Ch . 9 Agreem en t Protocols
9.1 In trodu ction
In d istribu ted system s, w h ere sites (or p rocessors) often com p ete as
w ell as coop erate to achieve a com m on goal, it is often requ ired th at
sites reach m u tu al agreem ent .
- Ex) In d istribu ted d atabase system s, d ata m an agers at sites m u st
agree on w h eth er to com m it or to abort a tran saction .
Th e form al settin g for a distribu ted agreem ent p rotocol is th e
followin g : Th ere are M p rocessors P=p 1 ,...,p M th at are tryin g to reach
agreem ent . A su bset F of th e p rocessors are fau lty, an d rem ainin g
p rocessors are n onfau lty . Each p rocessor p i P stores a valu e Vi .
Du rin g th e agreem ent p rotocol, th e p rocessors calcu late an agreem ent
v alu e A i . After th e p rotocol en d s, th e followin g tw o con d ition s sh ou ld
h old :
① For every p air p i an d p j of n on fau lty p rocessors, A i = A j . This
v alu e is th e agreem en t valu e.
② Th e agreem ent valu e is a fu nction of th e initial valu es {Vi } of th e
n onfau lty p rocessors (P - F).
9.2 Th e System Model
Agreem ent p roblem s h ave been stu d ied u n d er th e followin g system
m od el:
Th ere are n p rocessors in th e system an d at m ost m of th e p rocessors
can be fau lty .
Th e p rocessors can d irectly com m u nicate w ith oth er p rocessors by
m essage p assin g .
A receiver p rocessor alw ays kn ow s th e id entity of th e sen d er p rocessor
of th e m essage.
Th e com m u nication m ed iu m is reliable (i.e., it d elivers all m essages
w ith ou t introd u cin g an y errors) an d only p rocessors are p ron e to
failu re.
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9.2.1 Synchron ou s vs. Asynchron ou s Com p u tation s
Syn chron ou s com p u tation
A p rocess receives m essages
(1 rou n d ), p erform s a com p u tation
(2 rou n d ), an d sen d m essages to oth er p rocesses (3 rou n d).
Asynch ron ou s com p u tation
Th e com p u tation at p rocesses d oes n ot p roceed in lock step s. A
p rocess can sen d an d receive m essages an d p erform com p u tation at
any tim e.
9.2.2 Mod el of Processor Failu res
A p rocessor can fail in three m od es:
Crash fau lt : a p rocessor stop s fu nctionin g an d n ever resu m es
op eration .
Om ission fau lt : a p rocessor "om its" to sen d m essages to som e
p rocessors.
Maliciou s fau lt (Byzantin e fau lts): a p rocessor beh aves ran d om ly an d
arbitrarily .
9.2.3 Au th enticated v s. N on-Au th enticated Messages
Au th enticated m essage system
A (fau lty) p rocessor cann ot forge a m essage or ch an ge th e contents of
a received m essage.
A p rocessor can verify th e au th enticity of a received m essage.
An au th enticated m essage is also called a sign ed m essage.
N on-au th en ticated m essage system
A (fau lty) p rocessor can forge a m essage an d claim to h ave received it
from an oth er p rocessor or ch an ge th e contents of a received m essage
before it relays th e m essage to oth er p rocessors.
A p rocessor h ave n o w ay of verifyin g th e au th enticity of a received
m essage.
A n on-au th enticated m essage is also called an oral m essage.
9.2.4 Perform ance Asp ects
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Perform ance of agreem ent p rotocols
tim e: th e nu m ber of rou n d s
m essage traffic: th e nu m ber of m essages exch an ged to reach an
agreem ent
storage overh ead : th e am ou nt of inform ation th at n eed s to be stored at
p rocessors d u rin g th e execu tion of a p rotocol.
9.3 A Classification of A greem en t Protocols
Three w ell kn own p roblem s
① Byzantin e agreem ent p roblem
② Con sen su s p roblem
Problem
Wh o initiates
th e valu e
Fin al agreem ent
Byzantin e
Con sen su s
On e p rocessor
All p rocessors
Sin gle valu e
Sin gle valu e
Interactiv e
Con sistency
All p rocessors
A v ector of
v alu es
③ Interactive con sistency p roblem
Byzantin e agreem ent p rotocol
A sin gle valu e, w hich is to be agreed on, is in itialized by an arbitrary
p rocessor an d all n onfau lty p rocessors h ave to agree on th at valu e.
Con sen su s p roblem
Every p rocessor h as its ow n initial valu e an d all n onfau lty p rocessor
m u st agree on a sin gle com m on valu e.
Interactive con sistency p roblem
Every p rocessor h as its ow n initial valu e an d all n onfau lty p rocessors
m u st agree on a set of com m on valu es.
Th e th ree agreem ent p roblem s
9.3.1 Th e Byzantin e Agreem ent Problem
Three gen erals can n ot reach Byzan tin e agreem ent
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An arbitrarily ch osen p rocessor, called th e source processor, broad casts
its initial v alu e to all oth er p rocessors.
A solu tion to th e Byzantin e agreem ent p roblem sh ou ld m eet th e
followin g tw o objectiv es:
Agreem ent : All n onfau lty p rocessors agree on th e sam e valu e.
Valid ity : If th e sou rce p rocessor is n onfau lty, th en th e com m on agreed
u p on valu e by all n on fau lty p rocessors sh ou ld be initial valu e of th e
sou rce.
Tw o p oints sh ou ld be n oted :
① If th e sou rce p rocessor is fau lty, th en all n onfau lty p rocessors can
agree on any com m on valu e.
② It is irrelev an t wh at valu e fau lty p rocessors agree on or wh eth er th ey
agree on a valu e at all.
9.3.2 Th e Con sen su s Problem
Every p rocessor broad casts its initial v alu e to all oth er p rocessors.
Initial valu es of th e p rocessors m ay be d ifferent .
A p rotocol for reachin g con sen su s sh ou ld m eet th e follow in g
con d ition s:
Agreem ent : All n onfau lty p rocessors agree on th e sam e sin gle valu e.
Valid ity : If th e intial valu e of every n onfau lty p rocessor is v, th en
agreed u p on com m on valu e by all n on fau lty p rocessor m u st be v .
N ote th at if th e initial v alu es of n onfau lty p rocessors are d ifferent,
th en all n onfau lty p rocessors can agree on any com m on valu e.
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9.3.3 Th e Interactive Con sistency Problem
Every p rocessor broad casts its initial v alu e to all oth er p rocessors. Th e
initial valu es of th e p rocessors m ay be different .
A p rotocol for th e interactive con sistency p roblem sh ou ld m eet th e
followin g con d ition s:
Agreem ent : All n onfau lty p rocessors agree on th e sam e vector, (v 1 , v 2 ,
..., v n ).
Valid ity : If th e ith p rocessor is n onfau lty an d its initial valu e is v i ,
th en th e ith v alu e to be agreed on by all n onfau lty p rocessors m u st
be v i .
N ote th at if th e jth p rocessor is fau lty, th en all n onfau lty p rocessors
can agree on any com m on valu e for vj .
9.3.4 Relation s am on g th e Agreem en t Problem s
Th e Byzantin e agreem en t p roblem is p rim itive to th e oth er tw o
agreem ent p roblem s.
9.4 Solu tion s to th e Byzan tin e Agreem en t Problem
Th e Byzantin e agreem en t p roblem is also referred to as th e Byzan tin e
generals p roblem .
9.4.1 An Im p ossibility Resu lt
We n ow sh ow th at a Byzantin e agreem ent can n ot be reach ed am on g
three p rocessors, w h ere on e p rocessor is fau lty .
Con sid er a system w ith three p rocessors, p 0 , p 1 , an d p 2 . For sim p licity,
w e assu m e th at th ere are only tw o valu es, 0 an d 1, on w hich
p rocessors agree an d p rocessor p 0 initiates th e initial valu e.
Case Ⅰ: p 0 is n ot fau lty .
Since p 0 is n onfau lty, p rocessor p 1 m u st accep t 1 as th e agreed u p on
v alu e if con dition 2 is to be satisfied .
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Case Ⅱ: p 0 is fau lty
p 0 w ill agree on a valu e of 1 an d p 2 w ill agree on a valu e of 0,
w hich w ill violate con d ition 1 of th e solu tion .
9.4.2 Lam p ort-Sh ostak-Pease Algorithm
Lam p ort et al.' s algorithm , referred to as th e Oral Message algorith m
OM (m ), m >0, solves th e Byzantin e agreem en t p roblem for 3m +1 or
m ore p rocessors in th e p resence of at m ost m fau lty p rocessors. Let n
d en ote th e total nu m ber of p rocessors (clearly, n≥3m +1).
Algorithm OM (0)
1. Th e sou rce p rocessor sen d s its valu e to ev ery p rocessor .
2. Each p rocessor u ses th e valu e it receives from th e sou rce. (If it
receives n o valu e, th en it u ses a d efau lt valu e of 0)
Algorithm OM (m ), m >0.
1. Th e sou rce p rocessor sen d s its valu e to ev ery p rocessor .
2. For each i, let v i be th e valu e p rocessor i receives from th e sou rce.
(If it receives n o v alu e, th en it u ses a d efau lt valu e of 0.). Processor
i acts as th e n ew sou rce an d initiates Algorith m OM (m -1) w h erein it
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sen d s th e valu e v i to each of th e n-2 oth er p rocessors.
3. For each i an d each j ( i), let vj be th e valu e p rocessor i received
from p rocessor j in Step 2. u sin g Algorithm OM (m -1). Processor i
u ses th e valu e m ajority (v 1, v 2 , ..., v n - 1 ).
Th e m essage com p lexity of th e algorithm is O (nm ).
Exam p le 1
An execu tion of BG(2) on seven gen erals.
O i rep resents th e com m an d sent to Li , an d Li : O i is Li ' s rebroadcast
of its com m an d . Lj : Li : O i is Lj ' s rebroadcast of wh at Li said h is
ord er w as.
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9.4.3 Dolev et al.' s Algorithm
Th e algorithm requ ires u p to 2m +3 rou n d s to reach an agreem ent .
Data Stru ctu re
Th e algorithm u ses tw o thresh old s: LOW an d H IGH, wh ere
LOW :=m +1 an d HIGH :=2m +1.
Th e basic id ea is th at any su bset of p rocessors of size LOW w ill h av e
at least on e n onfau lty p rocessor .
Any su bset of p rocessors of size HIGH inclu d es a m ajority of
p rocessors, th at is, m +1, th at are n on fau lty .
Th e algorithm u ses tw o typ es of m essages: a "*" m essage an d a
m essage con sistin g of th e n am e of a p rocessor .
- Th e "*" d en otes th e fact th at th e sen d er of th e m essage is sen d in g a
v alu e of 1 an d th e n am e in a m essage d en otes th e fact th at th e
sen d er of th e m essage received a "*" from th e n am ed p rocessor .
W
i
x
: th e set of p rocessors th at h ave sen t m essage x to p rocessor i.
(N ote th at x is eith er a "*" or a p rocessor n am e.)
- Each p rocess m aintain s n +1 nu m bers of W sets.
- W xi : th e set of witnesses of m essage x for p rocessor i.
- A p rocessor j is a direct supporter for a p rocessor k if j directly
receives "*" from k .
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- Wh en p rocessor i receives th e m essage "k " from p rocessor j, it ad d s j
into
W
i
x
becau se j is a w itn ess to m essage "k ". Process j is an
indirect supporter for p rocessor k if | W jk | L O W;
- A p rocessor j confirm s p rocessor k if | W jk | H I GH ;
A p rocess i m aintain s a set, Ci , of confirm ed p rocessors.
Th e Algorithm
First rou n d : th e sou rce p rocessor sen d s a "*" m essage to all p rocessors
(inclu din g itself) if its valu e is 1. If its valu e is 0, it sen d n othin g in
th e first rou n d . If th e p rocessors fin ally agree on "*", th en th e agreed
u p on valu e is 1. Oth erw ise, th e agreed u p on valu e is 0.
Su bsequ ent rou n d s: a p rocessor sen d s its m essage to all p rocessors,
receives m essages from oth er p rocessors, an d th en d ecid es w h at
m essages to sen d in th e n ext rou n d .
initiation op eration
-
It initiates in th e secon d rou n d if it receiv es a "*" from th e sou rce
in rou n d 1.
- It initiates in th e K+1st rou n d if at th e en d of Kth rou n d th e
card in ality of th e set of th e con firm ed p rocessors (n ot inclu din g th e
sou rce) at least L O W+ m ax (0 ,
K
2
- 2) (referred to as th e condition
of initiation).
Fou r ru les
① In th e first rou n d , th e sou rce broad casts its valu e to all oth er
p rocessors.
② In a rou n d k>1, a p rocessor broad casts th e n am es of all p rocesses
for w hich it is eith er a d irect or in d irect su p p orter an d wh ich it
h as n ot p reviou sly broadcast . If th e con dition of initiation w as tru e
at th e en d of th e p reviou s rou n d , it also broadcasts th e "*" m essage
u nless it h as p reviou sly d on e so.
③ If a p rocessor confirm s H IGH nu m ber of p rocessors, it com m its to
a valu e of 1.
④ After rou n d 2m +3, if th e valu e 1 is com m itted , th e p rocessors agree
on 1; oth erw ise, th ey agree on 0.
Exam p le
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p rocessors, 3m +1
fau lty p rocessors, m
sou rce is n onfau lty .
Th e sou rce p rocessor broadcasts a "*" in th e first rou n d . In th e secon d
rou n d, 2m n onfau lty p rocessors w ill initiate (i.e., broadcast "*"). In th e
third rou n d, 2m +1 n onfau lty p rocessors (inclu d in g th e sou rce) will
broadcast m essages con tainin g th e n am e of th e p rocessors inform in g
th at th ey h ave w itn essed a "*" from 2m oth er n onfau lty p rocessors.
Thu s, in th e fou rth rou n d , th e witn ess set of all 2m +1 n onfau lty
p rocessors w ill contain all 2m +1 n onfau lty p rocessors an d th ey all w ill
com m it to a valu e of 1 in th e fou rth rou n d .
9.5 Ap p lication s of Agreem en t Algorithm s
Fau lt-Toleran t Clock Synchronization
Atom ic Com m it in DDBS
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