System Level Estimation of Interconnect Length in the Presence of IP Blocks

Transcription

1 System Leel Estmaton of Interconnect Lengt n te Presence of IP locks Tarane Taga, An aapetan, an aj Sarrafzae UCLA Computer Scence Department, Los Angeles, CA 995 {taga, an, maj}@cs.ucla.eu Abstract Wt te ncreasng sze an sopstcaton of crcuts an specfcally n te presence of IP blocks, new wrelengt estmaton metos are neee n te esgn flow of large-scale crcuts. Up to now, te propose tecnques for wrelengt estmaton n te presence of IP blocks approace ts problem eter n a flat framework base on te geometrcal structure of te crcut or n a erarccal framework base on unform strbuton property for stanar cells. In ts paper, we propose a tecnque for erarccal eraton of wrelengt estmaton n te presence of sngle an multple blockages usng Rent s parameter of te crcut by assumng non-unform probablty strbuton for stanar cells. To measure te accuracy of our estmaton, we compare our results wt te results of placement an routng usng a commercal CAD tool. Te results llustrate tat n te presence of multple IP blocks, te aerage error of our tecnque s less tan 8%, as compare to ts counterparts wt te aerage error of 35% an 5%. Keywors Wrelengt Estmaton, Rent s Rule, Herarccal Placement, Large-scale Crcuts, IP locks, on-unform Probablty Dstrbuton. Introucton Computer-ae esgn flow s experencng te tren of combnng front-en floor-plannng an back-en pyscal placement, wc s nee necessary to acee more effcent esgns. In ts process, a fast an yet accurate estmaton of system parameters suc as power, clock frequency, an wrelengt s crtcal to proe te front-en tool wt accurateenoug nformaton to ajust te early esgn ecsons before proceeng eep n te esgn flow. Early work on wrelengt estmaton s base upon an emprcal moel known as Rent s Rule [4]. Rent s Rule correlates te number of sgnal nput an output termnals T, to te number of gates C, n a ranom logc network as TAC P. A s often calle Rent coeffcent, wc s te aerage number of pns per cell. Te Rent exponent, P, s te feature parameter of te crcut [5] wc etermnes ts complexty. Te ger te Rent exponent, te more complex a crcut s. Usng Rent s Rule, te frst work on wrelengt estmaton s one by Lanman an Russo [6] wc was later mproe by Donat []. [] presente a new analyss of Donat s moel tat yels te lengt strbuton functons of te nterconnecton at bot te erarccal leel an system leel. ore recent work mproes te estmaton by conserng non-unform probablty [7, ] or recursely applyng Rent's rule on an entre monoltc system [8]. ost of te researc one on wrelengt estmaton s base on regularly place crcuts suc as stanar cell esgns. Wt te tren towar IP-block-base esgn, macro cells as blockage (sometmes referre to as obstacle, are more lkely to be present n te crcut [3]. Te blockage may be an on-cp memory, analog or RF blocks, or pre-esgne ar IP. Te presence of te blockage may sgnfcantly ncrease wrelengt an cause congeston [3]. Snce te presence of blockage makes te tratonal wrelengt estmatons far from realty, new tecnques soul be ere to aress te problem of wrelengt estmaton. Te frst work on te wrelengt estmaton n te presence of obstacles as been one by Ceng et. al. [3, 9]. In [3], te autors entfe two stnct effects of obstacles on nterconnecton lengt: ( canges ue to te restrbuton of nterconnect termnals an ( etours tat ae to be mae aroun te obstacles. Teoretcal expressons of bot effects for two-termnal nets ae been ere n a flat framework base on geometrcal caracterstcs of te crcuts. In [9], te autors represent a more complcate analytcal moel usng a polynomal generaton tecnque conserng te layout regon aspect rato an te presence of te blockages. Ter work, oweer, oes not conser te complexty of te crcuts nto account an ence te aerage wrelengt obtane form tese tecnques oerestmate te actual wrelengt for crcuts wt large cp area. In ter meto as long as two crcuts ae te same cp area, number of IP blocks an geometrcal parameters for te IP blocks, tey result n a unque estmaton for te lengt of wre for bot crcuts, een toug tese two crcuts may ae fferent number an caracterstcs for stanar cells an nterconnectons. Anoter work as been one n [] wc s an extenson to Donat s erarccal meto for conserng te effect of IP blocks. Te major plus of ts work s to conser te complexty of crcut (measure by ts Rent exponent n ts analyss along wt geometrcal caracterstcs of te crcut. ut, ts man rawback s tat t assumes a unform probablty strbuton for all te stanar cells wc s not a realstc assumpton. In te current researc, startng from Donat s erarccal tecnque [], s approac s extene to be able to conser obstacles n te placement area. It s sown ow to ere a close form expresson for te total wrelengt n te cases tat te cp area nclues eter a sngle blockage or multple blockages. In ts work, we assume a non-unform probablty strbuton for stanar cells oer non-blocke parts of te crcut wc was sown to be an accurate moel for crcuts n te real worl [7]. Smulaton results on te large crcuts confrm tat, n te presence of te obstacles, ts tecnque s more sutable to estmate te wrelengt tan non-erarccal tecnques or tecnques wt unform probablty assumpton. Te remaner of ts paper s organze as follows. In Secton, our metoology s explane. Wle Sectons 3 an 4 present a teoretcal analyss of aerage wrelengt n te presence of sngle or multple obstacles, Secton 5 sows te expermental setup an smulaton results of te propose tecnque. Conclung remarks an future work are presente n Secton 6.. etoology Smlar to [, ], our tecnque to estmate te aerage wrelengt s base on a top-own erarccal placement of te crcut nto a square anattan gr n te presence of obstacles. Te crcut s parttone erarccally nto four sub-crcuts. Ts erarccal parttonng s contnue untl te number of te stanar cells n all of te sub-crcuts s equal to or less tan β, were β s a preefne constant. At eac leel of erarcy, we euce te aerage number n of nterconnectons an te aerage lengt r of nterconnectons between eac two subcrcuts belongng to te same (+ leel of erarcy, but fferent leel of erarcy.

2 Gen te aboe moel for te crcut, te feature parameter of te crcut P wc s gen by Rent s rule, an te aboe parttonng sceme, we want to estmate te total nterconnecton lengt of te crcut n te presence of obstacles. Ts s one by calculatng te aerage number of nterconnectons n an te aerage lengt of te nterconnectons L at eery erarccal leel. Te total nterconnecton lengt oer all erarccal leels s Ltot nl ( were H s te fnest leel of erarcy. Snce at eery step of parttonng, eac sub-crcut s e by four, an n te last leel of erarcy te number of cells nse eac sub-crcut s less tan te factor β, te number of leels can be extracte from, C H log 4 ( ( Te aerage number of nterconnectons between te subcrcuts n eac leel of erarcy as been extracte usng Rent s exponent wc s expermentally proen to be a goo ncator of te complexty of te crcut []. Usng a smlar type of analyss as [3], te aerage lengt of nterconnecton between te sub-crcuts s calculate n eac leel of te erarcy. Ten, usng formula (, we estmate te total wrelengt by multplyng te aerage number of nterconnectons by te aerage lengt of nterconnectons for eac leel of te erarcy an summng all tese alues oer all te erarccal leels... Aerage umber of Interconnectons at Eac Leel of Herarcy In [], Donat sowe tat by apply Rent rule on eac leel of erarcy, te aerage number of nterconnectons at eac erarccal leel can be calculate from, P L( P n α AC( 4 4 (3 were C s te total number of cells, P s te Rent exponent, A s te Rent coeffcent an α s te fracton of te number of termnals for all te nterconnectons n one leel. Te alue α s ½ f eac net as just two termnals, an s somewat greater tan ½ but less tan for crcuts wt mult-termnal nets []. Parameter L sows te leel of erarcy... Te Aerage Lengt of Interconnectons at Eac Leel of Herarcy To start analyzng aerage wrelengt at eac leel of erarcy, we nee to make some assumptons an efne some termnologes at frst. Assumpton : To compute L we assume tat all of te nets ae two termnals. Ts smplfcaton s base on te knowlege tat tese nets are muc more tan all te oter nets n te crcut an tat mult-termnal nets can be moele as a collecton of two-termnal nets []. Te effect of mult-termnal nets s ncorporate nto our estmaton by usng ger alues for α n te calculaton of te aerage number of nterconnectons n as sown n te preous secton. Assumpton : We assume tat te aalable routng layers are suc tat te blockages are obstructons for bot placement an routng. Ts moel s base on wat commercal tools support for placement an routng of large-scale crcuts. Defnton : In leel of erarcy an ntra-bn wre s a wre tat ts termnals belong to te same bn,.e. same part of te cp area. Defnton : An nter-bn wre s a wre tat ts termnals belong to fferent bns n leel of erarcy, but to one bn n te leel (+ of erarcy. Defnton 3: In te presence of te obstacles, te transparentblock wrelengt, L T, s efne as te wrelengt wen te obstacle s assume to be transparent an wres can pass troug t. For two-termnal nets, transparent-block wrelengt s te anattan stance between tem. H Defnton 4: Detour wrelengt, L DT, s te etour lengt neee n a routng wre n te presence of te obstacles. In oter wors, L T L -L DT, were L s te Stener mnmal lengt of te net suc tat no part of te wre s route nse any of te obstacles. Fgure sows te ntra-bn transparent-block an etour efntons. In te next two sectons, we sow ow to obtan te aerage wrelengt n eac leel of erarcy n te cases tat we ae sngle blockage or multple blockages n te cp area. For te case tat tere s only a sngle blockage, we escrbe te researc one by [] wc assumes unform probablty strbuton for stanar cells an llustrate ow to exten ter meto for non-unform probablty strbuton to mproe te accuracy of estmaton. For te multple blockages case we just escrbe our meto wc assumes non-unform probablty strbuton for cells. It s sown ow to ere transparent-block an etour wrelengt for bot ntra-bn an nter-bn nets. ascally, te nter-bn aerage wrelengt analyss s te one we nee to use n our metoology. We ntrouce te analyss for ntra-bn aerage wrelengt to elp us ong te analyss for nter-bn aerage wrelengt. To obtan te aerage wrelengt, we ecompose t nto tree parts, namely transparent-block an etour n X an Y rectons suc tat, L L + L + L (4 T DT were L DT an L DT are te aerage etour n X an Y recton an LT s te aerage transparent-block wrelengt. 3. Teoretcal Analyss of te Aerage Wrelengt wt Sngle IP locks In ts secton, frst we escrbe te meto use n [] to analyze te aerage wrelengt n te presence of a sngle blockage by assumng a unform probablty strbuton for cells an ten we llustrate our approac to exten t by remong ts assumpton. 3.. Unform Probablty Dstrbuton A. Aerage Intra-bn Wrelengt If tere s no obstacle n te bn area, te aerage ntra-bn wrelengt can be easly obtane from (5, as sown n [3]. ( x x + y y x x y y Unform Lnt ra x xyy DT + 3 were subscrpt ntra enotes tat te aerage s taken oer all ntra-bn nets, n contrast to te nter-bn nets wc wll be scusse later. To obtan te aerage wrelengt n te presence of an obstacle, let us assume te obstacle s center s at poston (a, b an ts wt an egt are respectely W an H (see Fgure. ote tat n ts case P an P must be place outse of te obstacle,.e. P, P A S, were A s te set of all te ponts nse te bn an S s te set of all te ponts nse te obstacle. Wt te same type of analyss as for formula (5, we can calculate te aerage transparent-block wrelengt [3], (,, W, H, a, b/ (,, W, H, a, b were, (,, W, H, a, b ( x x + y y x x y y P, P A S ( 6 ( + + W H ( H + W 6HW( + ( a + b HW ( W + H HW ( a + b (,, W, H, a b x ( xyy WH P P AS,, L T, ntera (5, as (6a (6b

3 ascally, n equatons (6a an (6b, we get te ntegral oer te non-blocke area of te bn. y tat, we mean tat te termnals can be eerywere except te blocke part of te bn, but te nterconnectons can pass troug te blockage. For analyzng te ntra-bn aerage etour wrelengt n ertcal recton, we conser te etour alue as a ranom arable. For a ranom arable we ae E(Y E(Y X.Pr(X; so we ae,, ntra, ntra PrDT (7 were, ntra s te aerage etour n Y recton, gen tat a etour appens n tat recton. In [, 3] t was sown tat P (x,y H W (a,b P (x,y L T L DT L ntra Fgure : Defnton of ntra-bn transparent-block an etour wrelengt. DT, ntra s equal to /3H. On te oter an, Pr DT L s te probablty of occurrence of te etour n tat recton. Tus we ae, H( a W / H ( a W / L DT H, (8 ntra 3 WH (. Aerage Inter-bn Wrelengt Accorng to Defnton, a two-termnal nter-bn net s efne as a net wt one termnal n a bn an te oter termnal n te ajacent bn, eter orzontally, ertcally or agonally. In te followng, t s sown ow to ere expressons for calculatng aerage nter-bn wrelengt n te presence of an obstacle for orzontally ajacent bns. Te cases of ertcally an agonally ajacent bns are smlar to orzontally ajacent bns an so omtte from ts scusson for brety. A oreoer,, nter s te aerage etour lengt n Y recton gen tat a etour occurre n ts recton. Smlar to (8, L DT, nter equals to /3H. So, te aerage etour wrelengt n Y recton can be extracte from ( as, H( a W H( a W L / / DT,nter H ( 3 ( W H ( WH Te aerage etour wrelengt n X recton can be foun from,, nter PrDT., nter (3 were, nter can be calculate as, 4 mn( x ( a W + W W x + W x x P UA, P L /,, nter (4 x x + P UA P L x x, P LA, P U an Pr DT s te probablty of occurrng a etour n orzontal recton, wc s equal to, ( W ( b H / W ( b H / + W ( b H / W ( b H / PrDT (5 ( W H ( W H Hang a L T, L DT an L DT as te aerage transparent-block an etour wrelengt of orzontal ajacent bns A an, L ( A, n ts case can be extracte form ( on-unform Probablty Dstrbuton In te tecnque scusse n te last secton wc was base on te Donat s meto for analyss, eery erarccal leel s treate separately wt no knowlege of te lengt of nterconnectons from oter leels of erarcy. Howeer, te optmal placement tecnques try to place te nterconnecte blocks as close to eac oter as possble. In te erarccal moel, ts means tat an optmal placement tecnque wll place blocks tat are nterconnecte to a block of anoter square closer to te borer of te two squares as sown n Fgure 5. Te preous tecnque oes not conser ts nformaton. Defnton 5: We efne te nterconnecton lengt strbuton as te alue ncates tat for eac lengt l, ow many nterconnectons ae ts lengt. It was calle te occupancy probablty n [7]. A W AC W D W W. (a,b H A H. (a,b C D H CD Fgure : Two orzontal ajacent bns. Horzontally Ajacent ns Horzontally ajacent bns are sown n Fgure. In ts case, te aerage transparent-block wrelengt can be compute as [], (,, W, H, a, b (,, W, H, a, b (,, W, H, a, b L T, nter ( W H ( W H (9 were WW +W, b b b, a a-w/+w /, an a W / an s te same functon as n (6a. Te aerage etour wrelengt n Y recton can be expresse as, L Pr L ( DT, nter DT. DT, nter were PrDT s te probablty of occurrng a etour n Y recton wc can be expresse as, H ( a W / H ( a W / PrDT ( ( WH ( WH Ts probablty s equal to te porton of te area wc a etour can occur e by te non-blocke area of te two bns. Fgure 3: n A an D are agonally an bn an D are orzontally ajacent Accorng to [7] t can be seen tat for all but nearest-negbor ( 4P nets (l, te power-law functon of te form l can approxmate te occupancy probablty. Ts result was frst reporte n [] an re-ere n [7, 8] usng fferent tecnques. Ts result suggests tat, for any lengt of nterconnecton, te ger te lengt s te less te number of nets wt ts lengt can be, wc reflect te beaor of te optmal placement. A. Aerage Intra-bn Wrelengt ( 4P Conserng l as te occupancy probablty of ang a wre wt te lengt of l nstea of te unform probablty strbuton, te estmaton wll be mproe by multplyng ts occupancy probablty strbuton by te structural probablty strbuton wc we a before for te nterconnecton

4 estmaton. So, te formula (6a wll be transforme to L T, ntera, as (,, W, H, a, b/ (,, W, H, a, b were, non unform ( ( ( 3 P +,, W, H, a, b x x y y xxyy (6 P, P A S In orer to sole ts formula, accorng to Fgure 4, tere are 8 fferent regons for eac of te two ponts (x, y, an (x, y. So tere woul be 64 fferent ntegrals to conser for solng formula (6. Let us efne, ( x x y y ( 3 δ P + (7 Conser te followng equaton, δ xx yy xx yy + xx yy P, P A δ P A S, P S δ P, P S (8 + δ xx yy P, P A S formula (6 s te last term of te aboe formula. Te frst an tr terms n te formula (8 can be easly compute by an or by usng atematca equaton soler software. Te Secon term as 8 cases to conser, snce te frst pont moes nse te non-blocke part wc conssts of 8 regons an te secon pont moes nse te blocke area wc conssts of regon. So, te number of ntegraton s reuce to 8 +. Te formula for te case tat te frst pont s n regon an te secon pont s n blocke area can be obtane from, ( ( ( ( Hb H / Wa W / 3 P + + +, S,, W, H, a, b x x y y xxyy ( S Fgure 4: Integraton regons for ntra-bn transparent-block wrelengt An te formula for te case tat te frst pont s n regon an te secon pont s n blocke area can be obtane from, ( ( ( ( H b H / W x 3 P + +, S,, W, H, a, b x x y y xxyy ( ecause of te symmetry of te problem, te ntegraton for all of te oter regons can be calculate from (9 an ( usng fferent lmts for te ntegrals. ote tat te enomnator of te aerage transparent-block wrelengt soul be mofe to, nonunform ( ( ( 4 P +,, W, H, a, b x x y y xxyy ( on any specfe regon of ntegraton.. Aerage Inter-bn Wrelengt As t was sown n secton, all of te formulas for nter-bn transparent-block part of te wrelengt are usng te formulas for te ntra-bn transparent-block analyss. Hang mofe te formulas for transparent-block ntra-bn wrelengt, te formulas for transparent-block nter-bn wrelengt can be extracte accorngly wt no furter analyss. In te next part we sow ow to extract etour nter-bn wrelengt wt non-unform probablty... Horzontally Ajacent ns Let us conser Fgure 5, for calculatng te etour part of te wrelengt. Te beaor of an optmal placement necesstates ang ger ensty of termnals closer to te mle borer. A smple obseraton sows us tat te non-unform probablty strbuton of termnals oes not ae an effect on te ertcal etour of te ponts on te rgt an left ses of te obstacle, snce te concentraton of cells ffers n X recton. Howeer, te non-unform probablty strbuton as an effect on te orzontal etour of te ponts on te top an te bottom ses of te obstacle. Te effect of non-unform termnal strbuton on te total orzontal etour n X recton gen tat a etour appens woul be of te form of, ( ( 4P 4 ω( x, x x + x + y y xxyy, nonunform P UA P L (, ( ( 4 P 4 x + x + y y xxyy P UA P L Fgure 5: Te optmal placement beaor n presence of blockage ω ( x, x mn( x a W / + W W x + W. were (, As t can be seen, solng te nomnator of te aboe equaton manually s ffcult. umercal equaton solers can always elp n tese cases. 4. Teoretcal Analyss of te Aerage Wrelengt wt ultple lockages Te analyss of nter-bn aerage wrelengt n te presence of multple blockages can be performe by usng te analyss for aerage wrelengt n te presence of a sngle blockage. Te nter-bn aerage wrelengt for ertcally an agonally ajacent bns use te same type of analyss as orzontally ajacent bns an are omtte ere for brety. 4.. Horzontally Ajacent ns As sown n Fgure 6, n te presence of multple blockages, smlar to [3] te aerage transparent-block wrelengt can be calculate from, were, A A, ( A, ( A, + ( A, ( A, j ( A ( x x + y y j (3, xx yy (4 x, y A x, y (3 P were (x,y an (x,y are te coeffcents of te two ponts on eery rectangular regons A an, respectely. A A A 4 A A 3 Fgure 6: Horzontal ajacent bns wt multple blockages[] Te aerage etour wrelengt s more complcate n ts case. If te obstacles o not oerlap neter n X span nor n Y span, Ceng et. al. [3] sowe tat te effect of te obstacles on te aerage etour wrelengt s ate an ts problem can be treate as a combnaton of sngle blockage problems. ut for real crcuts ts assumpton seems to be too smplfyng an t mgt oerestmate te etour n some cases. Fgure (7 llustrate ts problem. Te etour for te termnals nse regon A an 3 5 4

5 equal to te maxmum etour aroun A an wc s less tan te summaton of te etour for tese two blocks. Instea we use a eurstc to estmate te amount of etour for ts case. Lemma. Eery two sjont blocks oerlap at most n eter X or Y spans. If two blocks oerlaps n bot spans, one coers all or part of te oter one; so we ecompose tem to seeral blocks wt no oerlap. Detour aroun eac blocks nmum etour aroun bot blocks Regon bounares n bounary Fgure 7: Horzontal ajacent bns etour wt multple blockages To tackle ts problem, frst we enumerate all te regons generate by blockages as sown n Fgure (7. If we ae n blockages, te number of tese regons equal to (n + snce eery blockages as two borer lnes n eac recton wc generates (n + regons n tat recton. Te man problem appens for te termnals nse te regons wc locate on te oerlappng parts of two or more blockages n eter X or Y spans. For tese regons we use a trck to transform tem to te stanar orzontal or ertcal etour problems. Algortm. As sown n Fgure (8 for regons A an, we cut tem nto two parts usng a ase lne. We efne te etour between (A, to be equal to mnmum of te etours of te top an bottom part of te otte lne an te aton of te etours for all te blocks generatng tese two regons. All tese etours can be sole usng te formula (. Smlar type of analyss ols for termnals n (A,, (A, an (A,. Snce te number of regons to conser s large, we perform te complete calculaton for te etour of θ% bggest blockages an a up te etour of te rest. y ajustng θ%, we can tune te accuracy an run tme trae-off. In our experments, we pcke θ. A A A Fgure 8: Algortm for etour wt multple blockage In [3], te autors nestgate te etour wrelengt for te case tat tere s one bg sngle blockage an numerous small blockages. Tere, tey sowe tat te aerage etour wrelengt s strongly correlate wt te geometrcal parameters of te bg blockage suc as wt, egt, splacement an aspect rato, an s not relate to te geometrcal parameters of oter small blockages. Ts motates us to estmate te etour wrelengt for tese types of crcuts by just conserng te bg blockage an gnorng te small blockages usng te exact meto presente n te preous secton. ut f all te blockages are of te same sze, te exact meto n preous secton oes not work an we use te eurstc presente n ts secton. Te analyss for te transparent-block an etour parts of te wrelengt woul be te same for ertcally-ajacent an agonally-ajacent bns an are so omtte from ts scusson for brety. 4. Aerage Wrelengt Hang a te aerage nter-bn wrelengt for orzontally, ertcally an agonally ajacent bns, te aerage nter-bn wrelengt can be obtane for eery leel of erarcy lke n []. For eery leel of erarcy, sown n Fgure 3, te aerage nter-bn wrelengt can be wrtten as, δ ( L nter ( A, Lnter ( C, D Lnter ( A, C Lnter (, D L (5 ntr, H 6 + δ L ( A, D + L (, C ( ( ntr ntr were,, an, respectely, enote tat te corresponng bns are orzontally, ertcally, or agonally ajacent. oreoer, δ s a parameter to capture te optmzaton beaor of placement algortm wc faors orzontally an ertcally ajacent bns to agonally ajacent bns. Placement algortms try to mnmze te lengt of wres as muc as possble. So, te probablty of ang a sort wre s more tan ang a long wre. Tat necesstates us to fferentate between orzontally/ertcally ajacent bns an agonally ajacent bns. To compute δ, we use te equaton ere n [, 7] for wre lengt strbuton for te entre crcut. In [] by usng smple teoretcal conseratons t s sown tat te normalze wrelengt strbutonε l for a goo two-mensonal placement s of te form of, ( l lmax ( l l γ ε l Cl (6 max wcγ s relate to Rent Exponent troug equaton, P + γ 3 (7 an l max s a constant rectly relate to te sze of te bn. Ts formula ncates tat te number of wres wt lengt l s ecreasng wt te factor of l ( P3 by ncreasng te lengt of te wre. Smlar to [], f we conser ε as te wrelengt / strbuton between orzontal or ertcal bns, an ε as te wrelengt strbuton between agonal bns, we ae, 4 ε / + ε (8 Snce te relatonsp between lengt of te orzontal/ertcal wres l / an agonal wres l for te bn confguraton n Fgure 3 s as, we can extract te lengt strbuton an, ε / l l /, anε from, 4ε / ε P3 + ( P3 ( ε / ε / (9 4ε / δ (3 5. Expermental Results We ae mplemente te tecnques propose n sectons 3 an 4 n C. We compare our wrelengt estmaton meto wt te meto n [], Ceng wrelengt estmaton meto [3], an te actual wrelengt of te crcut after te routng. For placement an routng, we use te agma lastfuson wc s a commercal CAD tool. For ong smulatons, we consere bot meum an large sze crcuts. In orer to erfy our teoretcal results, on te real-worl crcuts, we pcke most of our bencmarks from te ISPD 5 placement bencmark sute [3]. gblue3 s not use snce t as seeral moable macros wc soul go troug a process of floorplannng to fx te macros for runnng te estmaton algortms on t. So we exclue ts bencmark from te experments. gblue4 s a ery uge bencmark an we couln t run tem on our aalable systems. Te fle format of ISPD sute bencmarks s ookself. Te agma lastfuson tool accepts LEF/DEF format. We use te conerter from [5] to conert te ookself format to te LEF/DEF format to route te bencmarks wt ts tool. For te frst set of experments we use te bencmarks wt just a bg sngle IP block. To aapt te bencmarks to our

6 expermental purpose, we kept te bggest blockage an cange all te oter blockages to stanar cells. Te secon set of experments s for estmatng wrelengt n te presence of multple blockages. For ts set of experments we use te orgnal crcuts from ISPD 5 bencmark sute. Te number of cells an nets n eac of tese test crcuts are sown n Table. Te number of fxe cells s te number of obstacles nse te cp area, an oes not nclue te fxe I/O termnals. Te utlzaton s te percentage of area of stanar cells oer te non-blocke area of cp. For extractng Rent s exponent for eac bencmark, we use te same meto as [5] by performng a global placement of te bencmarks usng te Dragon placement tool [4] wc s an acaemc placement tool base on erarccal mn-cut parttonng wt termnal propagaton. Table : Specfcaton of te encmarks Test Crcut #Cells #Fxe Cells #et Utlzaton (% locke Area (% Test3, , Aaptec Aaptec 55, , Aaptec3 45, , Aaptec gblue gblue Table sows te total wrelengt estmaton n presence of a sngle blockage for Ceng s meto, te meto presente n [] an our meto. Actual wrelengt s reporte by lastfuson tool after te placement an routng. Table llustrates te same ata for wrelengt n presence of multple blockages. As t was sown n [], te reason tat Ceng s estmaton s far from actual wrelengt on te crcuts wt g complexty s tat t only consers te geometrcal caracterstcs of te crcut towar estmatng wrelengt at te top leel an gnores te complexty of te crcut totally. ot of te metos n [] an our meto, oweer, conser te complexty of te crcuts nto account troug calculaton of n k n eac leel of erarcy. As t can be seen, Ceng s meto results n better estmaton for te meum bencmark Test3. Te reason s tat on te meum-sze bencmarks wt lower complexty, wrelengt epens more on te geometrcal structure of te crcut an not on ts complexty. Table : Total Wrelengt Estmaton for Sngle lockages Estmate WL Actual Crcut eto Ceng Ours WL n [] Test Aaptec Aaptec Aaptec Aapcte Aerage Error (% Te reason tat bot approaces by Ceng an [] oerestmate te wrelengt s tat te probablty strbuton of te cells on te non-blocke porton of te cp area s consere as unform for bot approaces. Placement tools try to keep te connecte cells as close to eac oter as possble. So, te number of sort wres n te cp area s more tan te number of long wres after te placement. Tat s te reason tat unform strbuton probablty oerestmates te lengt of wres, snce t consers equal probablty of occurrence for eery lengt of wre. Our approac fxe ts problem by conserng nonunform probablty strbuton for cells. All of te tree tecnques perform wrelengt estmaton qute fast. For te largest test case gblue, our meto takes 85 Table 3: Total Wrelengt Estmaton for ultple lockages Crcut Estmate WL Actual eto Ceng Ours WL n [] Test Aaptec Aaptec Aaptec Aapcte gblue gblue Aerage Error (% secons, te meto n [] takes 8 secons an Ceng meto takes 75 secons. Tese alues nclue te runnng tme for processng te nput fles an bulng te ata structures. 6. Conclusons an Future Work In ts paper we propose a erarccal tecnque for wrelengt estmaton n te presence of blockages base on te assumpton of non-unform probablty for te strbuton of stanar cells. Smulaton results sow tat ts tecnque estmate te wrelengt muc more accurately as compare to ts oter counterparts suc as [, 3]. Te aerage error of ts tecnque for large crcuts s 4.9% for sngle blockage an 7.8% for multple blockages. Ts work can be use n early esgn stages to proe an estmaton of te en-pont wrelengt of a esgn. Ts meto can also be extene to perform erarccal probablstc congeston estmaton n presence of IP blocks. 7. References [] T. Taga an. Sarrafzae, Herarccal wrelengt estmaton for large-scale crcuts n te presence of IP blocks, submtte to IEEE Transacton on Very Large-Scale Integraton Systems, Specal secton on System Leel Interconnect Precton, aalable at ttp://er.cs.ucla.eu/ ~taga /tec_report_86.pf [] W. E. Donat, Placement an aerage nterconnecton lengts of computer logc, IEEE Trans. Crcuts Syst., ol. CAS-6, pp. 7 77, 979. [3] C.-K. Ceng, A.. Kang,. Lu, an D. Stroobant, Towar better wreloa moels n te presence of obstacles, n Proc. Asa an Sout Pacfc Desgn Automaton Conf., Feb., pp [4] H.. akoglu, Crcuts, Interconnectons, an Packagng for VLSI. Reang, A: Ason-Wesley, 99. [5] X. Yang, E. ozorgzae, an aj Sarrafzae, Wrelengt estmaton base on Rent exponents of parttonng an placement, n Proc. Int. Work. System-Leel Interconnect Precton,, pp [6]. S. Lanman an R. L. Russo, On a Pn ersus lock Relatonsp for Parttons of Logc Graps, IEEE Trans. on Computer, C-, 97, pp [7] D. Stroobant an J. V. 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[3] ttp:// [4] ttp://er.cs.ucla.eu/dragon [5] lsca.eecs.umc.eu/k/placeutls/