Laboratoire de l’Informatique du Parallélisme

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Laboratoire de l’Informatique du Parallélisme

École Normale Supérieure de Lyon

Unité Mixte de Recherche CNRS-INRIA-ENS LYON-UCBL n ^o 5668

A Repair Mehanism for Fault-Tolerane for

Tree-Strutured Peer-to-Peer Systems

Eddy Caron ,

FrédériDesprez ,

CharlesFourdrignier ,

Frank Petit,

CédriTedeshi

Ot2006

Researh Report N o

2006-34

École Normale Supérieure de Lyon

46 Allée d’Italie, 69364 Lyon Cedex 07, France

Téléphone : +33(0)4.72.72.80.37

Télécopieur : +33(0)4.72.72.80.80

Adresse électronique :

lipens-lyon.fr

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Peer-to-Peer Systems

EddyCaron ,FrédériDesprez , CharlesFourdrignier , Frank Petit, Cédri Tedeshi

Ot 2006

Abstrat

Faing the limits of traditional tools of resoure management within

omputational grids (related to sale, dynamiity, et. ofthe platforms

newly onsidered), new approahes, based on peer-to-peer tehnologies

are emerging. The resoure disovery and in partiular theservie dis-

overyisonerned bythisevolution. Amongthesolutions, apromising

one isthe indexing of resoures using trie strutures and more partiu-

larly prex trees. The major advantages of trie-strutured approahes

isthe apabilityto supportsearh querieson rangesof values withala-

teny growinglogarithmially inthenumberof nodesinthetrie. Those

tehniques are easy to extend to multiriteria searhes. One drawbak

of usingtries isits inherent poor robustness ina dynami environment,

where nodes joinand leave the network, leading to thesplit of thetree

intoaforest,whihresultsintheimpossibilitytorouterequests. Within

most reent approahes, the fault-tolerane is a prevention mehanism,

oftenrepliation-based. Therepliationanbeostlyintermofresoures

required. In this paper, we propose a fault-tolerane protool that re-

onnets subtrees a posteriori, after rashes, to have again a onneted

graphand thenreorder thenodesto rebuildaonsistent tree.

Keywords: Faulttolerane, peer-to-peer,prex trees

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les grillesdealul(mauvaispassage à l'éhelle,non prise enompte de

la dynamiité du réseau, et.), des alternatives fondées sur les tehno-

logies pair-à-pair sont en train d'émerger. La déouverte de ressoures

et en partiulier des servies de alul est touhée par ette évolution.

Parmiessolutions,ilexistedesapprohesprometteuses fondéessurdes

arbres lexiographiques. L'intérêt detellesapprohesreposesurlapossi-

bilité d'eetuer des requêtes surdes intervalles de valeurs ainsi que la

possibilitéderéaliserdel'autoomplétion surleshaînesdereherheen

temps logarithmiqueen latailledel'arbre. Cestehniquess'étendent fa-

ilement àdesreherhes multiritères. Cependantlastruture enarbre

est fragile et peut élater en une forêt si l'un des n÷uds vient à quit-

ter le réseau, rendant ainsi impossible le routage de ertaines requêtes

et n'orant au lient qu'une vue partielle des servies. Dans la plupart

de es approhes, la tolérane aux pannes, indispensable dans les envi-

ronnementsdynamiquesàlargeéhelle,estpréventive(réaliséeapriori)

et se fonde sur la répliation, qui est oûteuse en termes de ressoures

etde temps. Danse papier,nousprésentons unprotooletolérant aux

pannes de n÷uds,omplémentaire à larépliation, dans les arbres lexi-

ographiques. Ilsefonde surlareonnexionetlaréparationa posteriori

d'arbres quiont subilaperted'unou plusieurs n÷uds.

Mots-lés: Toléraneauxpannes, pair-à-pair, arbres de préxes

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1 Introdution

These last few years have seen the development of large sale grids onneting distributed

resoures(omputation resoures,storagefailities,omputation libraries, et.) inaseamless

way. This is now an eient alternative to superomputers to solve large problems suh as

high energy physis, simulation, bioinformati, et. However, existing middlewares used in

grids require most of the time a stable and entralized infrastruture. They usually loose

their performane on dynami and large sale platforms without entralized management of

resoures. To ope withthe harateristis of these emerging kind of platforms, it has been

suggestedto usepeer-to-peertehnologies within omputationalgrids [8 ℄.

Peer-to-peertehnologiesoer algorithmsallowingthesearh andretrievalofobjets over

thenet(dataitems, les,servies,et.). Among thesetehnologies, Distributed HashTables

(DHT) were initially designed for very large sale platforms, for example to share les over

the Internet. However, DHTs have several major drawbaks. Among them, their disovery

mehanismusually worksonexat searhes ofagiven key. Somework hasthenbeen done to

allowomplex requeststobesubmitted overDHTs or moregenerallyinstruturedpeer-to-

peer systems,i.e. systemsbased on request routing. Some of these worksare based on tries

(also alledprex trees). Atrie struture supports rangequeries inalogarithmi timeinthe

number ofnodesof thetrie.

Fault-tolerane is a mandatory feature for peer-to-peer systems to avoid the lossof data

storedonnodesandto allowaorretrouting ofmessages. Therashof oneor several nodes

inatrieleadstothelossofobjetsreferenesstored inthetrieandtothesplitofthetrieinto

several subtries, also alled a forest. Fault-tolerane within strutured peer-to-peer systems

usually uses repliation. Using suh an approah, eah node and eah link of thetrie would

have to be dupliated

k

^times,

k

^being ^the ^repliation ^fator. ^Keeping ^suh ^struture ^up ^is

ostly, mainly in terms of resoures used. Afterward, the purpose is to nd for the value of

k

^the ^right ^trade-o ^between ^the ^repliation ôst ând ^the ^robustness ôf ^the ^system. În ^this

paper, we study an alternative to the repliation approah based on the reonnetion of the

subtries and the a posteriori reordering of a onsistent trie. When the trie is disonneted,

a rst solution onsists in rebuilding a trie adding nodes of remaining subtries one by one.

Thisnaivemethodanleadtoaprohibitive ostwhenthenumberofremainingnodesislarge

(whihisusuallytheaseinpeer-to-peersystems). Forexample,loosingonenodeanleadto

aompletereonstrution ofthetrie. Aseondapproahonsistsinreonnetingthesubtries

to gettheoriginal trie bakat aminimum ost. Thisisthis kindof algorithm wedesribein

this paper ina distributed and asynhronous environment. It an also be used to omplete

the repliationproess.

Abriefhistoryofpeer-to-peertehnologiesisprovidedinSetion2,followedbytheformal

desriptionofthepartiulartriestrutureweuse(Setion3)andofthedistributedsystemwe

plae ourselves. We fousour study on fault-tolerane meanisms related to them. Then, in

Setion 4 we present the repair algorithm we designed and give its proof before a onlusion

and futureworkSetion.

2 Related Work

With the spread of the peer-to-peer tehnologies going along with the le sharing over the

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of unstrutured mehanisms, i.e., based on the ooding of searh requests [10 , 9℄. These

mehanisms resulted in overloading the network while providing non-exhaustive responses.

Addressingboth thesalabilityandtheexhaustivenessissueswithinpeer-to-peersystems,the

distributed hashtables [13 , 14, 18, 20 ℄, a.k.a., the strutured peer-to-peer group, are highly

salableinthesensethatthenumberoflogialhopsrequiredtorouteandtheloalstategrows

logarithmiallywiththenumberofnodespartiipatinginthesystem. Moreover,DHTsprevent

from loosing routing paths and objets' referenes by use of repliation and periodi sans.

Unfortunately, DHTs present several major drawbaks (homogeneous apaity assumptions,

topology awareness, et.). Amongthem,therigidityof therequestingmehanism, i.e., exat

math ona givenkeyhinders its useoverreal searhsystems.

A series of work givesthe opportunity to allow exible meanings of retrievalover stru-

tured peer-to-peer networks. First ahievement in this way has been the ability to desribe

resoures withsemi-strutured language, suh XML,as desribed in[3 ℄. [19℄enhanes DHTs

with traditional database operations. Several approahes, based on spaelling urves, suh

as Squid[15 ℄ or [17 ℄ support multi-dimensionalrange queries.[1℄ maps one-dimensionaldata

spae to d-dimensional Cartesian spae by using the inverse Hilbert mapping. Built on top

of multiple DHTs, SWORD [11 ℄ is an information servie aiming at disovering omputing

resoures on the gridbyansweringmulti-attribute range queries.

We fousinthisworkon trie-struturedretrievalsolutions, alsosupporting range queries

butoutperformingpreviousapproahesinthesensethatlogarithmi(oronstantifweassume

an upper bound on the depth of the trie) lateny is ahieved by parallelizing the resolution

of the queryinthe several branhes of thetrie. Prex HashTree(PHT) [12℄ builds a trie of

the entire key-spae on topof a DHT.Thepurposeof this arhiteture isto usethetrie asa

logial layerallowing omplex searheson top of anyDHT-like network. The arhiteture of

PHT results inthe multipliationof theomplexitiesof thetrie andof theunderlying DHT.

The Skip Graphs struture proposed in [2℄ is similar to a trie but is built with the skip

lists tehnology, allowing the use of their inherent fault-tolerane properties. But again, the

omplexity of the number of messages generated to proess range queries is in

O(m log(n))

^,

m

^being^the^number^of ^nodes^pertained^by^the^range^and

n

^the^total ^number^of ^nodesⁱⁿ^the

graph.

Other approahes proposeto relyon a triefor eah purpose, i.e.,indexing thekey-spae,

mapping the nodes of the trie on the network, and routing the requests. Among them,

Nodewiz [4℄ assumes a set of stati reliable nodes to host the trie, whih is unfortunately

hard toensureonpeer-to-peerplatforms. P-Grid[7℄buildsatrieonthewholekey-spae(i.e.,

thewholesetofpotentialkeys). Eahleafofthistrieorrespondstoasubsetofthekey-spae.

The fault-tolerane isahieved by probabilisti repliation.

Asa more general onsideration, none of these approahes addressthetopology/physial

loality awareness issue, i.e., no information about the underlying network is taken into a-

ount to build the logial (overlay) network, what an raise a signiant performane prob-

lem, physial loality being broken when the logial network is built. Moreover, the several

fault-tolerane solutions are mostly repliation-based, or DHT-based, also involving heavy

repliation mehanisms.

Initially designed for the purpose of serviedisovery over dynami omputational grids

and attempting to solve the above drawbaks of existing approahes, we reently developed

a novel arhiteture, based on a logial Greatest Common Prex Tree formally desribed in

Setion 3, that is dynamially built as objets (servies, but extensible to data items, les,

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3 Preliminaries

Greatest Common Prex Tree. Let an ordered alphabet

A

^be ^a ^nite ^set ^of ^letters.

Denote

≺

ân ôrder ôn

A

^. ^A ^non ^empty ^word

w

^over

A

îs â ^nite ^sequene ôf ^letters

a ₁ , . . . , a i , . . . , a l

^,

l > 0

^. ^The onatenation of two words

u

^and

v

^, ^denoted

u ◦ v

^or ^sim-

ply

uv

^,^is^equal^to^the^word

a ₁ , . . . , a _i , . . . , a _k , b ₁ , . . . , b _j , . . . , b _l

^suh^that

u = a ₁ , . . . , a _i , . . . , a _k

and

v = b ₁ , . . . , b _j , . . . , b _l

^. ^Let

ǫ

^be^the^empty ^word ^suh^that^for ^every^word

w

^,

wǫ = ǫw = w

^.

The length ofa word

w

^,^denoted^by

|w|

^,îs êqual^to ^the^number ôf^letters ôf

w

|ǫ| = 0

^.

A word

u

^is ^a ^prex (respetively, proper prex) of a word

v

îf ^there êxists â ^word

w

^suh ^that

v = uw

^(resp.,

v = uw

^and

u 6= v

^). ^The ^Greatest ^Common ^Prex ^(resp.,

Proper Greatest Common Prex) of a olletion ofwords

w ₁ , w ₂ , . . . , w i , . . .

⁽

i ≥ 2

^), ^denoted

GCP (w ₁ , w ₂ , . . . , w _i , . . .)

^(resp.

P GCP (w ₁ , w ₂ , . . . , w _i , . . .)

^), ^is ^the ^longest ^prex

u

^shared

by all of them (resp., suh that

∀i ≥ 1, u 6= w _i

^). Â ^[Propêr^℄ ^Greatest ^Common ^Prêx ^T^ree

([P℄GCPTree, alsoapartiular kindof trie)isa labeledrootedtreesuh thatboth following

properties aretruefor everynode ofthetree:

1. The node labelis aproperprex of anylabel initssubtree;

2. The node labelis theProperGreatestCommon Prexof all itssonlabels.

Inthefollowing we usetheword trieto designate our PGCP tree.

DistributedLexiographiPlaementTable. Thedistributed system onsideredinthis

paperonsistsofasetofasynhronous physial nodesorganizedina Distributed Hash Tables

(DHT).EahphysialnodemaintainsoneormorenodesofthelogialPGCPTree. Notethat

aDHTisused,butitan bereplaedbyanysystem,distributedor not,allowingtheretrieval

ofanynodefromanyothernode. We alsoonsiderthatthepotential existingfault-tolerane

mehanisms provided by this layer arenot usedwithin our arhiteture. We propose in this

papera fault-tolerane mehanismat thePGCP Treelevel.

Whenonewantstoinsertanobjetlabeled

o

înto^the^trie,â^messageîs^generated ôntain-

ing

o

^,âording^to^whih^the^messageîs^routed^within^the^trieûntil^reahing^the^node^labeled

v

^suh ^that

v

^is ^the ^smallest ^label ⁱⁿ^the ^trie ^that ^shares^with

o

^the^greatest ^ommon ^prex

ofanynode of thetrie with

o

^. ^More^formally^,^if

L

^denotes ^the^whole^set^of ^label^urrently ⁱⁿ

the trie, the set

U = {l ∈ L | GCP (l, o) = p}

^where

p = max _|m| {m = P GCP (l, o), l ∈ L)

^.

The label of the target node is

t = min _|w| {u ∈ U | u = pw}

^. ^One ^found, ^the ^target ^node

performs the insertion. If

t 6= o

^,^node(s) ^are^reated. ^If

o = tu

⁽

u 6= ǫ

^), ^a^new ^node ^labeled

o

is reated asa newson ofthe node labeled

t

^. ^If

t = ou

⁽

u 6= ǫ

^),â ^new^node îs^reated âs^the

fatherofthe nodelabeledby

t

^. ^Finally,îf^noneôf^theseônditions âre^satised,ît^means^that

o

^and

t

^must^be^siblings^but ^no^node ⁱⁿ^the^trie ^is ^labeled^by^their ^ommon ^prex. ^Thus ^two

nodes arereated, a node labeled

GCP (o, t)

^, ^father ^of ^the^node ^labeled ^by

t

^and ^also ^father

of the other newly reated node labeled by

o

^. ^The distributed routing algorithm (that also performsthereationandthemappingofnodes)requiresanumberofhopsboundedbytwie

thedepthofthe trie [5 ℄.

Physial nodesommuniate bymessage passing. Weassume two sendingfuntions. The

former,simplyreferred to SEND,is usedbyany physial node to senda message to another

node asynhronously, i.e., without waiting any aknowledgement. The latter, alled SYNC-

SEND, waits for an aknowledgement for eah message sent. We assume that eah physial

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4 Protool

In this setion, we give a detailed explanation of how the protool works. We divide the

algorithm ode intwo parts. The former showsthe rstphase developed withour tehnique

duringwhihauniquetrieisreoveredwithoutonsideringanylexiographiproperty. During

the seond phase, the trie is reorganized to eventually form a distributed greatest ommon

prex tree.

4.1 Trie Reovery

After a node

p

^detets ^the ^loss ^of ^its ^father ⁽

p.f ather

^), ^it ^searhes ^for ^a ^new ^father ^to ^link

on. Making a traversalof the DHT,Node

p

^ollets ⁱⁿ^Variable

P N

âll ^theâddressesôf êah

remaining physial node. Colleting the addresses in

P N

^,

p

^builds ^the ^set ^of ^logial ^nodes

^as^a ^temporary ^sonstored

in

p.tmpsons

^. ^Note ^that ^V^ariable

p.tmpsons

^is ^required ^to ^ompute

T

^using ^a ^PIF ⁱⁿ ^the

subtrieof

p

^. ^If

N \ T = ∅

^(i.e.,^there^is ^no^node ^for^whih

p

^may^link^on),^then

p

^is^onsidered

asthe root of the trie.

Theabovetehniquesuersofadrawbak: Severalnodeswithoutfathermaymakewhih

ould beome a bad hoie. In partiular, they an hoose as a temporary father a node

belongingtothesubtrieofanothernodebeinginthesamesituation. Bydoingthisinparallel,

yles mayappear. Ourstrategy isto detet andto break aposteriori suhyles asfollows.

After the hoie of its temporary father

tf

^,^a ^node

p

^sends ^a ^message ^HELLO ^with ^its

ID (

p.id

⁾ ^to

tf

^. ^In^the^next^step,

tf

^transmits^the ^message ^toîts ôwn^father, ând ^soôn. ^Step

bystep, one ofthe two following situationseventually arises:

1. The real root ofthe trie reeivesthe messageHELLO. Inthatase, therootnoties

p

^thatît îs^not învolved ⁱⁿâ^yle.

2. Themessageisreeivedbyafalse root,i.e.,anodehavingalso lostitsownfather. the

false root propagatesthemessage to itstemporary father.

Notethat, in the above latter ase, due to asynhrony of thenetwork, it ispossible thatthe

falseroot reeivesthe message HELLO sent by

p

^beforeîtêxeutedîts ôwn ^reovery ^phase.

Inthatase, the falserootisstill withoutatemporaryfather. ThemessageHELLO isthen

delayed until the false root hooses its owntemporary father.

Therefore,the messageHELLO sentby

p

^keepsîrulatingâmongîtsânestors,ârrying

the list of false roots' IDs whih were met during its traversal. Upon reeipt of a message

HELLO, iftherst itemofthelistarriedbythe messageisequal totheIDof thereeiver,

then a yle is deteted. In that ase, a leader eletion is omputed among the IDs of the

liste.g.,byhoosingthesmallest ID.Theleader beomestherootof thesubtrie, breaksits

linkwhihitsfather, andexeutesthe reovery phaseagain. (Theotherfalse rootsinvolved

in the yle remain onneted to the subtrie rooted by the leader.) Note that a yle may

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least one subtrie beomes thesubtrieof one false root. In other words, thenumber of yles

is periodially divided byat least

2

^. ^Therefore, ^the ^system êventually ôntains ône ^(rooted)

trie only.

4.2 Trie Reorganization

The trie reorganization is initiated one the trie reovery is done. Eah node

p

^having ^a

temporary son

q

^i.e.,

q

îsâ ^false^root ^withîts subtrieinitiatesa routing mehanismlosed to theoriginal key insertion [5℄. Letus onsiderthefollowing ases:

1. The value

p.val

îsâ^prex ôf^the^valueôf

q

^Figure^1,^Case

(i)

^. ^In^that^ase,

q

^(and ^its

subtrie) isplaed in thesubtrie of

p

^following ône ôf ^the ^four âses ^shown ⁱⁿ ^Figure^1,

Cases (

a

⁾ ^to⁽

d

^).

2. The value

p.val

îs ^not â ^prex ôf ^the ^value ôf

q

^. ^Then,

p

^moves

q

^to ^its ^father ^whih

nowhasthe responsibilityto plae

q

^.

p

q s s s

s i

^suh^that

q.val = pref ix(s ⁱ .val)

^.

p

q

newson

s s

s

1 k

i

(

c

⁾^There^exists

s ⁱ

^suh^that

P GCP (q.val, s i .val) > p.val

^.

p

s s s

q=s k+1 1 i k

(

d

⁾

p.val = pref ix(q.val)

^.

Figure 1: Afalse root

q

^is^linked ^to ^a ^node

p

^suh^that

p.val = pref ix(q.val)

^.

Notethatnewserviesmaykeepinsertingduringthetriereonstrution. So,anewsubtrie

mayhavebeen reatedatthe sameplaewherethefalserootinitiallywas. Thus,ourmethod

requirestotakeinaount thatanyfalserootbeingplaedinthetrieanmeet anodehaving

the same value. Inthat ase, the two tries mustbe merged. That isthe aim of themerging

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node

p

^exeutes ^Proedure

Gluing(q)

^, ^whih ^moves ^the ^sons ^of

q

^to

p

^before withdrawing

q

^from ^the ^trie ^(inluding ^the ^sons ^of

q

^'s ^father). ^Then, ^if ^neessary^,

p

^restarts ^reursively

merging andplaementsamong itssons,inorder to merge both subtries eventually.

4.3 Corretness Proof

In this subsetion, we disuss the orretness of our protool. In order to do this, we rst

need to make the realistiassumption thatundertheonsidered ontext, therash frequeny

islowenoughto make thetriefullybuiltsometime. (Intheoppositeway,thetrieouldnever

bebuiltand unusablemost ofthetime. Moregenerallyitisimpossibletosayanythingabout

termination otherwise.) In other words, we fairly assume that no rash ours after a rash

until the trie is fullybuilt, i.e., no two onseutive rashes interfere eah other, at one given

time.

Assuption 1 If a noderashes at time

t

^, ^then ^for ^every

t ^′ > t

^, ^no ^rash ^ours.

Lemma 2 Under Assumption 1, the reovery protool (Algorithm 1) terminates, and when

this ours,the system ontains onetrie only.

Proof. The validation mainly onsists inshowing that the protool terminates and that

the reorganizationof thetrie iseventually initiated (bysending amessage NOCYCLE).

Assume by ontradition that under Assumption 1, no node eventually sent a message

NOCYCLE.So,neitherLine

1.35

^nor ^Line

1.37

ⁱⁿÂlgorithm¹ îsêxeuted. ^Note^thatⁱⁿ^the

rst ase (Line

1.35

^), ^the ^node ^beomes ^the ^real ^node âfter ^the ^rash ôf îts ^father. ^So, ⁱⁿ

both ases, this means thatNOCYCLE never reahes thereal root of thetrie. The height

of thetrie being nite,thismeans thateveryMessage HELLO traversesyles only. When a

message HELLO is reeived by its initiator, the yleis broken bythe nodewhih is eleted

among the false roots partiipating in the yleLines

1.16

^to

1.21

^. ^Therefore, ^yles ^are

reated innitely often. Let

C

^be ^the ^number ôf ^reated ^yles. În ^the ^worst âse, â ^yle

ismade of at leasttwo nodes. So,

C

^is ^initially^bounded ^by

F/2

^,^where

F

^is ^the^number ^of

false root reated by the rash. When a yle is broken, at most one leader is eleted. So,

at most

C/2

^leaders âre âble ^to ^link ânother ^node âgain. În ^the ^next ^phase, ^the ^number ôf

yles is less than or equal to

C/2

^. ^Sine ûnder Âssumption ^1, ^yles ^may ^be ^reated ônly

whenfalse rootsarelinkedtoothernodes(exeuting Lines

1.10

^and

1.11

^),

C

^never ^gro^ws ^and

iseventually equalto

0

^. ^This ôntradits ^that^yles âre^reated înnitely ôften.

2

We nowonsiderthe phaseof trie reorganization showninAlgorithm2.

Lemma 3 Under Assumption 1 and assuming that the system ontains one trie only, the

reorganization protool (Algorithm 2) terminates, and whenthis ours, the trie is a

^and^its ^false ^son

f s

^:

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Algorithm 1 ReoveryProtoolfor eah node

F irst(list) = p.id

elseif

p.F ather 6=⊥

1

.

23

then send

<

^HELLO,

list>

^to

p.f ather

^;

1

.

24

elseif

p.tmpf ather 6=⊥

1

.

25

then

list := list + p.id

^;

1

.

26

send

<

^HELLO,

list>

^to

p.tmpf ather

1

.

27

elseif

.

46 tmpsons := tmpsons \ {q}

^;

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Algorithm 2 ReorganizationProtool foreah node

p

1

.

01

uponreeiptof

<

^MOVE,

f s>

^from

q

^do

1

.

02

if

f s.val = p.val

1

.

03

then //Isendto myselfthatafusionisneeded.

1

.

04

send

<

^MERGE,

f s>

^to

p

1

.

05

elseif

p.val = pref ix(f s.val)

1

.

06

then if

elseif

elseif

∃s ∈ p.sons | p.val < P GCP (s.val, f s.val)

1

.

14

then //

f s

^and

s

^have^a^PGCP^whih ^is^greater^than

.

20

else if

p.f ather 6=⊥

1

.

21

then send

<

^MOVE,

f s>

^to

p.f ather

1

.

22

else if

f s.val = pref ix(p.val)

1

.

23

then //Iaminthesubtrieof

t s [i].val = pref ix(t s [i + 1].val)

2

.

09

then send

<

^MOVE,

t s [i + 1]>

^to

t s [i]

^;

2

desribedinFigure1,

f s

îsêventually^plaedât^the^right^plaeⁱⁿ^the^subtrieôf

p

^refer

to Lines

1.06

^to

1.19

^. ^The ^resulting ^trie ^is^a

P GCP

^tree.

2. The valueof

p

îs^not â ^prex ôf

f s

q

^suh

that

q.val = P GCP (p, f s)

^Line

1.26

^. ^The^trie ^is^then^learly ^a

P GCP

^tree.

(b) Node

p

^has^a^father. ^Then,

f s

^is^moved^to^the^father^of

p

^Line

1.21

^. ^By^indution

^glued^together ^into

p

^:

1. There existsa pair of sons

s _i

^,

s _j

^of

p

^suh ^that

s _i .val

îs â ^prex ôf

s _j .val

^. ^Then,

s _j

^is

movedtoward

s i

^Lines

2.08

^to

2.11

^. ^Thisâseîs^similar^to ^theâbove^Case¹^(Cases⁽

a

⁾

or (

b

⁾ⁱⁿ^Figure ^1).

2. There existsapair of sons

s i

^,

s j

^of

p

^suh ^that

P GCP (s i , s j ) > p.val

^. ^Then,

s i

^and

s j

beome the two sons of a new son

q

^of

p

^suh ^that

q.val = P GCP (p, f s)

^Lines

2.12

to

2.16

^. ^Thisâse îsâlso ^similar^to ^theâbove ^Case ¹ ^(Case⁽

c

⁾ ⁱⁿ^Figure ^1).

3. There existsa pair of sons

s _i

^,

s _j

^of

p

^suh ^that

s _i .val = s _j .val

^. ^This^ase ^is ^solved ^by

initiatinga reursivemergingbetween

s _i

^and

s _j

^Lines

2.05

^to

2.07

^. ^This^ase^is^solved

byindutionon

s i

^and

s j

^.

4. There existsnopairofsons

s _i

^,

s _j

^of

p

^satisfying êither^Case ^1,^2,ôr ^3. În^thatâse,^the

subtrie of

p

^learly ^satises^the^properties^of ^a

P GCP

^tree.

2

From Lemmas2 and3 follows:

Theorem 1 UnderAssumption1, Algorithm1 andAlgorithm2 provide a

P GCP

^tree ^r^eon-

strutionafter the rash of a physial node.

5 Conlusion and Future Work

Inthis paper, wehave presenteda fault-tolerant protool inase ofnode rashesinaProper

CommonGreatestPrextree searhsystem. Thisprotool anbeoupled witharepliation

strategyto lowertheostsrelatedto highrepliationfators. Thisprotoolallows thereon-

netionandrepairofsubtriesaftertherashofoneormorenodes. Thisalgorithmguarantees

toreovera onsistent PGCP tree aftera nitetimeand thus to avoid partiallyrepliation.

Ourfutureworkwill onsistinonnetingthetwo mehanisms (repliationandrepair)in

orderto minimizethe ostoffault-tolerane on dynamiplatforms. Wewill alsodevelopand

validateexperimentallythemehanismsexposedinthispaperontheGrid'5000platformofthe

(13)

oftherepair algorithmandtoseeitsapaitytoanswerlients'requestsfaingdierentlevels

of dynamiity. Moreover, we will be able to see starting from whih level of dynamiity the

repair mehanism is no more eient alone, and then how we an progressively injet some

repliation asthe dynamiitylevel inreases.

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Laboratoire de l’Informatique du Parallélisme