Distributed Computing in Dynamic Networks: Towards a Framework for Automated Analysis of Algorithms

0.23 MB·English

by  Arnaud Casteigts

Distributed Computing inDynamicNetworks: Towards a Frameworkfor
AutomatedAnalysisofAlgorithms⋆
ArnaudCasteigts1,SergeChaumette2,andAfonsoFerreira3
1 SITE,UniversityofOttawa,Canada.
[email protected]
2 LaBRI,Universite´deBordeaux,France.
2
1 [email protected]
0 3 INRIA-MASCOTTE,SophiaAntipolis,France.
2
[email protected]

n
a
J

0 Abstract. Besidesthecomplexityintimeorinnumber ofmessages, acommon approach forana-
3 lyzingdistributedalgorithmsistolookattheassumptionstheymakeontheunderlyingnetwork.We
investigatethisquestionfromtheperspectiveofnetworkdynamics.Inparticular,weaskhowagiven

] property on the evolution of the network can be rigorously proven as necessary or sufficient for a
C
given algorithm.Themaincontributionofthispaperistopropose thecombination oftwoexisting
D
toolsinthisdirection:localcomputationsbymeansofgraphrelabelings,andevolvinggraphs.Such
. a combination makes it possible to express fine-grained properties on the network dynamics, then
s
c examinewhatimpactthosepropertieshaveontheexecutionataprecise,intertwined,level.Weillus-
[ tratetheuseofthisframeworkthroughtheanalysisofthree simplealgorithms,thendiscussgeneral

implicationsofthiswork,whichinclude(i)thepossibilitytocomparedistributedalgorithmsonthe
4
basisoftheirtopological requirements,(ii)aformalhierarchyofdynamicnetworksbasedonthese
v
9 requirements,and(iii)thepotentialformechanizationinducedbyourframework,whichwebelieve
2 opensadoortowardsautomatedanalysisanddecisionsupportindynamicnetworks.
5
5
. 1 Introduction
2
0
1 Thepastdecadehasseenaburstofresearchinthefieldofcommunicationnetworks.Thisisparticularly
1 true for dynamic networks due to the arrival, or impending deployment, of a multitude of applications
:
v involvingnewtypesofcommunicatingentitiessuchaswirelesssensors,smartphones,satellites,vehicles,
i orswarmsofmobilerobots.Thesecontextsofferbothunprecedentedopportunitiesandchallengesforthe
X
researchcommunity,whichisstrivingtodesignappropriatealgorithmsandprotocols.Behindtheapparent
r
a unityofthesenetworksliesagreatdiversityofassumptionsontheirdynamics.Oneendofthespectrum
correspondstoinfrastructurednetworks,inwhichonlyterminalnodesaredynamic–theseinclude3G/4G
telecommunicationnetworks,access-point-based Wi-Fi networks, and to some extentthe Internetitself.
At the other end lies delay-tolerantnetworks (DTNs), which are characterized by the possible absence
ofend-to-endcommunicationrouteatanyinstant.ThedefiningpropertyofDTNsactuallyreflectsmany
typesofreal-worldcontexts,fromsatellitesorvehicularnetworkstopedestrianorsocialanimalnetworks
(e.g.birds,ants,termites).In-betweenliesa numberofenvironmentswhosecapabilitiesandlimitations
requirespecificattention.
Aconsequenceofthisdiversityisthatagivenprotocolfordynamicnetworksmayproveappropriatein
onecontext,whileperformingpoorly(ornotatall)inanother.Themostcommonapproachforevaluating
protocolsindynamicnetworksistorunsimulations,anduse agivenmobilitymodel(orsetoftraces)to
generatetopologicalchangesduringtheexecution.Theseparametersmustfaithfullyreflectthetargetcon-
texttoyieldanaccurateevaluation.Likewise,thecomparisonbetweentwoprotocolsisonlymeaningful
⋆Apreliminaryversionofthispaperappearedin[9].
ifsimilartracesormobilitymodelsareused.Thisstateoffactsmakesitoftenambiguousanddifficultto
judgeoftheappropriatenessofsolutionsbasedonthesoleexperimentalresultsreportedintheliterature.
Theproblemisevenmorecomplexifweconsiderthepossiblebiasesinducedbyfurtherparameterslike
thesize ofthe network,the densityofnodes,thechoiceofPHY orMAClayers,bandwidthlimitations,
latency,buffersize,etc.
Thefundamentalrequirementofanalgorithmonthenetworkdynamicswilllikelybebetterunderstood
fromananalyticalstandpoint,andsomerecenteffortshavebeencarriedoutinthisdirection.Theyinclude
the worksby O’Dell etal. [23] and Kuhnetal. [18], in which the impactsof givenassumptionsonthe
network dynamics are studied for some basic problems of distributed computing (broadcast, counting,
andelection).Theseworkshaveincommonanefforttomakethedynamicsamenabletoanalysisthrough
exploitingpropertiesofastaticessence:eventhoughthenetworkispossiblyhighly-dynamic,itremains
connectedateveryinstant.Theapproachofpopulationprotocols[1,2]alsocontributedtomoreanalytical
understanding.Here,noassumptionsaremadeonthenetworkconnectivityatagiveninstant,butyet,the
samefundamentalideaoflookingatdynamicnetworksthroughtheeyesofstaticpropertiesisleveraged
bythe conceptofgraphofinteraction,inwhicheveryentityisassumedto interactinfinitelyoftenwith
itsneighbors(andthus,dynamicsisreducedtoaschedulingprobleminstaticnetworks).Besidesthefact
thattheaboveassumptionsarestrong–wewillshowhowstrongincomparisontoothersinahierarchy–,
we believethatthe veryattemptto flatten the time dimensiondoespreventfromunderstandingthe true
requirementsofanalgorithmonthenetworkdynamics.
As a trivialexample,considerthe broadcastingof a piece of informationin the networkdepictedin
Figure1.Thepossibilitytocompletethebroadcastinthisscenarioclearlydependsonwhichnodeisthe
initial emitter: a and b may succeed, while c cannot. Why? How can we express this intuitive property
the topology evolution must have with respect to the emitter and the other nodes? Flattening the time-
dimensionwithoutkeepinginformationontheorderingofeventswouldobviouslyloosesomeimportant
specificities,suchasthefactthatnodesaandcareinanon-symmetricalconfiguration.Howcanweprove,
more generally,that a given assumption on the dynamicsis necessary or sufficient fora given problem
(oralgorithm)?Howcanwefind(anddefine)propertythatrelatetofiner-grainaspectsthanrecurrenceor
moregenerallyregularities.Evenwhenintuitive,rigorouscharacterizationsofthiskindmightbedifficult
toobtainwithoutappropriatemodelsandformalisms–aconceptualshiftisneeded.
a b c a b c a b c
beginning movement end
Fig.1.Abasicscenario,whereanode(b)movesduringtheexecution.
We investigate these questions in the present paper. Contrary to the aforementioned approaches, in
which a given contextis first considered,then the feasibility of problemsstudied in this particularcon-
text,wesuggestthesomehowreverseapproachofconsideringfirstaproblem,thentryingtocharacterize
itsnecessaryand/orsufficientconditions(ifany)intermsofnetworkdynamics.Weintroduceageneral-
purpose analysis frameworkbased on the combinationof 1) local computationsby means of graph re-
labelings[19],and2)anappropriateformalismfordynamicnetworks, evolvinggraphs[15],whichfor-
malizestheevolutionof thenetworktopologyasan orderedsequenceofstatic graphs.Thestrengthsof
this combinationare several:First, the use of local computationsallows to obtain generalimpossibility
resultsthatdonotdependonaparticularcommunicationmodel(e.g.,messagepassing,mailbox,orshared
memory).Second,theuseofevolvinggraphsenablestoexpressfine-grainnetworkpropertiesthatremain
temporalinessence.(Forinstance,anecessaryconditionforthebroadcastproblemaboveistheexistence
ofatemporalpath,orjourney,fromtheemittertoanyothernode,whichstatementcanbeexpressedus-
2
ingmonadicsecond-orderlogiconevolvinggraphs.)Thecombinationofgraphrelabelingsandevolving
graphsmakesitpossibletostudytheexecutionofanalgorithmasanintertwinedsequenceoftopological
eventsandcomputations,leadingtoaprecisecharacterizationoftheirrelation.Theframeworkwepropose
shouldbeconsideredasaconceptualframeworktoguidetheanalysisofdistributedalgorithms.Assuch,
itisspecifiedatahigh-levelofabstractionanddoesnotimposethechoicefor,say,aparticularlogic(e.g.
first-ordervs.LMSO)orscopeofcomputation(e.g.pairwisevs.starwiseinteraction),althoughallourex-
amplesassumeLMSOandpairwiseinteractions.Finally,webelievethisframeworkcouldpavetheway
todecisionsupportsystemsormechanizedanalysisindynamicnetworks,bothofwhicharediscussedas
possibleapplications.
Localcomputationsandevolvinggraphsare firstpresentedinSection 2, togetherwith centralprop-
ertiesofdynamicnetworks(suchasconnectivityovertime,whoseintuitiveimplicationsonthebroadcast
problemwereexploredinvariouswork–seee.g.[3,6]).Wedescribetheanalysisframeworkbasedonthe
combinationofbothtoolsinSection3.Thisincludesthereformulationofanexecutionintermsofrelabel-
ingsoverasequenceofgraphs,aswellasnewformulationsofwhatanecessaryorsufficientconditionis
intermsofexistenceandnon-existenceofsucharelabelingsequence.Weillustratethesetheoreticaltools
inSection4throughtheanalysisofthreebasicexamples,i.e.,onebroadcastalgorithmandtwocounting
algorithms,oneofwhichcanalsobeusedforelection.(Notethatourframeworkwasrecentlyappliedto
theproblemofmutualexclusionin[16].)Therestofthepaperisdevotedtoexploringsomeimplications
oftheproposedapproach,articulatedaroundthetwomajormotifsofclassification(Section5)andmech-
anization(Section6).Thesectiononclassificationdiscusseshowthe conditionsresultingfromanalysis
translate into more general properties that define classes of evolving graphs. The relations of inclusion
between these classes are examined, and interestingly-enough, they allow to organize the classes as a
connectedhierarchy.We show how this classification can reciprocally be used to evaluate and compare
algorithmsonthebasisoftheirtopologicalrequirements.Thesectiononmechanizationdiscussestowhat
extentthe tasks related to assessing the appropriatenessof an algorithmin a givencontextcan be auto-
mated.Weprovidecanonicalwaysofcheckinginclusionofagivennetworktraceinallclassesresulting
fromtheanalysesinthispaper(inefficienttime),andmentionsomeongoingworkaroundtheuseofthe
coqproofassistantinthecontextoflocalcomputation,whichwebelievecouldbeextendedtoevolving
graphs.Section7eventuallyconcludeswithsomeremarksandopenproblems.
2 Relatedwork– the building blocks
Thissectiondescribesthebuildingblocksoftheproposedanalysisframework,thatare,LocalComputa-
tions to abstract the communicationmodel, Graph RelabelingSystems as a formalism to describe local
computations,andEvolvingGraphstoexpressfine-grainedpropertiesonthenetworkdynamics.Reading
thissectionisrequiredforaclearunderstandingofthesubsequentones.
2.1 Abstractingcommunicationsthroughlocalcomputationsandgraphrelabelings
Distributedalgorithmscanbeexpressedusingavarietyofcommunicationmodels(e.g.messagepassing,
mailboxes,sharedmemory).Althougha vastmajority of algorithmsis designedin oneof these models
– predominantly the message passing model –, the very fact that one of them is chosen implies that
the obtained results (e.g. positive or negativecharacterizationsand associated proofs)are limited to the
scopeofthismodel.Thisproblemofdiversityamongformalismsandresults,alreadypointedouttwenty
yearsagoin[20],ledresearcherstoconsiderhigherabstractionswhenstudyingfundamentalpropertiesof
distributedsystems.
Local computationsand Graph relabelings were jointly proposed in this perspective in [19]. These
theoreticaltools allow to representa distributed algorithmas a set of local interaction rules that are in-
dependentfromtheeffectivecommunications.Withintheformalismofgraphrelabelings,thenetworkis
3
represented by a graph whose vertices and edges are associated with labels that represent the algorith-
micstateofthecorrespondingnodesandlinks.Aninteractionruleisthendefinedasatransitionpattern
(preconditions,actions),wherepreconditionsandactionsrelatetotheselabelsvalues.Sincetheinter-
actionsarelocal,eachtransitionpatternmustinvolvealimitedandconnectedsubsetofverticesandedges.
Figure 2 showsdifferentscopes of computation,which are notnecessarily the same for preconditions
andactions.
(a) (b) (c) (d)
Fig.2.Differentscopesoflocalcomputations;thescopeofpreconditionsisdepictedinwhite(onleftsides),while
thescopeofactionsisdepictedinblack(onrightsides).Thedashedelementsrepresententities(verticesoredges)
thatareconsideredbythepreconditionsbutremainunchangedbytheactions.
The approach taken by local computations shares a number of traits with that of population proto-
cols, more recently introduced in [1,2]. Both approaches work at a similar level of abstraction and are
concernedwithcharacterizingwhatcanorcannotbedoneindistributedcomputing.Asfarasthe scope
ofcomputationis concerned,populationprotocolscanbeseenas aparticularcase oflocalcomputation
focusingonpairwiseinteraction(seeFigure2(c)).Themaindifferencebetweenthesetools(ifany,besides
that oforiginatingfromdistinctlines of research),hasmore to do with the rolegivento the underlying
synchronizationbetweennodes.Whilelocalcomputationstypicallyseesthisasanlowerlayerbeingitself
abstracted(wheneverpossible),populationprotocolsconsidertheexecutionofanalgorithmgivensome
explicitpropertiesof an interaction scheduler.This particularityled populationprotocolsto becomean
appropriatetooltostudydistributedcomputingindynamicnetworks,byreducingthenetworkdynamics
intospecificpropertiesofthescheduler(e.g.,everypairofnodesinteractinfinitelyoften).Severalvariants
ofpopulationprotocolshavesubsequentlybeenintroduced(e.g.,assumingvarioustypesoffairnessofthe
schedulerandgraphsofinteraction),howeverwebelievetheanalogybetweendynamicsandscheduling
has some limits (e.g.,in reality two nodesthat interactonce will not necessarily interact twice; and the
precise order in which a group of nodes interacts matters all the more when interactions do not repeat
infinitelyoften).We advocatelookingatthedynamicsatafinerscale, withoutalwaysassuminginfinite
recurrenceonthescheduler(suchaschedulercanstillbeformulatedasaspecificclassofdynamics),in
the purpose of studying the precise relationship between an algorithm and the dynamics underlyingits
execution.Toremainasgeneralaspossible,wearebuilding ontopoflocalcomputations.Onemayask
whetherremainingasgeneralisrelevant,andwhetherthevariousmodelsonFigure2areinfactequivalent
in power(e.g. could we simulate any of them by repetition of another?).The answeris negativedue to
differentlevelsofatomicity(e.g.models2(a)vs.2(c))andsymmetrybreaking(e.g.models2(c)vs.2(d)).
Thereaderisreferredto[14]foradetailedhierarchyofthesemodels.Notethattheequivalencesbetween
modelswouldhavetobere-consideredanywayinadynamiccontext,sincethedynamicsmaypreventthe
possibilityofapplyingseveralstepsofaweakermodeltosimulateastrongerone.
We now describe the graph relabeling formalism traditionally associated with local computations.
Let the networktopologybe representedby a finite undirectedlooplessgraph G = (V ,E ), with V
G G G
representing the set of nodes and E representing the set of communication links between them. Two
G
verticesuandv aresaidneighborsifandonlyiftheyshareacommonedge(u,v)inE .Letλ : V ∪
G G
E → L∗ beamappingthatassociateseveryvertexandedgefromGwithoneorseverallabelsfroman
G
alphabetL(whichdenotesallthepossiblestatestheseelementscantake).Thestateofagivenvertexv,
4
resp.edgee,atagiventimetisdenotedbyλ (v),resp.λ (e).Thewholelabeledgraphisrepresentedby
t t
thepair(G,λ),notedG.
According to [19], a complete algorithm can be given by a triplet {L,I,P}, where I is the set of
initial states, and P is a set of relabeling rules (transition patterns) representing the distributed inter-
actions – these rules are considered uniform (i.e., same for all nodes). The Algorithm 1 below (A for
1
short), gives the example of a one-rule algorithm that represents the general broadcasting scheme dis-
cussed in the introduction. We assume here that the label I (resp. N) stands for the state informed
(resp.non-informed).Propagatingtheinformationthusconsistsinrepeatingthissinglerule,starting
fromtheemittervertex,untilallverticesarelabeledI.4
Algorithm1Apropagationalgorithmcodedbyasinglerelabelingrule(r ).
1
initialstates:{I,N}(Ifortheinitialemitter,N foralltheothervertices)
alphabet:{I,N}
preconditions(r1):λ(v0)=I∧λ(v1)=N graphicalnotation:
actions(r1):λ(v1):=I I N I I
Let us repeat that an algorithm does not specify how the nodes synchronize, i.e., how they select
each other to perform a common computation step. From the abstraction level of local computations,
thisunderlyingsynchronizationisseenasanimplementationchoice(dedicatedproceduresweredesigned
to fit the various models, e.g. local elections [21] and local rendezvous [22] for starwise and pairwise
interactions,respectively).Adirectconsequenceisthattheexecutionofanalgorithmatthislevelmaynot
bedeterministic.Anotherconsequenceisthatthecharacterizationofsufficientconditionsonthedynamics
will additionally require assumptions on the synchronization – we suggest later a generic progression
hypothesisthatservesthispurpose.Notethatthethreealgorithmsprovidedinthispaperrelyonpairwise
interactions,buttheconceptsandmethodologyinvolvedapplytolocalcomputationsingeneral.
2.2 ExpressingdynamicnetworkpropertiesusingEvolvingGraphs
Inadifferentcontext,evolvinggraphs[15]wereproposedasacombinatorialmodelfordynamicnetworks.
Theinitialpurposeofthismodelwastoprovideasuitablerepresentationoffixedscheduledynamicnet-
works(FSDNs),inordertocomputeoptimalcommunicationroutessuchasshortest,fastestandforemost
journeys[6].Insuchacontext,theevolutionofthenetworkwasknownbeforehand.Inthepresentwork,
weuseevolvinggraphsinaverydifferentpurpose,whichistoexpresspropertiesonthenetworkdynam-
ics.Itisimportanttokeepinmindthattheanalyzedalgorithmsareneversupposedtoknowtheevolution
ofthenetworkbeforehand.
An evolving graph is a structure in which the evolution of the network topology is recorded as a
sequence of static graphs S = G ,G ,..., where every G = (V ,E ) corresponds to the network
G 1 2 i i i
topology during an interval of time [t ,t ). Several models of dynamic networks can be captured by
i i+1
this formalism, dependingon the meaning which is given to the sequence of dates ST = t1,t2,.... For
example,thesedatescouldcorrespondtoeverytimestepinadiscrete-timesystem(andthereforebetaken
fromatimedomainT⊆N),ortovariable-sizetimeintervalsincontinuous-timesystems(T⊆R),where
eacht isthedatewhenatopologicaleventoccursinthesystem(e.g.,appearanceordisappearanceofan
i
edgeinthegraph),seeforexampleFigure3.
We consider continuous-time evolving graphs in general. (Our results actually hold for any of the
abovemeanings.)Formally,weconsideranevolvinggraphasthestructureG = (G,SG,ST),whereGis
4Detectingsuchafinalstateisnotpartofthegivenalgorithm.Thereaderinterestedinterminationdetectionasa
distributedproblemisreferredto[17].
5
theunionofallG inS ,calledtheunderlyinggraphofG.Henceforth,wewillsimplyusethenotations
i G
V andE todenoteV(G)andE(G), thesetsofverticesandedgesoftheunderlyinggraph G.Sincewe
focus here on computation models that are undirected, we logically consider evolving graphs as being
themselvesundirected.The originalversionof evolvinggraphsconsidered undirected edges, as well as
possible restrictions on bandwidth and latency. Finally, we will use the notation G to denote the
[ta,tb)
temporalsubgraphG′ = (G′,SG′ ,ST′)builtfromG = (G,SG,ST)suchthatG′ = G,SG′ = {Gi ∈SG :
ti ∈[ta,tb)},andST′ ={ti ∈ST∩[ta,tb)}.
periodt0 →t1 periodt1 →t2 periodt2 →t3 periodt3 →t4
a e a e a e a e
c c c c
b d b d b d b d
G0 G1 G2 G3
(a) Sequenceofgraphsanddates
a e
[t0,t1) [t2, t4)
c
G = [t1,t3) [t0,t2) [t0,t3) [t,2t)4
b d
[t0,t1)
(b) Acompactrepresentation
Fig.3.Exampleofevolvinggraph.
2.3 Basicconceptsandnotations(givenanevolvinggraphG = (G,SG,ST)).
As a writing facility, we consider the use of a presence function ρ : E × T → {0,1} that indicates
whetheragivenedgeispresentatagivendate,thatis,fore ∈ E andt ∈ [ti,ti+1)(withti,ti+1 ∈ ST),
ρ(e,t)=1 ⇐⇒ e∈E .
i
A central concept in dynamic networks is that of journey, which is the temporal extension of the
conceptofpath.Ajourneycanbethoughtofasapathovertimefromonevertextoanother.Formally,a
sequenceofcouplesJ = {(e ,σ ), (e ,σ )..., (e ,σ )}suchthat{e ,e ,...,e } isa walkin Gand
1 1 2 2 k k 1 2 k
{σ ,σ ,...,σ }isanon-decreasingsequenceofdatesfromT,isajourneyinGifandonlyifρ(e ,σ )=1
1 2 k i i
foralli≤k.Wewillsaythatagivenjourneyisstrictifeverycouple(e ,σ )istakenfromadistinctgraph
i i
ofthesequenceS .
G
Let us denote by J∗ the set of all possible journeys in an evolving graph G, and by J∗ ⊆ J∗
(u,v)
those journeysstarting at node u and ending at node v. If a journey exists from a node u to a node v,
that is, if J∗ 6= ∅, then we say that u can reach v in a graph G, and allow the simplified notations
(u,v)
u v(inG),oru st vifthiscanbedonethroughastrictjourney.Clearly,theexistenceofjourneyisnot
symmetrical:u v <v u;thisholdsregardlessofwhethertheedgesaredirectedornot,becausethe
timedimensioncreatesitsownlevelofdirection–thispointisclearbytheexampleofFigure1.Givena
nodeu,theset{v ∈V :u v}iscalledthehorizonofu.Weassumethateverynodebelongstoitsown
horizonbymeansofanemptyjourney.HereareexamplesofjourneysintheevolvinggraphofFigure3:
– J ={(ab,σ ∈[t ,t )),(bc,σ ∈[σ ,t )),(ce,σ ∈[t ,t ))}isajourneyfromatoe;
(a,e) 1 1 2 2 1 2 3 2 3
– J ={(ac,σ ∈[t ,t )),(cd,σ ∈[σ ,t ),(de,σ ∈[t ,t ))}isanotherjourneyfromatoe;
(a,e) 1 0 1 2 1 1 3 3 4
6
– J ={(ac,σ ∈ [t ,t )),(cd,σ ∈ [t ,t ),(de,σ ∈ [t ,t ))}isyetanother(strict)journeyfrom
(a,e) 1 0 1 2 1 2 3 3 4
atoe.
We willsaythatthenetworkisconnectedovertimeiff∀u,v ∈ V,u v∧v u.Theconceptof
connectivityovertimeisnotnewandgoesbackatleastto[3],inwhichitwascalledeventualconnectivity
(althoughrecentliteratureonDTNsreferredtothistermsforanotherconceptthatwerenamed eventual
instant-connectivitytoavoidconfusioninSection5).
3 The proposed analysisframework
As a recall of the previous section, the algorithmic state of the network is given by a labeling on the
correspondinggraph G, then noted G. We denote by G the graph covering the period [t ,t ) in the
i i i+1
evolving graph G = (G,SG,ST), with Gi ∈ SG and ti,ti+1 ∈ ST. Notice that the symbol G was
usedherewithtwodifferentmeanings:thefirstasthegenericlettertorepresentthenetwork,thesecond
to denote the underlying graph of G. Both notations are kept as is in the following, while preventing
ambiguoususesinthetext.
3.1 Puttingthepiecestogether:relabelingsoverevolvinggraphs
For an evolving graph G = (G,SG,ST) and a given date ti ∈ ST, we denote by Gi the labeled graph
(G ,λ )representingthestateofthenetworkjustafterthetopologicaleventofdatet ,andbyG the
i ti+ǫ i i[
labeledgraph(Gi−1,λti−ǫ)representingthenetworkstatejustbeforethatevent.Wenote
Event (G )=G .
ti i[ i
Anumberofdistributedoperationsmayoccurbetweentwoconsecutiveevents.Hence,foragivenalgo-
rithmAandtwoconsecutivedatesti,ti+1 ∈ ST,wedenotebyRA[ti,ti+1) oneofthepossiblerelabeling
sequenceinducedbyAonthegraphG duringtheperiod[t ,t ).Wenote
i i i+1
RA[ti,ti+1)(Gi) = Gi+1[ .
Forsimplicity,wewillsometimesusethenotationri(u,v)∈RA[t,t′) toindicatethattheruleriisapplied
ontheedge(u,v)during[t,t′).Acompleteexecutionsequencefromt tot isthengivenbymeansof
0 k
analternatedsequenceofrelabelingstepsandtopologicalevents,whichwenote
X=RA[tk−1,tk) ◦Eventtk−1 ◦..◦Eventti ◦RA[ti−1,ti) ◦..◦Eventt1 ◦RA[t0,t1)(G0)
This combinationis illustrated on Figure 4. As mentioned at the end of Section 2.1, the execution of a
local computation algorithm is not necessarily deterministic, and may depend on the way nodes select
oneanotherata lowerlevelbeforeapplyinga relabelingrule.Hence,we denotebyXA/G thesetofall
possibleexecutionsequencesofanalgorithmAoveranevolvinggraphG.
3.2 Methodology
Belowaresomeproposedmethodsandconceptstocharacterizetherequirementofanalgorithminterms
oftopologydynamics.Moreprecisely,weusetheabovecombinationtodefinetheconceptoftopology-
relatednecessaryorsufficientconditions,anddiscusshowagivenpropertycanbeprovedtobeso.
7
t0 t1 t2 ... tk−1 tk
z G}|0 { z G}|1 { z Gt}k|−1 {
time
G0 G1[ G1 G2[ Gk−1 Gk[
R R R
[t0,t1) [t1,t2) [tk−1,tk)
start Evt1 Evt2 ... Evtk−1 end
Fig.4.CombinationofGraphRelabelingsandEvolvingGraphs.
Objectivesofanalgorithm GivenanalgorithmAandalabeledgraphG,thestateonewishestoreach
canbegivenbyalogicformulaP onthelabelsofvertices(andedges,ifappropriate).Inthecaseofthe
propagationscheme(Algorithm1Section2.1),suchaterminalstatecouldbethatallnodesareinformed,
P (G)=∀v ∈V,λ(v)=I.
1
TheobjectiveOAisthendefinedasthefactofverifyingthedesiredpropertybytheendoftheexecu-
tion,thatis,onthefinallabeledgraphGk.Inthisexample,weconsiderOA1 =P1(Gk).Theopportunity
mustbetakenheretotalkabouttwofundamentallydifferenttypesofobjectivesindynamicnetworks.In
theexampleabove,aswellasintheotherexamplesinthispaper,weconsideralgorithmswhoseobjective
is to reach a given propertyby the end of the execution.Anothertype of objectivein dynamicnetwork
is to considerthe maintenanceof a desired propertydespite the network evolution (e.g. coveringevery
connected componentin the network by a single spanning tree). In this case, the objective must not be
formulatedintermsofterminalstate,butratherintermsof satisfactorystate,forexamplein-betweenev-
erytwoconsecutivetopologicalevents,i.e.,OA = ∀Gi ∈ SG,P(Gi+1[).Thisactuallycorrespondstoa
self-stabilizationscenariowhererecurrentfaultsarethetopologicalevents,andthenetworkmuststabilize
in-betweenanytwoconsecutivefaults.Werestrictourselvestothefirsttypeofobjectiveinthefollowing.
Necessaryconditions GivenanalgorithmA, itsobjectiveOA andanevolvinggraphproperty CN,the
propertyCN isa(topology-related)necessaryconditionforOA ifandonlyif
∀G,¬CN(G) =⇒ ¬OA
Provingthisresultcomestoprovethat ∀G,¬CN(G) =⇒ ∄X ∈ XA/G | P(Gk). (Thedesiredstateis
notreachablebytheendoftheexecution(timek),unlesstheconditionisverified.)
Sufficient conditions Symmetrically, an evolving graph property CS is a (topology-related) sufficient
conditionforAifandonlyif
∀G,CS(G) =⇒ OA
Provingthisresultcomestoprovethat∀G,CS(G) =⇒ ∀X ∈XA/G,P(Gk).
Because the abstraction level of these computations is not concerned with the underlying synchro-
nization, no topologicalpropertycan guarantee,alone, that the nodeswill effectivelycommunicateand
collaboratetoreachthedesiredobjective.Therefore,the characterizationofsufficientconditionsrequires
additionalassumptionson the synchronization.We propose below a genericprogressionhypothesisap-
plicabletothepairwiseinteractionmodel(Figure2(c)).Thisassumptionmayormaynotbeconsidered
realisticdependingontheexpectedrateoftopologicalchanges.
ProgressionHypothesis1 (PH1). Inevery time interval [ti,ti+1), with ti in ST, eachvertex is ableto
applyatleastonerelabelingrulewitheachofitsneighbors,providedtherulepreconditionsarealready
satisfiedattimet (andstillsatisfiedatthetimetheruleisapplied).
i
8
Inthecasewhenstarwiseinteraction(seeFigure2(b))isconsidered,thishypothesiscouldbepartially
relaxedtoassumingonlythateverynodeappliesatleastoneruleineachinterval.
4 Examples ofbasicanalyses
Thissectionillustratestheproposedframeworkthroughtheanalysisofthreebasicalgorithms,namelythe
propagationalgorithmpreviouslygiven,andtwocountingalgorithms(onecentralized,onedecentralized).
Theresultsobtainedhereareusedinthenextsectiontohighlightsomeimplicationsofthiswork.
4.1 Analysisofthepropagationalgorithm
Wewanttoprovethattheexistenceofajourney(resp.strictjourney)betweentheemitterandeveryother
nodeisanecessary(resp.sufficient)conditiontoachieveOA .Ourpurposeisnotasmuchtoemphasize
1
theresultsthemselves–theyareratherintuitive–astoillustratehowthecharacterizationscanbewritten
inarigorousway.
Condition1 ∀v ∈V,emitter v
(Thereexistsajourneybetweentheemitterandeveryothervertex).
Lemma1 ∀v ∈ V : λt0(v) = N,λσ>t0(v) = I =⇒ ∃u ∈ V,∃σ′ ∈ [t0,σ) : λσ′(u) = I ∧u
vinG[σ′,σ)
(Ifanon-emittervertexhastheinformationatsomepoint,itimpliestheexistenceofanincomingjourney
fromavertexthathadtheinformationbefore)
Proof. ∀v ∈V :λt0(v)=N,(λσ>t0(v)=I =⇒ ∃v′ ∈V :r1(v′,v) ∈RA1[t0,σ))
(If a non-emitter vertex has the information at some point, then it has necessarily applied rule r with
1
anothervertex)
=⇒ ∃v′ ∈V,σ′ ∈[t0,σ):λσ′(v′)=I∧ρ((v′,v),σ′)=1
(AnedgeexistedatapreviousdatebetweenthisvertexandavertexlabeledI)
Bytransitivity, =⇒ ∃v′′ ∈V,∃σ′′ ∈[t0,σ):λσ′′(v′′)=I ∧v′′ vinG[σ′′,σ)
(AjourneyexistedbetweenavertexlabeledI andthisvertex) (cid:3)
Proposition1 Condition 1 (C ) is a necessary condition on G to allow Algorithm 1 (A ) to reach its
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objectiveOA .
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Proof. (usingLemma1).FollowingfromLemma1andtheinitialstates(I fortheemitter,N forallother
vertices),wehaveOA1 =⇒ C1,andthus¬C1 =⇒ ¬OA1 (cid:3)
Condition2 ∀v ∈V,emitter st v
Proposition2 UnderProgressionHypothesis1(PH ,definedintheprevioussection),Condition2(C )
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issufficientonG toguaranteethatA1willreachOA1.
Proof. (1):ByPH1,∀ti ∈ST\(tk),∀(u,u′)∈Ei,λti(u)=I =⇒ λti+1(u′)=I
Byiterationon(1):∀u,v∈V,u st v =⇒ (λ (u)=I =⇒ λ (v)=I)
t0 tk
Now,becauseλ (emitter)=I,wehaveC (G) =⇒ ∀X ∈X ,P (G ) (cid:3)
t0 2 A/G 1 k
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4.2 Analysisofacentralizedcountingalgorithm
Likethepropagationalgorithm,thedistributedalgorithmpresentedbelowassumesadistinguishedvertex
atinitialtime.Thisvertex,calledthecounter,isinchargeofcountingalltheverticesitmeetsduringthe
execution(its successive neighborsin the changingtopology).Hence, the countervertexhas two labels
(C,i),meaningthatitisthecounter(C),andthatithasalreadycountedivertices(initially1,i.e.,itself).
The otherverticesare labeled either F or N, dependingon whetherthey have already been counted or
not.Thecountingruleisgivenbyr inAlgorithm2,below.
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Algorithm2Countingalgorithmwithapre-selectedcounter.
initialstates:{(C,1),N}((C,1)forthecounter,N forallothervertices)
alphabet:{C,N,F,N∗}
ruler1:
C,i N C,i+1 F
Objective of the algorithm. Underthe assumption of a fixed numberof vertices, the algorithm reaches
a terminal state when all vertices are counted, which corresponds to the fact that no more vertices are
labeledN:
P =∀v ∈V,λ(v)6=N
2
TheobjectiveofAlgorithm2istosatisfythispropertyattheendoftheexecution(OA2 = P2(Gk)).We
prove here that the existence of an edge at some point of the execution between the counter node and
everyothernodeisanecessaryandsufficientcondition.
Condition3 ∀v ∈ V\{counter},∃ti ∈ ST : (counter,v) ∈ Ei, or equivalently with the notion of
underlyinggraph,∀v ∈V\{counter},(counter,v)∈E
Proposition3 ForagivenevolvinggraphGrepresentingthetopologicalevolutionsthattakeplaceduring
theexecutionofA2,Condition3(C3)isanecessaryconditiononGtoallowA2toreachitsobjectiveOA2.
Proof. ¬C (G) =⇒ ∃v ∈V\{counter}:(counter,v)∈/ E
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=⇒ ∃v ∈V\{counter}:∀ti ∈ST\{tk},r1(counter,v)∈/ RA2[ti,ti+1)
=⇒ ∃v ∈V\{counter}:∀X ∈X ,λ (v)=N
A2/G tk
=⇒ ∄X ∈XA2/G :P2(Gk) =⇒ ¬OA2 (cid:3)
Proposition4 UnderProgressionHypothesis1(notedPH below),C isalsoasufficientconditionon
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G toguaranteethatA2willreachitsobjectiveOA2.
Proof. C3(G) =⇒ ∀v ∈V\{counter},∃ti ∈ST :(counter,v)∈Ei
byPH1, =⇒ ∀v ∈V\{counter},∃ti ∈ST :r1(counter,v)∈RA2[ti,ti+1)
=⇒ ∀v ∈V\{counter},λ (v)6=N
tk
=⇒ ∀X ∈XA2/G,P2(Gk) =⇒ OA2 (cid:3)
4.3 Analysisofadecentralizedcountingalgorithm
Contrarytothepreviousalgorithm,Algorithm3belowdoesnotrequireadistinguishedinitialstateforany
vertex.Indeed,all verticesare initialized with the same labels (C,1), meaningthat theyare allinitially
countersthathavealreadyincludedthemselvesintothecount.Then,dependingonthetopologicalevolu-
tions,thecountersopportunisticallymergebypairs(ruler )inAlgorithmA .Intheoptimisticscenario,
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