An analysis of the single row transformation (SRT) approach to VLSI channel routing

AN ANALYSIS

OF THE

SINGLE ROW TRANSFORMATION (SRT)

APPROACH

TO

VLSI

CHANNE L ROUTING

BY

Kuma

r

Venguswamy. H.E.

A

th esis

suluuit.tcd

to

the School

of Graduate

Studies

ill

partial

fulfillment

ofthe

requirements

for

thedegreeof

Master

of Science

Department of

Ccinputel' Science

Mcuiorin

l

Universityof

Newfoun

dlnud

St..

John's

,

Newfoundland,Canada,

Ale

55,

11+1

Theauthorhas

granted

anIrrevocablenoR-._Iicence_ll1e

NationaIUbnlIy

ofCanadato

rerKOduoe.loan,

distribUteOfseD

copies

ofhts/he(

thesisby anymeansandIn anyformorformat.maI<ingthisthesisavai\able

to interested~ns: .

The author retains ownership of the copyright in tlislherthesis. Neither the

thesis

rior substantial extracts fromIt maybeprintedor otherwise reproduced without hislher per-mission.

L'auteura acoord6uoeIicenoeinttYocab&eat non excfusive permettanli\ IaBIbIiotMque nationaledu canadadereproduire.peAter. dlstribuerou vendredescopiesdesathOOe deQtJekIue maniersetsousQueIquefonne que ce soil pour mettre des exemplairesde cette these aIadisposition des persoones

interessees

.

l'8UteurcooseveIa proprieledudroitd'auteor QUiprotege sathese.NtIatheseni des eldralts substantiels de ceue-cr ne dolveot atrs imprim~ 0lJautrement reproduits sans son autorisation.

Ab stract

'I'Iristhe sislL(ld r('s sc~the problemordetailedrouting orVI~ryl.arw ' Sralc' lntcgrnted (VI,SI)clrcultsusing channel rout.ing.Recently,anovel al)prnac:hto dcla ilc..'CI VLSIrolltinghasbeenproposed hyWOllgandKwok [Wong88].Thill method, c1I11('d theSillylcflow1hlllS!Q/,/lw f io l1(SilT)approach,rl~pli\"l'Sthrc'l' ste ps,namely, For wardTransforma tion(FT),SingleRow HOIlHng(SHIl) 11I1dfi· nallyBackward Trausfoneatlon (BT ), Whenthisapparentl yelmple;Ip p roilch\\'as looked 1Itmoredosdy,itwasrealised that thisapproach has1I11lnyhi,ld/'l,prol» lcms,themost illlportiluthci ngthose posedby [11('crOSS(lVl'rsilltl...SlIlllayoul while"(ll'l'yilll; "ultIlt'liT step.T11Il"',althuugh SilT hilsf'flmntin gapP,'mnlt,u beIIpotcruinlill'l' nuwh,furt he rresearchI\"IISrequired.TIIf'p,o,,1of this 1,lwsis is toIllil1.,'il"Ifitlyilll,l ;llw lysisoftheSilT approach,l,Sit a lll,li,'stodU\tlllt'l,"" Illillp" sudtol1Sl.'1-111.'rt'.snll""rtllc ana lys is todl'Sigll1lchauuolrouter.

Altl lul1g hdiffcrcntFT·UT pairs111"(.'poss ible,only~tnlight f(Jn\"il l"ll111lil's nrcprecticul.finde\'I'UforsimpleFT-BTpair" theproblemsposedhy("I"OSSI)\'I'rsill l\wSHIlsohnionuco-ssh.a to a minimumcrossoverrouting , calling fllrt.ln-Ih~ip,1lof newulgorirluu«.hl'(";lIl"I.'mostofthe existingSIIIlalgorit hms!l;1\"(,1>l~'11,If'sigll1'd with IIII.'goalofIrerk optimisation,not crosso vertniuhnisation.To["Ill'ililllll' tln-desil; lI of1,111':<1,'algorhlnns,11tIlXOII0111Y ofSHHprohlomsi"IISI{1I1. 1\pl'f)I" ' s,'.! taxo no myclassifiesSHitproblems in t o billllrtill'uudnon-bipnttiteI'rul.l"Il1.~,ill ill furt herintojllTlllrll" li rJl/andmiJ'Cd51tllproblems.BlIsl'dOil IhislaXIJ1IOIlIY, var-ions nigorith illsfor1Ill'Iliffcrcnt clas ses of SHHproh ll'l11s hm'(, hl'f'lI11"\'('10[11'11. Since SIlllit'I'nt sso\"t'rsrWlYhc..'inevitable.dr,.'("lin'I:l'IJSSO\'c'rItillllllinl!.h,,.hllillll'~ are1l,~n'Ssary,To this1'1111different crossoverhandlingt.l'dlll ilp ll'S llTI' ,Iist"ll.s"..r] 11IId 11g"ll ",';,lisf'd fTossn\"('rhandlingh-chn illlll'ispropose d. TIll'Oll'p lin,timluf the iUllllysistothe sofl\I'lll'('designora challuelrouteraur]illlpll~Il Il'llt atil>nofthis software isalsoaddressed.

A

cknow

ledgements

JIIlll(Illitegr1lLdul totheDe part ment of ComputerScience audto the Schoo lofGrlldnaleStudiesofMemorial University ofNew found land,foradmitting meinto Illl'gl'lulliatcprogfilm inCom p uterSciencelind forprovidingfinanc ial sllpportilltlll~IonuofIIfellowshipandteachingassista nhhips.111111alsothankful totheFecnhy(IfEngillt't:ringand Ap p liedSciencesandto the Depn rt menlof Physics",hoseemp luYlIIenlswereor greatfil1<1ncia lhelp.

rexpressmj'sinceregratitu detomyadvisor,Dr. PaulGilInrtl, furhis SU I Je r v i~i()Jt,helpfuldlscusslons andencouragemen twhichhllw81'(,1Itl)'helpedme InClJlllpl"lpchistln-sis.~lysincereappreciationlindth ank s 10"il's.,)1111<'Fo lt z. 11";111of1111'lIcpnrt.un-ut, ofComputerScience. for allthe 11I·lpgenerouslye-xtended 111 11 111",1I,h-i,'('ollcrcdf.hronghout mygraduatestudy.

il!Jl<'assnciutiouandcutightcuingdiscuesious dl1l'ingthe early stllgeli ufth is llK'Sis work.11"\1'(111disclIssioliliwithDr.K.Vidyasankarwcroalsoappreciated.

TII,~slIl,porlfromthe systemsgroup.~was or greathelp. '1'11('prompt liI'rrin'IrourtI ll'A"III'r,,1olllcewasadelightandlilt'help of11111, 111.'secretaries \\'/1S Al'iltt'fll llr.u-knowlcdgcd.Filia lly,my thanks 1.0m)'famil y and all myFriendsfo r tllt'il'montIsu pportnudoncouragemcnt,

To the Memory of M

:IJ F

lLthc1

'

Contents

1 Introd uction 1.1 VLSI(]l's igll111I10Il1.,tl011 Li Aquick introdnctioutoVLSll'ol1tiug . 1.:1 Anouiliucoftlli.~thesiswork

AllOver view

or

DetailedRouting

l.l.I Enrl )'routers ..

2.1.2 AbriefOVI.'f\'icwof ChaOllclrollting••. • • •. '1. 1.:1 C:llillllldroll l itl&modcls

111 J:J 2.1A '\11Ull'r\"il'wof cxislil1& ChannelfOUlill1l,algorit hms I,) -)., Vi;,lIIinillli",.til>1Iinchilllllds.. ...•. 18

3 Anintrodu cti ontoSingleRow Routin g 23

:1.1 AlllulrmlllclililltotheSingleRowRout ingProblem(SH-HPj . :.H :1.1.1 Ori~illsilnd developments

or

theSlUt I'... ... .. :ti :1.2 :\lil XClll ll l11Ynflllgol'ilhmsfortheSIUtP•..

:1.2.1 Ikllrisl kalgorithmsfortheSRlt P.. .

:10 31

:1.2.2 EtuuncrntionalgorithmsforthoSHBJ>.

:1.2.:1 Concludingremarks ..

4 TheSin gleRow Trans format io n(S RT) Approach ~5

<1.1 ]m rerentFT·BT pai rs ·17

4.2 Some remarks011Wong'spaper. !j[

~.a Prol,lclllspo sedhy crossove rs !i;!

~A Crossovermnungcmontlcdmi (jllcs. !j."j

,1.5 Pseudosl rd,d H'sillSBn. !'"l'i

,Ui A Taxonomyor SRRproblems xn

5 Bipart-iteSingleRo w Rout ing 62

5.1 AllI!\...rvlewofITOSS(JI'CI'Freeroul ing Ii I

;j.J.[ l-orumlutiunsofthecrossoverfL',,,,,routingproblem. li!i

5.:! A Nell'approach totho viafreeroutlugo[ Pennutatlon Chllll lll'ls liS

5.2,] Pcrmuuuion Ch<ln ncl/Perunna ttonSRnpro h lf·m.~. 'j ~ l

5.2.2 Previous work011via Free routing of aPCHI' . in

5.2,:1 Somenewresults011viII fref'!'Out.ing,ofilI'CHI'. 7:1

5.2"1 Iksnipt io tlofthenewalgcrhluu if;

5.2." Exlf'nsi()n.~tothe proposed1I1gorilhll1 7!1

5.:1 MBS IlIl 1'fJllling.

5.a . 1 j\·[BSllltHillling usingbox Ilron ..lun-:;If'IISioll fl.;I,2 MBSHII mutin gusinginl/·d orkf..1sr-t .,prolu·h .

S.'I l\lflllilif,d'I'll rnS'l'algcruhm ~Ij

;'A.t f}dl\il"ofthenewalgorit hmTARNG·MO D

G Non.Bip:ut il eSingleRo wRouting

G.I Minimumcrossoverrouting

90

93

G.1 TOJlologiclI1"i"minimisAlionofchannels••. ... . 9S fi.2.l AIlCWheuristicforminimumnode deletion .101 Ci.:1 llolllillg of I'NSIlIlproblcms• . . .•. ... ••. . 10:1

1i.:1.1 I'NNSIWwithSNI+ RNRepproech. 105

fi.:!.:! EXh.'llsiullof Box-procedure toI) N N~)JUI. W!J

CD,:1 ExtensionofTARNG·MODto PNNSn.H. 110

(j.3A PNSIUl lI'ilhslraightnels(PN YSIUl) II;'

1;.:1.;' ~lllxilll lll llconflicting ove rlaptlt·Sm..·hcuristicfortill'

mini-1111111111mlt'(leldKmproblc11l ltn

li,:l,(i 1.(>~i(,;.1wiltingofI'NSRR•. 11.1

Ii..! 1I(1ll lirr~of 1oiNSI1Ilproblems

fiA.1 Ilo ut ingof;\1NSRRbygrollpiu& intoI'NSRR

Backw..rd Trn usfonuaticn

It; 12;

131 131

r

.

1.1 Urossovercollapsillg , , .. . .. ... 1,10

;.:1 Gri(l opening:Ageneralisedcrossoverhandlingtechnlquc. HI

8 Soft ware developmen t orall SRTbas edrouter 148

l:U.1 Generalrouter . 8.1.2 Specialrouters 8.2 Iml'lcmr.ntlllioudctAilsAnd fC5uIL.. 9 Conclusions viii 1·lll I.J!I 157

List of Figures

2.1 Exnmplo ofachanneland a swltchboxproblem 11

')'J Horizont a l111](I I"('r11C1I1constra intgraphsofa channel,. 12

2.:1 IliI fc1'l'lllroul.ingmodels 1·\

2.-[Constrained\'il1utluiuusatlouof a 2-111)'1-'1'channel. 1U

1.:' UuCtl1lsl.rilil1<:11viumini misat io n of II2-hl)'{'rchannel . 21

:1.1 All,'.lw lllpll·of unSIUllJl'Ob lcllI. 25

:1.2 1':);;11111'1..""fsome Il'S,,1andillegaln'il[isal iollsofan SIUlP. . 2» :1,:1 EXil111pksofcrossoversof differenttypes, 27 :1.-,1 Alllllt('1"\';,1representationandphysicalrealisationof anSIl RP. i!l

:1.5 llo11lillg a SIlRproblemwithTarug's algcnthm. :J<1 :I,(i ('Ollt1'l'lof[JlYHI/M. .. ... . . .. . .. . . . .. .. ..• 3.'; :1.7 A:!'grol1pSHU I'ro!JIClI1l'Out cdwithTa m g'salgcrit.luu• ;j(j a.,s Ai-grou pSltHprobh-m routedwith1)1I'salgorithut., ai :I,!J Iltten 'algtuphu-prcscntationandgrollps or 1111SHHproblem, :1!J

:I.I UFt'll~ i hlcorden' gencnulouforaBlIodt", ·11

·1.1 Chll11lldwillingII)' theSRTapproach ·16

,I.:! lJi rrl'I"(,1I1FT-IJ'J'pairarOI"theSIrl' Ol1sC(! rout ing ,18

1.3 Stepsof theabedtypeSRT. .I!)

1.1 An example of an caslty align ablecrossover. M

'.

5

ACI'MSOl'crwhichneeds acarefulalignllwlll :';0

4.GTranslation ofchannel nel sto SI1Rnets 58

,1.7 AtaxonomyofSRR problems

n

t

5.1 Someexamples ofSRRPnets. 1).2 Some examplesofInterlocking.

.),:1 A Permutation ChannelRouting Problem(pe RP)

IH

m

fi!1 .').,1PcnuntatlcnChannelrout ingbySRTapproach... ... ... 71

5.5 MlninnnanumborofIISCCI/t!i u.1Jc!W;IISilIlIJro a.ch j:l

5.G Propertiesofthe netsof apeRP m

6.7 EX1I111 plcof jvlJ]S RItrouting . Itl

!J.S "I'I~illstructureoflidsillthe.~ill1l(·lltm'lof 11hip,lftlu- Sittl X,I 5.!I Exa lllp l•.,;ofsillSl...undmulti-lute rlorkcdi\IBS R ll !l!'l 5. 10·1~nrug'sulgorltbm'sfailureincrossover fn'i'routing. 8S

5.1JTAIlNC:- ~'I ODalgorithmbasedroutingoraMBSHIl !l2

G.1 Hcelisntionwithdifferentcr0550VCl'1l!eliding10sameviiironut !lrj

fl.:! TopnlogiclllviRlIIinimisat ion or11 cllllllU d, Ill:!

6,3 Node delet ion ornPNSRRusingl\IOD11l'llri .~t. j (·, [(].I flo'! Node deletion ori1;\INSRRusing~101lhCllI'isl,k Ill!;

6,5 Routing or nI'NNSRRusingSNI+ ltNltapproach IU{j

G.G Placement or11rcsidualnct. IOil

6.8 PNNSIUlproblem wit hARRR crossing routedbybox proc edure 112 6.9 '{'lnn g'sorderbasedlevelassignmen t ... . ... .. • .•...114 (tl ORout illgAPNNSRRproblembyTARNG·"-IOD.... 116

0.11ROlltillg al' NYSqR problem 118

6.12APNYSRRproblem withmult iplecrossoversallth e st rllig htlIet 120

Cl.I: JAn examplewhere MODheurist icfllils. 121

6.101 LogicalrollLi"c:.;')raPNYSRRprob lem 125 6.15 Ml'iSIUll'outiugb)'grou ping. I:tU

i.1 Hl.'11h ](}n11('1\\'(.'1,.'11crossove randadjnc(mtnd!!·ClassA. l:Jot

i:.! Anexntup leof nli/!,llillgan easy crossover 135

i.:J Ildat io nbetweencrosso ver andadjar cllLncb·Class

n

137 iA All exa m pleofaligning an ea sy2·boUllflerosecver

ias

7..'; lIalldlillgofaclifficu!L crossoverbyspliL·"liglllC'chnwlllC'. I:J9

r.n

All.xmupl...of~(JliL·itli g ntha tneedsancxtcmel dogk,;

_1<"

r

.r

lI;n"lIingit crosso ver allastraightuct 14t

r.

s

Gridopc:niugtechniquetohandlecrossovers

,

<4

7.!J lJirrcrc litwayt; ofraiding backanSRRlayou t.. I<G

S,I SHflll'llrC'structureofageneralSRTmulct )·')0

:-l.t Suflwnl'C'structure forspeci a l dlal1Jl1:1reuters•• 152

:-l.:1 1\ tlll1l1ucI problemrouted bytheiurplcmcntcdrontcr 1M

Chapter 1

Introduction

1.1

V

LSI d

esign a

u t omation

Thehighlycomplexelect ro nicsystemsof todaynrc usuallybuihusing111I111111)<'r ofVeryLar ge Scale Integrated(V LS I)chi ps;the de;i gll

o

r

such VLS Ichipsis<I tiIlH;.'-(;O llsulIl ingand comp lextask,In fact,tlu' dcvelopuu'ru tim,>(01'1111d"C'ln mir prod uct containingVLSI co mponen t s may belongerthantl)('pruj'<t"\.c<!lifl'lillwof theproductandthe invos t.tuentinthe designprOl"t' SSmil)'no tIll'l','("Wt'1'<'(1hdon -theproductbecomesobsolete.Giventhissituation,costclk clh',· VI,SIrk-signs must beproducedas quic klyaspos sible,In' functionally'"OITl't:1. illtill'firstpass am]meet specificationswithoutlen g thytuningand ,kosigniterations .OIH'1\'11.1'ttl "d,i('\'('thesegoa lsis Laautomatethe VLSIdedgnpro rcss.

Sophisf.icatcdComputer AidedD{~i811(CAll)l<lols itN'lI('CO 'ss a ry II, I"UI'" withlhisCtlIll[llm: tusk orVLS[<lesig ll flllt OlIl<lliun .As illl p rlJ\'l,tl l"irn lilrilhril" il-lion technologies makehigher level sor integrationpossible.eM)tuo ls tosu ppor t thedesignandtestingorthesecompl ex chipsbecomemaudatcry, [IIfill :I"VI,SI designnutoumtlnn has stclldilyaccele ratedtokl~']l]Jill:"with•.I1l'illllllWII,iu lisin

an:h Hf:dnJ'cs lindadvances incircuit fabricaliontech n ologies illthe[lastdecad e ,

To d aythescow: ofVLSIDesignToo lsis verybroadand encompassesmuch of

tiledesig n processfromdesignspecifica tion, log icdesign ,partitioning,placement , routing, simu lat ion, and soonunti lthe finalmaskis realised,

Bueauseof thecomplexityinvolved,VLSIdesign isusua lly att e m pted in IIlIic rllrcliicillfllsldolland many ofthemodern CADtoolsarebased onandfully exploitsuchhierarchicalsche mes,Hie rarch icaldeco m position reduces theco m' plex it ybybreakingtheovcrnll circuitintoa numberof sma lle rsubcircnitssuch thatautomaticdesigniscom p ut ationally feasible,Top-downcircuit partitionin g 1111<1bol,tom- lipcircuitleyout isthetredit loual approac h, Thetop-down dece m-positionslopswhena subcircuit ca llbe dfed ivd )'laidoulbythedesig ners l1s iug intcrncfivo graphlcs ed itors01'untilthelayoutCIl Iibl'autonuu.lcajlygene rated by programs(~o-cilllcdmod ulegenerators01'cellcompilers].Thebcu om-uplayol1l processthentakes theset or layou tsfor thesebottomsubcircuitsamipli\l'l~sand intcrconucct.s them hierarchically,untilthecompletecircuitislaidout,

Alarge,olt on dominant partofllll'cost andtimereqllirl'lltodl'Sigllil 111111 p lc-'xchipisconsumedillthe ph ysica l t[tsi!)t!phase andinparticularinthe It,yo ulsl,ep which is thetas k or placinga set ormod ules(/,!accJttUllj01111chip amirealisinga sot ofinterconnectio ns between thesemodules(I'OIl1ill[l ), Ulcarly, pla CCI11Clltandrantingarcclosely relatedtoeachother andapoo rpla cemout could makethe subseque nt routin gdifllcult or impossi ble,Theidealway orso lving the layoutproblemasawholeistodevelop analgorlthm that simultane o uslyand glob allyconsiders the position of the modu les and niltilt'illlemrliollsbetween t,III'I'C11niTf'diutcrcouuec tions. Howe ve r,becauseorthetremendousco mplex it y ill\'ol\'I, I,thelayou tproblemisusuallysolvedusingthetwo scqucm.ielstepsof plac e11lent ruul rollting,tot'thepurpo ses orI,histhesis,theinteres tandthe foc us is,~o ld)'011the l'Outilig ste p,

1.2

A quick

int ro d uct io n to

VLSI routing

Autom ated rou t ing bad its origins in the designof PrintedCircuitBcerda(PCBs) whichinthe 70'shad typicallyhundredsofnets andlimitedrules{or inte rcon-nection. Aselect ronicdesign methods improved,theearlyautomated routinga

l-gorithmswere adapted andnumerous new algorithrm were developedtohandle multilayerPCB,LSIandVLSIrouting. Fu rthermore, theroutingtechniques were influencedbythe choice oftechnology (forexample,the numberof layersava ilable

for routing)aswell as bythe particulardes ign style- full customorsemi-custom using gate-array, standardcellorgeneralcellmethodologies.The cellbaseddesign approachis popu larand the entirechip can be considered as a tree madeup of blocks of decrea singcomplexity fromthe root to the leaf nodes.

Routing orinterconnection isone of theimporta ntsteps inthephysical designofVLSIcircuits.Onecan visualiseanIntegrated Circuit(lC) chip as made up of a numberofbuilding blocks which may be macrocells, stand ard cells,gate arrays,functionalblocks, ere.,depending uponthedesignstyle. Thesebu ilding blocks commun icatewitheachother viapinslocated on their boundaries .Thus, thero u t ing problem for anIe chipismainl y concerneclwithmakingthe inter con-nectionbetweenthe modules.Apartfromthe modules andthesignal net s whieh interconnectthesemodules,a chipwill ha.ve powerandgroundwiring whic hruns in a busstructure andfeeds eachof theblocks.Further, there isapedlramc which surrounds themodulesand providespadsfor connectionstotheexternalpinson thechip.Ingeneral,most ofthe chipareanotoccupiedbythemodul esis available

forinterconnectingthe modules andisingenera l,referredto as therouting area or rou ting dom ain.

In an integrat ed circuitchip,theremay bemany suchblocks an deach blockmay have many pins onit s bounda ries. Thus therearemanypins sca.ttered

Oilthesurfaceofthodlipwhichbelong to distinct sets, whereeachset ofpins re pres ent sII llct. Thepurposeof routingthe nis toconnect electricallythepins which makelipeachlidsuchthatall nets arcelectricallyisolatedfrom each ot he r. Eachnetwillbe made upofanumberofwiring segme n ts whichwillrun011one of thelayers avail ab leforrouting,subjectto same restr ict ions as dict ated by the l"Oll l iligIuodcls.\Vhcnthe adjacent segment s ofa netarerout ed in cliffel"Clitlayer s. "im.'lfeus ed1I.t tile pointswhere the layerchangeover occursso tha tthe en t irelid is electricallyeonnectod.

Hoeausc mnlingi.~a com p lex pro c essin itscll,iL isusuallyauomptod as n 2'ste pprocess,namelygff/bal'"O Il/ill!Janddr;l flilulmllfi ll!J.Global rout ing is ,1preliminary1l1aull ing stageforthe subsequent ,I<>t ilibl routi ng. III11H'globil l I'Onling phase, the overallcomplex shapedroutingHn'H [fln- wiringhHr l isbroken upiul"/I1iW)'smaller arcusof almpleand reg ular sll11pC'[sayrcctauglcs)amine-ts whichwi lluse theseIHCil Sfor rout ingare id ent ified. Ingeneral.the remar1)(' mote1111111onewayto splitthe tot a l rou t ingprobleminto II s{'Lor disjointdetailed routing problems .Thus ,lit theendofglobalronting(alsocalledlooserouti ng ) we 110lVCsubpro blems each orwhichis char acterisedbyaset ofpins ,liltinterconnect .ion

pettcru'Il lilan l\1'ea forroutingthe musingitchosen rout ingmod el.NoLI'tha I

t.lu-ronl.iugpaLhsofthe nets inside the routin gurcasis notdetcnuiucd yet. IIItill! subsequent phaseofdetailedrouting,theact ualwlrlngof(~ad lof theseS1I11I11il1'CilS isu-allsod suchtluu, allthebuildin g blocks 011lhr'chipgetinu-rconucctcd.

lJaS(.'(1allthe sha peofthe rout in gregiollandthe10c H I~,m111111nat u re of theterminals,thedcuulccl routing problemlUIS1)I'('ustudil'd IIsiliSSOIll ('popular 1l~llh'1s.IfI,ll('rcuungarea is arect a ngle(noolJsl ad esinsldc]wit h [lillslo ra\..l·don twoparallelsid(~sitis calledthe cha n nel routiug prob lem.'I'hcterm chfl/Illd usuall y r('(efl(10itrcclaug lilarstn lig lllch an nel,althoug h'1'.XorLshape d channelsare

ill ~() I'(l s>i i h ll~.Iftl'flnill,llsnre11110we(1 011al!fOIlI'sidesoftherectangular ro u t iug region,then it is calledthe uwitchboxroutingprolih-rn.Anothe r modelfor detailed

routingwhichispopular in PCB designistheSinsleRowRoutin!Problem (SRRP) whereallthepinstobe connectedlineup alon! asin!1erowand theroutin! area

iscebothsidesofthelineoCnodes.

1.3

An outlin

e

of this thes

is

work

This thesisaddressestheproblemof detailedrout ingof VLSIchips. Channel routingisthe mostcommonlyusedmethod forthedetailed rout ing phaseinmany oftheautomated VLSI designsystemsandvariouschan nelroutingalgorithTIlll existfor thispurpose.Recently,anovel approachto detailedVLSIroutingWlllI proposedbyWong and Kwok(Wong88!whichisapplicablefor rout ingchannels, switchboxesorany polygon shaped routing region.Thismethod,calledtheSingle Row Transfomotion(SRT) approach,requires3sle pt. Thefirststep,calledthe

Fonoard1hJn.sfo nnation(IT),converts theinput detailedrout ingproblem(aay achannel) into an equivalentsinglerow routingproblem. In the secondstep, theSingleRow Routin!solver,theSRRPis solvedand an SRR wiring IaYOlltls obtained. In the lut step,calledthe BackwardTransformation (BT),the SRR layout infoldedbackintoachannel layouttoobtain a realisationfor the original channel problem.Themain reasonforusing the 3-stepapproachinsteadofthe straightforwardsinglestep approach,asexplainedin(Wong88l, istoexploitthe efficientroutingalgorit hmsavailableforSRRP.Won!dal.(Wong88)claimthal thequalityof the solutioniscomparabletothatproduced byth,.directapproach. When thisapparently simple approachwaslookedat more closely by working out different examples,itwasrealisedthatthisapproach hasmanyhidden problems,themostimportantbeing the problemsposedby crossovers in theSRR layoutwhile carryingouttheBTstep.Thereasonforthese problemsisthefad that thisapproach attemplllto makea bridgebetweenthe channel routing pro blem

ilndthe SingleltcwIl onringpro blem which so farhasbeenst u died asdifferent problems withdiffcl'cll tunderlyingroutingmodels.To thebestoftheautho r's knowlodge,[\Voug88)istheonlywork rep ort edonSlIT basedrouterssofar.Thus, thegOlllorthisthesisistomakeadetailed stud yandana lysis of theSRTapproach a:-itappliestocha nn el routing,anatolisetheresultsofthe anal ysisto design a Chilllllelrouterbased011theSRTapproach.

Tilt:t.hesi,~is orga nised10 firs t present .the beckgrouudmate rial011cha n-nclrtltll;ugilllt! single1"011'rouriug, endthell 10 discuss theSilTapproach ,it s proh lt'll\sandSOI11Csolutions.Chapter2 gh'esahril{reviewofI.hl.'dclaill '!t rout-ingl.t.'c!luitlll(,S,t,,~pcciilllych a nnelrou ti ng ,highlight,ill,!;various routing lllodds an d illgo ri1.hms for solvingthe channelrouting pro blem. In Chapter3 the siug le row rollli ll/l;problem is in troducedand a ta xonomyofthe va ri ousalgo rithmstlla thave h<"'{~11report.edsofaris presen ted, Theinadequ acyofthecxlsuugalgorithmsfor thei1Jlptic n.tiollillSRThaSI.-clrouters isalsobrought. 0111..With this background . Chap ler'!introduces thcS RT approachtoVI.S II'011lillg.ThePT(llid l.lTsh'ps 1l1'I' disl.·IlSSI·11in1110 1"('detail b)' conside rin gdi[fefl'1l1IlOssih le FT·IlT pairs.Then,tIl{' hiddenprn b ll:ms(es p ec iallythe problems posedIJrcros sO\~.'''SillLlw SIIHSOl lll io ll) btIll'nppal't:nt ly simple

s

irr

approach arc lrlghlightcd.CI'OSSOI'CI"mauagomout. l1sill)!,!!I!'st eps(IfCI'OSSOI'C1"reductionandC1'OS SO \'(~ rha u d lingi.~propose-d.TIlt' ron-ccpt ofp.""IIIf)~/I'cle"r~illSitUnelsisintroducedand its 1lpp lica li o llsindicated. Filially,IIt.axonomyorSR f~Problemsinthelightofitsapplica tionillSIr!'basel! 1'0011.('rSisproposed .Chapt er5 considers thebipartitr- cast'S of SRH,llCl I11('I)',the 11crI1111(a(;o ll fJilmrtitcSnRandtheMixedDipal't;(eSRR.i\IlCIV0(11)algorithm, calledtheIm.rjllYJCcl/ul'cIVe ngDOj,for crossover FreeroutingofnPcrmutaf.inu Bipa r-tit,t:5RRpro blemand hencetltevlafree routillg

o

r

lh<..'corrcsp ondil1gPel'Jll1lla t ioll L'hnuucl, ispTlJpo~e(1.1"01'renting aMixedBip,'l.rlit(· Sit U IJI'OIlI(·I11.IIruodjfic dand t'II1tllll{"l~1vcrskmnfTnrng 'salgor ithm['JamS'I!,callr-d TJ\1tNG ·MODis proposed . Thelllllll 'tltldlUSt'Sa- colouringortheInterval Overlap Gl'aphoftile'SlUtCha p t N

G considers thenon-bipartite cases oftheSUR problem,namely,thel't'l'Il1ul;lLilJlI Non- BipartiteSHHandtheMixed Non-Bipart.itr-SRR problems.Fir~titdl'Lail{'(1

analys is oftheTopologicalviaminimis a t ionofchauuelsisIJrest'ut"dallilaII" W

heur is t icappr oachtothe minimumnode dclct.ioublpar j.itcsllbgn lphpwl ll1'11l.is proposed.Th eilvario usalgor ith ms arcPl'csl'lItL'dandCOI1lIlIlI"l~d10wul,I'tIll'

Per-mutat ionNon-Bipartit eSURproblemwith andwithout slrllightnets.III1,lll~«lSI'

ofthePermut ation Non-BipartiteSRRtheboxpro cedure uonuallyll1"111l un~~ nmin-imumcrosso versoluti on.For routingthe MixedNOll.B iparl ikprobk-rn, alIi\'id,·

;'11 <1conquerappro achusingthe conc e p tof!Jl'Qlllul[D uLi8 7]isIIWpos('(I,

wlu-rttlu-MixedNon-HiparuteSHRproblem is broken lur oPcrnuuationNIlII-llillfll"lill'S Ulf

prohlemsandrouted. Chapteridiscusses tlwII Hd,w iln l'l'rall~r{)nIWliulIsl" II,111'1"1'

differentcro SSOVCI'handl ing tec hniquesarc dis f lISSI',t,Ag"lwrnl(Tll~!'lJ\'I'rIHIIUllillp'

tec hnique , called91'i"O/lc rl,isrlescrihedwhichis11I1!'l'(1011,IIlsI'ndo"IOllp, 11\('(ll'l llll l'

lid,HIdlocalr-ivr-rrOlll illg,This techniqueis guaranned III[lrm illn'u fill..1 dnl1ll'.. 1

forilll)'type andnumberofIT(J~SO\'CI'S. Chauue-lwldth n"llld io ll1"d lllil [lIl'S an'

nlso discussed.Chapler S uses t.ltcanalrsi.~ral'l'iL-'dout intill'Ill'I'\'iollsl'llilllll'I'Sillill

presents1111:Ieamowork forthosoftwa redesigllof1\dUllllll'1rOllt t'!'h;\s..cl on th .·

SilT1I111)1'Oile ll ,lmplcmeutntlondetailsandCX1 )(~riIl1l'IIta l l'('Su llsnrc nlso[1l'1-'_"" 111"I! , Chaptel'!I pro v ides tileconcl usi on sandslIgg('l<h. dieectlousforfutun-work.

Chapter 2

An Overview of Detailed Routing

Thisc:lillplcl"·;in.'I\IIbro ad overview ofdetailedVLSIrouti ngwith<"lllphm;is 011 channe lrouting. To sta rt,the:ICOI}eofdetailed ro ulingillthl'VLSIlayout.pro -n·s.~i:. pt't-',wrt1.c.,<lllIulhistoririlJ detailedroilling1('(huiclIIPSan-111l'lll.io l)('cl.Tlu'li kl·ll l illl.l(J~Yrulatedtochanne l routing is]lI'{'SI'IIl,Nllll1l ldifferent.chunnclwilli ng 11Imidsarcdiscus s ed.Nextauoverview of(!ilfel"c1\lalgorithms forchannel ]"{luling ispn-';{·lIled.Ftually, theproblemofviamiulmisationillchannelsis discussed.

The pro!JlO::Iil ofVLSIcircuit layo utisusually[>il rtil iollcdintotwo s lIb-pruh lellls,nemcly,,Ittlei,,!}II set ofmodulesandmll/i,l!Ja sdof iutcrrouurx-tlons betweenthemodules.Placemen t.algorit hlllstry1.0place tlu-modulesill 11 mauucr l.hlll.willmini misetilecircuitareaendIacilltat eIll(' routingphase.Arlerplacl~ 1111"111,.LIn' ront.ingphaseconsists

or

two parts.Ilil lT1d y .y/almlmlllillYHudrlr/ui!u! mllli"l/ .dctuikx!routingsteptnkcs theinputfromtill'globalrout.lug.~l('(lallli carric's0111.th enctualrouting ornelswitilineach channeltlw rd lJ'l'csl1ltillgill Lln-pltys k 'lllayo u t..

2.1

Detailed routing techniques

The problem ofroutingor interconnectiontRkf.'~typi cally:10%o(till'totald.'sig ll time and50% ofthe chiparea lim! thusneedssp ci."i1l1a!.lclltio llilllhl~CAllul'\lLSI circuits.Thegenera l\ltS IroutingproblemRS\1'(·11aslIla lLYofil.s snhprohh-ms alld restrictedversionsarcNP-co mplete(Szym85,SIlrr87j prohlf'l1ls.1_{V"riollsrnlltillJ:;} modelshaveevolvedto Ionn nla tetheccnst relut » posedhytlu-lllilll u f ad n ri np,l ,·..h-nology and to casethedesign of CA D tools.Thus.([ifrer t'lIl,WillingII]gtll'il,IIIUliliS!'

difrf'l"!~l1t1llUlcrlyin groutih2;models .'l'hcIIr,;t.apllrnildu~10routingsl.ll!" l l·dwith the mazeandlinesea rchroute rs. butillthepastdl'C'il!!1'c:hillll1dfUlili ng1ii1.~ell'arl,\"

2.1.

1

E

arly

router

s

Th eenrllcst.mntc-rsiIT/~theso called(l1"fI1 IrJII/r ,.,..andL/~"s id !;llrilhill[I./'I'lil]was till 'carl il's1undurostgl!IH.'f illlllt'tlIOtlforrouttng,Itlilllls1111'shurll's tpalhIl(' -I.WCt'1i 1\1'0pinshyusingilII'HH' propngationilIltlInlll'lIi llgsdH'III".ifi\]lathl'xisls.

TIlI~me!udrawback is tlil' l' x(,!ssin'timeillld~,I OI'ilgt'rcqnjrr-nu-uts,TIll' mo.li-liedversions[Akcl'u"'i,Il ubi"'i'I]usedilfer{,lltsp(~~lllpt/~'lilliq11l'S(IIl'H I' I'lill)('llill~ SdIClllCS,di rccll'tlwm·cflulll.propagation (./('.) .~l'x t l'HllIl'lh eliiu.~rrlll·11rullll'l'.~ [lIig hll!lJwhichnrcsilllil fll'10r.l'{~mU1Cl'Sbut til> 1101l"ulIsi,l..r111"/'11111'<'1tll11r i xuf gridpoints0111111.'rOllli,lg!!.rid billusehnrizoutu]and\·<'I'I.it'illllw'S\\'t '{'p 1,'{·I'III'I 1Il';; instead,lht'l"i.'hyreducingLilli eandst o r agel'l'lJuirr' Ill"1I1S.However,1m ! hiln',l1IIId linerouters routeone net atatime anddonotclIll.,idc'I'IIll'Iwllisl,liSIIwhok-,H I'-cause of thislackof inform ationconcerningthr-inh'riu:li olll)O't'\\'/~'1Idilr/~J'l'lIt ue-t»,

INI"OOI11I'II,trlleSS llll" l1Is thal M thl' si1.l! ofthl'l'rnhl" ' ll( n ~oOlIIllO'n.'Ilt<'ofsil.. furlh""' lIItillJ4 ('rohl e-IIlI\'oultlbe the1I111l~'l'ro fncls) iIl C rl'm;('S,Ih" liIlM·r"' lui r",llosol ,·,· tJ"'IIrt,I,I"llI illr",',lS'''' ")(llf' lIrllli:l lly,Noi.lOl~' II0111i:l1lilll"algorithmfor auy1I1"IIIIM'r (If tl...\"1'·r"m J,]I'!<'flllllil)'11Il>; I".":llrrportcdyd,If "l'o))'lU>I"i:, 1cill. !IIlgorltll1l1,'xist.s Inso l,·..,Ull" u",,,,I"'r"r,h,'r;,,,,il~·,lh '·10 th,:t,~~~;, t'0I~· I1()lIliall.i l1\'~nfgorithmto!lOr.... "noll"'t1,,,~,,I" 'r,,of II",flllllil)·.

SOIllc:li1l11~1111l1.lre adyroutedndblocks the pathofasubseque ntnet.Moreover , clilrf~r('lItor,kr'agofnels as wellthe orderingof pinswithinthenetmay leadto difff~relltsolutions andtheseroutersmayfail{'I'CII inverysimplesilnillion,~,'l'bcsc drewbncks11....1 lotlw dc\'c101l111cn lofC/HlIlll cl IVlllel's.

2

.1.2

A

b

ri

ef

overview of C

hannel

rout

ing

Chunnolroutingis theworkhorse ofmany oftheprese ntdayCADsystemstlra t sup-purl illllollli1lecllnyolll.ThochannelroutingprohlcruWas Ionnuh.ted byIlashinlUto .lIulSII'I'l.'llSIlIash'llill1!J;I,whoalso gaveIll<'I.fjlEI/Yf :l1y"I'illl lll.'I'lu-rc nftcr, dillcrcutchannelillgo rithlllshave been reported[DI'II I ; fi .Hi\'c82. Yos !rS2,111\1',;8:1. .r(MIIISGj.Chaunc! routers are [Ja ral1e1routers: tit,')'{,(llIsill"l'tllt'illjllltIIl,tlistliSII wholeandthustheordering ofnetsilltheiuput. is1I0trele vant.Further.channel router sarc smarter inthesensethatthey capturetheintera ction betweentill'dif -fcrcutnotsllsingthe verticalconstraint graph(VC O) ,hl~f'JI1~proceeding withtill' df'laile drfJlIliug.Cy clesilltheVCG POSf:Jll'Ohlr~l\l.~.however.whichinsotuecases cuuldIJI~managedllsillg <Iogleggillg.Generally,IUUt;\'.routingisnotgua ruutced. GI'I~edychannel routersmakinglise ofheuristicrub ;searchforlocaluptiuuility, consequentlycreatingsit u a t ionswhere decisions made earliercause tIll'rhauucltc he unroutnblo or increasethe \\'idthorthe channel more1.111111Il<'cessary .1\fen'lIt 1,I'f-'lmistilnSI~knowledgeh/lsctl expertsrs[el11s[,JoobSr,]fur\ILSIrouting.

AI'lrlllwdnormallyreferstoar('(.'tallj!,lIhl l',~hapf'droutlugIH f.'i\whh st.raighr. edgf'snudtcnuinnlslocatedOiltwo of its sidesand110ohsladl'Sillsidethe roul ing110 1ll ilin.\Vhileotherco mplex Shllpl'San'possihll',til,'rect ang ular straig ht c1Hlllllf'1isthoIuudamcnta!shape. TheworddWlIl1tlIrom now onwardswillmea n a rcctnngularstraig htchannel,allexample of whichis shownillFig ur e 2.1.'l'hc swit chhoxis llllOtlwl'impor t antrouting problemwherepinslire allowed 1111<lllfour sidl'S(Ifthe rcctnuglc.The objectiveofthe cha nnelrouteris to lntcrceuucctthe

pins of the nets using the areainside the rectangle. The entire routingregion is assumedto be gridded,withterminalslocated at grid points,andthewiring segments can be eitherhorizontal orvertical.The terminalsarc generallyassumed to be fixed,although ill some models theyare assumed to be movable(slidable). Apartfrom achieving 100%connect ion, there areAfew factorswhich determine the qualityofrouting. The mostcommonlyusedfact ors areminimisationorthe total number ofhorizontaltracks used (becausethistraD91atcsdirectlyintochip area),minimisationof the total wirelength andminimisation ofthe number of vias used.Via minimisationisimportantbecausevias may decreasetheyield,and theirrelativelyhighresistancemaydegrade thespeedof operation of the circuit .

A rectangularstraight cha n nel

1

0t@,

_,

l' 3:

_

._,

"

,

9 ... ,.. 5 10 - : ' - 1 6 - -- : ' 6 8 - -

~

'_., : 7 - layer 1 9 8 7 3 5

--- -layer2 Aewit chbox problem • via

Figure2.1: ExampJeof a channeland a switchbox problem

Many of the routingalgorithmscapture the informationaboutthe nets and theirinteractionusingasuitable graph representationand process thesegraphs to arriveat the finalrouting. Two of the most commonly usedgraphrepresentations for channelsarethc so calledIIQrizontal const m int gruph(IICO)and the Vertical constraint graph(YCG).Figure 2.2depictsthese graphsfora channel.The lICO is basicallyan intervalgraph(theintervalsbeing the horizo ntalstretches ofthe ncte), where nodes of the graph representthe nets and the edgesofthe

n

e

G

represent the

inu.rsection ofll](~Intervale.Thatis, ifthere is1111edge inthe IICG hetwceu nodes (IandII,itl11ellllSthattill..'horizont alstretches ofthenetsNaIlfl(lN~intersect.

Anotherlmportnut property ofIIchannelisthefocaldensity.Thelocal denshyltlnnycolumnofachannelis equal10 the number ofnetswhichcut a\'Cftkallinedrawn atthatcolu m n. Theoveralldensityofa channelhs the maximum ofits local den sit ies. Thedensityisthelowerboundallthenumber oflIo1'i:wlI1111 tracksneede d(also calledchannelwid1h)in allYrealis ationof the dl;uJlld .The .•prlllofIInetrefersto tilehcrizmualstretc h of1\netinthecha nnel.

FlJl'llstrnigh t ne t. whichCOll11P.Ctsnodes illtill'SllllIecolu m nthespanis 1.1'1'0.1\Irft

IIl'I.connectsapill011thetopside10apin10itsIdl011thebottomside. Al'ifJl1I lid isddinedsimilarly.Allpinsofa(oca flIdIiI'011tile sa mesideofthechannel. Apermutntionuot111Is only2piuswhichare011tlu-opposite sides of1lil~rhuuuc]. '1'lins, l1penuutatjonc1lall udcanhave on lystudght.lcltuudriglltIl1'ls,whereas11 mixodclmnucl111lS1l111~or more lo calnets ill addition 10th<-'pcrmu tauounets.

y~,

~

.

1

-.

2

-.

·12

-

-

·' 1 .:. "j 0 t 6 1\c1111ll1ll"r)whlclll

f;1

'

~

5 . .J 3 I

,

CD

llcrizout al ,I 2 Constrahu

graph(lI('C:) Vertica l constraint Waph(Vl' G)

FiJ!,IIl'"2.2:Horizon talnudvortlcalconstmiutgm p hsofit1'111111111·1

Tlw

ve

e.:

1"111'1111"1'1'theordering infcnuatlo nbetweenthenels.TIll.'UOtll'!i oftill'VC(; representrho

m-rs

,

andtileedgesrepresenttheprtvcdcnce cOllsl ra inLs. 'I'hus1I11)'<"yell'illthevee:IIll' 1HtSthereis111IIlllrt'Soh'ahleconflictilltheordering 111\(1allllwronstrnintscannot.be satisfiedwitho lltbreakingtheloo p.Allintporteut

featureof the

yea

isth eleng t hofthelongestpa th,whichI' OSC:; IIlimitOil Llll' minim um numberofhorizontaltracks needed.III Ia rt.,fo rsim pIt'routingllI{)(ld~

thechannelwidth :::,max{channcldens it y,lengt h of longestpnt.hoftileVl :G},

2

.

1.3

C

ha nnel

rou

ting

mod

e

ls

Rou t ingmodelscanbeclassifiedbased on thenumberofla yersavailablefOl'rtJIILilig the wireseg m entsandtherestrict ionon theircourse ofrun,Themost(Otll1110U HI: 2-1ayc rschctuca,whilethree ormorelaye r sclwlHesIIrc !>1:COlllillgililporllll1l wilh theadvan cesinVLSI techno log y ,Thec\assificllt.io llher!.' ussumcs II~-layt'r111011..1, where2layers, called metal1andmetal2,incavaila ble Iorrouting.J\uo l lll'l'puint tonote isthatin some technologiesthepinsarclIccl.'ssilJlcFromnlltheI"Olltinll,layc'l's wher easinot he rsthepins areaccessible Iromonlyoue1'01ltiuglayer ,1"('llllil'iligtlli\l

thefirs tnndlll!' lastrout ingse gment sforellcli111'1IiI'illtill'Sil1l!,'1'1)1'1'.Till'mI t

ofthesc6'1\cnl!; couldruninot herlayer susingviaswherever1I,'c,. I.'l1. The/II'mode lisn 1'l".'Sel"vedlayer 1110111'1illwhich 1I111.hl'~, 'gll1l'lIl,~inUIlC' layer must runillthehorizontaldirectionwhile11111,111'segmentsintill'ol lll'l"lap '!" mustrunintheverticaldirect ion,TheHv -mcdcl slmpliflcsl!letuliling[oh1I11tl gua ra nteesasolutionif there Menoloopsilltilt'

vc

e;

butlut.roduecsmanyvins. Thcncxt modolisthef!-I~"I!JIIlOtic/ wh ichallowsbothhcu-izontul1I1l1!I"t'rt k illruns in bothla yel's.Thisreducesthenumberofvius signifkant l)', AFurtln-r rclnxatinn is theovcdapmQfldwhich allows pnrallclruus(1'1111alongIll...samelrcriaontnlor verticalgrid)oftwo01'morenelsill different layers. The ndvautage ofowdap is that it coul d result ina decreaseinchannel width,(w:::,rtf:!.),wln-rcIiis the chan ne l densi ty ,Thedisadv a ntage istha t10118rUIlSofoverlapcou ldresultin cross talkduelocupacit.ivccoupliug . Ila mbruschd,11.[l lallllJ8;J)Il"S"I'illf' illcl<'lnil thelowe rboundsforchnnucl with overlapmodelsfor1lI11Itilayt' "("i ISl' S.AnollWI'is thc4'11(1(;klcuccmodel,wheretwowir ingst'g nlt'liis (ofdiffert'II1Ill)"f~I'S)lire'lllllJwf~d

toshareagridpclnc.Algorithms an dsimplifil'(l proofsforchannelruu~illgill knock-kneemode have been reported(Me hl86].JII IIOIII('1lI0l1.-J:c11O~11O\wlilll(grill line sharingbytwonels ) andknock-k nee (r;ridIluinl lIhnrillr;h.rtwo'Wi ll)ill" allowed. RivestdId.(llivc8l)present provabl y good algorithlll:Cfur 1I1i"11I<11114. Figure 2,3 depicts realisations ofa drannclproblem IIsing thesedillcrcntII1lKIl4s, The solutionsdilTer intermsof the ch a nnelwidth,1I111111)("rof \'iasandtilt'total wire length.One importantthin gtokeep in mind when comparingthe1111alit yof routin gofthe sclutlousprod ucedby difle rcm ro uli ngIIlg ori t lllllsi.~thl'fl\l'~tllllt theun derlying rou ting modelin eachofthesealgorithm.~maybediITI'N'lItau.l ilis uotproper10compare direct.lytheresultsof/llgorithmswhichux..fluitt,lWf l'I'I'1I1 models,althoughsuchS\\'I.'Cllillgcomp arlsousart-11I11111',Ior")(illIlpl ,'inI\VlJlIl-\""S),

2

,1.4

A

n

ov

erv iew

o

f

e

xist

ing

C

ha nnel

ro

uting

a

lgo ri t hms

FromIlll' {'ilrlit'SlJefffl/9fIllgoril hmoflIas hillllllulllIIlSh '\, -ns(1Ii1~l.jl]\'Iri" us improvednud clficicnllllE,0rillll n!l(De lll;6,Ili\'to$'l,YIl"hs:!,llu rsS:1. JoulJ:«i]h"w h{.'<.'11proposed for channel routing,5011\('of lilt'S(,lllE,ofit h msIla\'{'bl'111-)(lI-n,I''11 to complexshaped chsuncls,channelswith1l1O\'llb~tcnuiuals,swilrhlm)( pmll' lcrus, multila)'crrouting,etc.The le ftedge ehalllLl'! router attemptstollIio ,illlisc' theplacement of horizontalsegments incad, Irilli;.TherouterIISC'SilI,ofl l'II'i.e-sortedorde rforthenet sand hen ceits IHllIIl',Itusc..,.thl'IIV'muddillIll always yieldsoptimumchannel wlrlthirthe reart'no "'rl iral co nslr a ints, 1I0 W I' \ W ,tIll' presenc eor wrtical const.ra iurs oflenyields sub-oprlmn lrl',,"ll.~,LIMlpsillIhI'\1(' (; result innon-routability"fthechanne land thislll',~.,.sillltl'"Sillillillg uf1I1'l.~1'<1111 '11

&'9Ic99;/l9,

IIItill:rCl.'ll'iclivfchannelroutiugprobb-m.rln-1' lIlin 'Il u r i ~.lJlI l;l1sq l,III"1I1 ora lIclhilSlillie IIIthe suiuclcvcl.TIleid"iIoftill'l/ogl19wlltl'l' \I'llSpruptlSl'l1by De lltsellIDl'llt 'ifit \\'l1ich n'IlHI\'('l'theheavy conslraiul1'1111'1'(1liy n-slril'li\,t·nllllins,

Herea nelillsplil betwee ndiflerenttUlck"i.c.,thelidoccup ies morethanone

ho ri ZOlllal t ritckandtheverticalpieces ofwireconnectingthedilTercnt ho ri7.onta l lICgmcnhlirerefer redtoI.,doglegs . Dogleg, ("linbeinternalor extern al. All cxlcrllaldoglegliesina colu m noutsi d e the spanofthenetand is needed ifAnet hasl/cJo. rll.Introdu ct ion of doglegsusuallyrcsnlts ina dcereascinchannel wkhh,

especiallywhenthereis~ngpa.th inthe

v

ea

ofthe channel. Further,cloglcWl

resolvethe1001111intI,e

vea

I\ndincrease thenmt Abilit),ofdll\nncls.Howeve r, eachdoglegintrod ucC'S 1\\'0 viesandtll;s ICAdsto increAsed area ,resistanc eand

tL'CllIcccllljl{'(.'C1 IImlreliahility.Thus,itis desirabletolimit thenumber ofdoglegs .

lJeutsdlProllOS('clnvery~illlpicandcRic;c'ntwartolutrodoccdoglegsorsjy /lltill'

cohnnuswherethetidhasitpin(exceptth elefl.andl"iglit pill).Lat e r,Deutsch

improvedhisnlgcrirhmbyintroducing acontrollingparametercalle dIYIlIgr:which

definesthe minimumllumhe rorsubnet stballmlstheassigned011the cllrrcnt t1'lHk Astilera nge parame terincrea ses ,fewerdoglegsareint rod uc('(1.

III recent)'''111"11heu risticshave bccu lItiliM't1 ill dlillHld aud!Switchbox TI,c' !/r(tll!J al.;oritlllllbyRi\"CSt(Rin.'8 1!\\·ll~thc'fir!Sl of thc'!:ll' aUl'llIllt.'4

to usc a few(k~'4than10)rules to impleme utIIdlllllllcl router,TI1C'tC arc!l('1"('ral lllO<lific'Clve rsionsofthe greedyalgorit hm. The.'.;rc...'tlycirauuclrouterS<.·lIn,,;the channelina tert.·to-rigllt,colum nby columnmanllC,.,C'Olllll!etiugtill'wiringwithin a columnbeforepro ceed ingto thenext .It mayplan.. anet onmorethanone track

andIIR!"(.· a vrrticalliuecrossingmore tha n011<'ho rizolltalllC"gIIl{.'111ofthc'!Saul('net. IIIeac!.column, theroute rperformsthe followinglIlc~pl{:

• IlrillS~intill:lletsillthetopand bottom ortill'chennol,usingtheshortest

\'CrticIIJIiIll',toeitherallempty rowOJ' IIrowthatcontainsthe net.

• Frcl"!!lipil~UlanytrackslISpossib leby Wilking\"1'rLicalconnectingjOg1tltllt

l"Ol1ilPSCndscu rrentlyoccup yingmore thnuOIU.'truck,

• Hc'Clun "!! tilt"<listance bet ween thetrAckll occnpic·d hynels still occupyin g IG

more th a nonetrack.

•Movesi\netup irits nextpinis on thelo por tileehauu clor,lowlIir ill<m-xt pinis 011the bottom.

•Add s ane wtrack ifllle chan nel isrnllilntlIIpill('Du Mnot('nlertill'dla l1l1d.

Thero u teru~u allycompletestherout ing,even intIll"pm;('1l0'orC)TIi Cli ln.,rtir.11

constrain ts,ofte n using nomor etha nOIlC extratr/lckthan lhedianne!1I'-lIl1it)· (minimu m number ortracb). However, thegreedy algoritlll llsulrl'rs rrulllSOInl' problem s. Since;1,ill grlx:dyitsearchesfor 101'11101'1hun,1'lltlSt't:IIll'lllly rn·alillj.\11;1· nations II'heredccisiol1.~madeearlierIllilymalwIll!' dWllIll'1lWro lllllhl" lll'infl1',llI" thewidthorthechannel.

1'11'0algorit h ms proposedbyYoshlmuea audKuhIYo1\1l82J.111ll-l1Ipl llu-placementofthenetson tracksinadifferentWCi}',TIlt'lintillgorit lllll,,1l"lIIllt liIn mini misethelongestpat hin the

v

ce

by coulbin ingIhi1M'"Iritd ,s tha I11I;lIillli",'till' palhth rough theVCO.Thisme rgingope rati onmo dHi('>!thl'VCGbytll~at iHglhulIl'

net s whichdo1I0tccnstralneachotherasalIiug if'node,I~ch sdofI1ltTS' '(!lIod(~ canbeOlss;gn ed Oilthesame trac k. Th esecondi\IStllilhlllarhi.·ws 10 11/1;."' 1pat h

minimisation t1lrou ghmAlchingtechniques011 ...hipl1l"l il("&r1lp1l.Bethiliguritlllll"

repo rtbeucr re:oUhlitha nllicdogleg router .

llicrnrchicallOull'!'{(Jllr~83JisbasedtillIIdiri,l,,II" /..."'I'II'· npl'l"tll'dl andwasthefirs troutertoillilolllillicalJ)' ('tllllpl" I!'D"uh,·!t·sdilli!'lll!dumlwl.,. ,\:-muplciuIIItracks.Ther,ellc rlll approach ill11;1'1';ll'dlin, 1 rol1l illS illIIIdivhh-til!'

routing urcu;11 10suhllrcl'seach orwhic h ill a"lx~grid.All1,1'l"lIlim,lll11l1'10 <'11 [.1·.1

inthe centeror the basicccll; theneitherli,lt"l"ll'intr-gerplllgrllm rniug or.1}'lIl1l1lk progra mmingisused10dc·ddcallop t ima lillll' rculJllI-diollpat t cr u h" lwl:"lI11'1"

minalscordiffcrentcells .The linearequationsart'n~I IlI.;I'(1sl1<"l1lhaltill'iUll'ger

There 1I1'e12,12and-Ipatterns fo r connectingsigll<llncls withtwo tennlnals,three termina lsanrl[ou rtermina lsrespectively in2x2recta ngu larcells. Thedivision processproceedsuntiltile singlecellrcsolut .iouisreachedcompleting therouting. lldiuelllclllisperformed011theroutingorallnetsiltevery levelorthehierarchy.

i\totallydifferent approachwastaken by.Ioobbani lindSiewiorck fJoob86] inthedevelopmentora channelroutercalled\Veilvel'. WCII\'ercombines algorith-mic approacheslike the verticalconst raint graph withsimpledeductiveandexport knowled ge. Sincetherearcmanymetries tha t need optimisation, Weave rpro vides anemcrt on each: constraint propaga tio n,wire length.con gest ion.ric,Tht'st· ex, pertsohs{~I'\'(!thcproblem andmakesuggesti onsb"Sl·{1011t.heirOWIlareaorconcern. Forexample-,1.11{~via eXf.!crtsuggests altcrnativcs to tlu-roul.ingofacompletednet undattellllltsto removeunnecessaryviesh~'changingtht, lilyers of some wire sl'g

-IllClIts.ThisexpertnotOli lyreducesthe1lI1111l>erof vias, hut.also freesupSOllie ro uuugSpl ice011thereassignedsegment'sor jgiuallayer,A ....Chl'I\l1lcr,whichisan expertsystemas well, then decideson the bestupplicutlon of illlth('snl;g{-'S1.io1\s received. III addition to its tell experts,wea ver providestheuser with theability toact!ISimaddltionulresource.Theuser canoverrideHill'systemdecisionand either pre-ro ute ordeletewiresegments.We averis ahle10 route Burstein's difficult swltchboxautomaticallyusingfewerviaslind lesswiringthan11 mauuelly guided solutionobtainedbythegreedy route r.

2

.2

Vi

a

mi

nimi sat

ion

in cha

nn

els

Therean-11uutubcr of reasonswhythenumber orvtasinIId1A11lll'1 111)'0l1t should hekept toIlmiuinuuu. IIIintegrated circu it pro ees siug ,1I10J"{'viasus uallyIC1\d toa loweryield. Fur t her, everyYiahas <Illassociatedresistan cewhich affects

t,llI:circuitperformance. TIlesizeofaviais usuallylargertita nthewidthof

thewires<locihence mere vilismean morefo ulingsll ~ ("('- 'I'ln-rcisan in\'('rsl' correlati onbctwoonthe 1lI1111be r ofviasusedbya routerandI,lll'1'n11lplt.'tiolll"lItl' uf therouter.Wit h thesedrawbacksinmiud,via-miuimisatlonalgori t.hm sha wlu-ou studiedextensivelyundertheruodclsofCOII~//lIill n/rillMil1illli.'II!iOIl(CVlI l)and

Um;(JlIslI~dllcllViIIMi/limil.'f1/io ll(UVl\I).

111constrainedvia minimisatlo u,allinhial1"~11l11isassun....tllnhi' uvnil-ab le and the goa lis toreduce tile 11111lllJ.crofvias in lheinputII\~lllllhy n'llssi/!,l1ill/!, so menels and net segments totheopposite1,,)'1'1',Thus,CVMdOl'Snot.Idlt'l"thl' layout of the in putsolution but attempts as('h'('lil"t~lnyer n·assip,lInll·lll.AllI' XH I Il-plo ofCV1\1inII2-la)'('1'channelis shownillFi/!,lll'l'1,1,11,llas!lillloluuudSlt 'I'I'liS [11<I5h71]ril'slformulatedthe(:V1\"problemfurtilePrinll·{lL'[rr-uit.lI(lHl'\1d"si/l,lI. Subseq ue ntly 1111111yalgorit hms ha vebeenproposed [l\lIji81J.(,i,'sSI]fill"('VMin a2·layercuvlronmcnt. The twolayerCV1\1problemis11(11NI' -(·IlI11I' II·II'ilwl1-'111 be solved in poly nomialtime,ChangandDo [ChD IISS]rllTllml;lll'llIIII'1,lIn',·lilyl 'r CVMproblemnud showedthatit islinNP-comp ld l'pl'OIJlc 11I lind pn's"lll,I'111111 heuristical~01'i Lh1l11ISwell.

The resultobtainedbyt.heCVlIIapproachis l1IillillHduul.\'withl"l"~PI'('t t.othegiven inpnt IIlYO\lt,1tHImaynot br.-thl'oprinmmaulutiunforlll" plOlJl"t11. For example,<IIIiuitia l layout wit h 10vlasshownillFigul"I'1,·I.~Iri\l1.~lalt 'sillln11 5-via solu tionalte rviaminimisati onwhere as.;111 inili HIlayoutwiUI11,'iilsshown ill Fig ure 2A.btransla tesinto a3-viasolut ion, Thus. I,{)ronsiderI III' pl"UIJII'lll asnwholeiustcndof acting 011adiscrete 1"calisatiunIIII'/!,oalof lopulu/!, indvia minirnisatlou\\'IISproposed.

TIleTOJloIQ!Ji CIIIViiiMi ll i misn/io ll,IIlso('Jlltl 'lllln('ull sl rll ill l'llVill :\l ill-irnisarion,considersrou ti ngandviamiuimiserion IlS linin!.(·g r1111·d sl.!·pHilt!was propo sedbyIIsu[lbu83], Here the routing or11r-lnuuu-ljxsplilin to 1,\\"0 sl,,·ps, na mely,/uJ!olO!Jiclllmtllill!J1111<1geometric/1111/'11/".1/.Till'IOl'olop,il"III'Olit illl;SII'J!

jlI]Etr

.

-

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Before'lin.mluimeetlou

(10vias]

ll

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(a

)

(b)

1!lliJ[

:

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.

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Aftervia minimisation

(5vi<l5)

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,

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,\I't.(',' \'ill lllillilllis'ltioll

Pd ;ls)

Fig u re'2.,1:Consu-alucdviamiuimis a ricn ofil2·Iilyc:rrhuuncl

atLl'lll plS tofind ll minimum viasolut ionfortb erhauucl.l;,kil1g1111.0accountonly

tlwtopologicalconstrnintsaudnotthegeomet ric;11rousrraiuts (imposedbythe

technology),whichliretakencareofby thegcomct.rlralmnpplug step. IIs lI ca

p-1.llr/' lthetopologi calconsf.rnints betweenthel1('lsillh'I"IlISoftheeivclc !in/ph(.'9

ofthochannel.(liven1\channel,itscircle graph(allIll'obt llil\<-,ditSfollo ws. First

obt.aiu till'circulorI'C/II'tI'Cllflllio u ofthechannel.bytritWrsillgthello de s ofthe channelin HcircularIashlon. llsillgthisrep rescutaf.km,l'\ "f'ryi-pillnet ofthe

rectang ularchannelwould trauslntc intoachordill'herjrr-ulurI't' P l"f·lWlll ill iuli.If

twonets crossiuthe:chuuucl, thentheccrrcs pondiug chordswillintersect.The

dnk !lmJlftisdrnwusuchthatits nodesrcpresellttil('chords[nets]andth<·edges l"iIpll1l'I·l.hl'intersection or cho rds(cross ing ofcbunuc lnets],itSillusLrill,cdillFigu re :L'i. Iftlll' circlegraphisbipllrtit cthenthec1111l1l11"'1CIUl1)1'ro uted withoutvias.

Otherwise, onlyits maximumbipartite subgr i\[lhcan1)(' routed with ou tviasand

therctnaiuing netsneedvias. Absoluteminimisa tion ofViilSmay necessitatemany

det oursinthediann e].Topologicalvia minlmise tiouofz-leyer.z-pinnet sha sbeen

:1 1 ·1 ,\Tnpnklgk lll Itl/IIL;II /;\1"111.lint'\"jll AChlllluclProblem

CHJ

I , 3 • 2

~I

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_. CiecleGraph 2 :1 I ·1 " Fhli11111.~·(JIIr

"nl

:

r

gl'O lI lc lrlftl1ll l' l'llI j:,

Figurv:!.5:Uuceustraiucdviam;nillli slIli"ll(If H1·111.\'1'1'1'I111I1lll'1

showntoIll! N'P,colllplclc[lIsl1S3,Sadw84]. Fo rthespecialcase

o

r

peruuuetjon

clrnnucls,polyno mial illgoriLhmshavebeenpropo sed [SalTS!I,RimS!lJ.Topological viaminimisationwillbe discussedillmere delailillChapte r G.

Ch

apter

3

A

n intr

oduction

to Single Row

Routing

IIIthischapt ertheSingle HOII"RoutingProblem (SltUP)i~iutrndurtxl.l\hlulIlll,h 11111U)'algoritluns fortlw SIUU'1111\-cbeenproposed. noworkIms1" '1'11 l"l'llur lt'd )"1' 1 011.,compara ti vestudy 1111<1a compr ehensive I.,XOIIOI1I)'oftIIPs(',lilli'n' ll!SIlI/lllg!,· rit.luus.The focusofthis chapterisataxonomyof :-illItI1I~oril l l1l1stl111111<11'('1""'11 reported todille, togetherwitha brierdescriptionof 111l'1i('nlgorhluus.Fnrtln-r, theilllld t'tlllllCyoftheexisting51lHalgorithmsto ,lclli"I'"tilt'm;" imllm('HM._IW , , " 11J1i1ill!ldl'l1lll11dt~dbytheSItTcuviromncntisindicntednud thisr-allsfill'I,hl' d"sign ofIIl'Worruo difl ca tlon of exist,ingSRn algcrltlnus.1'1l1'tirlll<lfanentlouis]l<I idIII and a dctullcdallll]y~i~is presente d forTamg'sillborilhlll['lj1fIl~'ll .

3.1

An Intr

oduction

to the Single Row Rout

ing

Problem (SRRP)

The SingleRowRouting Problem (SRRP) is a verymuch restricted subset ofthe generalroutingproblem.Itcontains a set V

=

{I,2, 3.. •n } of evenlyspaced nodal whichlie alonga single row. The set L

= {

NilN2••• •Nm}is called the ndlilll andeachnetNi(I~i~m) in L contains a set of twoor more of then nodes, which arcto be connect ed toget hertomake themelectrically com mon. Each node belongsto exactly onenet;l.e.,N;nNj

=

41ifi

i:

i-

For example,an SRRP withm

=

4 andn

=

8 (m

=

total number of nets and n

=

tot al numberof nodes) canhavethe netlist(orinterconnectio ns)Nt

=

(1,4 ),N2

=

(2, 7),N3

=

(3,6)and N4=(5,8). Arealisationofthe netlistL is to be madeon a singlelayerbyusing non-overlappingwires that are compo sed ofonly verticaland horizontal segments. One possible rea lisationofthenetlistgivenaboveis shown in Figure3.1.

Therow a n which the nodes lie is caned the node axis andthe area above the nodeaxisiscalledtheupper stredand the area below the node axisiscalled thelower sired. Themaximum number of horizontaltracksallowedinthe upper (lower) st reet iscalledthe upper (lower)sired capacitywhile theactualnumber of tracksused intheupper(lower)street in any particularrealisationis called the upper(lower)siredcongestion.The symbolsC".andC1•standforthe upper andthe lower street congestion respectively.Forexample, the realiution shown in Figure3.1hasC",

=1 and

C/.

=2.

In additionto usingonlynon-overlappingrectilinearsegments,a reali sa-tion usuallymustalso not have anyleft-rightzigzagging of the nets.This means that a verticalcut made at any node caninte rsectatmost one horizon t al segment fro m eachnet. fRagh83]refersto theleft-rightzigzagging as backwardmovesor backmoves.Only thosereali sations without backmoves are consideredas legal in

th isdiscussio n.Some cxmuplcs oflegalamiiIlcgaln'llli~aliullsurr-shownillFip, -ure 3,2,Howeve r,upanddown zigza ggi ngoft.11('1Il'Isi~"II(1II·,'d,i.I., Illl'lids 1'0111

changeoverIrom one street toanct ho.. inthejuter·1I0 dl~space.SurhInlll ~ililJlts

nn-ca lledcrossovers.Each netlila)'111I-\'C zero orlI1Ol'Pt:rosso\'(' r~Iln ll~A'ruormort-lwl s cancr05SOVCl'al thesamelutc r-no despace.Thomaximuml111 111lll'r ofl:rtlssO\'I'rs inany of theinter-nodespaceiscalled thecl"l),q.~()I 'r"bfJllllflnsunllydcnon-dilS1\', SOllierea lisa t ions Illustratingvarioustypes of crossoversan- shownillFignr l':\.:1.

Given11not.listLconta ini ngIIInets,1.I1l·n' 111"1'III!posslbk-rt'alis ill.io llS. Each rcali-o-tion111,1)'howea differcnt sl,rt'Ct("ollg,.oslinn111111a,liffl'r<'111 1111111111 '1'uf CI·OSSO\'crs.TheSRRPhasbeenst udieduralnlv\I'ilIIIlll'nl,j"r lin ' ofIllinililisi lll; IIU' t.rncksused.AlloptinnunrCillisalionis011':which 11IiuillliNI'.• 1111'S11l'l't ('lJlll;l'Sl io li inboth streets,Thus optimality wit hrespect 10I,l"ilr b IIsl·II(slln 'll'UIl~I'St iu l1) aimsallllillj'llisilll;the1'111ueofQ whereQ

=

llliIS{('"•.('/" I. :\U"lln'r I,"s.,illl,' op1.ill1alitycriterioucouldhethemiuimisntlonurII...nlJSSO\"'I'S.Littk-worklws beenreported in th isdirectionyetandII'l' willr-tnrntothisa"pl'din('1111[,11'1',I

and Chapter G.

~

N'

'

."'

,N

:

7 8

,

"

tV1=(1,01) N1=(2,i)

'va=(:IJi) ,VI

= (;..

..

$)

Figure:t.I:1\ 1Iexample of anSllllpml.l'·l l l

t\givenrouting probk-m(c.u,11dl1l11111'Iurilswi,dlhtl.~pm bklll )run ah\'ilr~he convertedintoaucqolvnlcnt SllIt !'.For r-x.nnph-, 11 rlUIIIIII,1l"tluM Ill'

casil)'COII\'I-'I'Lcd 1\/ 1'111 SIUl Pbynnfold ingthl'dl1l11I,..1andpl;II'inglilt,twoIUIIW'I

sides,sitlc II)'side.In11sanllur Will'.<lswitchbcxrnu1)('uufohb-dinto;1 singl,'row ornodes.Ouc ofthe advantegcs oftheSRH PwlU'1Irompan-dtoothorl'IJntilig

_-r=C=L

I:.! :1 ,I

r

,

G 7

8-~

Back move

Illega lrealisations

Vigu!"/.'a.:l:Examples of some legaland ilk'g1l1roalisatlo nsofallSRRP. prohlcllIs istheIflJ!0!ogicrl!jlljiditypossessed by theSRRP(TopologicalHuidhy is the ability todefer detailedroutingunt il the0\'1.'1'111]rcut uhilityis considered). '['Iu:SIl B.Phasalsobee n studiedunder50l1lCaddir.lunnlroustraints,SomeoftIl(' n'slrkll'dformsoftheSIlItPnrc:

•SHHPwilhprescribedslrl.oclcapacity:HI'n'tlll'l'l'isHlimit.(SIl.l'I,WO) Oilt)l(' 1II1 1l11wI'ofallowedtracks inuppe rlind10\1'('1' slrct'ls.ThisSil ltl'dot'Sflud lllJJllicllliutlsillPrinted CircuitBoard routing.hutitisno t.practicalfor VLSI nm!.iltg problems.

•SHil l'willioulCfOSSO\'Crs:Here the<lim is to produceacrossoverFree1"/." alisiltiollfor theSnB. proble m,even thoughitmay 1I0thave opti ma ltra ck congestion.Acrossoverfree soluti o ni~notpossible forullproblems. •SHill>withcrossoverboundK;This1i111i t~themaximum numberof wires

thaI, call changeover fromonestree tto auof.herillanyoftheinter-node SII;l("(~S ,Thenmnhcr ofcrossoversoneachnet.isunrestric ted as IOllg<ISneue ortil"inter-nodeSPilCC.'lhas morethan /\(l"OSS O I'( )l"S,

Figure3,3:Examples ofcrossoversof different types.

3.1.

1

Origin

s a

n d d

ev e

l

opments

o

f

th

e

SRR

P

Tlif~SHJll'wasone ofth eearliestrou tin gproblemsanditwas lnt.roducedby SO [SQ;'II,as a sulJprolJlcl11thatal"ise~'" solvingagCIICI'/l)mullilll)'crrout.iug problem hyn sys\.cmnticdecompositionintoa num berofindependentsinglerow single layer

problems , This appr oa ch involvesfive ste ps,lliUlH,lyviaasaigmncut..linearplace -mont ofvia columns,laYI~ritlg(alsocalled via miuimisaticn].sillgll'rowrout ing iUldli ll" ll)'viudilllill'lt ioll,Subsequently,algoritlllll.~withIIl'Cl'S.~II I")',111l1suJlidcllt ron.lit.ions In minimisethc11U111bCI"of trackswurcproposedI,yTilll,!;cttil,[TillgiG], 1\ 1111dill.[1\lIh.!))proposedthe1IIlel'vaf Gm/)It(IC)rcproseuuuiouoftheSRR P. TIll'.\'illsodcvclopcducccssnry and sufficientconditio nsforallopthuumrealisa tion, illl hollg l1[hfTdidnotl)l'f~S(,llt1111algorithm fol' SH IlPbased011theseccndluous. Hagh;ml1lcI(/1. IHag h8'2) propo sed an algorithm basedall/leiorde rCIUHlIC/"(I{ion techniquewhichIIO\ \'C\· CI' ,11'1\8notpract icalfor large street capacities. Ragha v1tll dIIf,(IlnghS:l]consideredSRRPwithbackmovcs,i11li1lysl'l.l

snu

p

with prescribed

streetcapacityamialsodiscussedanalgor it hmto generate11nos s()\'el'frc ~ 'Sli' lu t ion,ifoneexis ts,Tarugct el. [Tarn84]utilisedtheroudiriousforupt.iuuuu routingdev elopedby [Kuh79]andproposedaucffid en\.illgorit.llIllhilSl.-~llllIn »lm-plc heuristic,whichproducesalloptimalsolution

r

.JI'

manyIlrolJ\I)llls.I!llgllil\'1I11 etai,[Rilg h8'lj reporteddetniledresultsall t.hecomputarioual complexityofI,hl' snRP,Han&Salmi[llana.t],modifiedthebasiccuumcret.ioutechniqueglvcuill [R agh82Jfor speci a lcases of streetcapacit yand laterextendedthe ir work1,0deal witharbi trary streetcapac ity(Ha n 8 5],NOlle of tho IIlsuri1.11l11Scitedabove paid1111)' delai1cl:lattentionto crossov ermin imisation. Howeve r,Oilct"I,[nll~i'llll(Jdiril'll the notor!!I'r nnunu-ratlonlpdmiqlleof[llagll~2],lilkil1l;into ilITOUlIl111I'rrossnwr bound aswell. Du "lidLiu[DIILiS7!extended1111'Imsklll'urislicalgmi lhm of 'f"l'Ilgci11/.[Tal'll8~]by tilking intoaccoun tthl'yn JlI/li"f/ 1jfll'/,Illliltiad lill'y il" (lI,[Uhll tSS ]presentitgra p htheoretic approachtn,..;oll' illl;t.hc-SilltIImhll'lll,quill' recently ,ShcrwanicI/1/, [5her891havemodified till'algorithmill{l> uLi~i] lakiliH into accountthecliqueintersection,

Beforedescribingtheexistingalgontluus forsolving 1.11('SHIH'SOlll1' moreterminolo gyisrequired. Th eintervalgraph repl't'l'l:ntaLiun [1\lIhi'!Jj (also referredto asintervalrep resentation)of1111SHI1Pis111«:1-of//Ihurizontnlintervals rcproscntingthe111netstogether withanorder,whel't'III!such ordersilf('possihll', Figure3A.aillustratesone ofrhom!possible Interval representationsfor11gi\'l'll SnRP,TIll)intcrvalrepre scntaticuisacon('~ptllillr(,i1 ]i sali()11furwhichIII/'H'('xisl.~

11COI'l't-SP,)III]illgI1l1i(IUCphP 1rillroalisntionwhichrunI...obtaine-dh,\'a,Sill l]!! I' procedure.foirst .1 referenc elineisdrawnwhlrh('OIIIIl'l'1 s1111'lImh·sill slIlT I ' s s i nll Irorule ft. toright.Iftilereferenceline isst r<lighlc lll'd (J1I1.IIII'll1111'IIIliuriwlltill intervallinesarc mapped topologically into \'I'rlin, 111 lidhorizontulpaths.:-\1'1."1 illll! portionsofnetswhich lie above[below]thercfcrcnrcliuc an'IIlitl1pedintnpaths in theUPI)Cr(loll'cr)stree t ,Figure:JA.bdepicts tlH'phyaicnlrealisation('orrl'SIIUlldillp, toFigure;l,·l.a.Usingtheintervalrepres cu t atieu of uu SllHP,itisr-asytilidf' lItify

• :

N~

All

. .":' N, ~-' : Interval Representation

)

"if.

:

\

:

:

Z

:

(ConceptualReelisat lou] (a.)

' .

~;

•\ / •

~T

he"r"cn

"

J;n

c

(b)

Figure:JA:All intervalrepresentationandphysicalrealisat ion of anSIUlP. thecrossoversinthe corresponding realisation. 1\(,l"OS.~OWI· n:;;n!llIwhonoverthe refcrcuccliuccuts11netillthe intervalrepresentation.AlsoifthereareIimaximum or111lid s,then thellI11xillll1lllcrossoverbound1\',,,,,,.is obviously(III- 2)(Du871.

}\nothcl'il1ljlOI"La nlconceptoften usedillan.'illapistill' CIIIuumber: Ea chuorlc amIlid111ls a cut numberassociatedwithit,whichcallhefound From

theinterva lrepresentation.Consideranyintervalrepresentation of an SltRPand

drewanilllllgill1lryvertical line,perpendicular Lathenodeaxis, at CVI:J')'1I0 (\C, TI1l'1lthe11111 1111('1"oflids . excludingtheltel toWlliclitilt:lIodl'belongs.which '11'erillhy1,11<:\'l'l,tic~llineat nodei,iscalkxlIhr-cutnnmhurCi •Iortill!node i,Ci~fin dC';,aretl.,:cut numbers forthe nodei illthe uppe rIIIlUlo wer stn..ots respectlvely, Clearly, the total cutnumberfor thc nodeigivenbyC;

=

('i.

+

Gil

isall invariant property Ior agiven SRnproblem. C;uand(.';1obviouslydcperul onthe order of nets intheIGrepre se ntatio n . Witlltlwscdeflultlnns, theup per (lo wer)stree tconges t ionCu. (CI.)ca nbefoundas ('..ICt.)

=

max{C;.(C';Il) where1:5i$11.Theoverall congest ionQ;IImax{eu..C/,).

Thecutnumber ofanetNi ,denoted/ISfli.isthemuxinuuuofthccut numbers of all thenod es whichbelongtothene tN;.Let tIll'ncllistI, hep/lrlll iOlwd intotwo sublists1.1and1.1such th a tLl nL1=til111111"1 uJ.~=L TheIL!.Ill' internal cut numberiqj[Tnrn 84]ofthe netNjinI. with respectto1.1isdd illl~l l asthe cutIlumuc rorN;ill1~I''l'lms, tile11l1,I' I'Hillnil111111\1)<'1" llf;\ 11<"1 issimil.. r tothe tot alcut number ofthe net, exceptth<l!unlyaSllIJSl'1.ortIll'l'lll,il~'IId li!'t iscon~idercd "The residualCIl Lnumberrqjofthelid NJinL withrt·sp'~"I 1.{)1' 1 is defined asthe cutnumber ofNjin1,1U {NJ}.

3

.

2

A taxono my of

a

lgor

it hms for t

he SRRP

Because of thecrucialrole playedbytheSRHproblem,attelll pl..~haveIll,'ellIll/HII' to develo poptimum' algorit h msto solvethe SHill),Althoug htill' SHltl)is l\Sill l ill" andwellde finedsubproble mor the general routingproblem.ilisill!\llNP-WlIl pldl' [Ragh8.IJ.This hasled to the scerchfor efficient111:lIristi..algorithmstil suh'" 1.11<' SR RP,Whittdifferentiatesone intervalrcprcscntetion fromallol l l1,.istil"Ul"llf'l'il1!!, of nets.Th us,anyalgorithmtosolve theSRnPshouldnutput1111optitualordr-riug ofnet s which canth en be translated intoall optil1lallyl"H111!,1'Stt'lln-alisui.ion.

Tilealgorithmsthathavebeenreportedsofat

rOi'

MilvingtIll'SIlIII'r-an be gnlll)'cdunder1,1r0IIlainfillililie:s, The first f''' lIilymllsi!'ts

o

r

till'Ill'IIris 111 alg o rithm proposedbyTi ll "USdfl/,[Tal·u8'IJ.111111itsIliodiliull'i'llliollStl" "~rril>l'(l

byDuda/.[Dul,i87]and Sherwanietal. [She1'89]. ThesecondIaruilyisbased 011the net orderenumeration algorithmproposedby Raghavau clel.{Rllgh82j and it smodified versions reportedby HanandSalmi[lIan8~,HanSS]and Duet al. [111187].

3.

2.1

H

euristi c

al

gorithms

for th

e

SnRP

Tarng'salgo r ithm(Ta rnB4]

SincetheSIUll'isNlt-ccmple te[Ragh84].itisjust.iflableto lookfor efficient.

algo-ritlmlsbasedon simple heuristicsto solvetheSRRP.Tamgcltd.(1"'rn84]pl'Op05C{[ nne sudt algol'ithmwhich is easyto codeandoftcllleads to optimalsolutions.B e-sicully,the algorithm assigus net s withlargecut uurubersclosetotile1Ixisnudthe lIelNwith slilall cutnumbersfurtheraway fromt.lu-axis.Thus,<ISOil!.'proceeds ,l\\'IIYfro mthe1I00k·"xis,illeithe r direction ,lht·netswillhan'non-increasingcut numbers. Inorderto a chievc such lin eseignm-nt,IIII'IIl'1san' Ilrst groupedinto

Z/JIICI!. Netswithineach zonearethenorderedto get1111OI'el"lI11 orde ringof net s whichisthen usedto placethene ts ontracks.Tamg'salgorithm is now discussed

illmorodotujl.

Given1111instanceorallSRRP doscnbedbyIInetllst.1..the node cut 11I111IlJt'l'l< lI1llItllencLclltlllllllbcr s<lrefi rsldetcl"Ill illt'cl: tllenthenctsnrc part.it.ioucd iutnzonesIlnscdonthe irintern a l cut numbers.LetIfm" rbc themaximum netcut numhcr. Then a 11lid s withq

=

q" d% Meeeslg ucd to zoneZOoallnets with If

=

q""or-I are assigned to zone ZI andso on until nllthe netsare exhausted. Thisstt.'P is referred to asZO'I C(il/ottillg.Such wile aliollHt'nt!l(,ll's us to tn.'atnels

ofHIl'i'llIlUl'(~11 \.number as aunit.Forexample. I,lli'SlUt I'lllustratodinFigure :1.);hHi'l1/"",,,

=

:landsothere are J zones,Xu={I, 'I.;;.() .!)}.Xl

=

{2,;1,8}and

z~=

{rl

.where Ihl' numbe

rswithin braces idt'lllifyIl(·ts.III order 10Ilrhit,\,('LIIl' 3)

goalorkeepingthe nets wit hla rge cutnumbersclosetotheaxis,it,isdt'il l"t.hnt

net sorZoshoul dbe closertothe nodeaxisthanlid sor7.1illl tlsc on, Thenext step,called11eloly[erifl9,is tolindthe orderingorlI('[.Swithin eachzo ne,Firsttheintern alandres idualcut numberoreachnetinn.zo nelJ

with respect to£1={ZoUZI "UZi}isfound.Nelsinamilt'arethenordc~l"I'tl indecrea sing orderorinternal cutnum ber, lf1.\\'0nelshave tht'sam!.'lutcrual

cutnumber thenthey arcorde redbased0'1a dccrcastng rcskluulI'lltllllll1l"'I ',It

may hap pen1.11i1t somenetshave hoth theimcruulandI·t~ idll;l lrill1II11111u'rlht, sameandillLhat case the relativ e orderingorslidIlidsis arhitr a ry,Forexu.mph-, the orderornet s shown in Fig u re3,5, atthisstep\\1JllldIll'7.1""{!i,{I,·I, f"i,!I}},

Z~={(2,3),S}andZa={7},wherethe nctswithin( )rnu Iw\'C uum-hit.rsry order,teltheorder be {5, 1,4,6,O},{3,2,S}anti

Fl,

TheInststepis lnlck-fl$$iglllllcIIIillwhichtill'ordl.'!l'tl nl'1.sIrorn1.111'zun" Zu, ZI'" arc tilken andassignedto tracksto get 11rUllt·t'lllllilll"I',tlisilliUI1, I"irsl nets From Zliarctaken and put inficlit iollsr.rack'/~orany01.111'1"IT'Kk'1~snrh that

l

Ull,

the absolutevalue ofj, isminimi sed ,]~isn-icdflrst, IIII'll"+1 orT..I is usedendsoon.

or

course,only nou-ovcrlapphu;uplscall hI.'assigllC 'd1011ll' snmctrack.ArterIluishiugthe aeeigumcntoralllid sFromZo,.1similarpru(' ('llul'I' isFollowedfor thelidsor otherzonesbutwith till'1'1'S lril'liulilhilllIn'1ll' l suf a 111'\\'~OI1Cmustoccupyonlythose trackswhichan'nulinm-rIIItln-Ir1Id,,;IIs!'11 so Far. This meansthatthe nelsoftheIlC\\"ZIlIII'(";111 sl1ill"l' Iltl' oul."l'lIlosl trurks alre ady used,butmayIIOtusc anyinteriortracks,[-'IJI' example.,111'XOII('\SilltIll' exa m p le orFigure3,5 are allottedtotracks'J~,'/'-1 and'/+1, Nt'ltV:l railoccupy the track1'-1'The aboveprocessresul tsinIIsetoftruekswith111'1assigll"ll toir., andthiscoucc ptunlrealisatlouca nbe convertedtc auni,pll' layout(1"I';llisilli ulI). bydra wi ngtherefere ncelineandbytopclogirnll1lilppillg ,

From the descripti onofTarng's algorit hmsome prop erti escanbe de

-duocd;theirimplicationswillbe discussedinCha rl!cl'5.

• I'IV/icrly1:Whenever thereisa choicebetweenusing trackT+iorT_ifor

anot, thechoiceisirrelevant.The goalofTarng'salgorithm isto minimise

thedistance oftheIJeWfictitioustrackfr omtrack Toand itis immaterial

whetherthe newtrackisin the upwardsordownwardsdirection.

• PmflC1'ly2 :Wit hin eachZOIlC,ifso me nets havethesameinternalaswellas residua lcutnumbets,then theord er ofas signingthemtotracks isarblunry.

Oil'Salgot-itlnu[DuLiS" J

])\I' Snlgoritlnuis basically animprovedversionofTarng'salgorith m. Tarng 's

algorithmbeingaheuristicalgori t hmitisnotexpccted to producealloptimal

solut.icufol'allcases.DII cIttl.analysedthecas e sfOI'II'lddlTa rng'.~algo rit hm[ails

to produceallolltimalsolutionandintroduced the conceptof91YJ1IJl~.Theuotlist.

isdividedin tosubsets ,such thatnetsineachsubsetsatisfythe91'Ollp ill9property.

Maxima lse t

or

nets which cut acrossone ormorecommon node Ionna group, i.e,

there lsat leastone verticalline whichcursallthe nelsofagroup. Theconce p t of groupingisillust ra tedinFigure3.6.Heret.hutotlll uctlistI.

=

{Nt · ·.j\'r}eau hepMt.iti01w,Iinto twogroups. .rll={N" N2•:VI,N~ } <lJl(I.rh={N~. N!"No.N, } , withud.

N,

being commontobothgroups.

Dud III.ha veprovedthatTamg'shcnristic pl"indl' lc.'1I111'1I-p'produces

lH1optimallycongestedsolution aslong as1Il('input nets form<lsillyl r:9IYJ1111.III lheCl1SCofmlllli-glYJ1lpllcLSan optimal solut ioncannotheguar anteed. DucIal.

111'(J1}(1~'i 1tlmtumult.i-group

srrR

problemhebrokenilltoIInumber ofsiugb- group

SlUt problems.Thiltis,groupsshould beidentifiedandthelleach grouprout ed

iudcprndcru.lyusingTnmg'sapproach,resultinginan orderingofthe notswll,hill :13

Nets ,I,'1,3 ,4" ,2 ,3 Nodes 1 '13 4 5 6 :7 N,

:

~

~

:

:

tV

t : ~ Nil Nets Cut-Nu mber Znll l ' N] ,JV'h N~,JVo, Ng 3 7.1> N~,JV3, N8

.,

7.1 N, I 7.1

z

,~

{5, (1,·I,G,'1l

z,~

({2.31.") Z, =

I'J