Partitioning Algorithms. UCLA Department of Computer Science, Los Angeles, CA y Cadence Design Systems, Inc., San Jose, CA 95134

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1 On Implmnttion Chois for Itrtiv Improvmnt Prtitioning Algorithms Lrs W. Hgn y, Dnnis J.-H. Hung n Anrw B. Khng UCLA Dprtmnt of Computr Sin, Los Angls, CA y Cn Dsign Systms, In., Sn Jos, CA Astrt Itrtiv improvmnt prtitioning lgorithms suh s thos u to Fiui n Mtthyss (FM) [2] n Krishnmurthy [5] xploit n int gin ukt t strutur in slting mouls tht r mov from on prtition to th othr. In this ppr, w invstigt thr gin ukt implmnttions n thir t on th prformn of th FM prtitioning lgorithm. Surprisingly, sltion from gin ukts mintin s Lst-In-First-Out (LIFO) stks ls to signintly ttr rsults thn sltion from gin ukts mintin rnomly (s in [5] [7]) or s First-In- First-Out (FIFO) quus. Our xprimnts show tht LIFO ukts rsult in 35% improvmnt ovr rnom ukts n 42% improvmnt ovr FIFO ukts. Furthrmor, liminting rnomiztion from th ukt sltion is of grtr nt to FM prformn thn ing th Krishnmurthy gin vtor. By omining insights from th LIFO gin ukts with thos of Krishnmurthy's originl work, nw highrlvl gin formultion is propos. This ltrntiv formultion rsults in furthr 16% rution in th vrg ut ost whn ompr irtly to th Krishnmurthy formultion for highr-lvl gins, ssuming LIFO orgniztion for th gin ukts. 1 Prliminris This ppr isusss th prolm of iprtitioning iruit ntlist hyprgrph G =(V; E), whr th st of mouls V is ivi into isjoint U n W to minimiz th numr of signl nts from E tht ross th ut. In proution softwr for iruit prtitioning, itrtiv improvmnt is nrly univrsl pproh, ithr s postprossing rnmnt to othr mthos or s mtho in itslf. Itrtiv improvmnt is s on lol prturtion of th urrnt solution n n ithr gry (th Krnighn-Lin mtho [3] [9] n its lgorithmi spups y Fiui n Mtthyss [2], Krishnmurthy [5] n Dutt [1]) or hill-liming (th simult nnling pproh of Kirkptrik t l.[4], Shn [10] n othrs). Virtully ll implmnttions will lso us multipl rnom strting ongurtions (\multi-strt") [6] [11] in This rsrh ws support in prt y NSF grnt MIP n mthing funs from High-Lvl Dsign Systms. orr to yil pritl prformn (\stility"). This ppr will fous on itrtiv improvmnt lgorithms whih r s on th gry strtgy: strt with urrnt fsil solution n itrtivly prtur it into nothr fsil solution, opting th prturtion s th nxt solution only if it improvs th ost funtion. Th typ of prturtion us trmins topology ovr th st of fsil solutions, known s nighorhoo strutur. For th ost funtion to \smooth" ovr th nighorhoo strutur, th prturtion (lso known s nighorhoo oprtor) shoul smll n \lol". Usul nighorhoo oprtors for grph/iruit prtitioning involv swpping pir of mouls or shifting singl moul ross th ut. Erly gry improvmnt mthos pply suh oprtors, n quikly n lol minim whih usully orrspon to poor solutions. In 1970, Krnighn n Lin [3] introu wht is oftn sri s th rst \goo" grph prtitioning huristi. Th Krnighn-Lin (KL) lgorithm uss pir-swpping, n pros in psss. During h pss, vry moul is mov xtly on. At th ginning of th pss, ll mouls r \unlok" n th gin (i.., th rs in ut nts tht woul rsult from moving givn moul to th othr prtition) is lult for h of th n = jv j mouls. Thn, th pir of unlok mouls in U n W with highst omin gin is foun y srhing through th O(n 2 ) possil pirs. Aftr th slt mouls r swpp, thy om \lok" n th lgorithm upts oth th ost of th nw prtition n th gins of th rmining unlok mouls. This pross is itrt until ll th mouls r lok, t whih point th lowst-ost prtition nountr ovr th ntir pss is rstor n rturn. Anothr pss is thn xut using th rsult from th prvious pss s its strting point; th lgorithm trmints whn pss fils to improv th ost funtion. Th vntg of th KL lgorithm ovr gry pir-swpping is tht it is in som sns l to mov out of lol minim. This ours us th pir of mouls with highst omin gin is lwys swpp, vn if this omin gin is ngtiv. Howvr, if w onsir ll th

2 solutions tht r rhl within singl pss of th lgorithm to \nighors" of th strting solution, thn th KL lgorithm is still gry. Th min isvntgs of th KL lgorithm, s prsnt in [3], wr (i) tht it only works on grphs n (ii) tht it is omputtionlly xpnsiv. Although th numr of psss in most ss is rltivly low, th KL lgorithm rquirs vlution of O(n 2 ) swps for vry mov, rsulting in omplxity pr pss of O(n 2 log n). Shwikrt n Krnighn [9] xtn KL to hyprgrphs, ut i not improv th tim omplxity of th lgorithm. 1 Th FM Algorithm In 1982, Fiui n Mtthyss [2] prsnt KL-inspir lgorithm whih ru th tim pr pss to linr in th siz of th ntlist (i.., O(p) whr p is th totl numr of pins). Th Fiui- Mtthyss (FM) lgorithm is vry similr to KL: (i) FM lso prforms psss within whih h moul is mov xtly on; (ii) FM lso rors ll solutions nountr uring th pss n rturns th st on; n (iii) FM lso ontinus to prform psss until pss fils to improv th ost funtion. Th primry irn twn th KL n FM lgorithms lis in th nighorhoo oprtor. Inst of swpping pir of mouls, FM movs singl moul t tim. In othr wors, th gin lists r srh for singl moul whih hs highst gin. This sutl hng llows for signint improvmnt in runtim with littl loss in solution qulity. Fiui n Mtthyss mortiz th ost of upting th moul gins, suh tht th totl ost of ning th highst-gin moul is O(p) pr pss. Th nling t strutur is n rry of \gin ukts" whih groups th mouls of givn prtition oring to thir gins. Krishnmurthy's Extnsion to FM Ovr th pst, FM hs om prhps th singl most wily us n it prtitioning lgorithm in th VLSI CAD r. Mny works hv invstigt possil improvmnts n xtnsions. On ommonly-it xtnsion is tht of Krishnmurthy [5], who show how on oul intly introu \look-h" into th FM lgorithm to improv tirking whn th highst-gin ukt ontins mor thn on moul. Spilly, Krishnmurthy xtns th gin vlu of moul intoginvtor whih 1 Th rution from O(n 3 ) to O(n 2 log n) is hiv y mintining sort list of osts. Rntly, Dutt [1] prsnt spup of th originl KL lgorithm, ll QuikCut, whih uss n improv t strutur suh tht only O( 2 ) no pirs n to xmin to n th pir with mximum gin ( is th mximum no gr). As rsult, QuikCut hs tim omplxity of only O(mx(; log(n))), whr is th numr of gs in th grph. QuikCut urrntly works only on grphs, ut n xtnsion to hyprgrphs sms possil. givs squn of potntil gin vlus orrsponing to vrious numrs of movs into th futur. Krishnmurthy ns th ining numr U (s) of signl nt s with rspt to prtition U to th numr of unlok mouls of s in prtition U, unlss thr is lok moul of s in prtition U, in whih s U (s) =1. Intuitivly, th ining numr U (s) is msur of how iult it is to mov nt s out of prtition U. Th ining numr W (s) is similrly n. Th k-th lvl gin k (v i ) of moul v i 2 U is thn givn y 2 k (v i )= jfs 2 Ejv i 2 s; U (s) =k; W (s) > 0gj, jfs 2 Ejv i 2 s; U (s) > 0; W (s) =k, 1gj Eh lmnt k (v i ) in th gin vtor orrspons to th k-th lvl gin of moul v i. Not tht th rstlvl gin 1 (v i ) orrspons to th gin us in th FM lgorithm. Th intuition hin th highr-lvl gins k, with k>1, is tht th positiv trm ounts th numr of nts whih will hv ining numr k, 1 ftr th mov, whil th ngtiv trm ounts th numr of nts with urrnt ining numr k, 1whihwill hv ining numr qul to 1 ftr th mov. In othr wors, th positiv trm ounts nts with ining numr k, 1 tht r \rt" y th mov; th ngtiv trm ounts nts with ining numr k, 1 tht r \stroy" y th mov. (Th rt nts li on th si tht th moul is moving \from", n th stroy nts li on th si tht th moul is moving \to".) Krishnmurthy's mtho uss lxiogrphi orring of th vtors ( 1, 2, 3, :::) to rk tis whn n FM gin ukt ontins mor thn on moul. Krishnmurthy ompr his FM plus highr-lvl gin (FM+HL) lgorithm with th originl FM lgorithm n foun tht ing sonn thir-lvl gins improv th vrg solution qulity, with n omputtionl xpns of only O(kp), whr k is th numr of vlus mintin in (i.., th siz of) th gin vtor. This ws onrm y Snhis [7], who xtn FM+HL to multi-wy prtitioning. 2 Ti-Brking in th FM Algorithm During typil pss in th FM lgorithm, thr r usully mny tis (i.., th highst-gin ukt will ontin mor thn on moul). Figur 1 shows th numr of mouls in th highst-gin ukt t h mov throughout th rst pss of FM for th Primry1 tst s (w plot th vrg n mximum ovr 1000 runs). This stion invstigts how th mtho us 2 Th nottion us for th Krishnmurthy formul r pt from [5]. Not tht in orr to hnl 1-pin nts orrtly, th trm U (s) > 0 shoul to hng to U (s) > 1. Howvr, 1-pin nts n lso limint whil ring in th ntlist, oviting th n for suh hng.

3 to hoos ll (i.., moul) from th highst-gin ukt, n th mtho us to upt lls into gin ukts, will togthr t th prformn of th FM lgorithm mximum vrg Figur 1: Numr of mouls in th highst-gin ukt uring th rst pss of FM for tst s Primry1. Th vrg n mximum numrs wr gnrt from 1000 sprt FM runs. 2.1 Ti-Brking Shms In th originl ppr sriing th FM lgorithm [2], th gin ukts onsist of ouly-link lists. To intify th ll to mov, Fiui n Mtthyss onsir th rst ll in th highst-gin ukt of h prtition. Figur 2 rprous th \MAXGAIN" ukt us in th lgorithm sription of [2]; not tht this ukt hs only pointr to th h of th list. Thus, it is rsonl to infr tht whn slting ll to mov from th highst-gin ukt, Fiui n Mtthyss slt th ll t th h of th list. Suh n infrn is support y th ft tht this oprtion must prform in O(1) tim in orr for th omplxity of th lgorithm to rmin t O(p). Choosing th rst ll in th list stiss this omplxity rquirmnt. With rspt to insrting n upt ll into nw ukt, [2] rmovs ll from its urrnt list n movs it to th h of its nw ukt list. Consiring th rmovl n insrtion prours togthr, w s tht th gin ukts funtion s LIFO stks (rmov t h, insrt t h), ut oul just s sily funtion s FIFO quus (rmov t h, insrt t til) if pointr to th til of th list is inorport into th t strutur. Fiui n Mtthyss o not isuss th implitions of th hoi of ukt orgniztion on thir lgorithm's prformn. Howvr, s w shll show, this hoi hs signint t. Intrstingly, nithr Krishnmurthy nor Snhis points out ny hng in th ti-rking huristi us to slt mong lls with intil highr-lvl gins. Th nturl infrn is tht thir works lso +pmx MAX GAIN -pmx CELL ll# n ll#... Figur 2: Th gin ukt list strutur s shown in [2]. us LIFO mhnism, following th originl FM lgorithm sription. Howvr, in th o istriut y Snhis [8], it is lr tht th highst-gin moul is slt rnomly in th vnt of tis. Snhis in [7] nvr isusss th onsquns of this hng, ut writs: \W lso rnomiz ritrry hois in th lgorithm n prform numr of runs on h ntwork prtition t h irnt lvl" (pg 68, right olumn, rst prgrph). A similr sttmnt ismy Krishnmurthy [5] (pg 442, lft olumn, rst prgrph): \W osrv tht singl run woul not provi suint vin to ompr th rsults; for h of ths lgorithms, ing huristi in ntur involvs mking rtin ritrry hois, usully in th form of slting ny on lmnt from st ontining mor thn on lmnt. Thus, w rnomiz suh ritrry hois n prform numr of runs." Tht possily oth of ths works introu rnom ti-rking in ompring FM+HL to th originl FM is not trivil. Rnomiztion not only inrss th tim omplxity of th lgorithm ut, mor ritilly, pls into qustion th onlusions rwn from th xprimntl rsults. W hv xmin how th \ovious" ti-rking mhnism propos y Fiui n Mtthyss [2] omprs with ltrntiv shms. In prtiulr, w hv ompr LIFO sltion with rnom sltion (us y Snhis [7]) n FIFO sltion (n ltrntiv orgniztion with omplxity similr to LIFO). Our tst is th o istriut y Snhis [8] with pproprit moitions m for hnling LIFO n FIFO sltion. In ll of our xprimnts, w ssum h nohs unit r, n w onstrin th prtition sizs ju j n jw j to ir y t most 1.

4 2.2 Exprimntl Rsults Th thir olumn of Tl 1 lrly shows th ffts of th sltion mthoology. 3 Surprisingly, th FIFO shm is no ttr thn rnom sltion. Th LIFO shm givs onsirl improvmnt ovr rnom sltion. On possil xplntion my tht orgnizing th ukts suh tht \most rntly visit" mouls r pl nr th ginnings of th gin ukts impliitly uss nighorhoos (or prhps lustrs) of mouls to mov togthr. Furthrmor, sin thr r two gin struturs, on for h prtition, it is possil for h prtition to \pull" on irnt lustrs whil mintining th ln. If ths lustrs r non-intrfring, i.., wily sprt, mor of th rly movs will rsult in positiv gin, nling th urrnt pss to rh lowr-ost point in th solution sp. In othr wors, within h pss th solution ost urv will hv rltivly shrpr lin, n sty tlowr osts s it rturns k to th initil ost. 4 Ckt # Krishnmurthy lvls (#nos) Mtho k =1 k =2 k =3 k =4 Prim1 LIFO (833) RAND FIFO strut LIFO (1952) RAND FIFO Prim2 LIFO (3014) RAND FIFO iom LIFO (6514) RAND FIFO in2 LIFO (12637) RAND FIFO in3 LIFO (15433) RAND FIFO vq.sml LIFO (21918) RAND FIFO vq.lrg LIFO (25178) RAND FIFO % Impr. LIFO vs. RAND RAND FIFO Tl 1: Avrg utsiz rsults for 100 runs of FM (olumn 3) n Krishnmurthy highr lvl gins (olumns 4-6) using LIFO (Lst-In-First- Out), rnom n FIFO (First-In-First-Out) orgniztion shms for th gin ukts. 3 For sp rsons, ll tls givvrg utsiz rsults ovr 100 runs. Minimum utsiz rsults r qulittivly similr n r sprtly vill. 4 Not tht for iprtitioning, th ost t th n of th pss is xtly th sm s th ost t th ginning of th pss, mning tht improvmnt rsults from n initil rs in ost uring th pss, follow y orrsponing inrs in ost ltr in th pss. Columns 4-6 of Tl 1 show th ts of LIFO, rnom n FIFO sltion on highr-lvl gins s n y Krishnmurthy [5]. Introuing sonlvl (k = 2) gin n in som ss thir-lvl (k = 3) gin sms to improv th solution qulity for rnom n FIFO sltion. For LIFO sltion, w notth following: For onstnt k, th LIFO rsults r onsistntly ttr thn th rnom or FIFO rsults. For h of th tst ss, th k = 1 (FM) rsults using LIFO sltion r signintly ttr thn th rsults for ny k using rnom or FIFO sltion. In othr wors, th gin ukt orgniztion hs grtr t on solution qulity thn th numr of Krishnmurthy gin lmnts onsir. For som lrg tst ss (iom, inustry2 n vq.smll), th k = 1 (FM) rsults r ttr thn th k >1 rsults unr th LIFO shm. Rll tht th Krishnmurthy gin formul fvors moul in nt tht is lok to th si th moul is moving to, n isfvors moul in n unlok nt hving fw mouls on th si th moul is moving to. In som sns, th LIFO orgniztion hs similr funtion ut with no pnlty for moving moul tht longs to unlok nts. Tht Krishnmurthy gins osionlly prform wors thn LIFO FM suggsts tht following prviously mov mouls (i.., moving to th si to whih nt is lok) is mor importnt thn \stying wy from th minority" (i.., not moving to th si hving vry fw mouls of th inint nts). 3 A Krishnmurthy Vrint Th ov osrvtion { tht it my mor importnt tomov mouls whih r inint tolok nts { suggsts n ltrntiv multi-lvl gin formultion. If nt is ut, n only on prtition ontins lok mouls inint to this nt, w will giv highr priority to th mouls in th prtition with no lok mouls inint to th nt. W n implmnt this y inrsing th gin lmnts of moul h tim it is inint to nt whih oms lok to th opposit prtition. For instn, ssum moul is ing vlut for mov from prtition U to prtition W. If nt whih ontins moul hs t lst on moul lok in prtition W, n only fr mouls in prtition U, w will inrs ll k-th lvl gins y 1, whr k 2. W voi hnging th rst-lvl gin sin this shoul lwys rt th \tul" gin rsulting from mov of this moul. Howvr, w hoos to 1 to ll th othr gin lvls in orr to mk sur tht th inrs priority will hv n t on ll ti-rking instns.

5 Our ltrntiv gin formultion n xprss s follows for k 2: k (v i )= jfs 2 Ejv i 2 s; U (s) =k; W (s) > 0gj, jfs 2 Ejv i 2 s; U (s) > 0; W (s) =k, 1gj + jfs 2 Ejv i 2 s; 0 < U (s) < 1; W (s) =1gj s 1 : s 2 : Gin vtor of moul : in Krishnmurthy in our nw formultion (-1,0,0,0,1) (-1,0,0,0,1) (0,0,0,1,0) (0,1,1,2,1) Th rst two trms r intil to th formultion us y Krishnmurthy[5]. Th thir trm is nw n rprsnts th \ttrtion" to lok mouls. Figur 3 omprs th Krishnmurthy gin vtor with th gin vtor rsulting from our nw formultion. In th ginning, n unut nt ontins mouls ; ; ; n n oth gin vtors for moul r (,1; 0; 0; 0; 1). Aftr moul is mov to th othr prtition n oms lok, th gin vtor of moul is hng to (0; 0; 0; 1; 0) in Krishnmurthy's formultion, ut is hng to (0; 1; 1; 2; 1) in our formultion. Whn w rh th s whr moul is th only rmining moul (s 5), th gin vtors r (1; 0; 0; 0; 0) n (1; 1; 1; 1; 1) for Krishnmurthy's n our formultions, rsptivly. Not tht in this lst s, th Krishnmurthy gin vtor will not istinguish twn moul n som othr moul x hving gin vtor (1; 0; 0; 0; 0), whr non of th nts inint tox hv lok mouls. By ontrst, our gin formultion istinguishs twn mouls n x sin moul x will hv ginvtor (1; 0; 0; 0; 0)inourformultion. This is rguly n importnt irn: in most ss on woul prfr to \unut" th lok nt inint to moul for ommitting th unlok nt inint to moul x. Our xprimntl rsults lso sm to support this viw. Exprimntl Rsults W tst our nw gin formultion using th sm LIFO, rnom n FIFO sltion shms sri in Stion 2. Th rsults r shown in Tl 2. Not tht th thir olumn (pur FM) rsults r th sm s in th thir olumn of Tl 1 sin our nw formultion os not t th rst-lvl gin. As ws osrv with th Krishnmurthy formultion, th rsults using LIFO sltion shm with our nw formultion r signintly ttr thn th rsults using rnom or FIFO sltion shms. Howvr, th son-lvl gin rsults (olumn 4) using rnom n FIFO sltion shms show signint improvmnt ovr th pur FM rsults (olumn 3) with our nw formultion. This is in shrp ontrst to th rsults using th Krishnmurthy formultion, whih i not show muh improvmnt with highr-lvl gins using ithr rnom or FIFO sltion. It my tht our nw formultion tns to omput highr-lvl gins mor rfully, thus oviting th n for \goo" sltion shm (i.., th rsults for rnom n FIFO will mor losly mirror th rsults of LIFO s th lngth of th gin vtors inrss). Also, our nw formultion xpliitly givs highr priority to th nighors s 3 : s 4 : s 5 : (0,0,1,0,0) (0,1,2,1,1) (0,1,0,0,0) (0,2,1,1,1) (1,0,0,0,0) (1,1,1,1,1) Figur 3: Evolution of th gin vtor for moul oring to th Krishnmurthy lvl gin formultion n our nw gin formultion. of mov mouls, whih is similr to th t of th LIFO sltion shm. Tl 3 omprs LIFO rsults using our nw formultion ginst LIFO rsults using th originl Krishnmurthy formultion. In som ss our formultion ls to sustntil rution in th siz of th uts foun. 4 Conlusion W hv foun tht implmnttion hois ply n importnt rol for oth th FM [2] n Krishnmurthy [5] lgorithms. In prtiulr, sltion from gin ukts s on th impliit orring of link list rprsnttion is highly vntgous, n rsults in improv prtitioning solutions. W n tht liminting rnomiztion from th ukt sltion not only improvs th solution qulity, ut hs grtr impt on FM prformn thn ing th Krishnmurthy gin vtor. Orgnizing th gin ukts s LIFO (Lst-In-First-Out) stks ls to 35% improvmnt vrsus rnom ukt orgniztion n 42% improvmnt vrsus FIFO (First-In-First-Out) quus. W hv lso prsnt n ltrntiv highrlvl gin formultion, s on Krishnmurthy's pproh, whih inorports som of th intuition hin th LIFO orgniztion. This ltrntiv formultion rsults in furthr 16% rution in th vrg ut ost whn ompr irtly to th Krishnmurthyformultion for highr-lvl gins, ssuming LIFO orgniztion for th gin ukts. W liv thtmuh mor til stuy is nssry to ttr unrstn th t of \ovious" hois in th FM implmnttion on th solution

6 Ckt # Krishnmurthy lvls (#nos) Mtho k =1 k =2 k =3 k =4 Prim1 LIFO (833) RAND FIFO strut LIFO (1952) RAND FIFO Prim2 LIFO (3014) RAND FIFO iom LIFO (6514) RAND FIFO in2 LIFO (12637) RAND FIFO in3 LIFO (15433) RAND FIFO vq.sml LIFO (21918) RAND FIFO vq.lrg LIFO (25178) RAND FIFO % Impr. LIFO vs. RAND RAND FIFO Tl 2: Avrg utsiz rsults for 100 runs of our nw multi-lvl gin formultion (olumns 4-6) using LIFO, rnom, n FIFO orgniztion shms for th gin ukts. qulity n runtim. Thus, our futur work invstigts not only furthr ti-rking mhnisms, ut lso intrsting ts tht rsult from th orr impos y th ntlist rprsnttion n th list of fr mouls. 5 Stuis of th LIFO orgniztion shm in multi-wy prtitioning n in mor sophistit prtitioning pprohs suh sthtwo-phs FM mthoology r lso unr invstigtion. Rfrns [1] S. Dutt. Nw fstr krnighn-lin-typ grphprtitioning lgorithms. In Pro. IEEE Intl. Conf. Computr-Ai Dsign, pgs 370{377, [2] C. M. Fiui n R. M. Mtthyss. A linr tim huristi for improving ntwork prtitions. In Pro. ACM/IEEE Dsign Automtion Conf., pgs 175{181, [3] B. W. Krnighn n S. Lin. An int huristi prour for prtitioning grphs. Bll Syst. Th. J., 49(2):291{307, Ckt # Krishnmurthy lvls (#nos) Mtho k =1 k =2 k =3 k =4 Prim1 Ours (833) Krish strut Ours (1952) Krish Prim2 Ours (3014) Krish iom Ours (6514) Krish in2 Ours (12637) Krish in3 Ours (15433) Krish vq.sml Ours (21918) Krish vq.lrg Ours (25178) Krish % Impr. ovr Krish Tl 3: Rsults ompring our nw multi-lvl gin formultion with Krishnmurthy's multilvl gin formultion using LIFO orgniztion for th gin ukts. Avrgs r s on 100 runs. [4] S. Kirkptrik, C. D. Gltt, Jr., n M. P. Vhi. Optimiztion y simult nnling. Sin, 220:671{680, [5] B. Krishnmurthy. An improv min-ut lgorithm for prtitioning VLSI ntworks. IEEE Trns. on Computrs, 33(5):438{446, [6] T. Lngur. Comintoril Algorithms for Intgrt Ciruit Lyout. Wily-Tunr, [7] L. A. Snhis. Multipl-wy ntwork prtitioning. IEEE Trns. on Computrs, 38:62{81, [8] L. A. Snhis, prsonl ommunition, Mrh [9] D. G. Shwikrt n B. W. Krnighn. A propr mol for th prtitioning of ltril iruits. In Pro. ACM/IEEE Dsign Automtion Conf., pgs 57{62, [10] C. Shn. Plmnt n Glol Routing of Intgrt Ciruits Using Simult Annling. PhD thsis, Univ. of Cliforni, Brkly, [11] Y.-C. Wi n C.-K. Chng. Towrs int hirrhil signs y rtio ut prtitioning. In Pro. IEEE Intl. Conf. on Computr-Ai Dsign, pgs 298{301, Th input formt of ntlist is typilly funtion of how th othr vlopmnt tools rprsnt n output th iruit, n my group rlt nts or mouls togthr or fr prt. This rltnss/unrltnss will in turn rt within th t struturs us y FM to stor th ntlist informtion.