$(\ell,k)$-Routing on Plane Grids

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Omid Amini, Florian Huc, Janez Zerovnik

To cite this version:

Omid Amini, Florian Huc, Janez Zerovnik. (ℓ, k)-Routing on Plane Grids. [Research Report] RR-6480,

INRIA. 2008. �inria-00265297v2�

inria-00265297, version 2 - 25 Mar 2008

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Thème COM

(ℓ, k)-Routage dans les Grilles Planaires

Omid Amini — Florian Huc — Ignasi Sau

— Janez Žerovnik

N° 6480

OmidAmini

†

, FlorianHu

‡

,Ignasi Sau

‡ §

, Janez šerovnik

¶ k

ThèmeCOMSystèmes ommuni ants ProjetMASCOTTE

Rapportdere her he n°6480February200832pages

Résumé:Leproblèmederoutagedepaquetsjoueunrletrèsimportantdanslesréseaux detélé ommuni ation. Il onsisteàenvoyerdes donnéesentre les sommetsd'un réseauen untempsraisonable.Dansle

(ℓ, k)

-routage, haquesommetpeutenvoyerauplus

ℓ

paquets et enre evoir

k

. Leroutage depermutation orrespond au as

ℓ = k = 1

.Dans leroutage

r

- entral, tousles noeuds à distan e au plus

r

d'un sommetxé envoientun paquet à e sommet.Dans etarti lenousétudions lesproblèmespré édentsdanslesgrillesplanaires. Onutiliselemodèlestore-and-forward et

∆

-port. Nous onsidéronslesréseauxhalf et full-duplex.Lesrésultatsprin ipauxsont:

1. un algorithme tight pour le problème de routage de permutation dans les grilles hexagonalesfull-duplex,etdanslesgrilleshexagonalesettriangulaireshalf-duplex. 2. unalgorithmetightpourleroutage

r

- entraldanslesgrillestriangulairesethexagonales. 3. un algorithme tight pour le

(ℓ, ℓ)

-routage dans les grilles arrées, triangulaires et

hexagonales.

4. de bon algorithme d'approximation (en terme de temps de al ul) pour le

(ℓ, k)

-routagedanslesgrilles arrées,triangulairesethexagonales.Ondonneaussidenouvelles bornesinférieures surletemps d'éxé utiond'un algorithmequiutilise leroutage par leplus ourt hemin.

Tous esalgorithmessont omplètementsdistribués.

Mots- lés : Routage de paquets, algorithme distribué,

(ℓ, k)

-routage, grilles planaires, routagedepermutation,plus ourt hemin,algorithmesansmémoire

∗

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.

†

Max-Plan k-InstitutfürInformatik,Saarbrü ken,Germany

‡

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

§

GraphTheoryandCombinatori sgroup,MA4,UPC,Bar elona,Spain

¶

UniversityofLjubljanaFa ultyofMathemati sandPhysi s(IMFM),jubljana,Slovenia

k

Abstra t: Thepa ketroutingproblemplaysanessentialrolein ommuni ationnetworks. Itinvolveshowtotransferdatafromsomeoriginstosomedestinationswithin areasonable amountof time. In the

(ℓ, k)

-routingproblem, ea h node ansend at most

ℓ

pa kets and re eive at most

k

pa kets. Permutation routing is the parti ular ase

ℓ = k = 1

. In the

r

- entralroutingproblem,allnodesatdistan eatmost

r

fromaxednode

v

wanttosend apa ketto

v

.

Inthis arti lewestudy thepermutation routing,the

r

- entral routingandthegeneral

(ℓ, k)

-routingproblemson planegrids,that issquaregrids, triangulargridsandhexagonal grids.Weusethestore-and-forward

∆

-portmodel,andwe onsiderbothfullandhalf-duplex networks.Themain ontributionsarethefollowing:

1. Tightpermutationroutingalgorithmsonfull-duplexhexagonalgrids,andhalfduplex triangularandhexagonalgrids.

2. Tight

r

- entralroutingalgorithmsontriangularandhexagonalgrids. 3. Tight

(k, k)

-routingalgorithmsonsquare,triangularandhexagonalgrids.

4. Goodapproximationalgorithms(intermsofrunningtime)for

(ℓ, k)

-routingonsquare, triangularandhexagonalgrids,togetherwithnewlowerboundsontherunningtime ofanyalgorithm usingshortestpathrouting.

All these algorithms are ompletely distributed, i.e. anbeimplementedindependently at ea h node. Finally, we also formulate the

(ℓ, k)

-routing problem as a Weighted Edge Coloringproblem onbipartitegraphs.

Key-words: Pa ketrouting,distributedalgorithm,

(ℓ, k)

-routing,planegrids,permutation routing,shortestpath,obliviousalgorithm

1 Introdu tion

In tele ommuni ation networks, it is essential to be able to route ommuni ations as qui kly aspossible.Inthis ontext,thepa ket routing problemplaysa apital role.Inthis problem weare given anetwork and aset of pa ketsto berouted through thenodesand the edges of the network graph.A pa ket is hara terized by an originand a destination node,andtypi allyanedge an beusedbynomorethanonepa ketatthesametime.The obje tiveistondanalgorithmto omputeas heduletorouteallpa ketswhi hminimizes thetotaldeliverytime.Thisproblemhasbeenwidelystudiedin theliteratureundermany dierent assumptions. In 1988, Leighton, Maggs and Rao proved in their seminal arti le [31,29℄theexisten eofas heduleforroutinganysetofpa ketswithedge-simplepathson ageneralnetwork,in optimaltimeof

O(C + D)

steps.Here

C

isthe ongestion(maximum numberof paths sharingan edge) and

D

thedilation (length of a longest path)and it is assumedthatthepathsaregivenapriori.Theproofof[29℄usedLovászLo alLemmaand wasnon onstru tive.Thisresultwasfurtherimprovedin[28℄wherethesameauthorsgave anexpli italgorithm,using theBe k's onstru tiveversionoftheLo al Lemma.

Thesealgorithmsto omputetheoptimals heduleare entralized.Thenin[38℄Ostrovsky andRabanigaveadistributedrandomizedalgorithmrunningin

O(C +D +log

1+ǫ

_(n))

steps. Wegiveamoredetailed overviewinSe tion 1.1.

Although these resultsareasymptoti allytight,theydeal withageneralnetwork,and inmany asesitispossibletodesignmoree ientalgorithmsbylookingatspe i pa ket ongurations or network topologies. For instan e, is it natural to bound the maximum numberofmessagesthatanode ansendorre eive.Wefo usonthispointin Se tion1.2, wherewewillformallydenetheproblemunder studyinthispaper.

On theother hand,thenetworkunder studyplaysamajorroleonthequalityandthe simpli ity of thesolution. For example,in aradio wireless environment, ellularnetworks areusually modeled byahexagonal grid where nodesrepresentbase stations. The ellsof thehexagonalgrids havegood diameter to arearatio and still havea simplestru ture. If enters of neighboring ells are onne ted, the resulting graph is alled atriangular grid. Noti ethat hexagonalgrids aresubgraphs of thetriangular grid.We will talk aboutsu h networksin Se tion1.3.In this paperwefo uson thestudy of the

(ℓ, k)

-routingproblem in onvexsubgraphs, i.e. that ontainall shortest pathsbetween all pairsof nodesof the square,triangularandhexagonalgrid.

1.1 General Results on Pa ket Routing

In this se tion weprovidea fast overviewof the state-of-the-art of the generalpa ket routingproblem,in boththeo-line andon-linesettingsin Se tions1.1.2and1.1.3 respe -tively,fo usingmostlyonthelatter.Webeginbyre allingthree lassi allowerboundsfor thepa ketroutingproblem.

1.1.1 Classi al lower bounds

In the pa ket routing problem, there are three lassi al types of lowerbounds for the runningtimeof anyalgorithm:

1. Distan e bound :thelongestdistan e overthepathsofallpa kets(usually alled dilation, denoted

D

) onstitutes a lowerbound on the number of steps required to routeallthepa kets.

2. Congestion bound : the ongestion of an edge of the network is dened as the numberof paths usingthis edge. Then, the greatest ongestion overalltheedges of thenetwork(denoted

C

)isalsoalowerbound onthenumberofsteps,sin eat ea h stepanedge anbeusedbyatmostonepa ket.

3. Bise tion bound : Let

G = (V, E)

be the graphwhi h models the network, and

F

_{⊆ E}

a ut-set dis onne ting

G

into two omponents

G

1

and

G

2

. Let

m

be the number of pa kets with origin in

G

1

and destination in

G

2

. Then, the number of routingstepsusedbyanyalgorithmwillbeatleast

l

m

|F |

m

. 1.1.2 O-line routing

Givenasetofpa ketstobesentthroughanetwork,apathsystemisdenedastheunion ofthepathsthatea hpa ketmustfollow.Forageneralnetworkandanysetof

n

demands, wehaveseeninSe tion1.1.1thatthedilationandthe ongestionprovidetwolowerbounds fortheroutingtime.Thisprovesthatthe

dilation + congestion

ofapathssystemusedfor the routing pro edure is alower bound of twi e the routing time. In a elebrated paper, Leighton,MaggsandRaoprovedthefollowingtheorem :

Theorem1.1([31℄) Forany setof requests andapath systemfor these requests,thereis ano-lineroutingproto olthat needs

O(C + D)

stepstorouteallthe requests,where

C

is the ongestion and

D

isthe dilationof thepath system.

Inaddition,in[49℄theauthorsshowthat,giventhesetofpa ketstobesent,itispossible to nd in polynomialtime a path system with

C + D

within a fa tor 4of the optimum. Thus,Theorem1.1 anbeannoun edinamoregeneralway:

Theorem1.2([49℄) Foranysetofrequests,thereisano-lineroutingproto olthatneeds

O(C + D)

stepstoroutealltherequests,where

C + D

isthe minimum

congestion + dilation

overallthe possiblepath systems.

Furthermore,thisroutingproto ol usesxedbuersize,i.e. thequeuesize atallnodes is bounded bya onstantat ea h step.Nevertheless, itis importantto noti e that ahuge onstantmay be hidden inside the

O

notation. As we have said in the introdu tion, this resultwasfurtherimprovedin[28℄wherethesameauthorsgaveanexpli italgorithm.These algorithmsto omputetheoptimals heduleare entralized.Inadistributedalgorithmnodes mustmaketheirde isionsindependently,basedonthepa ketstheysee,withouttheuseof a entralizeds heduler. Thenin [38℄ OstrovskyandRabanigaveadistributed randomized algorithmrunningin

O(C + D + log

1+ǫ

_(n))

WerefertoS heideler'sthesis[45℄ fora omplete ompilationofgeneralpa ket routing algorithms.

1.1.3 On-line routing

Intheon-linesetting,theoldeston-lineproto olthatdeviatesonlybyafa torlogarithmi in

n

fromthebestpossibleruntime

O(C + D)

forarbitrarypath- olle tionsistheproto ol presentedbyLeighton,MaggsandRaointhesamepaper[31℄,runningin

O(C +D log(Dn))

stepswithhighprobability.Theauthors allthealgorithmon-line,ratherthandistributed. This s hedule assumes that the paths are given a priori, hen e it does not fo us on the problemof hoosing thepathstoroutethepa kets.

The resultsof[1℄ providearoutingalgorithm that is

log n

ompetitivewith respe t to the ongestion.Inotherwords,itisworsethananoptimalo-linealgorithmonlybyafa tor

log n

.Inthissettingthedemandsarriveonebyoneandthealgorithmroutes allsbasedon the urrent ongestion onthevariouslinks inthenetwork,sothis anbea hievedonlyvia entralized ontrolandserializingtheroutingrequests.In[3℄theauthorsgaveadistributed algorithm that repeatedly s ans the network so as to hoose the routes. This algorithm requiressharedvariablesontheedgesofthenetworkandhen eishardtoimplement.Note that thetwoon-line algorithmsabovedepend onthe demands andare thereforeadaptive. Re all that an oblivious routing strategy is spe ied by a path system

P

and afun tion

w

assigninga weight to everypathin

P

. This fun tion

w

hasthe propertythat for every sour e-destinationpair

(s, t)

,thesystemofowpaths

P

s,t

for

(s, t)

fullls

P

q∈P

s,t

w(q) = 1

. One anthink ofthis fun tion asafrequen ydistribution amongseveral pathsgoingfrom anorigin

s

toadestination

t

.Inadaptiverouting,however,thepathtakenbyapa ketmay alsodependonotherpa ketsoreventstakingpla einthenetworkduringitstravel.Remark thateveryobliviousroutingstrategyis obviouslyon-lineanddistributed.

The rstpaperto performa worst asetheoreti alanalysis onoblivious routingisthe paper of Valiant and Brebner [54℄, who onsidered routing on spe i network topologies su h asthe hyper ube.They givearandomized oblivious routingalgorithm.Borodinand Hop roft [6℄ and subsequently [22℄ have shown that deterministi oblivious routing algo-rithms annotapproximate welltheminimalloadonanynon-trivialnetwork.

In are ent paper, Rä ke[40℄ gavethe onstru tionof a polylog ompetitiveoblivious routingalgorithm for general undire ted networks. It seems truly surprising that one an ome losetominimal ongestionwithoutanyinformationonthe urrentloadinthenetwork. This resulthasbeenimprovedin [4℄. Lowerbounds onthe ompetitiveratio ofoblivious routinghavebeenstudiedforvarioustypesofnetworks.Forexample,forthe

d

-dimensional mesh, Maggs et al. [35℄ gavethe

ω(

C

_∗

d

(log n))

lowerbound on the ompetitiveratioof an obliviousalgorithmonthemesh, where

C

∗

istheoptimal ongestion.

Sofar,theobliviousalgorithmsstudiedintheliteraturehavefo usedonminimizingthe ongestionwhileignoringthedilation.Infa t,thequalityofthepathsshouldbedetermined by the ongestion

C

, and the dilation

D

. An open question is whether

C

and

D

an be ontrolledsimultaneously.Anappropriateparameterto apturehowgoodisthedilationof apath systemis thestret h, denedasthemaximumoverallpa ketsoftheratiobetween

the length of the path taken by the routing proto ol and the length of a shortest path from sour e to destination.In a re entwork,Bush et al. [8℄ onsidered again the ase of the

d

-dimensionalmesh. Theypresentedanon-linealgorithm forwhi h

C

and

D

areboth within

O(d

2

₎

ofthepotentialoptimal,i.e.

D =

O(d

2

_D

∗

)

and

C =

O(dC

∗

log(n))

,where

D

∗

is the optimal dilation(remark that by [35℄, it is impossibleto havea fa tor better than

ω(

C

∗

d

(log n))

).

There is asimple ounter-example network thatshowsthat in generalthe twometri s (dilation and ongestion) are orthogonal to ea h other : take an adja ent pair of nodes

u, v

and

Θ(

√

_n)

disjoint paths of length

Θ(

√

_n)

between

u

and

v

. For pa kets traveling from

u

to

v

, any routing algorithm that minimizes ongestion has to use all the paths, however,in this way somepa kets followlongpaths, giving high stret h. Nevertheless, in grids[8℄,andinsomespe ialkindofgeometri networks[7℄the ongestioniswithina poly-logarithmi fa tor from optimal and stret h is onstant (

d

the dimension). As mentioned beforeaninterestingopenproblemistondother lassesofnetworkswherethe ongestion andstret hareminimizedsimultaneously[2℄.Possible andidatesforsu hnetworks ouldbe forexamplebounded-growthnetworks,ornetworkswhosenodesareuniformly distributed in losedpolygons,whi h des ribeinteresting asesofwireless networks.

There entpaperofMaggs[34℄surveysa olle tionoftheoreti alresultsthatrelatethe ongestion and dilation of the paths taken by a set of pa kets in a network to the time requiredfortheirdelivery.

1.2 Routing Problems

Theinitialandnalpositioningofthepa ketshasadire tinuen eonthetimeneeded fortheirrouting.Consideringstati pa ket onguration,themoststudied onstraintsrefer tothemaximumnumberofpa ketsthatanode ansendandre eive.Duetotheirpra ti al importan e,someoftheseproblemshavespe i names:

1. Permutation routing: ea h nodeis theoriginandthedestination ofat mostone pa ket.Tomeasure the routing apabilityof aninter onne tion network,the partial permutation routing(PPR)problemisusuallyusedasthemetri .

2.

(ℓ, k)

-routing:ea hnodeistheoriginofatmost

ℓ

pa ketsanddestinationofatmost

k

pa kets.Permutation routing orresponds to the ase

ℓ = k = 1

of

(ℓ, k)

-routing. Anotherimportantparti ular aseisthe

(1, k)

-routing,inwhi hea hnode ansend atmostonepa ketandre eiveat most

k

pa kets.

3.

(1, any)

-routing : ea h node is originof at most one pa ket but there are no on-straintsonthenumberofpa ketsthatanode anre eive.

4.

r

- entralrouting :allnodesatdistan e atmost

r

ofa entralnodesend one mes-sagetothis entral node.

Inall these problems we are given an initial pa ket onguration,and theobje tive is to route allpa kets to theirrespe tive destinations minimizing thetotal routingtime, under the onstraintthatea hedge anbeusedbyat mostonepa ket atthesametime.

Besidesof the onstraintsabouttheinitialand nalpositionsofthepa kets, therealso exist dierent routing models at the intermediate nodes of the network.For instan e, in thehot potato model no pa ket anbestored at thenodes ofthenetwork,whereasin the store-and-forward at ea h step apa ket aneither stayat a nodeormoveto anadja ent node.Anotherwidelyused modelisthewormhole routing.

On theother hand,one an onsider onstraints onthe number ofin identedges that ea hnodeofthenetwork anusetosendorre eivepa ketsatthesametime.Inthe

∆

-port model[16℄,ea hnode ansendorre eivepa ketsthroughallitsin identedgesatthesame time.

In this arti le we study the store-and-forward

∆

-port model. In addition, we suppose that ohabitationofmultiple pa kets at thesamenode is allowed.I.e. aqueue isrequired forea houtgoingedgeatea h node.

The nature of the links of the network is another fa tor that inuen es the routing e ien y. The typeof links is usually one of thefollowing: full-duplex orhalf-duplex. In thefull-duplex asethereare twolinks betweentwoadja entnodes,onein ea h dire tion. Hen etwopa kets antransit,oneinea hdire tion,simultaneously.Inthehalf-duplex ase onlyonepa ket antransitbetweentwonodes,eitherinonedire tionoftheedgeorinthe other.Inthispaperwefo usonbothhalf andfull-duplexlinks.

1.3 Topologies

Wenowgiveabriefsummaryofvarious asesof

(ℓ, k)

-routingand

(1, any)

-routingthat have been studied for several spe i topologies. More pre isely, in Se tion 1.3.1 we list themostimportantresultsfor somenetworkswhi h haveattra ted agreatinterestin the literature, like hyper ubes and ir ulant graphs. Then we move to plane grids in Se tion 1.3.2. It is well known that there exist only three possible tessellations of the plane into regular polygons [55℄ : squares, triangles and hexagons.These graphsare those whi h we studyinthis arti le.

1.3.1 Dierent network topologies

In[20℄ theauthorsstudied thepermutation routingproblem in low-dimensional hyper- ubes (

d

≤ 12

). They gave optimal or good in the worst ase oblivious algorithms, i.e. algorithmsin whi h thepath used byapa ket isentirelydetermined byits originandits destination.An othernetworkwidelystudiedin theliteratureisthetwodimensionalmesh withrowand olumnbuses.Thisnetwork analsobediversieda ordingtothe apa ities of thebuses. In [51℄ Suel gavea deterministi algorithm to solvethepermutation routing problemin su hnetworks.Itgivesas hedule usingat most

n + o(n)

stepsandaqueueof size 2, where the queue is the maximum number of pa kets that have to be stored at an intermediatenode.Healsoproposedadeterministi algorithmfor

r

-dimensionalarrayswith busesworking in

(2

−

1

r

)n + o(n)

stepsandstillusingqueuesofsize 2.In[27℄,theauthors studiedthe

(ℓ, ℓ)

-routingprobleminthemeshgridwithtwodiagonalsandgave,for

ℓ

≥ 9

a deterministi algorithmusing

2ℓn

In[19℄,theauthors introdu edanalgorithm alled big foot algorithm.The ideaofthis algorithm isto identify twotypesof links and to movetowardsthedestination using rst thelinksofthersttypeandthenthoseofthese ondtype.Thealgorithmswedevelopwill usesu hastrategy.Theygiveanoptimal entralizedalgorithmforthepermutationrouting probleminfull-duplex2- ir ulantgraphsanddouble-loopnetworks(i.e.oriented2- ir ulant graphs).

Another network of great pra ti al importan e is thedouble-loop network : anetwork modeledbyagraphwithvertexset

v =

{v

0

, . . . , v

n−1

}

su hthat there aretwointegers

h

1

and

h

2

su hthat

E =

{v

i

v

i±h

1

, v

i

v

i±h

2

}

.Thepermutationroutingprobleminthisnetworkis studiedin[14℄.Theauthorsgaveanalgorithmforthepermutationroutingproblemwhi hin meanuses

1.12ℓ

steps(themeanbeingempiri allymeasured).In[15℄theauthorsdes ribed anoptimal entralizedpermutationroutingalgorithmin

k

- ir ulantgraphs(

k

≥ 2

),andin [42℄anoptimaldistributedpermutation routingin2- ir ulantgraphswasobtained.

Theproblemhasbeenalsostudied forpa ketsarrivingdynami ally.In[18℄,theauthor gave an optimal online s hedule for the linear array. He also gavea 2-approximation for ringsandshowthat,usingshortestpathrouting,nobetterapproximationalgorithmexists. In[21℄,theauthorsstudiedCubeConne tedCy les:

CCC(n, 2

n

₎

(hyper ubesofdimension

n

where ea h node isrepla edby a y le oflength

n

). Theygaveanalgorithm working in

O(n

2

₎

with

O(1)

buersfortheonlinepartialpermutationrouting(PPR).

1.3.2 Plane grids

Maybe the most studied networks in the literature are the two dimensional grids (or planegrids),andamong themin parti ularthesquare gridhasdeservedspe ial attention. Letus brieyoverviewwhat hasbeenpreviouslydoneon

(ℓ, k)

-routinginplanegrids.

In[32℄therstoptimalpermutationrouting(withrunningtime

2n

−2

),andqueuesofsize 1008appears.Thequeuesizeisredu edin[41℄to112andfurtherin[48℄to81.Furthermore, in[48℄theauthorsprovideanotheralgorithmrunningin near-optimaltime

2n +

O(1)

steps with a maximum queue size of only 12. [36℄ gives an asymptoti ally optimal algorithm for

(1, k)

-routing on plane grids, with queues of small onstant size.They introdu ed for therst timethe

(1, k)

-routingand the

(1, any)

-routingproblems. Thisresultwasfurther improved in [47℄, where theauthors gave near-optimal deterministi algorithm running in

√

k

n

₂

+

_O(n)

steps.Theygaveanotheralgorithm,slightlyworseintermsofnumberofsteps, but with queues of size only 3.They also studied the general problem of

(ℓ, k)

-routing in squaregrids.Theyproposedlowerboundsand near-optimalrandomizedanddeterministi algorithms.Theynallyextendedthemtohigherdimensionalmeshes.Theyperformed

(ℓ, ℓ)

-routingin

O(ℓn)

steps,the lowerbound being

Ω(

√

ℓkn)

for

(ℓ, k)

-routing.Finally, in [39℄, theauthorsgavedeterministi andrandomizedalgorithmsfor

(ℓ, k)

-routinginsquaregrids, with onstantqueue size.Therunning time is

O(

√

ℓkn)

steps,whi h isoptimal a ording totheboundof[47℄.Thiswork losedagapintheliterature,sin eoptimalalgorithmswere onlyknownfor

ℓ = 1

and

ℓ = k

.

Nodes in a hexagonalnetwork are pla edat the verti es of aregulartriangular tessel-lation,sothat ea h node hasup to six neighbors.Inother words,ahexagonalnetwork is

Fig.1Hexagonalnetwork(

△

)andhexagonaltessellation(

9

)

anitesubgraph ofthetriangular grids.These networks havebeenstudiedin avarietyof ontexts, spe ially in wireless and inter onne tion networks.The most known appli ation maybeto model ellularnetworkswithhexagonalnetworkswherenodesarebasestations. Butthese networkshavebeenalsoapplied in hemistry tomodelbenzenoidhydro arbons [52,25℄,in imagepro essingand omputergraphi s[26℄.

Inaradio ommuni ationwirelessenvironment[37℄,theinter onne tionnetworkamong base stations onstitutesahexagonalnetwork,i.e. atriangular grid,asit isshownin Fig. 1.

Tessellationof theplanewith hexagonsmaybe onsideredasthemostnaturalbe ause ellshave optimal diameter to area ratio. Hexagonal networks are nite subgraphsof the triangulargrid.The triangulargrid analsobeobtainedfrom thebasi 4-meshbyadding NE to SW edges, whi h is alled a6-mesh in [53℄. Here westudy onvex subgraphs, i.e. that ontain all shortest paths between all pairs of nodes, of the square, triangular and hexagonal grid. Summarizing, to the best of our knowledge the only optimal algorithms on erning

(ℓ, k)

-routingon plane grids (a ording to the lower bound of [47℄) have been foundonsquaregrids,butmoduloa onstantfa tor[39℄.Ontriangularandhexagonalgrids, thebest resultsarerandomizedalgorithmswithgoodperforman e[46℄.

1.4 Our Contribution

In this paperwestudy the permutation routing,

r

- entral and

(ℓ, k)

-routingproblems onplanegrids,thatissquaregrids,triangulargridsandhexagonalgrids.Weusethe store-and-forward

∆

-portmodel,andwe onsider bothfullandhalf-duplexnetworks.

WehaveseeninSe tion1.3.2thattheonlyplanegridforwhi hthereexistedanoptimal

(ℓ, k)

-routingis thesquaregrid. Inaddition, whatis importantisthat theresultsof these arti les on erning

(ℓ, k)

-routing in plane grids are optimal modulo a onstant fa tor. In this paperwe improve these resultsby giving tight algorithms in luding the onstant, in the ases of square, triangular and hexagonalgrids. It is important to stress that all the algorithmspresentedin thispaperaredistributed(ex epttheonegivenin Appendix B),

i.e. anbeimplementedindependentlyatea hnode.Ouralgorithmsonlyuseshortestpaths, thereforetheya hieveminimumstret h. Inaddition, thealgorithms areoblivious, sothey anbeusedin anon-lines enario,unless theperforman e guaranteesthat weproveapply onlytotheo-line ase.Themain newresultsofthisarti learethefollowing:

1. First tight (also in luding the onstant fa tor) permutation routing algorithms in full-duplexhexagonalgrids,andhalfduplextriangularandhexagonalgrids.

2. Firsttight(alsoin ludingthe onstantfa tor)

r

- entral routingalgorithms in trian-gularandhexagonalgrids.

3. First tight (also in luding the onstant fa tor)

(k, k)

-routing algorithms in square, triangularandhexagonalgrids.

4. Goodapproximationalgorithmsfor

(ℓ, k)

-routinginsquare,triangularandhexagonal grids.

Thispaperisstru turedasfollows.InSe tion2westudythepermutationroutingproblem. Althoughpermutationroutinghadalreadybeensolvedforsquaregrids,webegininSe tion 2.1byillustratingouralgorithmforsu hgrids.TheninSe tion2.2wegiveatight permuta-tionroutingalgorithmforhalf-duplextriangulargrids,usingtheoptimalalgorithm of[44℄. InSe tion 2.3 weprovideatightpermutation routingalgorithm for full-duplexhexagonal gridsandatightpermutationroutingalgorithmforhalf-duplexhexagonalgrids.InSe tion 3wefo uson

(1, any)

-routing,givinganoptimal

r

- entralroutingalgorithmsforthethree typesofgrids.WenallymoveinSe tion4tothegeneral

(ℓ, k)

-routingproblem.Weprovide adistributedalgorithmfor

(ℓ, k)

-routinginanygrid,usingtheideasoftheoptimalalgorithm forpermutationrouting.Wealsoprovelowerboundsfortheworst- aserunningtimeofany algorithmusingshortestpathrouting.Inaddition,theselowerboundsallowustoprovethat ouralgorithmturns outto betightwhen

ℓ = k

,yielding in this wayatight

(k, k)

-routing algorithm in square, triangular and hexagonal grids. We also propose in Appendix B an approa hto

(ℓ, k)

-routingintermsofagraph oloringproblem:theWeightedBipartite Edge Coloring.Wegivea entralizedalgorithmusingthisredu tion.

2 Permutation Routing

AswehavealreadysaidinSe tion1,inthepermutationroutingproblem,ea hpro essor is the originof at most onepa ket and thedestination of no more than onepa ket.The goalistominimizethenumberoftimestepsrequiredtorouteallpa ketstotheirrespe tive destinations. It orresponds to the ase

ℓ = k = 1

of the general

(ℓ, k)

-routing problem. This problem has been studied in a wide diversity of s enarios, su h as Mobile Ad Ho Networks[23℄,Cube-Conne tedCy le(CCC) Networks [21℄,Wirelessand RadioNetworks [10℄,All-Opti alNetworks[33℄andRe ongurableMeshes[9℄.

Inagridwithfull-duplexlinksanedge anbe rossedsimultaneouslybytwomessages, onein ea h dire tion.Equivalently, ea h edge betweentwonodes

u

and

v

is made of two independentar s

{uv}

and

{vu}

,asillustratedinFig.2a.

x

y

z

b)

u

v

uv

vu

a)

Fig.2a)Ea hedge onsistsoftwoindependentlinks.b)Axisusedinatriangulargrid

Remark2.1 If the network ishalf-duplex, itis easy to onstru t a 2-approximation algo-rithmfrom an optimal algorithm for the full-duplex aseby introdu ing odd-evensteps, as explainedfor examplein[14 ℄.

2.1 Square grid

Many ommuni ationnetworksarerepresentedbygraphssatisfyingthefollowing prop-erty:foranypairofnodes

u

and

v

,theedgesofashortestpathfrom

u

to

v

anbepartitioned into

k

disjoint lasses a ordingto awell-dened riterium. Forinstan e, on atriangular grid the edges of a shortest path an be partitioned into positive and negative ones [44℄. Similarly, on a

k

- ir ulant graph the edges anbe partitioned into

k

lassesa ording to theirlength.

Ingraphsthat satisfythis propertythere existsa naturalroutingalgorithm : routeall pa ketsalongone lassofedgesafteranother.Forhexagonalnetworksthisalgorithmturns outtobeoptimal[44℄.Optimalityfor2- ir ulantgraphsisprovedusingastati approa hin [19℄,andre entlyusingadynami distributedalgorithmin[42℄.In[19℄theauthorsintrodu e thenotionofbig-footalgorithmsbe ausetheiralgorithmroutespa ketsrstalonglonghops andthenalongshorthops ina2- ir ulantgraph.

On the square grid, thebig-foot algorithm onsists of two phases,moving ea h pa ket rsthorizontallyandthenverti ally.Inthiswayapa ketmaywaitonlyduringthese ond phase.Usingthatalldestinationsaredistin t,theoptimalityforsquaregridiseasytoprove. Summarizing,it an beprovedthat

Theorem2.1 Thereisatranslationinvariant obliviousoptimal permutationrouting algo-rithmfor full-duplexnetworks thatare onvexsubgraphsofthe innite squaregrid.

2.1.1 Regarding the queue size

Of ourse,thisisnottherstoptimalpermutationroutingresultonsquaregrids,asthe lassi al

x

− y

routing(rst route pa kets through the horizontal axis, and then through theverti alaxis)hasbeenusedforalongtime.Thus,anothermore hallengingissueisto redu e thequeuesize,aswehavealreadydis ussed in Se tion 1.3.2. Leightondes ribesin [30℄asimpleo-line algorithmforsolvinganypermutationroutingproblemin

3n

− 3

steps ona

n

× n

squaregrid,usingqueuesofsizeone.Sin ethediameterofa

n

× n

squaregridis

2n

−2

,thisalgorithmprovidesa

3

2

-approximation.Themaindrawba kisthatthisalgorithm iso-lineand entralized.In ontrast,ourobliviousdistributedalgorithmisoptimalinterms of running time, but it is easy to see that on a

n

× n

square grid, the queue size an be

n−1

2

.Up todate, thebest algorithmrunninginoptimaltimeto routepermutation routing instan eson squaregridsis thealgorithm ofSibeyn et al.[48℄, usingqueuesof size 81.So far,there isnoalgorithm that guaranteesoptimalrunningtimewith queuesofsize 1,and itisunlikelythatsu hanalgorithmexists.

Remark2.2 The same observation regarding the unbounded queue size applies to all the algorithmsdes ribedinthisarti le.However,ouraimistomat htheoptimalrunningtime, rather than minimizing the queue size. Additionally, it turns out that some appropriate modi ationsofthe permutationroutingalgorithmsthatweprovide forplane gridsallow us to nd oblivious algorithms whi h route any permutation within a fa tor 3 of the optimal runningtime, and usingqueues of size1(in fa t, we an saysomething stronger :wejust needmemory to keep 1 message atea h node). We do not des ribe these modi ations in thisarti le.

2.2 Triangular grid

We use the addressings heme introdu ed in [37℄ and used also in [44℄ : we represent anyaddressonabasis onsisting ofthreeunitary ve torsi,j, k onthedire tions ofthree axis

x, y, z

with a

120

degreeangle among them, interse ting on an arbitrary (but xed) node

O

. This node is the origin and is given the address O

= (0, 0, 0)

. This basis is represented in Fig. 2b. Thus, we an assume that ea h node

P

∈ V

is labeled with an addressP

= (P

1

, P

2

, P

3

)

expressed in this basis{i, j, k} withrespe t to theorigin

O

. At the beginning, ea h node

S

knowsthe address of the destination node

D

of the message pla edinitially at

S

,and omputestherelativeaddress

−→

SD =

D

−

S ofthemessage.Note thatthisrelativeaddressdoesnotdepend onthe hoi eoftheoriginnode

O

.Thisrelative addressistheonlyinformationthatisaddedintheheadingofthemessagetobetransmitted, onstitutinginthiswaythepa ketto besentthroughthenetwork.

Using thati

+

j

+

k

= 0

,it iseasyto see thatif

(a, b, c)

and

(a

′

_{, b}

′

_{, c}

′

₎

aretherelative addressesoftwopa kets,then

(a, b, c) = (a

′

_{, b}

′

_{, c}

′

₎

ifandonlyifthereexists

d

∈ Z

su hthat

a

′

_{= a + d}

,

b

′

_{= b + d}

,and

c

′

_{= c + d}

.

Wesaythatanaddress

−→

SD = (a, b, c)

isoftheshortestpathformifthereisapathfrom node

S

tonode

D

, onsistingof

a

units ofve tori,

b

unitsofve torj and

c

unitsofve tor k, andthispathhastheshortestlength.

Theorem2.2([37℄) Anaddress

(a, b, c)

isofthe shortestpathformif andonlyif atleast one omponentiszero, andanytwo omponents donot havethe samesign.

Corollary2.1 ([37℄) Anyaddresshas auniqueshortestpathform.

Thus,ea haddress

−→

SD

writtenin theshortestpathform hasatmosttwonon-zero ompo-nents,and theyhavedierent sign.In fa t,itis easyto nd theshortest pathform using thenextresult.

Theorem2.3([37℄) If

−→

SD = a

i

+ b

j

+ c

k,then

|

−→

SD

_{| = min(|a − c| + |b − c|, |a − b| + |b − c|, |a − b| + |a − c|).}

Permutationroutingonfull-duplextriangulargridshasbeensolvedre ently[44℄ attain-ingthedistan elowerboundof

ℓ

max

routingsteps,where

ℓ

max

isthemaximumlengthover theshortestpathsofallpa kets tobesentthroughthenetwork.

AssaidinRemark2.1,thenetworkishalf-duplex,one an onstru ta2-approximation algorithmfromanoptimalalgorithmforthefull-duplex asebyintrodu ingodd-evensteps. Thus, using this algorithm we obtainan upper bound of

2ℓ

max

for half-duplex triangular grids.

Let us show withan examplethat this naïvealgorithm is tight. That is, we shallgive an instan e requiring at least

2ℓ

max

running steps, implying that no better algorithm for a general instan e exists. Indeed, onsider a set of nodes distributed along a line on the triangulargrid. Wex

ℓ

max

andan edge

e

onthis line, and put

ℓ

max

pa kets at ea h side of

e

alongtheline, at distan eat most

ℓ

max

− 1

from anend-vertexof

e

. Forea hpa ket, ea hdestinationispla edattheothersideof

e

withrespe ttoitsorigin,atdistan eexa tly

ℓ

max

from the origin.It is easy to he k that the ongestion of

e

(that is, thenumber of shortestpaths ontaining

e

) is

2ℓ

max

, and thus any algorithm using shortestpath routing annotperforminlessthan

2ℓ

max

steps.On theother hand,

ℓ

max

is alowerbound forany distan e,yieldingthat theapproximationratioofouralgorithmisat most

2

.

Previousobservationsallowus tostatethenextresult:

Theorem2.4 Thereexistsatightpermutationroutingalgorithmfor half-duplextriangular gridsperforminginatmost

2ℓ

max

steps,where

ℓ

max

isthemaximumlengthovertheshortest paths ofall pa ketstobesent.This algorithm isa2-approximation algorithm for ageneral instan e.

2.3 Hexagonal grid

In a hexagonalgrid one an dene three typesof zigzag hains [50℄, represented with thi klinesinFig.3.Similarlytothetriangulargrid,inthehexagonalgridanyshortestpath between twonodes uses at most twotypes of zigzag hains [50℄. Let us now givea lower boundfortherunningtimeofanyalgorithm.Considertheedgelabeledas

e

inFig.3,and thetwo hains ontainingit(thoseshapingan

X

).Fix

ℓ

max

and

e

,andputonemessageon allnodespla edatboth hainsat distan eat most

ℓ

max

− 1

fromanendvertexof

e

.As in

the aseof thetriangulargrid, hoose thedestinations to bepla edontheother side of

e

alongthesamezigzag hainthantheoriginatingnode,atdistan eexa tly

ℓ

max

fromit.Itis learthatalltheshortestpaths ontain

e

.Itisalsoeasyto he kthatthe ongestionof

e

is inthis ase

4ℓ

max

− 4

, onstitutedofsymmetri loads

2ℓ

max

− 2

inea hdire tionof

e

.Thus,

2ℓ

max

− 2

establishesalowerboundfortherunningtimeofanyalgorithminthefull-duplex ase,whereas

4ℓ

max

− 4

isalowerbound forthehalf-duplex ase,undertheassumptionof shortestpathrouting.

e

Fig.3Zigzag hains inahexagonalgrid

Letusnowdes ribearoutingalgorithmwhi hrea hesthisbound.Wehavethree types of edges a ording to the angle that they form with any xed edge. Ea h edge belongs to exa tly two dierent hains, and onversely ea h hain is made of two types of edges. Moreover, in an innitehexagonalgrid any

2

hainsof dierenttype interse t exa tlyon oneedge.

Givenapairoforiginanddestinationnodes

S

and

D

,itispossibletoexpresstherelative address

D

− S

ountingthenumberofstepsusedbyashortestpathsonea htypeof hain. Inthiswayweobtainanaddress

D

− S = (a, b, c)

onageneratingsystemmadeofunitary ve tors following thedire tions of the three types of hains (itis not abasis in the stri t sense,sin etheseve torsarenotlinearlyindependentontheplane.However,wewill allit so).Choosethisbasissothatthethreeve torsformanglesof

120

degreesamongthem.As ithappensonthetriangulargrid[37, 43℄,there areatmosttwonon-zero omponents(see [50℄), andin that asetheymust havedierent sign.Nevertheless, nowthe addressis not unique,sin eanedgepla edatthebent (thatis,a hangefromatypeof hainto another) ofashortestpathispartofbothtypesof hains. Anyway,thisambiguityis notaproblem inthealgorithmthat propose,aswewillsee below.

Suppose rst that edges are bidire tional or, said otherwise, full-duplex. Roughly, the ideais to usethe optimalalgorithm for triangulargridsdes ribedin [43℄, and adapt it to hexagonalgrids. For that purpose welabelthe three types of zigzag hains

c

1

, c

2

, c

3

, and thethree typesofedges

e

1

, e

2

, e

3

. Without lossofgenerality, welabel them in su h away that

c

1

usesedgesoftype

e

2

and

e

3

,

c

2

uses

e

1

and

e

3

,and

c

3

uses

e

1

and

e

2

(seeFig.4).

e

c

c

c

1

2

3

1

e2

e3

Fig.43typesof hainsandedgesinahexagonalgrid

Forea htypeofedge,wedenetwophasesa ordingtothetypeof hainthatusesthis typeofedge.Thisdenestwoglobalphases,namely:duringPhase1,

c

1

uses

e

2

,

c

2

uses

e

3

, and

c

3

uses

e

1

. Conversely,duringPhase2

c

1

uses

e

3

,

c

2

uses

e

1

, and

c

3

uses

e

2

.

Wesupposethatatea hnodepa ketsaregroupedintodistin tqueuesa ordingtothe nextedge(a ordingtotherulesofthealgorithm)alongitsshortestpath.Giventherelative addresses

D

− S

intheform

(a, b, c)

,thealgorithm anbedes ribedasfollows.

Atea hnode

u

ofthenetwork:

1) Duringtherststep,moveallpa ketsalongthedire tionoftheirnegative omponent. Ifapa ket'saddresshasonlyapositive omponent,moveitalongthisdire tion. 2) Fromnowon, hangealternativelybetweenPhase1andPhase2.Atea h step(the

sameforbothphases):

a) Ifthere are pa kets with negative omponents,send them immediately alongthe dire tionofthis omponent.

b) Ifnot,forea houtgoingedgeorderthepa ketsinde reasingnumberofremaining steps,andsendtherstpa ketofea hqueue.

3) Ifatsomepoint,allthepa ketsin

u

haveremainingdistan eone,sendthem imme-diately.

Letusanalyzethe orre tnessandoptimalityofthisalgorithm.

In1)allpa kets anmove,sin einitiallythereisatmostonepa ketatea hnode.In2a), there anonlybeonepa ket withnegative omponentat ea h outgoingedge [43℄. In2b) the pa ketwithmaximumremaininglengthatea houtgoingedgeisunique,sin eallthese pa ketsaremovingalongtheirlastdire tion(theirnegative omponentisalreadynished, otherwisethey would be in 2a)) and ea h node is the destination of at most onepa ket. Hen e,usingthisalgorithmevery

2

steps(oneofPhase1andoneofPhase2)themaximum remaining distan e over all pa kets de reases by one. Moreover, during the rst step all pa kets de reasetheir remaining distan e byone.Be ause of this, after the

(2ℓ

max

− 3)th

stepthemaximumremainingdistan e hasde reasedatleast

1 +

2ℓ

max

−4

2

= ℓ

max

− 1

times, hen e the maximumremaining distan e is 1, and we are in 3). Sin e all destinations are dierent, all pa kets an rea h simultaneouslytheir destinations. Thus, thetotal running timeisatmost

2ℓ

max

− 3 + 1 = 2ℓ

max

− 2

,meetingtheworst aselowerbound.Again,

ℓ

max

isalowerboundforanyinstan e, hen ethealgorithm onstitutesa2-approximationfora generalinstan e.

Theorem2.5 Thereexistsatightpermutationroutingalgorithm for full-duplex hexagonal grids performing in

2ℓ

max

− 2

steps, where

ℓ

max

is the maximum length over the shortest pathsof allpa kets tobesent.

Remark2.3 The optimality stated in Theorem 2.5 is true only under the assumption of shortest path routing. This means that for ertain tra instan es the total deliver time may be shorter if some pa kets do not go through their shortest path. To illustrate this phenomenon, onsider the example of Fig.5. Node labeled

i

wants to send a message to node labeled

i

′

, for

i = 1, . . . , 8

. We have that

ℓ

max

= 5

,and thus our algorithm performs in

2ℓ

max

− 2 = 8

steps. It is lear that all shortest paths use edge

e

, and its ongestion bottlene ks the running time. Supposenow that we route the messages originating at even nodes through the path dened by the edges

{abcd}

, instead of

{fe}

,andkeep the shortest pathroutingformessagesoriginatingatoddnodes.One an he kthatwiththisroutingonly

7

stepsarerequired.

2

4

e

1

5

3

6

6'

1'

2'

3'

4'

5'

8

7

7'

8'

d

b

a

f

c

Fig.5Shortestpathisnotalwaysthebest hoi e

Inthehalf-duplex ase,justintrodu eagainodd-evenstepsinbothphases.Thus,wehave Phase1-even,Phase1-odd,Phase2-even,and Phase2-odd, whi h takepla esequentially. Now,1) onsistsobviouslyoftwosteps(even/odd). Usingthisalgorithm,every

4

stepsthe maximum remaining distan e de reases by one. In addition, during the rst

2

steps and duringthelast

2

stepsallpa ketsde reasetheirremainingdistan ebyone.Thus,thetotal runningtimeisatmosttwi ethetimeofthefull-duplex ase,thatis

2(2ℓ

max

−2) = 4ℓ

max

−4

steps,meeting again thelowerbound for therunningtime ofanyroutingalgorithm using shortest path routing. Again, this algorithm onstitutes a 4-approximation for a general instan e.

Theorem2.6 Thereexistsantightpermutationroutingalgorithmforhalf-duplexhexagonal grids performing in

4ℓ

max

− 4

steps, where

ℓ

max

is the maximum length over the shortest pathsof allpa kets tobesent.

Remark2.4 AsexplainedinAppendix A,thereexistsanembedding ofthe triangular grid into the hexagonal grid with load, dilation, and ongestion

2

. Using this embedding, any algorithmperformingon

k

stepsonthetriangulargirdperformson

2k

stepsonthehexagonal grid. Using this fa t, we obtain a permutation routing algorithm on full-duplex hexagonal gridsperformingon

2ℓ

max

steps.NotethattheoptimalresultgiveninTheorem2.5isslightly better.

Thesameapplies tohalf-duplexhexagonal networks,witharunningtimeof

4ℓ

max

using the embedding, in omparison to

4ℓ

max

− 4

stepsgiven byTheorem2.6 .

3

(1, any)

-Routing

Inthis asetheroutingmodelis thefollowing :ea hpa ket hasatmost onepa ket to send,but thereareno onstraintsonthedestination.That is,intheworst aseallpa kets anbesenttoonenode.Thisspe ial asewhereallpa ketswantto sendamessagetothe samenode in often alled gathering in theliterature[5℄. Noti ethat this routingmodel is on eptuallydierentfrom the

(1, k)

-routing,where themaximumnumberof pa kets that anode an re eiveisxedapriori.

Square grid Assume rst that edges are bidire tional. The modi ations for the half-duplex asearesimilartothoseexplainedin thepreviousse tion.

We willfo uson the asewhere allpa kets surroundinga givenvertexwantto send a pa kettothat vertex.We allthis situation entral routing,andifwewanttospe ifythat allnodesatdistan eatmost

r

fromthe enterwanttosendapa ket,wenoteitas

r

- entral routing.Notethat thissituationisrealisti inmanypra ti alappli ations,sin ethe entral vertex anplaytheroleofarouteroragatewayin alo al network.

Lemma3.1(Lower Bound) The number of steps required in a

r

- entral routing is at least

r+1

2

.

Proof: Letususethebise tionbound [17℄toprovetheresult.Itiseasyto ountthe num-berof pointsat distan eat most

r

from the enter, whi his

4

r+1

2

.Now onsiderthe ut onsistingofthefouredgesoutgoingfrom the entralvertex.Allpa ketsmusttraverseone oftheseedgestoarrivetothe entralvertex.This utgivesthebise tionboundof

4

r+1

2

/4

routingsteps.

2

Letusnowdes ribeanalgorithm meetingthelowerbound.

Proposition 3.1 Thereexists an optimal

r

- entral routingalgorithm on squaregrids per-forming in

r+1

2

routingsteps.

Proof: Express ea h node address in terms of the relative address with respe t to the entralvertex.Inthiswayea hnodeisgivenalabel

(a, b)

.Then,forea hpa ketpla edin anodewithlabel

(a, b)

ourroutingalgorithmperformsthefollowing:

•

If

ab > 0

,sendthepa ket alongtheverti alaxis.

•

If

ab < 0

,sendthepa ket alongthehorizontalaxis.

Queuesaremanagedsothatthepa ketshavinggreaterremainingdistan ehavepriority. This routing divides the square grid into 4subregions surrounding the entral vertex, asshown in Fig.6. Thetypeof routingperformedin ea h subregionissymbolized byan arrow.

Fig.6DivisionofthegridintheproofofProposition3.1

Letusnow omputetherunningtimeinthe

r

- entral ase.Itisobviousthatusingthis algorithmallpa ketsaresenttothe4axisoutgoingfromthe entralvertex.The ongestion oftheedgeintheaxis ontainingthe entralvertexalongea hlineis

1+2+3+. . .+r =

r+1

2

.

Sin eatea hsteponepa ketrea hesitsdestinationalongea hline,we on ludethat

r+1

2

isthetotalrunningtimeofthealgorithm.

2

Triangular grid Thesameideaofthesquare gridappliesto thetriangulargrid.Inthis ase, the number of nodes at distan e at most

r

is

6

r+1

2

. The ut is made of

6

edges. Dividingtheplaneonto6subregionsgivesagainanoptimalalgorithmperformingin

r+1

2

steps.

Hexagonal grid The same idea gives an optimal routing in the

r

- entral ase. In this ase the degreeof ea h vertex is 3, and then it is easy to he k (maybe a drawing using Fig.3 anhelp)that there are

3

r+1

2

nodesatdistan e atmost

d

that maywantto send amessage to the entral vertex,and the ut hassize 3.As expe ted, therunning time is again

r+1

2

. 4

(ℓ, k)

-Routing

Re all that in thegeneral

(ℓ, k)

-routingproblem ea hnode ansend atmost

ℓ

pa kets and re eive at most

k

pa kets. We propose a distributed approximation algorithm using

the ideasof the algorithms that wehave developed for the permutation routing problem. Wealso providelowersbounds for therunning time of any algorithm using shortestpath routing,that allowusto provethatouralgorithmistightwhen

ℓ = k

,onanygrid. Remark4.1 We also propose in Appendix B an approa h tond asolution of the

(ℓ, k)

-routing problem, on any grid, using the problem of Weighted Edge Coloring in a bi-partitegraph. Nevertheless,the algorithm obtainedusingthis approa h is entralized.

We start by des ribing the results for full-duplex triangular grids. The results an be ompletely adapted to square grids, but we fo us on the other grids sin ethere were few resultsin theliterature. Wealso showhowto adapt theresultsto hexagonalgridsand to

thehalf-duplexversion.Inthisse tionwedenote

c :=

l

max{ℓ,k}

min{ℓ,k}

m

=

max{

ℓ

k

,

k

ℓ

}



.Notethat

c

_{≥ 1}

. Lemma 4.1 and Lemma 4.2 provide twolowerbounds for therunning time of any algorithmusingshortestpaths.

Lemma4.1(First lowerbound) Theworst- aserunningtimeofanyalgorithmfor

(ℓ, k)

-routingon full-duplextriangular grids usingshortestpath routingsatises

Runningtime

≥ min{ℓ, k} · ℓ

max

Proof: Consideraset of

ℓ

max

nodespla edalong aline, pla ed onse utivelyat oneside ofadistinguishededge

e

.Ea hnodewantstosend

min

{ℓ, k}

messagesto thenodespla ed at theother side of

e

along theline, at distan e

ℓ

max

from it. Thenthe ongestion of

e

is

min

_{{ℓ, k} · ℓ}

max

,givingthebound.

2

Denition4.1 Given a vertex

v

, we all the re tangle of side

(a, b)

starting at

v

the set

R

v

a,b

=

{v + α

i

+ β

j

, 0

≤ α < a, 0 ≤ β < b}

We allsu hare tangle asquareif

a = b

.Noti e thatinthe triangulargridthenode setisgeneratedby

{

i

,

j

,

k

}

,wherek

=

−

i

−

j,aswehave explainedinSe tion2.2 .

Using standardgraphterminology,givena graph

G = (V, E)

andasubset

S

⊆ V

,the set

Γ(S)

denotesthe(open)neighborhoodin

G

oftheverti esin

S

.Thefollowingtheorem anbefound,forexample,in [13℄.

Theorem4.1(Corollary ofHall's theorem [13℄) Let

G = (V, E)

be abipartite graph, with

V = X

∪ Y

.If for allsubsets

A

of

X

,

|Γ(A)| ≥ c|A|

,then for ea h

x

∈ X

,thereexists

S

x

⊂ Y

su hthat

|S

x

| = c

,and

∀x, x

′

_{∈ X}

,

S

x

∩ S

x

′

=

∅

and

S

x

⊂ Γ(x)

. Weusethis theoremtoprovethefollowinglowerbound.

Lemma4.2(Se ond lower bound) The worst- ase running time of any algorithm for

(ℓ, k)

-routingon full-duplextriangular grids usingshortestpath routingsatises

Runningtime

≥

 max{ℓ, k}

4

·

 ℓ

max

+ 1

√

c + 1



,

where

c =

l

max{ℓ,k}

min{ℓ,k}

m

.

Proof: Supposewithoutlossofgeneralitythat

ℓ

≥ k

,otherwiserepla e

ℓ

by

max

{ℓ, k}

.Let

v

bea vertex, and onsider thesquare

R

v

d,d

, with

d :=

j

ℓ

max

_√

+1

c+1

k

.We laim that allnodes

insidethissquare ansend

ℓ

messagessu hthatalldestinationnodesareinthedestination set

D = R

v

d+ℓ

max

,d+ℓ

max

\R

v

d,d

.Let

S

bethesubgridgeneratedbypositivelinear ombinations of the ve torsi and j. More pre isely,

S :=

{v + α

i

+ β

j

, α

≥ 0, β ≥ 0}

. Fig.2b gives a graphi alillustration.

To provethis, we onsider a bipartite graph

H

on vertex set

R

v

d,d

∪ D

, with an edge betweena vertex of

R

v

d,d

anda vertex of

D

if theyare at distan e at most

ℓ

max

in

S

. To applyTheorem4.1,wehavetoshowthat anysubsetofverti es

A

⊂ R

v

d,d

hasatleast

c

|A|

neighbors in

H

. Theorem4.1 will thenensurethe existen eofafeasible repartitionofthe messagesfrom verti es of

R

v

d,d

to those of

D

su h that they alltravela distan e at most

ℓ

max

.

Given

A

⊂ R

v

d,d

, let us all

D

A

:=

{u ∈ D :

dist

S

(A, u)

≤ ℓ

max

}

, where dist

S

(A, u)

meanstheminimumdistan ein

S

from anyvertexof

A

tothevertex

u

.Forany

A

⊂ R

v

d,d

, weneedtoshowthat

|D

A

| ≥ c|A|

(1)

Without lossofgeneralitywesupposethat

A

ismaximal, in thesense that there isno set

A

′

stri tly ontaining

A

with

D

A

= D

A

′

. Insteadof onsideringall possiblesets

A

, we willshowbelowthatwe anrestri tourselvestore tangles.Hen egivenaset

A

,wedenote by

R

A

thesmallestre tangle ontainingthesubsetofverti es

A

.Werst laimthat

|D

R

A

\ D

A

| ≤ |R

A

\ A|

(2)

Indeed,thisequality anbeshownbyindu tionon

|R

A

\A|

.For

|R

A

\A| = 0

theequality istrivial.Supposethat itis truefor

|R

A

\ A|

. Theindu tion step onsists inshowingthat there is an element

x

in

R

A

\ A

su h that

|D

R

A

\ D

A∪{x}

| − |D

R

A

\ D

A

| ≤ 1

(note that

D

R

_A∪{x}

= D

R

A

):

•

Ifthereexists

x

su hthat

x +

jand

x

−

iarein

A

and

x

−

jisnotin

A

,thenwesele t this

x

.From

x

theonlynewvertexwemayaddto

D

A

is

x + ℓ

max

i.

•

Otherwise,ifthere exists

x

su h that

x

−

jand

x

−

i arein

A

and

x +

i isnotin

A

, thenwesele tthis

x

.Inthis asetheonlynewvertexwemayaddto

D

A

is

x

− ℓ

max

k.

•

If none ofthe previous asesholds, sin e

R

A

is thesmallest re tangle ontaining

A

, and

A

ismaximal,then ne essarilythere existsan

x

su h that

x +

i and

x

−

j arein

A

and

x

−

iisnotin

A

.Wesele tthis

x

,andtheonlynewvertexwemayaddto

D

A

is

x + ℓ

max

j.

Thus,inall asesthereexistsan

x

addingatmostoneneighborto

D

R

A

\ D

A

,whi hnishes the indu tion step and proves Equation (2). To nish the proof of the fa t that we an

restri tourselvesto re tangles, weshowthat, for any subset

A

, ifInequality (1) holdsfor

R

A

,thenitalsoholdsfor

A

.Indeed,Inequality(1) appliedto

R

A

gives:

c

_|R

A

| ≤ |D

R

A

| ,

whi hisequivalentto

c(

_{|A| + |R}

A

\ A|) ≤ |D

A

| + |D

R

A

\ D

A

|

(3)

UsingInequality(2)andthefa tthat

c

≥ 1

,Inequality(3) learlyimpliesthatInequality (1)holds.

Hen eforth we assumethat

A

is are tangle. Thelast simpli ation onsistsin proving thatwe anrestri tourselvestore tangles ontaining

v

.Inotherwords,itwillbesu ient toproveInequality(1)forallre tangles

R

v

a,b

.Givenare tangle

R

notpositionedat

v

,the re tangle

R

′

ofthesamesizepositionedat

v

haslessneighbors,hen eifInequality(1)holds for

R

′

, italsoholdsfor

R

.

Finally letusprovethat Inequality(1) holds forallre tangles

R

v

a,b

, with

1

≤ a, b < d

. Wehavethat

|R

v

a,b

| = ab

and

|D

R

v

a,b

| = (a + ℓ

max

)(b + ℓ

max

)

− d

2

.

Bythe hoi eof

d

, startingfromthe Inequality

d

2

_≤

(ℓ

max

+1)

2

c+1

andusing that

1

≤ a, b

, oneobtainsthat

d

2

_c

_{≤ (ℓ}

max

+ a)(ℓ

max

+ b)

− d

2

forany

1

≤ a, b < d

. Thisimplies, using

a, b < d

,that

cab

≤ (a + ℓ

max

)(b + ℓ

max

)

− d

2

forany

1

≤ a, b < d

,hen eInequality(1)(i.e.

c

_|R

v

a,b

| ≤ |D

R

v

a,b

|

)holds.

SobyTheorem4.1,ea honeofthe

d

2

nodesin

R

v

d,d

ansend

ℓ

messagestothenodesof

D

.Sin ethenumberofedgesgoingfrom

R

v

d,d

to

D

is

4d

− 1

,weapplythebise tionbound dis ussedinSe tion1.1.1to on ludethatthereisanedgeoftheborderofthesquare

R

v

d,d

with ongestionatleast

l

ℓ·d

2

4d−1

m

>



_ℓ·d

4



. Thisnishestheproofofthelemma.

2

Weobservethatthisse ond lowerbound isstri tlybetterthantherstoneifandonly if

c

√

c + 1

>

4ℓ

max

ℓ

max

+ 1

Ifboth

c

and

ℓ

max

arebig,the onditionbe omesapproximately:

max

_{{ℓ, k}}

min

{ℓ, k}

> 16

That is, the se ond lower bound is better when the dieren e between

ℓ

and

k

is big. Thisisthe aseofbroad astorgathering,wheremessagesareoriginated(ordestined)from (orto)asmallsetofnodesofthenetwork.

Lemma4.3(Combined lower bound) The worst- ase running time of any algorithm for

(ℓ, k)

-routingonfull-duplextriangular grids usingshortestpath routingsatises

Runningtime

≥ max



ℓ

max

· min{ℓ, k}, max{ℓ, k} ·

 ℓ

max

+ 1

4

√

c + 1



≈ ℓ

max

·max



min

_{{ℓ, k},}

max

{ℓ, k}

4

√

c + 1



Nowweprovideanalgorithmfromwhi hwederiveanupperbound.

Proposition 4.1(Upper bound (algorithm)) The algorithm for

(ℓ, k)

-routingon full-duplextriangulargridsisthefollowing:routeallpa ketsasinthepermutationrouting ase. Thatis, atea hnode sendpa ketsrstintheirnegative omponent,breakingtiesarbitrarily (there an be

ℓ

pa kets in oni t in a negative omponent). If there are no pa kets with negative omponents,sendanyofthe(atmost

k

)pa ketswithmaximumremainingdistan e.

Runningtime

≤

(

min

_{{ℓ, k} ·}

c(c−1)

₂

+ max

_{{ℓ, k} · (ℓ}

max

− c + 1)

,if

c

≤ ℓ

max

min

_{{ℓ, k} ·}

ℓ

max

(ℓ

max

+1)

2

,if

c > ℓ

max

Proof: Suppose again without loss of generality that

ℓ

≥ k

. We pro eed by de reasing indu tionon

ℓ

max

.Weprovethatafter

min

{ℓ, ℓ

max

k

}

steps,ea hpaquetwillbeatdistan e atmost

ℓ

max

− 1

ofitsdestination.Thisyields

Runningtime

(ℓ

max

)

≤ min{ℓ, ℓ

max

k

} +

Runningtime

(ℓ

max

− 1)

≤ min{ℓ, ℓ

max

k

} +

(

min

{ℓ, k} ·

c(c−1)

2

+ max

{ℓ, k} · (ℓ

max

− c)

,if

c

≤ ℓ

max

− 1

min

_{{ℓ, k} ·}

ℓ

max

(ℓ

max

−1)

2

,if

c > ℓ

max

− 1

≤

(

min

_{{ℓ, k} ·}

c(c−1)

₂

+ max

_{{ℓ, k} · (ℓ}

max

− c + 1)

,if

c

≤ ℓ

max

min

_{{ℓ, k} ·}

ℓ

max

(ℓ

max

+1)

2

,if

c > ℓ

max

Let us onsider the messages at distan e

ℓ

max

to their destinations. They are of two types,theonemovinga ordingtotheirnegative omponentandtheonemovinga ording totheirpositive omponent.

If

c

≤ ℓ

max

the rstones moveafter at most

ℓ

time steps.If

c < ℓ

max

theymovemore qui kly,indeedtheymoveatleaston eevery

ℓ

max

k

steps(

ℓ

max

k

≤ c · k = ℓ

).Thisisdueto thefa t that when

c < ℓ

max

at agivenvertex,at most

ℓ

max

k

messagesmayhavetomove a ordingtotheirnegative omponenttowardanodeatdistan e

ℓ

max

.

Aboutthemessageswhi h movea ordingto theirpositive omponent,sin eanodeis thedestinationofatmost

k

messages,theymaywaitatmost

k

steps.

Consequently,

ℓ

max

de reasesbyatleastoneevery

min

{ℓ, ℓ

max

k

}

steps,whi h givesthe

result.

2

This gives an algorithm whi h is fully distributed. Dividing the running time of this algorithmbythe ombinedlowerboundweobtainthefollowingratio:











min{ℓ,k}·

(

c

2

)

+max{ℓ,k}·(ℓ

max

−c+1)

ℓ

max

·max



min{ℓ,k} ,

max

{ℓ,k}

4

√

c+1



,if

c

≤ ℓ

max

min{ℓ,k}·(ℓ

max

+1)

2·max



min{ℓ,k} ,

max

{ℓ,k}

4

√

c+1



,if

c > ℓ

max

Weobservethatinall asestherunningtimeofthealgorithmisatmost

max

{ℓ, k}·ℓ

max

. Inparti ular, when

ℓ = k

(that is,

c = 1

) the runningtime is exa tly

max

{ℓ, k} · ℓ

max

=

min

_{{ℓ, k} · ℓ}

max

,andthereforeitis tight(seelowerbound ofLemma4.1).

Corollary4.1 There exists a tight algorithm for

(k, k)

-routing in full-duplex triangular grids.

Thepreviousalgorithms anbegeneralizedforhalf-duplextriangulargridsaswellasfor fullandhalf-duplexhexagonalgrids.Thegeneralizationtohalf-duplexgridsisobtainedby justaddingafa tor2inboththelowerboundandtherunningtimeofthealgorithm,aswe didforthepermutationroutingalgorithm.Thus,letusjust fo usonthe aseoffull-duplex hexagonalgrids,forwhi hwehavethefollowingtheorems:

Theorem4.2 There exists an algorithm for

(ℓ, k)

-routing in full-duplex hexagonal grids whoserunning timeisatmost:

Runningtime

≤

(

2 min

_{{ℓ, k} ·}

c(c−1)

₂

+ 2 max

_{{ℓ, k} · (ℓ}

max

− c + 1)

,if

c

≤ ℓ

max

2 min

_{{ℓ, k} ·}

ℓ

max

(ℓ

max

+1)

2

,if

c > ℓ

max

Lemma4.4(First lowerbound) Noalgorithm basedonshortestpathrouting anroute allmessagesusing lessthan

2 min

{ℓ, k} · ℓ

max

− min{ℓ, k}

stepsin theworst ase.

Denition4.2 Given a vertex

v

,we all the re tangleof the hexagonal grid of side

(a, b)

starting at

v

to the subset of the hexagonal grid

R

v

hexa,b

=

{v + α

i

+ β

j

+ γ

k

, 0

≤ α <

a,

_{−γ < β < b, 0 ≤ γ < b} ∩ H}

where

H

isthevertexsetofthe hexagonalgrid. We allsu h are tanglea square if

a = b

.

Thefollowinglemmagivesase ondlowerbound ontherunningtimeof anyalgorithm usingshortestpathroutingonfull-duplexhexagonalgrids.

Lemma4.5(Se ond lower bound) Theworst- aserunningtimeofanyalgorithmusing shortestpathrouting onfull-duplex hexagonalgrids satises:

Runningtime

≥



max

_{{ℓ, k}(2d +}

d

− 2

2d + 1

)



,

where

d =



√

73c+64ℓ

2

max

+121+144ℓ

max

8

√

c+1

−

3

8



and

c =

l

max{ℓ,k}

min{ℓ,k}

m

.

Noti e that when

ℓ

max

c

is big, this value tendsto

2 max

{ℓ, k}

ℓ

max

√

c+1

, obtaining a perfor-man earoundtwi e betterthanintriangulargrids.

Proof: Theproof onsists inshowingthattheverti esof

R

v

hexd,d

ansimultaneouslysend

max(ℓ, k)

messagestosomeverti esof

R

v

hexd+ℓ

max

,d+ℓ

max

\ R

v

hexd,d

.This isdoneasforthe triangulargrid,usingagainTheorem4.1.Wedonotgiveallthedetails,sin etheideabehind isthesameastheproofofLemma 4.2.

Sin e the number of verti es inside

R

v

hexd,d

is

4d

2

_{+ d}

− 2

, and the number of edges outgoing from

R

v

hexd,d

is

2d + 1

, the ongestion on these edges is

max

{ℓ, k}

4d

2

+d−2

2d+1

=

max

_{{ℓ, k}(2d +}

d−2

2d+1

)

.

2

5 Con lusions and Further Resear h

In this arti le we havestudied thepermutation routing, the

r

- entral routingand the general

(ℓ, k)

-routing problems on plane grids, that is square grids, triangular grids and hexagonal grids. We have assumed the store-and-forward

∆

-port model, and onsidered bothfullandhalf-duplexnetworks.Themainnewresultsof thisarti learethefollowing:

1. Tight (also in luding the onstant fa tor) permutation routing algorithms on full-duplexhexagonalgrids,andhalfduplextriangularandhexagonalgrids.

2. Tight(alsoin ludingthe onstantfa tor)

r

- entral routingalgorithmsontriangular andhexagonalgrids.

3. Tight(alsoin ludingthe onstantfa tor)

(k, k)

-routingalgorithmsonsquare, trian-gularandhexagonalgrids.

4. Goodapproximationalgorithmsfor

(ℓ, k)

-routinginsquare,triangularandhexagonal grids, together with new lower bounds on the running time of any algorithm using shortestpathrouting.

All these algorithms are ompletely distributed, i.e. anbeimplementedindependently at ea hnode.Finally,wehavealsoformulatedthe

(ℓ, k)

-routingproblemasaWeightedEdge Coloringproblem onbipartitegraphs.

There still remain severalinteresting open problems on erning

(ℓ, k)

-routingonplane grids.Of ourse,themost hallengingproblemseemstondatight

(ℓ, k)

-routingalgorithm foranyplanegrid,for

ℓ

6= k

.Anotherinterestingavenueforfurtherresear histotakeinto a ountthequeuesize.Thatis,todevise

(ℓ, k)

-routingalgorithmswithboundedqueuesize, orthatoptimizeboththerunningtimeandthequeuesize,undera ertaintrade-o.

A knowledgment.WewanttothankFrédéri Giroire,CláudiaLinhares-SalesandBru e Reedforinsightfuldis ussions.