of Minimizing Eletroni Routing
in Wavelength Routed Optial Rings
Rudra Dutta George N. Rouskas
TR-2000-11
September27,2000
Abstrat
Weonsidertheproblemofdesigningavirtualtopologytominimizeeletroniroutinginwavelength
routedoptialrings. ThefullvirtualtopologydesignproblemisNP-hardevenintherestritedasewhere
thephysialtopologyisaring,andvariousheuristishavebeenproposedintheliteratureforobtaining
goodsolutions,usuallyfordierentlassesofprobleminstanes. Wepresentanewframeworkwhihan
beusedtoevaluatetheperformaneofheuristis,andwhihrequiressigniantlylessomputationthan
evaluatingtheoptimalsolution. Thisframeworkisbasedonageneralformulationofthevirtualtopology
problem,anditonsistsofasequeneofbounds,bothupperandlower,inwhiheahsuessivebound
is at leastas strongas the previous one. Thesuessive bounds take larger amountsof omputation
toevaluate,andthenumberofbounds tobeevaluatedfor agivenprobleminstaneisonlylimitedby
theomputational poweravailable. Thebounds are basedondeomposingthe ringintosets of nodes
arrangedinapath,andadoptingthe loallyoptimaltopologywithineahset. Whileweonlyonsider
theobjetiveofminimizingeletroni routinginthis paper, ourapproahtoobtainingthesequeneof
boundsanbe appliedtomanyvirtual topologyproblemsonrings. Theupperboundsweobtain also
provideausefulseriesofheuristisolutions.
Keywords: VirtualTopology,LogialTopology,WavelengthDivisionMultiplexing(WDM),
WavelengthRoutedNetwork,Lightpath,All OptialNetwork,OptialRing,Bounds,Heuristis
DepartmentofComputerSiene
NorthCarolinaStateUniversity
Raleigh,NC27695-7534
Inreentyears,wavelengthrouted optialnetworkshavebeenseento beanattrativearhiteturefor the
nextgenerationofbakbonenetworks. Thisisduetothehighbandwidthinberswithwavelengthdivision
multiplexing (WDM) and the ability to trade o some of the bandwidth for elimination of eletro-opti
proessingdelaysusingwavelengthrouting[5℄. Ithasalsobeennotedin literaturethat,atleastintheshort
term,physialtopologiesintheformsofringsareofmoreinterestbeauseofavailablehigherlayerprotools
suhasSONET/SDH[7,6,13℄.
Of late, twoonerns havelearly emerged in the treatment of optial ringnetworks using WDM and
wavelength routing. First, it has been reognized that the the ost of network omponents, speially
eletro-opti equipment and SONETadd/drop multiplexers(ADMs), isa moremeaningful metrifor the
network ortopologyrather than thenumberof wavelengths. Seond, earlier studiesof wavelength routed
networkshadassumedthatindividualtraÆdemandsareomparabletothewavelengthbandwidth,andthis
assumptionwasreetedinthedesignoflogialtopologies[4,9,12,1,5℄. However,itisnowevidentthatthe
independent traÆstreams that wavelengthrouted networkswill arryare likely tohavesmall bandwidth
requirements ompared even to the bandwidth available in a single wavelength of a WDM system. This
assumptionis furthersupportedbythefatthat thenumberofdierenttraÆomponentsin anetworkis
likelytobemuhlargerthanthenumberofwavelengthsavailable. Thesetwoissuesgiverisetotheonept
oftraÆgrooming,whihreferstotehniquesusedtoombinelowerspeedtraÆomponentsontoavailable
wavelengthsinordertomeetnetworkdesigngoalssuhasostminimization.
Theproblem of designing logialtopologies that minimizeost asmeasured bythe amountof
eletro-opti equipmenthasreentlyreeivedmuh attentionin theliterature. In[11℄, theproblem ofwavelength
assignment to agiven set of lightpaths is onsidered. The disussion fouses on how limited wavelength
onversionaets the apability of the network, and spei asesare disussed with onstrutiveproofs.
Several dierent virtual topologiesfor ring networks are disussed in [7℄ in the light of the twin issuesof
traÆgroomingandnetworkost,andharateristimetrisarederived. Thestudyin[6℄omplements[7℄by
addressingonlythewavelengthassignmentissue,withthegoalofminimizingthenumberofSONETADMs.
Theoneptoflightpathsplittingin designingavirtualtopologyisdisussedandheuristisforwavelength
assignment are developed based on this onept. In [14℄, the problem of grooming traÆ is onsidered
both for unidiretional and bidiretional rings, with a primary goal of either minimizing the number of
SONET ADMs orthenumberof wavelengths. Thestrategypresentedis torstonstrut irlesfrom the
giventraÆ omponentsandthento groomtheseirles. AlgorithmsforexatsolutionsforuniformtraÆ
and heuristisfor theNP-hard non-uniformtraÆ ase arepresented. A similar approah of onstruting
irles rstis takenin [15℄, but nowthe resultantirlesare sheduledin asequeneof virtualtopologies.
Heuristialgorithms tominimizenetwork ostbygroomingarepresentedin[3℄, forspeial traÆpatterns
suh asuniform, ertain ases ofross-traÆ, andhub. In [2℄, theonept of at-allowabletraÆ pattern
is disussed, in whih the traÆ for eah node-pair is onstrained to be at most t. A bidiretional ring,
but with symmetri traÆ, is onsidered for dynami traÆ. A heuristi algorithm based on a bipartite
wellasthewavelength assignmentproblem. An Euler-traildeompositionoftheringnetworkispresented,
and a heuristi whih uses oneof the heuristis of [6℄and performs rerouting of lightpaths aswell, based
on the Euler-traildeomposition, is presented. Several ring arhitetures for dynami traÆ, onforming
to standard SONET self-healing rings, are reviewed in [13℄. The harateristis of the arhitetures are
disussedbothfortheaseoflimitedandunlimitednumberofavailablewavelengths.
As has been noted in literature [5, 14℄, the problem of logial topology design is NP-hard even for a
physial ring topology, and obtaining an exat solution requires signiant amount of omputation even
for modestsized rings. Heuristiapproahesare neededfor pratialpurposesand have been reported in
literature [14, 15, 3, 6, 8℄. To evaluate the performane of suh heuristi approahes when the optimal
solutionisnotavailable,ahievabilityboundsareuseful.
WepresentanewframeworkforomputingboundsfortheproblemofoptimaltraÆgroominginphysial
ringtopologies. Theframeworkisbasedontheideaofdeomposingtheringintopathsegmentsonsistingof
suessivelylargernumberofnodes. Thepathsegmentsaresolvedindependently,andtheindividualresults
areappropriatelyombinedtoobtainboundsontheoptimalvalueoftheobjetivefuntion. Inthismanner,
weobtainaseriesofbounds,bothlowerandupper,inwhiheahboundisatleastasstrongastheprevious
one. Therstfewboundsinthesequenerequiretrivialomputationaleort. Althoughsubsequentbounds
take suessivelylargeromputational eorts to determine, evenseverallater bounds requiresigniantly
lessomputationaleortthanthefullsolutiondoes,dependingontheprobleminstane. Weshowthatthis
result is due to the fat that solving apath segmentexatly takes an amount of time whih is orders of
magnitudelessthanthatrequiredforsolvingaringofthesamenumberofnodes. Theproblemweonsider
isverygeneral,aswedonotimposeanyonstraintsonthetraÆpatterns. Furthermore,theupperbounds
we derive are based on atual feasible topologies, so our algorithm for obtaining the upper bounds is a
heuristi forthe generalproblem of traÆ grooming. Finally, although weillustrate ourapproahusing a
spei formulationof theproblem, itis straightforwardtomodifyit to applyto awiderange ofproblem
variantswithdierentobjetivefuntionsand/oronstraints.
Therestofthepaperisorganizedasfollows. Theproblemofdesigningavirtualtopologythatminimizes
eletroni routing in a ring network is formulated as an Integer Linear Program (ILP) in Setion 2. In
Setion3wedesribethedeompositionoftheringnetworkintopathsegments,andweshowhowtoobtain
anoptimalsolutiontothepathsegmentsfromasimpliedversionoftheILP.Next,inSetion4wedesribe
howtoombineeÆientlythesolutionstoindividualsegmentstoyieldastrongsequeneoflowerandupper
0
1
(a)
i
i-1
i+1
.
.
.
N-1
.
.
.
from i-1
(b)
i
to i+1
Figure1: (a)AnN-nodeunidiretionalring,and(b)detailofanodewithaWADM
2 Problem Formulation
We onsideraunidiretionalring RwithN nodes 1
, numberedfrom0to N 1,asshownin Figure 1(a).
The berlink between eah pairof nodes ansupport W wavelengths, andarries traÆ in thelokwise
diretion; in other words, data ows from a node i to the next node i1 on the ring, where denotes
addition modulo-N. (Similarly weuse to denote subtrationmodulo-N.) The links ofRare numbered
from 0to N 1, suh that the link from nodei to node i1is numberedlink i. Eah node in the ring
is equipped with a wavelength add/drop multiplexer (WADM) (see Figure 1(b)). A WADM an perform
three funtions. It anoptially swithsome wavelengthsfrom the inoming link ofa nodediretly to its
outgoing link withouttheneedfor eletro-optionversionof thesignalarried onthewavelength. It an
alsoterminate(drop)somewavelengthsfromtheinominglinktothenode;thedataarriedbythedropped
wavelengths is onverted to eletroni form and undergoes buering, proessing, and possibly, eletroni
swithing at the node. Finally, the WADM an also add some wavelengths to the outgoing link; these
wavelengthsmayarrytraÆoriginatingatthenode,ortheymayarrytraÆoriginatingatpreviousnodes
in theringandeletroniallyswithedat thisnode.
Weassume that estimatesof the node-to-node traÆ are available,requiringthe designof avirtual or
logial topology (see the disussion below) onsisting of a set of stati lightpaths. These estimates may
1
Wedesribetheworkingpartofthering. Itisassumedthatthereisaprotetionpartforself-healingorfault-tolereneas
suhhangestakeplaeoverhumantime sales. Inthis paperwedonotonsider thedynamisenarioin
whihrequestsforlightpathsortraÆomponentsarereeivedontinuouslyduringoperation. Inthisase,
dynami groomingorlightpathalloation algorithmsare required, and theperformane metriof interest
is somefuntion of the bloking harateristisof the network. This latter porblem has been onsidered
elsewhere[16,17℄.
ThetraÆ demands betweenpairs of nodesin thering are given in thetraÆ matrixT = [t (sd)
℄. We
assume that the network supports traÆ streams at rates that are a multiple of some basi rate (e.g.,
OC-3). We let C denote the apaity of eah wavelength expressed in units of this basirate. Thus, C
denotesthemaximumnumberoftraÆunitsthatanbemultiplexedonaWDMhannel(wavelength). For
example, ifeahwavelengthruns at OC-48ratesand thebasirate isOC-3, then C =16. Eah quantity
t (sd)
2f0;1;2;gis also expressed in terms of thebasi rate, and it denotes thenumber of traÆ units
that originateatnodes andterminate atnoded.
Giventheringphysial topology,alogial topologyisdenedbyestablishinglightpathsbetweenpairsof
nodes. A lightpath isadiret optial onnetiononaertain wavelength. More speially, ifalightpath
spans morethan onephysial link in the ring, its wavelength is optially passed through by WADMs at
indermediatenodes, thus, thetraÆstreamsarriedbythelightpathtravelin optialformthroughout the
pathbetweentheendpointsofthelightpath. Weassumethat ringnodesarenotequippedwithwavelength
onverters,thereforealightpathmustbeassignedthesamewavelengthonallphysiallinksalongitspath.
An importantproblem thathasreeivedonsiderable attentionin theliterature isthedesignof logial
topologies that optimize a ertain performane metri. The performane metri of interest in this work
is the amount of eletroni forwarding(routing) of traÆ streams, sinesuh forwardinginvolves
eletro-opti onversionand addedmessagedelayandproessorloadat theintermediatenodes. There isalso the
possibilityof inreasedbuerrequirementandqueueing delay. Inaglobalsense, thismeansthat wewant
toreduethenumberoflogialhopstakenbytraÆomponents,individuallyorasawhole. Foreahnode,
it also means that wewantto redue theamount of traÆ that node hasto storeand forward. Thus we
havetwoalternativegoals,oneistominimizethetotaltraÆweightedlogialhopsin thenetwork,andthe
other is to minimize the maximum numberof traÆ omponents eletronially routed at anode. In this
paperwehavehosentoonentrateontheformer. AsingletraÆunitmaynotbebifurated,butdierent
traÆ units forthesamesoure-destination traÆomponentarelookedupon asseparatetraÆ andmay
beassigneddierentlogialroutes. Anotherpossiblequantityofinterestisthenumberofwavelengthseah
WADMisrequiredtoaddordropduetolightpathterminations. Thiswouldorrespondtotheminimizing
of the numberof SONET ADM's asin [14, 3, 6, 8℄. It should be noted that adding and dropping more
wavelengthswillresultinalargernumberoflogialhopsfortraÆ omponentsandthusthisonernisto
someextentsinorporatedintheeletroni forwardingissuewementionedabove.
Welett(l)denotetheaggregatetraÆloadonthephysiallinkl(fromnodeltonodel1)ofthering. The
valueoft(l)anbeeasilyomputedfromthetraÆmatrixT byaddingupalltraÆomponentst (sd)
suh
traÆ loadt(l)dueto thetraÆ fromsourenodesto destination nodedis denotedbyt (sd)
(l). Ifoneor
morelightpathsexist from node i tonodej in thevirtualtopology,the traÆ arriedbythose lightpaths
is denoted by t
ij
. The omponent of this loaddue to traÆ from soure node s to destination noded is
denotedbyt (sd)
ij
. Inourformulation,weforbidatraÆomponenttobearriedompletelyaroundthering
beforebeingdeliveredatthedestination,thuseahtraÆomponentantraverseagivenlinkatmostone.
This isreasonablebeausearryingthetraÆ omponent aroundtheringonsumesmorebandwidth,and
islikelytorequiremorelogialhops.
In our formulation we allow for multiple lightpaths with the samesoure and destination nodes. We
denotethelightpathountfromnodeitonodej byb
ij
,takingitsvaluefromf0;1;2;;Wg. Wealsodene
the potential lightpath set for alink to bethe set of lightpathsthat would passthrougha given link,and
denote itbyB(l)=f(i;j)j lightpath(i;j),ifitexisted, would passthroughlinklg.
With theabovenotation, weannow formulate theproblem ofdesigning avirtualtopologyfor aring
network suh that the total amount of eletroni routing at the ring nodes is minimized. The following
formulationasanIntegerLinearProblem(ILP)onsistsof O(N 4
+N 2
W)onstraintsandO(N 4
+N 2
W)
variables,where N isthenumberofnodesinthering,andW thenumberofwavelengths.
Given:
The physial topology, aunidiretionalringRofN nodes
The traÆmatrix T =[t (sd)
℄; s;d2f0(N 1)g
t (sd)
2f0;1;2;g
t (ss)
=0;8s
The wavelength limit W whihisthenumberofdistintwavelengthseahberlink anarry.
Find:
Virtualtopology, intermsoflightpathindiatorsb
ij
,lightpathwavelengthindiators (k )
ij
,andthetraÆ
routingvariablest (sd)
ij .
Subjet to:
TraÆ Constraints
t (sd)
ij
t (sd)
(i); 8(i;j);(s;d)
(1)
t (sd)
ij 2
f0;1;2;g; 8(i;j)
(2)
X
(i;j)2B(l) t
(sd)
ij =
t (sd)
(l); 8(s;d);l
t ij = sd t (sd) ij
; 8(i;j)
(4) t ij b ij
C ; 8(i;j)
(5) X j t (sd) ij X j t (sd) ji = 8 > > < > > : t (sd)
; s=i
t (sd)
; d=i
0; s6=i;d6=i 9 > > = > > ;
8i;(s;d) (6)
Wavelength Constraints X (i;j)2B(l) b ij W; 8l (7) X k (k ) ij = b ij
; 8(i;j)
(8) X (i;j)2B(l) (k ) ij
1; 8l;k
(9) Tominimize: min 0 X s;d;i;j2f0(N 1)g t (sd) ij X s;d2f0(N 1)g t (sd) 1 A (10)
Intheabove,thetraÆonstraint(1)ensuresthatalightpathanarrytraÆforasoure-destination
nodepaironlyifitisinthephysialrouteofthetraÆomponent. Constraint(3) statesthatthephysial
traÆonalinkduetoasoure-destinationnodepairmustbeequaltothesumofthetraÆonalllightpaths
passing throughthat link due to that nodepair. Constraints(4, 5) dene thetotal traÆ on alightpath
and relate it to the lightpath ount, respetively. Beause of the denition of the quantities t (sd)
(l), the
onstraints (1, 3) together ensure that no traÆ omponent an be routed ompletely around the ring
before beingdeliveredat thedestination node. Constraint(6) is anexpression of traÆ owonservation
at lightpath endpoints. Among the wavelength onstraints, onstraint (7) expresses the bound imposed
by numberof wavelengthsavailable, (8)relates thewavelength indiatorsto the lightpath ounts, and(9)
ensuresthatnowavelengthlashanour.
Thefull virtual topologyproblem dened above anbe thought of asonsisting of thefollowingthree
subproblems[5℄ 2
:
1. Topology Subproblem: Determine the virtual topology to be imposed on the physial topology,
thatisdeterminethelightpathsintermsoftheirsoureand destinationnodes.
2. Wavelength Assignment Subproblem: Determine the wavelength eah lightpath uses, that is
assignawavelengthtoeahlightpathinthevirtualtopologysothatwavelengthrestritionsareobeyed
foreahphysial link.
2
For the general virtual topology problem,lightpathrouting overthe physialtopology isanothersubproblem [5℄. Ina
unidiretionalringphysialtopology,thereisonlyonephysialroutingpossibleforeahlightpath,sothissubproblemistrivial
is,routethetraÆstreamsbetweensoureanddestination nodesoverthevirtualtopologyobtained.
Theframeworkwepresentbelowisbasedontheaboveformulationwehavehosen,butthisformulation
isnotessentialforit. Theframeworkanbeadaptedtomanyvariationsthatarepossibleintheformulation.
Theremaybemultipleberlinksbetweensuessivenodes,andthenodesmaybeequippedwithwavelength
routers insteadofWADMs. Hardwareforwavelength onversion,limitedorotherwise,maybe availableat
the nodes[11℄. A physial hoplimit forlightpathsmay be imposed onfeasibletopologies, dueto physial
berharateristisorOAMissues. Theringmaybebidiretional,eitherwithsomesimpleroutingstrategy
(suhasshortest-path,asin[14,6℄)thatallowsustoonsideritastwounidiretionalrings,orthelightpath
routing(eitherlokwiseorounter-lokwise)maybeintegratedaspartoftheoptimizationproess(asin
[8℄). Inalltheseases,theobjetivemaybetominimizeeletronirouting. Theframeworkwepresentmay
beextended, in astraightforwardmannerfor someofthese ases,and with someenhanementsin others.
When the objetive is not an additive funtion as in our formulation but some other funtion suh as a
min-max type(minimize eletroni routingat the node with maximumeletroni routing) ora quantied
version(minimizenumberofwavelengths added/dropped, numberofSONET ADMs, et.) ourframework
analsobeadaptedto obtainboundsfortheoptimalvalueoftheobjetivefuntion.
3 Path Deomposition of a Ring Network
3.1 Denition of Deomposition
WeonsideraringRwithN nodeslabeled0(N 1),inorder,andtraÆmatrixT. Wedeneasegment
of length n;1 n N, starting at node i;0 i < N, as the part of the ring R that inludes the n
onseutivenodesi;i1;i2;;i(n 1), andthelinksbetweenthem.
Wedene adeomposition ofringRaroundasegment oflengthnstartingatnode iasapathP (i)
n that
onsists ofn+2nodes andn+1links asfollows: thennodesand n 1links ofthesegmentofring Rof
lengthn startingatnodei, anewnodeS andalink fromS toi, andanewnodeD andalink fromnode
i(n 1) to D. Wealsoreferto P (i)
n
asann-nodedeomposition ofringRstarting at nodei. Figure 2
showssuhadeomposition.
AssoiatedwiththedeompositionP (i)
n
isanewtraÆmatrixT
P (i)
n =
h
t (sd)
P (i)
n i
,s;d2fi;i1;;i(n
0
1
i-1
i
i+1
N-1
i+n-1
i+n
(a)
..
.
...
.
.
. .
i
i+1
i+n-1
S
D
(b)
. .
.
Figure 2: Ann-nodedeomposition: (a)originalringRwithN nodes,and(b)adeompositionP (i)
n
ofthe
ringaroundasegmentoflengthnstartingat nodei
t (sd) P (i) n = 8 > > > > > > > > > > > > > > < > > > > > > > > > > > > > > : t (sd)
; if isdi(n 1)
P
j=2fi;i1;;d 1g t
(jd)
; if s=S;idi(n 1)
P
j=2fs1;s2;;i(n 1)g t
(sj)
; if d=D;isi(n 1)
t
pass thru
(i;n); if s=S;d=D
0; if 8 > > > > < > > > > :
s=d or
s=D or
d=S or
idsi(n 1)
(11)
where t
pass thru
(i;n)denotesthetraÆ oftheoriginalmatrixT that passes throughthesegmentof length
n startingat nodei, i.e., traÆ onringRthat usesthelinks of thesegmentbut doesnoteither originate
orterminateatanyofthenodesinthat segment. Weallthisthepass-through traÆ. Theamountofthis
traÆanbereadilyobtainedbyinspetionoftraÆmatrixT. Wehaveusedsdintheaboveexpression
to denotethat nodespreedesnodedin thedeompositionand sdto denotethat nodespreedesand
maybethesameasnodedin thedeomposition.
ThetraÆmatrixforthedeompositionisdenedsuhthatthetraÆowingfromanodestoanother
noded,suhthatsd,in thesegmentisthesameasthatintheoriginalringfortheorrespondingnodes
(see therst expressionin (11)). Thus anytraÆ omponentthe pathof whih is entirelyin the segment
the soureofall traÆomponentsin theoriginal matrixT originatedat anode outsidethesegmentand
destinedtoanynodeinthesegment(refertotheseondexpressionin(11)). NodeS alsoatsasthesoure
foralltraÆomponentsthatpassthroughthesegment,asthefourthexpressionin(11)indiates. Similarly,
nodeDis thesink fortraÆ originatingat anynodein thesegmentand terminatingatanodeoutsidethe
segmentintheoriginalring(seethethirdexpressionin(11)),aswellasforpass-throughtraÆ. Finally,any
traÆ omponentsin theoriginalmatrixT that donotoriginateorterminate atnodesofthe segment,or
donottraverseanylinksofthesegment,arenotinludedinthetraÆmatrixforthedeomposition. This
is apturedbythelast expressionin (11)whereitis shownthat notraÆ owsfromnodeD tonodeS in
thedeomposition.
Beause of the waythe traÆ matrix for thedeomposition is dened in (11), from the point of view
of any node k;i k i(n 1) in thesegment, the traÆ pattern in the new pathP (i)
n
is exatly the
same asin theoriginal ring. The newnodesS and D areintrodued inthedeompositionto abstrat the
interation of traÆ omponentsbetween nodes in and outsidethe segment. Speially, the newnodeS
hidesthedetailsofhowtraÆsouredatringnodesoutsidethesegmentandusingthelinksinthesegment
atually owsoverthe rest of thering, by providing asingle aggregation point for this traÆ. Similarly,
the newnode D providesasingleaggregationpointfortraÆ using thelinks ofthe segmentanddestined
to nodesoutsidethesegment,hidingthedetails ofhowthistraÆowsin therest ofthering. Finally,the
fatthatP (i)
n
isapath(i.e.,thatthereisnolinkfromnodeD tonodeS)meansthatthedetailsoftraÆin
theoriginalringthatdoesnotinvolveanynodesorlinks ofthesegmentarehiddenin thedeomposition.
3.2 Solving Path Segments in Isolation
Consider thetraÆ matrixT
P (i)
n =
h
t (sd)
P (i)
n i
of the deomposition P (i)
n
of asegmentof length nstarting at
nodei in the ringR, as givenin (11). This matrixanbethoughtof asrepresentingthetraÆ demands
in aring network onsisting of nodes S;i;:::;i(n 1);D, where there is simply notraÆ owingover
thelink fromnodeD tonodeS. Consequently,theILPformulationwepresentedinSetion 2anbeused
to obtainavirtualtopologythat minimizeseletroni routingfor thisringwith traÆ matrixT
P (i)
n
. Sine
theILPformulationdisallowstraÆroutingthat arriesatraÆomponentbeyonditsdestination andall
aroundthering,nolightpathsanbeformedtoarrytraÆoverthelinkfromD toS thatisabsentin the
deomposition. Thus, thetopologyobtainedinthismanneranbediretly appliedtothepathP (i)
n
. When
weusetheILP to ndanoptimal topologyforpath P (i)
n
we willsay that wesolvethe n-nodesegmentin
isolation.
Thetopologyobtainedbysolvingann-nodesegmentinisolationdoesnottakeinto aountthedetails
oftheoriginalringoutsideofthen-nodesegment. Suhatopologywillonlybeoptimalwithrespettothis
n-nodesegment,inthesensethatitwill minimizetheamountofeletroni routingwithinthesegment,but
withoutonsideringtheeetsthatdoingsowouldhaveontheamountofeletroniroutingatnodesofring
Routsidethesegment. Infat,this topologymaynotbeoptimalfortheringasawhole. Inotherwords,
0
1
2
n-2
n-1
Figure3: Aunidiretionaln-nodepathnetwork
n-nodesegmentwill bedierentthan thetopologyobtainedby solvingtheILP forP (i)
n
in isolation. Thus
it maynotbepossibleto ombineoptimalsolutionstodierentsegmentsin isolationintoa(near-)optimal
topologyfortheoriginalringR. Ourontributionisinprovingalooserresult: thatitispossibletoombine
optimalsolutionstodierentsegmentsinisolationtoobtainlowerandupperboundsontheoptimalsolution
to theringRasawhole.
Our motivation for using the path deomposition desribed in this setion is two-fold. First, as the
number n of nodes in a segment starting at some node i inreases, the resulting deomposition P (i)
n will
moreloselyapproximatetheoriginalring. Asaresult,theboundsweobtainwillbetighterwithinreasing
n. Seondand moreimportantly,apath deomposition signiantly alleviates theproblem ofexponential
growthin omputationalrequirementsforsolvingtheoriginal ILPfor ann-nodenetwork. Thisresultis a
diret onsequeneofthefollowinglemma.
Lemma3.1 A wavelengthassignment always existsfor afeasiblevirtual topology ona unidiretionalpath
network, anditan beobtainedin timelinearin thenumber oflinks andthe number ofwavelengths W per
link.
Proof. ConsiderapathnetworkwithnnodesasshowninFigure3. Consideranyfeasiblevirtualtopology
onthis network,that isanytopologyforwhih thenumberof lightpathsrossinganylink isnomorethan
W. Ifthere isanylink withlessthan W lightpaths,we addasmanysinglehoplightpathsonthat link as
neessarytomakethenumberoflightpathsonthatlinkequaltoW. Nowweanassignwavelengthsto all
thelightpathsin astraightforwardmanner. Attherstlink from node0tonode1,wehaveW lightpaths
to whihweanassignW wavelengthsin anyarbitraryfashion. Wethen examinetheselightpathsinsome
partiularorder. Consideralightpathassignedwavelengthwontherstlinkfromnode0tonode1. Ifthe
lightpathontinuesontotheseondlinkfromnode1tonode2,weassignthewavelengthwtothislightpath
ontheseondlink. Ifthelightpathterminatesatnode1,weknowthatanotherlightpathoriginatesatnode
1,sinetheloadoneahlink isthesame. Wepikonesuhlightpath,breakingtiesarbitrarily,andassign
itwavelengthw. Weontinuein thismanner,assigningwavelengthstothelightpathsoneahlinkuntilwe
reahthelastnodeandhaveoloredallthelightpaths. Weareguaranteedtobeabletoreahnoden 1on
eahoftheW wavelengths,beauseno lightpathpassesthroughororiginatesatnoden 1. Nowwehave
assigned awavelengthtoalltheoriginallightpathsofthefeasiblevirtualtopologywestartedwithandan
disardthesinglehoplightpaths(ifany)that wereaddedtobringthenumberof lightpathsateahlinkto
this is the value from whih we will obtainthe bounds. Sine we know that a wavelength assignment is
alwayspossibleand wearenotinterestedinthedetailsofthewavelengthassignment,weaneliminatethe
wavelengthassignmentfromourformulationaltogether. Thisimpliesthat weaneliminateSubproblem3,
the wavelength assignment subproblem, from the list of subproblems we enumeratedin Setion 2. Thus,
weeliminatethewavelengthvariables (k )
ij
andthe onstraints(8)and (9). The numberof variablesin the
formulationremains O(N 4
), but N 2
W variablesareeliminated, andthenumberof wavelengthonstraints
is redued from O(N 2
W)to O(N), though the totalnumber of onstraintsremains O(N 4
). This reates
a formulation that is smaller and requires dramatially less omputation to solve. In pratie, we have
found that eliminating the wavelength assignment subproblem an result in a redution in omputation
time by several orders of magnitude. For instane, ompletely solving a six-node ring network using the
original formulation (withwavelength assignment)requires between60 and 90 minutes onaSun Ultra-10
workstation. However, solving a six-node path network using the simplied formulation (no wavelength
assignment)requiresonlyafewseonds. Inbothases,weusedtheLINGO sientiomputationpakage
whihutilizesbranh-and-boundalgorithms tosolvetheILP.
AfurtherimprovementtotheILPformulationthat reduestherunningtimeforpathnetworksmaybe
obtained byexpliitly forbidding any lightpathsthat traversethe link from D to S, that is, by xing the
valuesofsomelightpathountstozero:
b
ij
= 0; 8b
ij
2 B(D;S) (12)
where i,j referto nodesinthedeomposition.
Wehavedevised awaytofurther reduetheomputationaleortsrequiredto solvetheILPforapath
segment; thetradeo is an inreasein theomplexity of theformulation. In this approah, weustomize
theformulationto inlude informationaboutombinationsoflightpathvariablesthat anproduefeasible
solutions,eliminatingombinationsweknowarenotonsistentbyappliationofproblem-speilogi. We
an also prune the searh spae by a largefator byreating a spae that weknow ontainsexatly one
optimal solution. The formulation has to be ustomized for eah value of n, where n is the number of
nodes in the deomposed segment. The amount of logi and extra information to be inluded in suh a
ustom formulation inreasesrapidly with n. We have arriedout suh ustomizationsfor 2- and 3-node
deompositions(theaseforasingle-nodedeompositionistrivial). Thegaininomputationaleortappears
tobemorethanosetbytheomplexityoftheformulationbeyondthispoint,andassuh,thismethodwas
notarriedoutanyfurther.
3.3 Interpretation of the Optimal Value for Deomposed Paths
We denote the optimal objetivevalue for the deomposition P (i)
n by
(i)
n
. That is, (i)
n
is the amountof
eletroni routingperformedbytheoptimalvirtualtopologyonthedeompositionP (i)
n
. Inthe
ofthedeomposition,donothaveanytraÆpassingthroughthematall,andhenetheydonotontribute
any eletroni routingin anyfeasible virtualtopology onthe deomposition. Thus, theoptimal objetive
value (i)
n
forthedeompositionP (i)
n
isontributedonlybythennodesabstratedfromthering. Sinethe
traÆ pattern seenbythese nodesis the sameaswhen they are inludedin the ring, (i)
n
also represents
theloallybest(lowest)amountofeletroniroutingatthissetofnodeswhenonsideredaspartofthering.
Inotherwords,theeletroniroutingatthisset ofnodesisminimized,irrespetiveofhowmuheletroni
routinghasto beperformedat othernodesoftheoriginalringasaresult.
Itmightappearthat asweinludemoreandmorenodes inthedeomposition,theoptimalsolutionto
the originalproblemwill beobtainedwhen allN nodeshavebeenrepresentedinthe deomposed network
(thatis,whenwehaveadeompositiononsistingofN+2nodes). However,beauseweonverttheringtoa
path,someinformationisnottakenintoaountatthepointwherewe\openup"theringwhenonsidering
anN nodedeomposednetwork.Asaresult,solvinganN-nodedeompositiondoesnotprovidetheoptimal
solutionfor theN-node ring. This islear from argumentsrelatedto this topi in [11, 8℄, for onveniene
wepresentthe argumentin the urrentontext here. Considerthe deomposition of an N-node segment
startingatnodei. NownotethatthedeomposedpathP (i)
N
ontainsallnodesandlinksoftheoriginalring
exeptthelinkfromnodei 1tonodei;thisiswheretheringis\openedup"tointroduethenewnodesS
andD. Consideralightpathintheoriginalringthatoriginatesatanodesi 1andterminatesatanode
dsuhthatid,andthereforeusesthelinki 1(fromnodei 1tonodei)inthering. Anysuhlightpath
will berepresentedbytwolightpathsin thedeomposition P (i)
N
: onefrom thenewnode S to node d,and
onefromthenewnodeD tonodes. Sinetheselightpathsarenotonstrainedtousethesamewavelength
in thedeomposednetwork,awavelengthassignmentfortheN-nodedeompositionmaynotbefeasiblefor
the N-node ring. Thus, the optimal solution to the deomposition may not bepossible to translate to a
solutionto the originalring, andwestill haveanoptimalvalue (i)
N
for theN-nodedeomposition,rather
thananoptimalvaluefortheringR.
4 Bounds
Inthissetionwedesribehowweanombinethe (i)
n
valueswegetfromn-nodedeompositionstoobtain
loweraswellasupperboundsonthetotalamountofeletroniroutingperformedintheoptimalasebya
virtual topologyon theoriginalring. The method ofombination isdierent forlowerand upperbounds.
We rstdisuss the asewhere we onlyonsider single-node deompositions, then moveonto thegeneral
asewherelargerdeompositionsareavailable.
4.1 Bounds Based on Single-Node Deompositions
A deomposition of an N-node ring around a singlenode i is shown in Figure 4. The next two setions
0
(a)
.
S
D
(b)
i
i+1
i-1
1
N-1
.
i
.
.
.
.
.
Figure 4: A singlenode deomposition ofaring: (a) originalring, and(b) singlenodedeompositionP (i)
1
aroundnodei
wholebyappropriatelyombiningtheoptimalILPsolutions (i)
1
forthepossiblesingle-nodedeompositions
P (i)
1
;i=0;;(N 1).
4.1.1 Lower Bound
Reall that (i)
n
represents the loally best amount of eletroni routing at the nodes in the segment of
lengthnstartingat nodei. Inpartiular, (i)
1
representstheloallybest amountofeletroniroutingthat
anbeahievedat node ionsidered in isolation. There may ormaynot be anoptimal(or even feasible)
virtualtopologyfortheringRthat ahievesthisvalueofeletroniroutingatnodei, butthere anbeno
topologywhihahievesanevenlowervalue. Thus, (i)
1
isalowerboundontheamountofeletronirouting
performedatnodeiforanyfeasiblevirtualtopology,andin partiular,fortheoptimalvirtualtopology.
Sine (i)
1
representsontributionto theeletroniroutingonlybynodei, weanaddtheontributions
together foreah nodetoobtainalowerboundonthetotaleletroni routingperformedforallnodesin a
feasiblevirtualtopology. Weallthislowerbound
1 :
1 =
N 1
X
i=0
(i)
1
(13)
Thequantity
1
4.1.2 UpperBound
Werstnote thatthevalueoftheobjetivefuntionforanyfeasiblevirtualtopologysetsanupperbound
ontheoptimalvalue,sineitorrespondstoanatualsolutionandtheoptimalsolutionanonlybebetter
than oras good as this solution. Thus, we onsider dierent ahievable topologies and we obtain upper
boundsfrom them.
First we onsider an upper bound we an obtaindiretly from thetraÆ matrix, without reourseto
deompositions. This bound orresponds to the simplest virtual topology possible, namely, the topology
onsisting onlyof single-hoplightpaths. Considernodei. Inthistopology,single-hoplightpathsfromnode
i 1arryingalltraÆto nodeiandbeyondterminateat nodei. Nodeieletronially swithesalltraÆ
forwhihitisnotthedestination,ombinesitwithitsownoutgoingtraÆ,andsouresanumberof
single-hop lightpaths(up to W)that arrythis traÆ to nodei1. Wewill allthisthe no-wavelength-routing
topology,sinenowavelengthsareoptiallyroutedatanynodeandeahlightpathspansexatlyonephysial
link. (This typeoftopologyisalled aPPWDM ringin [7℄.) Insuh atopology,eahnode iperformsthe
maximumpossibleamountofeletronirouting,whihwedenoteby (i)
. Quantities (i)
;i=0;;(N 1),
anbereadilyobtainedfromthetraÆmatrixT. Welet
0
denotethetotaleletroniroutingperformed
under theno-wavelength-routingtopology:
0 =
N 1
X
i=0 (i)
(14)
Sinethis isafeasibletopology,
0
isanupperboundontheoptimaleletroni routing,
Ingeneral,
0
isaratherlooseupperbound. Wenowonsiderhowwemightutilizesingle-node
deom-positionstoimproveupontheno-wavelength-routingtopologyandheneobtainatighterupperbound. To
this end, letusreferagainto Figure4(b),whihshowsasingle-nodedeomposition aroundnode i. Reall
nowthat,inderivingthebestloaleletronirouting (i)
1
atnodei,wemadetheassumptionthatalltraÆ
passingthroughnodeiisoriginatedbynodeS andterminatedbynodeD. Assumingfor themomentthat
nodes S and D are atual nodes in the ring, this is equivalent to saying that no lightpaths are optially
routedateithernodeS ornodeD,and,onsequently,bothnodeshavetoperformthemaximumamountof
eletroni routingpossible.
Let us all onentrator nodes those nodes whih do not perform any optial forwarding (i.e., they
terminateandoriginatealllightpaths). Intheno-wavelength-routingtopology,everynodeisaonentrator
node,andaordingtoourdisussioninthelast paragraph,nodesS andD anbeviewedasonentrator
nodesinthesinglenodedeomposition. Carryingthislineofthoughtonestepfurther,weareledtoonsider
anewvirtualtopologysuhthateveryothernodeisaonentratornode(performingthemaximumeletroni
routingpossible, (i)
)whiletheremainingnodesperformtheminimumeletroniroutingpossible, (i)
1 . This
topologyisillustratedin Figure5,wheretheeven-numberednodesareonentratornodes.
0
1
2
3
5
4
6
7
0
3
1
2
6
4
5
(a) N even
(b) N odd
max electronic routing (concentrator nodes)
min electronic routing (single-node decompositions)
Figure5: Virtualtopologywithalternatingsingle-nodedeompositionsandonentratornodes
dierentvirtualtopologieswhihyieldpotentiallydierentvaluesforthetotalamountofeletronirouting.
If thenumberofnodesN in theringis even,wehavetwopossibletopologies,dependingonwhether
even-numberedorodd-numberednodesare onentrators. If N is odd,anyvirtualtopologyonstruted in the
manner desribed abovewill havetwo onentrator nodes next to eah other at one position in the ring,
asillustratedin Figure5(b). Sinethere N waysofseletingthe position ofthese twoonentratornodes,
thereareN possiblevirtualtopologieswhenN isodd. Wetakethesmallestvalueoftotaleletronirouting
weanobtainfromthesetopologiesastheupperbound. Thisboundisindiatedby
1
,andinthegeneral
ase,itanbeexressedas:
1
= min
j2[0;N 1℄ 0
X
k 2f0;2;4;;2(b(N 2)=2)g
(j+k )
1 +
X
k2f0;2;4;;2(b(N= 2)=2)g (j+k )
1
A
(15)
Sinethisisafeasibletopologywhihinursthemaximumeletroniroutingonlyattheonentratornodes
while it inurs less at the others, theupperbound set by the objetivevalue of this topologymust be at
leastasstrongas
0
;thuswealsohavethat
1
0
In this setionweonsiderhowwemayombine deompositions ontainingmore thanasinglenode from
theringtoobtainasequeneofboundssimilartothose obtainedinthelastsetion.
4.2.1 Lower Bound
Inobtainingthebound
1
above,weremarkedthatweanaddthevarious (i)
1
quantitiestogethersinethey
eah representedeletroni routingat node ionly. Considerthe quantity (i)
2
: it representstheminimum
possibleamountofeletronirouting(bestase)atnodeiandnodei1takentogether. Weannotadd (i)
1
or (i1)
1
tothisquantityandstillhavesomethingthat isguaranteedtobealowerboundontheamountof
eletroniroutingthesenodestogetherperforminanyfeasibletopology,beausewearepotentiallyounting
the traÆ routed by a single node twie. However, we an add (i)
2
and (i2)
1
, sine the twoquantities
involve sets of nodes that are disjoint and therefore there is no potential double ounting of eletroni
routing. Generalizingthisnotion,wendthatweanaddthe (i)
n
valuesforanysetofdeompositionsthat
involvesegmentsthat aredisjointin the ring, and weare still guaranteedto obtaina lowerbound on the
objetivevalueforanyfeasibletopology. Weformalizethisinthefollowinglemma:
Lemma4.1 Let
n
be a set of segments of ring R whih partition the nodes of the ring in segments of
length nor smaller. Letl
k ;l
k
n; denotethe length (numberof nodes)of segment k;k=1;;j
n j, and
leti
k
denote itsstartingnode. The quantity
( n )= jnj X k =1 (i k ) l k (17)
is alower boundon the objetive valuefor any feasiblevirtual topologyon the ringR, andthereforeon the
optimal objetive value.
Wenowdene
n as: n =maxf( n )g (18)
where the maximumis taken overall partitions of the ring whih ontainsegmentswith n or lessnodes.
Figure 6showstwopartitions of thesamering, therst ontainingonly 1-and 2-nodesegments, and the
seond ontainingonly1-,2-and3-nodesegments.
Beause of thedenition (18),in omputingbound
n+1
wemust onsider all partitions (andbounds
derived therefrom) onsidered to ompute
n
, and, additionally, all partitions whih inlude one ormore
(n+1)-nodesegments. Speially, thesetof partitions
n+1
weonsider for
n+1
is apropersupersetof
the set of partitions
n
weonsider for
n
. Sine we draw themaximumbound from eah set asper the
denition inEquation18,thisallowsus todrawthegeneralonlusion that:
n+1
n
8n2f1(N 1)g (19)
Asaresult,thesequene
1 ;
2 ;;
N
,isastrongsequeneofboundsinwhiheahisatleastastightas
thepreviousone. Wenotethatourdenition of
1
(a)
(b)
Figure 6: Partitionsof thenodesofaringinto (a)segmentsofnomorethan2nodes,and(b)segmentsof
nomorethan3nodes
above. We were ableto express
1
in asimplerandmoreexpliit form beausethere is onlyonepossible
partition ofthenodesof aringintosingle-nodesegments.
Wedisusssomedetailsoftheomputationof
n
afterwedesribetheupperboundsinthenextsetion.
4.2.2 UpperBound
Itisnowstraightforwardtoobtainastrongsequeneofupperboundsalongthesamelines. InSetion4.1.2
weobtained the upper bound
1
by reatinga topologyin whih single-node segments (where eletroni
routingisminimized)alternatewithonentratornodes. Similarly,wenowdene
n
asthelowestobjetive
value we obtain for all topologies whih are reated by alternating onentrator nodes with segments no
larger than n nodesin size. Wean one again onsider this in lightof partitions of thenodes of a ring.
Now,however,thepartitionsareonstrainednotonlytousesegmentsofn-nodesorless,buteveryalternate
segmentmustontainexatlyonenode. Thesealternatesingle-nodesegmentsareusedasonentratornodes
in thetopologywereate, rather than assingle-nodedeompositions. One again, the form of this upper
bound isanalternate sumof (i)
x and
(i)
values,similartoexpression(15)for
1
;butfor
n
thevalueof
x isnotrestritedto1asfor
1
,insteaditantakeonanyvaluefrom1ton. Figure7showstwowayswe
anpartitionaringusingnolargerthan3-nodesegments,thusreatingtwotopologiesamongtheoneswe
wouldonsiderin omputing
3 .
We note that the bounds
0 and
1
(a)
(b)
Figure7: Twopartitionsofaringintoalternatingonentratornodesandsegmentsofnomorethan3nodes
Wealso notethat sineeverydeomposedsegmenthasto alternatewithaonentratornode, weanonly
useupto N 1nodedeompositions,and annotuseN-nodedeompositionsasforthelowerbounds. As
before, the set of alltopologieswe onsider in obtaining
n+1
isa superset of theset of alltopologieswe
onsider inobtaining
n
,thereforewemayassertthat:
n+1
n
8n2[0;N 2℄ (20)
givingusastrongsequeneofupperbounds.
Beause the bounds f
n
g are based on atual feasibly topologies, they also provide us with a useful
series ofheuristisolutionsto thering. Inthenextsetionwederivearesultwhih showsthetightnessof
theboundsandthusthegoodnessoftheheuristis,andweseeinSetion5thateventherstfewsolutions
of theseries anoutperformasimplistiheuristi. Thelatersolutionsin theseries anomparefavorably
with some heuristis reported in literature. Speially,
N 1
must be as good orbetter than a
single-hub arhiteture[7, 3, 2, 13℄, beauseit onsiders all topologieswith asingleonentratornode (whih is
equivalentto ahub node). For asimilar reason,
k
;k dN=2emustbeas good orbetterthan adouble
hubdesign,ifthehubsareonstrainedto bediametriallyoppositeinthering.
4.2.3 Tightness ofBounds
Consider the value (i)
(i)
1
for eah node of a ring, whih is the dierene betweenthe minimum and
minimumbenodej,andlettheorrespondingdierenebe ,sothat: (j) = N 1 min i=0 ( (i) (i) 1 ) (21)
Considerthetopologyinwhihnodejisaonentratornodeandtherestofthenodesfollowtheoptimal
topology forthe(N 1)-node deomposition P (j1)
N 1
. Theamountof eletroni routingperformedbythis
topologyisan upperbound ontheoptimalobjetivevalueforthering. Wedenote itby 0
andnote that
0 = (j) + (j1) N 1
. SinethisisatopologyusingasegmentofN 1nodesalternatingwithaonentrator
node,wemusthave
0
N 1
(22)
Nowweonsiderthepartitionoftheringintothesinglenodejandthe(N 1)-nodesegmentomprising
alltheothernodesagain,butthistimewithaviewtoobtainingalowerbound. Wedenotethislowerbound
by 0
and itis found by simplyadding the objetivevaluesfor the two orrespondingdeompositions, so
that 0 = (j) 1 + (j1) N 1
. Oneagain,this isoneofthepartitions intheset ofallpartitions ofthenodesof
theringintosegmentsofN 1nodesorless,so thatwehave
0
N 1
(23)
However,fromthedenition of 0
and 0
above,weknowthat
0 0 = (j) (24)
Combiningtheaboveresults(22,23,24),andonsideringthedenitionof (j)
,weanassertthat:
N 1 N 1 N 1 min i=0 ( (i) (i) 1 ) (25)
Of ourse,depending on thevalueof N and the omputationalpoweravailable,it may ormay notbe
pratial to ompute
N 1 and
N 1
; however, this is the theoretial limitation on the tightnessof the
frameworkofboundswepresent.
4.2.4 ComputationalConsiderations
The bounds
n (and
n
) forsuessivevalues of ninorporateprogressivelymoreinformation about the
problemandassuhrequireprogressivelymoreomputationaleorttodetermine. Thisinreasein
ompu-tationaleortmanifests itselfintwoways:
1. thealulationofthe (i)
n
valuesrequiredforagivenbound,and
2. theevaluationofallpartitionsoftheringinsegmentsoflengthatmostnbyanappropriateombination
ofthe (i)
n
n
makeareequallyappliableto thesequeneofupperbounds.
Theomputation ofabound utilizing aertainsize of deompositions requiresknowledgeof
deompo-sitions of all lower sizes as well. Thus, omputing
x
would require us to ompute (i)
n
for all values of
i2f0(N 1)g,andallvaluesofn2f1xg. However,theinrementalomputationofdeompositions
requiredtodetermine
x
onsistsonlyofdetermining (i)
x
forallnodesi,sine (i)
n
forn<xwouldalready
havebeenomputedwhendetermining
x 1
. Naturally,asx inreases,thisinrementaleortrequiredalso
inreases; as we have noted before the numberof variables and onstraintsinrease as O(n 4
). Thus the
maximumvalueofnforwhih weandeterminetheorrespondingbound islimitedbythis omputational
eort.
Regardingtheseondfatorthataetstheomputationtimerequiredtoobtainthebound
n
,wenote
that a straightforward approah would require us to enumerate all possible partitions of an N-node ring
into segmentsoflengthatmostn. Whileevaluatingeahpartition (i.e.,omputingthelowerbound forit)
takestimelinearinthenumberofsegmentsofthepartition(assumingthattheindividual (i)
x
;x2f1ng;
values are available), the number of possible partitions inreaseswith n. The total number of partitions
is maximum whenn =N, and this numberis easily seento be 2 N
1(beause eah link of the ringan
be broken to form partitions or not, with the single ase where no link is broken being exluded). For
smaller valuesof n, the totalnumber issmaller but still very large. We note that assuming node i is the
rst of a segment in thepartition (and thus exluding somepartitions), the total number of segments in
suh apartition mustalwaysbe atleast dN=nefor agiven valueofn, and sineeah segmentanonsist
of 1;2;;n nodes the totalnumber of suh partitions is at least n dN =ne
. Consideringn =2, we anset
a lowerlimit on this value, and thus say that for 2 n N, the total number of partitions is between
2 dN =2e
and2 N
1,andisthusexponentialinN. Thustheinremental numberofpartitionstoonsider for
agiven valueofnisalso exponentialin N in theworstase. Thusastraightforwardapproahto ombine
deompositionsinto bounds would severelylimitthemaximumvalueof N for whih weandeterminethe
orrespondingbounds.
However,byexploiting thepartiularharateristisof (i)
n
, wehavedevelopedapolynomial-time
algo-rithm to ompute
n
, assuming that theappropriate (i)
n
valuesare available. The algorithmis based on
inrementallybuilding thebest sumof (i)
n
aroundthe ring,and followingonly thebest partial sumat an
intermediate node. This algorithm is presentedas adynami programmingproblem in Appendix B, and
requires O(n 2
N) time to nd
n
given all (i)
x
values for x = 1;;n. For the largest number of total
partitions intheasen=N,this orrespondstoO(N 3
)insteadofO(2 N
)time, andin fatbeomeslinear
in N forasmallgivenvalueofn.
We anahieve animprovement also byusing the properties of (i)
n
values formalizedin thefollowing
lemma. Thesepropertiesintrodueaonstantfatorofimprovementtothedynamiprogrammingalgorithm
desribedin Appendix B.
Lemma4.2 An (x+y)-node deomposition yields at least as large an objetive value for the deomposed
x+y x + y
,if xandy arepositive andx+y<N.
Corollary 4.1 An x-node deompositionyields atleast aslargean objetive value for the deomposed
net-workas thesum ofobjetive valuesof anyombinationofsmallerdeompositions itanbepartitionedinto.
That is, (i) x (i) y 1 + (i+y1) y 2 + (i+y1+y2) y 3 ++ (i+y1+y2++yn 1) y n
,ifx=y
1 +y
2
++y
n .
Proof of Lemma4.2. Let us rst onsider the speial ase x =y =1. Consideratwo-node
deompo-sition for nodes i and r = i1 in the original ring network R. This yields avalueof (i)
2
for the node
pair deomposed. Now onsider anew ringnetwork R 0
with four nodes, whih hasthe samestruture as
the deomposed network above and has a traÆ matrix idential to the traÆ matrix T
P (i)
2
in the above
deomposition. It is easy to see that the optimum value of total eletroni routing for this new network
is (i)
2
, sine this is the best value for nodes i and r, and nodes S and D do not have any pass-through
traÆ and heneannot route any traÆ eletronially. Now onsider asingle-nodedeompositionof this
new network R 0 ,yielding (S) 0 1 , (i) 0 1 , (r) 0 1 and (D) 0 1
asthebest aseeletroni routingfor thefournodes.
One again, sinenodesS and D havenopass-throughtraÆ, (S) 0 1 and (D) 0 1
areboundto bezero. Now
the single-node deomposition matrixfor node i, T
P (i)
1
, is thesame nomatter whether the deomposition
is madefrom theoriginal network orthe newnetwork. Thus, thesinglenodebestaseeletroni routings
of thenodei isthe samein eitherase, in other words (i) 0 1 = (i) 1
. Similarly, (r) 0 1 = (r) 1
. Sine thesum
(S) 0 1 + (i) 0 1 + (r) 0 1 + (D) 0 1
isknowntoform alowerbound ontheoptimumvalueoftheeletronirouting
forthenewnetwork,weansaythat (i) 1 + (r) 1 (i) 2 .
Although in theaboveargument wehaveused speialvalues of1for eah of x and y,and hene2 for
x+y,theargumentsretaintheirvalidityifanyothervaluesareused. Thus, thelemmais true.
Theorollary followsimmediately from thelemma by repeated appliation within thesame
deompo-sition, and allowsusto disardpartitions in whih asmall segmentfollowsanother. Speially, ifweare
omputing
x
, weandisardapartition in whih ay
1
-node segmentis followedbyay
2
-nodesegment if
y
1 +y
2
x, beausethe partition we obtainby replaing these two segments by asingle (y
1 +y
2 )-node
segmentmustyieldahigherbound,andweareonlyinterestedin themaximumbound.
Theabovemethodsallowustoomputethef
n
gboundsbyombiningthe (i)
n
valuesinaninsigniant
fration ofthetimetakento omputethe (i)
n
valuesthemsleves. Inpratie,wefoundthat omputingthe
(i)
n
valuestook minutes andhours forinreasing n,while ombiningthem to form thef
n
gboundstook
milliseonds. Weonludethatthelimitingfatorindetermininghowmanyoftheseboundsanbeomputed
in areasonableamountof timeistheeortrequiredto solvetheILPfor n-nodedeompositionin orderto
omputeeahoftheN (i)
n
values.
Similar observations apply to the sequene of bounds f
n
gas the value of n inreases, and asimilar
algorithmanbeusedto omputeeÆientlyeahbound
n
giventhe (i)
n
Inthissetion,wepresenttheresultsofusingourframeworkfordierenttraÆmatries. First,wedisuss
the riteria we used to onstrut traÆ matries and to group results by these riteria; the results are
presentednext.
5.1 TraÆ Patterns
We rst reate the distintion betweensymmetri and asymmetri traÆ patterns. The term symmetri
applies to thering, rather than thetraÆ matrixitself. Weall atraÆ pattern symmetri ifthe traÆ
pattern from anynodeto the othersis repeated for allthe nodes. Inother words, fora symmetritraÆ
matrixT wehave
t (sd)
=t
(sx;dx)
8i;j2f0(N 1)g;81xN 1 (26)
This typeof traÆ pattern isof interestsinethe traÆ pattern lookssimilar from dierent nodeson the
ring. If the aboverelation holds exatly, the optimal topology is likelyto be a regular topology. Rather
thanonneourattentiontothosetraÆmatriesin whih(26)holdsstritly,weonsiderthegeneralase
wheretraÆomponentsoftheformt
(sx;dx)
foragivensanddarealldrawnfromthesamedistribution.
If thevarianeof thisdistributioniszero, then(26)holds exatly, andweallthe resultingtraÆpattern
stritly symmetri, otherwise weall it statistially symmetri. Ifa traÆmatrixis highly asymmetri,so
that thetraÆ patternsseenbydierent nodesofthe ringareverydierent,thentheoptimal topologyis
likelytoperformwellin thesetionsoftheringwherethereislessongestionandpoorlyin thesetionsof
high ongestion. This isalso likelyto bethe asefor anyfeasible topology, in other words, thedierene
betweenthebest and worst may beomparativelyless. Forthisreason, wehavehosento onentrateon
statistially symmetritraÆ matriesforallourresults.
Havingsaidthat thetraÆpatternlookssimilarfromeahnode,weturn ourattentiontowhatitlooks
like from a given node. We onsider three simple ases whih are represented shematially in Figure 8.
First,thetraÆfromagivennodeitoallothernodesmaybethesame,werefertothisasuniformtraÆ.
Forastritlysymmetrimatrix,thisresultsineahelementofthematrixhavingthesamevalue(otherthan
thediagonalelements,whihmustbezero). WhenthetraÆomponentstoalltheothernodesarenotthe
same,weanspeakofafallingtraÆpatterninwhihthetraÆfromnodestonodesxdereaseslinearly
asxinreases. Similarlywespeakofarisingpattern. Again,weintroduetheoneptofstatistialvariation
sothat atualmatrixelementsvary from thesepatternsstatistially anddo notvary stritlylinearlyasin
Figure 8.
Inorder to haveagood basisforomparison for theabovethree typesof traÆ patterns, we fous on
theoneptofharateristiphysial loadofthetraÆ matrix. GivenatraÆmatrix,weanomputethe
total traÆowingoveragivenlink of theringin astraightforwardmanner. Fortheproblem instaneto
befeasible,this traÆmustbelessthanorequalto WC,where W isthenumberofwavelengthsandC is
theapaityinunits oftraÆofeah wavelength. Ifthematrixisstatistiallysymmetri,theloadoneah
1
2
3
. . . . .
N-1
Traffic
Distance to Destination Node
falling
uniform
rising
Figure8: DierenttraÆpatterns.
We allthis valuetheharateristiphysialloadofthe matrixandobtainitby takingtheaverageof the
physialloadoneah link,andexpress itasaperentageofWC. Forthesamepattern,theharateristi
loadsaleswiththematrixelements. Forexample,bymultiplyingeahelementofamatrixofharateristi
load50%byafatorof1.5,itisonvertedintoamatrixofharateristiload75%butofthesamepattern.
5.2 Results
We present resultspertaining to 8-node and 16-node rings. For most of our results, the value of W was
taken to be between 16 and 20 and the value of C around 48. We used randomly generated statistially
symmetritraÆmatriesforalltheruns. AdisreteuniformprobabilitydistributionwasusedforalltraÆ
generation. Wefousonharateristiphysialloadvaluesof50%and90%.
Beause of dierent traÆ patterns and dierent harateristiload values, the absolute values of the
dierentbounds plottedare noteasytoompare. Itisneessarytoexpress themin termsofsome
hara-teristi oftheproblemthat makesitsensibleto omparethem. Weonentrateonthequantity
0 , whih
denotes theamountofeletroni routingperformedbyatopologythatdoesnotemployoptial forwarding
at all. This isoften atuallyused in networks at transitorystages in whih the bermedium isemployed
with WDM, but nowavelengthroutingis employed[5,10℄. Weanonsider thisaseto orrespondtono
grooming,thatis,noeorthasbeenmadetogroomindividualtraÆomponentsintolightpaths. Theother
extreme (not neessarily ahievable) is omplete grooming, in whih all traÆ is groomed into lightpaths
and noeletroniroutingis performed. Theatualamountofeletroniroutingperformedbyanyfeasible
topology falls between these limitsand may be expressed asa fration of
0
to indiate the eetiveness
of grooming. Thus1indiatesthat allthetraÆhasbeenleftungroomed,while 0orrespondstothebest
situation in whih no traÆ is left to be groomed. The valuesf
n
theheuristisolutionsreated. Thevaluesf
n
grepresentlowerbounds onthegroomingeetivenessthat
anbereahedinpriniple,thatis,intheoptimalase. Inourplots,wenormalizeeahsetofresultstothe
orrespondingvalueof
0
andplotthegroomingeetivenessratiosasabove. Otherquantitiesin theplot
whihwedisussbelowaresimilarlynormalized.
Wepresenttwobroadsetsofdata. Intherst,ordetailedsetion,wepresent
n and
n
forvaluesofn
upto 7,forN =8;16,forthetwoharateristiloads andstatistially uniform,risingand fallingpatterns.
Figures 9-14presentthedatafor8-noderingsandFigures15-18presentthedatafor16-noderings. For
N =8,therisingandfalling matriesweresogeneratedthat thelowesttraÆomponentin arow(traÆ
to nearestnodeforrisingpattern,traÆtofarthest nodeforfallingpattern)areloseto zero. ForN=16,
wepresentdataonly fortwotypesof fallingtraÆ patterns. Oneis astatistially fallingmatrix in whih
the traÆfalls to zeroat thefarthest nodeaswiththe 8-nodease. Wealsohaveadata set in whih the
traÆ fromnode sfallsto zeroat nodes8,thatis, halfwayaroundthering. Suh apattern ouldbeof
interestifabidiretionalringisdeomposedintotwounidiretionalringsbyadoptingshortestpathrouting
foralltraÆomponentsapriori,aswementionedin Setion2. Nodatasetforarisingpatternoruniform
pattern isinludedforN =16.
Figures9,10showtheresultsfor8-noderings,statistiallyuniformtraÆ, 50%and90%harateristi
loadrespetively. Figures11,12similarlyshowtheresultsforstatistiallyfallingtraÆ,50%and90%load,
and gures13, 14 the resultsfor statistially falling traÆ, 50% and 90% load, respetively. We observe
that all the gures look similar. There is a sharp derease from
0 to
1
and more moderate derease
thereafter. In all ases,there is a marked derease for
N 1
, and thegroomingeetiveness for
N 1 is
between0.1and0.2inallases. Thegrowthof
n
isomparativelyless. Figures15,16showdetailedresults
for16-noderings,statistiallyfallingtraÆ,fallingtozeroattheendofthering,50%and90%harateristi
loadrespetively. Figures17,18showsimilarresultsforstatistiallyfallingtraÆ,fallingto zeroatapoint
halfwayaroundthering. Theseshowsimilarharateristisastheresultsfor8-noderings. Eventhoughthe
largest valueof n forwhih
n
is obtained, 7, isnowless thanhalf of N, weobservethat
7
one again
representsgroomingeetivenessbetween0.1and0.2inalltheases. Thuswegenerallyobservethatweget
goodgrommingeetivenessandthat thelowerboundsareomparativelylessinmagnitude. Thisvalidates
our approah of desribing thevalues ofthe bounds with respet to the no-optial-forwardingaserather
thantheoptimalvalue,beauseitindiatesthatahighvalueofeletroniroutingforsomefeasibletopology
is more likelyto result from lak of grooming (and anbe orretedby proper grooming) than being the
inevitableonsequeneofahighoptimalvalue.
Inthisdetailed setion,wealsoplotthreeotherquantities(thesedonotvarywithnumberofnodeslike
n and
n
). Therstisalowerbound omputedafterthefashionof theMooreboundfoundin literature
(for example, [12℄) byonsidering the numberof lightpathendpoints available to traÆ, while theseond
is based onaper-nodeonsideration of thelightpathendpoints, derived after thefashion of [12℄. Bounds
of thistypehavebeendevelopedbyonsiderationofgeneraltopologiesanditisexpetedthat ourbounds,
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
Figure9: DetailedresultsforN=8,Statistiallyuniformpattern, 50%load
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
Figure11: DetailedresultsforN=8,Statistiallyfallingpattern,50%load
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
Figure13: DetailedresultsforN =8,Statistiallyrisingpattern,50%load
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
Figure15: DetailedresultsforN =16,Statistiallyfalling pattern,50%load,fallingtoend
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
Figure17: DetailedresultsforN =16,Statistiallyfalling pattern,50%load,fallingtoN=2
0
1
2
3
4
5
6
7
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Decomposition Size n
Grooming Effectiveness
Ψ
n
Φ
n
Lightpath Endpoint
Node−specifc Lightpath Endpoint
2−hop lower bound
5
10
15
20
25
30
0
500
1000
1500
2000
2500
3000
3500
4000
4500
Data Sets
Electronically Routed Traffic
Ψ
1
Ψ
2
Ψ
3
Φ
6
2hop
Figure19: EnsembleresultsforN =8,Statistiallyuniformpattern, 90%load,eletroni routing
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data Sets
Grooming effectiveness
Ψ
1
Ψ
2
Ψ
3
Φ
6
2hop
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data Sets
Grooming effectiveness
Ψ
1
Ψ
2
Ψ
3
Φ
6
2hop
Figure21: Ensembleresultsfor N=8,Statistiallyfalling pattern,50%load,(normalized)
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data Sets
Grooming effectiveness
Ψ
1
Ψ
2
Ψ
3
Φ
6
2hop
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data Sets
Grooming effectiveness
Ψ
1
Ψ
2
Ψ
3
Φ
5
2hop
Figure23: EnsembleresultsforN =8,Statistiallyrising(step)pattern,50%load,(normalized)
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Data Sets
Grooming effectiveness
Ψ
1
Ψ
2
Ψ
3
Ψ
5
Φ
5
2hop
end ofthering,(gures 15,16)therstbound hasaomparablevalueto thelargest
n
wehaveobtained
(in oneaseitisnearlyequalandin theotherasedistintlylarger).
Finally,weplotaneasytoomputelowerboundontheperformaneofasimpleheuristiwhihisbased
on solvingthe problemoptimallybut using onlysingle-hopand two-hoplightpaths. Forevenvaluesof N,
the simpliityofthis heuristiisespeially attrativesineawavelengthassignmentis alwayspossibleand
thusneednotbeperformed,aresultforwhihweomittheformalstatementandproofhere. Alowerbound
ontheperformaneofsuhaheuristiiseasyto obtainbyonsidering thatatraÆomponentfromnode
s to node sm must be eletroniallyrouted atleast b(m 1)=2 times,for m >2. Weallthis bound
the 2-hop lower bound. Sine
0 and
1
are obtainedfrom topologiesthat an by denition ontain no
lightpathslonger thantwohops, theobjetivevalue oftheoptimal two-hoptopologywill bydenition be
equaltoorlessthanthese. Thisisborneoutbytheresults. However,ineahaseweseethatall
n values
forn>1arelowerthanthe2-hoplowerbound. Thuseventherstfewsolutionsprovidedbyourframework
anoutperformsimplistiheuristissuhasthetwohopoptimaltopology.
In the seond set of data, we present dierent runs in eah of whih results are plotted for 30 traÆ
matriesofthesamepatternandsamevalueofN (either8or16). Foreahmatrix,
n and
n
areplotted
forvaluesofnuptoamaximum,the2-hoplowerboundisalsoplotted. Figures19-24presenttheseresults.
Onlythe
n
valueswhihprodueappreiableimprovementsoverthepreviousonesareplottedforimproved
readability. Similarly,onlythehighest
n
valueobtainedisplotted. The2-hoplowerboundisalsoplotted.
Figures19,20bothplottheresultsforthesameensemble,8-noderingswithstatistiallyuniformtraÆ
of 90%harateristiload. Figure19showstheatualvaluesof eletroniroutingwhilegure20expresses
the samedatanormalizedto produegroomingeetivenessvaluesasdesribed above. All the restof the
gures use the normalizedvalues. Figures 21, 22 representensembles of 8-node rings, statistially falling
traÆ matries of50%and 90%harateristiloadvaluesrespetively. Figure 23showstheresultsfor an
ensembleof8-noderings,50%load;thetraÆpatternisstatistiallyasteprise,traÆtonearnodesbeingthe
sameonaverage,whilethetraÆtofurthernodesisagainthesamebutahighervalue. Figure24represents
an ensemble of 16-noderings, statistially uniform traÆ with 50%load. Forthe last two ensembles the
highestvalueofnforwhih
n and
n
areplottedis5,fortheearlieroneitis6.
Theensembleresultsonrmthedetailed resultsweobtainedearlier. Beausewehaveobtainedbound
valuesupto asmallervalueofn, wedonotsee thelowvaluesofgroomingeetiveness weenountered in
the detailed results,but thevaluesofthe earlierboundsindiate that thesameharateristiarelikelyto
emerge. Wenotefromthenormalizedgraphsthat
1
islikelytoahieveagro