A Fully Polynomial Time Approximation Scheme for Timing Driven Minimum Cost Buffer Insertion

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1 27. A Fully olyomial Time Approximatio Scheme for Timig Drive Miimum Cost Buffer Isertio ABSTRACT Shiya Hu Dept. of Electrical ad Computer Egieerig Michiga Techological Uiversity Houghto, Michiga 4993 shiya@mtu.edu As VLSI techology eters the aoscale regime, itercoect delay has become the bottleeck of the circuit timig. As oe of the most powerful techiques for itercoect optimizatio, buffer isertio is idispesable i the physical sythesis flow. Bufferig is kow to be N-complete ad existig works either explore dyamic programmig to compute optimal solutio i the worst-case expoetial time or desig efficiet heuristics without performace guaratee. Eve if buffer isertio is oe of the most studied problems i physical desig, whether there is a efficiet algorithm with provably good performace still remais ukow. This work settles this ope problem. I the paper, the first fully polyomial time approximatio scheme for the timig drive miimum cost buffer isertio problem is desiged. The ew algorithm ca approximate the optimal bufferig solutio withi a factor of + ruig i O(m 2 2 b/ b 2 / time for ay 0 <<, where is the umber of cadidate buffer locatios, m is the umber of siks i the tree, ad b is the umber of buffers i the buffer library. I additio to its theoretical guaratee, our experimets o 000 idustrial ets demostrate that compared to the commoly-used dyamic programmig algorithm, the ew algorithm well approximates the optimal solutio, with oly 0.57% additioal buffers ad 4.6 speedup. This clearly demostrates the practical value of the ew algorithm. Categories ad Subject Descriptors B.7.2 [Itegrated Circuits]: Desig Aids - lacemet ad Routig; J.6 [Computer-aided Egieerig]: Computeraided Desig Geeral Terms Algorithms, erformace, Desig Keywords Buffer Isertio, Fully olyomial Time Approximatio Scheme, N-complete, Cost Miimizatio, Dyamic rogrammig ermissio to make digital or hard copies of part or all of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. DAC 09, July 26-3, 2009, Sa Fracisco, Califoria, USA Copyright 2009 ACM /09/ Zhuo Li ad Charles J. Alpert IBM Austi Research Laboratory 50 Buret Road Austi, Texas {lizhuo, alpert}@us.ibm.com. INTRODUCTION As VLSI techology eters the aoscale regime, itercoect delay has become the bottleeck of the circuit timig sice devices scale much faster tha itercoects. As oe of the most effective itercoect timig optimizatio egies, buffer isertio is idispesable i the physical sythesis flow [, 2, 3]. It is demostrated i [4] that i two recet IBM ASIC desigs, over oe-fourth gates are buffers. Buffer isertio is oe of the most studied problems i physical desig. Existig works iclude [5, 6, 7, 8] o explorig dyamic programmig techiques with advaced data structures to compute optimal timig drive bufferig solutios. Sice buffers themselves are a drai o power, it is highly desirable to use as little bufferig resources as possible i buffer isertio. Bufferig with cost (power or area miimizatio has bee cosidered i [6]. It proposes a dyamic programmig algorithm which rus i pseudo-polyomial time. This surprises o oe as the miimum cost timig drive bufferig problem is N-complete [9]. Derived from this classic problem, bufferig techiques have bee developed for various scearios. For example, there are works [6, 0,, 2] hadlig slew, oise ad/or variatios, ad works explorig the iteractio of bufferig with routig [3], placemet [4], ad floorplaig [5, 6]. Despite the fact that may bufferig techiques have bee developed, the uderlyig bufferig problem is still less studied especially i theory. The dyamic programmig ca compute the optimal solutio but ot rus i polyomial time, while heuristic algorithms ca ru fast but without ay performace guaratee. hether there is ay efficiet algorithm with provably good performace still remais ukow. This work aims to settle this ope problem ad advace the uderstadig of miimum cost timig drive bufferig problem from a theoretical poit of view. Yet, our ew algorithm is highly practical. I this paper, we propose a fully polyomial time approximatio scheme (FTAS for the N-complete timig drive miimum cost bufferig problem. I our cotext, a fully polyomial time approximatio scheme refers to a bufferig algorithm which is able to compute a solutio with the cost at most + times worse tha the cost of the optimal bufferig solutio for ay >0. It rus i time polyomial i the iput size of the problem istace ad /. Give a N-complete problem, a FTAS is geerally regarded as a ultimate solutio i theory. The mai cotributio of this paper is summarized as follows. A FTAS algorithm is proposed to approximate the optimal bufferig solutio withi a factor of + i 424

2 O(m 2 2 b/ b 2 / time for ay 0 <<adi O(m 2 2 b/ + m 2 b + 3 b 2 time for ay, where is the umber of cadidate buffer locatios i the tree, m is the umber of siks i the tree, ad b is the umber of buffers i the buffer library. This work presets the first provably good approximatio algorithm o the timig-drive miimum cost bufferig problem. The proposed FTAS is motivated from the algorithms i [7, 8] for the layer assigmet problem. Nevertheless, our FTAS features ovel techiques such as double- oracle based solutio search ad timig-cost approximate dyamic programmig algorithm. latter rus i O( m2 The time to compute a bufferig solutio with the cost at most ( + ad the timig at most ( + T where refers to the optimal cost ad T refers to the timig costrait. The ew FTAS algorithm is highly practical. Experimetal results o 000 idustrial ets usig a buffer library cosistig of 48 buffer types demostrate that the FTAS approximates the optimal solutio by oly 0.57% additioal buffers with 4.6 speedup compared to the dyamic programmig algorithm. 2. RELIMINARIES A routig tree T =(V,E is give as a iput to the miimum cost timig bufferig problem, where V = {s 0} V s V,adE V V. As i may previous works [9, ], the routig tree is assumed to be biary i this paper. Trees i other topologies ca be easily coverted to a biary tree [7]. Vertex s 0 is the root/driver of T, V s is the set of sik vertices, ad V is the set of cadidate buffer locatios. For each sik s V s, there is a sik capacitace C(s ad a required arrival time (RAT. The driver s 0 has a arrival time, deoted by AT (s 0. A et satisfies the timig costrait if the arrival time is o greater tha the required arrival time at driver. By subtractig AT (s 0 ad each sik RAT by AT (s 0, oe ca modify the et such that AT (s 0 = 0 ad all RAT are positive. Note that this will ot impact bufferig solutios. Defie the timig costrait T to be the maximum RAT after the above modificatio. A buffer library B cotaiig all buffer types which ca be assiged to cadidate buffer locatiosisalsogive. NotethatB icludes both o-ivertig buffers ad ivertig buffers. As is commoly used i physical sythesis, the Elmore delay model is adopted. The Elmore delay o a edge e = (v i,v j is computed by D(e =R(e C(e + C(v 2 j,where C(e,R(e,C(v j refer to the edge capacitace, the edge resistace ad the dowstream capacitace viewig at v j,respectively. For a buffer b placed at vertex v j, its buffer delay is computed by D(b =R(b C(v j+k(b, where R(b ad K(b refer to the drivig resistace ad the itrisic delay of buffer b, respectively. Each buffer b has also a iput capacitace C(b adacostw(b. I this paper, the buffer area is used as buffer cost to illustrate our ew algorithm. A buffer assigmet γ is a mappig γ : V B {b} where b deotes the case with o buffer iserted. The total cost of a bufferig solutio γ for the tree T is defied as the sum of the costs over all iserted buffers. The timig drive miimum cost bufferig problem, kow as N-complete [9], ca be formulated as follows. Timig Costraied Miimum Cost Bufferig: Give a biary routig tree with cadidate buffer locatios ad a buffer library with b buffer types, to compute a buffer assigmet solutio such that the timig costrait is satisfied ad the total buffer cost is miimized. 3. ALGORITHMIC FLO Our FTAS algorithm for timig costraied miimum cost bufferig problem is motivated from [7, 8] for a layer assigmet problem. Let deote the cost of the optimal bufferig solutio. At a high level, the FTAS works i the followig ituitive framework. It first makes a guess x o ad uses a procedure called oracle to check whether such a guess is good, i.e., sufficietly close to or ot. If this is the case, retur x. Otherwise, make a ew guess o. This procedure is iterated util a good guess is made. Certaily, there are two algorithmic desig challeges i the above framework. First, oe does ot kow the optimal cost, which is our target, the how to decide whether x for ay x? Moreover, oe eeds to perform this check efficietly. Secod, if the curret guess is ot good, how to fid a possibly better guess? The first difficulty eeds to a saliet desig of the oracle ad the secod difficulty eeds the usage of a efficiet oracle based solutio search. These two key compoets will be described i this paper. 4. DOUBLE- ORACLE SEARCH 4. The Oracle Give ay positive umber x, the oracle ca efficietly decide whether x, where is the total buffer cost of the optimal bufferig solutio. I fact, the decisio is aswered approximately depedig o, meaig that the aswer is either x or < ( + x. A key compoet i the oracle is a polyomial time timigcost approximate dyamic programmig algorithm which will be described i Sectio 5. This algorithm has three critical properties. First, the algorithm ca be performed efficietly, i time polyomial i = /. Secod, it will either retur a bufferig solutio with the cost o greater tha a cost budget value, or coclude that this cost budget is too low to fid ay bufferig solutio (approximately satisfyig the timig costrait. Third, the dyamic programmig algorithm will retur a solutio with timig slightly larger tha the timig costrait T with cotrolled error. recisely, the timig of the solutio is bouded by ( + T where is the target approximatio ratio. This is acceptable especially cosiderig that i early stage of physical sythesis flow (or whe chip is i the prototype stage, the timig costrait is ofte set accordig to the desiger experiece ad thus i geeral the timig costrait is ot striget. I additio, varyig, our algorithm ca provide more desig flexility (i terms of timig ad cost tradeoff to the circuit desigers. Eve i the late physical desig stage where timig costrait is striget, i practice, oe ca still rip-up ad rebuffer the ets with timig violatios usig [6] to compute the optimal bufferig solutios. e call this procedure timig recovery. Eve if FTAS with timig recovery is ot guarateed to ru i polyomial time, i practice it would still ru much faster tha usig [6] aloe. This is the case sice there are very few ets which violate the timig costraits by our FTAS as idicated i the experimets. Give a timig-cost approximate dyamic programmig algorithm, we are ready to preset the oracle. First, for ay 425

3 positive umber, each buffer cost w is scaled by the factor of x w followed by dow-roudig, i.e., w becomes. x The dyamic programmig is performed to the scaled ad rouded bufferig problem with the cost budget set to /. There are two possible decisio results i a oracle query.. By dyamic programmig, a bufferig solutio with timig o greater tha ( + T is foud for the total buffer cost /. This meas that usig the uscaled ad urouded costs, the cost of the obtaied bufferig solutio will be smaller tha x + x =(+x. This is the case sice the roudig error i cost is at most x by otig that the roudig error is at most x at each buffer ad there are oly cadidate buffer locatios. Therefore, there is a bufferig solutio with cost smaller tha ( + x ad the timig at most ( + T i the origial bufferig problem. e coclude that < ( + x. 2. By dyamic programmig, there is o bufferig solutio with cost w = / which ca satisfy eve the relaxed timig costrait ( + T. This meas that the origial uscaled bufferig problem does ot have a bufferig solutio withi the cost x = x satisfyig ( + T ad thus T. e coclude that x i the origial bufferig problem. Sice both timig ad cost are rouded, our FTAS actually computes the ( + approximatio to the bufferig problem with the logest path delay bouded by ( + T. Accordig to Lemma 2 i Sectio 5, the proposed dyamic programmig algorithm rus i O( m2 + m 2 b time which gives the time for a oracle query. 4.2 The Double- OracleBasedSolutioSearch After obtaiig the oracle, a ( + approximatio could be computed as follows. Start with a iitial lower boud l ad a iitial upper boud u o the optimal bufferig cost. For example, u could correspod to always usig largest buffer at every cadidate buffer locatio ad l could be set to the cost of the sigle buffer with smallest cost i the buffer library. Oe just eeds to compare the timig of the give ubuffered et to the timig costrait to accout for the case where o buffer eeds to be iserted. Subsequetly, a biary search is performed withi these bouds. That is, each time x = l + u is used to query the oracle 2 ad depedig o the decisio results, l ad u will be updated accordigly, i.e., update u =(+xi Case ad update l = x i Case 2. The complexity of this algorithm certaily depeds o the gap betwee the iitial upper ad lower bouds. Oe ca make the time complexity idepedet of the iitial boud values as i [8]. This is accomplished by utilizig the fact that a oracle query takes the time iversely proportioal to. recisely, a larger leads to a coarser approximatio but rus faster while a smaller leads to fier approximatio but rus slower. This provides the opportuity i varyig approximatio ratio adaptively to accelerate the whole procedure of oracle based solutio search. Istead of stickig to the target approximatio ratio, oe ca use larger iitially ad gradually reduce it to the target. he these form a decreasig geometric sequece (e.g.,...,27, 9, 3,, /3,...,, the total asymptotic rutime will be bouded by the last query sice the a oracle query takes the time proportioal to /. Our oracle based solutio search is similar to the oe i [8] with a importat differece as follows. Sice their algorithm oly rouds oe parameter while our approach rouds two parameters (cost ad timig Q, applyig the techique i [8] leads to adaptively settig both roudig parameters. This is ot desired for roudig Q sice the timig error will ot be cotrolled by. Thatis, i first few iteratios durig oracle based search, the timig of the solutios may be sigificatly larger tha ( + T sice approximatio ratios there would be big (e.g., the first few approximatio ratios i..., 27, 9, 3,, /3,..., are much bigger tha the target. This meas that by dyamic programmig, oe oly kows whether the cost budget / is sufficiet for computig a solutio with much larger timig, which may lead to the wrog decisio i arrowig dow the gap betwee upper ad lower bouds. It motivates us to propose to oly chage the correspodig to cost but ot the correspodig to timig. e call it double- oracle based solutio search. Thisiswhytherearetwo ad i the dyamic programmig algorithm i Sectio 5. Namely, correspods to the approximatio ratio o cost ad correspods to the approximatio ratio o timig Q. If we fix both of them to, oe caot reduce the total rutime to be idepedet of the iitial bouds. Our idea is to fix at while adaptively chagig. I details, let u,i deote the upper boud ad deote the lower boud after i-th iteratio i the oracle based solutio search. Iitially, u,0 = u ad l,0 = l. A geometric sequece of approximatio ratios i covergig to will be used i oracle based solutio search for (but ot sice it is fixed at. Followig [8], set i = Ê u,i. ( I each iteratio, r u,i x = (2 + i is used to query the oracle. It ca be proved that after i-th oracle query, u,i+ + =( u,i 3/4, (3 sice for Case u,i+ =(+ ix = u,i 3/4 /4, + =, adforcase2u,i+ = u,i, + = x = u,i /4 This process is iterated util the ratio betwee upper ad lower boud is o greater tha 2. Let i be the first i such that u,i 2. Accordig to Lemma 2 i Sectio 5, a oracle query takes O( m2 time. e first boud the time util i,whichiso( m2 + m 2 b + i i m2 2 i 2 i i 2 i 3 b 2. Sice i i i i ad u,i > 2, = i i meas i 2 ( Thus, O( m2 2 u,i =O( m2 2 2 i i i 2 i i i i i + i i i É u,i. (4 u,i 3/4. 426

4 It is show i [8] that i i = u,i i i ( u,i ( 4 3 i i = 0 j<i ( ( 4 3 j, (5 u,i (ote that j starts from 0, ad l,t u,t < by otig that 2 3/4 i is the first i such that u,i 2. Subsequetly, i i = u,i 0 j<i ( u,i (4/3 j < (4/3 j. (6 2 3/4 0 j<i The last term is the sum of a mootoically decreasig geometric sequece which is certaily bouded by O(. Thus, O( m2 2 Similarly, sice i i É u,i O( [8], we have O( m2 b =O( m2 2. (7 2 i i i 2 < 0 j<i 0.59 /2 (4/3j = i i i = O( m2 b ad O( 3 b 2 i i i =O( 3 b 2. The total rutime for oracle based solutio search is bouded by O( m2 2 b + m 2 b + 3 b 2, (8 sice is fixed at. After i oracle queries, the ratio betwee upper ad lower bouds is at most 2. ith this better startig poit, the timig-cost approximate dyamic programmig ca be efficietly performed with both ad set to. First set x to the curret lower boud. Subsequetly, scale ad roud the cost w of each buffer to w where is the target x. Note that is o greater tha u,i 2,which meas that there is at least oe solutio with scaled cost o greater tha 2/. Otherwise, similar to Case 2, there is o bufferig solutio withi cost 2 x =2x u,i which is a cotradictio. This solutio is the ( + approximatio which will be retured by our FTAS. Similar to the argumet i Case (, scalig the result back by the factor of x forms a lower boud o ad the maximum roudig error is x = x = l,t. Thus, the cost of the obtaied bufferig solutio is at most ( +. This last step uses the dyamic programmig which takes O( m2 time by otig that 3 both ad are set to. The efficiet computatio is due to the fact that the ratio betwee the curret upper ad lower bouds has bee reduced to 2. Together with the time for oracle based solutio search i Eq. (8, the total time is O( m2 3 b +m 2 b+ 3 b 2 which ca be simplified to O(m 2 2 b/ b 2 / for 0 <<adtoo(m 2 2 b/ + m 2 b + 3 b 2 for. e reach the followig theorem. Theorem : A(+ approximatio to the timig costraied miimum cost bufferig problem ca be computed i O(m 2 2 b/ b 2 / time for ay 0 <<adi O(m 2 2 b/ + m 2 b + 3 b 2 timefor, where is the umber of odes i the tree, m is the umber of siks i the tree, ad b is the umber of buffers i the buffer library. 5. OLYNOMIAL TIME TIMING-COST A- ROIMATE DYNAMIC ROGRAMMING 5. Boud Distict Cost ad RAT Q A careful ivestigatio i Lillis algorithm [6] would reveal that the umber of solutios is ot polyomially bouded which is why Lillis algorithm is a pseudo-polyomial algorithm. To desig a efficiet algorithm, we certaily eed to boud the umber of solutios durig solutio propagatio. A major iovatio i the proposed FTAS algorithm is a dyamic programmig algorithm with polyomially bouded ad Q. As a result, the umber of solutios will also be polyomially bouded (sice there is oly oe possible o-domiated solutio with each pair of ad Q, amely, the oe with the smallest C. First ote that we have two i the algorithm, oe beig which refers to the approximatio i cost ad the other beig which refers to the approximatio i timig. is varyig while is fixed to i the fast double- oracle based solutio search. To boud, recall that i Sectio 4, oe first scales ad rouds each buffer cost w to a iteger as w = w x.after that, the oracle oly wats to kow whether there is a solutio (approximately satisfyig the timig costrait with cost up to /. Let = /. The oracle uses this cost boud to perform the dyamic programmig. Thus, wheever there is a solutio with cost greater tha, it will be elimiated from the solutio set. Cosequetly, there are at most + distict (0,,..., ataylocatiodurig solutio propagatio. Our ew dyamic programmig algorithm is as follows. First, it always works with cost bis ( -bi sice the oracle scales ad rouds each buffer cost before performig the dyamic programmig algorithm. I dyamic programmig, right before a brach merge, all the solutios are also discretized ito timig bis (Q-bi ad the the solutio pruig is performed. This allows us to boud Q as well. Cosequetly, the umber of o-domiated solutios durig solutio propagatio is bouded. To boud the umber of distict Q, right before each brach merge, for all Q 0, roud up Q of each brach to the earest value i {0,T/m,2T /m,..., T }, wherem is the umber of the siks. e uderestimate the delay by roudig. For example, whe = 0.5 ad m = 2, Q =0.7T ad will be up-rouded to 3T/m.Thesolutios with Q<0 will be prued sice the arrival time at driver is T T/m 0. Thus, there are at most +=m/2 + distict Q after roudig. Together with the fact that there are at most O( distict at ay poit, at most O(m/= O( m o-domiated solutios ca be obtaied after ay brach merge (the umber of o-domiated solutios before brach merge will be discussed soo. This is due to the fact that there is oly oe solutio for each pair of Q,, amely, the oe with miimum C. Note that the roudig error i timig at each brachig poit is at most T/mad thus at most T forthewholetreewithm brachig poits. Note that the timig of the obtaied solutio (i the scaled problem is at most T but it is with the rouded Q. This meas that after roudig Q back, we obtai a bufferig solutio with timig at most ( + T for the origial bufferig problem. The time complexity aalysis is as follows. After performig a brach merge, there are at most O( m odomiated solutios. After that, solutios are propagated 427

5 i its upstream brach. A add wire operatio does ot itroduce ew solutio. After performig buffer isertio operatio at a ode v, for solutios with ew buffers iserted at v, there are oly b distict C (where b is the umber of buffer types. Sice the umber of is always bouded by O(, there are at most O(b=O(b/ o-domiated buffered solutios. This is due to the fact that there is oly oe solutio for each pair of C,, i.e., the oe with maximum Q. e are to boud the time for computig these O(b= O(b/ o-domiated buffered solutios. Give a solutio, buffer isertio at v leads to b possible ew solutios. Sice there are at most O(m/( + 2 b/ odomiated solutios aywhere (see below, computig odomiated buffered solutios at ode v takes O(mb/( + 2 b 2 / time. This is due to that withi a sigle bi, (Q, C based pruig takes liear time i the umber of geerated solutios which is the same as the pruig without cosiderig i [5]. Note that cross -bi pruig is ot performed sice this will ot improve the asymptotic complexity. Together with those O(m/ ubuffered solutios (which are propagated by add wire from the last brach merge, there are at most O(m/ + bodomiated solutios. he these solutios are propagated alog this brach, there are at most O(m/ + b= O(m/( + 2 b/ o-domiated solutios at ay poit before ext brach merge sice there are at most cadidate buffer locatios ad Q is ot rouded util the brach merge. Before brach merge, all solutios are first put ito cost bis ( -bi ad timig bis (Q-bi. That is, they are placed ito the bis with iteger costs from 0 to /. I each cost bi, there are m/ + timig bis with the timig as 0,T/m,2T /m,..., T by up-roudig each Q. By a liear traversal of all bis, domiated solutios will be prued. The whole process certaily takes time liear i the umber of solutios, i.e., O(m/( + 2 b/ time. I solutio pruig, sice is always a iteger, there are + possible mergig results of left ad right brach solutios for each. For example, to obtai merged cost =3,the four possibly combiatios of left-brach solutio γ ad right-brach solutio γ 2 are ( (γ =0,(γ 2=3,(2 (γ =,(γ 2=2,(3 (γ =2,(γ 2 =, ad (4 (γ =3,(γ 2 = 0. For a fixed combiatio o,there are a list of solutios with distict Q alog each brach. For each Q-bi, the time complexity ca be easily bouded sice all the solutios i each brach are o-domiated ad thus for the same, C are icreasigly sorted. Oe just eeds to correspodigly merge two solutios with the same Q. Table : A example for brach merge. Left brach (Q,, C C =0 = =2 =3 Q = Q = T/m Q =2T/m Right brach (Q,, C C =0 = =2 =3 Q = Q = T/m Q =2T/m It is helpful to look at a example to illustrate the above aalysis. Refer to Table. To obtai the merged cost of 3, suppose that we are mergig solutios with =2ithe left brach ad solutios with = i the right brach. The correspodig are show with arrows. For Q =0 after brach merge, the miimum C is C = = 7. For Q = T/m after brach merge, the miimum C is C = = 37. For Q =2T/m after brach merge, the miimum C is C = = 90. Oe the also eeds to cosider the other three mergig possibilities for the merged cost to be 3, i.e., ( (γ =0,(γ 2=3, (2 (γ =,(γ 2 = 2, ad (4 (γ =3,(γ 2=0. Subsequetly, the solutio with the miimum C for each pair of, Q is picked. This process takes O(m/ time for each merged sice there are oly O(m/distict Q. Summig over all till, the time complexity for brach merge ad solutio pruig i brach merge is O(m/ ( 2 =O(m 2 /(. Together with the time for puttig the solutios ito bis before brach merge, the total rutime is O(m/( + 2 b/ + m 2 /( for a sigle brach merge. 5.2 Time Complexity As metioed above, a sigle buffer isertio together with pruig takes O(mb/( + 2 b 2 / time. Summig over cadidate buffer locatios, total buffer isertio takes O(m 2 b/( + 3 b 2 / time. This certaily upper bouds the time for add wire. A sigle brach merge takes O(m/( + 2 b/ + m 2 /( time. Summig over all m brachmerges,o(m 2 /( +m 2 b/ + m 2 2 /( time is eeded. Thus, the dyamic programmig will ru i O( m2 (9 2 time to compute a solutio with the cost at most ( + optimal cost ad with timig at most ( + T. e reach the followig lemma. Lemma 2: The timig-cost approximate dyamic programmig algorithm ca compute a timig drive bufferig solutio with the cost at most (+ optimal cost ad with timigatmost(+ 2T i O( m2 time for ay, > 0, where is the umber of cadidate buffer locatios, m is the umber of siks, ad b is the umber of buffers i the buffer library. 6. EERIMENTAL RESULTS e compare the proposed FTAS for the timig drive miimum cost bufferig problem to the dyamic programmig algorithm [6] which computes optimal bufferig solutio. The experimets are performed o a set of 000 ets at various scales extracted from a idustrial ASIC chip. The buffer library cosists of 48 buffer types icludig buffers ad iverters. The buffer cost is measured by buffer area i this paper. However, other metric ca be easily hadled i FTAS. Refer to Table 2 for the compariso. Cost Ratio ad Speedup are computed by comparig to the total buffer cost ad the rutime of dyamic programmig algorithm. # Vio. specifies the umber of ets with timig violatios. The results with small <are show. This rage of is desired i practice sice oe always wishes to compute solutios close to the optima. e make the followig observatios. The dyamic programmig i [6] computes the optimal solutio. Total buffer cost is , o et has timig violatio, ad CU time is secods. 428

6 Table 2: Compariso of the dyamic programmig [6] ad the FTAS algorithm o 000 idustrial ets. I dyamic programmig solutio, total buffer cost is , ad CU is secods. # Vio. specifies the umber of ets with timig violatios. Cost Ratio ad Speedup are computed by comparig to dyamic programmig. FTAS #Vio. Total Cost CU(s Cost Ratio Speedup % % % % % % % 5.7 Average 5.2 Table 3: The obtaied timig o the ets violatig the timig costraits for FTAS with =0.0. FTAS with =0.0 Net Timig Costrait Actual Delay Timig Violatio % % % Our FTAS works very well i practice. Compared to the dyamic programmig solutios, there are oly slight solutio degradatios i total buffer costs while o average over 5 speedup is obtaied. For example, whe the target approximatio ratio is set to = 0.0, the actual approximatio ratio (cost ratio is oly 0.57% while 4.6 speedup is achieved. The speedup is so sigificat sice the total umber of solutios at driver over 000 ets is 66,676 for FTAS with = 0.0 (for all iteratios i performig double- oracle based solutio search while it is,22,604 i the dyamic programmig algorithm [6]. It is importat to ote that the cost ratio is theoretically guarateed to be o greater tha. I practice, it is much smaller as is show i Table 2. This clearly demostrates the effectiveess of our FTAS algorithm. Larger leads to more speedup while smaller leads to less solutio quality degradatio. This is agai as guarateed theoretically. Sice timig is rouded i our timig-cost approximate dyamic programmig, there are timig violatios i the obtaied bufferig solutios. However, it is clear that this happes with very small probability i practice as idicated by our experimetal results. he =0.0, oly 3 out of 000 ets have timig violatios. I additio, the obtaied timig is theoretically guarateed to be withi ( + T where T is the timig costrait. For example, the ets with timig violatios for FTAS with =0.0 are show i Table 3. Their actual delays are clearly bouded by the ( + T =.0T. he the timig costraits are striget, oe may eed every et to satisfy the timig costrait. For this, the followig timig recovery procedure could be performed. Those ets with timig violatios ca be ripped up ad rebuffered usig optimal dyamic programmig [6]. The overall rutime would still be much better tha [6] sice few ets eed to be rebuffered. For the ets without timig violatios, the approximatio ratio is bouded by +. For the ets with timig violatios, the optimal solutios are computed i rebufferig. Thus, the approximatio ratio is still bouded by + after timig recovery. Refer to Table 4 for the results. The cost is icreased compared to FTAS without timig recovery sice roudig o timig i FTAS uderestimates delay ad may make the cost of the obtaied bufferig solutio smaller tha the optimal cost (esp. for may of the ets with timig violatios eve if roudig o cost icreases it. Empirically, FTAS with = 0.0 gives the best performace sice it has fewest ets which eed rebufferig. Table 4: The results of FTAS with timig recovery. FTAS with Timig Recovery #Vio. Total Cost CU(s Cost Ratio Speedup % % % % % % % 2.4 Average REFERENCES []. Saxea ad N. Meezes ad. Cocchii ad D.A. Kirkpatrick, Repeater scalig ad its impact o CAD, TCAD, vol. 23, o. 4, pp , [2] J. Cog, A itercoect-cetric desig flow for aometer techologies, roceedigs of the IEEE, vol. 89, o. 4, pp , 200. [3] Z. Li, C. Alpert, S. Hu, T. Muhmud, S. Quay, ad. Villarrubia, Fast itercoect sythesis with layer assigmet, ISD, [4].J. 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