Estimation of Wiring Area for Hierarchical Design

Transcription

1 0 Estimation of Wiing Aea fo Hieachica Design Bend Schümann Gehad Zimmemann SFB-124 Repot-No. 24/92 Fachbeeich Infomatik Univesität Kaisesauten Ewin-Schödinge-Staße D-6750 Kaisesauten

2 Estimation of Wiing Aea fo Hieachica Design 1 Estimation of Wiing Aea fo Hieachica Design Abstact The estimation of the aea of ces befoe ayout, especiay of shape functions, is vey impotant in a hieachica design pocess and a necessity fo top down design styes. The majo pobem is the estimation of wiing space. Fo the intena wiing between the subces eative good modes exist. Pobems occu fo the wiing space to connect the pins of a ce with its subces. This pape gives a mathematica desciption of the wiing modes fo both cases and especiay teats the pin connection pobem. Ou goa is to find the simpest mode that expains the expeiments and to intoduce ony a sma numbe of mode paametes. A set of vaues fo these paametes is an impotant esut which is suppoted by new measuements fo standad ce bocks. 1. Intoduction The focus of inteest in CAD fo VLSI has shifted fom individua agoithms to compete CAD systems. This shift has esuted fom inceasing demands on the compexity, pedictabiity, and efficiency of the design pocess and on the quaity of the poducts. Due to the compexity of the task the design pocess has been spit into steps between eves of a pat-of hieachy and between desciption domains, oughy the behavioa, stuctua, and physica domain /GaK83/. These steps coespond to individua toos and agoithms within the toos. One equiement to combine the toos into a CAD system is the CAD famewok. Because of the stong dependency between the individua design steps, thee is a second equiement fo a CAD system, which we wi ca the gue. We can view the design of a digita system as a panning pocess with stepwise efinement of an initia specification /GiO84/. In each efinement step decisions ae made on the basis of the state of the cuent design (as a esut of design decisions of the past) and on the basis of a pediction of futue design decisions. This pediction is the gue between past and futue and thus gues a steps of the design pocess togethe. If the futue design steps woud be fuy pedictabe, as fo exampe the size of an unfoded PLA, the gue pocess woud be a - not necessaiy simpe - cacuation with known paametes. In the VLSI design pocess this is typicay not the case. As ong as thee is fee design space, futue decisions ae not fuy pedictabe and have to be estimated, as fo exampe the esut of pacement and the esuting wiing space.

3 Estimation of Wiing Aea fo Hieachica Design 2 Because of this, the gue pays an impotant oe in a muti-step design pocess, moe so in top-down than in bottom-up design styes. Fo exampe, high-eve synthesis is cuenty hampeed by vey uneiabe cost and pefomance abstactions as objective functions. The goa of ou eseach is an impovement in a cases whee gue is necessay. So fa we have speciaized on aea estimations /Zim88/ and the wok epoted hee is pogess in this diection. Ou next goa is pefomance estimation and has, in the case of CMOS, to ey heaviy on pecise aea and wie ength estimations. Estimation cannot be pecise in the sense that the fina esut and the estimation fuy match. Thee wi aways be a toeance. The moe pedictabe the futue design pocess is, the smae the toeance wi be. Thus a eduction of pediction toeances can ony be achieved by making the design steps moe pedictabe. In addition to the toeance, estimation eos ae possibe. Ou goa is the eduction of these eos. A second goa is the quantitative detemination of estimation toeances and eos. Fo the use of the gue, a designe o a too, it is impotant to know how eiabe the estimations ae. The eiabiity can be eithe expessed by uppe and owe bounds o by pobabiistic measues. This is a new but impotant fied of eseach whee we can epot ony peiminay esuts. Many diffeent appoaches have been pubished fo aea estimation. Reis was one of the fist with an empiica method. Hee we do not want to go into futhe detais. Expeience with ou shape function geneato /Zim88/ has unveied some inteesting aspects. A compaison between pedicted and ea foopan shapes ed us to the beief that the pedicted aea is cose to the owe bound of a the foopans that can be geneated /Zim90/. Just by changing the oientation of the fist cut ine of foopans, changes in aea of ove 70% have been found. Othe dependencies of this kind wi be shown fo standad ce bocks in this pape. We theefoe have to be caefu about what we want to pedict. Pedictions can aso be used in the sense of what-if toos. A designe may not want to compete a design, but evauate diffeent options. The knowedge in this case is again detemined by the past decisions and by defaut assumptions about the futue. Such assumptions aso pay an impotant oe in gue appications of pedictions and the defaut paametes we found ae theefoe, besides the estimation method, an impotant esut. 2. The Aea Estimation Mode 2.1 Shape Functions The goa of ou method is the estimation of shape functions of ectangua ayouts. Shape functions fom the bode between feasibe and infeasibe shapes. Figue 1 shows the shape function of a fixed macoce A. The back dot (cone) defines the x and y dimensions of the ectangua ce A. A othe points on the cuve and in the hatched aea ae made feasibe by adding empty space to A on any of the fou sides o within A. It can be used fo wiing.

4 Estimation of Wiing Aea fo Hieachica Design 3 y A Figue 1: Maco shape function x IF A has diffeent shapes, fo exampe (A 1,..., A 4 ) by contoing the ayout of A with diffeent paametes, then a muti-maco shape function as in figue 2 evoves. Pe definition as a owe aea bound, the shape function is monotonic and A 4 coud be constucted by adding empty aea to A 2 o A 3. The shape function in figue 2 is theefoe fuy defined by the cone coodinates A 1, A 2, A 3. y A 1 A 2 A 4 A 3 Figue 2: Muti-maco shape function x A 4 may have popeties that neithe A 2 o A 3 have. Fo exampe, A 4 may have a pins on one side, wheeas A 2 and A 3 have pins on a fou sides. Thus, in a specific foopan, A 4 may, despite age aea, poduce a smae foopan o a shote citica deay path. Theefoe, we do not aways enfoce monotony. Fo fexibe ces, ony estimated shape functions can exist, if the ayout stye offes design decisions that infuence the ayout aea. Thus each cone point has toeances in x and y and the shape function has a toeance band aound the aveage cuve as shown in figue 3. In the exteme the shape function can be a continuous cuve, but due to the toeances it can aways be appoximated by a suitabe numbe of cones. Modue geneatos can be eithe modeed by muti-maco shape functions if they geneate fuy pedictabe ayouts o by fexibe ce shape functions othewise.

5 Estimation of Wiing Aea fo Hieachica Design 4 y x Figue 3: Fexibe ce shape function If we assume that a ayout geometies fo which we seek shape functions can be appoximated by sicing topoogies, these can be epesented by sicing tees. If shape functions fo the eaves of a tee ae known and the oientations of a sices in the tee ae detemined, the shape functions can be easiy added up the tee and thus fo the oot node which epesents the ce of inteest /Ott83/. Figue 4 shows this coespondence. c A B D c C E D F G F C E G B A Figue 4: Sicing stuctue and tee If the oientations of the sices ae not known, optima oientations can be detemined by adding monotonic shape functions hoizontay and veticay and seecting the cones with minima aea of both esuts. This esuts in a monotonic shape function at each intena node of the tee with the attibute vetica o hoizonta at each cone.

6 Estimation of Wiing Aea fo Hieachica Design 5 The esuting shape function estimates the tota aea ony if no additiona wiing space is equied. We theefoe extended the mode by tying to estimate this space /Zim88/. This ed to a five coo mode. 2.2 The Five Coo Mode Let us ook at two eves of a muti-eve pat-of hieachy. This invoves thee eves of ces, 1,, and + 1.The eve of a ce s denoted by a supescipt and the ce instance by a subscipt. The ode of eves is such that is composed of ces which ae composed of ces (figue 5). Let us futhe assume that we seek the shape function of. Each ce has pins at its peimete which we denote. Nets ae gobay defined fo a ce eves. A net can theefoe exist at many eves. Fo the pupose of estimation we divide nets into segments that ae fuy contained in one eve ony. Such a segment of net n m of ce is denoted 1 by n m and is defined as a set of pins { }{ }. 1 j c + 1 p i, k p i, k p ij, o n m ed box c c 2 1 p 1, 1 ed pink box c 1 1 c c 11 pink 0 c 12 c 3 1 Figue 5: Two eve hieachy exampe A segment can be futhe divided into a pink pat n p m and a ed pat n m (figue 5). The 1 j pink pat connects ony the pins of the subces (dotted ines). Pink segments ae intena connections of a ce and exist if moe than one intena pin exists. Red segments connect pink segments o the ony one intena pin of a net to an extena pin (soid ines). Seen fom ce, the whoe segment n m c + 1 of the supece c + 1 can be seen as extena to. We assign the coo geen to extena net segments. With this notion of coos of nets we define wiing aea w p, w and w g fo ce. Accodingy we define encosing ectanges fo each coo, caed the pink, ed, and geen box

7 Estimation of Wiing Aea fo Hieachica Design 6 (figue 5). In addition we define a bue box fo the case that a nets of the cuent eve ae disegaded. Each box has an aea, fo exampe a p, and x and y dimensions x p, y p. In addition we have empty (back) aea, denoted a e 1. If we assume ed subces, the aeas of the fou box types of ce ae: j a b j a 1 j + a e (1) a p a b + w p (2) a a p + w (3) a g a + w g (4) w g has to be expained in moe detai. w g is the shae of ce of the pink and ed wiing aea of the supece. We need this notion duing foopanning of ce, because w c + 1 and w p c + 1 ae useess paametes fo the pacement of ces. Since w c w p c + 1 w g, (5) we deive by appying equation (3) to and using equations (1), (4), and (5) a c + 1 a g c + a e c + 1. (6) i Thus, besides the empty space a e c 1, the geen aeas of the ces competey fi the c + 1 foopan of. c + 1 c + 1 i i c The ed boxes ae the outines of the ces afte ayout with pins at the fame. The ed boxes can be defined ecusivey using equations (1), (2) and (3): a j a 1 j w p w a e (7) The disadvantage of this definition is the fact that a net may pass though sevea eves of the hieachy without connecting moe than one intena pin, that means without pink segments. No good mode is known fo the estimation of a chain of ed segments ove sevea eves. Figue 6 demonstates this pobem. At this point an expanation of the hieachy is necessay. We stated with a pat-of hieachy with two o thee eves at most, and a hunded to a thousand ce instances at each eve. With this notion we can easiy hande 10 5 to 10 7 pimitive ces. This numbe of eves woud not justify the above definitions. Ou shape function estimation pocedue, as aeady mentioned,

8 Estimation of Wiing Aea fo Hieachica Design 7 ed net segment pink net segment Figue 6: Chain of ed net segments makes use of a sicing stuctue which can be expessed by the sicing tee. This tee aso defines a hieachy of ces with two subces pe ce in ou method o up to five as epoted by othe authos to aow fo pin whees. If we extend the oigina hieachy by the sicing hieachy, the assumption of many eves is justified. 2.3 The Pink Mode Because of the disadvantage of the ecusive ed mode, we use the foowing ecusive pink mode. In ode to get id of the ed net segments, we add a ed segments to the next pink segment on a highe eve. Figue 7 expains this notion. The pink net segments and the pink boxes ae shown by dotted ines. The extension of the pink segment yieds an incease of the pink wiing aea pe net. We denote this new aea by w p. Figue 7: Pink net mode

9 Estimation of Wiing Aea fo Hieachica Design 8 The pink ecusion can be descibed by the aea equation ã p j ã p 1 j w pc + i + a e This means that the pink aea of ce does not depend on the nets without a pink segment at eve. If we need the ed aea of a ce, equation (3) has to be modified to (8) a ã p + w c i (9) w c i now incudes a ed segments of nets at a owe eves of the hieachy that wee eft out in ã p. Aso, the bue box must be newy defined: ã b j ã p 1 j + a e (10) We mode the pink wiing aeas of a net as x and y additions to the ce dimensions: xw pn m t x p α x n m (11) t x t is the tack demand facto, meaning the potion of a tack equied fo a singe wie net. is the facto fo vetica tacks. is futhe spit into and fo the cases of a vetica o hoizonta sicing ine diection. These cases equie diffeent amount of wiing space. We now have the fou factos,,, and. αx is the geometic width of the net n.the y dimensions ae handed equivaenty. Figue 8 defines the used dimensions. As ong as we do not know the oientation of ces, we cannot distinguish between eft/ight and top/bottom (/, t/b). We theefoe use and yw yt + yb. p t xv c p t xh t x p p t yv p t yh p t xv p t xh xw x + x y b y p y x b x p x Figue 8: Dimensions

10 Estimation of Wiing Aea fo Hieachica Design 9 We can cacuate xw and yw by buiding the sum of a individua contibutions of nets that ae counted. Fo the pink wiing aea we ony count the nets that have pink segments in the cuent ce. We ca this set of nets N p c. xw pc i m N p c xw pn m (12) The y dimensions ae defined accodingy. The dimensions of the bue box ae the esut of the sum of the shape functions ỹ b f x b 1 ( ) of the subces j of ce, esuting in ã b x bc i ỹ b (13) Two difficuties aise. Fist, eafces in the tee ae ed and second, the empty space has to be handed. Let us dea with the fist pobem fist. Thee ae two soutions. In the fist soution we educe the ed dimensions to pink by subtacting the appopiate ed wiing aea. The disadvantage of this soution is the fact that we thow the exact dimensions of the eaf ces away and epace them by dimensions based on the ed wiing aea estimation. The othe soution is a second set of tack demand factos fo the case of ed subces. We t xv t xh ca these factos,,, and. These wi be smae than the pink ones because a ed segments of nets within the subces ae aeady incuded in the ce aea. Now the question aises, which coo has a ce composed of two ed subces and so foth. We defined a edness facto between 0 and 1 which defines the popotion of the pink and ed tack demand factos used. Red ces have the facto 1. Each time, two ces ae composed, the aveage of the factos is taken and a eduction facto (<1) is subtacted unti 0 is eached. We impemented this soution, which aso has its disadvantages. It is vey difficut to find the ight eation between ed and pink factos and the vaue fo the ed eduction facto. We theefoe favo the fist soution now. 2.4 Feedthoughs The second pobem, the empty space, occus because of the cones in the shape functions. If we compose two ces on top of each othe as shown in figue 9, the composite ce gets the maximum of both x dimensions and the sum of the y dimensions. The diffeence in x esuts in empty space. This empty space coud be popagated up in the mode. We use a diffeent soution. We assume that empty space inceases the tanspaency of a ce. Tanspaency is modeed by vetica and hoizonta feedthoughs. Standad ces, fo exampe, have buit-in feedthoughs pependicua to the ow. We measue vetica feedthoughs by the wiing space xf in the x-dimension of a ce. The empty space of ce B in figue 9 is modeed as wiing space and thus x e t yv t yh a e a e

11 Estimation of Wiing Aea fo Hieachica Design 10 B a e A Figue 9: Empty space xfĉ B xfc B + x e c B (14) ĉ denotes the ce with empty space added. The question now is: What is the vetica tanspaency of AB. We use the weighted sum of two extemes. One exteme takes the aveage of the tanspaencies of AB, the othe the minimum. This esuts in xfc AB w xfĉ A + xfĉ B ( 1 w) min ( xfĉ 2 A, xfĉ B ). (15) The hoizonta tanspaency in this case is simpy, because no empty space has to be consideed:. (16) The case of ces with a vetica sicing ine is handed appopiatey. In the cacuation of wiing space the feedthough dimensions ae subtacted up to the additiona wiing space fo nets. Any emaining feedthough space is popagated up in the sicing tee. This pink mode has poved its quaity fo standad ce and fexibe ce exampes. The esuts fo standad ces ae shown in chapte 3. Ou shape function geneato theefoe sticty uses the pink mode fo the compete sicing and hieachy tees. In ode to use the shape function, the additiona ed wiing space has to be added at the eve whee the shape function is used. We ca this additiona space the ed shift. 2.5 Red Shift yfc AB yfc A + yfc B The ed shift is handed as an addition to the x and y dimensions of the pink ce. Let π be the numbe of one wie pins of ce (coesponding to the numbe of nets ). If we assume an equa distibution of the pins on the fou sides of the ce, we can cacuate fou vaues, π, π t, and π b. This is simia to the assumption that the pins can feey foat aound the peimete of the ce. If we have moe knowedge about the positions of the pins, fo exampe afte goba outing duing foopanning, the fou vaues can be detemined moe pecisey. If the capacity of the sides of the ed box is age than the associated numbe of pins, we can cacuate the dimensions of the ed box as: N π

12 Estimation of Wiing Aea fo Hieachica Design 11 x x p + max( ( π t + π b ) t xv + ( π + π ) t xh xf, 0) (17) Fo bevity we have eiminated the ce identifie and have assumed a unit gid. is the ed shift tack demand facto fo vetica nets and is the facto fo the vetica pat of hoizonta nets. The feedthoughs xf ae ony subtacted up to a vaue which eaves the ed shift positive (max-function). The y-dimension is handed accodingy. The capacity of a side depends on the design stye. Let us assume bocks of hoizonta standad ce ows as an exampe. It is easonabe to have no pins to the side of ce ows. Theefoe, the capacity of the vetica sides is equa to the y-dimension minus the accumuated height of the ows. The atte is equivaent to of the bue box. Theefoe, we have the additiona conditions fo : y π + y b and y π + y b (18) Fo the x-dimension the conditions ae diffeent. We assume that the intena pin positions of the nets enteing fom the bottom ae equay distibuted. Fo R ows, π b ( R 1) R nets coss the bottom ow. The numbe of feedthoughs of this ow is assumed to imit the capacity. None of the buit-in feedthoughs of the standad ces of this ow is used fo intena wiing. T x Let be the aveage tanspaency fo vetica wies of a standad ces in, defined as numbe of vetica feedthoughs divided by the x-dimension of the ce, then we aive at the conditions: y b t xh x π b ( R 1) and x π t ( R 1) (19) T R x T R x y t xv In the case the conditions ae not met, we add one gid unit to the x- o y-dimension fo each pin that vioates the condition. 3. Expeimenta Resuts Hee we estict ouseves to standad ce bocks. The exampes ae taken fom a age design with neay standad ces. The design was boken down into thee eves of a hieachy and the exampes ae taken fom the owest eve. The stuctue of the design was geneated with the high-eve synthesis system MIMOLA which esuted in a high connectivity. Theefoe the eative wiing aea is vey age compaed to othe benchmaks. We used a commecia standad ce ibay ACMOS 4H fom Siemens AG. This is a 1.25µ m CMOS technoogy with an aveage tanspaency T x 0.6. Fo the ayout of the standad ce bocks we used ou own impementation of simuated anneaing simia to TimbeWof SC /SeL87/ and a heuistic channe oute that coud oute most channes in density. Two wiing ayes ae assumed.

13 Estimation of Wiing Aea fo Hieachica Design 12 We pefomed thee kinds of expeiments. In the fist expeiment we compaed the estimated ed shape function fo ces which had unesticted pin intevas with synthesized ayouts. Then, we deeted the extena pins fom the netist of these ces fo measuing the pink boxes. The second esut was a set of ayouts of a mutipexe that occued sevea times on the chip. The diffeent sizes of these ayouts show that the estimation of the ed shift is vey impotant. The thid expeiment was the quantitative detemination of the ed shift fo ces with esticted pin intevas. A expeiments have shown that the tack demand factos stongy depend on the estictions on the pins. In the case of no estiction (ou fist expeiment), we found and to be zeo. The outing is neay staight fom the intena pins to the ce fame. Thee is no need fo additiona tacks in the othogona diection. Fo ou standad ce synthesis too, the vaue seems to be easonabe. Regading the detaied outing esut, it seems that shoud have the same o a sighty ess vaue. We coud not measue this vaue exacty because the tanspaency xf educed the hoizonta ed shift to zeo in a cases. Tabe 3 shows the expeimenta esuts of 8 diffeent ces. Ou cicuits consist of 166 to 827 standad ces with 62 to 423 pins (tabe 1). We measued the pink and ed boxes fo a wide aspect atio ange (diffeent numbe of ows). In a cases thee wee no estictions on the pins. t xh t xv t yv t yh 0.3 numbe of cename subces nets fame pins au32bit au32_ct nmux05x nmux08x pe1_ pe2_ sioo3x16th sn Tabe 1: Exampe cicuits cut net-to-cut tack demand oientation oientation facto hoizonta paae 0.5 hoizonta othogona 0.1 vetica paae 0.1 vetica othogona 0.4 Tabe 2: Pink tack demand factos Coumns 1 and 2 of tabe 3 show the ce name and the numbe of ows of ou measuements. Coumns 3 and 4 contain the aeas of the smaest pink ayouts (which we got by deeting the fame pins fom the netist) and the deviations of the pink ayouts fom the pink shape function. The shape function was computed by using the sizing paametes of tabe 2. The ast two coumns show the minimum aeas and the deviations of the ed boxes. Except fo two exampes (pe1_1536 and sioo3x16th), the eos of the ed shape functions wee ess than 10%. This shows that we can estimate vey we the smaest ayouts of a ce which usuay wi be synthesized without pin estictions. The two ces with the age estimation eos wee investigated in moe detai. We found that these ces have many age nets with many connections. These ces cannot be handed accuatey by ou standad ce pacement too.

14 Estimation of Wiing Aea fo Hieachica Design 13 Fo instance, the smaest manua ayout of ce sioo3x16th was about 30% smae (2.6mm 2 ) than the smaest ayout ou too achieved. It seems that the esuts of synthesis toos which can hande these nets bette wi be cose to the shape function. The estimation eos of the pink ayouts ae mosty due to the synthesis too, too. This too geneay yieds not vey good esuts when no foces to the ce fame exist. Fo these cases the cost function shoud be changed. The penaty fo oveapping ces must be inceased. On the othe hand, geneating a good pink ayout is of no pactica inteest. Ou second expeiment shows vey ceay that the ayout in a top-down design depends agey on the positions of the fame pins. Figue 10 shows the ed shape function and a ayouts of a mutipexe (nmux05x32) which occued sevea times on the chip. The shape function was computed with the tack demand facto 0.3 fo ces without pin estictions. The figue shows two esuts. Fist, the shape function coesponds to the owe bound of the measued vaues and second, the diffeent heights of ayouts with the same numbe of ows. The diffeent y-dimensions fo neay the same x-vaue esut fom diffeent pin inteva estictions as a esut of foopanning. We can see that this infuence is extemey age and thus the estimation of the ed shift is vey impotant fo a pecise foopanning at the next uppe hieachy eve. pink boxes ed boxes cename ow aea min ave. eo aea min ave. eo numbes [mm 2 ] [%] [mm 2 ] [%] au32bit au_ct nmux05x nmux08x pe1_ pe2_ sioo3x16th sn Tabe 3: Aveage aea estimation eo fo pink boxes and ed boxes of ces without pin estictions t yh In ou thid expeiment, we investigated this ed shift fo ces with esticted pin positions. We esticted ouseves to exampes whee the capacities of the sides wee age than the numbe of pins. Fo these sampes we tied to find tack demand factos fo the enteing nets. We esticted the pins to a singe side and assigned the pins andomy to the sides. The tack demand factos shoud be appoximatey doube of these fo unesticted pin positions because the aveage net ength wi be twice as age due to the estictions (if we assume a andom pacement fo both cases). Tabe 4 shows the deviation of the ed shape function to the sampe ayouts by

15 Estimation of Wiing Aea fo Hieachica Design 14 Figue is missing Figue 10: Estimated shape function and ayout esuts with diffeent pin positions using the tack demand facto t yh 0.6. The facto t xv is supefuous fo ou expeiments because the buit-in feedthoughs of ou standad ces educe the hoizonta ed shift x to zeo. cename ow aea min ave. eo numbes [mm 2 ] [%] au32bit au_ct nmux05x nmux08x pe1_ pe2_ sioo3x16th sn Tabe 4: Aveage aea estimation eo fo ed boxes. The pins wee andomy assigned to one side each. Expeiments with futhe estictions on the fame pins have shown that the ce aea wi not incease in a cases. We had sevea exampes whee the aea sighty deceased when the pins wee esticted to smae intevas. In othe cases, the aea inceased. Fo a sma numbe of sampes the vaiations of the ce aeas wee moe than 30%.

16 Estimation of Wiing Aea fo Hieachica Design 15 We obseved a simia esut by anothe expeiment, too. Fo this expeiment we ony intechanged the intevas of the pins. This was done fo the nmux08x32 mutipexe. Since a input pots and the output pot had the same width (32 bit), the ed shift must be unchanged athough we intechanged the assignment of the pots to the intevas. A ed shift computation that does not anayze the contents of a ce (i.e. the netist and/o the behavio) can ony depend on the ocations of the pins. In ou expeiment, the numbe of pins assigned to one inteva was the same fo a pemutations. Howeve, the vaiations of the ce aeas wee about 15%. Both expeiments show that it is not possibe to estimate pecisey the ed shift of a ce with abitay pin estictions. It has no sense to ook fo a moe compex ed shift mode that sha impove the esuts of tabe Concusion The most difficut pat of the shape function computation is an accuate wiing aea estimation. The aim of this pape was to descibe in detai the wiing aea estimation of ou shape function geneation method. We intoduced a mathematica five coo mode. Each coo is eated to a cetain pat of the wiing aea. Since it is not possibe to estimate accuatey the wiing space of the net segments connecting the fame pins of a ce, we descibed a ecusive pink mode that consides the inteconnection of the subces ony. The emaining wiing space needed to connect the fame pins has to be added ony on the hieachy eve whee the shape function is used. Inaccuacies of the ed shift wi not be mutipied with the numbe of hieachy eves of a cicuit. Resuts taken fom a age numbe of sampe ayouts have shown that the accuacy of the pink mode is sufficient fo an eay aea estimation and fo foopanning. The same accuacy was obtained fo the ed shape function compaed to the owe bound of a ayouts of a ce. Howeve, the wiing aea we need to connect the fame pins cannot be estimated with the same accuacy. The intechange of the association of the pins and intevas aeady esuts in aea diffeences of moe than 10%. Unti now we esticted ou measuements to standad ce bocks whee the capacities of the sides wee age than the numbe of pins. Ou next aim is to impove ou ed shift mode fo the case that the pins ae not equay distibuted ove the ce fame. This is necessay fo the top-down foopanning whee the goba oute decides the pin positions. Then, we wi impove ou aea estimation mode fo the uppe genea ce eves and we wi ty to detemine the mode paametes fo these eves moe pecisey.

17 Estimation of Wiing Aea fo Hieachica Design Refeences /Gak83/ D. Gajski, R. Kuhn, New VLSI Toos, IEEE Compute, Vo.16, No 12, 11-14, 1983 /GiO84/ /Ott83/ /SeL87/ /Zim88/ /Zim90/ L. van Ginneken, R. Otten, Stepwise Layout Refinement, Poc. Int. Conf. on Compute Design, Pot Cheste, 30-36, 1984 R. Otten, Efficient Foopan Optimization, Poc. Int. Conf. on Compute Design, Pot Cheste, , 1983 C. Sechen, K. Lee, An Impoved Simuated Anneaing Agoithm fo Row-Based Pacement, Poc. Int. Conf. on Compute Aided Design, Santa Caa, , 1987 G. Zimmemann, A New Aea Shape Function Estimation Technique fo VLSI Layouts, Poc. 25th Design Automation Confeence (DAC), Anaheim, 60-65, 1988 G. Zimmemann, VLSI Aea Estimation Toeances - Shape Function Geneation vs. Foopanning, Poc. of the EURO ASIC 90, Pais, , 1990