Multi-Way VLSI Circuit Partitioning Based on Dual Net Representation

Transcription

1 Multi-Wy VLSI Ciruit Prtitioning Bs on Dul Nt Rprsnttion Json Cong Dprtmnt of Computr Sin Univrsity of Cliforni, Los Angls, CA Wilurt Lio n Nrynn Shivkumr Dprtmnt of Computr Sin Stnfor Univrsity, Stnfor, CA Astrt In this ppr, w stuy th r-ln multi-wy prtitioning prolm of VLSI iruits s on nw ul ntlist rprsnttion nm th hyri ul ntlist (HDN), n propos gnrl prigm for multi-wy iruit prtitioning s on ul nt trnsformtion. Givn ntlist w first omput K-wy prtitioning of nts s on th HDN rprsnttion, n thn trnsform th K-wy nt prtition into K-wy moul prtitioning solution. Th min ontriution of our work is in th formultion n solution of th K-wy moul ontntion (K-MC) prolm, whih trmins th st ssignmnt of th mouls in ontntion to prtitions whil mintining usr-spifi r rquirmnts, whn w trnsform th nt prtition into moul prtition. Unr nturl finition of ining funtion twn nts n mouls, n prfrn funtion twn prtitions n mouls, w show tht th K-MC prolm n ru to min-ost mx-flow prolm. W prsnt n ffiint solution to th K-MC prolm s on ntwork flow omputtion. W pply our ul trnsformtion prigm to th wll-known K-wy FM prtitioning lgorithm (K-FM) n show tht th nw lgorithm, nm K-DulFM, rus th nt utsiz y 20% to 31% ompr with th K-FM lgorithm. W lso pply th sm prigm to th K-MFFC-FM lgorithm, K-FM lgorithm s on mximum fnout-fr on (MFFC) lustring rport in [10], n show tht th rsulting lgorithm, K-DulMFFC-FM rus th nt utsiz y 15% to 26% ompr with K-MFFC-FM. Furthrmor, w ompr th K-DulFM lgorithm with EIG1[18] n Proli [26], two rntly propos sptrl-s iprtitioning lgorithms. W show tht K-DulFM rus th nt utsiz y 56% on vrg whn ompr with EIG1 n prous omprl rsults with Proli. 1. Introution Th K-wy prtitioning prolm is on of prtitioning th mouls in ntwork into K susts (prtitions) of "pproximtly" th sm siz whil minimizing th numr of intronntions twn th K prtitions. This prolm hs mny pplitions in VLSI iruit sign rnging from iruit lyout to logi simultion n multion. Th xisting prtitioning lgorithms in th litrtur n lssifi into two-wy prtitioning (iprtitioning) lgorithms n multi-wy prtitioning lgorithms. Th iprtitioning lgorithms inlu th itrtiv improvmnt mthos, th nlytil mthos, th min-ut s mtho, n th nt-s prtitioning mtho. Som of th st known itrtiv improvmnt s prtitioning mthos inlu th Krnighn-Lin (KL) lgorithm [22], th Fiui-Mtthyss (FM) lgorithm [15], th FM-lgorithm with look-h shm [24], n th simult nnling pproh [23, 17]. Th nlytil mthos inlu oth th us of linr plmnt formultion with th qurti ojtiv funtion, whih is solv y omputing th son smllst ignvtor of th Lplin mtrix of th givn ntwork [14, 2, 4, 18], n th us of th linr plmnt formultion with linr ojtiv funtion, whih is solv y n itrtiv mtho in [26]. Th min-ut s mtho uss th mximum flow lgorithm to omput sris of minimum uts in th givn iruit in

2 -2- orr to otin n r-ln ut with smll ut siz [29]. Th nt-s prtitioning pproh first omputs iprtitioning of th nts, n thn trnsforms th nt prtitioning solution into moul prtitioning solution [20, 9]. Th multi-wy prtitioning lgorithms inlu th rursiv iprtitioning y Krnighn n Lin [22], gnrliztion of th FM-lgorithm with lookh y Snhis [27], th priml-ul lgorithm [30], n gnrliztion of th grph sptrl-s prtitioning mtho to multi-wy rtio-ut y Chn, Shlg, n Zin [5]. To ru th omputtionl omplxity for prtitioning vry lrg iruits, lustr-s prtitioning mthos hv n introu. In this pproh lustrs r intifi n ollps, n th rsulting lustr ntwork is prtition using xisting prtitioning mthos. Clustring mthos inlu rnom-wlk lustring [8, 19], multiommoity-flow s lustring [31], liqu s lustring [13], gomtri ming with min-imtr lustring [1], n lustring s on mximum fnout-fr ons (MFFCs) [10]. Prtitioning with moul rplition [25, 21] n th ommunition-omplxity s prtitioning mtho [3] hv lso n propos to furthr ru th mount of intronntions. Sin th ojtiv of th prtitioning prolm is to minimiz th numr of nts to ut, w liv tht ssigning nts, inst of mouls, to prtitions will l to ttr prtitioning solutions in gnrl. Th nt-s iprtitioning lgorithm y Cong, Hgn, n Khng [9] is thrfor of prtiulr intrst to us. This lgorithm first omputs iprtitioning of th nts using th grph sptrl mtho, n thn trnsforms nt iprtitioning solution into moul iprtitioning solution y solving th moul ontntion prolm (to isuss in Stion 3 in til). It ws shown tht th moul ontntion prolm for iprtitioning n solv optimlly y omputing minimum vrtx ovring in iprtit grph, n vry nourging xprimntl rsults wr rport. Howvr, th minimum vrtx ovring formultion for th moul ontntion prolm is inhrnt to iprtitioning n nnot sily gnrliz to multi-wy prtitioning. In this ppr, w introu nw ul ntlist rprsnttion nm hyri ul ntlist (HDN) n propos gnrl prigm for multi-wy iruit prtitioning s on ul trnsformtion. Givn ntlist, w first omput K-wy prtition of th nts s on th HDN rprsnttion, n thn trnsform th K-wy nt prtition into K-wy moul prtition. Th min ontriution of our work is th formultion n solution of th K-wy moul ontntion (K-MC) prolm. W introu ining funtion twn mouls n nts, n prfrn funtion twn mouls n prtitions in trmining th st ssignmnt of mouls in ontntion to prtitions. W show tht th K-MC prolm n formult s min-ost mx-flow prolm, n w prsnt two ffiint ntwork flow s lgorithms for solving th K-MC prolm unr stti prfrn funtions n ynmi prfrn funtions. Th rst of th ppr is orgniz s follows. W prsnt th prolm formultion n trminologis in Stion 2. Stion 3 prsnts th hyri ul ntlist rprsnttion n our K-wy prtitioning lgorithms s on th ul trnsformtion prigm. Stion 4 prsnts xprimntl rsults. W onlu th ppr in Stion 5 with som osrvtions n irtions for futur work. An xtn strt of this ppr ws prsnt in th 1994 Intrntionl Confrn for Computr Ai Dsign (ICCAD 94) [12]. 2. Prolm formultion Givn ntlist NL to prtition into K prtitions, w us M = {m 1, m 2,..., m p } to not th st of mouls in NL, N = {n 1, n 2,..., n q } to not th st of nts in NL, n P 1, P 2,..., P K to not th K prtitions, whr p is th numr of mouls, n q is th numr of nts in NL. Th mouls my hv iffrnt rs.

3 -3- An optiml r-ln K-wy prtitioning solution of givn ntlist NL stisfis th following onitions: (i) (ii) Eh moul is ssign to xtly on prtition. Th totl r of th mouls in h prtition r within th usr-spifi r ouns, i.. (1 α). K A Ai (1 +α). K A for h prtition P i, whr A is th totl r of ll th mouls in NL, A i is th totl r of ll th mouls in prtition P i, n α is usr-spifi prmtr ontrolling th llowl slk in th r onstrint. (iii) Th numr of nts ut is minimiz. Givn ntlist NL (for xmpl, shown in Fig. 1()), w introu th following finitions: (i) (ii) (iii) Ntlist Hyprgrph: NHG = (V (NHG ), H (NHG )), whr h vrtx in V (NHG ) rprsnts moul m i (1 i p ) n h hyprg in H (NHG ) rprsnts nt n j (1 j q ) (s Fig. 1 ()). Nt Intrstion Grph (NIG): NIG =(V (NIG ), E (NIG )), whr h no in V (NIG ) rprsnts nt n i (1 i q ), n thr is n g in E (NIG ) twn n i n n j iff n i n j φ (i.. th two nts shr ommon mouls). Not tht NIG is grph inst of hyprgrph (s Fig. 1 ()). Dul Ntlist Hyprgrph (DNHG): DNHG =(V (DNHG ), H (DNHG )) whr h no in V (DNHG ) rprsnts nt n h hyprg in H (DNHG ) rprsnts N(m i ), th st of nts inint to moul m i (1 i p ) (s Fig. 1 ()). i 1 2 g 3 j i g j h h 6 f f () Ntlist (NL) () Ntlist Hyprgrph (NHG) i g j i g j h h f f () Nt Intrstion Grph (NIG) () Dul Nt Hyprgrph (DNHG) Figur 1 Diffrnt Ciruit Rprsnttions

4 -4-3. Multi-wy iruit prtitioning s on ul trnsformtion In this stion, w first rviw IG-MATCH, th iprtitioning lgorithm s on ul trnsformtion y Cong, Hgn, n Khng [9] n isuss its limittions for multi-wy iruit prtitioning. Thn, w sri th K-DulFM lgorithm in til in Sustions 3.2 to 3.6 n us it to illustrt how to pply our nw ul trnsformtion thniqu to multiwy prtitioning lgorithm. Finlly, in Sustion 3.7, w prsnt th K-DulMFFC-FM lgorithm, whih pplis our ul trnsformtion thniqu to th K-MFFC-FM lgorithm, K-FM s lgorithm with MFFC lustring [10] Rviw of th IG-MATCH lgorithm Th IG-MATCH lgorithm prtitions th NIG using sptrl s mtho, n thn trnsforms th nt prtitioning to moul prtitioning solution. Givn ntlist shown in Fig. 2(), n nt iprtitioning solution P 1 n P 2 of th NIG shown in Fig. 2(), IG-MATCH ssigns ll th mouls in nts n (i.. mouls 1 n 4) to P 1, n ll th mouls in nts h n j (i.. mouls 3 n 7) to P 2. This ssignmnt ssurs tht nts,, h n j will not ut in th rsulting moul prtitioning solution. Not tht ths thr nts r not inint to ny g ut in th prtitioning of NIG. Eh of th rmining unssign mouls r prt of nts tht intrst (i.. shr mouls with) nothr nt in th othr prtition. Whn two (or mor) nts rsi in iffrnt prtitions n shr som moul m k, th qustion riss s to whih prtition th ontn moul m k shoul ssign to. This is th moul ontntion (MC) prolm: in Fig 2(), mouls 2, 5 n 6 r in ontntion sin thy r shr y nts on oth sis. A iprtit grph B ws onstrut in [9] s follows. Th nos in B rprsnt th nts on th ounry of th prtitioning solution of NIG. For h nt n i in P 1 n nt n j in P 2 shring som moul m k, w introu n g twn n i n n j. For xmpl, Fig. 2() shows th iprtit grph B, su-grph of NIG, for th nt iprtitioning of NIG in Fig. 2(). It ws shown tht for ny ssignmnt of mouls in ontntion, th st of nts ing ut in th rsulting moul prtitioning solution form vrtx ovr in B. Thrfor, th moul ontntion prolm is ru to th on of omputing minimum vrtx ovr in iprtit grph, whih n solv optimlly in polynomil tim. It is sy to s tht nts n g form th minimum vrtx i 1 2 g 3 j i P1 P2 g j h h 6 f f () Givn Ntlist () NIGs Nt Prtitioning i g P1 2 P f () Moul Contntion Rsolution () Finl Moul Prtitioning Figur 2 Illustrtion of th IG-MATCH Algorithm

5 -5- ovr of th iprtit grph shown in Fig. 2(). Thrfor, w i to llow nts n g to ut whil kping nts i,,, n f intt in solving th moul ontntion prolm, whih ls to th moul iprtitioning solution shown in Fig. 2(). It is lr tht th minimum vrtx ovring formultion for th moul ontntion prolm is inhrnt to iprtitioning n nnot sily gnrliz to multi-wy prtitioning. Also, th minimum vrtx ovr formultion nnot hnl th r ln onstrint fftivly. Th ojtiv of [9] ws to minimiz rtio-ut siz, n thrfor, th r ln onstrint ws not n issu in thir formultion. Our work to prsnt in th rminr of this stion givs novl n mor gnrl formultion of th moul ontntion prolm for r-ln multi-wy prtitioning tht ls to ffiint solutions. In th nxt fw sustions, w shll prsnt th K-DulFM lgorithm in til n us it to illustrt our ul trnsformtion thniqu for multi-wy iruit prtitioning Ovrviw of K-DulFM lgorithm Our ul ntlist s K-FM prtitioning lgorithm, nm K-DulFM, onsists of th following phss: (1) W first onvrt th nt hyprgrph to ul nt rprsnttion. Th ul nt rprsnttion us in K-DulFM, nm hyri ul ntlist (HDN ), is omintion of NIG n DNHG. (2) Assign nts to prtitions. W us th K-FM prtitioning lgorithm [27, 15] to prtition th nts into K prtitions. (3) W trnsform th nt prtitioning solution into moul prtitioning solution y solving th K-wy moul ontntion prolm (K-MC). W show tht th K-MC prolm n formult s min-ost mx-flow prolm n w prsnt two ffiint lgorithms to solv th K-MC prolm. (4) W furthr improv th moul prtitioning solution using gin th K-FM itrtiv improvmnt lgorithm. Th susqunt sustions sri ths phss in til Gnrting ul ntlist rprsnttions Th nt intrstion grph (NIG ) ws us in [9] sin th grph sptrl s lgorithm us in thir nt prtitioning lgorithm pplis only to grphs n nnot us for hyprgrphs. Howvr, w noti tht for mny tst iruits, thr r numr of nts inint to th sm moul (s Fig. 3()), n ths nts will form lrg liqu (omplt grph) in th NIG (s Fig. 3()). In this s, th mmory rquirmnt for storing NIG is high n prtitioning NIG lso tns to mor iffiult n tim intnsiv. Morovr, sin NIG n vry ns whn lrg nts xist in th iruit, itrtiv improvmnt s prtitioning lgorithms my sily trpp in lol optim. On th othr hn, th ul nt hyprgrph (DNHG ) (Fig. 3()) fin in Stion 2 is mor onomil in trms of mmory rquirmnt whn ompr to th NIG. Howvr, our stuy shows tht us of DNHG irtly s th ul rprsnttion os not giv th st prtitioning rsults sin DNHG rprsnttion os not istinguish th numr of nts ontning for moul. To voi ths prolms, w introu thrshol prmtr CF whn onstruting th nt intrstion grph. Whn th numr of nts inint to th sm moul is mor thn CF, w onnt ths nts y hyprg inst of lrg liqu. Th rsulting ul ntlist rprsnttion is ll th hyri ul ntlist (HDN ) rprsnttion. Not tht if w st CF to 2, thn th HDN is th sm s th DNHG. In gnrl, HDN (shown in Fig. 3()) is omintion of NIG n DNHG. Our xprimntl rsults onfirm tht nt prtitioning s on HDN prous ttr rsults thn thos s on NIG or DNHG rprsnttions. Th HDN ws onstrut in two stps:

6 -6- (i) (ii) Construt th Dul nt hyprgrph. For h hyprg with no mor thn CF nos, rpl it with liqu. In our implmnttion, CF ws hosn to 5. Not tht th HDN is hyprgrph in gnrl. Sin w us th K-FM lgorithm for nt prtitioning (s nxt su-stion), hyprgrph rprsnttion prsnts no prolm to us Prtition of ul ntlist rprsnttion Aftr onstruting th HDN hyprgrph, w us th K-FM lgorithm to omput K-wy prtitioning of HDN to otin K-wy prtitioning of th nts in th originl ntlist. W wnt to minimiz th numr of gs ut in HDN so tht th susqunt moul ontntion is sir to solv. For xmpl, "" nt prtitioning shown in Fig. 4() my rsult in ll th mouls in ll nts ing in ontntion. Howvr, for goo nt prtitioning s shown in Fig. 4(), mouls in nts, f, n g n ssign oring to th nt prtitioning, n only th shr mouls in nts,, n will in ontntion. Thrfor, it is vry importnt to otin goo nt prtitioning s on th K-wy prtitioning of HDN. In our lgorithm, w omput n initil nt prtitioning, using th following simpl trministi, gry lgorithm. (i) (ii) Sort th nts in sning orr of thir sizs (in trms of th numr of mouls). Assign th lrgst unssign nt to th prtition whr most of its ssign mouls rsi (without violting r onstrints). (iii) Mov th unssign mouls tht r prt of th slt nt to th sm prtition trmin in stp 2. g h g h f f () Prtil Ntlist f g () Nt Intrstion Grph f g h h () Dul Nt Hyprgrph () Hyri Dul Ntlist (CF=5) Figur 3 Avntg of Hyri Dul Ntlist

7 -7- (iv) Rpt stps 2 n 3 until ll nts r ssign. In ition, w lso gnrt numr of rnom initil nt prtitions if suffiint CPU tim rmins. W thn pply th K-FM lgorithm to th nt prtitions (gry n rnom) to improv th K-wy prtitions of HDN Solution to th K-Wy Moul Contntion prolm (K-MC) Th min ontriution of this ppr is th gnrl formultion of th K-wy moul ontntion (K-MC) prolm n th ffiint mthos of solution, whih llow us to trnsform K-wy nt prtition (otin y ny prtitioning lgorithm) to K-wy moul prtition suh tht th numr of nts ut is minimiz. W shll isuss our formultion n th mthos of solution in til in this sustion Prolm sttmnt W sy tht moul m is in ontntion if thr xist two nts n i n n j ontining m suh tht n i n n j r in two prtitions in th nt prtitioning solution. W us M ont to not th st of mouls in ontntion. Th K-MC prolm is to ssign mouls in M ont to propr prtitions so tht th totl numr of nts ing ut is minimiz. If w strt with goo nt prtitioning (whih is usully th s ftr pplying K-FM lgorithm on HDN ), th siz of M ont is muh smllr thn th numr of mouls in th originl ntlist. From our xprimnts, w s tht for th MCNC nhmrks, th prntg of mouls in ontntion rngs from 50-62% for K = 2, 45-60% for K = 3, 40-50% for K = 4, n 30-42% for K = 5. Thrfor, th K-MC prolm is muh simplr thn th originl K-wy prtitioning prolm, n juging y th trn of our rsults, it gts simplr with inrsing K Bining funtion n prfrn funtion Goo solutions to th K-MC prolm shoul minimiz th numr of nts ing ut unr th r onstrint. Sin this prolm is NP-Hr in gnrl, w rsort to ffiint huristi lgorithms. W introu funtion to stimt th numr of nts ut whn moul is ssign to prtition P i. Intuitivly, nt n j hs high ffinity for moul m k in ontntion if it hs high proility of ing stisfi (unut) ftr ttrting m k into its prtition, n low ffinity if it is most likly to ut vn ftr otining m k. W introu ining funtion (f ) to msur this ffinity twn f g f g () () Figur 4 Comprison of two nt-prtitioning solutions

8 -8- nt n moul. Lt n j nt n m k n j moul in ontntion. Th ining funtion f (n j, m k ) shoul pn on th following ftors: (i) (ii) (iii) S(n j ), th numr of mouls in n j : As th numr of mouls in nt inrss, th proility of th nt ing stisfi (unut) is ru. So th f (n j, m k ) shoul invrsly proportionl to th nt siz S(n j ). C(n j ), th numr of mouls in ontntion in nt n j :IfC(n j ) is high, th proility of th nt ing stisfi is low. Hn, th f (n j, m k ) shoul invrsly proportionl to C(n j ). S (n j ) C (n j ), th numr of mouls of nt n j in its prtition lry, i.. th numr of mouls in n j not in ontntion. Th f (n j, m k ) shoul irtly proportionl to this ftor sin th proility of th nt ing stisfi inrss s this numr inrss. Thrfor, w onsir th rtios S (n j ) C (n j ) n S (n j ) C (n j ) to of primry importn in trmining th S (n j ) C (n j ) ining funtion. From th two rtios, w fin th ining f untion of nt n j for moul m k to f (n j, m k ) = ( S (n j ) C (n j )) 2 S (n j ) C (n j ) W fin tht f (n j, m k ) = 0ifm k is not in n j. Also, if two mouls m k n m l in nt n j r lry ssign to two iffrnt prtitions (i.. n j is lry ut), thn f (n j, m i ) = 0 for ny m n j. Bs on th finition of th ining funtion, w fin th prfrn funtion pf (P i, m k ) twn moul m k in ontntion n prtition P i s follows: pf (m k, P i ) = Σ f (n, m k ) n P i Tht is, th prfrn funtion twn moul m k n prtition P i is th sum of ining funtion vlus twn m k n ll nts in prtition P i. Our ojtiv is to fin n optiml ssignmnt of th mouls in M ont to th prtitions suh tht th umultiv prfrn ovr ll ssignmnt gs is mximiz Flow-s formultion of th K-MC prolm W us th min-ost mx-flow lgorithm[16, 11] to omput th optiml moul ssignmnt. First, w onstrut n ssignmnt ntwork (AN) s follows. W onstrut iprtit grph in whih th nos rprsnt th mouls in M ont n th prtitions in P n h irt g (m k, P i ) onnts moul m k to prtition P i. Thn, w sour no s to AN n onnt it to vry moul no m k in AN. Similrly, w sink no t to AN n onnt vry prtition no P i to th sink t. Fig. 5 shows n xmpl of th ssignmnt ntwork. For h g in th ssignmnt ntwork, w fin its pity p ( ) n ost ost ( ) s follows: (i) if =(s, m k ), p ( ) = 1, ost ( ) = 0; (ii) (iii) if = (P i, t ), p ( ) = p (P i ), ost ( ) = 0, whr p (P i ) is th numr of mouls tht prtition P i n pt without violting its r onstrints. if = (m k, P i ), p ( ) = 1, ost ( ) = MAX pf (P i, m k ), whr MAX is positiv onstnt lrgr thn ny prfrn funtion vlu.

9 -9- Mouls M Prtitions P 1 s 2 t 3 4 Figur 5 Assignmnt Ntwork (AN) Lt m = M ont. In gnrl, w hv Σ q p (P i ) m ont (= M ont ). i =1 (C1) Tht is, th totl xss pity in ll prtitions is lrgr thn th numr of mouls in ontntion. (It is sy to show tht whn ll mouls r uniform in siz, onition C1 is lwys tru.) Whn moul sizs vry signifintly, w n only us p (P i ) to stimt th mximum numr of mouls llow in P i without violting th r ln onstrint, n suh stimtion usully tns to onsrvtiv (s on th lrgst or vrg moul siz). In prinipl, if w rlx th r slk prmtr α s fin in Stion 2, w n lwys stisfy onition C1. In th ss whr C1 is not stisfi, w shll prsnt ynmi r upting shm in Stion tht stisfis th r ln onstrint. Lmm 1 Th vlu of th mximum flow in th ssignmnt ntwork is m ont whn onition C1 is tru. Proof Aoring to th min-ut mx-flow thorm, w n only to show tht th siz of min-ut in th ssignmnt ntwork is m ont. First, w show tht th siz of min-ut is t lst m ont. Lt (X, X) min-ut whr s X n t X. Lt M X th st of moul nos in X n M X th st of moul nos in X. Lt m X = M X. Cs 1: Suppos tht thr xists prtition no P i in X. Thn, th totl pity of th gs twn P i n nos in M X is m X n th totl pity of th gs twn th sour n th gs in M X is m ont m X. Thrfor, th ut siz of (X, X) is t lst m X + (m ont m X ) = m ont. Cs 2: Suppos tht thr is no prtition no in X. Thn, th min-ut must ut twn th sink no n th rmining nos in AN. Sin onition C1 is stisfi, th ut siz is t lst m. Hn, in oth ss, th ut siz of (X, X) is t lst m. Morovr, it is sy to s tht th siz of min-ut is no mor thn m sin th ut twn th sour n th rmining nos is of siz m. Thrfor, w onlu tht th siz of ny min-ut in AN is m whn onition C1 is stisfi. Thorm 1 Th min-ost mx-flow in th ssignmnt ntwork inus moul ssignmnt whos totl prfrn funtion is mximum, whn onition C1 is tru. Proof Aoring to Lmm 1, whn onition C1 is tru, th vlu of mx-flow in N is m. Givn mx-flow f,w n inu moul ssignmnt s follows: Bus of th pity onstrints in N n th intgr flow proprty, it is sy to show tht mx-flow f of vlu m stisfis th following proprty: for h m i, thr is xist uniqu P j suh

10 -10- tht f (m i, P j ) = 1 n f (m i, P k ) = 0 for k j. W ssign moul m i to prtition P j if f (m i, P j ) = 1, n otin moul ssignmnt solution. Morovr, th moul ssignmnt solution stisfis th prtition r onstrints u th pity onstrints on gs (P j, t ) s. On th othr hn, givn moul ssignmnt solution stisfying th prtition r onstrints, w n riv mxflow f s follows: If moul m i is ssign to prtition P j, w fin f (s, m i ) = f (m i, P j ) = 1 n f (m i, P k ) = 0 for k j. Furthrmor, w fin f (P j, t ) to th sum of ll inoming flows t no P j. It is sy to vrify tht f is mx-flow of vlu m. Thrfor, h mx-flow f in N inus K-MC moul ssignmnt solution stisfying th prtition r onstrints, n vi vrs. Morovr, th ost of th mx-flow f is ost (f ) = m. MAX m M ont Σ pf (m, p )., p P, f (m,p )=1 Sin MAX is onstnt, mximizing th totl prfrn in moul ssignmnt solution is quivlnt to minimizing ost (f ). Hn, min-ost mx-flow in N inus moul ssignmnt with th mximum totl prfrn funtion. W us th ugmnting pth lgorithm [16, 11] for omputing minimum-ost mximum-flow in th ssignmnt ntwork. W strt with flow of vlu zro. At h stp, w omput th minimum ost ugmnting pth in th rsiul grph of th ssignmnt ntwork. Thn, w ugmnt th flow vlu y on, n upt th rsiul grph of th ssignmnt ntwork. Th ugmnttion pross stops ftr m stps. It n shown tht th tim omplxity of th minimumost mximum-flow omputtion is O (K. m 2 + m 2. logm ) using Fioni hps. Our implmnttion hs tim omplxity O (K. m 3 ) using simpl shortst pth lgorithm. Aftr w otin min-ost mx-flow, w n trmin th ssignmnt of mouls in ontntion in linr tim. Whn onition C1 is not stisfi (it ours in rr ss whn th r slk prmtr α is vry smll n/or th moul sizs vry signifintly), th mx-flow in th ssignmnt ntwork hs vlu lss thn m, whih mns tht som mouls in M ont r lft unssign. On option woul to romput th tul rsiul pitis t th trmintion of th flow lgorithm, n rpt th min-ost mx-flow lgorithm until ll mouls r ssign. In thory, w my hv to go through svrl suh itrtions of flow omputtion until ll mouls r ssign. In th nxt su-stion, w prsnt ynmi r upting shm tht rquirs xtly on flow omputtion Dynmi upting of r onstrint Th flow-s lgorithm for th K-MC prolm prsnt in th prvious su-stion omputs th pity of th g from prtition to th sink using th numr of mouls tht prtition oul ommot s on th vrg moul siz. This shm works wll whn ll mouls r mor or lss uniform in siz. Howvr, in som iruits (suh s Tst02 in th MCNC nhmrk suit), thr xist fw mouls (suh s mro loks) whos rs r muh lrgr thn th rst of mouls. In orr to fftivly hnl th s whn moul sizs vry onsirly, w introu n r lning shm whih first rmovs n ssign th mouls of xssivly lrg rs from ontntion, n thn ssigns th rmining mouls in ontntion s on n urt r pity onstrint whih is ynmilly upt uring flow omputtion.

11 -11- Lt A th umultiv r of ll mouls, K th numr of prtitions n α th usr-spifi slk. W first omput Ex = {m r (m ) (1 +α). K A }, whih is th st of mouls tht nnot ommot vn into th lrgst llow prtition siz. W thn ssign h of ths mouls to singl prtition sin omining ny of thm with othr mouls in th sm prtition will furthr violt th r ln onstrint 1. Th rsiul r (totl r of rmining mouls), not y A rs,isa r (Ex ) whr r (Ex ) is th umultiv r of th mouls in Ex. Our nw r ln onstrint will thn rquir tht th K Ex prtitions stisfy (1 α). A rs Ai (1 +α) K Ex. A rs K Ex It is sy to s tht Ex <K whn α > 0. Osrv tht h ugmnting pth in our min-ost mx-flow omputtion ssigns on mor moul to prtition with possil r-ssignmnt of som othr mouls. For xmpl, Fig. 6 () shows simpl ugmnting pth in th rsiul grph of th ssignmnt ntwork, whih ssigns moul to prtition y. Fig. 6 () shows mor gnrl ugmnting pth, whih ssigns moul to prtition y n rssigns moul from prtition y to prtition x. Hn, ftr h ugmnting pth omputtion, w know th ssignmnt of th nw moul n (possil) rssignmnts of othr mouls. Thrfor, w n upt th r pity onstrints of ll fft prtitions. W moify our flow lgorithm s follows. W us th sm finitions for pity n flow s rlir, ut w r sltiv out ugmnting flow long min-ost ugmnting pths in th rsiul grph. W fin two xtr rrys urrnt [P i ] n pity [P i ] to not th tul totl r of th mouls ssign to P i so fr, n th tul mximum llowl r of P i, rsptivly. Th nw min-ost mx-flow lgorithm from onptul point of viw n sri s follows: Whil thr xists n ugmnting pth in th rsiul grph Fin th min-ost ugmnting pth p = (s, m i 1, P j 1, m i 2,..., m it, P jt, t ) suh tht for ll ssignmnts m i P j spifi y p,(urrnt [P j ] + r (m i )) > pity [P j ] hols Augmnt flow long pth p n upt th rsiul grph Upt urrnt [P j ] for ll fft prtitions s follows if (m i, P j ) p thn urrnt [P i ] = urrnt [P i ]+r (m i ) if (P j, m i ) p thn urrnt [P i ] = urrnt [P i ] r (m i ) Upt pity [P j ] for ll fft prtitions oringly En-whil Tht is th moifi min-ost mx-flow lgorithm pts ugmnting pths spifying ssignmnts tht stisfy th tul prtition r onstrints y mintining two rrys rflting th rs u to th urrnt moul ssignmnt. On onrn is th numr of ugmnting pths tht w hv to rjt for w fin on whih stisfis th r onstrint. In ft our lgorithm inorports hking of r onstrint into th st-first srh pross for omputing th min-ost ugmnting pth, so tht w gnrt only th min-ost ugmnting pth whih stisfis th r onstrint. This is s on simpl osrvtion tht thr r thr typs of gs in th rsiul grph of th ssignmnt ntwork: 1 Givn n r ln onstrint spifi y slk prmtr α s shown in Stion 2, it is NP-omplt to i if thr is prtitioning solution stisfying suh onstrint whn th mouls r not uniform in siz. As rsult, it is not lwys possil for th usr to spify slk prmtr whih gurnts n r-ln prtitioning solution. Th st ny prtil prtitioning lgorithm n o is to try to minimiz th violtion of r ln onstrint.

12 -12- y x s t s y t () Simpl Assignmnt of Moul () Assignmnt of Moul y Rssignmnt of othr mouls Figur 6 Augmnting pths in th rsiul grph of th ssignmnt ntwork (i) (ii) (iii) th gs twn th sour n moul nos th gs twn moul nos n prtition nos th gs twn prtition nos n th sink Th min-ost ugmnting pth lgorithm is s on th Dijsktr s shortst pth lgorithm [11], whih prforms st-first srh strting from th sour n kps xpning th no rhl from th sour with th minimum pth ost in th rsiul grph. A min-ost sour-to-sink pth is foun whn th sink is rh in th st-first srh pross. In our moifi ugmnting pth lgorithm, w follow th gs of typ (i) n typ (ii) in th rsiul grph uring st-first srh, ut not th gs of typ (iii). Inst, w us th vlus mintin in pity n urrnt rrys whn iing if w n xtn th urrnt min-ost pth from prtition no to th sink. If xtning th pth to th sink (whih will form n ugmnting pth) os not violt th r ln onstrint of th prtitions involv in th pth, w otin min-ost ugmnting pth whih stisfis th r ln onstrint. Othrwis, w ontinu to pross th nxt no in th st-first srh orring. With this moifition, w n gurnt to gnrt min-ost ugmnting pth whih stisfis th tul r pity onstrint for h prtition n th omplxity of th lgorithm is th sm s tht for rgulr min-ost mx-flow using th shortst pth lgorithm Dynmi upting of ining funtions uring flow omputtion On prolm with th flow-s solution prsnt in n rlir su-stion is tht th stti prfrn funtions r not rfltiv of th hngs of th ining funtions of th nts s mor n mor mouls r ssign uring flow omputtion. In Fig. 7, whn m 1 is ssign to P 2 s on n 3 s strong ffinity, it is lr tht f (n 4, m 2 ) shoul inrs sin th proility of its nt ing stisfi is inrs, n hn th pf (P 2, m 2 ) inrss. Also, f (n 1, m 2 ) shoul ssign zro sin n 1 is ing ut, n pf (P 1, m 2 ) shoul upt oringly. Bining funtions n upt ffiintly s follows ftr moul m k is ssign to prtition P i. Lt N (m k ) = N 1 N 2, whr N 1 is th sust of nts tht r pl in P i n N 2 r th sust of nts tht r not in P i. Lt M ont th st of mouls tht r still in ontntion. (i) Rmov moul m k from th st of mouls in ontntion. M ont M ont {m k }

13 -13- P1 P2 n1 m1 n3 n2 n4 m2 Figur 7 Exmpl of ynmi upt of ining ftors (ii) For h nt n N 1, romput th f s n upt th pf s ssoit with P i inrmntlly. for h nt n N 1 for ll mouls m j n, n m j M ont pf (P i, m j ) = pf (P i, m j ) (f ol (n, m j ) f nw (n, m j )) (Not tht f ol (n, m j ) <f nw (n, m j ) sin C (n ) is ru y on) (iii) For h nt n N 2, ssign its f s to zro, n upt th pf s of ll th prtitions ut P i inrmntlly s follows: for h nt n N 2 for h m j N 2, n m j M ont pf (P i, m j ) = pf (P i, m j ) f ol (n, m j ) (Not tht f nw (n, m j ) = 0) Suh ynmi upting of ining funtions n prfrn funtions n sily inorport in our min-ost mxflow lgorithm ftr h flow ugmnttion. As not rlir, h flow ugmnting pth ssigns on mor moul to prtition n lso possily spifis r-ssignmnt of som othr mouls (s Figurs 6 () n ()). Thrfor, ftr fining h ugmnting pth Q i, w n upt th ining funtions rlt to th mouls in Q i, n thn upt th rlt prfrn funtions oringly. Thrfor, th g osts in th ssignmnt ntwork my hng ftr h flow ugmnttion. Evn with ynmi upting of g osts in th ssignmnt ntwork, w n still show tht flow ugmnttion stops ftr m stps. So, w hv th following rsults: Thorm 2 Assum tht h moul hs onstnt vrg gr in th ntlist. With ynmi upting of r onstrints, ining funtions n prfrn funtions, th K-MC prolm n solv in O (K. m 2 + m 2. logm ) tim s on min-ost mx-flow omputtion in th ssignmnt ntwork, whr K is th numr of prtitions n m is th numr mouls in ontntion. Proof Sin w o not upt g pity in flow omputtion, oring to Lmm 1, th vlu of mximum flow is still m n th lgorithm stops ftr m flow ugmnttions. Aftr h flow ugmnttion, w n to upt th prfrn funtions of th mouls long th ugmnting pth, whih tks t most O (K. m ) tim Th ovrll omplx-

14 -14- ity 2 is thn O (m. (K. m + m 2. logm + K. m )), whih is still O (K. m 2 + m 2. logm ). In rl-lif CMOS VLSI signs, mouls r onnt on vrg to fw nts n hn our ssumption of onstnt vrg gr for mouls is vli. Th K-DulFM lgorithm with ynmi upting of r onstrints, ining funtions in th flow omputtion is not s K-DulFM/DF, n th flow omputtion with stti ining funtions is not s K-DulFM/SF. From th rsults shown in th nxt stion, w shll s tht in gnrl K-DulFM/DF prous ttr prtitioning solutions ompr to K-DulFM/SF. Th inrs in omputtion tim u to ynmi upting of g osts is ngligil u to th inrmntl upting Rfinmnt of moul prtitioning solution Aftr solving th K-MC prolm w pply nothr pss of th K-FM prtitioning lgorithm to furthr rfin th moul prtitioning solution. Howvr, w osrv tht in ll tst ss th K-FM s rfinmnt stp onvrgs vry quikly with vry fw moul movs, whih is strong inition tht th K-FM moul prtitioning solution otin from K- wy nt prtitioning n moul ontntion rsolution is of vry high qulity K-DulMFFC-FM lgorithm Sustions 3.2 to 3.6 sri th K-DulFM lgorithm in til n illustrt how to omin our ul ntlist trnsformtion s multi-wy prtitioning prigm with th K-FM lgorithm. To furthr monstrt th powr n fftivnss of our multi-wy iruit prtitioning prigm s ul ntlist rprsnttion, w prsnt in this sustion how to pply our prigm to th K-MFFC-FM lgorithm y Cong, Li n Bgroi [10]. To ru th omputtionl omplxity of prtitioning vry lrg iruits, n to tk th vntg of th signl irtion informtion, Cong, Li n Bgroi propos lustring mtho s on mximum fnout-fr ons (MFFCs) [6, 7] s prprossing stp to thir prtitioning lgorithm. Th MFFC omposition thniqu ws first propos for omintionl iruits [6] for uplition-fr thnology mpping of lookup-tl s FPGAs. Lt input (v ) not th st of nos whih r th fnins of no v, n lt output (v ) not th st of nos whih r th fnouts of no v. For no v in th ntwork, on of v not C v, is sugrph of logi gts (xluing primry inputs (PIs)) onsisting of v n its prssors suh tht ny pth onnting no in C v n v lis ntirly in C v. W ll v th root of C v. A fnout-fr on (FFC) t v, not y FFC v, is on of v suh tht for ny no u v in FFC v, output (u ) FFC v. Th mximum fnout fr on (MFFC) of v, not y MFFC v, is FFC of v suh tht for ny non-pi no w,ifoutput (w ) MFFC v, thn w MFFC v. It is not iffiult to show tht MFFC is uniqu for vry no, n ny FFC of v is ontin in MFFC v. Clrly, if gt u is in MFFC v, its vlu is us solly for gnrting th output t gt v (n its snnts). Thrfor, it is vry nturl to lustr u n v togthr. In gnrl, ll th gts in singl MFFC v n onsir to losly rlt, sin thy r us solly for omputtion of v. Th MFFCs lustring lgorithm onsirs oth signl irtion n logi pnny, n prous nturl lustring solution in linr tim. Th K-MFFC-FM lgorithm rport in [10] pplis th K-FM lgorithm irtly on th lustr iruits to gt vry promising rsults. W ppli our ul nt trnsformtion prigm to K-MFFC-FM n otin th K-DulMFFC-FM lgorithm. In th K-DulMFFC-FM lgorithm, mouls wr lustr using th MFFC lgorithm, n thn nt prtitioning of th HDN of th lustr ntwork ws otin using th K-FM lgorithm. Thn, th K-MC prolm ws solv using th 2 This omplxity nlysis is s on using Fioni hp. It is sy to s y similr nlysis tht ynmi upting os not inrs th om-

15 -15- min-ost mx-flow thniqu, n th solution ws furthr improv using th K-FM lgorithm. By our onvntion, th lgorithm with ynmi upting of ining funtions is nm K-DulMFFC-FM/DF n th lgorithm with stti ining funtions is nm K-DulMFFC-FM/SF. Th xprimntl rsults for oth K-DulFM n K-DulMFFC-FM r rport in th nxt stion. 4. Exprimntl rsults W hv implmnt oth th K-DulFM/SF n K-DulFM/DF lgorithms on SUN SPARC worksttions. W ompr th two lgorithms with th onvntionl K-FM lgorithms on st of MCNC nhmrk iruits (Tst02-06, PrimGA1, PrimGA2) n 5 lrg iruits provi y th Hwltt-Pkr Rsrh L (CPU, GA, FPU, GA2, FPU2). Ciruits CPU, GA n FPU onsist of lookup-tls (for multi-fpga implmnttion). Ciruits FPU2 n GA2 r th originl ntlists of FPU n GA for thnology mpping. Tl 1 shows th hrtristis of th nhmrk iruits, inluing th numr of mouls (gts n I/O ps), numr of nts, mximum moul r vrsus totl moul r, th mximum numr of mouls in nt (mx nt), n th mximum numr of nts inint to th sm moul (g). Tls 2()-2() show th omprison of K-DulFM/SF n K-DulFM/DF with th K-FM lgorithm for K rnging from 2 to 5. For h xmpl, th K-FM lgorithm ws run 20 tims, h on rnom initil moul prtitioning. In orr to otin fir omprison, w mk sur tht th runtims of K-DulFM/SF n K-DulFM/DF r omprl with tht of th K-FM lgorithm. As rsult, th K-DulFM lgorithms wr run on with th gry nt prtition, n thn pproximtly 10 tims 3, h on rnom initil nt prtitioning of th ul ntlist rprsnttion (HDN). Th r slk prmtr α ws st to 10% in oth K-FM n K-DulFM lgorithms. Ciruit # Mouls * # Nts Mx moul/ Totl r Mx nt Dg Tst / Tst / Tst / Tst / Tst / PrimGA / PrimGA / pu / GA / FPU / GA **1/ FPU **1/ Tl 1. Chrtristis of th Tst Ciruits. plxity of min-ost mx-flow vn if rgulr hps or linr srhs r us for hoosing th min-ost no to xpn. *Th numr of mouls inlu th I/O ps. **For GA2 n FPU2 moul rs wr not givn. Th rs wr ssum to Th numr of runs vris from xmpl to xmpl in orr to mth th runtim of 20-run K-FM lgorithm on th sm xmpl in orr to otin fir omprison. Th MCNC iruits n iruits CPU, GA, n FPU from HP usully rng from runs, whil th unmpp iruits FPU2 n GA2 from HP rng twn 5 n 8 runs.

16 -16- Ciruit K-FM K-DulFM/ SF K-DulFM/ DF Tst Tst Tst Tst Tst PrimGA PrimGA pu GA FPU GA FPU ovrll Tl 2 () Comprison of K-DulFM ginst K-FM for K = 2 Ciruit K-FM K-DulFM/ SF K-DulFM/ DF Tst Tst Tst Tst Tst PrimGA PrimGA pu GA FPU GA FPU ovrll Tl 2 () Comprison of K-DulFM ginst K-FM for K = 3 Ciruit K-FM K-DulFM/ SF K-DulFM/ DF Tst Tst Tst Tst Tst PrimGA PrimGA pu GA FPU GA FPU ovrll Tl 2 () Comprison of K-DulFM ginst K-FM for K = 4

17 -17- Ciruit K-FM K-DulFM/ SF K-DulFM/ DF Tst Tst Tst Tst Tst PrimGA PrimGA pu GA FPU GA FPU ovrll Tl 2 () Comprison of K-DulFM ginst K-FM for K = 5 On n s from Tls 2() - 2() tht th K-DulFM/DF lgorithm onsistntly outprforms th K-FM lgorithm y signifint mrgin, from out 20% to 31% rution in nt utsiz for K = 2 through 5. Th K-DulFM/SF lgorithm prous rsults with out 12% to 21% utsiz rution s ompr to K-FM for K = 2 through 5. Th K-DulFM/DF lgorithm in gnrl outprforms th K-DulFM/SF lgorithm for ll th iruits tst, n inits tht ynmi upting of ining funtions is in usful. In trms of ffiiny, th K-DulFM/SF lgorithm is gnrlly fstr thn th K-DulFM/DF lgorithm, n th iffrn inrss s th numr of prtitions inrss. Eh run of K-DulFM/SF for FPU2 took roun 6500 sons n 8200 sons for K = 3 n K = 4, rsptivly on SUN SPARC10 worksttion, whil h run for K-DulFM/DF took roun 6900 sons n 9000 sons, rsptivly. Thr r lso improvmnts, though lss rmti, whn th rution in vrg nt utsiz is onsir s oppos to th minimum nt utsiz. For instn, th K-DulFM/DF lgorithm rus th vrg nt utsiz (whn ompr to th K-FM lgorithm) y 12% for th MCNC iruits for K = 4. Similrly, th K-DulFM/SF rus th vrg nt utsiz y 10%. Th stnr vitions of th nt utsizs from th thr lgorithms r omprl (of 9% - 13%). Th CF prmtr ws st to 5 for oth K-DulFM lgorithms for ll th rsults shown hr. Th rsults my vry onsirly if w hng th vlu of CF. As w hng th CF prmtr from 2 to 10, th utsiz otin y th K-DulFM lgorithms vris y 18% on vrg, with stnr vition 12%. Howvr, th rng of th CF prmtrs is vry smll sin it is rlt to th gr of th mouls in ntlist. For instn, 99% of th mouls in ll of th MCNC nhmrk n HP Rsrh L iruits hv gr of lss thn 9. Thus on n sily fin goo rng for th CF vlu. W ompr th K-DulFM lgorithm with oth EIG1 [18] n Proli [26], two rntly rport sptrl-s mthos spiliz to solv th iprtitioning prolm. EIG1 uss qurti ojtiv funtion, whil Proli onsirs th linr ojtiv funtion n otins imprssiv improvmnt ovr EIG1 with onsirl longr omputtion tim. W prsnt omprisons of K-DulFM with ths two lgorithms in Tl 3 for iruits in th ACM/SIGDA nhmrk suit s on th rsults rport in [26]. Th r slk ws st to 10%, n th numr of runs for K- DulFM ws st to 10 sin it ls to omprl runtim onsum y EIG1. W s tht in gnrl th K-DulFM lgorithms prou iprtitioning solutions with omprl nt utsizs s Proli whil onsuming muh lss omputtion tim, n onsistntly outprform EIG1 y signifint mrgin (56% nt utsiz rution on vrg) whil onsuming th sm mount of tim. Morovr, K-DulFM hs th following vntgs ovr EIG1 n Proli: (i) Th

18 -18- r ln onstrint nnot mol nturlly in th EIG1 n Proli formultion, ut is sily ommot in K-DulFM. (ii) EIG1 n Proli nnot hnl som prtil onstrints suh s th pr-ssignmnt of rtin mouls to prtitions, ut K-DulFM n ommot suh onstrints sily y pr-ssigning ths mouls in th initil prtitioning solution n mrking thm lok throughout ll phss of th prtitioning. (iii) Furthrmor, EIG1 n Proli r sign spifilly for iprtitioning whil K-DulFM is pplil to gnrl multi-wy prtitioning. Finlly, w rport th rsults otin y th K-DulMFFC-FM lgorithm, whih ws sri in Stion 3.7. Th K-MFFC-FM lgorithm ws vlop to prtition iruit to multipl prossors for prlll logi simultion. Sin th gol is to minimiz intr-prossor ommunition, Cong, Li n Bgroi [10] moifi thir ojtiv funtion to minimiz th numr of nts ut whil ignoring th onntions twn gts n input/output ps. (This is us in thir pplition primry input vril n m vill for vry prossor, n gnrting n output to n output p is quivlnt to writ oprtion to lol isk with no ommunition ovrh.) Our K-DulMFFC-FM lgorithm ws moifi oringly to us th sm ojtiv funtion for fir n urt omprison. Tls 4()-4() show th omprison of th K-DulMFFC-FM lgorithms with th originl th K-MFFC-FM 4 [10] unr th sm ojtiv funtion for th ICSAC85 nhmrk suit (whih onsist of only omintionl iruits n r suitl for MFFC lustring). Th r slk ws st to 5%, n K-MFFC-FM ws run 20 tims, n th K-DulMFFC-FM 5 lgorithms wr run from tims for omprl runtims. Th β vlu tht ouns th mximum siz of th lustr ws st to 1/3 for oth lgorithms sin highr vlus (lik 1/2) somtims i not prou r-ln prtitions for th K- MFFC-FM lgorithm. From Tls 4 () n (), w s tht th K-DulMFFC-FM lgorithms hv on th vrg 15 to 26% smllr utsiz thn K-MFFC-FM for th ICSAC85 nhmrk suit. 5. Conlusion n possil futur xtnsions Th rsults in this ppr show onviningly tht nt prtitioning s mthos prou ttr solutions to th multiwy iruit prtitioning prolm thn irt moul prtitioning. Our formultion n solution to th K-wy moul ontntion prolm provi gnrl n fftiv frm-work to onvrt K-wy nt prtitioning solution to K-wy moul prtitioning solution. Both th K-DulFM/SF n K-DulFM/DF lgorithms n xtn sily to hnl mny prtil onstrints, suh s I/O oun onstrint on h prtition, pr-spifi ssignmnt of mouls to Ciuit EIG1 Proli K-DulFM/ SF K-DulFM/ DF lu strut primry iom s s s inustry Tl 3 Comprison of K-DulFM ginst EIG1 n Proli for K = 2 4 K-MFFC-FM is rvit to K-MFM in Tl 4. 5 K-DulMFFC-FM is rvit to K-DulMFM in Tl 4.

19 -19- Ciuit K-MFM K-DulMFM/ SF K-DulMFM/ DF ovrll Tl 4 () Comprison of K-DulMFFC-FM ginst K-MFFC-FM for K = 4 Ciuit K-MFM K-DulMFM/ SF K-DulMFM/ DF ovrll Tl 4 () Comprison of K-DulMFFC-FM ginst K-MFFC-FM for K = 8 prtitions, t. Our prtitioning rsults for th tst iruits, rnging from 1000 to 31,000 mouls, prov tht our K-DulFM lgorithm is sll in trms of th siz of th iruit. Othr prtitioning lgorithms (suh s th grph sptrl s mtho or th rnom-wlk s mtho) my fil to prou solutions for lrg iruits u to high mmory usg (.g. to stor th Lplin mtrix or rnom-wlk of qurti lngth in trms of th numr of mouls) or sp inffiiny. Th mmory n sp ffiiny of th K-DulFM lgorithm nls us to hnl prolms of muh lrgr sizs. Whn th prolm siz is not too lrg, mor lort K-wy prtitioning lgorithms othr thn th simpl K-FM lgorithm n lso us to prou ttr nt prtitioning of th ul ntlist rprsnttion. Improvmnt on th nt prtitioning solutions usully ls to improvmnt on th rsulting moul prtitioning solution. In this ppr, w onntrt primrily on minimizing th numr of nts ut sujt to r onstrints. Our gnrl frmwork is pplil to othr ojtiv funtions lso. For xmpl, our mthos n xtn to hnl th following two ojtivs: (F1) (F2) Th wirility ojtiv funtion: This is populr ojtiv funtion us to minimiz th numr of intr-hip wir rossings. Our ul ntlist pproh n us to optimiz this funtion y stting ining funtions twn nt n ny of its ontn mouls to on. In this s, on n show tht min-ost mx-flow in th ssignmnt ntwork orrspons to moul ssignmnt solution with th minimum totl wir rossing. Prformn-s ojtiv funtion: Som rntly propos ojtivs optimiz for systm prformn unr timing onstrints. For xmpl, Shih, Kuh n Tsy[28] propos n lgorithm to minimiz th wirility ojtiv funtion sujt to thrml, I/O pin, n timing onstrints in ition to r onstrints. In (F1) w show how to minimiz th wirility funtion. Th thrml n I/O pin onstrints n hnl in similr wy s th

20 -20- r onstrint. Hn, th lgorithm propos in Stion n moifi to ommot ths two onstrints (with two itionl rrys). Th timing onstrints n hnl y ssigning lotions to prtitions suh tht h pir of prtitions will t rtin istn from h othr. W n thn nsur tht timing onstrints r stisfi whn th prfrn funtions r ing upt in th lgorithm (s xplin in Sustion 3.5.5, Stps (ii) n (iii)). In prtiulr, on moul, longing to nt n, is ssign to prtition P, th unssign mouls of n r onstrin to pl in prtitions tht r suffiintly "los" to P. This onstrint n stisfi y mking th ost of th gs from ths unssign mouls to "istnt" prtitions prohiitivly high. W lso osrv tht th hoi of CF prmtr fin in Sustion 3.3 my fft th prtitioning rsults onsirly. W woul lik to vlop fftiv lgorithms to utomtilly omput th vlu of CF prmtr s on th hrtristis of th iruits so tht th K-DulFM lgorithm n onsistntly prou ttr prtitioning rsults. 6. Aknowlgmnts Th uthors woul lik to thnk Phil Kuks n Grg Snir t Hwltt-Pkr Lortory for proviing nhmrk iruits. This work is prtilly support y ARPA/CSTO unr ontrt J-FBI , th Ntionl Sin Fountion Young Invstigtor Awr wr numr MIP , n grnts from AT&T Bll Lortoris, Hwltt-Pkr, Xrox Fountion, Xilinx unr th Cliforni MICRO progrm n NY1 Awr mthing progrm. Rfrns [1] C. J. Alprt n A. B. Khng, Gomtri Emings for Fstr (n Bttr) Multi-Wy Ntlist Prtitioning, Pro. ACM/IEEE Dsign Automtion Conf., pp , Jun [2] E. R. Brns, An Algorithm for Prtitioning th Nos of Grph, SIAM J. Alg. Dis. Mth., Vol. 3, pp , [3] M. Brsl n A. Sngiovnni-Vinntlli, Huristi Mthos for Communition-Bs Logi Prtitioning, 4th ACM/SIGDA Physil Dsign Workshop, pp , April [4] R. Boppn, Eignvlus n Grph Bistion: An Avrg-Cs Anlysis, IEEE Symp. on Fountions of Computr Sin, pp , [5] P. Chn, M. Shlg, n J. Zin, Sptrl K-Wy Rtio-Cut Prtitioning n Clustring, Pro. 30th ACM/IEEE Dsign Automtion Conf., Jun [6] J. Cong n Y. Ding, On Ar/Dpth Tr-off in LUT-Bs FPGA Thnology Mpping, Pro. 30th ACM/IEEE Dsign Automtion Conf., pp , Jun [7] J. Cong n Y. Ding, On Ar/Dpth Tr-off in LUT-Bs FPGA Thnology Mpping, IEEE Trns. on VLSI Systms, Vol. 2, pp , Jun [8] J. Cong, L. Hgn, n A. B. Khng, Rnom Wlks for Ciruit Clustring, IEEE 4th Int l ASIC Conf., pp. P14-2.1, Spt [9] J. Cong, L. Hgn, n A. B. Khng, Nt Prtitions Yil Bttr Moul Prtitions, IEEE 29th Dsign Automtion Confrn, pp , Jun [10] J. Cong, Z. Li, n R. Bgroi, Ayli Multi-Wy Prtitioning of Booln Ntworks, Pro. ACM/IEEE 31st Dsign Automtion Conf., pp , Jun 1994.

21 -21- [11] T. Cormn, C. Lisrson, n R. Rivst, Algorithms, MIT Prss, Cmrig, MA (1990). [12] J. Cong, W. Lio, n N. Shivkumr, Multi-Wy VLSI Ciruit Prtitioning Bs on Dul Nt Rprsnttion, Pro. IEEE Int l Conf. on Computr-Ai Dsign, pp , Nov Also vill s UCLA Computr Sin Dprtmnt Th. Rport CSD [13] J. Cong n M. Smith, A Bottom-up Clustring Algorithm with Applitions to Ciruit Prtitioning in VLSI Dsigns, ACM/IEEE Dsign Automtion Conf., pp , Jun [14] W. Donth n A. Hoffmn, Lowr Bouns for th Prtitioning of Grphs, IBM J. Rs. Dv., pp , [15] C. Fiui n R. Mtthyss, A Linr Tim Huristi for Improving Ntwork Prtitions, ACM/IEEE Dsign Automtion Conf., pp , [16] L. R. For n D. R. Fulkrson, Flows in Ntworks, Printon Univ. Prss, Printon, N.J. (1962). [17] J. Grn n K. Supowit, Simult Annling without Rjt Movs, Pro. Int l Conf. on Computr Dsign, pp , [18] L. Hgn n A. B. Khng, Fst Sptrl Mthos for Rtio Cut Prtitioning n Clustring, Pro. IEEE Int l Conf. on Computr-Ai Dsign, pp , [19] L. Hgn n A. B. Khng, A Nw Approh to Efftiv Ciruit Clustring, Int l Conf. on Computr- Ai Dsign, pp , Nov [20] L. Hgn n A. B. Khng, Nw Sptrl Mthos for Rtio Cut Prtitioning n Clustring, IEEE Trns. on CAD, pp , Spt [21] J. Hwng n A. El Gml, Optiml Rplition for Min-Cut Prtitioning, Int l Conf. on Computr-Ai Dsign, pp , Nov [22] B. Krnighn n S. Lin, An Effiint Huristi Prour for Prtitioning of Eltril Ciruits, Bll Systm Thnil J., F [23] S. Kirkptrik, C. D. Gltt, n M. P. Vhi, Jr., Optimiztion y Simult Annling, Sin, Vol. 220, pp , My [24] B. Krishnmurthy, An Improv Min-Cut Algorithm for Prtitioning VLSI Ntworks, IEEE Trns. on Computrs, Vol. 33, pp , [25] C. Kring n A. R. Nwton, A Cll-Rpliting Approh to Minu-Bs Ciruit Prtitioning, IEEE Int l Conf. on Computr-Ai Dsign, pp. 2-5, Nov [26] B. M. Riss, K. Doll, n F. M. Johnns, Prtitioning Vry Lrg Ciruits Using Anlytil Plmnt Thniqus, Pro. ACM/IEEE 31st Dsign Automtion Conf., Jun [27] L. Snhis, Multipl-Wy Ntwork Prtitioning, IEEE Trns. on Computrs, Vol. 38, pp , [28] M. Shih, E. Kuh, n R. Tsy, Prformn-Drivn Systm Prtitioning in Multi-Chip Mouls, Pro. ACM/IEEE Dsign Automtion Conf., pp , Jun [29] H. Yng n D. F. Wong, Effiint Ntwork Flow Bs Min-Cut Bln Prtitioning, Pro. IEEE Int l Conf. on Computr-Ai Dsign, pp , Nov

22 -22- [30] C. W. Yh, C. K. Chng, n T. T. Lin, A Gnrl Purpos Multipl-Wy Prtitioning Algorithm, Pro. 28th ACM/IEEE Dsign Automtion Conf., Jun [31] C. W. Yh, C. K. Chng, n T. T. Lin, A Proilisti Multiommoity-Flow Solution to Ciruit Clustring Prolms, Int l Conf. on Computr-Ai Dsign, pp , Nov