DTIC ADA461741: An Algorithm for Multipath Computation Using Distance-Vectors With Predecessor Information

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by  Defense Technical Information Center

1
An Algorithm for Multipath Computation using Distance-Vectors with
Predecessor Information
SRINIVAS VUTUKURY J.J. GARCIA-LUNA-ACEVES
[email protected] [email protected]
ComputerSciencesDepartment ComputerEngineeringDepartment
UniversityofCalifornia UniversityofCalifornia
SantaCruz,CA95064 SantaCruz,California95064
NetworkingandSecurityCenter
SunMicrosystemsLaboratories
PaloAlto,California94303
Abstract—RoutingalgorithmsintheIPInternetprovideasinglepathbetweeneach With the exception of DASM [17], all of the above routing algo-
source-destinationpairandwheremorethanonepathisprovided, theyarepaths rithmsfocuson theprovision of asingle pathtoeach destination. A
ofequallength. Single-pathroutingisinherentlyslowinrespondingtocongestion drawback of DASM, however, is that it uses multi-hop synchroniza-
andtemporarytrafficbursts;multiplepathsarebettersuitedtohandlecongestion.
tion, which limitsitsscalability. Recently, wepresented MPDA [16]
AlsothepathsprovidedinRIPandOSPFarenotfreeofloopsduringtimesofnet-
whichisthefirstroutingalgorithmbased onlink-statesthat provides
worktransition, whichcanbedebilitatingtonetworkperformance. Wepresenta
distributedroutingalgorithmforcomputingmultiplepathsthatneednothaveequal multipleloop-freepathsusingone-hopsynchronization. Inthispaper,
lengthbetweeneachsource-destinationpairinacomputernetworksuchthatthey wepresentavariantofMPDAcalledMPATH,whichisthefirstrouting
areloop-freeateveryinstant—insteadystateaswellasduringnetworktransitions. algorithmbasedondistancevectorsthat(a)providesmultiplepathsof
Thealgorithmisscalabletolargenetworksasitusesonlyone-hopsynchronization unequalcosttoeachdestinationthatarefreeofloopsateveryinstant
whichisunlikediffusingcomputationsthatrequireinternodalsynchronizationspan-
— in steady stateas well as during network transitions, and (b) uses
ningmultiplehops.Thesafetyandlivenesspropertiesofthealgorithmareprovenand
a synchronization mechanism that spans only one hop, which makes
itscomplexityisanalyzed.
itmorescalablethanroutingalgorithmsbased ondiffusingcomputa-
tionsspanningmultiplehops. MPATHisapath-findingalgorithm,and
I. INTRODUCTION differs from prior similar algorithms in the invariants used to ensure
multipleloop-freepathsofunequalcost. Thepeculiardifferencesbe-
The most popular routing protocols used in today’s internets are
tweenMPATHandMPDAisaresultofthedifferencesinthekindof
based on the exchange of vectors of distances (e.g., RIP [7] and
informationthatnodesexchange.
EIGRP[2])ortopologymaps(e.g.,OSPF[11]). RIPandmanyother
Section II describes MPATH. Section III presents the correctness
routing protocols based on the distributed Bellman-Ford algorithm
proofs showing that MPATH is loop-free at every instant, safe, and
(DBF) for shortest-path computation suffer from the bouncing effect
live. Section IV analyzes thecomplexity of MPATH.SectionV pro-
and thecounting-to-infinity problems, whichlimitstheir applicability
videsconcludingremarks.
tosmallnetworksusinghopcount asthemeasureofdistance. OSPF
andalgorithmsbasedontopology-broadcast(e.g.,[15],[12])incurtoo
muchcommunicationoverhead,whichforcesthenetworkadministra- II. DISTRIBUTEDMULTIPATHROUTINGALGORITHM
torstopartitionthenetworkintoareasconnectedbyabackbone. This
A. ProblemFormulation
makesOSPFcomplexintermsofrouterconfigurationrequired.EIGRP
usesaloop-freeroutingalgorithmcalledDUAL[3],whichisbasedon Acomputernetworkisrepresentedasagraph where
internodalcoordinationthatcanspanmultiplehops. issetofnodes(routers)and isthesetofedgesG(lin=ks()Nco;nLn)ectingthNe
InadditiontoDUAL,severalalgorithmsbasedondistancevectors nodes.AcostisassociatedwLitheachlinkandcanchangeovertime,but
have been proposed to overcome the counting-to-infinity problem of isalwayspositive. Twonodesconnectedbyalinkarecalledadjacent
DBF[14],[10],[9],[17]. Allofthesealgorithmsrelyonexchanging nodesorneighbors.Thesetofallneighborsofagivennode isdenoted
queriesandrepliesalongmultiplehops,atechniquethatissometimes by i. Adjacentnodescommunicatewitheachotherusinigmessages
calleddiffusingcomputations,becauseithasitsorigininDijkstraand andNmessagestransmittedoveranoperationallinkarereceivedwithno
Scholten’sbasicalgorithm[1]. errors, inthepropersequence, andwithinafinitetime. Furthermore,
Acoupleofroutingalgorithmshavebeenproposedthatoperateus- such messages are processed by the receiving node one at a time in
theorder received. Anodedetectsthefailure, recoveryandlinkcost
ingpartialtopology information[4], [6]toeliminatethemainlimita-
changesofeachadjacentlinkwithinafinitetime.
tionoftopology-broadcastalgorithms.Furthermore,severaldistributed
shortest-pathalgorithms[8],[13],[5]havebeenproposedthatusethe Thegoalofourdistributedroutingalgorithmistodetermineateach
distanceandsecond-to-lasthoptodestinationsastheroutinginforma- node the successor set of for destination , which we denote by
tionexchangedamongnodes. Thesealgorithmsareoftencalledpath- i i i,suchthattheroiutinggraph j consistingoflinkset
findingalgorithmsorsource-tracing algorithms. Allthesealgorithms Sj(t)(cid:18)N m isfreeofSloGopj(sta)teveryinstant ,even
eliminateDBF’scountingtoinfinityproblem,andsomeofthem[5]are fw(hmen;nli)njnk c2osStsj a(rte);chman2ginNggwith time. The routing graph t
moreefficientthatanyoftheroutingalgorithmsbasedonlink-statein- forsingle-pathroutingisasink-treerootedat ,becausethesuSccGesjs(ot)r
formationproposedtodate.Furthermore,LPA[5]isloop-freeatevery sets i haveatmostonemember.Inmultipjathrouting,therecanbe
instant. moreStjh(atn)onememberin ;therefore, isadirectedacyclic
i
graph with as the sink nSojd(te). There areSpGotje(ntt)ially several
foreachdesjtination ;however,theroutinggraphweareinterSesGtejd(ti)s
ThisworkwassupportedinpartbytheDefenseAdvancedResearchProjectsAgency(DARPA)under
grantsF30602-97-1-0291andF19628-96-C-0038. definedbythesuccejssorsets i k i i ,
Sj(t) = fkjDj(t) < Dj(t);k 2 N g
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1. REPORT DATE 3. DATES COVERED
1999 2. REPORT TYPE 00-00-1999 to 00-00-1999
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
An Algorithm for Multipath Computation using Distance-Vectors with
5b. GRANT NUMBER
Predecessor Information
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
University of California at Santa Cruz,Department of Computer REPORT NUMBER
Engineering,Santa Cruz,CA,95064
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)
11. SPONSOR/MONITOR’S REPORT
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12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
14. ABSTRACT
15. SUBJECT TERMS
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a. REPORT b. ABSTRACT c. THIS PAGE 6
unclassified unclassified unclassified
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
2
ProcedureINIT-PATH Procedure
Invokedwhenthenodecomesup. CalledNbTyUPATHtoprocessanevent.
f1.Initializealltables. g f1.IfeventisamessageMfromneighgbor ,
2.Run algorithm. a.Foreachentry[ , , ]inM//Note k , .)
k k
EndINIT-PAPTAHTH Set j adndp . d=Dj p=pj
i i
b.ForeachDdjekst inadtion pwjikth anpentryinM,
AlgorithmPATH Removeexistingjlinks( , )in andaddnew
i
InvokedwhenamessageMisreceivedfromneighbor , link( , , )to ,whnerej Tk
i i i
foranadjacentlinkto haschangedorwhenanodeisk and m j d. Tk d=Djk(cid:0)Dmk
i
initialized. k 2.Iftheeventmis=anpajdkjacentlink-statuschange,update and
1.Run g toupdateneighbortables. clearneighbortablesof ,iflinkisdown. lki
2.RunNTU toupdatemaintables. End k
3.ForeMachTdUestination markedaschanged, NTU
Addupdateentry[j, , ]tothenewmessage .
i i 0 Fig.2. NeighborTableUpdateAlgorithm
4.WithinfiniteamountjofDtimjep,jsendmessage to M
0
eachneighbor. M
EndPATH by and isthepredecessorto ontheshortestpathfrom to
i
asknotifipejdkby . j k
Fig.1. ThePATHAlgorithm 4. jTheNeighborLkinkTable istheneighbor ’sviewofthenet-
i
work as known to and cTonktains link informkation derived from
thedistanceandpreidecessorinformationintheneighbordistance
where istheshortest distance of node todestination . Wecall table.
i
suchaDrojutinggraphtheshortestmultipathifordestination j. 5. Adjacent Link Table stores the cost of adjacent link to each
i
After a seriesof link cost changes whichleave thenetjwork topol- neighbor .Ifalinkisdownitscostislkinfinity.
ogyinarbitraryconfiguration,thedistributedroutingalgorithmshould Nodesexchakngeinformationusingupdatemessageswhichhavethe
worktomodify insuchawaythatiteventuallyconvergestothe followingformat.
shortest multipatShGojf the new configuration, without ever creating a 1. Anupdatemessagecanoneormoreupdateentries. Anupdate
loopin duringtheprocess. entry is a triplet [ , , ], where is the distance of the node
BecaSusGej is node ’s local variable, its value has to be explic- sendingthemessagjedtodpestinationd and isthepredecessoron
k
itly or implDicjitly commkunicated to . If is the value of thepathto . j p
i k
as known to node , the problem niow becDojmkes one of computDinjg 2. Eachmessjagecarriestwoflagsusedforsynchronization: query
i . However, becauseofnon-zeroprop- andreply.
i i i
Sagja(tti)on=deflkajyDs,jkdu(tr)in<g nDetjw(ot)rkgtransitions there can be discrepancies
ifnortmheinvalue o.fTDojkpraevnednittslocooppsy,tDhejirkefaotrei,,awdhdiictihonmalaycocnasutrsaeinlotsopmsutsot C.ACsommpenuttiionngeDdjiearlier, our strategy isto firstdesign ashortest-path
beimposSeGdwj hencomputing i.Weshowlaterthatifthesuccessorset routing algorithmand thenmake the multipathextensions toit. This
at each node for eachdestiSnajtion satisfycertainconditions called subsection describes our shortest-path algorithm PATH and the next
loop-freeinvairiantconditions,thenthjesnapshotattime oftherouting subsection describes the multipath extensions. Figure 1 shows the
graph impliedby i isfreeofloops. Ourstolution tothis pseudocode of PATH. INIT-PATHis called at node startup to initial-
problemSGcoj(nts)istsoftwopaSrtjs(:t()1)computing usingashortest-path ize the tables; distances are initialized to infinity and node identities
i
routingalgorithmcalledPATHand(2)extendiDngjittocompute such toanullvalue. PATHisexecutedinresponsetoaneventthatcanbe
i
thattheysatisfyingloop-freeinvariantconditionsateveryinstaSntj. eitherareceiptofanupdatemessagefromaneighbor ordetectionof
anadjacent linkcost or linkstatus(up/down) change. PATHinvokes
B. NodeTablesandMessageStructures procedureNTU,describedinFigure2, whichfirstupdatestheneigh-
bor distance tables and then updates withlinks where
AsinDBF,nodesexecutingMPATHexchangemessagescontaining i
and . PATTHktheninvokes(mpr;ocned;udre)MTU,
distances todestinations. In additiontothe distancetoa destination, i i i
dsp=eciDfiendki(cid:0)nFDigmukre5, wmhi=chpcnoknstructs bymergingthetopologies
nodesalsoexchangetheidentityofthesecond-to-lastnode,alsocalled i
andtheadjacentlinks . T
predecessor node, which is the node just before the destination node i i
TkThe merging process ilskstraightforward if all neighbor topologies
on the shortest path. In this respect MPATH is akin to several prior
containconsistent linkinformation, but when twoor moreneigh-
algorithms[5],[13],[8],butdiffersinitsspecification,verificationand i
Tbokrslinktablescontain conflictinginformationregarding aparticular
analysisand,moreimportantly,inthemultipathoperationdescribedin
link, the conflict must be resolved. Two neighbor tables are said to
thenextsection.
contain conflicting information regarding a link, if either both report
Thefollowinginformationismaintainedateachnode :
thelinkwithdifferent cost or one reportsthelinkand theother does
1. TheMainDistanceTable contains and , whiere isthe
i i i not. Conflicts are resolved as follows: if two or more neighbor link
distanceofnode todestination andDj isthepjpredecessDorjtodes-
i tables contain conflicting information of link , then isup-
tination onthesihortestpathfrojm topj.Thetablealsostoresfor datedwithlinkinformationreportedbythene(igmh;bonr) thatoTfifersthe
eachdesjtination , thesuccessor seit ji, feasibledistance i, shortestdistancefromthenode totheheadnode kofthelink,i.e.,
2. rTephoerMtedaidnisLtainnkceTjaRbDleji andistwthoeflnaogdsec’Sshjavniegwedoafntdherenpeotrwt-oitr.FkDanjd ltekin+tmDamninker=;omneinwfalykii+stoDbmirkeajkkit2iesNailgw.ayTsieisnafraevobmrrookfelnowineraacdodnrseisss-
i
containslinksrepresenTtedby where isalink neighbor. Because itselfistheheadofthelinkforadjacentlinks,any
withcost . (m; n; d) (m; n) informationaboutainadjacentlinksuppliedbyneighborswillbeover-
3. TheNeigdhbor DistanceTable for neighbor contains and riddenbythemostcurrentinformationaboutthelinkavailabletonode
i
where isthedistanceofneighbor tok ascommDunjikcated .Figure4showsthesignificanceofthetie-breakingrule.
i i
pjk Djk k j i
3
Procedure
i p q 1.CleaMrliTnUktable .
i
2 1 1 1 1 1 1 3 2.Foreachnode T occurringinatleastone i,
a.Find j 6=i . Tk
p q x i y x i y i i i
b.Let MbeIsNuc hthmatinfDjk+lkjk2N g.Tiesare
1 4 3 1 i i
broknenconsistentlyM.NIeNigh=bo(rDjnis+thlenp)referredneighbor
q j j p
fordestination .Foreachlinkn( , , )in ,
(a) i
Addlink j to . j v d Tn
i
1 p 1 x 3 3.Update iw(ijt;hve;acdh)linkTi.
i 2 1 1 j 4.RunDijTkstra’sshortestpatlhkalgorithmon ito
1 1 q 3 y 4 5.fiFonrdenaecwhDdejis,tiannadtiopnij. ,if or changeTdfrom
i i
(b) previousvalue,setchjangeDdjandprjeport-itflagsfor .
End j
Tableshowingthepreferredneighbors.
Destination p q x y j MTU
Distance 2 1 2 3 5 Fig.5. MainTableUpdateAlgorithm
Pref.Nbr p q q p q
Fig.3. Exampleillustratingthemaintableupdateprocedure. (a)Showstheadjacent Let , called the feasible distance, be an ’estimate’ of the dis-
nlienikgshbanodrtnaebilgehsbortablesofnodei. (b)Showsthemainlinktableiaftermergingthe tanceoFfnDojide tonode inthesensethat i isequalto i when
the network isiin stablejstate, but to prevenFtDlojops during pDerjiods of
networktransitions,itisallowedtobetemporarilydifferfrom .
i
p x Dj
i j Loop-freeInvariantConditions(LFI)[16]:
q y
(a)
(1)
i k i
FDj(t) (cid:20) Dji(t) k2N
i p q (2)
i i i
Sj(t) = fkjDjk(t)<FDj(t)g
Theinvariantconditions (1)and(2)statethat,foreachdestination
p q x i y x i y ,anode canchooseasuccessorwhosedistanceto ,asknownto ,
jislessthani thedistanceofnode to thatisknowntojitsneighbors. i
q j j p i j
(b) Theorem1: [16] If theLFI conditions are satisfiedat any time ,
the impliedbythesuccessorsets isloop-free. t
i
p x SPGrjo(otf): Let thenfrom (2)Swje(th)ave
i
i j k2Sj(t)
q y
(3)
(c) i i
Djk(t) < FDj(t)
Atnode ,becausenode isaneighbor,from (1)wehave
Fig.4. Significanceofthetie-breakingRule.(a)Anexamplenetworkwithunitlinkcosts.
k i
(b)Node hasthecostsofitsadjacentlinksandtheshortestpathtreesofitsneighbors
and .Tihedistancesofnodes and from isidenticalthroughbothneighbors
pand .q(c)IfMTUbreakstiesinaxrbitraryymanneirwhileconstructing i,itmaychoospe k i (4)
asqthepreferredneighborfornode andchoose aspreferredneiTghborfornode , FDj(t) (cid:20) Djk(t)
rpesultinginagraphthathasnopathxfrom to . Tqies,therefore,cannotbebrokenyin Combining (3)and (4)weget
arbitrarymanner. i j
(5)
k i
Aftermergingthetopologies,MTUrunsDijkstra’sshortestpathal- FDj(t) < FDj(t)
gorithmtofindtheshortestpathtreeanddeletesalllinksfrom that Eq.(5)statesthat,if isasuccessorofnode inapathtodestina-
i
arenotinthetree. BecausetherecanbemorethanoneshorteTst-path tion , then ’s feasiblekdistance to isstrictlyilessthan the feasible
tree,whilerunningDijkstra’salgorithmtiesareagainbrokeninacon- distanjceofnkode to . Now,ifthesujccessorsetsdefinealoopattime
sistentmanner. Thedistances andpredecessors canthenbeob- withrespecttoi,thjenforsomenode ontheloop,wearriveatthe
i i
tainedfrom . ThetreeiscoDmjparedwiththeprevpijousshortestpath tabsurdrelation j . ThpereforetheLFIconditionsare
i p p
treeandonlyTthedifferencesarethenreportedtotheneighbors.Ifthere sufficientforlooFpD-frje(etd)o<m.FDj(t)
arenodifferences,noupdatesarereported. Eventuallyalltablescon- TheinvariantsusedinLFIareindependentofwhetherthealgorithm
vergesuchthat givetheshortestdistancesandallmessageactivity uses link states or distance vectors; in link-state algorithms, such as
i
willcease.TheDprojofsaregiveninsectionIII. MPDA,the arecomputed locallyfromthelink-statescommuni-
i
cated by theDnjekighbors while in distance-vector algorithms, like the
D. Computing i MPATHpresentedhere,the i aredirectlycommunicated.
InthissubsecStjion,thefinaldesiredroutingalgorithmMPATHisde- Theinvariants(1)and(2)Dsjukggestatechniqueforcomputing
i
rivedbymakingextensionstoPATH.MPATHcomputesthesuccessor suchthatthesuccessor graph fordestination isloop-frSeje(ta)t
sets byenforcingtheLoop-freeInvariant conditionsdescribedbe- everyinstant.ThekeyisdetermSinGinjg(t) inEq.(1)j,whichrequires
i i
lowaSnjdusinganeighbor-to-neighborsynchronization. node toknow ,thedistanceFfrDomj(t)tonode inthetopology
k
i Dji(t) i j
4
ProcedureINIT-MPATH cannot send any update messages or add neighbors to any successor
set. Afterreceivingrepliesfromallitsneighbors thenodeisallowed
Invokedwhenthenodecomesup.
f1.InitializetablesandrunMPATHg. tomodifythesuccessorsetsandreportanychangesthatmayhaveoc-
curredsincethetimeithastransitionedtoACTIVEstate,andifnone
EndINIT-MPATH
ofthedistancesincreasedbeyondthereporteddistance,thenodetran-
Algorithm sitionstoPASSIVEstate. Otherwise, thenode sendsthenext update
InvokeMdPwhAeTnHamessageMisreceivedfromneighbor , message withthe query bit set and becomes ACTIVE again, and the
foranadjacentlinkto haschanged. k wholecycle repeats. If anode receives amessagewiththe query bit
1.Run toupdatkeneighbortablges. setwheninPASSIVEstate,itmodifiesitstablesandthensendsback
2.RunNTU toobtainnew and . anupdatemessagewiththereplyflagset. Otherwise,ifthenodehap-
3.IfnodMeTisUPASSIVEornodDeijisACTpIijVE lastreplyarrived, penstobeinACTIVEstate,itmodifiesthetablesbutbecausethenode
Resetgoactiveflag. ^ isnotallowedtosendupdateswheninACTIVEstate,thenodesends
backanemptymessagewithnoupdatesbutthereplybitset. Ifare-
Foreachdestination markedasreport-it,
a. j plyfromaneighborispendingwhenthelinktotheneighborfailsthen
b.FIfDji minf,DSejit;gRoDacjitgiveflag. animplicitreplyisassumed,andsuchareplyisassumedtoreportan
c. Dji >RDji infinitedistancetothedestination. Becauserepliesaregivenimmedi-
d.RAdDdji[ , Dji , ]tomessage . atelytoqueriesandrepliesareassumedtobegivenuponlinkfailure,
i i 0 deadlocksduetointer-neighborsynchronizationcannotoccur.Eventu-
e.ClearjrepRoDrt-jitpfljagfor . M
ally,allnodesbecomePASSIVEwithcorrectdistancestodestinations,
Otherwise,thenodeisACTIVEjandwaitingformorereplies,
whichweproveinthenextsection.
Foreachdestination markedaschanged,
f. j
4.ForeachFdeDstjiin atiomninfmDarjik;eFdDasjigchanged, III. CORRECTNESSOFMPATH
a.Clearchangedflagjfor The following properties of MPATH must be proved: (1) MPATH
b. j eventuallyconvergeswith givingtheshortestdistancesand(2)the
i i i i
5.FoSreja chfnkeijgDhjbkor<,FDjg successorgraph islooDpj-freeateveryinstantandeventuallycon-
a. . k vergestotheshorSteGstjmultipath.PATHworksessentiallylikePDA[16]
00 0
b.MIfeve ntMisaqueryfrom ,Setreplyflagin . exceptthatthekindofupdateinformationexchangedisdifferent;PDA
00
c.Ifgoactiveset,Setquerykflagin . M exchangeslink-statewhilePATHexchangesdistance-vectorswithpre-
00
d.If non-empty,send to M. decessor information. The correctness proof of PATH is identical to
00 00
6.IfgoaMctiveset,become M k,otherwise PDA and are reproduced here for correctness. The convergence of
become . ACTIVE MPATH directly follows from the convergence of PATH because ex-
End PASSIVE tensionstoMPATHaresuchthatupdatemessagesinMPATHareonly
MPATH delayedafiniteamountoftime.
Fig.6. Multi-pathLoop-freeRoutingAlgorithm
Definitions: The -hop minimum distance of node to node in a
networkisthemininmumdistancepossibleusingapathiof hops(jlinks)
orless.Apaththatoffersthe -hopminimumdistanceisncalled -hop
table thatnode communicatedtoneighbor .Becauseofnon-zero
propaTgaiktion delay,i is a time-delayed versiokn of . We observe minimumpath. Ifthereisnopnathwith hopsorlessfromnodento
that,ifnode delaysTuikpdatingof with until Tinicorporatesthe thenthe -hopminimumdistancefromn to isundefined. An i-hopj
distance iinitstables,then FDsajitisfiesDthjieLFIckondition. minimumntreeofanode isatreeinwhiichjnode istherootanndall
i i paths of hops or less firom the root to any otheir node is an -hop
PseudoDcjode for MPATH isFsDhojwn in Figure 6. MPATH enforces
minimumnpath. n
theLFIconditionsbysynchronizingtheexchangeofupdatemessages
Let denotethefinaltopologyofthenetwork,asseenbyanom-
amongneighborsusingqueryandreplyflags. Ifanodesendsames- G
niscient observer, after all link changes occurred. (We use bold font
sage withaquery bitset, thenthenode must wait untilareply isre-
torefer toquantities in ). Without lossof generality, assume is
ceived from all its neighbors before the node is allowed to send the G G
connected; if isdisconnected, theproof appliestoeach connected
nextupdatemessage. ThenodeissaidtobeinACTIVEstateduring G
componentindependently.
thisperiod. Theinter-neighborsynchronizationusedinMPATHspans
Wesaythatarouter knowsatleastthe -hopminimumtree,ifthe
onlyonehop,unlikealgorithmsthatusediffusingcomputationthatpo-
treecontainedinitsmaiinlinktable isantleastan -hopminimum
tentiallyspanthewholenetwork(e.g.,DASM[17]). i
treerootedat in andthereareatlTeast nodesin nthatarereach-
AssumethatallnodesareinPASSIVEstateinitiallywithcorrectdis- ablefromtheirootG.Notethat issuchthnatthelinksTwiithheadnodes
tancestoallothernodesandthatnomessagesareintransitorpending i
thataremorethani hopsawayTfrom mayhavecoststhatdonotagree
tobeprocessed. Thebehavior ofthenetworkwhereeverynoderuns
withthelinkcostsnin . i
MPATHissuchthatwhenafinitesequenceoflinkcostchangesoccurs G
in the network within a finite time interval, some or all nodes to go
Theorem2: Ifnode hasadjacentlinkcoststhatagreewith and
through a series of PASSIVE-to-ACTIVEand ACTIVE-to-PASSIVE foreachneighbor , irepresentsatleastan -hopminGimum
i
statetransitions,untileventuallyallnodesbecomePASSIVEwithcor- tree,thenaftertheekxeTcuktionofMTU,theminim(unm(cid:0)co1s)ttreecontained
rectdistancestoalldestinations.
in isatleastan -hopminimumtree.
i
LetanodeinPASSIVEstatereceiveaneventresultinginchanges T Proof: TheprnoofisidenticaltotheproofofLemma1in [16]and
initsdistancestosomedestinations. Beforethenodesendsanupdate
isprovidedintheappendixforconvenientreference.
messagetoreportnewdistances,itchecksifthedistance toanydes-
i
tination hasincreasedabove thepreviouslyreported dDisjtance . Theorem3: Afinitetimeafterthelastlinkcostchangeinthenet-
i
Ifnoneojfthedistancesincreased,thenthenoderemainsinPASRSIDVjE work, the main topology at each node gives the correct shortest
i
state. Otherwise, thenode setsthequery flagintheupdate message, pathstoallknowndestinatTions. i
sendsit,andgoesintoACTIVEstate. WheninACTIVEstate,anode Proof: TheproofisidenticaltotheproofofTheorem2in [16]
5
andisprovidedintheappendixforconvenientreference. IV. COMPLEXITYANALYSIS
A node generates update messages only to report changes in dis-
ThemaindifferencebetweenPATHandMPATHisthattheupdate
tancesandpredecessor,soafterconvergencenomessageswillbegen-
messagessentinMPATHaredelayedafiniteamountoftimeinorderto
erated. Thefollowingtheoremsshow thatMPATHprovidesinstanta-
enforcetheinvariants.Asaresult,thecomplexityofPATHandMPATH
neousloop-freedomandcorrectlycomputestheshortestmultipath.
areessentiallythesameandarethereforecollectivelyanalyzed.
Theorem4: ForthealgorithmMPATHexecutedatnode ,let be The storage complexity is the amount of table space needed at a
thetimewhen isupdatedandreportedforthe -thtimie. Ttnhen, node. Each one of the neighbor tables and the main distance ta-
i i
thefollowingcoRnDdijtionsalwayshold. n blehassizeoftheorderN andthemainlinktable cangrow,
i
during execution of MTUO,(jtNojs)ize at most times T . The
i
storagecomplexityisthereforeoftheorder jN j .O(jNj)
(6) i
FDji(tn) (cid:20) minfRDji(tn(cid:0)1);RDji(tn)g ThetimecomplexityisthetimeittakesfoOr(tjhNenjejNtwjo)rktoconverge
i i (7) afterthelastlinkcostchangeinthenetwork. Todeterminetimecom-
Proof: FFrDomj(tt)hew(cid:20)orkiFngDojf(tMn)PATHti2nF[ting;.t6n,+w1e)observethat plexityweassumethecomputationtimetobenegligibleascomparedto
is updated at line 3c when (a) the node goes from PASSIVE- thecommunicationtimes.If isthetimewheneverynodehasthe -
Rto-DAjiCTIVE because of one or more distance increases (b) the node hopminimumtree,becauseetvnerynodeprocessesandreportschangnes
receives the last reply and goes from ACTIVE-to-PASSIVEstate (c) infinitetime isbounded. Let forsome
thenode isinPASSIVEstateandremainsinPASSIVEstatebecause finiteconstantjtn.+F1ro(cid:0)mttnhjeorem3,theconvejrtgne+n1ce(cid:0)titmnejc(cid:20)an(cid:18)beatmost
thedistancedidnotincreaseforanydestination(d)thenodereceives and,henc(cid:18)e,thetimecomplexityis .
thelastreplybutimmediatelygoesintoACTIVEstate. Thereported jNTj(cid:18)he computation complexity is the tiOm(ejNtajk)en to build the node’s
FdfoisDlltoajiwnicsse.uRWpdDhaejtinerdethmaetalniinonsdeeu3niascehianacnAhgCetidTmIdeVuRrEinDpghjitahisseeu,ApCdaTteIVdmEaatyplhianalesseo3.cbB,eEemcqa.ou(ds6ie-) wshiiothtrateTksekitspinaftohrmtrieaetioinnTisiOfr(ojNmitj.hjNeThnj)eerioegpfhoebrroearttihotaenbcaleonsmdTprkuui.ntantUiinopgndaDctoiinmjkgpstlorefaxTiotyni
fiedbythestatementonline3f,whichimpliesFEDq.ji(7). Tis iO(jN jjNjilog(jNj)) .
OT(hjeNcojmjNmju+nicjNatiojjnNcjolomgp(ljeNxijt)y)is the number of update messages
Theorem5: (Safety property) At any time , the successor sets required for propagating a set of link-cost changes. The analysis for
computedbyMPATHareloop-free. t multiplelink-costchangesiscomplexbecauseofthesensitivitytothe
i
Sj(t)Proof: Theproofisbasedonshowingthatthe and com- timingofthechanges. So,weprovidetheanalysisonlyforthecaseof
i i
putedbyMPATHsatisfytheLFIconditions. Let FbDetjhetimSejwhen singlelink-costchange. Anoderemovesalinkfromitsshortestpath
isupdatedandreportedforthe -thtime. Thtenproofisbyinduc- tree if only a shorter path using two or more links is discovered and
i
RtioDnjontheinterval . LetthenLFIconditionbetrueuptotime once discovered the path is remembered. Therefore, a removed link
,weshowthat [tn;tn+1] will not be added again to the shortest path which means that a link
tn canbeincludedanddeletedfromtheshortest pathbyanodeatmost
(8)
i k onetime. Becausenodesreporteachchangeonlyoncetoeachneigh-
FDj(t) (cid:20) Dji(t) t2[tn;tn+1]
FromTheorem4wehave bor,anupdatemessagecantravelonlyonceonalinkandthereforethe
number ofmessages sent byanodecanbeat most . Forcer-
(9)
FDji(tn) (cid:20) minfRDji(tn(cid:0)1);RDji(tn)g taintopologiesandsensitivelytimedsequenceoflinkOc(ojsEtjc)hangesthe
i i i (10) amountofcommunicationrequiredbyPATHcanbeexponential.Hum-
FDj(tn+1) (cid:20) minfRDj(tn);RDj(tn+1)g
(11) blet [8] provides an example that exhibits such behavior, and though
i i
FDj(t) (cid:20) FDj(tn) t2[tn;tn+1) PATHisdifferentfromtheshortest-pathalgorithmpresentedinthatpa-
Combiningtheaboveequationsweget
per,wenotethatPATHisnotimmunefromsuchexponentialbehavior.
(12) However,webelievesuchscenariosrequiresensitivelytimedlink-cost
i i i
FDj(t) (cid:20) minfRDj(tn(cid:0)1);RDj(tn)g t2[tn;tn+1] changes which are veryunlikely tooccur inpractice. If necessary, a
Let be the time when message sent by at is received and
0 smallhold-downtimebeforesendingupdatemessagesmaybeusedto
processted byneighbor . Becauseofthenon-izerotpnropagationdelay
preventsuchbehavior.
acrossany link, issukch that andbecause is
0 0 i
modifiedat antdremainsunchtanng<edtin< tn+1 weget RDj
tn (tn;tn+1) V. CONCLUDINGREMARKS
(13)
i k 0 Wehavepresentedthefirstroutingalgorithmbasedondistancein-
RDj(tn(cid:0)1) (cid:20) Dji(t) t2[tn;t)
i k 0 (14) formationthat providesmultiplepathsthat neednot haveequal costs
RDj(tn) (cid:20) Dji(t) t2[t;tn+1] and that are loop-free at every instant, without requiring inter-nodal
FromEq.(13)and(14)weget
synchronizationspanningmorethanonehop. Theloop-freeinvariant
i i k (15) conditions presented here arequite general and can be used with ex-
Fromm(1in2)faRnDd(j1(5tn)(cid:0)th1e);inRdDucjt(ivtne)sgtep((cid:20)8)foDllojiw(ts).Betc2au[sten;tn+1] istinginternetprotocols. ThemultiplesuccessorsthatMPATHmakes
i
at initialization, from induction we have that FDj(t0)(cid:20) availableateachnodecanbeusedfortrafficload-balancing,whichas
Djki(t0)forall . GiventhatthesuccessorsetsarecomputeFdDbjia(ste)do(cid:20)n wehaveshown using other algorithms(MPDA[16]) isnecessary for
k minimizingdelaysinanetwork. MPATHcanthereforebeusedasan
Dji(t,)itfollowtsthattheLFIconditionsarealwayssatisfied.According
i alternativetoMPDAtogetsimilarperformance. Inafutureworkwe
FtoDthjeTheorem1thisimpliesthatthesuccessorgraph isalways
intend to compare the performance of the three multipath routing al-
loop-free. SGj
gorithmsMPATH,MPDAandDASM[17]intermsofcontrolmessage
Theorem6: (Livenessproperty)Afinitetimeafterthelastchange overheadandconvergencetimesandanalyzetheirrelativemerits.
in the network, the give the correct shortest distances and
Dji. Sji = REFERENCES
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[3] J.J.Garcia-Luna-Aceves. Loop-FreeRoutingUsingDiffusingComputations. IEEE/ACMTrans. element i. Let be the highest priority neighbor for which
Networking,1:130–141,February1993. j 2 Mn K . Atmost nodesin
i i i i i i
[4] J.J.Garcia-Luna-AcevesandJ.Behrens.Distributed,scalableroutingbasedonvectorsoflinkstates. DcajnKha+velKles=semr oirnefqDujakl d+isltkajnkce2thNang which immpli(cid:0)es1path TK
IEEEJournalonSelectedAreasinCommunications,October1995. existswithatmost hops.Let bejtheneighborof in K.T;henj
[5] J.J.Garcia-Luna-AcevesandS.Murthy.Apath-findingalgorithmforloop-freerouting.IEEE/ACM i
Trans.Networking,February1997. thepath m(cid:0)ha1satmost v hops.Because j isTaKtleasta
[6] J.J.Garica-Luna-AcevesandM.Spohn. Scalablelink-stateinternetrouting. Proc.International -Khop;mvin!imujmtree,thelimnk(cid:0)1 mustagreeTwKiith . Since
ConferenceonNetworkProtocols,October1998. (n(cid:0)1) , from indvuc!tionj hypothesis thereGis apath
[7] C.Hendrick.RoutingInformationProtocol.RFC,1058,june1988. i i i i
[8] P.A.Humblet. AnotherAdaptiveDistributedShortestPathAlgorithm. IEEETrans.Commun., DvK +inlK <suDchjtKha+tthleKlengthisatmost .
39:995–1003,June91. i;Novw wGeinneedtoshow thatthepreferredDnein;ivghbor for isalso ,
[9] J.M.JaffeandF.H.Moss.AResponsiveDistributedRoutingAlgorithmforComputerNetworks.
IEEETrans.Commun.,30:1758–1762,July1982. sothatthelink willbeincludedintheconstructionvof i,thKus
[10] P.M.MerlinandA.Segall. AFailsafeDistributedRoutingProtocol. IEEETrans.Commun., ensuringtheexivst!encjeofthepath in . IfsomeneigGhbnor
27:1280–1287,September1979. otherthan isthepreferredneighib;orfjor GthinenoneofthefollowiKng0
[11] J.Moy.OSPFVersion2.RFC,1247,August1991.
two conditKions should hold: (a) v or (b)
[[1132]] BRA..uPtRoeanrjloammgaoonpu.asFlaSanyuslattn-etdmolMse.r.aInnFttaebirmrnoeaatndw.coaArsktiRonfgers:opRuoetnisnseigaverinchDfoiarsmntrdiabtEiuoxtnepd.erCSieohnmocrpetue,ts2et:-r5PN1a–eth6tw9Ro,orMkustairancnghdA1I9Slg9Do1Nr.i,th7m,1w98it3h. Dvi.K0+lKi 0 =DviK+lKi andprioDrivitKy0of+Kl0Kii0sg<reaDterviKtha+nplKriiorityof
[14] A.Segall. Optimaldistributedroutingforvirtualline-switcheddatanetworks. IEEETrans.Com- KCase(a):Because itfollowsthatthepath
[15] Jm.uSnp.,in2e7l:l2i0a1n–d2R09.,GJaalnluagareyr.1E97v9en.tDrivenTopologyBroadcastwithoutSequenceNumbers. IEEE in isgreaDtejirKth+anlKico(cid:20)stoDfjiK0+lKii0n whichimpliesthat
Trans.Commun.,37:468–474,1989. v ;isjnotTaKi0 hopminimumtreev–!acojntradGictionofassumption.
[16] S.VutukuryandJ.J.Garcia-Luna-Aceves. ASimpleApproximationtoMinimumDelayRouting. TTKhie0refore (n(cid:0)1) .
Proc.ofACMSIGCOMM,1999. i i i i i
[17] W.T.ZaumenandJ.J.Garcia-Luna-Aceves.Loop-FreeMultipathRoutingUsingGeneralizedDif- Case (bD):vKLe+t lKb=emthiensfeDt vokf+neilgkhjkbo2rsNthagt give the minimum
fusingComputations.Proc.IEEEINFOCOM,March1998. distancefor ,i.e.Q,fjoreach ,
i i i i
. Similajrly, let be skuc2hQthjatDfojrke+aclhk =minf,Djk+lkjk 2
APPENDIX i i i
N g Qv . If and k 2 Q,vthDenvkit+folllkow=s
i i i
ProofofTheorem2: mfroimnfsDamvke+arglkujmken2t aNs ign casek(a2) tQhavt k 2=inQj is greater than
letLetHbinedtheenosetetoanfnno-dheosptmhaitnaimreuwmitthreineroohtoepdsaftronmodeiininG. aLnedt –coastcoofnvtra!dictjioinnoGf aimsspulmyipntgioTn.ki Bisencoautsave(;n(cid:0)hja1s)-thhTeokiphimghineismtupmriotrrietye
Mdinenotethedistanceof to in .Lent bethecoistofHthinelink among all members of and Kand , also has
Din;j .Node iscalledthehieadjoftHheinlink dij .Thenotation thehighestpriorityamoQngjallmQemvb(cid:18)ersQofj . kTh2ereQfovreK .
iin!dicjatesapaithfrom to ofzeroormoreil!inkjs;ifthepathhais;zeroj Also, fromthesameargument itcanbeinfQervredthat Qv (cid:18). TQhjis
links,then . Theilenjgthofpath isthesumofcostsofall provesthat willbeincludedintheconstructionoKf 2i.QBvecause
linksintheipa=th.j i ; j Dannid;vle+ngdthvojvf=!Djni;jiinnG,iwshleesrsetdhavnjoisrtehqeuafilntaolcostofrfoGlminnkinvdu!ctiojn,
Property1: From the principle of optimality (the sub-path of a hypothesis,wie;havvelenGginthof in lessthDanni;vorequalto .
shortestpathbetweentwonodesisalsotheshortestpathbetweenthe Thisprovespart1ofthetheoreim;. j Gin Din;j
endnodesofthesub-path),if and aretwo -hopminimumtrees
0
rootedatnode and and HareseHtsofnodesnthatarewithin hops ProofofTheorem3:
0
ftForoormeia;ijc.ihnFjHor2anMidinHt,0hMreesil;epjnegctthiMveoilf;yjp,.aththeniM;=jiMnb0o=thMHinanadndHM0iinsne(cid:21)qunal. iin,TtThiheiesnpaetrtwoleooafrskitsnhba-yhsoianptdmmucointsiitomnuomnttrnele,i.nthkBesegcwlaohubesareeltthimelieosnwnguhemesntblefoororopef-afnrcoehedneposadtihne
DLent i h(cid:21)n Dhi,(cid:20)whDerne i isthesetofnodesin i. Because thenetwork, isthetimNe(cid:0)wh1eneverynodeNhastheshortestpathto
isaAtlea=stSank2NiAk -hopmiAnikmumtreeandnode cTankappearat everyothernotNde(cid:0).1Weneedtoshowthat isfinite.Thebasecaseis
i
Tmkostonceineach(nof(cid:0)1),each hasatleast uniiqueelements. ,thetimewheneverynodehas1-hopmtNin(cid:0)i1mumdistanceandbecause
i i
Therefore, hasatlAeakst Aeklements. n(cid:0)1 tth1eadjacentlinkchangesarenotifiedwithinfinitetime, . Let
i
Let Abethesetof n(cid:0)1nearestelementstonode in . That forsome . Giventhatthepropagationdelta1y<sar1efinite
i i
is, Mn , n(cid:0),1andforeach and i A , tenac<hn1odewillhavnee<achNofitsneighbors -hopminimumtreeinfinite
i i i i i i
Mn (cid:18)A jMnj=n(cid:0)1 j 2Mn . v2A (cid:0)Mn timeafter . FromTheorem2wecansneethatthenodewillhaveat
i i i i i i
miTnofDprjokve+thlekjtkhe2orNemgit(cid:20)ismsuifnfifciDenvtkto+plrkojvke2thNefogllowing: leastthe tn -hopminimumtreeinfinitetimeafter . Therefore,
1. Let representthegraphconstructedbyMTUonlines2and3. (n.+Fro1m) inductionwecanseethat .tn
i
(i.e.,GbneforeapplyingDijkstrainline4). Foreach there tn+1 <1 tN(cid:0)1 <1
isapath in suchthatitslengthisatmostj 2 M. ni ProofofTheorem6:
2. Afterrunin;ingjDijkGstinraon online4inMTU,theDresni;ujltingtree TheconvergenceofMPATHfollowsdirectlyfromtheconvergence
i
isatleastan -hopminimuGmntree. of PATH because the update messages in MPATH are only delayed
Letusfirstassumnepart1istrueandprovepart2becauseitissimple. a finite time as allowed at line 4 in algorithm PATH. Therefore, the
From the statement in part 1 for each node there is a path distances inMPATHalsoconvergetoshortestdistances. Because
i i
iDB;eijckasjutrsiaen,tGwheinerewcaaintrheilnefnegrtthhaentroemdieosssatinDpaint;jh.ii,Int;htehetjrreewjesiuct2hlotnilnsMegtnrgtunrtcehteeadatfhtmearosrsauttnDlneianin;sjgt. crahtaendgebsytatoDfhteeDjrnjiceoaignrehvbearolgwresanyicnse.trheFeprioorrmtteadblilnteoest3ihnaeifinnneMiitgePhAtbioTmrHes,aDwndjiekaorb=eseinrDvceojkrtphfooa-rt
nodes includingnn(cid:0)od1e . FromMprnoperty 1, it follows that the tree kwh2enNnoide becomespassive holdstrue.Becauseallnodes
nconstructedisatleastan i-hopminimumtree. arepassiveiatconvergenceitFfoDllojiw=stDhajit
We now prove part 1.nOrder the nodes in in non-decreasing i i i
i . Sj = fkjDjk < FDj;k 2
order. Theproof isbyinductiononthesequenMcenofelementsin . i k i i
i N g=fkjDj <Dj;k2N g
Thebasecaseistruebecausefor ,thefirstelementof , Mn
i i
and m1 AsinductionhypothMesis,llmet1th=e
mstaitnemflekinjkth2oldNfiogrthefilmirs1t =Di1;mel1e:mentsof . Considerthe -th
i
m(cid:0)1 Mn m