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How Graph Theory can help Communications
Engineering
David Coudert, Xavier Munoz
To cite this version:
David Coudert, Xavier Munoz. How Graph Theory can help Communications Engineering. Broad
band optical fiber communications technology (BBOFCT), Dec 2001, Jalgaon, India. pp.47-61.
�inria-00429187�
D. Coudert X. Mu~noz
Abstra t
Wegiveanoverviewofdi erentaspe tsofgraphtheorywhi h anbeappliedin ommuni a-tionengineering,nottryingtopresentimmediateresultstobeappliedneithera ompletesurvey of results,but to givea avorof how graphtheory anhelp this eld. We dealin this paper withnetworktopologies,resour e ompetition,statetransitiondiagramsandspe i modelsfor opti alnetworks.
1 Introdu tion
It is not news that the development of s ien es and te hnology makes it impossible to be aware of all eldsof knowledge, andspe ialization annot be avoided inthe s ienti resear h wherethe rapid advan esmake it diÆ ulteven to followthenew resultsintheownspe i dis ipline.
This unavoidable fa t turns outto bea stronglimitation in the development of s ien e, sin e manya hieved results insome dis iplines that ould be useful inother eldsof knowledge are in most ases unknownbys ientists.
It isdesirable that resear hers know at least the existen e of other eld's advan es. Although notbeingspe ialistontheseotherareas,theyshouldknowwheretolookforinformationinbene t of theirown resear h.
Multidis ipline teamsandmultidis iplineareasarewordsthatnowadays seemto beimportant inthes ienti resear h. Indeed ollaborationwithothergroupsisalwaysfruitfulnotonlybe ause ea hdis ipline anhelptheotherswithknownresultsbutalsobe ause di erentpointsofview an helpone ea h other to better understandingtheinsight oftheproblems.
This is even more important if we fo us on an illae s ientiae, i.e. s ien es whi h are used as tools in other s ien es su h asmaths is with respe t to engineering. Mathemati ians are usually looking for new problems and engineers are usually looking for new solutions. Dialogue of both areas isinteresting forboth ommunities.
Nevertheless dialogue is not always easy and requires some e ort from both, mathemati ians and engineers. The translation of a problemarising from engineeringto amathemati al language requires a deep knowledge of the dis ipline in whi h the problem is ontextualized. Setting of assumptionsandhypothesisofthemodelisalwaysa ompromisebetweenrealism andtra tability. Finally,theresultsgivenbythemathemati almodelmustbevalidated withsomeexperimentalor simulation results.
One annot expe tto have a libraryof "plug and play"models fornew problems, but theart of reating a mathemati al model is one of the examples in whi h ooperation of engineers and mathemati iansis ru ial inorderto get valuable results.
In this paper we give an overview of di erent aspe ts of graph theory whi h an be applied in ommuni ation engineering, not trying to present immediate results to be applied neither a ompletesurvey ofresults, butto give a avorof howgraphtheory an helpthis eld.
basi on epts of graph theory. Se tion 3 is devoted to graphsas models for network topologies. Di erentproblemsrelatedwith ommuni ationsinwhi hGraphTheorymightbeusefularetreated inSe tion4. Finally,Se tion5dealswithspe i problemsthatappearinOpti al ommuni ations, and a detailedexampleis also showninthat se tion.
2 Graph theory on epts
AgraphG=(V;E)isanorderedpair onsistingofasetV =V(G)ofelements alledverti es and a setE =E(G)of unorderedpairsof verti es alled edges.
Twoverti esu;v2V(G)aresaidto beadja ent ifthereisanedge(u;v) 2E(G)joiningthem. The edge (u;v) 2 E(G) is also said to be in ident to verti es u and v. The degree of a vertex v, Æ(v) is the numberof verti es inV(G)adja ent to it. The minimum and maximum degrees of thegraph,Æ andrespe tively, arede nedastheminimumandmaximumoverthedegreesof all verti es of thegraph.
A path w(v 0 ;v n ) is a sequen e of verti es v 0 ;v 1 ;:::;v n su h that (v i 1 ;v i ) 2 E(G), i = 1::n. Su hapathissaidtobeoflengthn. Agraphissaidtobe onne ted ifthereisapathbetweenany pair of verti es. If the graph is onne ted, the distan e between two verti es d(a;b) is the length of the shortest path joining a and b and the diameter of the graph is the maximum among the distan esbetweenall pair ofverti es ofthegraph.
A graph an be de ned and represented in di erent ways. One typi al representation is by meansofadrawinginwhi hverti esaredrawnaspointsintheplaneorinthespa eandedgesare drawnaslinesjoiningpoints. Inspiteofbeinganintuitiveviewofagraph,itseldomleadsto some insight on itsproperties. Graphs analso be hara terizedbyitsadja en y matrix, Theadja en y matrix of a graph with nverti es is a nnmatrix A =(a
ij
) with a ij
=1 if(i;j) 2E(G)and 0 otherwise. Noti ethat theadja en ymatrixisalways asymmetri matrix. Anotherwayto de ne a graph is giving the adja en yrules on the set of verti es. Graphs that an be de nedthat way areusually\well stru tured",and useto haveni e properties.
Sin e agraphis abinaryrelationshipwithinelementson ageneri set, it isnotsurprisingthat graphs appear, at least impli itly, in many di erent ontexts of the s ienti knowledge su h as so iology,e onomy,orengineering.
Graphs an begeneralized to other relatedstru tures thathave beenalsowidely studied: Multigraphsandweightedgraphs. Insome asesitisusefulto onsideredgeshavingaweight,
usuallya numeri al value, or eitherthatmorethanone edge an jointhesametwo verti es. Digraphs. A digraphordire ted graph is agraph inwhi h we onsideredges ( alled ar s in the ontext ofdigraphs)having adire tionassigned. Inotherwords,a digraphG=(V;A)is anorderedpair onsistingofaset V =V(G)of verti es anda setA=A(G)ofordered pairs of verti es alledar s.
Hypergraphs. Instead of onsidering edges as binary relations between verti es, hyperedges are de nedas subsetsof verti es (with nota determined size). A hypergraphG=(V;H) is thende nedasan orderedpair onsistingofasetV =V(G)ofverti es anda setH=H(G) of subsetsof verti es alledhyperedges.
Dire ted hypergraphs. A dire ted hypergraphG=(V; ~
H) is thende nedas an orderedpair onsisting of a set V =V(G)of verti es and a set
~ H =
~
H(G)of ordered pairs of subsets of verti es alled hyperar s.
reader an ndother related on epts and de nitionsin[5,12 ℄.
3 Graphs and network topologies
Con erningto inter onne tionnetworksitisstraightforwardthata networktopology anbe mod-eled by a graph in whi h the verti es of the graph play the role of nodes in the network and the edges model thepoint-to-point links between di erent nodes. The graph an be undire tedinthe ase of full-duplex ommuni ation links, ordire ted in ase the links are unidire tional orin ase of highly asymmetri ommuni ation proto ols su h asADSL. Alternatively, ifsome nodes share a ommon physi al or logi al bus, the bus an be modeled by a hyperedge, and hen e the whole network by a hypergraph. This might be the ase of networksthat use CSMA/CD-like proto ols su h asEthernet.
Yet in the early 50's graph theory was used to model telephony networks, and from the very beginninginterestingresults amefromthesestudieslike,forinstan e,thestudiesfornon-blo king multistage swit hingnetworks(the Clos Network, [17 ℄). Sin e thenlots of studies have beendone ingraph theorywhen onsideringgraphsastopologi al modelsfor di erent kindsof networks.
Lots of aspe ts have to be onsidered in the design of an inter onne tion network and many of them are highly dependent on the topology and an be studied in terms of Graph Theory. In most ases animportantrestri tionisthemaximumnumberoflinksthat anbe onne tedto ea h node(maximumdegree ofthegraph). In ase thatitis notpossible (or onvenient)to have alink betweenanypairofnodes, amultihopsolution mustbeadopted, i.e. amessage musthop through intermediate nodes before rea hing its destination. It is desirable, however, to have the number of hopsbetween nodes assmall aspossible (smalldiameter of thegraph). The problemof nding graphswith xed degreeand numberof verti es having thesmallestpossible diameterisknownas the(;n){problem, and hasbeenwidelystudiedinthe literature ([12,7 ℄).
The designershouldinsome asesforeseethatthenetworkmight growinthefuturebyadding new nodes. It is onvenient that the network size an be in reased without loosing its stru tural properties. There aremanygraphsthat have this property(s alability).
Anotherimportantitemofinter onne tion networkisroutingand ommuni ationsalgorithms. In ase that the graph is well stru tured, eÆ ient algorithms an be spe i ally design for that network topology. TheeÆ ien y of arouting proto ol dependson thesimpli ityofthe pro essing intheintermediatenodesandonthememoryneededinthesepro essorstohandlethedelivery(su h assmallroutingtables). Routinghasalsobeenwidelystudiedforspe i networktopologiesunder di erent ommuni ation models (virtual paths, store{and{forward, wormhole routing, de e tion (hot potato) routing,...). EÆ ien y of a routing an also be measured in terms of throughput whi hisrelatedwiththenumberofroutesusing thesamelinkornode(forwardingindex[16 ,30 ℄). Among the ommuni ationalgorithms,broad asting and dissemination(gossiping) of informa-tion are of spe ial interest sin e they appear often inmany ommuni ation pro esses. They have also beenwidely studiedunder di erent modelsand assumptions[31 ℄.
It an be also interesting to know the behavior of topology and routing when some of the omponentsof thenetworkfail. Fault toleran e an measuredindi erent ways: bythein reasing of the diameterwhen removing links ornodes(;D
0
;s){problem[36 ℄); by thenumberof links or nodesthatmayfailkeepingthenetworkstill onne ted( onne tivity[26 ℄)orintermsofrouting,by thea essibilityofnodeswithoutre on guringtheroutingtablesundernetworkfailures(surviving graph[23 ℄).
beobtainedfor generi oreven unknowngraphs[22 ℄.
We have fo used onsomeimportant parametersto be onsideredinorder to hoosea network topology. There are, however, many other aspe ts we have not onsidered here and may also be important insome ases, like,for instan e,the eÆ ient layout of thenetwork inthe aseof VLSI systems(planarity, rossingminimization,[5 ℄)ortheeÆ ient embedding oflogi al topologies onto a spe i physi al one in the ase of parallel implementation of algorithms that must run over a spe i network [34 ℄ (thetypi alinstan e forthislast exampleisthe ase ofparallelizing theFFT algorithm onthehyper ube network).
4 Other appli ations of graph theory
Besidestheimportan eandusefulnessofusing graphstomodelthetopology ofthenetwork,there are someother ways inwhi hgraph theory an help ommuni ation engineers. In this se tion we showtwodi erent examples:
4.1 State transition diagrams
Another appli ation of graph theory in ommuni ations engineering is the one in whi h verti es modelstatesofsomepro essandar s(dire tededges)modelthepossibletransitionsbetweenthese states. The mostimportant exampleforthis kindof modeling arethewell knownMarkov hains, butthey arenottheonly ones.
An interestingareainwhi hGraphTheory anhelptoobtainresultsinthesekindofpro esses is the ase of onvolutional odes or the morere ent time-spa e odes ([40 ℄) that are de oded by meansof maximumlikelihood,su h astheway theViterbi algorithmworks.
Thesymboltobetransmittedinthesekindof odesdependsnotonlyonthesymbolwantedto bede odedatthere eiver,butalsoonthestatethetransmissionis. Moreover,boththetransmitted symboland thepresent state determinea new state inthetransmission.
The usual transition of statesis given bya shiftregisteron thestate, i.e. states aresequen es onsisting on the last transmitted symbols. Shift register leads to a very well known family of dire ted graphs alledDeBruijndigraphs [28 ℄.
Transition diagrams given byShift Register have not been proved to be optimum in terms of ode gain(gain interms of errorprobabilityinde oding), and othertransition diagrams ouldbe onsideredinorder to onstru t better odes.
4.2 Resour e ompetition
Indeedgraphs aneasilymodelanysituationinwhi hbinaryrelationshipsout omelikeforinstan e the ompetitionof agentstowards thesameresour es: verti es ofthegraphrepresent thedi erent agents, and two verti es are adja ent ifthey want to use the sameresour e. Inorder to ompute the minimumnumber of resour es needed one an use the resulting \in ompatibility" graph. In fa t,thisproblemturnstobeastandardproblemingraphtheory alledvertex oloring: i.e. assign olorsto theverti esof thegraphsu hthatnottwoadja entverti es musthave thesame olor [5 ℄ In the remaining of this se tion two examples will be shown for better understanding of this kindofgraphs.
The frequen y assignment problem for a ellular network has been thoroughly dis ussed in the literatureowing,inpart,totheimportante onomi reper ussionsthatithasfornetworkoperators. Thelargest outlayto be onfrontedbyanoperatoristhe li ensefee fortheallo ation of theradio frequen y spe trum. As this spe trum is limited, the pro tability of the network will depend on thenumberofstations requiredto satisfytheusersdemand.
In thedesignof a ellularmobile ommuni ationsystem, aregionisdivided into seriesof ells, ea h one overed by a base station. Ea h station works at a ertain frequen y, and the same frequen y an be used in di erent ells. However, the reuse of the same frequen y is limited by themaximumtolerable interferen e between ells. The frequen y assignment problem onsists on nding an assignment of frequen ies to the base stations using the smallest possible number of frequen ies onsideringtheinterferen e onstrains.
A ellularnetwork anbedes ribedbyagraphwherethenodesrepresentthe ellsandtheedges joint nodes orrespondingto ells that annot usethesamefrequen y be ause of interferen es.
Thismodel analsobere ned onsideringdi erentlevelsofinterferen es,di erenttransmitters per station, or some xed restri tions in the assignment. See [1 ℄ for some re ent results and algorithmsto solve that problem.
4.2.2 Con i t graph in wavelength assignment for virtual paths
Let us onsider in this se tion WDM opti al networks onsisting of routing nodes inter onne ted bypoint-to-point ber-opti links whi h an support a ertain numberof wavelengths. Swit hing will be supposedto bealso opti al, and pa kets are supposed not to hange of wavelength at the intermediate nodes. When a ommuni ation mustbe establishedbetweento nodesof thenetwork a virtualpathis reserved and a wavelengthis assigned to thattransmission.
An important restri tion inthis model is that two di erent virtual paths sharing at least one ommon link must have di erent wavelengths, as if they did not, it would not be possible to distinguishbothsignalsat the nextintermediatenode.
To omputetheminimumnumberofwavelengths requiredtoestablishasetofvirtualpathswe de ne the Con i t Graph the following way: Verti es of the graph stand for the di erent virtual paths and twoverti es willbe adja ent ifthe orresponding virtualpathshare at leastone link.
As in the previous se tion, the wavelength assignment problem an be solved in the Con i t Graph by assigning olors (wavelengths) to its verti es in su h a way that adja ent verti es have di erent olors[4 ℄.
One an obje t that this a priori planing is not realisti , and it is true. But this stati and simple setting of the problem an be improved by using more realisti dynami approximations based onon-line algorithms[1℄.
Noti ethattheexamplesshowninthisse tionhaveidenti altranslationintoa oloringproblem of graphs.
Related to thethe wavelengthassignment problem,it isinterestingto point a questionwith a (maybe)surprisinganswer.
The re ent Dense WDM (DWDM) allows to use a onsiderable numberof wavelengths inthe same ber. Is it equivalent to use a single ber with DWDM that using multiple bers with onventionalWDMtojointwo nodesinthenetwork(supposingthatbothsolutionsallowthesame total numberofwavelengths perlink)?
Manyfa tors shouldbe onsidered to de ide whi h solution is better. But someGraphtheory ba kground an help torealize that bothsolutionsare nottopologi ally equivalent [11 ℄. Infa t, it
theusage ofthe resour es(wavelengths), asshown inthefollowingexample:
Suppose a simple network onsisting on four nodes onne ted in a star topology as shown in Figure1and imaginewe wantto establishthreesimultaneous ommuni ationsbetweennodes1{2, 2{3 and 1{3. Thegoal will beassigning wavelengths to these transmissionrequirementsusing the minimumnumberofwavelengths.
1-3 2-3 1-2 J J J J J b b b 3 2 1 4 b b Z Z Z b b
Figure1: A networkand the on i t graphfor3 requirements.
Iflinks onsiston two bersea h,therequirement anbesatis edbyusingone wavelengthper ber, while two wavelengths per ber are not enough to solve the assignment with a single ber perlink. This an beeasily seenintermsofgraph olorings,sin ethe on i tgraphinthisformer ase isa triangle,and verti es ina triangle annot be oloredwith two olors.
5 Graph theory and all-opti al networks
In the pre eding se tions we have reviewed di erent aspe ts in whi h Graph Theory an help Communi ationsengineering. We wanttofo usinthis se tiononopti alnetworks. At a rstsight one might thinkthat, from the graph theoreti point of view,there is notany di eren ebetween opti alnetworksand generi inter onne tion networks, sin egraphtheorydo not areaboutwhat arethewiresmade of.
Butbesidesthegeneralresultsand modelsforinter onne tion networksthat anbealso useful inour ontext,newspe i problemsappearedinthestudyofallopti alnetworks. Theseproblems arenotex lusive of this eldbuthave nowa parti ularinterest.
Inparti ular,wearenot on ernedinthisse tionwith lassi alnetworksinwhi hsomeele tri al wiresarerepla ed byopti al bers andat ea h endpointthere is aele tri -to-opti al shift, buton networks inwhi h swit hing is alsodoneusing lightwave te hnology.
Some of the properties of all-opti al networks, that leadto newproblems inGraph theoryare thefollowing:
While ele tri al bu eringis te hnologi ally easy to implement,this isnotthe ase of opti al systems. Routing must be donewithout storing the information in the routers and pa kets annotwaitto be delivered throughalink without opti alto ele tri al onversion [25 ℄. Fast routing algorithms must be realized (su h as MPLS [2 , 3℄) and, in ase of network ongestion,de e tion routing[38 , 14 ℄ ouldbe agoodsolution.
Newrouting onstraintsappearwhenusingWDM,di erentfromthose inTDM.Anexample of problemsarising fromWDM wasgiven intheprevious se tion.
Newmodelsfornetworksarebeingdevelopedduetothenewdevi esinopti alte hnology. In this ontext there have appearedmanystudieson busnetworks[6, 18 ℄inwhi h thenetwork topology is nomoremodeledbya graphbutbyahypergraph.
we model anetwork bya dire ted hypergraphusing existingopti al devi es.
5.1 Using dire ted hypergraphs with OTIS and OPS ouplers
Topologiesallowingone-to-many ommuni ationsinasingle ommuni ationstep have been exten-sivelystudied. Thebuildingblo kistheOpti alPassiveStar(OPS) oupler[27 ℄whi h anbeused asasingle-OPSbroad ast-and-sele tnetworkinwhi hanin omingopti alsignalisbroad asttoall output ports [10 ℄. Although agreat deal ofresear h e ortshave been on entrated on single-OPS topologies [24 , 15 ,37 ℄, theOPS ouplerpresentsaseveredrawba k sin eits size is te hnologi ally limitedbyitssplitting apabilities. Thatisthereasonwhymulti-OPStopologiesseemsmoreviable and s alable [6, 18 ℄.
In this se tion, we present some multi-OPS topologies with the single-hop Partitioned OPS networks (POPS) [13 ℄ and the multi-hop sta k-Kautz network [18 ℄. We also present the sta k-graphs,afamilyofdire tedhypergraphswhi hisveryusefultomodelandmanipulatedmulti-OPS networks. Finally, we present the OTIS ar hite ture [35 ℄, an opti al devi e whi h was used to implementthe opti alinter onne tions ofthe POPSand thesta k-Kautz.
5.1.1 Opti al passive star
Anopti alpassivestar oupler(OPS)isasinglehopone-to-manyopti altransmissiondevi e[24 ,37 ℄ whi h an be view as an opti al bus. An OPS(s;z) hass inputsand z outputs. When s=z,the OPSissaid tobeofdegree s(seeFigure2). Whenone oftheinputnodessendsamessage through anOPS oupler, thesoutputnodeshave a essto it. An OPS ouplerisapassiveopti al system, i.e. it requires no external power sour e. It is omposed of an opti al multiplexer followed by an opti al berora free opti alspa e and a beam-splitter[29 ℄ that dividesthe in ominglight signal into s equal signals of a s-th of the in oming opti al power. Finally, in a single-wavelength OPS ouplers of degree s, only one opti al beam an be guided through ea h devi e. Consequently, no two nodes an have on urrent a essto anyOPS.
An OPS oupler of degree s an be modeled by a hyperar linking two hypergraph nodes omposed of s verti es ea h [10 ℄, meaning that the pro essors of one set (the OPS sour es) an sendmessagesonlythroughthehyperar ,whereastheotherset(theOPSdestinations) an re eive messagesonlythroughthesamehyperar . Figure2showsan OPS ouplermodeledbyahyperar .
hyperarc
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Optical Passive Star coupler
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Figure 2: Modelingan opti al passive star oupler ofdegree 4 bya hyperar .
5.1.2 Sta k-graphs
Thesta k-graphswerede nedin[10 ℄asaparti ular lassofdire tedhypergraphs. Informally,they anbeobtainedbymeansofpilingups opiesofadigraphGandsubsequentlyviewingea hsta k of snodesas a hypergraphnode and ea h sta k of sar s asa hyperar . The value sis alledthe
fa tor s, obtainedfromthe digraphG.
Sta k-graphsare not optimum in terms of the (;D){problem, and other families of dire ted hypergraphs ouldbe also onsidered[6℄. Butsta k-graphsareeasy todeal withandwewill show inthefollowing, throughexamples,thatare very eÆ ient to modelmulti-OPSnetworks.
5.1.3 Partitioned Opti al Passive Stars network
The Partitioned Opti al Passive Stars (POPS) topology, introdu edin [13 ℄, is an inter onne tion ar hite ture that uses multiple non-hierar hi al OPS to a hieve single-hop networks. The POPS network POPS(t;g),is omposedofN =tgnodesandg
2
OPS ouplersofdegreet. Thenodesare dividedintoggroupsofsizet(seeFigure3). Ea hOPS ouplerislabeledbyapairofintegers(i;j), 0i;j<g. TheinputoftheOPS(i;j) is onne tedtothei-thgroupofnodes,and theoutputto the j-th group of nodes. This network bene ts of therouting fa ilities of singlehop networks and of thebroad ast apabilitiesof OPS-basednetworks.
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Figure3: POPS(4;2) witheight nodes, modeled as&(4;K 2 ).
Sin eanOPS ouplerismodeledbyahyperar ,thePOPSnetworkPOPS(t;g) anbemodeled asadire ted hypergraph,and infa titis notdiÆ ultto seethattheunderlyingtopology onsists ona sta k-K + g (or&(t;K + g
),forshort)ofsta king-fa tort, whereK + g
isthe ompletedigraphwith loopshaving g nodesand g
2
ar s (seeFigure 3),as proposedin[9 , 27 ℄.
Using thesta k-graphmodel,embeddingofrings,tori,deBruijnandotherwellknowstopology on POPSwereeasily obtained[9, 8℄.
5.1.4 Sta k-Kautz
We saw with the POPS networks that sta k-graphs represent a powerful tool for building multi-OPSnetworksbasedongraphspresentinggoodtopologi al hara teristi s. TheKautzdigraph[33 ℄ isone su hgraph.
The Kautz digraph K(d;k) of degree d and diameter k has d k 1
(d+1) verti es. A vertex is labeledwitha wordoflengthk,(x
1 ;;x
k
),on thealphabet=f0;;dg, jj=d+1,inwhi h x
i 6= x
i+1
, for 1 i k 1. There is an ar from a vertex x = (x 1 ;;x k ) to all verti es y su h that y = (x 2 ;;x k ;z), z 2 , z 6=x k
. The Kautzdigraph an also be de ned interms of line digraphiteration [28 ℄ and we have K(d;k) = L
k 1 (K d+1 ), where K d+1
denotes the omplete digraphwithoutloops. ItisbothEulerianandHamiltonianandoptimalwithrespe ttothenumber of nodes if d > 2 [33 ℄. A shortest path routing algorithm (every path is of length at most k) is
0 1 2
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Figure 4: Sta k-Kautznetwork SK(3;2;2).
indu edbythelabels of thenodes. It an beextended to generatea pathof length at most k+2 whi h survivesd 1 linkornodefaults[32℄. Also,node disjoint paths an be easily generated.
The sta k-Kautz were rst proposedin [18 ℄ as a multi-hop alternative ofthe POPS networks. Theyaremodeled byasta k-graphbuildfromtheKautzdigraph,asfollows. LetK
l
(d;k) denotes theKautzdigraphinwhi hea h vertex re eive a loop and hen ehasdegree d+1.
De nition 1 The sta k-Kautz SK(s;d;k) is the sta k-graph &(s;K l
(d;k)) of sta king-fa tor s, degree d+1 and diameter k (see Figure4).
The sta k-Kautz inherits most of the properties of the Kautz digraph, like simple and fault tolerant routing algorithms, as des ribed in [18 ℄. Also broad asting is very eÆ ient Furthermore, thePOPS network isa sta k-Kautzof diameterone,POPS(t;g)=SK(t;g 1;1).
5.1.5 OTIS
The Opti al Transpose Inter onne tion System (OTIS) ar hite ture, was rst proposed in [35 ℄. It provides a large numberof inter onne tions ina free opti al spa e (no ber). OTIS(p;q) is an opti alsystemwhi hallowspoint-to-point(1-to-1) ommuni ationsfrompgroupsofq transmitters
lenses
Transmitters
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Figure5: OTIS(3;12).0ip 1,0jq 1,tothere eiver(q j 1;p i 1). Opti alinter onne tionsintheOTIS ar hite ture arerealized witha pair of lenslet arrays ina free spa e of opti al inter onne tionsas shown inFigure5.
A family of digraphs obtained from OTIS(p;q) were de ned in [20 ℄ as given in de nition 2. It has been shown in [20 ℄ that this family of digraphs ontains the Kautz digraphs. This result was used in [19 ℄ to propose an implementation of the opti al inter onne tions of Kautz, POPS and sta k-Kautz networks using OTIS. Furthermore, an optimalimplementation of thede Bruijn network with OTIS,interms ofnumberof lenses,wasshownin[20℄.
De nition 2 The H(p;q;d) digraph is de ned from the OTIS(p;q) ar hite ture. It has degreed, su h that d divides pq, and n =
pq d
verti es and m = pq ar s. Its verti es are integers modulo n and its neighborhood is de ned as follows:
+ H (u)= v= 1 d (pq 1) du+ q +1 p(du+ ) ; 0 d 1
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Optical
Multiplexers
Sources
Destinations
OTIS(4,2)
OTIS(2,4)
OTIS(2,2)
Optical Interconnection Network
Beam Splitters
Light Direction
Figure6: Opti alinter onne tionsofPOPS(4;2) using theOTISar hite ture.
Another familyof networks, all OTIS-networks, inwhi h bothopti s and ele troni s ommu-ni ationslines areused,wasde ned andstudied in[41 , 39 ,21 ℄.
6 Con lusions
WepresentedsomeideasfromGraphTheorythat anbeusefulinordertosolvedi erentproblems inCommuni ation Engineering.
Even in the ase that a problemarising from engineering has not easy solution in the Graph Theory ontext, makingthe e ortoftranslating it maybeusefulbe ause:
Translating theproblemto Graphs an give someinsight on it.
A problem translated to maths is understandable by other s ienti ommunities that an helpsolvingit.
Problems omingfrom otherdis iplinesmayhavethesametranslation and mighthave been solved before.
solve many problems on Graphs by providing heuristi algorithms that give solutions with someboundederror.
Keeping both ommunities,engineersand mathemati ians,workingtogetherwill help develop-ing solutions in engineering and providing new hallenging and more lose to reality problemsto mathemati ians.
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