Election in the Complete Graph Paola Flocchini Election in the Complete Graph 3 6 8 2 1 5 7 4 Paola Flocchini

Ask neighbours one at a time Paola Flocchini Trivial Algorithm. Ask neighbours one at a time 3 2 I am 1 You win 5 1 7 6 Paola Flocchini

Ask neighbours one at a time Paola Flocchini Trivial Algorithm. Ask neighbours one at a time 3 2 5 1 7 6 Paola Flocchini

Paola Flocchini Message Complexity O(n2) Paola Flocchini

- Territory acquisition (capture neighbours) Paola Flocchini 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 Paola Flocchini

A node attacks another node, if successful Paola Flocchini 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 Paola Flocchini

Paola Flocchini 4 3 8 territories Paola Flocchini

Bigger territories win over smaller ones (i.e. higher stages) Paola Flocchini The Attack Bigger territories win over smaller ones (i.e. higher stages) In case of tie, smaller Ids win 3 8 8 4 Paola Flocchini

Bigger territories win over smaller ones (i.e. higher stages) Paola Flocchini The Attack Bigger territories win over smaller ones (i.e. higher stages) In case of tie, smaller Ids win 3 8 8 4 Paola Flocchini

Send capture message to one neighbour Paola Flocchini 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 Paola Flocchini If still candidate, 3 becomes passive

Paola Flocchini Attacking a passive 3 8 Paola Flocchini

Increase territory and stage (+1) Paola Flocchini Attacking a passive captured 3 8 Increase territory and stage (+1) Paola Flocchini

3 becomes passive, if still candidate Paola Flocchini 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 Paola Flocchini

Increase territory and stage (+1) Paola Flocchini 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 Paola Flocchini Increase territory and stage (+1)

3 becomes passive, if still candidate Paola Flocchini 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 Paola Flocchini 3 becomes passive, if still candidate

Paola Flocchini passive candidate captured Paola Flocchini

When does a candidate become a leader ? Paola Flocchini 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 Paola Flocchini

The territories of any two candidates are disjoint Paola Flocchini 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 Paola Flocchini 4

Paola Flocchini Paola Flocchini

We need: number of stages and messages per stage Paola Flocchini We need: number of stages and messages per stage Paola Flocchini

candidate --- candidate 2 msgs Paola Flocchini Messages per attack candidate --- candidate 2 msgs 2 msgs candidate --- passive candidate --- captured 4 msgs At most 4 messages per attack Paola Flocchini

A candidate with n/2 +1 captured nodes becomes leader and notify Paola Flocchini Number of stages A candidate with n/2 +1 captured nodes becomes leader and notify n/2 +1 stages Paola Flocchini

How many candidates in each stage ? Paola Flocchini 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 Paola Flocchini

 4 n/i O( ) O( ) Message Complexity At most 4 messages per attack Paola Flocchini 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) Paola Flocchini

Election in a Complete Graph with Chordal Sense of Direction Paola Flocchini 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. Paola Flocchini

(Id:3, originLink: 1) 8-1=7. the message was originated from Paola Flocchini (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 Paola Flocchini

(Id:3, originLink: 1) (Id:3, originLink: 2) 3 6 8 2 1 5 7 4 Paola Flocchini (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 Paola Flocchini

Modulo 8 (Id:3, originLink: 2) Which means my link 8-2=6 3 6 8 2 1 5 7 Paola Flocchini 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 Paola Flocchini

Similarly for the other candidates Paola Flocchini 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 Paola Flocchini

Similarly for the other candidates Paola Flocchini 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 Paola Flocchini

Similarly for the other candidates Paola Flocchini 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 Paola Flocchini

Similarly for the other candidates Paola Flocchini 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 Paola Flocchini

Message Complexity Planar graph - 2 messages per edge Paola Flocchini Message Complexity Planar graph - 2 messages per edge O(n) [because a planar graph has O(n) edges] Paola Flocchini