1. Election in the Complete Graph
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Election in the Complete Graph 3 6 8 2 1 5 7 4
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2. Ask neighbours one at a time
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Trivial Algorithm.
Ask neighbours one at a time 3 2
I am 1
You win 5 1 7 6
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3. Ask neighbours one at a time
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Trivial Algorithm.
Ask neighbours one at a time 3 2 5 1 7 6
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4. Paola Flocchini
Message Complexity
O(n2)
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5. - Territory acquisition (capture neighbours)
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Better Algorithm
Ideas:
- In stages
- Territory acquisition (capture neighbours)
ensuring that a node is captured by at most
one candidate in the same stage
- Disjoint territories
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6. A node attacks another node, if successful
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A node attacks another node, if successful
it captures the node increasing the size of its territory
(= stage number)
Defeated nodes become captured (belonging
to a owner) and stop attacking
CANDIDATE: still playing trying to increase the territory
PASSIVE: transitional phase, will not attack anymore,
will eventually become captured
CAPTURED: belong to a territory, owned by a candidate
An attack could reach candidate passive captured
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7. Paola Flocchini 4 3 8
territories
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8. Bigger territories win over smaller ones (i.e. higher stages)
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The Attack
Bigger territories win over smaller ones (i.e. higher stages)
In case of tie, smaller Ids win 3 8 8 4
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9. Bigger territories win over smaller ones (i.e. higher stages)
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The Attack
Bigger territories win over smaller ones (i.e. higher stages)
In case of tie, smaller Ids win 3 8 8 4
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10. Send capture message to one neighbour
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Attacking a candidate
Send capture message to one neighbour
(stage,Id:3) 3 8
captured 8 3
Increase territory and stage (+1)
(stage,Id:3) 3
reject 8 8 3
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If still candidate, 3 becomes passive

11. Paola Flocchini
Attacking a passive 3 8
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12. Increase territory and stage (+1)
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Attacking a passive
captured 3 8
Increase territory and stage (+1)
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13. 3 becomes passive, if still candidate
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Attacking a captured 3 8 no
If I know I belong
to a bigger territory
(stage,Id:3) 8 3
3 becomes passive, if still candidate 3 8
Not sure, ask my owner
(stage,Id:3) 8 3
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14. Increase territory and stage (+1)
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Attacking a captured
If I know I belong
to a bigger territory
(stage,Id:3) 8 3 8 no 3
Not sure, ask my owner
yes
(stage,Id:3) 8 8 3 3
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Increase territory and stage (+1)

15. 3 becomes passive, if still candidate
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Attacking a captured
If I know I belong
to a bigger territory
(stage,Id:3) 8 3 8 no 3
Not sure, ask my owner no
(stage,Id:3) 8 8 3 3 3
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3 becomes passive, if still candidate

16. Paola Flocchini
passive
candidate
captured
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17. When does a candidate become a leader ?
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When to terminate ?
When does a candidate become a leader ?
When it captures more than n/2 nodes
If a candidate has captured more than n/2 nodes
nobody else can become leader
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18. The territories of any two candidates are disjoint
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Important
The territories of any two candidates are disjoint
Because at any time, any node has only ONE owner. 3
Each territory is
rooted in its owner 6 8 2 1 5 7
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19. Paola Flocchini
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20. We need: number of stages and messages per stage
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We need: number of stages and messages per stage
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21. candidate --- candidate 2 msgs
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Messages per attack
candidate --- candidate
2 msgs
2 msgs
candidate --- passive
candidate --- captured
4 msgs
At most 4 messages per attack
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22. A candidate with n/2 +1 captured nodes becomes leader and notify
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Number of stages
A candidate with n/2 +1 captured nodes becomes
leader and notify
n/2 +1 stages
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23. How many candidates in each stage ?
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How many candidates in each stage ? i
Stage i ---> territory of size i i i i i
With disjoint territories
There cannot be more than n/i
candidates in stage i
ni  n/i
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24.  4 n/i O( ) O( ) Message Complexity At most 4 messages per attack
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Message Complexity
At most 4 messages per attack
ni  n/i
Messages in stage i  4 n/i
Harmonic number Hn/2
= O(log n)
 4 n/i 1
n/2 O( )
n/2 O(
4 n  1/i ) = 1
M(completeElect) = O(n log n)
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25. Election in a Complete Graph with Chordal Sense of Direction
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Election in a Complete Graph with Chordal Sense of Direction 5 1 4 1 2 5 3 3 2 4 5 4 1 2 3 3 2 4 5 1 4 3 3 2 5 1 2 4 1 5
Any ring algorithm
IDEA: Put information in messages. At the next step, use a smaller ring.
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26. (Id:3, originLink: 1) 8-1=7. the message was originated from
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(Id:3, originLink: 1) 1 7 3
8-1=7.
the message
was originated from
my neighbour 6 2 1 7 5 6 4 3 6 5 8 2 4 7 3 3 1 5 4 2 6 7 6 1 2 3 5 2 4 1 4 3 5 2 1 6 7 3 4 6 2 5 1 5 4 7 3 5 6 2 7 4 7 3 5 6 1 2 1 7 4
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27. (Id:3, originLink: 1) (Id:3, originLink: 2) 3 6 8 2 1 5 7 4
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(Id:3, originLink: 1) 3 1 7 6 2 1 7
(Id:3, originLink: 2) 5 4 3 6 6 8 2 5 4 3 1 7 3 2 5 4 6 7 6 1 2 3 5 2 4 1 4 3 5 2 1 6 7 3 4 6 2 5 1 5 4 7 3 5 6 2 7 4 7 3 5 6 1 2 1 7 4
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28. Modulo 8 (Id:3, originLink: 2) Which means my link 8-2=6 3 6 8 2 1 5 7
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Modulo 8 3 1 7 6 2 1 7
(Id:3, originLink: 2) 5 4 3 6 6 8 2 5 4 3 1 7 3 2 5 4 6
Which means
my link 8-2=6 7 6 1 2 3 5 2 4 1 4 3 5 2 1 6 7 3 4 6 2 5 1 5 4 7 3 5 6 2 7 4 7 3 5 6 1 2 1 7 4
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29. Similarly for the other candidates
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Similarly for the other candidates
Modulo 8 3 1 7
(Id:2, originLink: 6) 6 2 1 7 5 4 3 6 6 8 2 5 4 3 1 7 3 2 5 4 6 7 6 1 2 3 5 2 4 1 4 3 5 2 1 6 7 3 4 6 2 5 1 5 4 7 3 5 6 2 7 4 7 3 5 6 1 2 1 7 4
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30. Similarly for the other candidates
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Similarly for the other candidates
Modulo 8
(Id:2, originLink: 6+6=4) 3 1 7 6 2 1 7 5 4 3 6 6 8 2 5 4 3 1 7 3
(Id:2, originLink: 6) 2 5 4 6 7 6 1 2 3 5 2 4 1 4 3 5 2 1 6 7 3 4 6 2 5 1 5 4 7 3 5 6 2 7 4 7 3 5 6 1 2 1 7 4
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31. Similarly for the other candidates
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Similarly for the other candidates
Modulo 8 3 1 7 6 2 1 7 5 4 3 6 6 8 2 5
Which means
my 8-4=4 4 3 1 7 3 2 5 4 6 7 6 1 2 3 5 2 4 1 4 3 5 2 1 6 7 3 4 6 2 5 1 5 4 7 3 5 6 2 7 4 7 3 5 6 1 2 1 7 4
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32. Similarly for the other candidates
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Similarly for the other candidates
Modulo 8 3 1 7 6 2 1 7 5 4 3 6 6 8 2 5 4 3 1 7 3 2 5 4 6 7 6 1 2 3 5 2 4 1 4 3 5 2 1 6 7 3 4 6 2 5 1 5 4 7 3 5 6 2 7 4 7 3 5 6 1 2 1 7 4
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33. Message Complexity Planar graph - 2 messages per edge
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Message Complexity
Planar graph - 2 messages per edge
O(n) [because a planar graph has O(n) edges]
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