A Mathematical Analysis of Prophet Dynamic Address Allocation

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A Mathematical Analysis of Prophet Dynamic Address

Allocation

Cédric Lauradoux, Marine Minier

To cite this version:

Cédric Lauradoux, Marine Minier. A Mathematical Analysis of Prophet Dynamic Address Allocation.

[Research Report] RR-7085, INRIA. 2009, pp.15. �inria-00429480�

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A Mathematical Analysis of

Prophet Dynamic Address Allocation

Cédric Lauradoux — Marine Minier

N° 7085

Unité de recherche INRIA Rhône-Alpes

655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France)

Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52

Prophet Dynami Address Allo ation

Cédri Lauradoux, Marine Minier

ThèmeCOMSystèmes ommuni ants ProjetSWING

Rapportdere her he n°7085O tober200915pages

Abstra t: Prophetisadynami addressallo ationproto oldes ribedatINFOCOM2003. Thisproto olisbaseduponafamilyofpseudo-randomgenerators. ThegoalofProphetisto establishanaddressess hemefreeof oni t. Theaddressing apabilitiesofProphetdepend ontheunderlyingpropertiesofthepseudo-randomgenerators. Thedierentpseudo-random generators proposed in Prophet are analyzed and the limits of the s heme are exhibited. Mostnotably, theperiodsof thegeneratorslimittheaddressing apabilities ofanodeand thefa tthat Prophetis ollision-free. Inthis resear hreport,weshowthattheunderlying assumptionsmadeinProphet annotbemetbypseudo-randomgenerators.

Prophet Dynami Address Allo ation

Résumé: Prophetest unproto oled'allo ation dynamiqued'addresseprésenté à INFO-COM 2003. Prophet permet de générer des adresses libres de ollision sans né éssité de ommuni ationsadditionnellesque elle liées àl'allo ationpropredes adresses. Ce proto- oles'appuiesurdeuxhypothèsesimportantes: l'existen edegénérateurspseudo-aléatoires degrandepériode etlafaible probabilitéd'obtenir deuxfoislamêmevaleurdansdeux sé-quen es pseudo-aléatoires. Les apa itésd'adressagede Prophetdépendent despropriétés sous-ja entes des générateurs employés. Pour implémenter leur proto ole, les auteurs de Prophetontproposés l'emploi de générateurs pseudo-aléatoiresde typelinéaire ongruen-tiel. Les s hémas proposés par les auteurs sontanalysés. Il en résulte que Prophetn'est pas apable de vérierl'hypothèse surla période des séquen es générées diérentes. Ce i impliquequeProphetn'estpas apabledesupportertoutesles ongurationspossiblesd'un réseau dynamique sans sa rier l'uni ité des adresses générées. Notre on lusion est que leshypothèsesutiliséesdansProphetnesontpas ompatiblesave l'emploi desgénérateurs pseudo-aléatoires.

1 Introdu tion

Dynami addressallo ationisafundamentalprobleminmobilead-ho networks(MANETs). Addressallo ationpre eedsanyfurtheroperationsinnetworking. This taskisparti ularly di ult when there is no a ess to a entralized infrastru ture, like in MANETs. The hallenge onsistsin providing unique addressesto thenetwork nodesusingparalleland independentpro esses. Severalpropositionshavebeenmadetosolvethisproblem. A om-pletesurveyonthisproblem anbefoundin[BCM09℄. Four lassesofalgorithmshavebeen identied sofar: oni t-dete tion allo ation[PRD00℄, oni t-freeallo ation [MDMD01℄, best-eortalgorithms[NP02℄andto on ludetheProphetalgorithm[ZNM03℄. Prophetisa parti ularlyintriguingalgorithmbe auseitappearsasasigni antbreakthroughinmobile networking.Itprovidesanaddressesallo ations hemefreeof oni twithalow omplexity, alow ommuni ationoverheadandalowlaten y.

The Prophet proto ol is investigated in depth in this paper. Prophetis based on as-sumptionsrelatedtorandomnumbergenerationandtheauthorsmadeanintensiveusedof linear ongruentialgenerators(LCGs). Then,weshowthat theunderlyingideaof Prophet issimilartotheproblemofparallelpseudo-randomgenerators[Dur89℄. Thisproblem omes from theparallel omputingand it is morespe i ally relatedto distributed Monte Carlo simulation[Nie92℄. Indeed, oneof the fundamental assumption ofProphet onsists in us-ing sequen es oflarge period as in parallel omputing. Moreover,theauthors of Prophet haveindeed used twowell-known strategies to transforman LCG into aparallel random-numbergenerator. Unfortunately,thereexiststwofundamentaldieren esbetweenthegoal ofProphetandtheproblemdis ussedin parallel omputing. First,theparametersused in theLCGsforProphetare hosendynami allyandrandomlywhiletheyare hosenstati ally inparallel omputing. Thisfundamentaldieren eallowstohavesome ontrolonthe prop-ertiesoftherandomnumbersprodu edinparallel omputing. Inthe aseofProphet,thisis learlynotthe ase. Se ond,Prophetassumesthattheparallelsequen esare ollisionfree. Thisstrongassumption notfoundin parallel omputing annotbeunfortunatelysatised usingparallelpseudo-randomgenerators. Therefore,weshowthatProphet annota hieve itsmaingoal: uniqueaddressesallo ation(seeFigure1).

3

5 6 8

8 12 11 11 6 4 7 3

Figure1: AnexampleofaddressesassignedwithProphetfor

N = 12

.

A denition of dynami address allo ation is given in Se tion 2. The prin iples and the assumptions used in the design of Prophetare des ribed in Se tion 3. Then, wend parti ularly useful to analyzed thetoy example used by theauthors to present Prophet (Se tion4). Thekeydenitionsofpseudo-randomgeneratorsareremaindedinthisse tion. Thepra ti alpseudo-randomgeneratorsproposedforProphetandthepossibilitytoa hieve ollision-freedynami addressallo ationaredis ussedinSe tion5. Finally,we on ludethe paper.

2 Denitions

Thegoalofthisse tionistoestablish learlythegoalofadynami addressallo ation algo-rithm. Thedenitionofsu hanalgorithmandofanaddressgrapharegiven. Theaddress graph des ribes the address allo ation s heme. A dynami address allo ation algorithm mustbeabletoprovideaddressesforanypossibleaddressgraph. Thosedenitionsarevery importantto understandthelimitationsofProphet.

Denition1 An address graph is an a y li oriented graphs of arbitrary indegree and outdegree.

Theedgesoftheaddressgraphrepresentstheaddressesdependen y. Ifthereisanedge from A to B, it means that A has assigned an address to B. There is at least anetwork onne tionbetweentwonodes onne tedbyanedge at thetimeof theaddressallo ation. But two dis onne ted nodes of the address graph an have a network onne tion. The assignments hemefortheaddressesmayberelatedtoanothernetworkme hanism.

Theindegree

d

i

ofanode orrespondstothenumberofaddressesassignedtothis parti -ularnode. Agivennode anhaveanarbitrarynumberofaddresses. Theoutdegree

d

o

isthe numberof timesthatagivennodehasassignedaddresses. Forsimpli ity andwithoutloss ofgenerality,onlyaddressgraphsforwhi htheindegreeofanynodeis

d

i

≤ 1

are onsidered inthefollowing.

Denition2 A dynami addresses allo ation algorithm onsists in giving a unique labeltothe

N

nodesof anypossible address graphs.

Therepresentationoftheaddressassignmentasana y li orientedgraphisveryuseful to apturethe omplexityofdynami addressesallo ationandthelimitationsofthe urrent approa hestosolvetheproblem. Inanetwork,severalnodes anstartto assignaddresses to other nodes. These parti ular nodes are alled root and they haveanindegree

d

i

= 0

. Thetwoextremesetupsforadynami networkare:

•

An addressgraph with anode A(0,

N − 1

) (Figure 2(a)).

N − 1

nodes request an addresstoasingleroot. This ase orrespondstothe lassi aladdressallo ationwith a entralizedinfrastru ture.

•

An address graphin whi h allthe nodes havean indegreeand outdegree equalto 0 (Figure2(d)). Ea hnode hoosesforitselfagivenaddress,i.e. ea hnodeisaroot.

Betweenthosetwoextremesetups,any ongurationispossible. Forinstan einwireless sensorsnetworks(WSNs), asingleroot anassigntheaddressesofallthenodeswhi hare distantfromonehop(seeFigure2(a)and( )). Inanad-ho networks,itispossibletohave

k

rootsinordertoredu ethedelaytoobtainanaddress. Inthisparti ular ase,theoverall graph an be viewed as a singlenetwork or as

k

networks (sub-graph) with one network asso iated to ea h root. If thegraph is onsidered asasingle network, it impliesthat all the sub-networks are merged. Therefore, our denition is enoughto overoperations like merging. Intheirpaper[ZNM03℄,theauthors providematerials foraddressgraphswith a singleroot.

(a) (b) ( ) (d) A(0,2) B(1,0) C(1,0) A(0,1) C(0,0) B(1,0) A(0,0) B(0,0) C(0,0) A(0,1) B(1,1) C(1,0)

Figure 2: Dierent setups for an a y li oriented graphsof indegree

d

i

≤ 1

with

N = 3

, A(

i

,

j

)with

d

i

= i

and

d

o

= j

.

3 Prophet: prin iple and hypothesis

3.1 Ar hite ture of the s heme

In Prophet, ea h node re eived a bla k box (see Figure 3) allowinghim to generate new addressesuponrequest. Thisbla kboxisinitializedwiththenodeaddress

a

andaseed

x

0

. Theseedis usedtoinitializeaninternal state

x

t

whi his updatedbythefun tion

g

. This fun tion

g

anbea ounter (in rementation)asanyother fun tion. Theinternal state

x

t

andtheaddress

a

areused toprodu enewaddressesusingtheaddressderivationfun tion

f

. It should be notedthat theauthors in their paper[ZNM03℄ donot mentionexpli itly theexisten eofanupdatefun tion. However,itsimpliesgreatlyouranalysis.

Usingthisrepresentation,onemayndtheproblem onsideredbyZhouetal. very sim-ilartotheproblemofparallelrandomgenerators[Dur89,Hal89℄. Distributed omputations likesimulationsmakeanintensiveuse ofrandomnumbers. Ea h threadsof thesimulation may needto haveitsown streamof randomnumberssu h that allthestreams ofrandom numbersareun orrelated. However,thisproblemisessentiallystati (everythingisknown inadvan e: numberof omputers,topology).

A rst limitation of Prophet is the la k of initialization algorithm. Prophet an not beused when ea h node hoosesfor itself an address(allthe nodeshave anindegree and outdegreeequalto 0). Prophetanwerstheissueofaddressderivationanditisnotsuitable

a0,1

a0,i+1

g

g

x

0

x

0

a

0,0

x

1

x

i

f(x

0

, a

0,0

)

f

(x

1

, a

0,0

)

a0,2

f(x

i

, a

0,0

)

Figure3: AviewofProphet.

when severalroots assign addresses. It annot be used formerging networks ontrary to theauthors laim. Inthefollowing,asinglenodewith

d

i

= 0

isassumed.

3.2 Update fun tion and address derivation fun tion

Themain assumptions onthe update fun tion and onthe addressderivation fun tion are givenhereasin theoriginalpaper:

Assumption1 The interval betweentwoo urren esof the samenumberin asequen eis extremelylong.

Assumption2 Theprobabilityofmorethanoneo urren eofthesamenumberinalimited numberof dierent sequen esinitiatedby dierentseedsduringsomeintervals isextremely low.

TheAssumption1 orrespondswiththerequirementofpseudo-randomgenerators: the period

T

ofasequen e

S

mustbeaslargeaspossible. Thereadersmayndhelpful infor-mationsonpseudo-randomnumbersin lassi altextbooks[Nie92,Knu97℄.

TheAssumption2isapropertythat mustbesatisedby omparingseveralsequen es. Itmeansthattherepetitionofagivenvalueinseveralsequen esmustbereallyimprobable. Theremainingpartsofthepaper on ernthedesignof

f

and

g

su hthatAssumption1 and2areveried. First,thetoyexamplegiveninSe tionIII.Boftheoriginalpaper[ZNM03℄ isdisse ted. Then,wedealwiththepra ti aldesignof

f

and

g

proposedbytheauthors.

4 Prophet: the toy example

Toillustratethebasi ideaoftheirpaper,Zhouetal.[ZNM03℄proposetousethefolowing fun tions:

f

(a, x

t

) = a

× x

t

× 11 mod 7

x

t+1

= g(x

t

)

= f (a, x

t

)

(1)

with

a

agivenaddress. Whenanewaddressisprovided, anew seedis omputed. Inthis example, the newseed is the addressitself. Here, theupdate fun tion

g

and the address derivationfun tion are thesame. Fortheaddressgraphdepi tedin Figure 4(a), Prophet allo ationwithEquation1providestheaddressesshowninFigure4(b). Theaddressesare generatedbytwopseudo-randomgenerators(Figure4( )).

(3) (4) (1) (5)

f

(3, x) = 33x mod 7

f

(1, x) = 11x mod 7

(b) ( ) (a)

Figure4: Exampleofallo ationwithaninitialaddress

a

= 3

.

It must be noti ed that the multipli ation by eleven is a simple permutation of the address

a

.

f

anbewritten:

f

(γ, x

t

) =

γ

× x

t

mod 7

γ

=

a

× 11 mod 7

Inthislaterform

f

(γ, x

t

)

,thosefun tionsare learlyidentiedasthemultipli ationlinear ongruentialgenerators(MLCGs). Thisisaspe ial aseofthelinear ongruentialgenerators denedbyLehmer[Leh51℄.

4.1 Linear Congruential Generators

Alinear ongruentialgeneratorofmultiplier

b

,modulus

m

andin rement

c

isdened by:

x

t+1

= b

× x

t

+ c mod m.

(2) ThisfamilyofgeneratorshasbeenwellstudiedandnumerousworksexistonLCGs[PM88℄. Mostnotably,theyhavestatisti alweaknesseswhi hmadethemunse urefor ryptographi appli ations[Knu85,Ste87℄. Somebasi propertiesofMLCGs arenowremained.

Forthe MLCGs (

c

= 0

), it is well known that the period

T

, i.e. the smallest integer su h that

x

t+T

= x

t

,isthesmallestinteger

k

su hthat:

b

k

_{≡ 1 mod m.}

If

a

and

m

are oprime,westatethatusingtheEuler-Fermattheorem:

b

φ(m)

_{≡ 1 mod m}

(3)

where

φ(m)

istheEuler'stotientfun tion,thenumberofpositiveintegerslessthanorequal to

m

thatare oprimeto

m

. Therefore,theperiod

T

annotex eed

φ(m)

whi hismaximal if

m

isaprime,i.e.

φ(m) = m

− 1

. An integer

b

verifyingEquation3is alledaprimitive rootmodulo

m

. There is

φ(m

− 1)

primitiverootsmodulo

m

.

4.2 Analysis

In this example, ea h node is assigned at the same time with an address, a seed and a parti ularinstan eof aMLCGwithmodulus

m

. Thiste hniqueisknownasthedierent multipliers strategyin parallel omputing [Dur89℄. The period

T

represents the number of addresses that an be allo ated by a node. If the period

T

is not large enough, the Assumption1 annotbemet.

Thisstrategyfailedinthe ontextofProphetbe ausethemultipliersare hosenrandomly over

[1, m

− 1]

. Theprobabilitytohave

T

= m

− 1

isequalto:

P

[T = m

_{− 1] =}

φ(m

− 1)

m

_{− 1}

.

Inthegivenexample(

m

= 7

),thisprobabilityisonly

1

3

andmoregenerallyfor

m

− 1 > 6

:

√

m

_{− 1}

m

_{− 1}

≤ P [T = m − 1] ≤ 1 −

√

m

_{− 1}

m

_{− 1}

.

Assumingthat wehavesu essfullyallo ated

i

− 1

addresses, i.e theaddressesare unique and the period asso iated to an address is always maximal, the probability for the

i

-th allo ationto su eedis:

P

[T

i

= m

− 1] =

φ(m

− 1) − i

m

_{− 1}

with

T

i

theperiodasso iatedtothe

i

-thaddress. Thisresultstronglylimitstheaddressing apabilityofthes heme. Themoreaddressesareallo atedwith su has heme, thehigher is theprobabilitythat weget adegenerated fun tion

f

. Prophet anprovide oni t-free addressesforalltopologiesifanodehasanaddressthatimpliesalowperiodpseudo-random generator

A se ond problem ariseswith xedpoints. Ifthe Equation1is usedwith

m

= 7

, itis easily seenthat the initialvalue 2alwaysprodu es

2

asan address. This problem anbe solvedifanodewhi h re eivedanaddresssear hesandpreventstheuseofaxedpoint.

A third problemarises whenthe Assumption2is onsidered. Let onsider theaddress graphdes ribedinFigure5(a). Inthis ase,thepseudo-randomgeneratorsused toassign theaddresses(Figure5( ))havefullperiod(seeFigures6and7). However,westillhavea ollisionontheaddressassigned(Figure5(b)).

(3)

f

(3, x) = 33x mod 7

(b) ( )

f

(6, x) = 66x mod 7

(5) (a) (1) (4) (6) (4)

Figure5: Exampleofallo ationwitha ollision.

Infa t,theAssumption2 anbewritten:

∄i < σ,

with

∀x, y ∈ [1, m − 1], ∀x

′

_{, z}

_{∈ [1, m − 1], x 6= x}

′

and

z

6= y

i

_{mod m}

su hthat:

x

× y

i

≡ x

′

_{× z mod m}

where

σ

is axed threshold. Theauthorsdonotseeanyotherwayto verify thisproperty thanexhaustivesear h(

O(m

4

₎

). Thispropertyisgenerallynotveriedbyparallel pseudo-randomgenerators.

To on ludethis analysis,theuseof MLCGs for theimplementation of Prophetis not re ommended be ause the periods of the dierent generatorsare di ult to ontrol (As-sumption1)andbe ausenoexpe tationontheprobabilityof ollisionsbetweenthedierent streams anbefound(Assumption2).

Figure 6: Behaviour of the generator for

a

= 3

.

Figure 7: Behaviour of the generator for

a

= 6

.

5 Analysis of the pra ti al s heme

5.1 Des ription and Analysis

Theauthorsproposedanother s hemeto implementProphet. Intheirpra ti al onstru -tion,theupdate fun tion

g

andtheaddressderivationfun tion

f

aredierent:

f

a

(x

t

) = [a + h(x

t

) mod m] + 1

h(x

t

) =

n

Y

i=1

p

x

i

t

i

x

t+1

= g

j

(x

t

)

= (x

1

t

, x

2

t

,

· · · , x

j

t

,

· · · , x

n

t

).

(4)

The fun tion

h

is the produ t of the

n

rst prime numbers

p

i

put to the power

x

i

t

. The internalstate

x

t

isave torof

n

integers

x

1

t

, x

2

t

,

· · · x

n

t

. Thisve tor anbeviewedasavirtual oordinate systemin theaddressgraph. Theupdate fun tion

g

j

onsists in in reasingby onethe oordinate

x

j

t

. Thevalue

j

isxedforagivennode. Themodulus

m

istheboundon thenumberofaddresseswhi h anbeallo ated. Whenanaddressisrequested,theinternal stateis rstupdatedand then, thefun tion

f

i

isapplied. On e anewaddress

b

= f

a

(x

t

)

is produ ed by anode

a

the urrent state of thenode is sentas aseed forthe new node and the newvalue

j

b

= j

a

+ 1

. Theroot node hasfor address

a

and its internal state is

∀i ∈ [1, n], x

i

0

= 0

. Anexamplefor

n

= 4

andwith

x

0

= (0

k 0 k 0 k 0)

isgiveninFigure8.

[a, (1, 0, 0, 0)], j = 1

[a, (2, 0, 0, 0)], j = 1

[a + 5, (2, 0, 0, 0)], j = 2

[a, (0, 0, 0, 0)], j = 1

[a + 3, (1, 0, 0, 0)], j = 2

[a + 3, (1, 1, 0, 0)], j = 2

[a + 7, (1, 1, 0, 0)], j = 3

Figure8: Exampleofallo ationwithaninitialaddress

a

.

j

Fun tion

x

t+j

Period

T

1

f

a

(x

t+j

) = (a + 2

j

_{mod m) + 1}

_(j

k 0 k 0 k 0 k 0)

16 2

f

a

(x

t+j

) = (a + 2

× 3

j

_{mod m) + 1}

₍₁

k j k 0 k 0 k 0)

51 3

f

a

(x

t+j

) = (a + 6

× 5

j

_{mod m) + 1}

₍₁

k 1 k j k 0 k 0)

76 Table1: Periodofseveralfun tionsfor

m

= 151

.

Despiteitsapparent omplexity,thisperiodofthesequen egeneratedisrelativelyeasy toobtain. Thekeyderivationfun tion

f

a

anbewrittenfor

j

= 1

:

f

a

(x

t+k

) = [a + 2

k

mod m] + 1.

Theperiod

T

isgivenby

T

= k

− i

where

i

isxedintegerand

k

isthesmallestintegersu h that:

2

k

≡ 2

i

_{mod m}

and

k > i.

(5)

Thisresult anbeeasilyextendtoothervaluesof

j

anddierentinternalstates

x

t

. For instan ewehave omputedtheperiod ofseveralfun tionsfor

m

= 151

.

The previous table learly show that we obtainas in Se tion 4a family of generators withaperioddi ultto ontrol. TheAssumption1 annotbemetbythisgenerator.

As shown previously, the period

T

of the generator is exa tly the same than in the previous. In fa t,in thislast ase, weare looking forpowers ofprime numbersthat have a maximal period, i.e. that are primitive root mod

m

. In the sameway, the number of primitiverootmod

m

is

φ(m

− 1)

andstronglydepends onthede ompositionof

m

− 1

.

IntheProphetpra ti als heme,with

m

aprimenumber,thefournumbers2,3,5,7must haveamaximalorder. Duetotheprobabilities omputedinSe tion4,thisishardtorea h. Withtheprimenumber

m

= 1031

,

φ(m)

isequalto

408

,theorderof2,3and5is515andthe orderof7is206. So,ndingan

m

valueforwhi hthe4rstprimeintegershaveamaximal orderhasareallylowprobability. Andthegeneralprobabilityto obtainaprimitiverootis exa tlythesamethantheonegivenin Se tion4.

5.2 Parameters hoi es and examples

Theimplementationof thiss hemerequiretodenethree parameters,(1) themodulus

m

, (2)therstaddress

a

,and(3)

n

thenumberofprime numbersusedandtheinternalstate. Inthe original paper [ZNM03℄, there is no dis ussionaboutthis three parametersand no pra ti alparametersareprovided. Later in [Zho08℄, Zhou provide details onthe hoi e of

m

and

n

.

Choi efor

m

. Themodulus

m

mustbe hosen arefullyregardingthesizeofthenetwork

N

and the previous remarks on the period. The original paper, it is onsidered that the modulus

m

shouldbetheaddressrange+1.. In[Zho08℄,itis onsideredthat

m

mustbethe highestprime numberlessor equalthat theaddressrange. However,itis straightforward

3 5 6 8 12 7 10 3 2 13 9 1 3 5 6 8 (a) (b) 8 12 11 11 6 4 7 3 5 6 8 3 11 8 12 4 7 ( ) 11 6 6

Figure9: AddressassignmentobtainedwithProphetfor

N = 12

.

that this solutionis ill-fated whenthe addressrange mat h the network size

N

: it is not possibletoassign

N

uniqueaddressesfrom omputationsintherange

[0, m

− 1]

if

m <

N

. Clearly,abetter hoi ewould havebeento onsider that

m

isthe smallestprime number greaterorequal

N

.

Choi e for

a

. This parameter anae t the probability of ollisions sin e the value

a

anappearlaterin anotherpseudo-randomsequen e. Weassumethat thisvalueis hosen randomlyin

[0, m

− 1]

.

Choi efor

n

and internal state. Thenumberofprimenumbers

n

usedintheinternal statedependsonthemaximalpossibledepthforanaddressgraph. Foranaddressgraphof size

N

,themaximaldepth is

N − 1

:

n

≥ N − 1

. In[Zho08℄, thevalue

n

= 209

isusedfor

N = 50

. It should benoti ed that theinitialization of theinternal stateof theroot node (address

a

)hasnoimpa tontherespe tiveperiodofthegenerator(see Equation5).

To on lude,theFigure9showsseveraladdressesassignmentperformedbyProphetfor

N = 12, n = 11, m = 13

anddierenttopologies.

6 Con lusion

We have seen the limits of the proposition of Prophet. Prophet is based on automaton whi h must be hara terized by their inner behaviour(Assumption 1) and by their outer behaviour(Assumption 2). While the ore ideaof Prophet are sound, the appli ation of pseudo-randomgeneratorsleadsto as hemewithmany ollisions. Thes hememayworks for some address graphs but many ollisions an be expe ted for some setups. The key questionofProphetishowtodesignthese fun tionsin ordertoallow ollisionfreeaddress allo ation.

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