremetalty ad tmg aware optmzato, etc., ad s the method of choce dustral tmg aalyzers. We dstgush our approach from block-based SST [], [6] whch assumes that the dstrbutos ad ther correlatos are kow a pror. Ths s usually ot the case, but the bouds or rages of parameters are easer to obta. Further, certa parameters such as V dd ad Mller Couplg Factors (MCF) are ot statstcal ature, ad are therefore aturally descrbed terms of rages. Slack values are ormalzed wth respect to the FO4 delay of a verter, throughout the paper. % of Paths 00% 80% 60% 40% 0% Normalzed Delta Slack 688

2 Recetly, a block-based statc tmg algorthm that also works wth parameter rages was preseted [3]. The prmary goal of that work was to preserve the accuracy at oly the worst-case corer. To acheve ths goal, the arrval tme sestvtes to the parameters were adjusted durg the output arrval tme computato. s a result, the arrval tmes at o-worst case settgs were ot accurate (we show ths Table 7 of Secto 4). Our goal ths work s dfferet: we wsh to compute accurate tmg at all pots, ot just the worst case pot. Ths eables us to compute the sestvty of a tmg parameter by evaluatg the tmg at two dfferet pots the parameter space ad computg the chage the tmg quatty. Ths paper s orgazed four sectos as follows. I the ext secto we establsh some otato ad backgroud for our method, wth commets o a couple of papers relevat to our work. I Secto 3, we troduce the varous algorthms for propagatg the arrval tmes through the crcut. We preset expermetal results Secto 4 based o a 45m commercal mcroprocessor desg.. Prelmares The tmg graphs for a verter ad a two-put NND gate are show Fg. The label o the edge s the delay of the putoutput trasto of the gate. We also assocate the oto of a arrval tme at every ode the graph. For a sgle put gate, such as a verter, as show Fg (a), the output arrval tme s gve by: = + d () where s the put arrval tme ad d s delay of the arc betwee ad. For a two put NND gate show Fg (b), the output arrval tme 3 s gve by: 3 = max( + d3, + d 3 ) () whch geeralzes a obvous way for gates wth more tha two puts. tmg path s a sequece of odes such that a delay arc exsts betwee two successve odes the sequece. The frst ode the sequece s the put ode of the path ad the last ode the sequece s the output ode of the path. Fg : Tmg graph of (a) verter, ad (b) NND gate Suppose the delay of a gate depeds o parameters. ssumg a frst order varato model, the delay ca be wrtte as: d = d + = a X where a are the delay sestvtes ad X wth =,...,. We refer to (3) as the delay hyperplae. We assume that the physcal parameters such as L, V dd, etc. have bee trasformed to the abstract parameters X by meas of the affe trasformato as descrbed [3],[5]. We use X to deote the set of pots the hypercube defed by X. Sce the arrval tme at the output ode of a tmg path s smply the summato of delays alog the path, we express the arrval tme at the output ode, as the arrval tme hyperplae: = + 3 = d b X d 3 d 3 3 (a) (b) (3) (4) where s the omal arrval tme. Thus, the arrval tme at the output ode of a path has a smple represetato that fathfully captures the sestvty of the path to the parameters, gve by the b terms (4). However, such a smple lear represetato of arrval tmes s ot helpful for computg sestvtes whe may paths coverge o a ode. Cosder the scearo depcted Fg 3(a) whch shows four dfferet paths wth dfferet arrval tmes (deoted as,, 3, ad 4 respectvely), ad dfferet sestvtes to some parameter X. If these four paths coverge at the same ode, the arrval tme at that ode s gve by the maxmum of the arrval tme of the four paths (geeralzed from ()) ad ulke (4), the arrval tme s a olear fucto of the parameters. The maxmum arrval tme at the omal value of X s gve by path (hyperplae ). However, f the parameter chages by -0.5, path s the domat path whereas f the parameter chages by +0.7, path 3 s domat. Therefore, we eed a represetato of a arrval tme that s fathful to the fact that dfferet paths domate for dfferet settgs of the parameters, ad as a result gves the correct arrval tme for ay settg of the parameters. Oe represetato of the arrval tmes the presece of a max fucto s to use a boudg hyperplae as show by the dotted le Fg 3(b) ( oe dmeso t s a le) [3]. The authors propose a algorthm that provdes a tght upper boud o the worst case arrval tme value at the ode. The advatage of ths method s that the represetato remas lear ad caocal, ad that t esures that the worst case delay of the crcut s a upper boud o the true delay of the crcut. However, as metoed the troducto, mcroprocessor desg we are terested the sestvtes aroud a desg pot rather tha the worst case delay. s the fgure shows, the boudg hyperplae s sgfcatly accurate at o-worst case settgs of the parameters, partcularly aroud the omal value. Further, o formato as to whch path s domat ad uder what codtos s provded. ddtoally, sce the method artfcally rases the delay hyperplaes durg the MX computato, the sestvty of the delay wth respect to the parameters s ot preserved. However, f we relax the requremet that we propagate oly a sgle arrval tme hyperplae, the a pecewse-plaar represetato of the MX fucto s possble (see Fg 3(c)). We allow a set of hyperplaes such that each ca be a maxmum hyperplae for some settg of the parameters wth the allowed rages. Itutvely, each hyperplae represets a path up to that ode the tmg graph X X - X (a) (b) (c) Fg 3(a) MX arrval tme (T) of 4 paths usg: (b) a boudg hyperplae usg [3], ad (c) a exact pece-wse plaar represetato More formally, for a set of m arrval tme hyperplaes gve by: j = j + bjx, j =,, m = We say that a hyperplae j s pruable f ad oly f: m max = max (,, j, j,, m ) (6) + = 4 T 3 For example, Fg 3(c), 4 ca be prued because t does ot cotrbute to the MX fucto (show by the dotted le). set of hyperplaes s called rreducble f o hyperplae the set s 4 T 3 4 (5) T 3 689

3 pruable. pplyg ths defto to the four paths Fg 3(a), we see that the three hyperplaes {,, 3 } form a rreducble set ad the MX s show by the dotted le Fg 3(c) whch forms a pece-wse lear represetato (pece-wse plaar hgher dmesos). Whle ths rases the possblty of expoetal blowup the umber of hyperplaes uder the worst case, sce every path (there are expoetal umber of paths the worst case) could become crtcal at some settg of the parameters, we show Secto 4 that ths s ot the case o practcal dustral desgs. I the ext secto, we also descrbe a algorthm that trades accuracy for ru-tme ad avods the potetally expoetal ru-tmes. We meto passg that recetly [5], a brach ad boud method was proposed to compute the exact worst case path delay usg the method of [3] to provde the bouds eeded to prue the search space. Whle the method could be adapted to compute the crcut delay at ay settgs of the parameters, t volves searchg through the path space ay tme the parameter settg chages. Further, the ru tme depeds o the qualty of the upper bouds. s show Fg 3(b), the upper boud ca be very loose at oworst case settgs of the parameters, partcularly at the omal corer. Fally, ths method does ot explctly provde us wth a techque for determg uder what codtos a partcular path could become the most crtcal, somethg that s useful for the desgers to kow. 3. Propagato of rrval Tmes I ths secto, we descrbe techques to propagate ad prue the arrval tmes o the tmg graph. The basc operatos of statc tmg are the SUM ad the MX. Cosderg the verter Fg (a), the SUM operato s defed as: = { j + d j } (7) where s the set of put hyperplaes ad s the set of output hyperplaes. I the rest of ths secto, we focus our atteto o the more complex MX operato. Gve a set of arrval tmes at the puts of a gate, the arrval tme at the output of the gate s the uo of the sets of arrval tmes at the puts. We refer to the set of hyperplaes at the output ode as U= {,..,m}. We eed to perform a prug operato o U to determe U (deally would be rreducble). Such a prug operato s ecessary order to esure that the umber of hyperplaes o every ode does ot crease expoetally as we perform a forward propagato alog the tmg graph. 3. Parwse Prug lgorthm We frst descrbe a smple algorthm that compares the arrval tmes a parwse maer. Gve two hyperplaes ad, we wrte f 0 for all values of X X. That s (usg (4)): ( ) + ( b b ) X 0 (8) = Sce the parameters have all bee ormalzed to le betwee - ad +, (8) s always true f: ( ) ( b b ) (9) = Note that gve two hyperplaes ad, we have oe of:. whch case we prue,. whch case we prue, The algorthms descrbed the paper ca be adapted for the MIN operato a straghtforward maer. 3. ether or ad we keep them both. Gve a set of hyperplaes U={,.., m } the prug algorthm s outled Fg 4. The PIRWISE algorthm begs wth the set U, ad tally assumes that all the hyperplaes U are opruable. hyperplae U s compared agast all other opruable hyperplaes U ad f t s ot pruable, t s added to the set. The overall ru-tme for ths operato s O (m ). PIRWISE(I:U;Out:) //U={,, m} ={}; Mark all hyperplaes o-pruable; for =:m f ( s marked prued) cotue; for j=:m f ( j s marked prued) cotue; f( j ) mark j prued; ed //for j=:m ed //for =:m dd all o-pruable hyperplaes to Fg 4: PIRWISE prug algorthm 3. Necessary ad Suffcet Codto for Prug The PIRWISE algorthm does ot guaratee that the set s rreducble. Ths ca be llustrated by Fg 3 where the PIRWISE algorthm does ot mark 4 as o-pruable sce (9) does ot hold for ay hyperplae that s compared wth 4. Therefore, the codto descrbed (9) to prue a hyperplae s suffcet but ot ecessary. I order to determe f a hyperplae U s pruable or ot, t must be smultaeously compared wth all other o-pruable hyperplaes U. Thus, to determe f a hyperplae j U ca be prued, the followg codto must be satsfed: j k k =,..., m, k j (0) X, =,, has o feasble soluto (0) s a ecessary ad suffcet codto to determe f a hyperplae ca be prued. The FESCHK algorthm that performs prug based o (0) s show Fg 5. Sce feasblty checkg s doe by all LP solvers, we use the commercal optmzato package CPLEX [6] whch performs feasblty check effcetly. We ote that the algorthm s heretly parallelzable sce the feasblty check for each hyperplae ca be performed parallel, f a mult-processor mache were avalable. lso, ulke the method [5], FESCHK ca be easly adapted to fd a pot X where a gve hyperplae (path) s o-pruable (crtcal). FESCHK (I:U;Out:) //U={,, m} ={}; Mark all hyperplaes o-pruable; for j=:m f ( j s marked prued) cotue; formulate (0), check for feasblty; // oly clude k ot marked prued f(soluto to (0) s feasble) = U j; else mark j as prued ed //for j=:m Fg 5: FESCHK prug algorthm Whle the feasblty checkg ca be doe tme polyomal the sze of (0), the theoretcally expoetal umber of hyperplaes that are possble at a ode makes ths expoetal. 690

4 However, as we show the ext secto, FESCHK s practce effcet o realstc crcuts. I order to optmze the ru-tme spet determg the set of hyperplaes to propagate, FESCHK algorthm ca be appled selectvely. If the umber of hyperplaes o the ode exceeds a certa user-specfed threshold, we apply FESCHK; else the PIRWISE algorthm s used to prue the hyperplaes. Ths mples that some redudat hyperplaes that ca be prued are carred forward, utl the threshold s reached. Ths algorthm s deoted as PIRWISE_FESCHK_THRESH. 3.3 Explorg Ru-tme ccuracy Trade-offs Whle the algorthms descrbed the prevous subsectos are exact, the ru-tme depeds o the ature of the logc coe ad the umber of parameters cosdered. I the worst-case, the ru-tmes could be expoetal sce expoetal umber of hyperplaes may be carred. We ow descrbe two methods whch trade-off accuracy for rutme Soft-Prug Strategy s explaed the prevous secto, the PIRWISE prug strategy ca ofte be effcet, leadg to a expoetal blow-up of the umber of o-pruable hyperplaes, f most of them caot be prued usg (8). Istead, f we relax (8) as follows, the addtoal prug ca be acheved: ( ) + ( b b ) ε, ε < 0 () = Itutvely, ths mples that we mark as pruable eve f t ca exceed by a small amout. I order to accout for the fact that prug may lead to a accurate tmg estmate at some settg of the parameters, we rase hyperplae by creasg ts omal arrval tme, by the mmum amout requred for (8) to be satsfed. Ths ot oly allows us to prue but may also allow several other plaes to be prued by the rased hyperplae of, thereby cosderably decreasg the umber of hyperplaes that eed to be propagated. There s a tradeoff betwee the umber of hyperplaes prued ad the pessmsm the actual arrval tme umbers due to rasg some of the hyperplaes, based o the value of. However, practce, a small value of, (-0.5% of based o our expermets) provdes a cosderable speedup wthout sgfcatly overestmatg the arrval tmes. I our mplemetato, we use ths dea of a soft threshold for parwse-prug our hyperplaes as a preprocessg stage to reduce the cardalty of U, before applyg FESCHK. Each hyperplae s allowed to be rased at most oce durg PIRWISE prug f () s true but (8) s stll false. Ths algorthm s referred to as PIRWISE_SOFT_PRUNE_FESCHK ad the results are show Secto Shrkg Hypercube Method We ow descrbe a method that allows accuracy to be traded-off for ru-tme by lmtg the umber of hyperplaes carred. Ths dea s explaed Fg. 6 where four hyperplaes a rreducble set are show. X s [-, ] as before. However, f X s restrcted to le [-0.5, 0.5], ad 4 are pruable as show (b), whle at the omal value of X (terval sze s zero), 3, ad 4 are all pruable. Thus, by reducg the sze of the rage of parameters, fewer hyperplaes ca be propagated. Hyperplaes that are ot pruable outsde the rage are replaced wth a boudg hyperplae usg the method of [3]. Whle ths method s pessmstc outsde the reduced rage of X, t s faster sce fewer hyperplaes are propagated. Further, by preservg accuracy wth the reduced rage whch s cetered o the omal pot, the arrval tmes aroud the omal are stll calculated accurately. 3 4 Fg 6: (a) Four o-pruable hyperplaes (,, 3, 4) (b) Shruk hypercube for the four plaes (a), (gve by -0.5 X 0.5) such that oly ad 3 are o-pruable More formally, for every ode we have a trple cosstg of the o-pruable hyperplaes, the hypercube where the hyperplaes are o-pruable, ad a boudg hyperplae whch s a upper boud of all the hyperplaes at that ode. Cosder a m-put gate: t the th put we have the trple: <, X a, B > where s a rreducble set, X a s the set of pots the reduced hypercube gve by a X j a ( 0 a, j=,,) ad B s the boudg hyperplae. To compute the trple at the output, we start wth the tal trple <U, X 0, U b > where U = m, X 0 s the smallest hypercube from the puts, ad s gve by X0 = X a Xam, ad U b = { B,, B m }. We prue U such that the umber of o-pruable hyperplaes s less tha the user specfed threshold. We do ths by teratvely shrkg X 0 f ecessary (by some delta), as show the algorthm SHRINK_HYPERCUBE Fg 7. We also compute the boudg hyperplae B o U b, usg [3]. SHRINK_HYPERCUBE(I:U,X 0,U b,n;out:,x,b)) //U={,, m}, U b={b,,b m} //N = maxmum umber of o-pruable hyperplaes allowed //X 0 = Ital hypercube pply FESCHK algorthm wth bouds o X from X 0, to obta the rreducble set of. whle (sze() > N) Shrk hypercube X 0 by delta pply FESCHK wth ew bouds o X to obta the ew rreducble set of. ed Compute boudg hyperplae B o U b Fg 7: SHRINK_HYPERCUBE algorthm The SHRINK_HYPERCUBE method s equvalet to FESCHK whe X s [-,], ad reduces to the method [3] f N =. Ths algorthm ca be exteded to shrk each dmeso by dfferet amouts ad to also use bary search the whle loop Fg Results (a) I ths secto, we preset the smulato results obtaed o a 45m based commercal mcroprocessor desg. Global varatos four dfferet parameters types, amely supply voltage (V dd ), Mller Couplg Factor (MCF), chael legth of NMOS trasstors (L ), ad chael legth of PMOS trasstors (L p ), are cosdered. L ad L p are each dvded to two dfferet types, based o whether the devce s omal or low power, ad further to three types based o layout depedet formato. Each of the dvdual L parameters s assumed to vary depedetly of each other, thereby resultg 4 dfferet parameters ( for L, MCF, ad V dd ). The rages of these parameters are show Table. (b) 69

5 Table : Rage of varatos for parameters Parameter (total of 4) V dd 3 Rage of Varatos 0 to -8% L ad L p ( dfferet types) ±0% MCF ±33% The bouds o these parameters are provded as a put to the tmg ege. We foud that the delay s lear the parameter varatos wth these rages. The lbrary characterzato flow has bee ehaced to compute the delay sestvtes o all tmg arcs wth respect to each of the above parameters as a fucto of put slopes ad output loads. The prug algorthms descrbed Secto 3 are appled o four dfferet desg blocks. Table presets formato about the bechmark crcuts. The tmg ege s mplemeted C++, wth a terface to CPLEX [7], to perform FESCHK prug. The arrval tmes are computed for RISE ad FLL trastos at the MX ad MIN modes as s typcal a statc tmg tool. Table : Bechmark formato Block Block Block3 Block4 # of regsters # of odes # of tmg arcs Ru-tme Comparsos The ru-tmes for performg a forward propagato o the tmg graph computg the set of rreducble arrval tme hyperplaes o every ode are show Table 3. The ru-tme umbers are relatve to the tmg ru where a sgle parameterzed hyperplae (the hyperplae wth the largest (smallest) arrval tme at the omal pot for MX (MIN) aalyss) s propagated. The results dcate sgfcat dfferece betwee the ru-tmes o Block versus the other blocks. s show Fg 8, ths s because there are a large umber of recoverget paths Block ad cosequetly a larger fracto of odes that cota 00 or more hyperplaes. It s also terestg to see that, for Block, PIRWISE takes a order of magtude more rutme tha FESCHK although the complexty of PIRWISE s less tha that of FESCHK, whch ca be explaed as follows. Table 3: Ru-tmes (relatve to omal) wth 4 parameters Method Block Block Block3 Block4 PIRWISE - 4.x.5x.69x FESCHK 4.x.67x.40x.74x PIRWISE_FESCHK_ 4.44x.x.6x.76x THRESHOLD (N=50) Sce PIRWISE s a suffcet but ot a ecessary codto for prug, t carres forward a sgfcat umber of pruable hyperplaes, whch has a cascadg effect as the hyperplaes are propagated through the crcut. FESCHK o the other had does more work to fd truly pruable hyperplaes at every ode ad the umber of hyperplaes t carres forward s therefore sgfcatly reduced. For example, there were 93 odes that had more tha 000 hyperplaes wth PIRWISE whereas there were oly 5 such odes wth FESCHK. 3 The ormalzed parameter for V dd s assumed to vary from [-,0] wth - represetg the case where V dd s at -8%. The prug algorthms are modfed to hadle ths specal case, accordgly. 4 Ths ru dd ot fsh due to suffcet memory. To summarze, our expermets o the four blocks dcate that PIRWISE_FESCHK_THRESH provdes sgfcatly better ru-tme performace over the PIRWISE method for crcuts wth large umber of hyperplaes (as see Block ). t the same tme t performs better tha FESCHK o Blocks -4. Thus, the rutme of these methods s very depedet o the topology of the crcuts. Sce Block has a large umber of equally crtcal paths, we focus o that block the rest of the secto. cdf % of odes 0% 00% 80% 60% 40% 0% 0% Number of hyperplaes Block Block Block 3 Block 4 Fg 8: cdf of the umber of hyperplaes o the four blocks for MX operatos, performed usg FESCHK 4. pproxmate Methods I order to explore ru-tme accuracy trade-offs, the SHRINK_HYPERCUBE method, descrbed Secto 3.3. s appled o Block for dfferet values of N, where N deotes the maxmum umber of hyperplaes that ca be propagated at every ode. The ru-tmes ad the smallest sze of the hypercube, computed across the puts of the 63 sequetal gates the desg, for the case of 4 parameters, are show Table 4. Table 4: SHRINK_HYPERCUBE method o Block Number of hyperplaes allowed Ru-tme relatve to FESCHK Sze of hypercube x x x x x 0.75 step-sze of 0.5 s used to shrk the hypercube each terato of the whle loop Fg 7. The ru-tmes are compared wth respect to the FESCHK ru-tme Table 3. The results dcate a good trade-off betwee the ru-tmes, the sze of the hypercube (deoted Secto 3.3. by a, where a X j a, j =,, ), ad the maxmum umber of hyperplaes allowed (N). Table 5: Dstrbuto of the hypercube sze Block Sze of Hypercube No. of coes Cumulatve % 0.5.9% % % % cdf of the sze of the hypercube for each of the tmg coes Block s show Table 5, for the case where N was set to 00, the SHRINK_HYPERCUBE algorthm. The results dcate that more tha 95% of the tmg coes have a hypercube of sze, mplyg less tha 00 hyperplaes o them, ad hece the arrval tmes computed o all these coes are exact, for ay settg. To further explore ru-tme accuracy trade-offs, the PIRWISE_SOFT_PRUNE_FESCHK algorthm, explaed Secto 3.3. was appled o Block. The results are compared 69

6 wth FESCHK algorthm Table 6. The table shows a reducto the maxmum umber of hyperplaes o a ode by a factor of three, whe compared wth FESCHK. ccordgly, a 33% speedup over FESCHK s obtaed at the expese of a small overestmato (maxmum of.6%) the omal arrval tmes. Table 6:PIRWISE_SOFT_PRUNE_FESCHK o Block FESCHK PIRWISE_SOFT_ PRUNE_FESCHK Ru-tme 4.x 9.58x Number of hyperplaes o the largest coe Slack Computato We brefly expla how we compute the slacks at the puts of all regsters the blocks. Cosder the coe show Fg 9. I our framework, the arrval tmes at the data ad clock puts of the samplg regster are rreducble sets of hyperplaes deoted as d ad c, respectvely. The requred arrval tme at the data put of the samplg regster s gve by: R d = { j + T S jc c} () c where T s the cycle tme ad S s the setup tme. We do ot cosder setup tme varatos ths work. O all our bechmarks, the cardalty of c was oe sce there was o fa the clock etwork. The marg at the data put of the regster s gve by: M R, R R } (3) d = { d jd jd d d d Clock logc Tmg Coe Fg 9: tmg coe Thus, (3) ca be computed O ( d. R d ) tme. M d may be prued further usg the techques of Secto 3. I order to evaluate the sestvty of the slack of the varous paths to parameter varatos, we frst compute the set of rreducble slack hyperplaes at the data put of each of the regsters o Block, for the case of 4 parameters. We ow cosder the coe wth the hghest umber of rreducble slack hyperplaes o Block (cosstg of 948 hyperplaes). Table 7 shows the slacks (computed as a mmum of the margs of the 948 rreducble hyperplaes) at dfferet settgs of the parameters, (oe of whch are worst case): ) omal (Nomal), ) all devces 5% faster (Fast L), 3) low V dd, hgh MCF (Low V dd, Hgh MCF), 4) certa layout type devces beg 5% slower (Slow Layout) 5) low power devces beg 5% slower (Slow Low Power) ad 6) all parameters at ther worst values (Worst Case). The slack at each of these settgs s sgfcatly dfferet from the omal slack, demostratg that dfferet paths have dfferet sestvtes ad the ablty of our method to predct that. We also compute the upper boud o the arrval tmes (T) at each of these settgs usg the upper boudg hyperplae method [3] order to determe the extet of pessmsm duced by usg such a upper boudg method, ad the results are show the last colum Table 7. Expectedly, at the worst case corer settg, the arrval tme computed usg [3] s exact, ad there s o overestmato, whereas at other settgs of the parameters, d c Samplg Regster partcularly at the omal, the arrval tmes computed usg [3] are hgher by as much as 0-30%. Table 7: Slacks at dfferet settgs of the parameters Settg Normalzed Slack Delta Slack w.r.t. Nom. T Overestmato usg [3] Nomal % Fast L % Low V dd, Hgh % MCF Slow Layout % Slow Low % Power Worst Case % Fg 0 shows the plot of the omal slacks versus the ew slack for the 948 hyperplaes at a dfferet settg X. Ths ew settg s computed such that the path marked wth a P the fgure, whch has a large omal slack, becomes the most tmg crtcal at that settg. The settg correspoded to a 9% droop V dd, worst case MCF, certa layout trasstor types beg fast, others beg slow. The set of paths that are ecrcled are the most sestve whe the parameters are at ths partcular settg. Note that ths formato s ot obtaed curret tmg flows based o omal slacks. I ths case, the path marked wth P would ot have bee cosdered crtcal. However, our flow we ca compute the slack at ay settg of the parameters, thus eablg a what-f aalyss. ew slack.5.5 P omal slack Fg 0: Slacks at a dfferet settg of X s.t. the path (marked P the fgure) wth a large omal slack becomes the most tmg- crtcal CONCLUSION We preset a block-based framework for computg the arrval tmes ad slacks at all settgs of the parameters, where oly rages o the parameter varatos are kow. We descrbe varous prug techques ths paradgm. Results o dustral crcuts show the vablty of our approach. CKNOWLEDGMENTS We thak Prof. Fard Najm (Uv. of Toroto, Caada) ad Drs. Noel Meezes ad Rafael Ros (Itel Corporato) for helpful dscussos o the approach. REFERENCES [] H. Chag ad S. Sapatekar, Statstcal Tmg alyss uder Spatal Correlatos, TCD, 4(9):467-48, 005. [] K. Kllpack, C. V. Kashyap, ad E. Chprout, Slco Speedpath Measuremet ad Feedback to ED Flows, DC, pp , 007. [3] S. Oass, ad F. N. Najm, Lear-Tme pproach for Statc Tmg alyss Coverg ll Process Corers, ICCD, pp. 7-4, 006. [4] L. Scheffer, Why are Tmg Estmates so Ucerta? What could we do about ths? TU workshop, 006. [5] L. G. Slva, M. Slvera, ad J. R. Phllps, Effcet Computato of the Worst-Delay Corer, DTE, pp. -6, 007. [6] C. Vsweswarah et al, Frst Order Icremetal Block Based Statstcal Tmg alyss DC, pp , 004. [7] CPLEX