Leader in a clique in O(nlog(n)) messages By Pierre A. Humblet and the help of Miki Asa.

Leader in a clique in Leader in a clique in O(nlog(n)) messages O(nlog(n)) messages

By Pierre A. HumbletBy Pierre A. Humblet

and the help of Miki Asaand the help of Miki Asa

Rules of the gameRules of the game Initialization : Initialization :

Each node has {1 . . N-1} edges + edge (–1) to Each node has {1 . . N-1} edges + edge (–1) to itself.itself.

master(id) = id, state = active.master(id) = id, state = active. Node attempt to capture other nodes Node attempt to capture other nodes

(increase the (increase the sizesize of their domain). of their domain). A node that has been captured stop capturing A node that has been captured stop capturing

and it’s state = stopped.and it’s state = stopped. Node can be captured by many other nodes Node can be captured by many other nodes

but has only one but has only one mastermaster the last one that the last one that captured it.captured it.

Node that captured all the nodes Node that captured all the nodes leaderleader

How it’s doneHow it’s done2 messages2 messages TEST(size,id)TEST(size,id)

An attempt to capture other node.An attempt to capture other node. A TEST from a node I didn’t captured A TEST from a node I didn’t captured

forward to my master.forward to my master. A TEST from a node I captured A TEST from a node I captured

answer it with WINNER(id).answer it with WINNER(id). WINNER(id)WINNER(id)

The result of the fight The result of the fight id won the id won the fight.fight.

Determined by who got bigger domain Determined by who got bigger domain (if equal bigger id)(if equal bigger id)

A B

State = activeMaster = A

State = activeMaster = B

A B

B winsAB : TEST(size(A),id(A))BB : TEST(size(A),id(A))BB : WINNER(B)

B winsB winsWaiting

B winsAB : TEST(size(A),id(A))BB : TEST(size(A),id(A))BB : WINNER(B)

A B


State = activeMaster = B

B wins

A B

A winsAB : TEST(size(A),id(A))BB : TEST(size(A),id(A))BB : WINNER(A)BA : WINNER(A)

A winsA wins

A winsAB : TEST(size(A),id(A))BB : TEST(size(A),id(A))BB : WINNER(A)BA : WINNER(A)

A B


State = stopMaster = A

A wins

C B

A

State = activeMaster = C



C B

A

A winsCB : TEST(size(C),id(C))BA : TEST(size(C),id(C))AB : WINNER(A)

A wins

Waiting

C B

A

A winsCB : TEST(size(C),id(C))BA : TEST(size(C),id(C))AB : WINNER(A)




A wins

C B

A

C winsCB : TEST(size(C),id(C))BA : TEST(size(C),id(C))AB : WINNER(C)BC : WINNER(C)

C wins

C B

A

C winsCB : TEST(size(C),id(C))BA : TEST(size(C),id(C))AB : WINNER(C)BC : WINNER(C)


State = stopMaster = C


C wins

VariablesVariablesSTATE :STATE : active – won all the fights active – won all the fights

stopped – lost a fight stopped – lost a fight won’t be a leader won’t be a leader

SIZE :SIZE : number of nodes I captured (send me winner (ID)) number of nodes I captured (send me winner (ID)) including me including me

(size of my domain).(size of my domain).

MASTER : the id of my master.MASTER : the id of my master.

EDGE(id) : the edge leading to node idEDGE(id) : the edge leading to node id

Dealing with messagesDealing with messages

PENDING_SET : messages waiting to be forward to the masterPENDING_SET : messages waiting to be forward to the master

PENDING : number of messages in PENDING_SETPENDING : number of messages in PENDING_SET

InitializationInitialization

Each awaken node with ID receive winner(ID) on edge > 0.Each awaken node with ID receive winner(ID) on edge > 0.

Receiving WINNER(id) on edge eReceiving WINNER(id) on edge e

If (id = ID) If (id = ID) // I won the fight// I won the fight{{

if (e > 0 & state = active)if (e > 0 & state = active) // fight that I initiated// fight that I initiated{{SIZE ++SIZE ++if SIZE = N if SIZE = N // I capture all the nodes// I capture all the nodes

stopstop // announce elected// announce electedelse else

send TEST(SIZE,ID) on edge SIZE //continue send TEST(SIZE,ID) on edge SIZE //continue }}

elseelse // I didn’t initiate the // I didn’t initiate the fightfight

do nothingdo nothing}}

Receiving WINNER(id) (cont.)Receiving WINNER(id) (cont.)

ElseElse // id != ID// id != ID{{

if (MASTER != id)if (MASTER != id) // my master lost // my master lost {{

MASTER = idMASTER = id // change the master// change the masterSend WINNER(id) on EDGE(id) // inform the Send WINNER(id) on EDGE(id) // inform the

new masternew master}}Else Else // my master won// my master won{{

do nothingdo nothing}}

PENDING - -PENDING - - // continue as a slave// continue as a slaveif PENDING > 0if PENDING > 0

send from PENDING_SET on EDGE(MASTER)send from PENDING_SET on EDGE(MASTER)

}}

Receiving TEST(size,id) on edge eReceiving TEST(size,id) on edge eIf e < SIZEIf e < SIZE // message from my domain// message from my domain

{{

if (size,id) > (SIZE,ID) if (size,id) > (SIZE,ID) // I lost this fight // I lost this fight

{{

STATE = stoppedSTATE = stopped // out of the game // out of the game

send WINNER(id) on EDGE esend WINNER(id) on EDGE e // inform the new master// inform the new master

}}

else else // I won // I won

send WINNER(ID) on EDGE esend WINNER(ID) on EDGE e // inform my slave that I // inform my slave that I wonwon

}}

Else Else // not from my domain // not from my domain

{{

EDGE(id) = eEDGE(id) = e // update my data-base // update my data-base

PENDING ++PENDING ++ // send TEST to the master // send TEST to the master

If PENDING = 1If PENDING = 1

send TEST(size,id) on EDGE (MASTER)send TEST(size,id) on EDGE (MASTER)

elseelse

put TEST(size,id) on EDGE (MASTER) on PENDING_SETput TEST(size,id) on EDGE (MASTER) on PENDING_SET

}}

1

6

5

4

3

2

stateSize

Master

active13

active12

active11

active16 active

15

active14

1 3

stateSize

Master

stop13

6

5

4

2active

23

active12

active16 active

15

active14

1

8

4

stateSize

Master

active24stop

14 1 3

stop13

6

5

4

2active

23

active12

active16 active

15

active14

Waiting

stateSize

Master

1 3

stop13

6

5

4

2active

23

active12

active16

active14

Waiting

active15

stateSize

Master

1 3

stop13

6

5

4

2active

23

active12

active26

stop16

Waiting

active15

stateSize

Master

1 3

stop13

6

5

4

2active

23

active12

active26

stop16

Waiting

active15

Waiting

stateSize

Master

1 3

stop16

6

5

4

2stop

23

active12

active36

stop16

Waiting

active15

Waiting

stateSize

Master

1 3

stop16

6

5

4

2stop

23

active12

active36

stop16

Waiting

stop13

stateSize

Master

1 3

stop16

6

5

4

2stop

26

active12

active46

stop16

Waiting

stop13

stateSize

Master

1 3

stop16

6

5

4

2stop

26

active12

active46

stop16

Waiting

stop13

Waiting

stateSize

Master

1 3

stop16

6

5

4

2stop

26

active16

active56

stop16

Waiting

stop13

Waiting

stateSize

Master

1 3

stop16

6

5

4

2stop

26

active16

active66

stop16

Waiting

stop16

Waiting

TerminationTermination

Each node can’t send more then N-1 Each node can’t send more then N-1 TEST.TEST.

Each TEST cause at the most 3 more Each TEST cause at the most 3 more messages.messages.

CorrectnessCorrectness

Lemma 1Lemma 1

At the end of the algorithm there will be a At the end of the algorithm there will be a node with size = N.node with size = N.

ProofProof Size only grows with time.Size only grows with time. For each TEST sentFor each TEST sent

size++ orsize++ or Node is “Waiting”Node is “Waiting”

Therefore Therefore

Enough to prove there is no deadlockEnough to prove there is no deadlock

LivenessLiveness Suppose there is a deadlockSuppose there is a deadlock Lets take the node with the biggest Lets take the node with the biggest

(size,ID) (size,ID) it’s active. it’s active. Eventually it will get an answer for Eventually it will get an answer for

the last test it initiated. Winner(ID).the last test it initiated. Winner(ID). Once he’ll get that answer Once he’ll get that answer size++ size++

Size = N Size = N leader leader Send TEST for the next nodeSend TEST for the next node

complexitycomplexityMy domain:My domain: the nodes from which I the nodes from which I

received winner(ID) while I was active + received winner(ID) while I was active + myself.myself.

Lemma 1Lemma 1SIZE = size of my domain.SIZE = size of my domain.


if (e > 0 & state = active)if (e > 0 & state = active) // fight that I // fight that I initiatedinitiated

{{SIZE ++SIZE ++

Got winner(ID) from other node & state = Got winner(ID) from other node & state = active active all conditions fulfill all conditions fulfill size++ size++

Size++ Size++ all conditions fulfill all conditions fulfill Got winner(ID) Got winner(ID) from other node & state = activefrom other node & state = active

Lemma 2Lemma 2If SIZE(A) at time Ta = SIZE(B) at time Tb If SIZE(A) at time Ta = SIZE(B) at time Tb

then the domains at Ta & Tb are disjoint.then the domains at Ta & Tb are disjoint.Proof:Proof: Assume domain(A) & domain(B) are joint at Ta or Assume domain(A) & domain(B) are joint at Ta or

Tb Tb They are joint at max(Ta,Tb) They are joint at max(Ta,Tb) There is a Tm < max(Ta,Tb) s.t. There is a Tm < max(Ta,Tb) s.t.

At Tm a node from domain(A) join domain(B) At Tm a node from domain(A) join domain(B) After Tm domain(A) doesn’t grow After Tm domain(A) doesn’t grow At Tm Size(A) >= SIZE At Tm Size(A) >= SIZE Tm > Ta Tm > TaAfter Tm domain(B) > domain(A) >= SIZE After Tm domain(B) > domain(A) >= SIZE Tm > TbTm > TbContradictionContradiction

Lemma 3Lemma 3

Rank the nodes in decreasing order by Rank the nodes in decreasing order by their SIZE at termination.their SIZE at termination.

SS11, S, S22, S, S33, . . . , S, . . . , Snn-1-1, S, Snn

Then SThen Skk <= <= N/N/kk

ProofProof

SS11 – S – Skk-1-1 had once the size of S had once the size of Skk

there all disjoint (lemma 2)there all disjoint (lemma 2)

conclusionconclusion

There is only one node with SIZE=N There is only one node with SIZE=N one leaderone leader

Number of TEST sentNumber of TEST sent


if (e > 0 & state = active)if (e > 0 & state = active) // fight that I // fight that I initiatedinitiated

{{SIZE ++SIZE ++if SIZE = N if SIZE = N // I capture all the nodes// I capture all the nodes

stopstop // announce elected// announce electedelse else

send TEST(SIZE,ID) on EDGE(SIZE) //continue send TEST(SIZE,ID) on EDGE(SIZE) //continue }}

For each TEST sent For each TEST sent SIZE++ SIZE++Each TEST cause at the most 3 more Each TEST cause at the most 3 more

messages messages

Total number of messages is no Total number of messages is no more thenmore then

4*(N/1 + N/2 + N/3 + . . . + N/K)4*(N/1 + N/2 + N/3 + . . . + N/K)

= O(N log(K))= O(N log(K))

K = the number of nodes awake.K = the number of nodes awake.

ImprovementsImprovements The bound for the number of The bound for the number of

messages can be tightened by messages can be tightened by noticing that the first TEST received noticing that the first TEST received by a generates at the most one more by a generates at the most one more message.message.

The algorithm can stop as soon as The algorithm can stop as soon as SIZE = N/2 (then announce elected).SIZE = N/2 (then announce elected).

Leader in a clique in O(nlog(n)) messages By Pierre A. Humblet and the help of Miki Asa.

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Transcript of Leader in a clique in O(nlog(n)) messages By Pierre A. Humblet and the help of Miki Asa.