Traffic Grooming in Unidirectional WDM Rings with Bounded Degree Request Graph

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Traffic Grooming in Unidirectional WDM Rings with

Bounded Degree Request Graph

Xavier Muñoz

To cite this version:

Xavier Muñoz. Traffic Grooming in Unidirectional WDM Rings with Bounded Degree Request Graph.

[Research Report] RR-6481, INRIA. 2008. �inria-00265565v2�

inria-00265565, version 2 - 20 Mar 2008

a p p o r t

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Thème COM

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Groupage de Trafic Dans les Anneaux

Unidirectionnels WDM avec Graphe de Requêtes de

Degré Borné

Xavier Muñoz — Ignasi Sau

N° 6481

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

WDM ave Graphe de Requêtes de Degré Borné

∗

Xavier Muñoz

†

, IgnasiSau

‡ §

ThèmeCOMSystèmes ommuni ants ProjetMASCOTTE

Rapportdere her he n°6481February200814pages

Résumé:Legroupagedetra estundesproblèmeslesplusimportantsdanslesréseaux optiques.Il onstisteàgrouperdessignauxdebasdébitdansdesuxdeplusgrande apa ité, ave l'obje tifderéduirele oûtduréseau.DanslesréseauxSONETWDM, e oûtestdonné prin ipalementpar le nombred'ADMs. Nous onsidérons l'anneauunidire tionnel omme graphephysique. En termesdethéorie desgraphes,le groupagedetra revientàtrouver unepartition des arêtesdugraphe derequêtes ensous-graphesave unnombremaximum d'arêtes,enminimisantlenombretotaldesommetsdanslapartition.

Nous onsidéronsungraphederequêtesdedegrémaximum

∆

,etlebutestde on evoir unréseauquisoit apabledesatisfairetouslesgraphesderequêtestelsque haquesommet peutétablire auplus

∆

ommuni ations.Un modèlepermettant ette exibilité n'existait pas dans la littérature.Nous formalisonsle problèmeet trouvons la solution exa te pour

∆ = 2

et

∆ = 3

(sauf le as

C = 4

). Nous donnons des bornes inférieures et supérieures pourle asgéneral.

Mots- lés : Réseaux optiques, SONET, groupage de tra , ADM, dé omposition des graphes,graphe ubique,graphessansbridges.

∗

This workhas been partially supported byEuropean proje t IST FETAEOLUS, PACAregion of Fran e,MinisteriodeEdu a iónyCien iaofSpain,EuropeanRegionalDevelopmentFundunderproje t TEC2005-03575,CatalanResear hCoun ilunderproje t2005SGR00256,andCOSTa tion293GRAAL, andhasbeendoneinthe ontextofthe r CorsowithFran eTele om.

†

GraphTheoryandCombinatori sgroup,MA4,UPC,Bar elona,Spain

‡

Mas otteProje t-I3S(CNRS/UNSA)andINRIA-Sophia-Antipolis,Fran e

§

Bounded Degree Request Graph

¶

Abstra t: Tra groomingis amajorissuein opti al networks. It referstogroupinglow rate signalsinto higher speed streams, in order to redu e theequipment ost. In SONET WDMnetworks,this ostismostlygivenbythenumberofele troni terminations,namely ADMs.We onsiderthe asewhenthetopologyisaunidire tionalring.Ingraph-theoreti al terms,thetra groomingprobleminthis ase onsistsinpartitioningtheedgesofarequest graphintosubgraphswithamaximumnumberofedges,whileminimizingthetotalnumber ofverti esofthede omposition.

We onsiderthe asewhentherequestgraphhasboundedmaximumdegree

∆

,andour aim is to design a network being able to support any request graphsatisfying the degree onstraints.Theexisting theoreti al models in the literatureare mu hmore rigid, anddo notallowsu h adaptability.Weformalizethe problem, andsolvethe ases

∆ = 2

(forall valuesof

C

)and

∆ = 3

(ex eptthe ase

C = 4

). Wealsoprovidelowerandupperbounds forthegeneral ase.

Key-words: Opti alnetworks,SONEToverWDM,tra grooming,ADM,graph de om-position, ubi graph,bridgelessgraph.

1 Introdu tion

Tra groomingisthegeneri termforpa kinglowratesignalsintohigherspeedstreams (see the surveys [3, 9, 15, 16, 20℄). By using tra grooming,it is possible to bypassthe ele troni satthenodeswhi harenotsour esordestinationsoftra ,andthereforeredu ing the ostofthenetwork.Typi ally,inaWDM(WavelengthDivisionMultiplexing)network, insteadofhavingoneSONETAddDropMultiplexer(ADM)oneverywavelengthat every node, it may be possible to have ADMs only for the wavelengths used at that node (the otherwavelengthsbeingopti allyrouted withoutele troni swit hing).

Theso alledtra groomingproblem onsistsinminimizingthetotalnumberofADMs tobeused,inordertoredu etheoverall ostofthenetwork.Theproblemiseasilyseentobe NP- ompleteforanarbitrarysetofrequests.See[11,10,1℄forhardnessandapproximation resultsoftra groominginrings,treesandstarnetworks.

Here we onsider unidire tionalSONET/WDMringnetworks.Inthat asetherouting isuniqueandwehavetoassign toea hrequestbetweentwonodesawavelengthandsome bandwidthonthiswavelength.Ifthetra isuniformandifagivenwavelength an arryat most

C

requests,we anassigntoea hrequestatmost

1

C

ofthebandwidth.

C

isknownas thegroomingratioorgroomingfa tor.Furthermoreifthetra requirementissymmetri ,it anbeeasilyshown(byex hangingwavelengths)thattherealwaysexistsanoptimalsolution in whi h the samewavelengthis given to apairof symmetri requests. Thenwithout loss of generality we will assign to ea h pair of symmetri requests, alled a ir le, the same wavelength. Then ea h ir le uses

1

C

of the bandwidth in the whole ring. If the two

end-nodesare

i

and

j

, weneedoneADMat node

i

andone at node

j

.Themain pointis that iftworequestshavea ommonend-node,they anshare anADMif theyare assignedthe samewavelength.

Thetra groomingproblemforaunidire tionalSONETringwith

n

nodesanda groom-ingratio

C

has been modeled asagraphpartition problem in both[2℄ and [14℄ whenthe requestgraphisgivenbyasymmetri graph

R

.Toawavelength

λ

isasso iatedasubgraph

B

λ

⊂ R

in whi hea h edge orresponds to apairof symmetri requests (thatis, a ir le) and ea h node to an ADM. The grooming onstraint, i.e. the fa t that awavelength an arryatmost

C

requests, orrespondstothefa t thatthenumberofedges

|E(B

λ

)|

ofea h subgraph

B

λ

isatmost

C

.The ost orrespondstothetotalnumberofverti esusedinthe subgraphs,andtheobje tiveisthereforetominimizethisnumber.

This problem has been well studied when the network is aunidire tional ring [3, 4,7, 8, 9,14, 12, 13,15, 18, 19℄. With theall-to-allset of requests, optimal onstru tionsfor a givengroomingratio

C

wereobtainedusingtoolsofgraphanddesigntheory,inparti ular forgroomingratio

C = 3, 4, 5, 6

and

C ≥ N (N − 1)/6

[3℄.

Most of the resear h eorts in this grooming problem have been devoted to nd the minimumnumberof ADMs requiredeither for agiven tra pattern orset of onne tion requests(typi allyuniformall-to-all ommuni ationpattern), oreither forageneraltra pattern. Howeverin most asesthetra patternhasbeen onsidered asaninputforthe problemforpla ingtheADMs.Inthispaperwe onsiderthetra groomingproblemfrom

adierentpointofview:Assumingagivennetworktopologyitwouldbedesirabletopla e theminimumnumberofADMs aspossibleat ea h nodein su hawaythatthey ouldbe onguredto handledierenttra patterns orgraphsofrequests. One annot expe t to hangetheequipmentofthenetwork ea htimethetra requirements hange.

Withoutanyrestri tionin thegraphofrequests,thenumberofrequiredADMsisgiven by the worst ase, i.e. when the Graph of Requests is the omplete graph. However, in many asessomerestri tionsonthegraphofrequests mightbeassumed.Fromapra ti al pointofview,itisinterestingto designanetworkbeingabletosupportanyrequestgraph withmaximumdegreenotex eedingagiven onstant.Thissituationisusualinrealopti al networks,sin eduetote hnology onstraintsthenumberofallowed ommuni ationsforea h nodeisusuallybounded.Thisexibility analsobethoughtfromanotherpointofview:if wehavealimitednumberof availableADMs to pla eat thenodesofthenetwork,thenit isinterestingto knowwhi h isthemaximumdegreeofarequestgraphthat ournetworkis abletosupport, depending onthegroomingfa tor.Equivalently, givenamaximumdegree andanumberofavailable ADMs, itisuseful toknow whi h valuesofthegroomingfa tor thenetworkwill support.

The aim of this arti le is to provide a theoreti al framework to design su h networks withdynami ally hangingtra .Westudythe asewhenthephysi alnetworkisgivenby anunidire tionalring,whi h isawidelyusedtopology(forinstan e,SONET rings).In[6℄ theauthors onsiderthisproblemfromamorepra ti alpointofview:they all

t

-allowable atra matrixwherethenumberof ir uitsterminatedatea hnodeisatmost

t

,andthe obje tiveisalsotominimizethenumberofele troni terminations.Theygivelowerbounds onthenumberofADMsandprovidesomeheuristi s.

In addition,wealsosupposethat ea h pairof ommuni atingnodesestablishes a two-way ommuni ation.Thatis,ea hpair

(i, j)

of ommuni atingnodesin theringrepresents tworequests:from

i

to

j

,and from

j

to

i

.Thus, su h apairusesalltheedgesofthering, thereforeindu ingoneunityofload.Hen e,we anusethenotationintrodu edin [4℄ and onsiderea hrequestasanedge,andthenagainthegrooming onstraint,i.e.thefa t that awavelength an arryatmost

C

requests, orrespondstothefa tthatthenumberofedges

|E(B

λ

)|

of ea h subgraph

B

λ

is at most

C

. The ost orresponds to the totalnumber of verti esusedin thesubgraphs.

Namely,we onsidertheproblemofpla ingtheminimumnumberofADMsinthenodes oftheringin su hawaythatthenetwork ouldsupportany requestgraphwithmaximum degreeboundedbya onstant

∆

.Notethatusingthisapproa h,asfarasthedegreeofea h nodedoesnotex eed

∆

, thenetwork ansupport awiderangeof tra demands without re onguring the ele troni s pla ed at the nodes. The problem an be formally stated as follows:

Traffi GroominginUnidire tionalRingswithBounded-DegreeRequestGraph Input : Three integers

n

,

C

,and

∆

.

Output : An assignmentof

A(v)

ADMstoea hnode

v ∈ V (C

n

)

,in su h awaythat foranyrequestgraph

R

withmaximumdegreeatmost

∆

,itexistsapartitionof

E(R)

into subgraphs

B

λ

,

1 ≤ λ ≤ Λ

,su hthat:

(i) |E(B

λ

)| ≤ C

forall

λ

;and

(ii)

ea hvertex

v ∈ V (C

n

)

appearsin atmost

A(v)

subgraphs. Obje tive : Minimize

P

v∈V

(C

n

)

A(v)

,andtheoptimumisdenoted

A(n, C, ∆)

.

When the request graph is restri tedto belong to a sub lass of graphs

C

of the lass of graphs with maximum degree at most

∆

, then the optimum is denoted

A(n, C, ∆, C)

. Obviously,foranysub lassofgraph

C

,

A(n, C, ∆, C) ≤ A(n, C, ∆)

.

Inthisarti lewesolvethe ases orrespondingto

∆ = 2

and

∆ = 3

(givinga onje ture forthe ase

C = 4

),andgivelowerboundsforthegeneral ase.Theremainderofthearti le isstru turedasfollows:inSe tion2wegivesomepropertiesofthefun tion

A(n, C, ∆)

,to beusedin thefollowingse tions. InSe tion 3wefo us onthe ase

∆ = 2

,givinga losed formulaforallvaluesof

C

.InSe tion4westudythe ase

∆ = 3

,solvingall asesex eptthe ase

C = 4

,forwhi hwe onje turethesolution.Finally,Se tion5isdevotedto on lusions andopenproblems.

2 Behavior of

A(n, C, ∆)

Inthisse tionwedes ribesomepropertiesofthefun tion

A(n, C, ∆)

. Lemma2.1 The following statements hold :

(i)

A(n, C, 1) = n

. (ii)

A(n, 1, ∆) = ∆n

. (iii) If

C

′

_{≥ C}

,then

A(n, C

′

_{, ∆) ≤ A(n, C, ∆)}

. (iv) If

∆

′

_{≥ ∆}

,then

A(n, C, ∆

′

_{) ≥ A(n, C, ∆)}

. (v)

A(n, C, ∆) ≥ n

for all

∆ ≥ 1

.

(vi) If

C ≥

n∆

2

,

A(n, C, ∆) = n

. Proof:

(i) The request graph an onsist in a perfe t mat hing, soany solution uses1 ADM pernode.

(ii) A

∆

-regular graph an be partitioned into

n∆

2

disjoint edges, and we annot do better.

(iii) Anysolutionfor

C

isalsoasolutionfor

C

′

. (iv) If

∆

′

≥ ∆

, the subgraphswith maximum degreeat most

∆

are a sub lass of the lassofgraphswithmaximumdegreeatmost

∆

′

. (v) Combine(i)and(iv).

(vi) Inthis asealltheedgesoftherequest graphtintoonesubgraph.

2

Sin e we are interestedin the numberof ADMs requiredat ea h node, letus onsider thefollowingdenition:

Denition2.1 Let

M (C, ∆)

bethe least positive number

M

su hthat,for any

n ≥ 1

,the inequality

A(n, C, ∆) ≤ M n

holds.

Lemma2.2

M (C, ∆)

isanatural number.

Proof: Firstofall,weknowbyLemma2.1that,forany

C ≥ 1

,

A(n, C, ∆) ≤ A(n, 1, ∆) =

∆n

. Thus

A(n, C, ∆)

is upper-bounded by

∆n

. On the other hand, sin eanyvertex may appearin therequestgraph,

A(n, C, ∆)

islower-boundedby

n

.

Supposenowthat

M

isnotanaturalnumber.Thatis,suppose that

r < M < r + 1

for somepositivenaturalnumber

r

.Thismeansthat, forea h

n

,thereexistsatleastafra tion

r

M

oftheverti eswithatmost

r

ADMs.Forea h

n

,let

V

n,r

bethesubsetofverti esofthe requestgraphwithatmost

r

ADMs.Then,sin e

r

M

> 0

,wehavethat

lim

n→∞

|V

n,r

| = ∞

. Inotherwords,there isanarbitrarilybigsubsetofverti eswithatmost

r

ADMs per ver-tex. But we an onsider arequest graph with maximumdegree at most

∆

on the set of verti es

V

n,r

, andthis means that with

r

ADMspernode is enough,a ontradi tion with

theoptimalityof

M

.

2

Iftherequestgraphisrestri tedtobelongtoasub lassofgraphs

C

ofthe lassofgraphswith maximumdegreeatmost

∆

,thenthe orrespondingpositiveintegerisdenoted

M (C, ∆, C)

. Again,foranysub lass

C

,

M (C, ∆, C) ≤ M (C, ∆)

.

CombiningLemmas2.1and2.2,weknowthat

M (C, ∆)

de reasesbyintegerhopswhen

C

in reases.Onewouldliketohaveabetterknowledgeofthosehops.Thefollowinglemma givesasu ient onditiontoassurethan

M (C, ∆)

de reasesbyatmost1when

C

in reases by1.

Lemma2.3 If

C > ∆

,then

M (C + 1, ∆) ≥ M (C, ∆) − 1

.

Proof: Suppose that

M (C + 1, ∆) ≤ M (C, ∆) − 2

, and let us arrive at ontradi tion. Beginningwithasolutionfor

C + 1

,wewillseethatadding

n

ADMs(i.e.in reasing

M

by1) weobtainasolutionfor

C

,a ontradi tionwiththeassumption

M (C, ∆) ≥ M (C +1, ∆)+2

.

Therequestgraphhasatmost

∆n

2

edges,andtheninasolutionfor

C + 1

thenumberof subgraphswithexa tly

C + 1

edgesisatmost

∆n

2(C+1)

.Allthesubgraphswith

C

orlessedges analso beused in asolution for

C

. We removean edge from ea h one of thesubgraphs with

C + 1

edges,obtainingat most

∆n

2(C+1)

edges,orequivalentlyat most

∆n

C+1

additional ADMs.Wewantthis numbertobeat most

n

, i.e.

∆n

C + 1

≤ n ,

whi h isequivalentto

∆ ≤ C + 1

,andthisistruebyhypothesis.

2

Weprovidenowalowerboundon

M (C, ∆)

.

Proposition 2.1(General Lower Bound)

M (C, ∆) ≥



C+1

C

∆

2

 .

Proof: Sin ewehaveto onsiderallthegraphswith maximumdegreeatmost

∆

, we an restri tourselvesto

∆

-regulargraphswith girthgreaterthan

C

.Then, thebestone ould dois to partitionthe edgesof therequest graphinto treeswith

C

edges. Inthis ase,the sumof thedegreesof all theverti esin ea h subgraph is

2C

.Thus, the average degreeof theverti esin all thesubgraphsis atmost

2C

C+1

, hen eitexists at leastonevertex

v

with averagedegreenotgreaterthan

2C

C+1

. Therefore,

v

must appear in at least

M

v

subgraphs, with

2C

C+1

M

v

≥ ∆

.Thus,

M (C, ∆) ≥



C+1

C

∆

2



.

2

Corollary2.1 If the setof requestsisgiven bya

∆

-regulargraphof girth greater than

C

, then

M (C, ∆) ≥

 ∆

2



Proof: TrivialfromProposition2.1.

2

Ifthevalueof

C

islargein omparisonto

n

thenumberofADMsrequiredpernodemay belessthan

M (C, ∆)

asstatedin thefollowinglemma:

Lemma2.4

A(n, C, ∆) ≤



n∆

2C

 n.

Proof: Thenumberofedgesofarequestgraphwithdegree

∆

isatmost

n∆

2

.We an parti-tionthisedgesgreedilyintosubsetsofatmost

C

edges,obtainingatmost



n∆

2C



subgraphs.

Thus, in this partition ea h vertex appears in at most



n∆

2C



subgraphs, as we wanted to

prove.

2

Noti e that this is notin ontradi tion with Corollary 2.1, sin e the inequality of the denitionof

M (C, ∆)

mustholdforall valuesof

n

.

3 Case

∆ = 2

Proposition 3.1

A(n, C, 2) = 2n − (C − 1).

Proof: Considerthe ase whenthe requestgraphis

2

-regular andhas girthgreaterthan

C

.Then,afeasiblesolutionis obtainedbypla ing

2

ADMsat ea h vertex. Whatwedois to ountin howmanyADMs we anassurethatwe anpla eonlyoneADM.

Letusseerstthatwe annotuse

1

ADMinmorethan

C −1

verti es.Supposethis,i.e.that wehave

1

ADMin

C

verti esand

2

inalltheothers.Then, onsideraset ofrequestsgiven bya y le

H

oflength

C + 1

ontainingallthe

C

verti eswith

1

ADM insideit,andother y les ontainingtheremainingverti es.Inthissituation,wearefor edtouse

2

subgraphs fortheverti esof

H

,andatleast

2

verti esof

H

mustappearinbothsubgraphs.Hen ewe willneedmorethan

1

ADMinsomevertexthathadinitially only

1

ADM.

Now, let us see that we an always save

C − 1

ADMs. Let

{a

0

, a

1

, . . . , a

C−2

}

be the set of verti eswith only

1

ADM, that we an hoose arbitrarily. We will see that we an

de omposethesetofrequestsinsu hawaythattheverti es

a

i

alwayslieinthemiddleofa pathora y le, overinginthiswaybothrequestsofea hvertexwithonly

1

ADM.Indeed, supposerst thattwoofthese verti es(namely,

a

i

and

a

j

)do notappear onse utivelyin oneof the disjoint y les of the set of requests. Let

b

i

be the nearest vertex to

a

i

in the y lein thedire tion of

a

j

, and onverselyfor

b

j

(

b

i

maybeequalto

b

j

if

a

i

and

a

j

dier only onone vertex). Then, onsider twopaths (eventually, y les)of the form

{b

i

, a

i

, . . .}

and

{b

j

, a

j

, . . .}

,toassurethatboth

a

i

and

a

j

liein themiddleofthesubgraph.Wedothe same onstru tionforea hpairofnon- onse utiveverti es.

Now, onsideralltheverti es

{a

0

, . . . , a

i

, . . . , a

t−1

}

whi hareadja entinthesame y le oftherequest graph,with

t ≤ C − 1

. Let

b

0

bethe nearestvertexto

a

0

dierent from

a

1

, and let

b

t−1

be the nearestvertexto

a

t−1

dierentfrom

a

t−2

. Then, onsider asubgraph withthepath(or y le, if

b

0

= b

t−1

)

{b

0

a

0

a

1

. . . a

t−1

b

t−1

}

.

2

4 Case

∆ = 3

We studythe ases

C = 3

and

C ≥ 5

in Se tions4.1 and 4.2, respe tively. Wedis uss theopen ase

C = 4

in Se tion5.

4.1 Case

C

= 3

We study rst the ase when the request graph is abridgeless ubi graphin Se tion 4.1.1, andthenthe aseofageneralrequestgraphin Se tion4.1.2.

4.1.1 BridgelessCubi RequestGraph

Wewill need somepreliminarygraphtheoreti al on epts.Let

G = (V, E)

be agraph. For

A, B ⊆ V

,anA-B path in

G

isapathfrom

x

to

y

,with

x ∈ A

and

y ∈ B

.If

A, B ⊆ V

and

X ⊆ V ∪ E

aresu hthateveryA-B pathin

G

ontainsavertexoranedgefrom

X

,we saythat

X

separates thesets

A

and

B

in

G

. Moregenerallywesaythat

X

separates

G

if

G − X

isdis onne ted,thatis, if

X

separatesin

G

sometwoverti esthat arenotin

X

.A separatingset ofverti esisaseparator.Avertexwhi h separatestwoother verti esofthe same omponentisa ut-vertex,andanedgeseparatingitsendsisabridge.Thus,thebridges in agrapharepre isely those edgesthat do notlieon any y le.A set

M

ofindependent edgesin agraph

G = (V, E)

is alled amat hing. A

k

-regularspanning subgraphis alled a

k

-fa tor.Thus, asubgraph

H ⊆ G

isa1-fa torof

G

ifandonlyif

E(H)

isamat hingof

V

.Were allawellknownresultfrom mat hingtheory:

Theorem4.1(Petersen, 1981) Every bridgeless ubi graph hasa1-fa tor.

Then,ifweremovea1-fa torfroma ubi graph,whatitremainsisadisjointsetof y les.

Corollary4.1 Everybridgeless ubi graphhasade ompositionintoa1-fa toranddisjoint y les.

1

2

3

4

5

6

7

8

9

10

11

12

13

14

A

B

C

D

E

F

G

a)

b)

u

w

v

c)

Fig.1a)De omposition of abridgeless ubi graphintodisjoints y lesand a1-fa tor. b) De omposition of a bridgeless ubi graphinto paths of length

3

. ) Cubi bridgeless graphusedin theproofof Proposition4.1

Anexampleofade ompositionofabridgeless ubi graphintodisjoints y lesanda1-fa tor isdepi tedinFig.1a.

Proposition 4.1 Let

C

bethe lass of bridgeless ubi graphs. Then,

M (3, 3, C) = 2.

Proof: Letusproofthatwe analwayspartitiontherequestgraphintopathswith3edges in su h away that ea h vertex appears in 2 paths. To do so, we take the de omposition given byProposition 4.1, together with a lo kwise orientation of theedges of ea h y le. With this orientation,ea h edgeof the1-fa torhastwoin oming andtwooutgoing edges ofthe y les.Forea hedgeofthe1-fa torwetakeitstwoin omingedges,andforminthis wayapathoflength

3

.Itiseasytoverifythatthisisindeedade ompositioninto pathsof lengththree.Forinstan e, ifwedothis onstru tionin thegraphof Fig.1a, andwelabel theedgesofthe1-fa toras{A,B,...,G}andtheonesofthe y lesas{1,2,...,14}(seeFig. 1b),weobtainthefollowingde omposition:

{1, A, 6}, {5, B, 2}, {3, C, 8}, {7, D, 9}, {14, E, 11}, {10, F, 12}, {4, G, 13}

Nowletus see that we annot dobetter, i.e.with

2n − 1

ADMs. If su h asolutionexists, there would be at least onevertex with only

1

ADM, and the average of the number of ADMs of all theother verti esmust not ex eed

2

. In order to see that this is notalways possible, onsiderthe ubi bridgelessgraphon

10

verti esof Fig.1 . Let

w

bethevertex withonly

1

ADM.Thisgraphhasnotrianglesex eptthose ontaining

w

.Sin ewe anuse only

1

ADMin

w

,wemusttakeallitsrequestsin onesubgraph.Itisnotpossibleto over the

4

remaining requests of the nodes

u

and

v

in one subgraph, and thus without lossof generalitywewill need

3

ADMs in

u

.With these onstraints,one an he kthat thebest

solutionuses

20

ADMs,that is

2n > 2n − 1

.

2

Taking alookat theproofweseethat theonlypropertythatweneedfrom thebridgeless ubi graph is that we an partition it into a1-fa tor and disjoint y les. Hen e,we an relaxthehypothesisofProposition 4.1toobtainthefollowing orollary:

Corollary4.2 Let

C

be the lass of graphsof maximum degreeat most3 that anbe par-titionedintodisjoints y lesanda1-fa tor. Then

(i)

M (3, 3, C) = 2

;and

(ii)

M (C, 3, C) ≤ 2

for any

C ≥ 4

.

4.1.2 General RequestGraph

It turns out that when the request graph is not restri ted to be bridgelesswe have that

M (3, 3) = 3

.

Proposition 4.2

M (3, 3) = 3

.

Proof: By (ii) and (iii) of Lemma 2.1 we know that

M (3, 3) ≤ 3

. We shall exhibit a ounterexample showing that

M (3, 3) > 2

, proving the result. Consider the ubi graph

G

depi ted in Fig. 2a. We will prove that it is not possible to partition the edges of

G

into subgraphswith at most3 edgesin su h awaythat ea h vertexappearsin at most 2 subgraphs.

a

A

e

0

A

A

a

a

a

2

1

1

1

e

e

2

2

3

3

3

a)

b)

c)

H

G

0

a

e

Fig.2a) Cubi graph

G

that annot be edge-partitionedinto subgraphs with atmost 3 edges in su h a way that ea h vertex appears in at most 2 subgraphs. b) Graph that annotbepartitionedinto2 onne tedsubgraphswithatmost3edges. )Counterexample ofProposition4.2showingthat

M (3, 3) = 3

Indeed,supposetheopposite, i.e.thatwe anpartitiontheedgesof

G

into ( onne ted) subgraphs

B

1

, . . . , B

k

with

|E(B

i

)| ≤ 3

in su hawaythat ea h vertexappearsin atmost 2subgraphs,andletusarriveat a ontradi tion.

FollowingthenotationillustratedinFig.2a,let

A

1

, A

2

, A

3

bethe onne ted omponents of

G\{e

1

, e

2

, e

3

}

.Letalso,withabuseofnotation,

a

i

= A

i

∩e

i

,

i = 1, 2, 3

,and

a

0

= e

1

∩e

2

∩e

3

.

Claim1 There exist an index

i

∗

_{∈ {1, 2, 3}}

and a subgraph

B

k

∗

ontaining

a

0

, su h that

B

k

∗

∩ A

i

∗

= {a

i∗

}

.

Proof: Among allthesubgraphs

B

1

, . . . , B

k

involvedin thede ompositionof

G

, onsider the

ℓ

subgraphs

B

j

1

, . . . , B

j

ℓ

overingthe edges

{e

1

, e

2

, e

3

}

. If

ℓ = 1

, then the subgraph

B

j

1

isastarwith threeedges and enter

a

0

, andthen

B

j

1

∩ A

i

= {a

i

}

forea h

i = 1, 2, 3

. If

ℓ ≥ 3

, then the vertex

a

0

appears in 3 subgraphs, a ontradi tion. Hen e it remains to handlethe ase

ℓ = 2

.If the laimwasnottrue,it would implythat forea h

i = 1, 2, 3

it would exist

j

f(i)

∈ {j

1

, j

2

}

su h that

B

f(i)

∩ A

i

ontainsat least oneedge. In parti ular, thiswould implythatthegraphdepi tedinFig. 2b ouldbepartitionedinto 2 onne ted

subgraphswithatmost3edges,whi his learlynotpossible.

2

Suppose withoutlossofgeneralitythat theindex

i

∗

givenbyClaim 1isequalto 1.Thus,

a

1

appearsinasubgraph

B

k

∗

thatdoesnot ontainanyedgeof

A

1

.Therefore,theedgesof

A

1

mustbepartitionedinto onne tedsubgraphswithat most3edges,insu hawaythat

a

1

appearsinonly1subgraph,andallitsotherverti esinatmost2subgraphs.Letusnow seethatthis isnotpossible,obtainingthe ontradi tionwearelookingfor.

Indeed,sin e

a

1

hasdegree2in

A

1

andit anappearinonlyonesubgraph,itmusthave degreetwointhesubgraphinwhi hitappears,i.e.inthemiddleofa

P

3

ora

P

4

,be ause

A

1

istriangle-free.Itiseasytoseethatthisisequivalenttopartitioningtheedgesofthegraph

H

depi tedin Fig. 2 into onne tedsubgraphswith atmost 3edges,in su h awaythat thethi kedge

e

appearsin asubgraphwithatmost2edges,andea hvertexappearsinat most2subgraphs.Observethat

H

is ubi andtriangle-free.Let

n

1

bethetotalnumberof verti esofdegree1in allthesubgraphsofthede ompositionof

H

.Sin eea hvertexof

H

anappearinatmost2subgraphsand

H

is ubi ,ea hvertex anappearwithdegree1in atmost1subgraph.Thus,

n

1

≤ |V (H)| = 6

.

Sin ewehavetouseatleast1subgraphwithatmost2edgesand

|E(H)| = 9

,thereare at least

1 +



9−2

3

 = 4

subgraphs in thede omposition of

H

.But ea h subgraphinvolved in thede omposition of

H

has at least 2verti es of degree 1, be ause

H

is triangle-free.

Therefore,

n

1

≥ 8

,a ontradi tion.

2

4.2 Case

C

≥ 5

For

C ≥ 5

we aneasily provethat

M (C, 3) = 2

, makinguseof a onje ture madeby Bermondet al.in1984[5℄andprovedbyThomassenin 1999[17℄:

Theorem4.2([17℄) Theedgesofa ubi graph anbe2- oloredsu hthatea h mono hro-mati omponent isapathof lengthatmost5.

Alinear

k

-forest isaforest onsistingofpathsoflengthatmost

k

.Thelinear

k

-arbori ity ofagraph

G

istheminimumnumberoflinear

k

-forestsrequiredto partition

E(G)

, andis denotedby

la

k

(G)

[5℄.Theorem4.2isequivalenttosayingthat,if

G

is ubi ,then

la

2

(G) = 2

. Letusnowseethat Theorem4.2impliesthat

M (C, 3) = 2

forall.Indeed,allthepaths ofthelinearforestshaveatmost

5

edges,and ea hvertexwillappearin exa tly

1

ofthe

2

linearforests, sothe de omposition given byTheorem [17℄ is apartition of theedges of a ubi graphintosubgraphswithatmost

5

edges,insu hawaythatea hvertexappearsin

at most

2

subgraphs. Infa t the resultof [17℄ isstronger, in thesense that

G

anbe any graphofmaximumdegreeatmost3.Thus,wededu ethat

Corollary4.3 For any

C ≥ 5

,

M (C, 3) = 2

.

Thomassen also proved [17℄ that 5 annot be repla ed by 4 in Theorem 4.2. This fa t do not imply that

M (4, 3) = 3

, be ause of the following reasons : (i) the subgraphs of the de omposition of the request graph are not restri ted to be paths, and (ii) it is not ne essaryto beabletonda2- oloringof thesubgraphsofthede omposition (a oloring in this ontext meansthat ea h subgraphre eivesa olor,and 2subgraphswiththe same olormusthaveemptyinterse tion).

5 Con lusions

Wehave onsideredthetra groomingprobleminunidire tionalWDMringswhenthe requestgraphbelongstothe lassofgraphwithmaximumdegree

∆

.Thisformulationallows the network to support dynami tra without re onguring theele troni equipment at thenodes.Wehaveformallydenedtheproblem,andwehavefo usedmainlyonthe ases

∆ = 2

and

∆ = 3

, solving ompletely the former and solving all the ases of the latter, ex eptthe asewhen thegroomingvalue

C

equals

4

.Wehaveprovedin Se tion4.1.2that

M (3, 3) = 3

, andin Se tion 4.2that

M (C, 3) = 2

for all

C ≥ 5

. Be ause oftheintegrality of

M (C, ∆)

andLemma2.1,

M (4, 3)

equals either2or3.We onje turethat

Conje ture 5.1 The edges of a graph with maximum degree at most 3 an bepartitioned into subgraphs with at most 4 edges, in su h away that ea h vertex appears in at most 2 subgraphs.

If Conje ture 5.1 is true, it learlyimplies that

M (4, 3) = 2

. Corollary4.2 states that

M (4, 3, C) = 2

,

C

beingthe lassofbridgelessgraphsofmaximumdegreeatmost3. Never-theless,ndingthevalue of

M (4, 3)

remainsopen. Wehavealso dedu edlowerandupper boundsinthegeneral ase(anyvalueof

C

and

∆

).Tab.1summarizesthevaluesof

M (C, ∆)

thatwehaveobtained.

C \ ∆

1

2

3

4

5

6

. . .

∆

1

1

2

3

4

5

6

. . .

∆

2

1

2

3

4

5

6

. . .

∆

3

1

2

3

≥ 3

≥ 4

≥ 4

. . .

≥



2∆

3



4

1

2

2??

≥ 3

≥ 4

≥ 4

. . .

≥



5∆

8



5

1

2

2

≥ 3

≥ 3

≥ 4

. . .

≥



3∆

5



. . .

1

. . .

. . .

. . .

. . .

. . .

. . .

. . .

C

1

2

2

≥



C+1

C

2



≥



C+1

C

5

2



≥



C+1

C

3



. . . ≥



C+1

C

∆

2



Tab.1Valuesof

M (C, ∆)

. The ase

C = 4, ∆ = 3

isa onje turedvalue

This problem an ndwide appli ationsin thedesignof opti alnetworks using WDM te hnology.Itwouldbeinterestingto ontinuethestudy forlargervaluesof

∆

, whi h will ertainly relyon graphde omposition results. Anothergeneralization ould be to restri t therequestgraphtobelongtoother lassesofgraphsforwhi hthereexistpowerful de om-positiontools,likegraphswithbounded tree-width,orfamiliesofgraphsex ludingaxed graphasaminor.

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