Decentralized Consensus Proto cols 1 Goals of the lecture � Decentralized Consensus Proto cols � V eri�cation of Synchronous Proto cols � Algo rithms fo r computing functions of global state. Bermond, Konig, and Ra ynal � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 2 Consensus Proto cols T ransfo rmation � Systems Reactive � � � X X X X � � n no des � connected top ology � bi-directional channels � m channels � D diameter � no sha red mem o ry/ clo ck � message-based comm unication � reliable delivery � each no de kno ws its identit y and channels adjacent to it � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 3 Consensus Proto cols @ @ @ @ @ @ � � � � � � � initial data distributed on the no des � required symmetric algo rithm � aim is to compute a global function/p redicate such p roto cols a re called Consensus p roto cols. c � Vija y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 4 Ideas in the algo rithm � Computation in phases: Init, phase , phase , � � � ; phase , term. k 1 2 � Logical synchronization induced � w ak eup on receivinga message � T ermination : no de iterates phase so long as it receives new info rmation ) di�erent no des ma y terminate at di�erent times If D is kno wn, the algo rithm stops after D phases. c � Vija y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 5 Filtering Notions 1. In phase p send only the new info rmation that is received in phase p � 1 . 2. if sent ( c ) = r eceiv ed ( c ) then p ro cesses connected thru that channel can never lea rn any new info rmation along that chan- nel. 3. A t phase p : � P lea rns r eceiv ed ( c ) � sent ( c ) . � Send \end" message if this is already kno wn. � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 6 Algo rithm to compute the routing table Pro cess P : � D kno wn � p : numb er of the current phase � c : any channel incident on P Inf : global info rmation kno wn b y P f identities of the no des fo r which P kno ws a sho rtest route g New : new info rmation obtained since the b eginnin g of this phase. sen t( c ) : message sent on channel c at the current phase. receiv e( c ) : message received through channel c . � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 7 Algo rithm [Contd.] Init p 0 Inf f identit y of the no de g sent( c ) Inf fo r all c Phases while p < D do p p + 1 send h sent ( c ) i on all channels c New � F o r every channel c do receive h received ( c ) i on c 8 y 2 received( c ) � Inf � New : Rout ( c ) Rout ( c ) [f y g New New [ (received( c ) � Inf ) Inf Inf [ New sent( c ) New � received( c ) T erm Rout : minimum routing table Inf : identities of all no des � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 8 General Algo rithm � D not kno wn � OPEN : set of channels still op en Init : p 0 ; Inf f initial data g New Inf ; OPEN set of all channels 8 c : received( c ) � Phases : while OPEN 6 = � do p p + 1 8 c 2 OPEN do sent( c 0 New � received( c ) send h send ( c ) i on c New � 8 c 2 OPEN do received h received ( c ) i on c if (received( c ) = send( c )) then OPEN OPEN �f c g New New [ (received( c ) � Inf ) call compute Inf Inf [ New c � Vija y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 9 Pro of Idea � During phase p a no de P receives the info rmation contained in the no des at distance exactly p from itself. ! p � 1 p � 1 � closed( c ) at the end of phase p � T ( P ) = T ( Q ) i � T ( P ) = set of no des at distance at most i from P . c � Vija y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 10 Pro of [Contd.] Notation : i N ( P ) = set of no des at distance i from P . i i T ( P ) = [ N ( P ) j � i c = channel ( P ; Q ) � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 11 8 c 2 op en : sent ( c ) new � r ecd ( c ) p p � 1 p � 1 p � 1 8 c 2 op en : received ( c ) sent ( c ) p p p � 1 op en op en � f c j sent ( c ) = received ( c ) g p p p p � 1 new [ received ( c ) � inf p p p � 1 inf inf [ new p p � 1 p Theo rem : p new = N ( P ) p p � 1 p � 2 sent ( c ) = N ( P ) � N ( Q ) p p � 1 p � 2 received ( c ) = N ( Q ) � N ( P ) p p � 1 p � 1 op en = f ( P ; Q ) j T ( P ) 6 = T ( Q ) g p p inf = T ( P ) p � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 12 p � 1 p � 1 Lemma closed = f ( P ; Q ) j T ( P ) = T ( Q ) g (B) p p � 1 p � 2 p � 1 p � 2 ) Given N ( P ) � N ( Q ) = N ( Q ) � N ( P ) . t t p-1 t t p-2 p-3 p-2 t t P Q p � 1 p � 2 p � 1 p � 2 T ( P ) = T ( Q ) [ N ( P ) [ N ( P ) � � p � 2 p � 1 p � 2 p � 2 = T ( Q ) [ N ( P ) � N ( Q ) [ N ( P ) � � p � 2 p � 1 p � 2 p � 2 = T ( Q ) [ N ( Q ) � N ( P ) [ N ( P ) p � 2 p � 1 = T ( Q ) [ N ( Q ) p � 1 = T ( Q ) � Vija c y K. Ga rg Distributed Systems F all 94
Decentralized Consensus Proto cols 13 ( � � p � 1 p � 1 Given T ( P ) 6 = T ( Q ) p � 1 p � 2 p � 1 p � 2 T o sho w that N ( P ) � N ( Q ) = N ( Q ) � N ( P ) P Q t t p-1 p-2 x p-1 x t x � Vija c y K. Ga rg Distributed Systems F all 94
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