A Fast On-Chip Decoupling Capacitance Budgeting Algorithm Using Macromodeling and Linear Programming

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1 1. A Fas On-Chp Decouplng Capacance Budgeng Algorhm Usng Macromodelng and Lnear Programmng Mn Zhao, Rajendran Panda, Savhr Sundareswaran, Shu Yan and Yuhong Fu Freescale Semconducor, Inc. ABSTRACT We propose a novel and effcen charge-based decouplng capacance budgeng algorhm. Our mehod uses he macromodelng echnque and effecve radus of decouplng capacance o reduce he sze of he problem. We formulae he nonlnear opmzaon no a lnear program (LP) by negrang he nodal equaons across a me perod of neres and hrough ceran approxmaons. To reduce he error caused by lnearzaon, we do mulple eraons of he lnear program. Expermenal resuls demonsrae ha, wh he proposed algorhm, even very large power neworks (eg. 5 mllon nodes) can be opmzed n a couple of hours wh 1- ransen analyses. Comparson of our algorhm wh anoher heursc mehod shows area effcency and run me advanage of our mehod. Caegores and Subjec Descrpors B.. [Hardware]: INTEGRATED CIRCUITS layou General Terms Algorhms, Performance, Relably, Verfcaon Keywords Decouplng capacance, budgeng, macromodelng, sequence of lnear programmng 1. INTRODUCTION Wh he ncrease n VLSI crcu frequency and supply volage scalng, desgnng a robus power dsrbuon nework has become a challengng ask. Sac IR drop s usually addressed hrough ncreased mealzaon (reducon of ressance), pad placemen, opology opmzaon and power densy aware floorplannng. However, he chef echnque for lmng dynamc volage flucuaons s o place decouplng capacors (known as decap, nanabbrevaed manner) close o problem spo. As he fabrca- Permsson o make dgal or hard copes of all or par of hs work for personal or classroom use s graned whou fee provded ha copes are no made or dsrbued for prof or commercal advanage and ha copes bear hs noce and he full caon on he frs page. To copy oherwse, o republsh, o pos on servers or o redsrbue o lss, requres pror specfc permsson and/or a fee. DAC 00, July 8, 00, San Francsco, Calforna, USA. Copyrgh 00 ACM /0/000...$5.00. on process moves up he echnology nodes ladder, he ncreased wre ressance s aggravang he supply nose problem. However, he hgh leakage curren n hese echnology nodes dscourages addon of abundan decouplng capacance. Decouplng capacance also uses up de area and affecs de yeld adversely. For hese reasons, dynamc supply nose needs o be addressed wh a mnmum amoun of decap whch needs o be placed opmally. In hs paper, we address hs mporan problem. We formulae he problem o mnmze he oal decap subjec o volage consrans on he nework nodes, and consrans on he decap amoun ha can be added a varous places. The soluon ams a realzng dynamc volages beer han a user-specfed hreshold level a all mes. Decap budgeng and placemen s a non-lnear opmzaon problem [1]. Recenly, several sensvy-based echnques were proposed. The auhors of [1] proposed a sensvy-based nonlnear program, whch s solved by quadrac programmng and usng a compressed pecewse lnear form o sore adjon sensvy. [] used he conjugae graden(cg) mehod, wh a merged adjon sensvy heursc o speed up he sensvy calculaon. [] reduced he problem sze usng he geomerc mulgrd concep and hen used a sequenal quadrac program on he reduced grd. Is applcaon s hus lmed o regular mesh srucures. In [], L, e.al. used dvde-and-conquer o reduce he sze of he sensvy-based opmzaon. [5] solved he nonlnear opmzaon hrough a sequence of lnear programmng mehod. Sensves o decaps were used as lnear consrans n he opmzaon. All hese mehods requre calculaon of sensves wh respec o decouplng capacance locaon. The adjon mehod of sensvy calculaon [ 8] needs o sore waveforms a every node n boh he orgnal and he adjon neworks, whch may exhaus he memory resource for large neworks. Moreover, he foresad mehods add decap and recompue sensvy n an erave procedure. As he complexy of one adjon sensvy compuaon s he same as one ransen analyss, he erave nonlnear opmzaon procedure becomes que expensve. Charge-based decap esmaon echnques have also been proposed [9, 10] n he conex of power supply nose-aware floorplannng. An approxmae lumped decouplng capacance s esmaed for each floorplan module, assumng full VDD volage as he nal volage of decouplng capacance. Our approach dffers from he prevous works n many respecs: (1) The charge bass of consrans, formed hrough negraon of he nodal equaons over a me perod of n- 1

2 eres, smplfes compuaons subsanally. Our approach frs lnearzes he charge-based consrans hrough ceran approxmaons (o be explaned furher) and hen performs a small number of eraons o accoun for he non-lneary. A small lnear program s solved n each eraon. () The conceps of macromodelng [11] and effecve radus of capacance are used for problem sze reducon. A small macromodel wh few pors, whch are he poenal ses for decap connecon, s creaed by reducng he nework. () I handles he decap pre-exsng n he nework and he decap o be added n dfferen ways. Inrnsc decap of devces and nerconnec and pre-placed decap cells are modeled nsde he macromodel hrough her companon models placed n he nework before he macromodelng procedure. Addonal decap ha s opmzed s exernal o he macromodel, and s conneced a he model s pors.. BACKGROUND.1 Overvew of Power Grd Smulaon A chp s power dsrbuon sysem s modeled as a lnear RLC nework wh ndependen me-varyng curren sources modelng he swchng currens of he ranssors. Smulang he nework requres solvng he followng sysem of dfferenal equaons, whch s formed by a Modfed Nodal Analyss (MNA) [1] approach: G x()+c x () =b(), (1) where G s a conducance marx, C s an admance marx resulng from capacve and nducve elemens, x() s he me-varyng vecor of volages a he nodes, and currens hrough nducors and volage sources, and b() s he vecor of ndependen me-varyng currens and volages from ndependen sources and companon models. Ths dfferenal sysem s very effcenly solved by reducng o a lnear algebrac sysem (G + C/h) x() =b()+c/h x( h), () usng Backward Euler (BE) echnque. A fxed me sep, h, s used o make he LHS marx saonary so ha s facors can be reused n a ransen smulaon.. Macromodelng A par of he power grd or he enre power dsrbuon nework can be modeled as a mul-por lnear elemen wh curren ransfer characerscs gven by I = A V + S, I, V, S R m,a R m m () where m s he number of pors n he model, A s he por admance marx, V s he vecor of por volages, I s he vecor of currens hrough he pors from an exernal nework no he model, and S s a vecor of curren sources conneced beween each por and he reference node. S essenally brngs he effec of movng all nernal curren sources o he pors. Macromodelng s he procedure of dervng Equaon () from he modfed nodal equaons () of a power grd, ncludng he nrnsc and already placed decouplng capacors modeled hrough her companon models usng backward Euler echnque.» G11 G 1 G T 1 G» U V =» J 1 J + I () where I s he vecor of currens hrough he nerface, and U and V are volages a he nernal nodes and pors respecvely. G 1, G 11, G are pars of he admance marx whch ncludes he companon conducance C/h of equaon (). J 1 and J are curren sources a he nernal nodes and pors respecvely, and nclude he companon currens C/h x( h) of equaon (). The dealed macromodelng procedure s descrbed n [11].. DECAP BUDGETING.1 Problem Formulaon We se ou o mnmze he oal capacance o be added a he nework nodes, subjec o he consrans ha () he dynamc node volages are beer han a specfed hreshold, () he capacance ha can be added a a node s bounded, and () he node volages sasfy he MNA equaons. mnmze subjec o P C () V k V hre, () C C max,, () V k should sasfy MNA a me pon k, for all power grd nodes, and me pons k Here, V hre s specfed by he user and C max, s se dependng on local congeson and he decap amoun realzable per un area.. Flow Oulne Overall flow of he proposed echnque s oulned below. Sep 1 Run ransen analyss of he power nework, wh nrnsc capacances and decouplng capacances ha have already been placed. Sep Inspec he ransen volage waveforms and deermne () he regons where he volage consrans are volaed (called volaon regons), () he me wndows durng whch volaons occur (called he volaon me wndows), and () a se of nodes o be used for macromodelng and opmzaon (called he samplng nodes). Ths sep s dealed ou n Secon.. If no volaon regon s found, hen opmzaon s compleed. Oherwse, connue wh sep. Sep For each volaon regon, deermne he opmal amoun of decap o be added a he samplng nodes. Ths s done by frs macromodelng he nework usng he samplng nodes n he volaon regon as he pors, and hen runnng few eraons of a lnear program descrbed n Secon and 5 on hs model. For volaon regons ha overlap each oher, he opmzaon for hese regons s done smulaneously. Sep Dsrbue he decap of he samplng nodes evenly no he respecve samplng regons and go o sep 1.. Volaon Regon, Samplng Nodes, and Volaon Tme Wndow A volaon regon s llusraed n Fgure 1. Frs, he de s paroned no unformly szed les usng pre-defned x and y pches. A se of conguous les (he cenral crcular regon plus he darkly shaded regon) wheren one or more nodes volae he volage consran defnes he core volaon regon. The core volaon regon s hen expanded n all drecons by a pre-deermned dsance (known as he 18

3 A le The core volaon regon The volaon regon Fgure 1: An llusraon of he volaon regon effecve radus) o nclude addonal les (lghly shaded les surroundng he darkly shaded ones). Ths expanded regon s called he volaon regon. The expanson s based on he fac ha decap added ousde he expanded regon has lle or no mpac on he volages n he core regon, and vce versa [1, ]. Moreover, he expanded regon provdes more space for decap addon. The large chunk of nework ousde he expanded regon can be absraced away by reducng hose nodes n he macromodel. There can be more han one volaon regon, hough only one s shown n he Fgure 1. By runnng he opmzaon ndependenly on dfferen volaon regons, we are hus able o reduce he problem sze sgnfcanly. Ths also brngs n an opporuny for parallelzaon. If wo or more volaon regons overlap each oher, hen hese regons are opmzed ogeher. Opmzng hem separaely n such cases wll resul n sub-opmal resuls as we wll hen gnore he srong neracon of capacances and volages across hese regons. The effecve radus for he decap depends on he effecve ressance whch s deermned by he echnology node, he curren varaon, and he mealzaon used n each layer. I s very easy o come up wh a conservave esmaon of how far he core regon should be expanded based on one-me expermens wh represenave desgns n a gven echnology, or hrough approxmae calculaons. Samplng nodes are represenave nodes sampled from each le havng a volage volaon. A few (less han 10) nodes per le wh wors volage volaon are sampled o represen he behavor of all nodes n hose les.we keep he le small enough for hs assumpon o be vald. The res of he nodes are absraced away by macromodelng. The LP formulaon assumes ha all he decap s added a he samplng nodes. Afer solvng he LP, however, we dsrbue he decap a he samplng nodes evenly among he nodes n he le regons ha he samplng nodes belong o. Vdd Vhre Wors Volage 0 1 Fgure : A volaon me wndow [, 1] The volaon me wndows for a samplng node are deermned from he volage waveform a ha node obaned hrough he dynamc analyss of he power nework. A volaon me wndow s marked by me nsances:,he me nsance of maxmum volage ha occurred before a volaon, and 1, he me nsance when he volage recovers back o V hre afer a volaon. An example of a volaon me wndow s shown n Fgure. The volage a s assumed o be he nal volage of he decap before dscharge. If a volaon regon has mulple volaon me wndows, he opmzaon s done sasfyng he charge ransfer consrans (dscussed n Secon.1) for each of hese wndows.. LP-BASED DECAP BUDGETING.1 Charge Transfer Equaons of a Macromodel The charge ransfer equaons correspondng o a volaon me wndow are derved by negrang equaon () over he volaon me wndow, [, 1]:» " R 1 # " R 1 # G11 G 1 Ud R G T 1 G 1 = 0 J R 1d 1 Vd J d + R 1 Id Now he macromodelng procedure as saed n Secon. can be appled o oban a charge-based macromodel: where Q s R 1 and B s (G T 1G 1 11 Q = A W + B (5) R Id, A s G G T 1G 1 11 G1, W s 1 Vd, R R 1 J 1d 1 J d). Noe ha Q represens he oal charge flowng hrough he pors from he decap o he nework durng he me perod [, 1]andW represens he average volages of he pors mulpled by he me duraon [, 1]. Now our sysem consss of he nework wh pre-exsng capacors and nducors as a macromodel, and he decap o be added as exernal elemens conneced o he pors of he macromodel, as shown n Fgure. The problem hus reduces o opmzng he decap added a he pors whle provdng suffcen charge o pull up he por volages above V hre durng he volaon me wndows. decaps Q sample nodes macromodel Q = A W + B Fgure : Charge macromodel wh decaps. Volage Based Consrans Fgure depcs he volage waveform a a node before and afer addng decap. The objecve of addng decap s o pull up he volage above he V hre level as shown n ha fgure. Vdd Vhre Volage Vo 0 W The volage curve afer addng decap The volage curve before addng decap 1 Fgure : Volage waveforms wh & whou decap Applyng he charge equaons of he macromodel (5) o he case where decap has been R added (.e. he upper volage waveform n Fgure, W s 1 Vd, and represens he shaded area shown n ha fgure. Approxmang he upper volage waveform o a sragh lne beween and 1,we can wre he volage consrans as: W (V o, + V hre ) ( 1 )/, SP, () 19

4 R where SP s he se of samplng nodes (pors), W s 1 V d for he samplng node, andv o, s he volage of he node a he sar of he wndow. The above volage based consrans make anoher approxmaon, vz. he V o obaned from he ransen analyss before addon of decap has no changed apprecably wh he addon of decap. In he charge equaons of he macromodel (5), Q s he charge flow from he decap node o he macromodel. In order o keep he supply nework volage above V hre, enough charge should be released from he decap. To keep he decap node self above V hre, he maxmum charge ha can be released from he decap s C (V o V hre ). Therefore, we have he consrans: where M = V o,1 V hre V o, V hre V o,m V hre M C A W + B () 5, C =[C1,C,.., Cm]T where C s he decap value a he node, V o, s he orgnal volage of he node. The operaon represens Hadamard s produc,.e, he enry-wse produc of vecors M and C.. Consrans on Capacance The decouplng capacance has o sasfy he capacance consrans, whch are specfed for he le regons. Afer we deermne he decap a he samplng nodes, we evenly dsrbue hem whn he le regons ha he samplng nodes belong o. Therefore, f C max, s he maxmum amoun of decap allowed n he le regon, we wll have X C k C max, (8) k le C max, s decded by he capacance densy possble for a gven echnology and he amoun of whe space avalable n he le. When a block consss of mulple les and he decap placemen s used for floorplan purpose, a more complex se of consrans on capacance can be derved from he floorplan consrans.. The Complee LP Formulaon Combnng he charge ransfer consrans (), volagebased consrans (), consrans on capacance (8), he decap budgeng problem for a volaon regon can be formulaed no a lnear programmng formulaon mnmze subjec o P SP C (9) M C A W + B W L P k le C k C max,, le volaon regon, where m = SP, C =[C 1,C,.., C m] T, M = V o,1 V 1 V o, V V o,m V m 5, L = (V hre + V o,1) ( 1 )/ (V hre + V o,) ( 1 )/ (V hre + V o,m) ( 1 )/ where SP s he se of samplng nodes. If we assume ha V reaches he hreshold volage a 1,henV s V hre and M s he same as M gven n (). Noe ha, n he above 5, formulaon, he elemens of vecor C are he only varables of opmzaon. 5. ENHANCEMENTS 5.1 Sequence of Lnear Programmng In Secon., we assumed ha he maxmum charge ha could be dscharged from a decap s C (V o, V hre ). Ths s rue only f he node s wors volage a 1, denoed as V, s V hre. Acually, V self s dependen on C. Therefore, consrans () should acually be rewren as (V o, V ) C X 1 k m A k W k + B, SP (10) Snce boh V and C are varables, consran (10) s no lnear anymore. To handle hs nonlneary, we use he followng erave procedure, wheren he lnear programmng opmzaon s solved n each eraon. 1. Se p = 0 and he nal value V p based on he ransen analyss resuls. If V p <V hre,sev p = V hre.. Se V p V, and deermne he decap budge C p by solvng he LP problem (9).. Se C p C, and ge he new node volage V by subsung W by (V + V o,1) ( 1 )/ andhen solvng he lnear equaons (V o,1 V 1) C 1 (V o, V ) C (V o,n V m) C m 5 = A (V 1 + V o,1) T (V + V o,) T (V m + V o,m) T 5 + B where ( 1 )/ st. Noe ha he lef hand sde of he above equaons mply Backward Euler formulaon of he decap, C. Here, V C s equvalen o C/h x() of equaon and V o, C s equvalen o he C/h x( h) ofequaon.. Updae V p+1. Here, we se a sep lengh σ o lm he soluon space o he neghborhood of V p.ifv s below V hre,weseobev hre. The procedure can be descrbed as follows. f V V p s sasfacorly small, sop; else f V <V hre, V p+1 else f V else f V else V p+1 = V p = p +1 = V hre V p σ, V p+1 V p + σ, V p+1 = V p = V p σ + σ Here, we could also use he frs order Tayler seres approxmaons of he nonlnear consrans (10) o replace he formulaon (9), as done n he radonal successve lnear programmng (SLP) [1]. Our comparson of he wo mehods showed he resuls o be smlar. Therefore, we use he above descrbed nuve, erave, procedure. 5. Relaxaon of Capacance Consrans Durng he erave lnear programmng procedure, he volage of he samplng nodes may bounce up and down before selng down. Bu, f he consrans on capacance are gh, hs may lead he opmzaon no he nfeasble 0

5 regon whou a chance for recovery. Therefore, we added a relaxaon facor o relax he capacance consrans. Wh hs, he objecve of LP formulaon (9) becomes mnmze X SP C + β C relax and he consrans on capacance become X C k C max, + C relax, le volaon regon k le where C relax s he relaxaon facor and β s a wegh facor. Generally, β s se o a very large number o force he capacance consrans o be sasfed. By seng he β, he relaxaon facor could also provde a rade-off beween he capacance consrans and he oal capacance requred. 5. Nonunform Tlng When here are mulple volaon regons n a nework, unform sze lng may cause problem. Typcally, some volaon regons occupy a large area wh a slow volage graden across he regon and oher volaon regons occupy very lle area wh a large volage graden across he regon. Gven wo nodes n1 andn, he volage graden s defned as V (n1) V (n) dvded by he dsance of wo nodes. When graden s large, he volage and curren drawn a wo closely locaed nodes could be que dfferen. The wo nodes hen can no be represened well by he samplng node anymore. Therefore, a very fne gran lng may become necessary. However, he fne gran lng wll cause large volaon regons o generae a very large LP problem. In vew of hs, we use non-unformed lng. Fne gran lng s appled o he small volaon regons wh hgh volage graden, whle coarse gran lng s used for large volaon regons wh small volage graden.. COMPLEXITY ANALYSIS AND EXPERIMENTAL RESULTS The proposed echnques were mplemened as par of an exsng n-house power grd analyss ool [1]. An effcen drec lnear solver based on Cholesky facorzaon was used for macromodelng and a publc doman lnear programmng solver, lp solve [15], was ulzed o solve he lnear programmng problem. All he expermens were performed on Lnux machnes wh.ghz CPU and 1GB-GB memory. Table 1: Benchmark Deals Chp #nodes Supply Wors #me Analyss vol(v) Vol(v) pons CPU(s) Chp-1 198, Chp-,88, The proposed mehod was benchmarked usng global power neworks from real desgns. The number of nodes, supply volage, wors volage, number of me seps smulaed and CPU me (n Seconds) for he dynamc analyss are lsed n Table 1. Ideally, we would lke o compare our resuls wh he exsng sensvy-based mehods ha could handle very large power neworks wh non-unform meshes. However, for example, [] requres abou 000s on a GHz Lnux machne o solve a non-unform power nework of abou 1 mllon nodes. Due o he absence of common benchmarks and he effor nvolved n mplemenng he complcaed sensvy-based algorhms whn he framework of our ool, we wll compare he expermenal resuls wh a smpler heursc mehod, and also presen a complexy analyss of our mehod..1 Complexy Analyss The varous approxmaons dscussed before usually cause he decap o be under/over-esmaed slghly. So, as saed n Secon., we erae beween he opmzaon and ransen analyss verfcaon (seps -) unl he volage consrans are sasfed. From column 11 of Table, we can see ha consrans are sasfed ypcally n 1- eraons. For each such (ouer) eraon, one sequence of lnear programmng opmzaon and one ransen analyss are execued. Suppose n, m, l, N and h l are he number of nodes, samplng nodes, me pons, volaon regons and lnear program eraons respecvely. Le O lp (m) beheme complexy of LP. Le n 1.5 l be he me complexy of he ransen analyss, as assumed n []. Then he me spen per ouer eraon s n 1.5 l + n 1.5 N + N O lp (m) h 1,where n 1.5 N s me spen on macromodelng of volaon regons. Generally he sze of samples m s several orders smaller han he orgnal nework sze, n. Mos mporan of all, by adjusng he lng granulary and usng non-unform lng, he sze of m can be kep whn conrol, ndependen of he sze of he power grd and he number of he volaon nodes. In our mplemenaon, we adjus he lng such ha m 000. Also, we se h l 10. If he volages of he samplng nodes do no converge whn 10 LP runs, we connue wh he nex sep of flow. Overall, he opmzaon me s lmed compared o ha of one ransen analyss of he enre nework, especally for large neworks. Ths can be seen from columns 10 and 11 of Table. For example, n case of Chp- wh 1.85V specfcaon, he me spen on ransen analyss runs (one for sep 1 and one for sep of Secon.) s 90s, whle he me spen on opmzaon s only 8s.. Comparson wh a Bnary Search based Greedy Mehod To benchmark he qualy of our LP based decap opmzaon, we compared he resuls wh anoher nuve mehod whch s a greedy mehod based on bnary search. The bnary search mehod places decap unformly across he core volaon regon. For each core volaon regon, he amoun of decap placed s decded hrough bnary search. The search begns by placng 0.1ff/um area capacance n he volaon regons. The capacance s ncreased by 10 mes n each sep unl he frs successful soluon s found. From hen on, he bnary search s used o deermne a mnmum capacance ha sasfes he volage consrans. Table compares he resuls of our mehod wh ha of he bnary search mehod. The volage specfcaon and number of volaon nodes are shown n columns and respecvely. The wors volage afer opmzaon, amoun of decap added, he CPU me, and he number of eraons are gven n columns - for he Bnary Search mehod and n columns 8-11 for our mehod. The CPU me ncludes he enre decap budgeng flow as gven n Secon., ncludng he pre-opmzaon ransen analyss, sequence of LP, and he pos-opmzaon ransen analyss verfcaon. The number of eraons refers o he number of opmzaonverfcaon eraons. The oal number of ransen analyss runs s hus #er +1. 1

6 Table : Comparson wh he Bnary Search Mehod Chp Spec #Vo he Bnary Search Mehod Our Mehod Vol(v) Nodes Vol(v) Decap CPU(s) #er Vol(v) Decap CPU(s) #er Chp nf nf nf nf nf nf nf nf nf nf 9.9 Chp pf pf pf pf pf pf pf pf 11. From Table, we can see ha our mehod always uses less decap han he bnary search based mehod. The larger he volaon regons, he more gans our mehod shows. Ths s because he proposed opmzaon places he decap a he mos needed locaon. In addon, our mehod always consumes much less CPU me. Noe ha, n order o speed up he greedy mehod, we dd he bnary search of all he volaon regons smulaneously. Oherwse, he CPU me per eraon wll be #voregon mes he CPU me per eraon lsed n Table. The drawback wh he smulaanous bnary search s ha a capacance value ha orgnally pulled he volages up successfully above he hreshold level may fal because of he capacance reducon of he neghborng volaon regons. In hs case, he bnary search has o be rese. Moreover, s dffcul o apply he greedy echnque o floorplan or placemen purpose due o he nflexbly n placng dfferen decap a dfferen locaons of one volaon regon based on floorplan/placemen consrans. To llusrae how he erave LP procedure mproves he qualy of opmzaon, Table shows he resuls of he frs eraons of 5 volaon regons of Chp-1 a.0v specfcaon. Column s he rao of he volaon regon area o he oal area, and column s he number of samplng nodes. The decap added n each of he eraons are repored under columns -. We can see ha only one eraon of LP may sgnfcanly overesmae he decap value. Afer a couple of eraons, he decap value becomes sable. In our mplemenaon, we se change n volage dv 0.00 as he condon of convergence. Wh only LP eraons, he volaon regons,, and 5 converged. Table : The frs four LP eraons Vo Area #Sample 1s nd rd h regon rao nodes (nf) (nf) (nf) (nf) Vo Vo Vo Vo Vo CONCLUSION In hs paper, we have proposed a novel and effcen charge-based decouplng capacor budgeng algorhm. We proposed several echnques o reduce he problem sze and also o have he problem sze scale well w.r.o he nework sze and he number of volaon nodes. We used macromodelng, and he localy effecs of he decouplng capacance, along wh non-unform lng. By negrang he nodal equaons over he volaon me perod, we replaced he me consumng sensvy calculaons and nonlnear programmng opmzaon wh a much cheaper sequence of lnear programmng procedure. Acually, each ndvdual echnque proposed here can be combned wh oher exsng approaches. Expermenal resuls were gven o demonsrae ha he proposed algorhm s capable of opmzng power neworks wh 5 mllon nodes usng 1- ransen analyss runs, and n a couple of hours. Comparson of our algorhm wh a greedy heursc shows he advanage of area effcency and run me effcency of our mehod. By modfyng he capacance consrans, he mehod can easly be ncorporaed no physcal desgn flows. 8. REFERENCES [1] H. Su, S. S. Sapanekar, and S. R. Nassf, An algorhm for opmal decouplng capacor szng and placemen for sandard cell layous, n ISPD, pp. 8, 00. [] J.Fu,Z.Luo,X.Hong,Y.Ca,S.X.D.Tan,andZ.Pan, A fas decouplng capacor budgeng algorhm for robus on-chp power delvery, n ASP-DAC, pp , 00. [] K. Wang and M. M. Sadowska, On-chp power supply nework opmzaon usng mulgrd-based echnque, n DAC, pp , 00. [] H.L,Z.Q,S.X.D.Tan,L.Wu,Y.Ca,andX.Hong, Paronng-based approach o fas on-chp decouplng capacor budgeng and mnmzaon, n DAC, pp , 005. [5] Z.Q,H.L,J.Fan,S.X.D.Tan,Y.Ca,andX.Hong, On-chp decouplng capacor budgeng by sequence of lnear programmng, n Proceedngs of h Inernaonal Conference on ASIC, 005. [] S. W. Drecor and R. A. Rohrer, The generalzed adjon nework and nework sensves, IEEE Transacons on Crcu Theory, vol. 1, pp. 18, Aug [] P. Feldmann, T. V. Nguyen, S. W. Drecor, and R. A. Rohrer, Sensvy compuaon n pecewse approxmae crcu smulaon, TCAD, vol. 10, pp , Feb [8] L. T. Pllage, R. A. Rohrer, and C. Vsweswarah, Elecronc and Sysem Smulaon Mehods. New York: McGraw-Hll, [9] S. Zhao, K. Roy, and C. K. Koh, Decouplng capacance allocaon for power supply nose suppresson, n ISPD, 001. [10] H. H. Chen and D. D. Lng, Power supply nose analyss mehodology for deep-submcron VLSI chp desgn, n DAC, pp. 8, 199. [11] M. Zhao, R. V. Panda, S. S. Sapanekar, and D. T. Blaauw, Herarchcal analayss of power dsrbuon neworks, TCAD, vol. 1, pp , Feb. 00. [1] C. Ho, A. Ruehl, and P. Brennan, The modfed nodal approach o nework analyss, IEEE Trans. Crcus and Sysems, vol. CAS-, no., pp , 195. [1] M. S. Bazaraa, H. D. Sheral, and C. M. Shey, Nonlnear Programmng. John Wley and Sons, Inc., 199. [1] A. Dharchoudhury, R. Panda, D. Blaauw, R. Vadyanahan, B. Tuuanu, and D. Bearden, Desgn and analyss of power dsrbuon neworks n PowerPC mcroprocessors, n DAC, pp. 8, [15] M. R. C. M. Berkelaar and e. al., LP SOLVE 5.5 Users Manual, 005.