A Timing-constrained Algorithm for Simultaneous Global Routing of Multiple Nets

Transcription

1 L # A iming-constraind Algorithm for Simultanous Global Routing of Multipl Nts Jiang Hu Sachin S. Sapatnkar Dpartmnt of ECE, nirsity of Minnsota, Minnapolis, MN 5555, SA ABSRAC In this papr, w propos a nw approach for VLSI intrconnct global routing that can optimiz both congstion and dlay, which ar oftn compting objctis. Our approach proids a gnral framwork that may us any singl-nt routing algorithm and any dlay modl in global routing. It is basd on th obsration that thr ar sral routing topology flxibilitis undr timing constraints. hs flxibilitis ar xploitd for congstion rduction through a ntwork flow basd hirarchical bisction and assignmnt procss. Exprimntal rsults on bnchmark circuits ar quit promising.. INRODCION As intrconnct is bcoming on of th dominant factors affcting VLSI prformanc in dp submicron ra, th rquirmnts on th quality of intrconnct routing ar bcoming strictr, and th routing problm is consquntly growing mor difficult to sol. In global routing, a gin st of global nts ar routd coarsly, in an ara that is concptually diidd into small rgions calld routing clls. For ach nt, a routing tr is spcifid only in trms of th clls through which it passs. h numbr of allowabl routs across a boundary btwn two nighboring clls is limitd. On fundamntal goal of global routing is to rout all th nts without orflow, i.., th numbr of wirs across ach boundary dos not xcd its supply. Various works ha bn proposd, for xampl, squntial approach [], rip-up-and-rrout tchniqu [], multicommodity flow basd algorithm [] and hirarchical mthods [, 5]. hn intrconnct bcoms a prformanc bottlnck in dp submicron tchnology, mrly minimizing congstion is not adquat. In latr works [6, 7, 8], intrconnct dlays ar xplicitly considrd during global routing. In [6], ach nt is initially routd in SER-C [9], aftr which th congstd ara is rippd up and rroutd by locally applying a multicommodity flow algorithm. In [7], bginning with a st of routing trs satisfying timing constraints for ach nt, a multicommodity flow mthod is applid to choos a singl routing tr for ach nt, such that th congstion is minimizd. At placs whr orflow occurs, th wirs ar rippd up and rroutd through maz routing in which th timing objcti is combind with wirlngth and congstion. For global routing on standard cll dsigns, th work of [8] incorporats th timing issu with an itrati dltion tchniqu. In [], timing constraints ar combind with a top-down hirarchical bisction and assignmnt mthod for FPGA routing whr th switch dlay dominats and wir dlays ar nglctd. In global routing, congstion and dlay ar oftn compting objctis. In ordr to aoid congstion, som wirs must mak dtours, and th signal dlay may consquntly suffr. In this papr, w propos a nw approach to global routing such that both congstion and timing objctis can b optimizd at th sam tim. his work is supportd in part by th NSF undr contract R and th SRC undr contract 98-DJ-69. On ky obsration is that thr ar sral routing topology flxibilitis that can b tradd into congstion rduction whil nsuring that timing constraints ar satisfid. xprss ths flxibilitis through th concpts of a soft dg and a slidabl Stinr nod and xploit thm in global routing through hirarchical bisction and assignmnt as in [5, ]. Howr, du to intrdpndnc on timing slack consumption and th prsnc of slidabl Stinr nods, th assignmnt is not straightforward as in [5, ]. propos a ntwork flow formulation so that th timing slack consumptions ar adapti to th congstion distributions. Finally, a timingconstraind rip-up-and-rrout procss is prformd to orcom any inabilitis of th hirarchical approach in satisfying congstion constraints. Sinc th timing prformanc of initial routing solution can b prsrd, our mthod proids a gnral framwork that can accommodat any singl-nt routing schm and can b applid on any dlay modl.. PRELIMINARIES.. Problm background and congstion mtrics ar gin a st of nts, with ach nt bing dfind by a st of pins or drir is dnotd as, whr th sourc. considr routing in two layrs, on for horizontal wirs and th othr for rtical wirs. As in conntional global routing, w tssllat th ntir routing rgion into an array of uniform rctangular clls. rprsnt this tssllation as a grid graph "! #!%, whr &! to th st of grid clls, and a grid dg )*, ' ' ( corrsponds! corrsponds to th boundary btwn two adjacnt grid clls. will rfr to a grid dg simply as a boundary. h numbr of wirs that ar allowd to cross a boundary is limitd by an uppr bound, which is calld th supply of th boundary and xprssd as -./). During th routing, th numbr of wirs that ar routd across a boundary ) is dsignatd as th dmand?> &/). h orflow 5/) at boundary ) is 687 9"/&/);:<-=/). h dmand dnsity for a boundary ) is dfind &/)BC-=/). us th mtrics of th maximum dmand 687 and th total orflow MON HI=J K /) to aluat th congstion rduction... Soft dgs A routing tr P is #R S Q dscribd by a st of nods and a st of dgs S (. h location for a nod is spcifid by its coordinats and #. An dg in is uniquly idntifid by th nod pair V S or th notation V intrchangably, whr is th upstram nd of this dg. Routing in th rctilinar spac rquirs that ach dg has a fixd orintation, ithr horizontal or rtical. For xampl, whn w considr th connction btwn C and in Fig. (b), w usually choos an uppr L-shapd or a lowr L-shapd connction, both of which ar indicatd in =X th dottd. X lins. In ach cas, a bnd (dgr-two Stinr) nod or is inducd. Sinc thr ar many uncrtaintis at th global routing stag, i.., th dtaild routs ar not dtrmind, th spcifications on dlays nd to captur th natur of th dlay functions without bing compltly xact. In this spirit, ths two routs and many multi-bnd monoton

2 V V uppr-l lowr-l uppr-l (a) lowr-l (b) Figur. Routing with soft dgs. routs conncting and C can b rgardd to ha sam dlay prformanc, if th xtra dlay from a small numbr of ias can b nglctd for th sam rason. Howr, ths routs may ha diffrnt influncs on th congstion distribution whn w considr multipl nts in global routing. Bfor ths diffrnt influncs bcom clar, it is bttr to kp th flxibilitis on routs rathr than to mbd thm into th rctilinar spac prmaturly. Basd on this obsration, w may connct and with a soft dg, which is dfind as follows. Dfinition : A soft dg is an dg conncting two nods?v, such that:. and,. its dg lngth V is fixd,. th prcis dg rout btwn V and is not dtrmind. will rfr to th traditional dgs in a rctilinar tr with fixd orintations as solid dgs. h soft dg connction btwn and S is shown as a solid cur in Figur (b). By kping dg soft, w can maintain th flxibility on routs conncting and until w considr congstion in global routing with othr nts. In S Figur (c), in th prsnc of anothr nt, a Z-shapd rout for is chosn to rduc congstion without hurting th dlay. In fact, th concpt of soft dg is also usful in singl-nt routing. Considr th procss of constructing th Stinr minimum tr in Fig. (b) in a mannr similar to Prim s minimum spanning tr algorithm. If w bgin by conncting sink to sourc, and arbitrarily choos th uppr-l connction, th Stinr minimum tr will not b rachd. Instad of fixing th dg orintation immdiatly, w connct S thm with soft dg, as shown in Fig. (a). hn, nod S is joind to dg at th closst connction point. h S closst connction point btwn a nod * and an dg V AS is dfind by V its coordinats AS and V such that "* C and Figur (b), Stinr nod is introducd at th Stinr minimum tr is obtaind. %/ %/ (c).*. In point and th.. Dlay proprtis and slidabl Stinr nods o masur th signal dlay of an intrconnct, w mploy th Elmor dlay modl. Although occasional larg rrors mak Elmor dlay unsuitabl for critical nts [], it has a rol in global routing bcaus of its fidlity [9] and simplicity, and is a rasonabl modl considring that th routing in global stag is coars and th numbr of nts may b ry larg. i (a) k j Figur. Connction btwn nod * and soft dg S V Latr in our algorithm, w will pnaliz th xcssi us of ias. i (b) k. j For a gnral form of a partially constructd routing tr, shown in Figur (a), lt us considr th procss of obtaining an optimal connction btwn nod * S and dg V. h dashd lins ar othr nods and dgs of this routing tr, and rprsnts th closst connction point btwn * S and V. wish to sarch for an optimal connction point within th bounding box dfind by and. Suppos w connct * S to V at point /, as indicatd in Figur (b). Lt : :. If th dlay at an arbitrary sink E is E. and th its rquird arrial tim is P E, thn th dlay slack -= E timing slack P for a routing tr P P, E : E. h on th nt is th minimum dlay slack among all th sinks in this nt. If th objcti is to minimiz wir cost subjct to timing constraints, th optimal connction (Stinr) point hr is a point with a non-ngati nt timing slack, lying as clos to as possibl. h optimal connction point is likly to b a non-hanan point such as in Fig. (b). A similiar conclusion is drid in []. A carful obsration tlls us that thr ar oftn many Stinr nod locations for a spcific alu of. h st of locations for a gin alu of form a locus as illustratd by th thicknd sgmnt in Figur (b). hn w slid th Stinr nod along this locus, th lngths of its incidnt dgs ar prsrd and so is th dlay at ach sink. Similar to th rational for soft dgs, w only spcify this locus instad of a point for this Stinr nod and call it as slidabl Stinr nod (SSN).. ALGORIHM.. Algorithm oriw his algorithm includs thr phass: () prformanc drin routing for ach nt, () HBA: hirarchical biscting of routing rgions and assigning soft dgs to boundaris along th bisctor, and () RR: timing-constraind rip-up-and-rrout. In phas, ach nt is routd to mt its timing constraints without considring congstion. Any singl-nt prformanc drin routing mthod,.g., P-tr [], RAS tr [] or MVER [], can b applid hr. Bsids satisfying timing constraints, ach routing tr should b soft. his can b achid through utilizing soft dgs during routing as in th xampl of Fig. or rplacing L-shapd connctions in th rsults with soft dgs. hus, at th nd of phas, timing-constraind routing trs ar gnratd along with topology flxibilitis to b xploitd in th subsqunt phass. " b b b 5 Figur. An xampl of bisction. sourc sink Stinr nod In phas, a routing rgion is rcursily bisctd into subrgions in a top-down mannr. At th topmost ll, th whol routing rgion is bisctd into lft(uppr) and right(lowr) hals with th sam or similar siz by a bisctor lin which is formd by a column(row) of conscuti rtical(horizontal) grid cll boundaris. For xampl, in Fig., th thicknd bisctor lin is composd

3 . Each soft dg that intrscts this bisctor is assignd to a boundary. Aftr th assignmnt, a psudo-pin is insrtd into th soft dg at th assignd boundary, and thrfor this soft dg is split into two nw soft dgs that blong to two sparat subrgions. On assignmnt for th xampl in Fig. is shown in Fig.. In th nxt hirarchical ll, bisctions and assignmnts ar applid on th lft(uppr) and right(lowr) half rgion along an orthogonal orintation. his procss is rpatd until th subrgion is a singl grid cll or a pair of nighboring grid clls. hus, at th nd of this procss, th rout for ach soft dg is spcifid to th dtaild ll of grid clls it gos through. of thr boundaris, ), ) and ) 5 sourc sink Stinr nod Figur. An assignmnt rsult from ntwork flow solution. h crucial part is to dtrmin how to assign th soft dgs to th boundaris on th bisctor lin. h basic goal is to assign all of th soft dgs without xcding any boundary supply and without causing any dlay iolations. In ordr to mak th assignmnt fasibl, somtims it is ncssary to allow som wirs to dtour, which initably incrass dlay, i.., som timing slack is consumd. In addition to nsuring absnc of dlay iolations, it is naturally dsirabl that th consumption of th timing slack is minimizd, sinc th timing slack may b ndd in th subsqunt lls of bisction and assignmnt. hs objctis ar achid through a min-cost ntwork flow formulation. h hirarchical bisction and assignmnt in phas is a mthod of diid-and-conqur that has th adantag of simplifying th problm natur. It rducs a two-dimnsional problm into on dimnsion. Howr, a dcision at a highr hirarchical ll may orlook th nds at a lowr ll. In phas, any soft dg that could not b assignd in th ntwork solution is tmporarily assignd to a boundary such that th maximum dmand dnsity is minimizd and no dlay iolation is incurrd. hs rsidual orflows will b cland in phas. h third phas is a timing-constraind rip-up-and-rrout procss. It is similar to traditional rip-up-and-rrout xcpt that a constraint on dg lngth is imposd to nsur no timing iolation. It rips up th dgs on a st of most congstd boundaris and rrouts thm through maz routing. h cost in maz routing is dfind as th summation of dmand dnsitis or all boundaris that a soft dg passs through... Basic ntwork formulation Aftr on bisction, th assignmnt problm is formulatd as: Assignmnt problm: Gin a bisctor lin composd of a st of conscuti boundaris ) ), and a st of soft dgs # S V S V intrscts, assign ach soft dg to a boundary )* such that thr is no orflow on any boundary )* or no dlay iolation on any routing tr P which has at last on, and th timing slack consumption is mini- S V soft dg mizd. s # b b b Figur 5. Ntwork formulation of th xampl in Fig. without considring SSN. h numbr on ach arc is its capacity. sol this problm through a formulation of th ntwork flow problm and applying a min-cost max-flow algorithm [5] on it. h ntwork ; is a dirctd graph consisting of a st of rtics and arcs. h rtx st includs all boundaris in and soft dgs in #, plus a sourc - and targt. For th bisction in Fig., its corrsponding ntwork is illustratd in Fig. S 5. do not us SSN at this momnt for simplicity and only in P is includd in th ntwork. h usag of SSN will b introducd in sction... hr ar thr typs of arcs: () from sourc - to ry boundary rtx, () from som boundary rtics to som soft dg rtics, () from ry soft dg rtx to th targt. Each arc has a cost and a capacity associatd with it. For ach typ arc, its cost is and its capacity is th corrsponding boundary supply. In this xampl, w assum that ach boundary has a supply of. For ach typ arc, its capacity is and its cost will b dfind latr. For ach typ arc, its capacity is and its cost is. (a) (b) (c) Figur 6. Rlati positions of a boundary and a soft dg. An arc from a boundary rtx to a soft dg rtx implis a candidat assignmnt btwn thm. Not ry pair of boundary and soft dg rtics is automatically qualifid for constructing a typ arc btwn thm. For any boundary and any soft dg, thr ar thr rlati positions btwn thm as shown in Fig. 6. In Fig. 6(a), th boundary lis ntirly within (th bounding box of) th soft dg. If w choos an assignmnt of th soft dg to this boundary, thr will b no chang in th lngth of th soft dg, and two ias ar inducd. If a boundary lis partially within th bounding box of a soft dg, as in Fig. 6(b), w ha an L- intrsction btwn th boundary and th soft dg, whr no chang in th soft dg lngth is rquird and on ia is inducd. In ithr of ths two cass, w can always st up an arc btwn thm without affcting th dlay. hs arcs ar calld basic arcs, and thy ar th solid typ arcs in Fig. 5. h third situation is shown in Fig. 6 (c), whr th soft dg dos not intrsct with th boundary. In this cas, an assignmnt on this pair will rquir a wir dtour, and w nd to chck whthr or not this may caus t

4 ) any dlay iolation. An arc can b constructd for such a pair only if th assignmnt on this pair will not caus any dlay iolation. For th xampl in Fig., if S th timing slack of P rmains nonngati whn th soft dg gos through boundary ), thn an arc (a dashd lin) btwn thm is constructd in Fig. 5. call such a construction as a soft dg xpansion and ach xpansion implis a timing slack consumption. catgoriz th trs across th bisctor lin into singlcrossing trs and multi-crossing trs, which ar th trs that cross only onc (such as P in Fig. ) and mor than onc (such as P in Fig. ), rspctily. Initially, w construct all th basic arcs for all th soft dgs in # and prform an xpansion for all th soft dgs that blong to singl-crossing trs. h xpansions of dgs in multi-crossing trs will b discussd in th nxt sction. h cost of a typ arc is dfind according to th timing slack of its corrsponding tr, sinc on major objcti is to minimiz timing slack consumption. If th timing slack of tr P is S P bfor th assignmnt, and is P if its soft dg V is assignd to boundary )*, thn w dfin th arc cost as: S V P () - /)* P It can b sn that if a soft dg intrscts with a boundary ntirly or partially, its corrsponding typ arc has a cost of unity. As a scondary objcti, w hop to rduc th numbr of ias in th wiring. hrfor, for th situation > in Fig. 6(b), w rduc its cost by a small usr-spcifid offst... Construction of arcs for multi-crossing trs Gnrally spaking, adding a typ arc btwn a boundary rtx and a soft dg rtx may incras th liklihood of obtaining a fasibl ntwork flow solution. Hnc, a soft dg xpansion is usually dsird as long as no dlay iolation is incurrd. On issu that was not discussd in th last sction is th procdur for thos soft dgs that blong to multi-crossing trs, such as P in Fig.. h difficulty hr is that th timing slack consumptions for th soft dgs ar corrlatd. For som spcifid timing constraints, whthr a soft dg can b xpandd, or how far it can b xpandd, dpnds on whthr othr crossing dgs in th sam tr ar xpandd, and how far thy ha bn xpandd. For x- S X ampl, in Figur, th xpansion of S. dpnds on whthr has bn xpandd and how far, i.., to ) or to ). In fact, ths soft dgs compt with ach othr on a common timing slack rsourc, which must b allocatd proprly. sol this difficulty by idntifying th ncssary xpansions through th min-cut mthod. It is wll known that th max-flow quals th forward capacity of th - : min-cut in a ntwork flow problm[5]. In th bginning, w run a max-flow algorithm on th partially constructd ntwork to obtain an -,: min-cut -,. h forward capacity of this cut is dnotd by %D. If %D #, thn it is guarantd that ry soft dg can b assignd to a boundary without any orflow, and thus, no mor xpansion is ncssary. Othrwis, th maximum fasibl flow is lss than th numbr of soft dgs to b assignd, thus w nd to incras th capacity of th min-cut through additional soft dg xpansions. In th xampl for Fig., bfor th xpansion for multi-crossing trs, th mincut is indicatd in th dashd cur in Fig. 5, whr th rtics in ar in th shadd rgion and rtics in ar unshadd. can s that th forward capacity D whil thr ar 5 soft dgs that nd to b assignd, thus, w nd to xpand som soft dg(s) from th multi-crossing tr P if possibl. h min-cut rsult shows us not only whthr mor xpansions ar ncssary but also th congstion distribution information or whr to mak th xpansion. Ery forward arc in th min-cut S= X S and must b saturatd [5],.g., S - ar saturatd. If a soft dg rtx V is in, its downstram arc must b saturatd and thrfor, it can always b assignd to a boundary without inducing orflow, i.., it is not in a congstd ara. On th othr hand, if a boundary rtx )* is in (and not all of its downstram arcs ar saturatd), its upstram arc must b saturatd and th soft dgs corrsponding to its downstram rtics ar locatd in a congstd ara. Adding an arc from a boundary rtx V )* matchs a soft dg in a congstd ara to an uncongstd boundary. Lmma : h ncssary and sufficint condition to incras th max-flow DFEG of a ntwork is to add a forward arc btwn and to a soft dg rtx S with D DFEG. for ry min-cut mak a swp among all th soft dgs in multi-crossing trs and pick at most on soft dg from ach tr to xpand in ordr to incras th capacity of min-cut. Mor prcisly spaking, for ach multi-crossing tr P V pairs, w choos on with minimum cost to add an arc btwn thm if no dlay iolation is inducd. Aftr on itration of xpansions, w run th max-flow min-cut algorithm again to rpat this procss until %D # or no mor fasibl arc can b found. Not that th timing slack computation in a latr itration of xpansions should account for any wir dtour in othr soft dgs of th sam tr in prious xpansions. In th xampl in Fig. 5,, from all th )*, and S can mak an xpansion btwn ) S and if no dlay iolation is inducd, and thn th ntwork problm bcoms fasibl. h itrati min-cut and xpansion tchniqu maks th allocation of timing slack in multi-crossing trs adapti to th congstion distribution, and xpansions ar mad only whn ncssary, without wast... tilization of slidabl Stinr nods (SSN) In phas, if w us th MVER algorithm togthr with soft dgs, w can ha a slidabl Stinr nod that proids xtra flxibility in routing. h appaling fatur of SSN is that whn w slid it along its locus, th timing prformanc is prsrd. s b b b /.5 /.5 / p / t capacity/gain Figur 7. Ntwork formulation considring SSN. h positions of a SSN within a grid cll do not affct wir congstion distributions, hnc w can considr on arbitrary position for a SSN within a grid cll. For ach SSN whos locus intrscts with, w considr only two candidat positions, "! ach on a diffrnt sid of th bisctor lin, such as and in Fig.. nd to considr candidat positions on both sids of, sinc thy rsult in rmarkably diffrnt intrsctions btwn thir incidnt soft dgs and th bisctor lin. On ach sid of, w only considr th grid cll that has a boundary in such that this boundary intrscts th locus of th SSN, sinc th SSN position in this grid cll can proid th maximum orlap btwn its incidnt soft S "! dg(s) and. For xampl, in Fig., intrscts with two boundaris ) and ) S "!!, whil would intrsct only with ). It

5 L is idnt that a largr orlap implis a largr numbr of basic arcs which ar prfrrd as thy will not consum timing slacks. "! S "! For and, all thr associatd soft dgs S "!, S and ar includd S in th rtics in th ntwork as shown in Fig. 7. Obiously, S "! cannot b assignd simultanously with S "! or. his xclusinss constraint can b instantiatd through adding a psudo-rtics and formulating a gnralizd ntwork flow modl [5], whr ach arc has a gain factor associatd > with it. For xampl, th amount of flow will rduc aftr passing through an arc with gain factor of >. sol this gnralizd ntwork flow problm through ayn s algorithm [6]. Aftr th assignmnt, only on of th candidat SSN positions is slctd. h locus of th SSN is truncatd at th intrsction with, and th part whr th slctd position locatd would b rtaind, as shown in Fig... EXPERIMENAL RESLS h xprimnts aim to tst th ffct of th proposd algorithm on both timing and congstion. raditional rip-upand-rrout(rr) and timing-constraind rip-up-and-rrout(rr) mthods ar tstd togthr with our algorithm(hbarr) on th sam st of circuits. h circuits that w tstd blong to th CBL/NCS bnchmark suit whos statistics ar shown in abl. h initial routing trs ar obtaind through MVER[] algorithm so that thy must satisfy timing constraints. h rsults ar listd in abl. abl. Bnchmark circuits. Circuit # moduls # nts # pins apt ami 85 8 ami xrox 696 h congstion rsults ar xprssd in trms of total orflow and th maximum dmand DFEG. h congstion rsults from rip-up-and-rrout(rr) ar gnrally good. h congstion rsults from RR ar much wors than RR, bcaus th algorithm may gt stuck in a dadlock and fail to find a solution undr timing constraints. A nai combination of timing constraints with rip-up-and-rrout dos not work, and a craftd approach is ncssary to optimiz ths two compting objctis simultanously. abl also shows th congstion rsults from HBA for rfrnc. En though thy ar usually bttr than RR, thy ar not idal yt and not comparabl with thos of RR, and should b considrd as intrmdiat rsults. hn w combin RR with HBA, th congstion rsults ar found to b good and ar mostly bttr than n RR, which dos not satisfy th timing constraints. Sinc hirarchical approach is bttr at a global planning ll whil rip-up-and-rrout is spcializd to find local and mor dtaild routs, it is natural that a combination of ths two complmntary approachs can yild a good rsult on congstion rductions. hn w compar th timing rsults, it is not surprising that only RR causs dlay iolations whil thr is no dlay iolation in th rsults from RR or our algorithm. h prcntag of nts with dlay iolations from RR ar listd in column 6, and rangs from :. h total CP tim for thr phass of our algorithm on ach circuit ar listd in th rightmost column in sconds. Sinc ach circuit has diffrnt numbr of nts and th numbr of pins on on nt may b btwn two and sral dozns, it would b mor intrsting to aluat th arag CP tim on ach -pin nt as a normalizd comparison. h third column, # gis th total numbr of soft dgs from th initial routing trs in ach circuit. It is conciabl that th formulation of soft dgs is quialnt to a dcomposition to -pin nts. Basd on this data, th arag CP tim is found to b.6 scond/-pin-nt in th worst cas. 5. CONCLSION propos a nw approach to timing-constraind global routing. formaliz th routing tr topology flxibilitis undr timing constraints through th concpts of a soft dg and a slidabl Stinr nod, and trad ths flxibilitis into congstion rduction whil th timing constraints ar satisfid. Exprimntal rsults show that our proposd algorithm can achi good congstion rsults whil satisfying timing constraints. REFERENCES [] C. Chiang, C. K. ong and M. Sarrafzadh, A wightd Stinr tr-basd global routr with simultanous lngth and dnsity minimization, IEEE rans. on CAD, Vol., No., pp. 6-69, Dc., 99. [] B. S. ing and B. N. in, Routing tchniqus for gat array, IEEE rans. on CAD, Vol. CAD-, No., pp. -, Oct., 98. [] R. C. Cardn IV, J. Li and C.-K. Chng, A global routr with a thortical bound on th optimal solution, IEEE rans. on CAD, Vol. 5, No., pp. 8-6, Fb., 996. [] M. Burstin and R. 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Lillis, C. K. Chng,.. Lin and C.. Ho, Nw prformanc drin routing tchniqus with xplicit ara/dlay tradoff and simultanous wir sizing, Proc. DAC, pp. 95-, Jun [] J. Cong and C. K. Koh, Intrconnct layout optimization undr highr-ordr RLC modl, Proc. IAD, pp. 7-7, 997. [5] R. K. Ahuja,. L. Magnanti and J. B. Orlin, Ntwork flows: thory, algorithms, and applications. Prntic Hall, ppr Saddl Rir, NJ, 99. [6] K. D. ayn and L. Flischr, Fast and simpl approximation schms for gnralizd flow, Proc. Symposium on Discrt Algorithms, pp , 999. abl. Exprimntal rsults. Rip-up-and-rrout RR HBA HBARR Circuit Grid DV % CP apt ami ami ami ami xrox xrox xrox