A Repair Mechanism for Fault-Tolerance for Tree-Structured Peer-to-Peer Systems

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A Repair Mechanism for Fault-Tolerance for

Tree-Structured Peer-to-Peer Systems

Eddy Caron, Frédéric Desprez, Franck Petit, Cédric Tedeschi, Charles

Fourdrignier

To cite this version:

Eddy Caron, Frédéric Desprez, Franck Petit, Cédric Tedeschi, Charles Fourdrignier. A Repair

Mecha-nism for Fault-Tolerance for Tree-Structured Peer-to-Peer Systems. [Research Report] RR-6029, LIP

RR-2006-34, INRIA, LIP. 2006, pp.15. �inria-00115997v3�

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Thème NUM

A Repair Mechanism for Fault-Tolerance for

Tree-Structured Peer-to-Peer Systems

Eddy Caron — Frédéric Desprez — Charles Fourdrignier — Franck Petit — Cédric Tedeschi

N° 6029

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EddyCaron , Frédéri Desprez , Charles Fourdrignier, Fran k Petit, Cédri

Tedes hi

ThèmeNUM Systèmes numériques ProjetGRAAL

Rapportde re her he n°6029 O t 2006 16 pages

Abstra t:

Fa ing the limits of traditional tools of resour e management within

omputa-tional grids (related to s ale, dynami ity, et . of the platforms newly onsidered), newapproa hes, basedon peer-to-peerte hnologiesareemerging. The resour e dis- overyandinparti ular theservi edis overyis on ernedbythisevolution. Among

thesolutions, a promising one is theindexing of resour es usingtrie stru turesand more parti ularly prex trees. The major advantages of trie-stru tured approa hes isthe apabilitytosupportsear hqueriesonrangesofvalueswithalaten ygrowing

logarithmi ally in the number of nodes in the trie. Those te hniques are easy to extend to multi riteria sear hes. One drawba k of using tries is its inherent poor robustnessinadynami environment,where nodesjoinand leavethenetwork,

lead-ing to the split of the tree into a forest, whi h results inthe impossibility to route requests. Within most re ent approa hes, thefault-toleran e isa prevention me h-anism, often repli ation-based. The repli ation an be ostly in term of resour es

required. In this paper, we propose a fault-toleran e proto ol that re onne ts sub-trees a posteriori, after rashes, to have again a onne ted graph and then reorder thenodes torebuild a onsistent tree.

Key-words: Faulttoleran e, peer-to-peer, prex trees

Thistextisalsoavailableasaresear hreportoftheLaboratoiredel'Informatiquedu Paral-lélismehttp://www.ens-lyon.fr/LIP.

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pair-à-pair de dé ouverte de servi es

Résumé :

Fa eauxlimitesdesoutilstraditionnelsdegestionderessour esdanslesgrillesde al ul(mauvais passageà l'é helle,nonprise en omptede ladynami itéduréseau, et .),desalternativesfondéessurleste hnologiespair-à-pairsontentraind'émerger.

La dé ouverte de ressour es eten parti ulier des servi es de al ulest tou hée par ette évolution. Parmi es solutions, il existe des appro hes prometteuses fondées sur des arbres lexi ographiques. L'intérêt de telles appro hes repose sur la

possi-bilité d'ee tuer des requêtes sur des intervalles de valeurs ainsi que la possibilité de réaliserde l'auto omplétion surles haînes de re her he en temps logarithmique en la taille de l'arbre. Ces te hniques s'étendent fa ilement à des re her hes

mul-ti ritères. Cependant la stru ture en arbre est fragile et peut é later en une forêt si l'un des n÷uds vient à quitter le réseau, rendant ainsi impossible le routage de ertaines requêtes et n'orant au lient qu'une vue partielle des servi es. Dans la

plupart de es appro hes, la toléran e aux pannes, indispensable dans les environ-nementsdynamiquesàlarge é helle, estpréventive(réaliséea priori)etsefondesur larépli ation,qui est oûteuseen termesderessour eset detemps. Dans epapier,

nous présentons un proto ole tolérant aux pannes de n÷uds, omplémentaire à la répli ation, dans les arbres lexi ographiques. Il se fonde sur la re onnexion et la réparation a posteriori d'arbresqui ont subi laperte d'unou plusieurs n÷uds.

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1 Introdu tion

Theselast few years have seen thedevelopment of large s ale grids onne ting

dis-tributed resour es ( omputation resour es, storagefa ilities, omputation libraries, et .) in a seamless way. This is now an e ient alternative to super omputers to solve large problems su h ashigh energy physi s, simulation, bioinformati , et .

However, existing middlewares used ingrids require most of the time a stable and entralized infrastru ture. They usually loose their performan e on dynami and large s ale platforms without entralized management of resour es. To ope with

the hara teristi s ofthese emerging kindof platforms,ithasbeensuggested touse peer-to-peerte hnologies within omputationalgrids [8℄.

Peer-to-peer te hnologies oer algorithms allowing the sear h and retrieval of obje ts over the net (data items, les, servi es, et .). Among these te hnologies, DistributedHashTables(DHT)wereinitiallydesignedforverylarges aleplatforms,

for example to share les over the Internet. However, DHTs have several major drawba ks. Amongthem,theirdis overyme hanismusuallyworksonexa tsear hes of a given key. Some work has then been done to allow omplex requests to be submitted over DHTs or more generally in stru tured peer-to-peer systems, i.e.

systems based on request routing. Some of these works are based on tries (also alledprex trees). A trie stru ture supportsrange queries inalogarithmi time in thenumberof nodesof the trie.

Fault-toleran e isa mandatoryfeature forpeer-to-peersystems toavoidtheloss

ofdatastored onnodesandtoallowa orre trouting ofmessages. The rashofone orseveralnodesinatrie leadsto thelossofobje tsreferen es storedinthetrie and tothesplitofthetrieintoseveralsubtries,also alledaforest. Fault-toleran ewithin

stru tured peer-to-peer systems usually uses repli ation. Using su h an approa h, ea hnodeandea hlinkofthetriewouldhavetobedupli ated

k

times,

k

beingthe repli ationfa tor. Keepingsu h stru ture upis ostly, mainly intermsof resour es

used. Afterward, thepurposeistondfor thevalueof

k

theright trade-obetween the repli ation ost and the robustness of the system. In this paper, we study an alternativeto therepli ationapproa hbasedonthere onne tionofthesubtriesand thea posteriori reorderingofa onsistenttrie. Whenthetrieisdis onne ted,arst

solution onsistsinrebuilding atrie addingnodesofremaining subtries one byone. Thisnaivemethod anleadtoaprohibitive ostwhenthenumberofremainingnodes islarge(whi hisusually the aseinpeer-to-peersystems). Forexample,loosingone

node an leadto a omplete re onstru tion of thetrie. A se ond approa h onsists inre onne tingthesubtries togetthe originaltrie ba kat aminimum ost. Thisis

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this kind of algorithm we des ribe inthis paper in a distributed and asyn hronous

environment. It an also be usedto omplete therepli ationpro ess.

A brief history of peer-to-peer te hnologies is provided in Se tion 2, followed by the formal des riptionof the parti ular trie stru ture we use (Se tion 3) and of the distributed system we pla e ourselves. We fo us our study on fault-toleran e

me anisms relatedto them. Then, inSe tion 4 we present the repair algorithm we designed and give its proofbeforea on lusionand futurework Se tion.

2 Related Work

With the spread of the peer-to-peer te hnologies going along with the le sharing overtheInternet,purelyde entralizedsear hsystemshaveemerged. Su htoolsrst tooktheshapeof unstru tured me hanisms,i.e.,basedon theoodingofsear h

re-quests[10,9℄. Theseme hanismsresultedinoverloadingthenetworkwhileproviding non-exhaustiveresponses. Addressingboththes alabilityandtheexhaustiveness is-sueswithin peer-to-peersystems,the distributedhashtables[13 , 14,18 ,20℄, a.k.a.,

thestru tured peer-to-peer group, arehighly s alable inthe sense that thenumber of logi al hops required to route and thelo al state grows logarithmi ally withthe number of nodes parti ipating in the system. Moreover, DHTs prevent from

loos-ing routing paths and obje ts' referen es by use of repli ation and periodi s ans. Unfortunately, DHTs present several major drawba ks (homogeneous apa ity as-sumptions, topology awareness, et .). Among them, the rigidity of the requesting

me hanism, i.e.,exa tmat honagivenkeyhindersitsuseoverrealsear hsystems.

A series of work gives the opportunity to allow exible meanings of retrieval overstru tured peer-to-peer networks. First a hievement in this way has been the

abilityto des riberesour es withsemi-stru tured language,su hXML,asdes ribed in[3℄.[19℄enhan esDHTswithtraditionaldatabaseoperations. Severalapproa hes, based on spa e lling urves, su h as Squid [15 ℄ or [17 ℄ support multi-dimensional

rangequeries.[1℄mapsone-dimensionaldataspa etod-dimensional Cartesianspa e byusing the inverse Hilbertmapping. Builton topof multiple DHTs,SWORD [11 ℄ is an information servi eaiming at dis overing omputing resour es on the grid by

answeringmulti-attribute rangequeries.

Wefo usinthisworkontrie-stru turedretrievalsolutions,alsosupportingrange queries but outperforming previous approa hes in the sense that logarithmi (or onstant ifwe assumean upperbound on thedepth of thetrie) laten yis a hieved

byparallelizingtheresolution ofthequeryintheseveralbran hesofthetrie. Prex Hash Tree (PHT) [12 ℄ builds a trie of the entire key-spa e on top of a DHT. The

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purpose of this ar hite ture is to use the trie as a logi al layer allowing omplex

sear hes on top of any DHT-like network. The ar hite ture of PHT results in the multipli ationof the omplexities ofthetrie and oftheunderlying DHT.

The Skip Graphs stru ture proposed in [2 ℄ is similar to a trie but is built with theskiplistste hnology,allowingtheuseoftheir inherentfault-toleran e properties. But again, the omplexity of the number of messages generated to pro ess range

queries isin

O(m log(n))

,

m

being the numberof nodespertainedby therangeand

n

thetotal numberof nodesinthegraph.

Other approa hes propose to rely on a trie for ea h purpose, i.e., indexing the key-spa e, mapping the nodesof the trie on thenetwork, and routing therequests.

Among them, Nodewiz [4℄ assumes a set of stati reliable nodes to host the trie, whi h is unfortunately hard to ensure on peer-to-peer platforms. P-Grid [7℄ builds a trie on the whole key-spa e (i.e., the whole set of potential keys). Ea h leaf of

thistrie orrespondsto asubset ofthekey-spa e. Thefault-toleran e isa hieved by probabilisti repli ation.

As a more general onsideration, none of these approa hes address the topol-ogy/physi al lo alityawarenessissue,i.e.,noinformation about theunderlying

net-work is taken into a ount to build the logi al (overlay) network, what an raise a signi antperforman eproblem,physi allo alitybeingbrokenwhenthelogi al net-work isbuilt. Moreover, the several fault-toleran e solutions aremostly

repli ation-based, or DHT-based,also involvingheavy repli ationme hanisms.

Initiallydesignedforthepurposeofservi edis overyoverdynami omputational gridsandattemptingtosolvetheabovedrawba ksofexistingapproa hes,were ently developed a novel ar hite ture, based on a logi al Greatest Common Prex Tree

formally des ribed in Se tion 3, that is dynami ally built as obje ts (servi es, but extensibleto dataitems, les,et .) are de lared.

3 Preliminaries

1 , . . . , a

i

, . . . , a

l

,

l > 0

. The on atenation of two words

u

and

v

,denoted

u ◦ v

or simply

uv

,isequal to the word

a

1 , . . . , a

i

, . . . , a

k

, b

1 , . . . , b

j

, . . . , b

l

su hthat

u = a

₁

, . . . , a

i

, . . . , a

k

and

v = b

1 , . . . , b

j

, . . . , b

l

. Let

ǫ

be the empty word su h that foreveryword

w

,

wǫ = ǫw = w

. The length ofaword

w

,denotedby

|w|

,isequal to thenumberof lettersof

w

|ǫ| = 0

.

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Aword

u

isaprex (respe tively,properprex)ofaword

v

ifthereexistsaword

w

su hthat

v = uw

(resp.,

v = uw

and

u 6= v

). TheGreatest CommonPrex (resp., Proper Greatest Common Prex) ofa olle tion 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 byall ofthem(resp., su h that

∀i ≥ 1, u 6= w

i

). A[Proper℄Greatest CommonPrexTree ([P℄GCPTree,alsoaparti ularkindoftrie)isalabeledrooted treesu hthatboth following properties aretruefor everynode ofthetree:

1. Thenodelabelisa properprexof any labelinits subtree;

2. ThenodelabelistheProperGreatestCommon Prex ofall itsson labels.

In the following we usethe word trieto designateour PGCP tree.

Distributed Lexi ographi Pla ement Table. Thedistributed system

onsid-ered in this paper onsists of a set of asyn hronous physi al nodes organized in a Distributed Hash Tables(DHT).Ea hphysi alnodemaintains oneormore nodesof thelogi alPGCP Tree. NotethataDHTisused,but it an berepla edbyany

where

p = max

|m|

{m = P GCP (l, o), l ∈ L)

. The label of the target node is

t =

GCP (o, t)

,fatherofthe node labeledby

t

andalso fatheroftheothernewly reated node labeledby

o

. The distributed routing algorithm (that also performs the reation and the mapping of nodes) requiresa numberofhopsbounded bytwi ethedepth ofthetrie [5℄.

Physi al nodes ommuni ate bymessage passing. We assumetwo sending fun -tions. Theformer, simplyreferredto SEND,isusedbyanyphysi al nodeto senda

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messagetoanothernodeasyn hronously,i.e.,withoutwaitinganya knowledgement.

Thelatter, alledSYNC-SEND,waitsforana knowledgementforea hmessagesent. We assume that ea h physi al node may rash. So, when a physi al node rashes, one or morelogi al nodes arelost.

4 Proto ol

Inthisse tion, we give adetailedexplanation ofhowtheproto olworks. Wedivide the algorithm ode in two parts. The former shows the rst phase developed with our te hnique during whi h a unique trie is re overed without onsidering any

(Propagation of Information with Feedba k) Proto ol [6 , 16 ℄,

p

omputes

T

The above te hnique suers of a drawba k: Several nodes without father may make whi h ould be ome a bad hoi e. In parti ular, they an hoose asa tem-porary father a node belonging to the subtrie of another node being in the same

situation. By doing this in parallel, y les may appear. Our strategy is to dete t and tobreak a posteriori su h y les asfollows.

1. Thereal rootofthetriere eivesthemessageHELLO.Inthat ase, theroot

noties

p

thatitisnot involved ina y le.

2. Themessage is re eived by afalse root,i.e., a node having alsolost its own father. the falseroot propagatesthemessage to its temporary father.

Notethat, inthe above latter ase, dueto asyn hrony of thenetwork,it ispossible

that the false root re eives the message HELLO sent by

p

before it exe uted its own re overy phase. Inthat ase, thefalse root isstill without a temporary father. ThemessageHELLO isthendelayeduntilthefalseroot hoosesitsowntemporary

father.

Therefore,themessageHELLO sentby

p

keeps ir ulatingamongitsan estors, arryingthelistoffalseroots'IDswhi hweremetduringitstraversal. Uponre eipt

of a message HELLO, if the rst item of the list arried by the message is equal to theID ofthe re eiver, thena y le isdete ted. In that ase, a leader ele tion is omputed among the IDs of theliste.g., by hoosing the smallest ID. The leader be omes the root of the subtrie, breaks its link whi h its father, and exe utes the

re overy phase again. (The other false roots involved in the y le remain on-ne tedtothe subtrierootedbytheleader.) Notethata y lemaybe reated again. However, inthe worst ase, at ea h relaun hing of there overy phase, at least one

subtrie be omes thesubtrie of one false root. Inother words, thenumber of y les is periodi ally divided byat least

2

. Therefore, the systemeventually ontains one (rooted) trieonly.

4.2 Trie Reorganization

The trie reorganization is initiated on e the trie re overy is done. Ea h node

p

having a temporary son

q

i.e.,

q

isa falseroot withits subtrieinitiatesa routing me hanism losed to the original key insertion [5℄. Let us onsider the following ases:

1. Thevalue

p.val

is a prex ofthe value of

q

Figure1, Case

(i)

. In that ase,

q

(and its subtrie) ispla ed inthe subtrieof

p

following one of thefour ases showninFigure1,Cases (

a

) to(

d

).

2. Thevalue

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p

q

s

1 i

k

(

i

)

p.val

= pref ix(q)

and

p.val

= P GCP (s

1 , . . . , s

k

)

.

p

q

s

1 i

k

(

a

)Thereexists

s

i

su hthat

s

i

.val

= pref ix(q.val)

.

p

q

s

1 i

k

(

b

)Thereexists

s

i

su hthat

q.val

= pref ix(s

i

.val)

.

p

q

newson

s

1 k

i

(

c

)Thereexists

s

i

su hthat

P GCP

(q.val, s

i

.val) > p.val

.

p

s

q=s

1 i

k

k+1

(

d

)

p.val

= pref ix(q.val)

.

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pla ed in the trie an meet a node having the same value. In that ase, the two

tries must be merged. That is the aim of the merging proto ol, initiated by the sending of a message MERGE. Upon re eipt of this message, a node

p

exe utes Pro edure

Gluing(q)

, whi h moves the sons of

q

to

p

before withdrawing

q

from thetrie (in luding the sons of

q

's father). Then, if ne essary,

p

restarts re ursively mergingand pla ementsamong itssons,inorderto merge both subtries eventually.

4.3 Corre tness Proof

Inthissubse tion,wedis ussthe orre tnessofour proto ol. Inorderto dothis,we rst need to make the realisti assumption that under the onsidered ontext, the rashfrequen yislowenough tomakethetriefullybuiltsometime. (Intheopposite

way, the trie ould never be built and unusable most of the time. More generally it is impossible to say anything about termination otherwise.) In other words, we

fairly assume that no rash o urs after a rash until the trie is fullybuilt, i.e., no two onse utive rashes interfereea h other, atone given time.

Assuption 1 If a node rashes at time

t

,then for every

t

′

_{> t}

, no rash o urs.

Lemma 2 UnderAssumption1,there overyproto ol(Algorithm1)terminates,and whenthis o urs, the system ontainsone trieonly.

Proof. The validation mainly onsists inshowing that the proto ol terminates andthat the reorganizationof thetrie iseventually initiated (by sendinga message

NOCYCLE).

Assume by ontradi tion that under Assumption 1, no node eventually sent a

messageNOCYCLE.So,neitherLine

1.35

norLine

1.37

inAlgorithm1isexe uted. Note that in the rst ase (Line

1.35

), the node be omes the real node after the rash ofits father. So,inboth ases,this means thatNOCYCLEneverrea hesthe

real root of the trie. The height of the trie being nite, this means that every Message HELLO traverses y les only. When a message HELLO is re eived by its initiator, the y leis broken by the node whi h is ele ted among thefalse roots parti ipatinginthe y leLines

1.16

to

1.21

. Therefore, y lesare reatedinnitely often. Let

C

be the number of reated y les. In theworst ase, a y le ismade of atleasttwonodes. So,

C

isinitiallyboundedby

F/2

,where

F

isthenumberoffalse root reated by the rash. When a y le is broken, at most one leader is ele ted.

So,at most

C/2

leadersare ableto linkanother node again. Inthenext phase,the number of y les is less than or equal to

C/2

. Sin e under Assumption 1, y les

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Algorithm 1 Re overy Proto ol forea h 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

to

p.tmpf ather

;

1

.

42 p.tmpf ather :=⊥

;

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Algorithm 2 ReorganizationProto olfor ea hnode

p

1

.

01

uponre eiptof

<

MOVE,

f s>

from

q

do

1

.

02

if

f s.val = p.val

1

.

03

then //Isendtomyselfthatafusionisneeded.

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

haveaPGCPwhi hisgreaterthan

.

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 //Iamin thesubtrieof

[i].val = t

_s

[i + 1].val

2

.

06

then send

<

MERGE,

t

_s

[i + 1].val)

2

.

09

then send

<

MOVE,

t

_s

[i + 1]>

to

t

_s

[i]

;

2

.

10 p.sons := p.sons \ {t

_s

[i + 1]}

;

2

.

11 i := i + 1

2

.

12

elseif

p.val < P GCP (t

s

[i].val, t

s

[i + 1].val)

2

.

13

then

p.sons := p.sons ∪ N ewnode(P GCP (t

s

[i].val, t

s

[i + 1].val),

2

.

14 t

s

17

endif

2

18

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maybe reatedonlywhenfalserootsarelinkedto othernodes(exe uting Lines

1.10

and

1.11

),

C

never grows and iseventually equal to

0

. This ontradi ts that y les

are reated innitelyoften.

2

Wenow onsider the phaseoftrie reorganization showninAlgorithm2.

is aprex of

f s

's valueLine

1.05

. In that ase, following the four ases des ribed in Figure 1,

f s

is eventually pla ed at theright pla e in the subtrie of

p

refer to Lines

1.06

to

1.19

. The resulting trie is a

P GCP

1.26

. The trie is then learly a

P GCP

tree.

p

and

f s

merge,thentherearefour asesto onsiderafter

p

and

f s

gluedtogether into

p

:

1. Thereexistsapairofsons

s

i

,

s

j

of

p

su hthat

s

i

.val

isaprexof

s

j

.val

. Then,

s

j

is moved toward

s

i

Lines

2.08

to

2.11

. This ase is similar to the above Case 1(Cases (

a

) or (

b

) inFigure1).

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2. There exists a pair of sons

s

i

,

s

j

of

p

su h that

P GCP (s

i

, s

j

) > p.val

. Then,

s

i

and

s

j

be ome the two sons of a new son

q

of

p

su h that

q.val =

P GCP (p, f s)

Lines

2.12

to

2.16

. This aseisalsosimilartotheaboveCase1 (Case(

c

) inFigure1).

3. There existsa pair of sons

s

i

,

s

j

of

p

su h that

s

i

.val = s

j

.val

. This ase is solvedbyinitiatingare ursivemergingbetween

j

of

p

satisfyingeither Case1,2,or3. Inthat ase, the subtrieof

p

learly satisesthe properties ofa

P GCP

tree.

2

From Lemmas2 and 3follows:

Theorem 1 Under Assumption 1, Algorithm 1 and Algorithm 2 provide a

P GCP

tree re onstru tion after the rash of a physi al node.

5 Con lusion and Future Work

Inthis paper,we have presented afault-tolerant proto ol in aseof node rashes in aProperCommon GreatestPrex treesear h system. Thisproto ol anbe oupled

witharepli ationstrategytolowerthe ostsrelatedto highrepli ationfa tors. This proto olallowsthere onne tionandrepairofsubtriesafterthe rashofoneormore nodes. This algorithm guarantees to re over a onsistent PGCP tree after a nite

timeandthus to avoidpartially repli ation.

Our futurework will onsist in onne tingthe two me hanisms (repli ation and repair) in order to minimize the ost of fault-toleran e on dynami platforms. We willalso develop andvalidateexperimentally theme hanisms exposedinthis paper on the Grid'5000 platform of the fren h ministry of resear h. The aim of su h

experimentation willbetoseetheperforman eoftherepairalgorithm andtoseeits apa ityto answer lients' requests fa ing dierent levels of dynami ity. Moreover, we willbeable to seestartingfrom whi h levelof dynami itytherepair me hanism

isnomoree ient alone, andthen howwe anprogressivelyinje t somerepli ation asthedynami itylevelin reases.

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