Area Minimization of Power Distribution Network Using Efficient Nonlinear. Programming Techniques *

Transcription

1 Area Mnmzaon of Power Dsrbuon Newor Usn Effcen Nonlnear Prorammn Technques * Xaoha Wu 1, Xanlon Hon 1, Yc Ca 1, C.K.Chen, Jun Gu 3 and Wayne Da 4 1 De. Of Comuer Scence and Technoloy, Tsnhua Unversy, Bejn, Chna, De. Of Comuer Scence and Enneern, Unv. Of Calforna a San Deo, La Jolla, Calforna 3 De. Of Comuer Scence, Hon Kon Unv. Of Scence and Technoloy, Hon Kon 4 De. Of Comuer Enneern, Unv. Of Calforna a Sana Cruz, Calforna E-mal:{wuxh, honxl, cayc}@er.cs.snhua.edu.cn ABSTRACT: Ths aer deals wh area mnmzaon of ower dsrbuon newor for VLSIs. A new alorhm based on effcen nonlnear rorammn echnques s resened o solve hs roblem. The exermen resuls rove ha hs alorhm has acheved he objecs ha mnmze he area of ower/round newors wh hher seed. 1. INTRODUCTION Power/Ground (/ nes are always very moran n desn of VLSIs because hey affec he erformance of crcus serously. Snce her wre wdhs are much wder han snal nes, / nes usually cover a lare oron of ch area. So hey are ofen ven he frs rory n roun rocess. There are wo basc consrans n desn and omzaon of / nes. The frs s he undesrle wear-ou of meal wrn caused by elecromraon, and he second s he narrown marns caused by volae dros [7][8]. Alhouh ncreasn wre wdh can solve hese roblems, would cos oo much wrn resources. Consequenly, s necessary o mnmze he area of / nes under he wo consrans [8]. Generally sean, he desn of / nes consss of wo man ses. Frsly, a ooloy for / nes s consruced, and can be rees [4][10] or eneral rah [1][][8]. Secondly, wre wdhs of / nes are mnmzed. I s no dffcul o mnmze wre wdhs of rees branches because her curren drecons are fxed. However, o mnmze wre wdhs of a eneral rah s much more dffcul because curren drecons are unceran before branch wdhs are deermned. In mos cases, s a roblem of non-lnear omzaon subjec o non-lnear consrans. Moreover, he of varles s usually remendous because of lare scale of modern VLSIs. Several sudes on area omzaon for / nes have been ublshed. The alorhm resened by [8] s based on feasble drecon mehod [1]. However, feasble drecon mehod may brn ou he roblem of zzan [5], whch leads o oor converence. Aumened Laranan funcon s used by [1]. In order o avod comuaon of aral dfferenal of currens subjec o ressance, boh currens and ressance are used as varles and Krchoff s law s rearded as a consran, whch resuls n redundan searchn sace and ncreasn he scale of roblem. In 1999, Xandon Tan and C.-J. Rchard Sh roosed an neresn alorhm [9]. The basc dea s o resul a consraned nonlnear rorammn roblem no a sequence of lnear rorammn. Le [1], volaes and currens are also used as varles. To solve hese roblems, we resen a new alorhm based on enaly mehod, conjuae raden mehod and crcus sensvy analyss. Only conducance s used as varles. As a resul, hs alorhm s le o deal wh real desns of IC ndusry.. PROBLEM FORMULATION In he laer dscusson, we assume ha every comonen sorbs s mos curren, as s so called wors case. [] has roved ha f he consrans were no volaed n wors cases hey would be sasfed n oher cases. For he smlary of ower and round nes, we wll only descrbe he alorhm for ower nes.durn he descron of roblem formulaon and soluon mehod, follown noaons wll be used: E node :se of ndces of all nodes n / ne. E bch :se of ndces of all branches n / ne. E leaf :se of ndces of all ower ns n E node. E vdro :se of ndces of he nodes whch volae volae dro consrans n E node. E vem :se of ndces of he branches whch volae elecromraon consrans n E bch. E ne ( :se of ndces of node s nehborn nodes n E node. * Ths wor s suored by Naonal Scence Foundaon (No and 973 Naonal Key Projec (No

2 ρ :shee ressance. l, w, :lenh, wdh and conducance of branch (,..1 The Objecve Funcon The objecve funcon s he area of / nes. a l w ρl bch bch Se α ρl a, α bch. The Consrans 1 Volae dro consrans The dfference beween nu volae v dd and ower ns volaes mus be smaller han allowle bes volae dro u, v dd Curren densy consrans leaf ( (1 v u E (3 / w σ v v ρl σ E (4 q 3 Krchoff s law Krchoff s law can be descrbed by he node volae equaons se: ( v q qne ( v bch, q E In hs alorhm Krchoff s law s sasfed naurally and doesn emere exlcly. 4 Mnmal wdh consrans Own o echnolocal resrcon, ower branches have o be wder han mnmal meal lne wdh, whch can be rearded as a consran. However, a more smle way s o se he lnes ha do no sasfy hs consran as mnmal meal lne wdh, whch s more effcen and leads o beer resuls. 3. SOLUTION METHOD 3.1 Formulaon of Penaly Funcon Penaly mehod s he manframe of hs alorhm and enaly funcon s defned as below, s f a w s leaf bch (6 node 0 f v vdd u E (7 leaf v ( vdd u else 0 f ρ lσ > v (8 Ebch ρlσ v else Where a s defned by ( and w s enaly arameer. Se w s (9, enaly funcon can leaf bch be rewren as f a (10 3. Penaly mehod A frs, enaly arameer w s ven an nal value, hen w s ncreased and new enaly funcon s omzed aan unl (5 all consrans are sasfed. In hs way, ornal roblem s ransferred no a sequence of unconsraned mnmzaon roblems. The soluon mehod can be descrbed as below, 1. Se enaly arameer w as an nal value, nal conducance vecor as G (0 and error bound as ε 1 > 0.. Solve unconsraned mnmzaon roblem, oban curren conducance vecor G (l. mn f a a w s (11 leaf bch 3. If < ε 1, hen so, else ncrease w, and o o se. In alorhm based on enaly funcon, he ey s o fnd an effcen mehod o solve he unconsraned mnmzaon roblem. In our alorhm, FR conjuae raden mehod s aled. Is ses are shown as follows: 1. Suose nal conducance vecor as G (l and error bound as ε > 0, nal descen drecon s se as ( l ( l neave drecon of raden, P f ( G (1. Deermne a nonneave scalar λ whch mnmze f. Le G ( be he conducance vecor a he h eraon, f ( G λ P mn f ( G λp (13 ( 1 3. Udae conducance vecor G G λp (14 ( 1 4. If f ( G < ε, hen so; else chose a new descen drecon P (1 whch sasfes he condon below, ( 1 f ( G β ( f ( G ( 1 ( P f ( G 1 β P ( (15 5. Go o se. 3.3 Lne Search Afer en descen drecon, we need o deermne nonneave scalar λ whch mnmze objec funcon, ha s, f ( G λ P mn f ( G λp Ths rocess s wha so called lne search, whch s used o reduce objecve funcon alon descen drecon as much as ossble. Snce dervave of enaly funcon subjec o λ s dffcul o e, quadrac nerolaon mehod s aled for only requres he value of enaly funcon. 3.4 Resze Objecve Funcon and Udae enaly arameer In omzaon of / nes, he numercal dfference beween objecve funcon and enaly erm s very lare, whch would resul n over consderaon of objecve funcon and mae he rocess of omzaon dffcul o connue. Increasn enaly arameer wll lead o ll-condonn roblem, a beer soluon s o resze objecve funcon by a scalar γ. Defne w as he rao of enaly erms o objecve funcon and mae fxed, γ can be exressed by w as,

3 w γ α γ w bch bch α (16 In radonal enaly mehod, w s always udaed by mn a fxed udan rao, whch may le exermen resuls rely on udan rao. To solve hs roblem, we use almos he same way whch deermnes γ o udae enaly arameer w. Se enaly arameer w n las unconsraned omzaon s w old, he new one w new can be calculaed by he consan w as below, w w new wold w γ a wnew wold (17 γ a 4. GRADIENT CALCULATION I s necessary o calculae he raden of enaly funcon o e descen drecon n conjuae raden mehod. The follown noaons wll be used n he descron. N: ornal newor. N(, N(, N(: Ns adjon newors used o calculae raden of volaes of nodes, and q. n node : nodes of N or s adjon newors. n bch : branches of N or s adjon newors. M: coeffcen marx of N or s adjon newors. V(, V(, V(: vecors formed by node volaes of N(, N( and N( resecvely. B(, B(, B(: rh sde of node volae equaon ses derved from N(, N(, N(. v, v, v q : volaes of node, and q n N. G: conducance vecor of N or s adjon newors. : conducance of branch (a, b n N or s adjon newors, (a, b bch v a, v b : he volaes of branch (a, bs wo end nodes n N. v a (, v b (: he volaes corresondn o v a and v b n N(. v a (, v b (: he volaes corresondn o v a and v b n N(. v a (, v b (: he volaes corresondn o v a and v b n N(. Snce v and v q have been nown before each eraon, solue value s unnecessary. Suose v >v q, (7 and (8 can be rewren as, s v ( vdd u ρl σ ( v Evdro E vem (18 From (10, he aral dfferenal of enaly funcon f subjec o conducance can be exressed as, a (19 Wh (, (9 and (18, (19 can be rewren as, v s (0 f vdro α w vq v vem (0 s dffcul o calculae for node volaes canno be exressed by conducance n exlc form. To solve hs roblem, we use he mehod of adjon newor, whch s resened by Drecor and Rhorer n 1969 [3][8]. Nex we wll descrbe how o calculae raden of node s volae subjec o conducance vecor. The frs se s o buld adjon newor N(, whch has he same ooloy and conducance vecor wh N. However, all sorbn curren of N( s leaf nodes are se as 0 exce node, whch sorbn curren s se as 1A. Besdes, all nu volaes are se as 0. The second se s o form node volae equaons se for N(. Snce N and N( have he same ooloy and conducance vecor, hey share he same coeffcen marxes. Ther dfference only resdes n he rh sde: all elemens of rh sde of N( s node volae equaons se are zero exce he elemen corresondn o node s 1, ha s, [ 0,0, L,0, 1,0,, ] T B( L 0 Where neave one aears a ndex. So v can be v calculaed as, ( va ( va ( ( (1 Wh (1, (0 can be rewren as, α va ( s ( va ( va ( ( vdro vem w ( v a ( s ( ( ( vdro vem Thus, he raden of enaly funcon f subjec o conducance vecor G can be exressed as follow, f ( G,, L,, L, 1 ( shows ha we may have o solve an ornal newor and many adjon newors n each eraon o e f (G, whch wll cos oo much CPU me. Forunaely, we fnd a way o avod solvn each adjon newors resecvely, V( can be o by solvn node volae equaons se MV ( B(, v a ( and v b ( are wo elemens of V(, le vecor C a [ 0,0, L,0,1,0, L,0] and C b [ 0,0, L,0,1,0, L,0] where 1 aears a ndex a and b resecvely, we have, nbch va ( CaV (, ( CbV ( Then we can e s V ( and vdro T vem solvn he follown lnear equaon resecvely, M M sv ( s B( vdro vdro ( V ( V ( vem E vem Add (3 and (4, calculaed by solvn, (3 ( B( B( vdro vem ( V ( V ( by (4 s V ( ( V ( V ( can be 3

4 M vdro vdro sv ( s B( vem vem ( V ( V ( ( B( B( (5 Use X o relace he vecor on he lef sde of (5 and B new o relace he vecor on he rh sde of (5, (5 s rewren as, MX B new (6 Then, we exress ( as α w ( va ( Ca Cb X (7 To oban he raden of enaly funcon subjec o G, we should mere adjon newors by formn a new rh sde accordn o (5, hen solve (6, and fnally use (7 o e f (G. Thus he raden of enaly funcon subjec o conducance vecor can be obaned a one me. Snce he coeffcen marx derved from node volae equaons se s symmerc and osve defne, Incomlee Cholesy Decomoson Conjuae Graden (ICCG [6][11] s used o solve node volae equaons se. In ICCG, comun re-condoned marxes wll coss mos me. As he coeffcen marxes of ornal and adjon newor are he same, hey can share he same re-condoned marx. Besdes, mos / nes have very secal ooloy; we can use he equvalen crcu echnque o accelerae / ne solver s seed. 5. Analyss of Tme Comlexy To dscuss me comlexy, we have such noaons: N er :Number of eraon. N lne ( :Number of lne searchn n eraon. T lne (, j :Tme of solvn a newor durn lne searchn j n eraon. T re ( :Tme of comun re-condoned marx for ICCG n eraon. T so ( :Tme of solvn ornal newor wh re-condoned marx n eraon. T sa ( :Tme of solvn (6 wh re-condoned marx n eraon. In each eraon we need solve ornal newor and (6 o e new searchn drecon, hen do lne searchn. So me comlexy can be exressed as below, T ( (, j (8 N er Nlne ( Tre Tso( Tsa ( Tlne 1 j 1 Accordn o (8, he alorhm s effcency deends on he seed of / ne solver. Generally sean, he comlexy of solvn a lnear equaons by eraon mehod s O(#er N, where N s he rane of coeffcen marx. However, he coeffcen marx of node volae equaons se s symmerc, osve defne and very sarse. So ICCG can solve much faser. Besdes, mos / nes have a very secal ooloy descrbed n secon 4, whch can accelerae / ne solver s seed. So s dffcul o e he exac comlexy of solvn a / newor. Insead, we resen several examles o llusrae emrcal rend of our / nes solvers effcency as le 1 shows, Tle 1. Examles o llusrae / ne solver s effcency crcu name node me(s es es es es es es es es es crcu name Tle. Comarson of our alorhm aans Tan-Sh alorhm our alorhm Tan-Sh alorhm node branch area area me (s reduced me (s reduced seed u (% (% 4x x x x Tle 3. The exermen resuls of larer crcus crcus name node sras runs area(µm me(s branch-and-bo und mehod our alorhm es es es es es es

5 es es es EXPERIMENT RESULTS Ths alorhm has been develoed on sun ulra_sarc 50M worsaon wh lanuae C and C. For comarson, we have used he es cases whch [9] resened. The resuls of comarson are shown as le. In le, he column of area reduced(% s reduced wrn area of ornal area n ercenae, whch shows ha all / nes omzed by our alorhm occuy smaller area han [9]. The column of seed u says ha o lare crcus le adjon newors o mnmze he area of / nes whle sasfyn he consrans. The exermen resuls show ha hs roram s very robus wh hh seed and low wrn resource consumon. Furher more, has he caly of omzn lare-scale crcus. REFERENCE [1] S. Chowdhury and M. A. Breuer, Mnmal area desn of ower/round nes havn rah ooloes, IEEE Trans. on Crcus and Sysems,.1441~1451, December [] S. Chowdhry and M. A. Breuer, Omzaon Desn of 100x100 our alorhm s much faser han Tan-Sh alorhm. Relle IC Power Newors Havn General Grah There are wo reasons for area mrovemen. Frsly, only conducance s looed as he varles, whch avods redundan searchn sace. Secondly, here s no any assumon such as fxn currens or volaes le Tan-Sh alorhm. We have also esed hs alorhm wh much larer crcus from IC ndusry. To comare wh hs alorhm, we have also develoed an alorhm based on branch-and-bound mehod o omze / nes area. Tle 3 lss he resuls. Tooloes, Proc. 6h DA Conf.,. 787~790, [3] S. W. Drecor and R. A. Roher, A Generalzed Adjon Newor and Newor Sensves, IEEE Trans. on Crcus and Sysems, Vol. CT-16,. 318~33, Auus [4] K-H. Erhard, F. M. Johannes and R. Dachauer. Tooloy Omzaon Technques for Power/Ground Newors n VLSI, Proc. Euroean Desn Auomaon Conference,. 36~367, 199. Comarn wh oher alorhm, our alorhm s le o deal wh lare crcus. There are 1,618,06 comonens n our lares es crcu. The runnn me s second. From le 3, we can see ha area obaned by he alorhm roosed n hs aer s much smaller han ha derved from branch-and-bound mehod. [5] [6] [7] R. Flecher, Praccal Mehod of Omzaon, Vol., New Yor: Wley, Gene H. Golub and Charles F. Van Loan, Marx Comuaons, Johns Hons Unversy Press, 1983 F. M. D Heurle, Elecromraon and falure n elecroncs: An nroducon, Proc. IEEE, vol. 59,. 1409~1418, Oc Fure 1 shows how he of he nodes, whch volae he volae dro and he elecromraon consrans, decreases wh eraon (ae crcu es1 as examle. The converence characersc of hs alorhm s also descrbed by fure 1. The [8] Msuhash T. & Kuh E. S, Power and Ground Newor Tooloy Omzaon for Cell Based VLSIs, roceedns of 9h ACM/IEEE Desn Auomaon Conference,. 54~59, 199. vercal lnes say ha enaly arameer s udaed and a new unconsraned omzaon bens. A hese mes he of volan nodes decreases more sharly and a las converes o zero. [9] Xan-Don Tan, C.-J. Rchard Sh, Draos Luneanu, Jyh-Chwen Lee and L-Pen Yuan, Relly-Consraned Area Omzaon of VLSI Power/Ground Newors Va Sequence 60 of Lnear Prorammns, DAC 99 New Orleans, Lousana, , [10] WU Xao-ha, QIAO Chan-e, YIN L, HONG Xan-lon 40 Desn and Omzaon of Power/Ground Newor for 30 BBL-Based VLSIs Auus 000, Aca Elecronca Snca. [11] Xaoha Wu, L Yn, Xanlon Hon, A Power and Ground 0 Newor Solver wh he Mehod of Incomlee Cholesy 10 decomoson Conjuae Graden, March 000, Chnese Journal of Semconducors. 0 [1] G. Zouendj, Mehods of Feasble Drecons, Amserdam, of volan nodes of eraon F. 1. Volan nodes reducon wh he of eraons The Neherlands: Elsever, CONCLUSION AND FUTURE WORK In hs aer we resen a new alorhm based on enaly mehod, conjuae raden mehod, crcus sensvy analyss and mere of 5